Dreptul
de copyright:
Cartea
downloadata de pe site-ul http://www.mateinfo.ro/ nu poate fi publicata pe un alt site si nu
poate fi folosita in
scopuri comerciale fara specificarea
sursei si acordul autorului
Adrian Stan
1. Multimea
numerelor reale
1.. Scrierea in baza zece:
abcd
¯
=a⋅
10
3
+b⋅
10
2
+c⋅10+d
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWGHbGaamOyaiaadogacaWGKbaaaiabg2da9iaadggacqGHflY1caaIXaGaaGimamaaCaaaleqabaGaaG4maaaakiabgUcaRiaadkgacqGHflY1caaIXaGaaGimamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadogacqGHflY1caaIXaGaaGimaiabgUcaRiaadsgaaaa@4E10@
a-cifra miilor; b-cifra sutelor; c-cifra zecilor; d-cifra
unitatilor;
a,efg
¯
=a⋅10+e⋅
10
−1
+f⋅
10
−2
+g⋅
10
−3
=
=a⋅10+e⋅0.1+f⋅0.01+g⋅0.001
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@71EA@
e-cifra
zecimilor; f-cifra sutimilor; g-cifra miimilor.
2. Fractii
-Fractii zecimale
finite:
a,b
¯
=
ab
¯
10
;
a,bc
¯
=
abc
¯
100
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWGHbGaaiilaiaadkgaaaGaeyypa0ZaaSaaaeaadaqdaaqaaiaadggacaWGIbaaaaqaaiaaigdacaaIWaaaaiaacUdadaqfGaqabSqabeaaa0qaaaaakmaanaaabaGaamyyaiaacYcacaWGIbGaam4yaaaacqGH9aqpdaWcaaqaamaanaaabaGaamyyaiaadkgacaWGJbaaaaqaaiaaigdacaaIWaGaaGimaaaacaGG7aaaaa@4849@
-Fractii zecimale
periodice:- simple:
a,(b)
¯
=
ab
¯
−a
9
;
a,(bc)
¯
=
abc
¯
−a
99
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWGHbGaaiilaiaacIcacaWGIbGaaiykaaaacqGH9aqpdaWcaaqaamaanaaabaGaamyyaiaadkgaaaGaeyOeI0IaamyyaaqaaiaaiMdaaaGaai4oamaavacabeWcbeqaaaqdbaWaaubiaeqaoeqabaaaneaaaaaaaOWaa0aaaeaacaWGHbGaaiilaiaacIcacaWGIbGaam4yaiaacMcaaaGaeyypa0ZaaSaaaeaadaqdaaqaaiaadggacaWGIbGaam4yaaaacqGHsislcaWGHbaabaGaaGyoaiaaiMdaaaGaai4oaaaa@4D9F@
mixte:
a,b(c)
¯
=
abc
¯
−
ab
¯
90
;
a,b(cd)
¯
=
abcd
¯
−
ab
¯
990
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWGHbGaaiilaiaadkgacaGGOaGaam4yaiaacMcaaaGaeyypa0ZaaSaaaeaadaqdaaqaaiaadggacaWGIbGaam4yaaaacqGHsisldaqdaaqaaiaadggacaWGIbaaaaqaaiaaiMdacaaIWaaaaiaacUdadaqfGaqabSqabeaaa0qaaaaakmaanaaabaGaamyyaiaacYcacaWGIbGaaiikaiaadogacaWGKbGaaiykaaaacqGH9aqpdaWcaaqaamaanaaabaGaamyyaiaadkgacaWGJbGaamizaaaacqGHsisldaqdaaqaaiaadggacaWGIbaaaaqaaiaaiMdacaaI5aGaaGimaaaacaGG7aaaaa@544C@
3.. Rapoarte si proportii
a
b
se
numeste
raport
∀b≠0;
a
b
=
a⋅n
b⋅n
=k,
n∈
Q
*
,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaadaWfGaqaaaWcbeqaaaaakiaadohacaWGLbWaaCbiaeaaaSqabeaaaaGccaWGUbGaamyDaiaad2gacaWGLbGaam4CaiaadshacaWGLbWaaCbiaeaadaqfGaqabSqabeaaa0qaaaaaaSqabeaaaaGccaWGYbGaamyyaiaadchacaWGVbGaamOCaiaadshadaWfGaqaaaWcbeqaaaaakiabgcGiIiaadkgacqGHGjsUcaaIWaGaai4oamaaxacabaaaleqabaaaaOWaaSaaaeaacaWGHbaabaGaamOyaaaacqGH9aqpdaWcaaqaaiaadggacqGHflY1caWGUbaabaGaamOyaiabgwSixlaad6gaaaGaeyypa0Jaam4AaiaacYcadaWfGaqaaaWcbeqaaaaakiaad6gacqGHiiIZcaWGrbWaaWbaaSqabeaacaGGQaaaaOGaaiilamaaxacabaaaleqabaaaaaaa@6056@
k se numeste coeficient de proportionalitate
;
Proprietatea fundamentala a proportiilor:
a
b
=
c
d
⇒a⋅d=b⋅c
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaacqGH9aqpdaWcaaqaaiaadogaaeaacaWGKbaaaiabgkDiElaadggacqGHflY1caWGKbGaeyypa0JaamOyaiabgwSixlaadogaaaa@464C@
4. Proportii derivate:
a
b
=
c
d
⇒{
b
a
=
d
c
sau
d
b
=
c
a
sau
a
c
=
b
d
a
a±b
=
c
c±d
sau
a±b
b
=
c±d
d
a
b
=
a+c
b+d
sau
a
b
=
a−c
b−d
sau
a
2
b
2
=
c
2
d
2
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@899A@
5. Sir de
rapoarte egale:
a
1
b
1
=
a
2
b
2
=.........=
a
n
b
n
=
a
1
+
a
2
+
a
3
+....+
a
n
b
1
+
b
2
+
b
3
+.....+
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOyamaaBaaaleaacaaIXaaabeaaaaGccqGH9aqpdaWcaaqaaiaadggadaWgaaWcbaGaaGOmaaqabaaakeaacaWGIbWaaSbaaSqaaiaaikdaaeqaaaaakiabg2da9iaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiabg2da9maalaaabaGaamyyamaaBaaaleaacaWGUbaabeaaaOqaaiaadkgadaWgaaWcbaGaamOBaaqabaaaaOGaeyypa0ZaaSaaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadggadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaGGUaGaaiOlaiaac6cacaGGUaGaey4kaSIaamyyamaaBaaaleaacaWGUbaabeaaaOqaaiaadkgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaamOyamaaBaaaleaacaaIZaaabeaakiabgUcaRiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaey4kaSIaamOyamaaBaaaleaacaWGUbaabeaaaaaaaa@6886@
;
(
a
1
,
a
2
,
a
3
,......
a
n
)şi(
b
1
,
b
2
,
b
3
,....
b
n
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadggadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyyamaaBaaaleaacaaIZaaabeaakiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaGaam4xbiaadMgadaqadaqaaiaadkgadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamOyamaaBaaaleaacaaIYaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaiodaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaaa@551E@
sunt direct
proportionale
⇔
a
1
b
1
=
a
2
b
2
=..=
a
n
b
n
=k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aaCbiaeaaaSqabeaaaaGcdaWcaaqaaiaadggadaWgaaWcbaGaaGymaaqabaaakeaacaWGIbWaaSbaaSqaaiaaigdaaeqaaaaakiabg2da9maalaaabaGaamyyamaaBaaaleaacaaIYaaabeaaaOqaaiaadkgadaWgaaWcbaGaaGOmaaqabaaaaOGaeyypa0JaaiOlaiaac6cacqGH9aqpdaWcaaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9iaadUgaaaa@4ABD@
.
(
a
1
,
a
2
,
a
3
,......
a
n
)şi(
b
1
,
b
2
,
b
3
,....
b
n
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadggadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyyamaaBaaaleaacaaIZaaabeaakiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaGaam4xbiaadMgadaqadaqaaiaadkgadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamOyamaaBaaaleaacaaIYaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaiodaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaaa@551E@
sunt invers
proportionale
⇔
a
1
⋅
b
1
=
a
2
⋅
b
2
=..=
a
n
⋅
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaamyyamaaBaaaleaacaaIXaaabeaakiabgwSixlaadkgadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaeyyXICTaamOyamaaBaaaleaacaaIYaaabeaakiabg2da9iaac6cacaGGUaGaeyypa0JaamyyamaaBaaaleaacaWGUbaabeaakiabgwSixlaadkgadaWgaaWcbaGaamOBaaqabaaaaa@4F18@
6. Modulul
numerelor reale Proprietati:
| a |
def
¯
¯
{
a,
a〉0
0,
a=0
−a,
a〈0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbaacaGLhWUaayjcSdWaaCbiaeaaaSqabeaaaaGcdaadbaqaaiaadsgacaWGLbGaamOzaaaadaGabaabaeqabaWaaCbiaeaaaSqabeaaaaGccaWGHbGaaiilamaaxacabaaaleqabaaaaOGaamyyaiabgQYiXlaaicdaaeaadaWfGaqaaaWcbeqaaaaakiaaicdacaGGSaWaaCbiaeaaaSqabeaaaaGccaWGHbGaeyypa0JaaGimaaqaaiabgkHiTiaadggacaGGSaWaaCbiaeaaaSqabeaaaaGccaWGHbGaeyykJeUaaGimaaaacaGL7baaaaa@4ECA@
1.
| a |≥0,
∀a∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbaacaGLhWUaayjcSdGaeyyzImRaaGimaiaacYcadaqfGaqabSqabeaaa0qaaaaakiabgcGiIiaadggacqGHiiIZcaWGsbaaaa@419C@
; 2.
| a |=0,
⇔a=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbaacaGLhWUaayjcSdGaeyypa0JaaGimaiaacYcadaqfGaqabSqabeaaa0qaaaaakiabgsDiBlaadggacqGH9aqpcaaIWaaaaa@41CD@
;
3.
| a |=|
−a
|,
∀a∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbaacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaacqGHsislcaWGHbaacaGLhWUaayjcSdGaaiilamaavacabeWcbeqaaaqdbaaaaOGaeyiaIiIaamyyaiabgIGiolaadkfaaaa@4517@
; 4.
| a |=| b |,
⇔a=±b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbaacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaacaWGIbaacaGLhWUaayjcSdGaaiilamaavacabeWcbeqaaaqdbaaaaOGaeyi1HSTaamyyaiabg2da9iabgglaXkaadkgaaaa@4737@
;
5.
|
a⋅b
|=| a |⋅| b |
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbGaeyyXICTaamOyaaGaay5bSlaawIa7aiabg2da9maaemaabaGaamyyaaGaay5bSlaawIa7aiabgwSixpaaemaabaGaamOyaaGaay5bSlaawIa7aaaa@488D@
; 6.
|
a
b
|=
| a |
| b |
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWcaaqaaiaadggaaeaacaWGIbaaaaGaay5bSlaawIa7aiabg2da9maalaaabaWaaqWaaeaacaWGHbaacaGLhWUaayjcSdaabaWaaqWaaeaacaWGIbaacaGLhWUaayjcSdaaaaaa@4419@
;
7.
|
| a |−| b |
|≤|
a±b
|≤| a |+| b |
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaabdaqaaiaadggaaiaawEa7caGLiWoacqGHsisldaabdaqaaiaadkgaaiaawEa7caGLiWoaaiaawEa7caGLiWoacqGHKjYOdaabdaqaaiaadggacqGHXcqScaWGIbaacaGLhWUaayjcSdGaeyizIm6aaqWaaeaacaWGHbaacaGLhWUaayjcSdGaey4kaSYaaqWaaeaacaWGIbaacaGLhWUaayjcSdaaaa@554D@
;
8.
| x |=a,
⇒x=±a,
a〉0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG4baacaGLhWUaayjcSdGaeyypa0JaamyyaiaacYcadaqfGaqabSqabeaaa0qaaaaakiabgkDiElaadIhacqGH9aqpcqGHXcqScaWGHbGaaiilamaavacabeWcbeqaaaqdbaaaaOGaamyyaiabgQYiXlaaicdaaaa@48BC@
;
9.
| x |≤a,
⇔x∈[−a,a],
a〉0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG4baacaGLhWUaayjcSdGaeyizImQaamyyaiaacYcadaqfGaqabSqabeaaa0qaaaaakiabgsDiBlaadIhacqGHiiIZcaGGBbGaeyOeI0IaamyyaiaacYcacaWGHbGaaiyxaiaacYcadaqfGaqabSqabeaaa0qaaaaakiaadggacqGHQms8caaIWaaaaa@4C3D@
;
10.
| x |≥a,
⇔x∈[−∞,−a]∪[a,+∞],
a〉0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG4baacaGLhWUaayjcSdGaeyyzImRaamyyaiaacYcadaqfGaqabSqabeaaa0qaaaaakiabgsDiBlaadIhacqGHiiIZcaGGBbGaeyOeI0IaeyOhIuQaaiilaiabgkHiTiaadggacaGGDbGaeyOkIGSaai4waiaadggacaGGSaGaey4kaSIaeyOhIuQaaiyxaiaacYcadaqfGaqabSqabeaaa0qaaaaakiaadggacqGHQms8caaIWaaaaa@550F@
.
7. Reguli de
calcul in R
1.
(
a+b
)
2
=
a
2
+2ab+
b
2
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbGaey4kaSIaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabg2da9iaadggadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaamyyaiaadkgacqGHRaWkcaWGIbWaaWbaaSqabeaacaaIYaaaaOGaai4oaaaa@44E3@
2.
(
a−b
)
2
=
a
2
−2ab+
b
2
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbGaeyOeI0IaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabg2da9iaadggadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIYaGaamyyaiaadkgacqGHRaWkcaWGIbWaaWbaaSqabeaacaaIYaaaaOGaai4oaaaa@44F9@
3. (a+b)(a-b= a
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabeaacaaIYaaaaaaa@36DC@
-b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabeaacaaIYaaaaaaa@36DC@
;
4.
(
a+b+c
)
2
=
a
2
+
b
2
+
c
2
+2ab+2bc+2ca
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbGaey4kaSIaamOyaiabgUcaRiaadogaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadogadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaamyyaiaadkgacqGHRaWkcaaIYaGaamOyaiaadogacqGHRaWkcaaIYaGaam4yaiaadggaaaa@4F84@
5.
(
a+b
)
3
=
a
3
+3
a
2
b+3a
b
2
+
b
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbGaey4kaSIaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaakiabg2da9iaadggadaahaaWcbeqaaiaaiodaaaGccqGHRaWkcaaIZaGaamyyamaaCaaaleqabaGaaGOmaaaakiaadkgacqGHRaWkcaaIZaGaamyyaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaacaaIZaaaaaaa@4970@
;
6.
(
a−b
)
3
=
a
3
−3
a
2
b+3a
b
2
−
b
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbGaeyOeI0IaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaakiabg2da9iaadggadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaIZaGaamyyamaaCaaaleqabaGaaGOmaaaakiaadkgacqGHRaWkcaaIZaGaamyyaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGIbWaaWbaaSqabeaacaaIZaaaaaaa@4991@
;
7.
a
3
−
b
3
=(a−b)(
a
2
+ab+
b
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCaaaleqabaGaaG4maaaakiabgkHiTiaadkgadaahaaWcbeqaaiaaiodaaaGccqGH9aqpcaGGOaGaamyyaiabgkHiTiaadkgacaGGPaGaaiikaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGHbGaamOyaiabgUcaRiaadkgadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaa@484B@
;
8.
a
3
+
b
3
=(a+b)(
a
2
−ab+
b
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCaaaleqabaGaaG4maaaakiabgUcaRiaadkgadaahaaWcbeqaaiaaiodaaaGccqGH9aqpcaGGOaGaamyyaiabgUcaRiaadkgacaGGPaGaaiikaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGHbGaamOyaiabgUcaRiaadkgadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaa@4840@
.
8. Puteri
cu exponent intreg
a
n
def
¯
¯
a⋅a⋅a⋅......⋅a
︸
n
factori
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCaaaleqabaGaamOBaaaakmaameaabaGaamizaiaadwgacaWGMbaaamaavacabeWcbeqaaaqdbaaaaOWaaGbaaeaacaWGHbGaeyyXICTaamyyaiabgwSixlaadggacqGHflY1caGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacqGHflY1caWGHbaaleaacaWGUbWaaCbiaeaaaWqabeaaaaWccaWGMbGaamyyaiaadogacaWG0bGaam4BaiaadkhacaWGPbaakiaawIJ=aaaa@55F1@
1.
a
o
=1;
a
1
=a;
0
n
=0;
5.
(
a
m
)
n
=
a
m⋅n
2.
a
m+n
=
a
m
⋅
a
n
6.
a
−n
=
1
a
n
,a≠0
3.
(a⋅b)
n
=
a
n
⋅
b
n
7.
(
a
b
)
n
=
a
n
b
n
,b≠0
4.
a
m
a
n
=
a
m−n
;a≠0
8.
a
m
=
a
n
⇔m=n.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@ACB9@
9. Proprietatile radicalilor de ordinul doi
1.
a
2
=| a |≥0,∀a∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaaqabaGccqGH9aqpdaabdaqaaiaadggaaiaawEa7caGLiWoacqGHLjYScaaIWaGaaiilaiabgcGiIiaadggacqGHiiIZcaWGsbaaaa@442B@
2.
a⋅b
=
a
⋅
b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGHbGaeyyXICTaamOyaaWcbeaakiabg2da9maakaaabaGaamyyaaWcbeaakiabgwSixpaakaaabaGaamOyaaWcbeaaaaa@3F8C@
3.
a
b
=
a
b
,b≠0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaadaGcaaqaaiaadggaaSqabaaakeaadaGcaaqaaiaadkgaaSqabaaaaOGaeyypa0ZaaOaaaeaadaWcaaqaaiaadggaaeaacaWGIbaaaaWcbeaakiaacYcacaWGIbGaeyiyIKRaaGimaaaa@3F3A@
4.
a
n
=
(
a
)
n
=
a
n
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGHbWaaWbaaSqabeaacaWGUbaaaaqabaGccqGH9aqpcaGGOaWaaOaaaeaacaWGHbaaleqaaOGaaiykamaaCaaaleqabaGaamOBaaaakiabg2da9iaadggadaahaaWcbeqaamaalaaabaGaamOBaaqaaiaaikdaaaaaaaaa@407F@
,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F2@
5.
a±
b
=
a+
a
2
−b
2
±
a−
a
2
−b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGHbGaeyySae7aaOaaaeaacaWGIbaaleqaaaqabaGccqGH9aqpdaGcaaqaamaalaaabaGaamyyaiabgUcaRmaakaaabaGaamyyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadkgaaSqabaaakeaacaaIYaaaaaWcbeaakiabgglaXoaakaaabaWaaSaaaeaacaWGHbGaeyOeI0YaaOaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamOyaaWcbeaaaOqaaiaaikdaaaaaleqaaaaa@49EE@
unde
a
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
-b=k
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
.
10. Medii
Media aritmetica
m
a
=
x+y
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGHbaabeaakiabg2da9maalaaabaGaamiEaiabgUcaRiaadMhaaeaacaaIYaaaaaaa@3CB0@
Media geometrica
m
g
=
x⋅y
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGNbaabeaakiabg2da9maakaaabaGaamiEaiabgwSixlaadMhaaSqabaaaaa@3D6D@
Media ponderata
m
p
=
p⋅x+q⋅y
p+q
;p,q−ponderile
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGWbaabeaakiabg2da9maalaaabaGaamiCaiabgwSixlaadIhacqGHRaWkcaWGXbGaeyyXICTaamyEaaqaaiaadchacqGHRaWkcaWGXbaaaiaacUdacaWGWbGaaiilaiaadghacqGHsislcaWGWbGaam4Baiaad6gacaWGKbGaamyzaiaadkhacaWGPbGaamiBaiaadwgaaaa@5205@
Media armonica
m
h
=
2
1
x
+
1
y
=
2xy
x+y
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGObaabeaakiabg2da9maalaaabaGaaGOmaaqaamaalaaabaGaaGymaaqaaiaadIhaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaamyEaaaaaaGaeyypa0ZaaSaaaeaacaaIYaGaamiEaiaadMhaaeaacaWG4bGaey4kaSIaamyEaaaaaaa@44F7@
.
Inegalitatea mediilor
2xy
x+y
≤
xy
≤
x+y
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaGaamiEaiaadMhaaeaacaWG4bGaey4kaSIaamyEaaaacqGHKjYOdaGcaaqaaiaadIhacaWG5baaleqaaOGaeyizIm6aaSaaaeaacaWG4bGaey4kaSIaamyEaaqaaiaaikdaaaaaaa@44CA@
11. Ecuatii
a⋅x+b=0⇒x=−
b
a
,a≠0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgwSixlaadIhacqGHRaWkcaWGIbGaeyypa0JaaGimaiabgkDiElaadIhacqGH9aqpcqGHsisldaWcaaqaaiaadkgaaeaacaWGHbaaaiaacYcacaWGHbGaeyiyIKRaaGimaaaa@48EA@
x
2
=a⇒x=±
a
,
a≥0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabg2da9iaadggacqGHshI3caWG4bGaeyypa0JaeyySae7aaOaaaeaacaWGHbaaleqaaOGaaiilamaaxacabaaaleqabaaaaOGaamyyaiabgwMiZkaaicdaaaa@4591@
;
a⋅
x
2
+b⋅x+c=0⇒
x
1,2
=
−b±
b
2
−4ac
2a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgwSixlaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGIbGaeyyXICTaamiEaiabgUcaRiaadogacqGH9aqpcaaIWaGaeyO0H4TaamiEamaaBaaaleaacaaIXaGaaiilaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacqGHsislcaWGIbGaeyySae7aaOaaaeaacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGinaiaadggacaWGJbaaleqaaaGcbaGaaGOmaiaadggaaaaaaa@5556@
.
a≠0,
b
2
−4ac≥0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaaicdacaGGSaWaaCbiaeaacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGinaiaadggacaWGJbGaeyyzImRaaGimaiaac6caaSqabeaaaaaaaa@42D8@
| x |=a,
a≥0⇒x=±a.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG4baacaGLhWUaayjcSdGaeyypa0JaamyyaiaacYcadaWfGaqaaaWcbeqaaaaakiaadggacqGHLjYScaaIWaGaeyO0H4TaamiEaiabg2da9iabgglaXkaadggacaGGUaaaaa@484D@
x
=a,
a≥0⇒x=
a
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWG4baaleqaaOGaeyypa0JaamyyaiaacYcadaWfGaqaaiaadggacqGHLjYScaaIWaGaeyO0H4TaamiEaiabg2da9iaadggadaahaaWcbeqaaiaaikdaaaaabeqaaaaaaaa@4384@
[ x ]=a⇒a≤x〈a+1⇔x∈[a,a+1)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWG4baacaGLBbGaayzxaaGaeyypa0JaamyyaiabgkDiElaadggacqGHKjYOcaWG4bGaeyykJeUaamyyaiabgUcaRiaaigdacqGHuhY2caWG4bGaeyicI4Saai4waiaadggacaGGSaGaamyyaiabgUcaRiaaigdacaGGPaaaaa@4F81@
.
12. Procente
p % din N =
p
100
⋅N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGWbaabaGaaGymaiaaicdacaaIWaaaaiabgwSixlaad6eaaaa@3C44@
D =
S⋅p⋅n
100⋅12
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGtbGaeyyXICTaamiCaiabgwSixlaad6gaaeaacaaIXaGaaGimaiaaicdacqGHflY1caaIXaGaaGOmaaaaaaa@4347@
…. Dobânda obtinuta prin depunerea la banca
a unei sume S de bani pe o perioada
de n luni cu procentul p al dobândei anuale acordate de banca
.
Cât la suta reprezinta
numarul a din N.
x % din N =a
⇒x=
a⋅100
N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4TaamiEaiabg2da9maalaaabaGaamyyaiabgwSixlaaigdacaaIWaGaaGimaaqaaiaad6eaaaaaaa@4095@
.
13. Partea
intreaga
1.
x=[ x ]+{ x }
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9maadmaabaGaamiEaaGaay5waiaaw2faaiabgUcaRmaacmaabaGaamiEaaGaay5Eaiaaw2haaaaa@3EF5@
,
∀x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaadkfaaaa@3A1B@
,
[ x ]∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWG4baacaGLBbGaayzxaaGaeyicI4SaamOwaaaa@3B45@
si
{ x }∈[0,1)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWG4baacaGL7bGaayzFaaGaeyicI4Saai4waiaaicdacaGGSaGaaGymaiaacMcaaaa@3E56@
2.
[ x ]≤x<
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWG4baacaGLBbGaayzxaaGaeyizImQaamiEaiabgYda8aaa@3C98@
[ x ]+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWG4baacaGLBbGaayzxaaGaey4kaSIaaGymaaaa@3A7F@
[ x ]=a⇒a≤x<a+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWG4baacaGLBbGaayzxaaGaeyypa0JaamyyaiabgkDiElaadggacqGHKjYOcaWG4bGaeyipaWJaamyyaiabgUcaRiaaigdaaaa@444A@
3.
[ x ]=[ y ]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWG4baacaGLBbGaayzxaaGaeyypa0ZaamWaaeaacaWG5baacaGLBbGaayzxaaaaaa@3CD8@
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
∃K∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqIaam4saiabgIGiolaadQfaaaa@39FB@
a. i.
x,y∈[
k,k+1
]⇔|
x−y
|<1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaeyicI48aamWaaeaacaWGRbGaaiilaiaadUgacqGHRaWkcaaIXaaacaGLBbGaayzxaaGaeyi1HS9aaqWaaeaacaWG4bGaeyOeI0IaamyEaaGaay5bSlaawIa7aiabgYda8iaaigdaaaa@4A66@
4.
[
x+k
]=k+[ x ]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWG4bGaey4kaSIaam4AaaGaay5waiaaw2faaiabg2da9iaadUgacqGHRaWkdaWadaqaaiaadIhaaiaawUfacaGLDbaaaaa@407B@
,
∀k∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaam4AaiabgIGiolaadQfaaaa@3A16@
,
x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadkfaaaa@394B@
5.
{
x+k
}={ x }
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWG4bGaey4kaSIaam4AaaGaay5Eaiaaw2haaiabg2da9maacmaabaGaamiEaaGaay5Eaiaaw2haaaaa@3F27@
,
∀x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaadkfaaaa@3A1B@
,
∀k∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaam4AaiabgIGiolaadQfaaaa@3A16@
6. Daca
{ x }={ y }⇒x−y∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWG4baacaGL7bGaayzFaaGaeyypa0ZaaiWaaeaacaWG5baacaGL7bGaayzFaaGaeyO0H4TaamiEaiabgkHiTiaadMhacqGHiiIZcaWGAbaaaa@44FE@
7. Daca
x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadkfaaaa@394B@
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3850@
[
[ x ]
]=[ x ]∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaadaWadaqaaiaadIhaaiaawUfacaGLDbaaaiaawUfacaGLDbaacqGH9aqpdaWadaqaaiaadIhaaiaawUfacaGLDbaacqGHiiIZcaWGAbaaaa@412C@
[
{ x }
]=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaadaGadaqaaiaadIhaaiaawUhacaGL9baaaiaawUfacaGLDbaacqGH9aqpcaaIWaaaaa@3CD3@
,
{
[ x ]
}=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaadaWadaqaaiaadIhaaiaawUfacaGLDbaaaiaawUhacaGL9baacqGH9aqpcaaIWaaaaa@3CD3@
,
{
{ x }
}={ x }
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaadaGadaqaaiaadIhaaiaawUhacaGL9baaaiaawUhacaGL9baacqGH9aqpdaGadaqaaiaadIhaaiaawUhacaGL9baaaaa@3F86@
8. Identitatea
lui Hermite
[ x ]+[
x+
1
2
]=[
2x
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWG4baacaGLBbGaayzxaaGaey4kaSYaamWaaeaacaWG4bGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaaaiaawUfacaGLDbaacqGH9aqpdaWadaqaaiaaikdacaWG4baacaGLBbGaayzxaaaaaa@43CD@
,
∀x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaadkfaaaa@3A1B@
9.
[
x+y
]≥[ x ]+[ y ]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWG4bGaey4kaSIaamyEaaGaay5waiaaw2faaiabgwMiZoaadmaabaGaamiEaaGaay5waiaaw2faaiabgUcaRmaadmaabaGaamyEaaGaay5waiaaw2faaaaa@4349@
,
∀x,y∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiaacYcacaWG5bGaeyicI4SaamOuaaaa@3BC9@
10.
Prima zecimala, dupa virgula, a
unui numar N este data
de
[
10⋅{ N }
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaIXaGaaGimaiabgwSixpaacmaabaGaamOtaaGaay5Eaiaaw2haaaGaay5waiaaw2faaaaa@3EA8@
sau
[
(
N−[ N ]
)⋅10
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaadaqadaqaaiaad6eacqGHsisldaWadaqaaiaad6eaaiaawUfacaGLDbaaaiaawIcacaGLPaaacqGHflY1caaIXaGaaGimaaGaay5waiaaw2faaaaa@41B2@
2.
Inegalitati
1.
a>1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg6da+iaaigdaaaa@389C@
a
k−1
<
a
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCaaaleqabaGaam4AaiabgkHiTiaaigdaaaGccqGH8aapcaWGHbWaaWbaaSqabeaacaWGRbaaaaaa@3CAF@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
k≥1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgwMiZkaaigdaaaa@3964@
a∈(
0,1
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiopaabmaabaGaaGimaiaacYcacaaIXaaacaGLOaGaayzkaaaaaa@3C0B@
a
k
<
a
k−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCaaaleqabaGaam4AaaaakiabgYda8iaadggadaahaaWcbeqaaiaadUgacqGHsislcaaIXaaaaaaa@3CAF@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
k≥1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgwMiZkaaigdaaaa@3964@
2.
0<a≤b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgYda8iaadggacqGHKjYOcaWGIbaaaa@3B33@
⇒(
a
m
−
b
m
)(
a
n
−
b
n
)≥0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H49aaeWaaeaacaWGHbWaaWbaaSqabeaacaWGTbaaaOGaeyOeI0IaamOyamaaCaaaleqabaGaamyBaaaaaOGaayjkaiaawMcaamaabmaabaGaamyyamaaCaaaleqabaGaamOBaaaakiabgkHiTiaadkgadaahaaWcbeqaaiaad6gaaaaakiaawIcacaGLPaaacqGHLjYScaaIWaaaaa@47FC@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
m,n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaacYcacaWGUbGaeyicI4SaamOtaaaa@3ADF@
3.
a+
1
a
≥2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgUcaRmaalaaabaGaaGymaaqaaiaadggaaaGaeyyzImRaaGOmaaaa@3BEE@
(
∀
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqGHaiIiaiaawIcacaGLPaaaaaa@384C@
a>0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg6da+iaaicdaaaa@389B@
a+
1
a
≤−2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgUcaRmaalaaabaGaaGymaaqaaiaadggaaaGaeyizImQaeyOeI0IaaGOmaaaa@3CCA@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
a<0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgYda8iaaicdacaGGUaaaaa@3949@
4.
1
2
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmamaakaaabaGaam4AaaWcbeaaaaaaaa@3885@
<
1
k
+
k−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaWGRbaaleqaaOGaey4kaSYaaOaaaeaacaWGRbGaeyOeI0IaaGymaaWcbeaaaaaaaa@3B68@
=
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGRbaaleqaaaaa@36FE@
-
k−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGRbGaeyOeI0IaaGymaaWcbeaaaaa@38A6@
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F2@
1
2
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmamaakaaabaGaam4AaaWcbeaaaaaaaa@3885@
>
1
k
+
k+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaWGRbaaleqaaOGaey4kaSYaaOaaaeaacaWGRbGaey4kaSIaaGymaaWcbeaaaaaaaa@3B5D@
=
k+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGRbGaey4kaSIaaGymaaWcbeaaaaa@389B@
-
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGRbaaleqaaaaa@36FE@
.
5.
a
2
+
b
2
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaaaOqaaiaaikdaaaaaaa@3B54@
≥
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImlaaa@37B9@
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F2@
(
a+b
2
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWcaaqaaiaadggacqGHRaWkcaWGIbaabaGaaGOmaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@3BE0@
≥ab
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImRaamyyaiaadkgaaaa@3986@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
a,b∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4SaamOuaaaa@3ACB@
6.
a
2
+
b
2
a+b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaaaOqaaiaadggacqGHRaWkcaWGIbaaaaaa@3D47@
≥
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImlaaa@37B9@
a+b
2
≥
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbGaey4kaSIaamOyaaqaaiaaikdaaaGaeyyzImlaaa@3B34@
ab
≥
2
1
a
+
1
b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGHbGaamOyaaWcbeaakiabgwMiZoaalaaabaGaaGOmaaqaamaalaaabaGaaGymaaqaaiaadggaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOyaaaaaaaaaa@3EBC@
,
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
a,b>0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyOpa4JaaGimaaaa@3A32@
7.
a
2
+
b
2
+
c
2
≥ab+bc+ca
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGJbWaaWbaaSqabeaacaaIYaaaaOGaeyyzImRaamyyaiaadkgacqGHRaWkcaWGIbGaam4yaiabgUcaRiaadogacaWGHbaaaa@4639@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
a,b,c∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaaiilaiaadogacqGHiiIZcaWGsbaaaa@3C63@
8.
3(
a
2
+
b
2
+
c
2
)
≥
(
a+b+c
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaabeaabaGaamyyamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqacaqaaiaadogadaahaaWcbeqaaiaaikdaaaaakiaawMcaaaGaayjkaaGaeyyzIm7aaeWaaeaacaWGHbGaey4kaSIaamOyaiabgUcaRiaadogaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@4841@
∀a,b,c∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamyyaiaacYcacaWGIbGaaiilaiaadogacqGHiiIZaaa@3C5C@
R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaaaa@36CA@
9.
a
2
+
b
2
+
c
2
a+b+c
≥
1
3
(
a+b+c
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadogadaahaaWcbeqaaiaaikdaaaaakeaacaWGHbGaey4kaSIaamOyaiabgUcaRiaadogaaaGaeyyzIm7aaSaaaeaacaaIXaaabaGaaG4maaaadaqadaqaaiaadggacqGHRaWkcaWGIbGaey4kaSIaam4yaaGaayjkaiaawMcaaaaa@4B1E@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
a,b,c∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaaiilaiaadogacqGHiiIZcaWGsbaaaa@3C63@
10.
a+b+c
≥
3
3
(
a
+
b
+
c
)∀a,b,c≥0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGHbGaey4kaSIaamOyaiabgUcaRiaadogaaSqabaGccqGHLjYSdaWcaaqaamaakaaabaGaaG4maaWcbeaaaOqaaiaaiodaaaWaaeWaaeaadaGcaaqaaiaadggaaSqabaGccqGHRaWkdaGcaaqaaiaadkgaaSqabaGccqGHRaWkdaGcaaqaaiaadogaaSqabaaakiaawIcacaGLPaaacqGHaiIicaWGHbGaaiilaiaadkgacaGGSaGaam4yaiabgwMiZkaaicdaaaa@4BDC@
11.
(
n−1
)(
a
1
2
+...+
a
n
2
)≥2(
a
1
a
2
+...
a
1
a
n
+
a
2
a
3
+...+
a
n−1
a
n
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@61A5@
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F2@
12.
n(
a
1
2
+...+
a
n
2
)≥
(
a
1
+...+
a
n
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaabmaabaGaamyyamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadggadaqhaaWcbaGaamOBaaqaaiaaikdaaaaakiaawIcacaGLPaaacqGHLjYSdaqadaqaaiaadggadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkcaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@4DA1@
,
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolaad6eaaaa@393D@
13.
a
n
+
b
n
2
≥
(
a+b
2
)
2
,∀n∈N,a,b>0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbWaaWbaaSqabeaacaWGUbaaaOGaey4kaSIaamOyamaaCaaaleqabaGaamOBaaaaaOqaaiaaikdaaaGaeyyzIm7aaeWaaeaadaWcaaqaaiaadggacqGHRaWkcaWGIbaabaGaaGOmaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaGGSaGaeyiaIiIaamOBaiabgIGiolaad6eacaGGSaGaamyyaiaacYcacaWGIbGaeyOpa4JaaGimaiaac6caaaa@4DEA@
14.
0<
a
b
<2⇒
a
b
<
a+r
b+r
,∀r>0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgYda8maalaaabaGaamyyaaqaaiaadkgaaaGaeyipaWJaaGOmaiabgkDiEpaalaaabaGaamyyaaqaaiaadkgaaaGaeyipaWZaaSaaaeaacaWGHbGaey4kaSIaamOCaaqaaiaadkgacqGHRaWkcaWGYbaaaiaacYcacqGHaiIicaWGYbGaeyOpa4JaaGimaiaac6caaaa@4B06@
1<
a
b
⇒
a
b
>
a+r
b+r
,∀r>0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgYda8maalaaabaGaamyyaaqaaiaadkgaaaGaeyO0H49aaSaaaeaacaWGHbaabaGaamOyaaaacqGH+aGpdaWcaaqaaiaadggacqGHRaWkcaWGYbaabaGaamOyaiabgUcaRiaadkhaaaGaaiilaiabgcGiIiaadkhacqGH+aGpcaaIWaaaaa@4899@
15.
| x |
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG4baacaGLhWUaayjcSdaaaa@3A12@
≤a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyizImQaamyyaaaa@388E@
(
a>0
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbGaeyOpa4JaaGimaaGaayjkaiaawMcaaaaa@3A24@
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
−a≤x≤a.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamyyaiabgsMiJkaadIhacqGHKjYOcaWGHbGaaiOlaaaa@3DC5@
16.
|
a±b
|≤| a |+| b |
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbGaeyySaeRaamOyaaGaay5bSlaawIa7aiabgsMiJoaaemaabaGaamyyaaGaay5bSlaawIa7aiabgUcaRmaaemaabaGaamOyaaGaay5bSlaawIa7aaaa@4778@
,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiilaaaa@36A3@
a,b∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4SaamOuaaaa@3ACB@
sauC
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaadggacaWG1bGaam4qaaaa@3993@
.
17.
|
a
1
±
a
2
±...±
a
n
|≤|
a
1
|+...+|
a
n
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaeyySaeRaamyyamaaBaaaleaacaaIYaaabeaakiabgglaXkaac6cacaGGUaGaaiOlaiabgglaXkaadggadaWgaaWcbaGaamOBaaqabaaakiaawEa7caGLiWoacqGHKjYOdaabdaqaaiaadggadaWgaaWcbaGaaGymaaqabaaakiaawEa7caGLiWoacqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkdaabdaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakiaawEa7caGLiWoaaaa@566C@
,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiilaaaa@36A3@
in
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaad6gaaaa@37D4@
R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaaaa@36CA@
sau
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaadggacaWG1baaaa@38CB@
C
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaaaa@36BB@
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOlaaaa@36A5@
18.
|
| a |−| b |
|≤|
a−b
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaabdaqaaiaadggaaiaawEa7caGLiWoacqGHsisldaabdaqaaiaadkgaaiaawEa7caGLiWoaaiaawEa7caGLiWoacqGHKjYOdaabdaqaaiaadggacqGHsislcaWGIbaacaGLhWUaayjcSdaaaa@49A4@
in
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaad6gaaaa@37D4@
R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaaaa@36CA@
sau
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaadggacaWG1baaaa@38CB@
C
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaaaa@36BB@
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOlaaaa@36A5@
19.
1
n
2
=
1
n⋅n
≤
1
(
n−1
)n
=
1
n−1
−
1
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOBamaaCaaaleqabaGaaGOmaaaaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGUbGaeyyXICTaamOBaaaacqGHKjYOdaWcaaqaaiaaigdaaeaadaqadaqaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaaGaamOBaaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGUbGaeyOeI0IaaGymaaaacqGHsisldaWcaaqaaiaaigdaaeaacaWGUbaaaaaa@4D53@
1
n!
<
1
(
n−1
)n
=
1
n−1
−
1
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOBaiaacgcaaaGaeyipaWZaaSaaaeaacaaIXaaabaWaaeWaaeaacaWGUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaiaad6gaaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOBaiabgkHiTiaaigdaaaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamOBaaaaaaa@4653@
20.
a,b∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4SaamOwaaaa@3AD3@
,
m,n∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaacYcacaWGUbGaeyicI4SaamOwaaaa@3AEB@
,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiilaaaa@36A3@
m
n
∉Q
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaadaWcaaqaaiaad2gaaeaacaWGUbaaaaWcbaaaaOGaeyycI8Saamyuaaaa@3A69@
⇒|
m
a
2
−n
b
2
|≥1.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H49aaqWaaeaacaWGTbGaamyyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaad6gacaWGIbWaaWbaaSqabeaacaaIYaaaaaGccaGLhWUaayjcSdGaeyyzImRaaGymaiaac6caaaa@452A@
21. Numerele pozitive
a,b,c
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaaiilaiaadogaaaa@3A08@
pot fi lungimile laturilor unui triunghi daca si numai daca
∃
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqcaaa@36C8@
x,y,z∈
R
+
*
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaaiilaiaadQhacqGHiiIZcaWGsbWaa0baaSqaaiabgUcaRaqaaiaacQcaaaaaaa@3E65@
a.i
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaac6cacaWGPbaaaa@3879@
a=y+z,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaadMhacqGHRaWkcaWG6bGaaiilaaaa@3B6E@
b=x+z,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2da9iaadIhacqGHRaWkcaWG6bGaaiilaaaa@3B6E@
c=x+y.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2da9iaadIhacqGHRaWkcaWG5bGaaiOlaaaa@3B70@
22.
(
a
b
)
a−b
≥1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWcaaqaaiaadggaaeaacaWGIbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaamyyaiabgkHiTiaadkgaaaGccqGHLjYScaaIXaaaaa@3ECB@
a≠b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaadkgaaaa@3987@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
a,b>0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyOpa4JaaGimaaaa@3A32@
,
23.
a,b,c∈
R
+
*
⇒
a+b
c
+
b+c
a
+
c+a
b
≥6.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaaiilaiaadogacqGHiiIZcaWGsbWaa0baaSqaaiabgUcaRaqaaiaacQcaaaGccqGHshI3daWcaaqaaiaadggacqGHRaWkcaWGIbaabaGaam4yaaaacqGHRaWkdaWcaaqaaiaadkgacqGHRaWkcaWGJbaabaGaamyyaaaacqGHRaWkdaWcaaqaaiaadogacqGHRaWkcaWGHbaabaGaamOyaaaacqGHLjYScaaI2aGaaiOlaaaa@5078@
24. Daca
x
1
,...,
x
n
≥0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamiEamaaBaaaleaacaWGUbaabeaakiabgwMiZkaaicdaaaa@3FFD@
si
x
1
+...+
x
n
=k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadIhadaWgaaWcbaGaamOBaaqabaGccqGH9aqpcaWGRbaaaa@3FD7@
constant atunci produsul
x
2
⋅
x
2
...
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaakiabgwSixlaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGUaGaaiOlaiaac6cacaWG4bWaaSbaaSqaaiaad6gaaeqaaaaa@404D@
e maxim când
x
1
=...=
x
n
=
k
n
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabg2da9iaac6cacaGGUaGaaiOlaiabg2da9iaadIhadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaadUgaaeaacaWGUbaaaiaac6caaaa@41D4@
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F2@
25. Daca.
x
1
,...,
x
n
<0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamiEamaaBaaaleaacaWGUbaabeaakiabgYda8iaaicdaaaa@3F3B@
si
∏
i=1
n
x
i
=k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHpis1aOGaamiEamaaBaaaleaacaWGPbaabeaakiabg2da9iaadUgaaaa@3FE7@
constant
⇒
x
1
+...+
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4TaamiEamaaBaaaleaacaaIXaaabeaakiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadIhadaWgaaWcbaGaamOBaaqabaaaaa@4034@
e minima atunci când
x
1
=...=
x
n
=
k
n
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabg2da9iaac6cacaGGUaGaaiOlaiabg2da9iaadIhadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaGcbaqaaiaadUgaaSqaaiaad6gaaaGccaGGUaaaaa@41E9@
26. Daca
x
1
,...,
x
n
≥0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamiEamaaBaaaleaacaWGUbaabeaakiabgwMiZkaaicdaaaa@3FFD@
si
x
1
+...+
x
n
=k=
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadIhadaWgaaWcbaGaamOBaaqabaGccqGH9aqpcaWGRbGaeyypa0daaa@40DD@
constant atunci
x
2
p
1
⋅
x
2
p
1
...
x
n
p
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDaaaleaacaaIYaaabaGaamiCamaaBaaameaacaaIXaaabeaaaaGccqGHflY1caWG4bWaa0baaSqaaiaaikdaaeaacaWGWbWaaSbaaWqaaiaaigdaaeqaaSWaaWbaaWqabeaaaaaaaOGaaiOlaiaac6cacaGGUaGaamiEamaaDaaaleaacaWGUbaabaGaamiCamaaBaaameaacaWGUbaabeaaaaaaaa@4658@
este maxim când
x
1
p
1
=
x
2
p
2
=...
x
n
p
n
=
k
p
1
+...+
p
n
,
p
i
∈
N
*
,i=
1,n
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5B2F@
27. Teorema lui Jensen:
Daca
f:
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdaaaa@379C@
Ι
→R,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4QaamOuaiaacYcaaaa@3967@
(Ι interval) si
f(
x
1
+
x
2
2
)
≤
(
≥
)
f(
x
1
)+f(
x
2
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaWaaSaaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamiEamaaBaaaleaacaaIYaaabeaaaOqaaiaaikdaaaaacaGLOaGaayzkaaGaeyizIm6aaSbaaSqaamaabmaabaGaeyyzImlacaGLOaGaayzkaaaabeaakmaalaaabaGaamOzamaabmaabaGaamiEamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiabgUcaRiaadAgadaqadaqaaiaadIhadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaeaacaaIYaaaaaaa@4D9F@
∀
x
1
,
x
2
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyicI4maaa@3CD4@
Ι
⇒f(
x
2
+...+
x
n
n
)
≤
(
≥
)
f(
x
1
)+...+f(
x
n
)
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4TaamOzamaabmaabaWaaSaaaeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaaiOlaiaac6cacaGGUaGaey4kaSIaamiEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6gaaaaacaGLOaGaayzkaaGaeyizIm6aaSbaaSqaamaabmaabaGaeyyzImlacaGLOaGaayzkaaaabeaakmaalaaabaGaamOzamaabmaabaGaamiEamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadAgadaqadaqaaiaadIhadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaaaeaacaWGUbaaaaaa@56C9@
∀
x
i
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEamaaBaaaleaacaWGPbaabeaakiabgIGiodaa@3A68@
Ι
,
i=
1,n.
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2da9maanaaabaGaaGymaiaacYcacaWGUbGaaiOlaaaaaaa@3B08@
28. Inegalitatea mediilor
n
1
a
1
+...+
1
a
n
≤
a
1
...
a
n
n
≤
a
1
+...+
a
n
n
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGUbaabaWaaSaaaeaacaaIXaaabaGaamyyamaaBaaaleaacaaIXaaabeaaaaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaaaaaGccqGHKjYOdaGcbaqaaiaadggadaWgaaWcbaGaaGymaaqabaGccaGGUaGaaiOlaiaac6cacaWGHbWaaSbaaSqaaiaad6gaaeqaaaqaaiaad6gaaaGccqGHKjYOdaWcaaqaaiaadggadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkcaWGHbWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaacaGGUaaaaa@542A@
29.
(
a
1
+
a
2
+...+
a
n
)(
1
a
1
+...+
1
a
n
)≥
n
2
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaIYaaabeaakiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadggadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaadaqadaqaamaalaaabaGaaGymaaqaaiaadggadaWgaaWcbaGaaGymaaqabaaaaOGaey4kaSIaaiOlaiaac6cacaGGUaGaey4kaSYaaSaaaeaacaaIXaaabaGaamyyamaaBaaaleaacaWGUbaabeaaaaaakiaawIcacaGLPaaacqGHLjYScaWGUbWaaWbaaSqabeaacaaIYaaaaOGaaiOlaaaa@5133@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
a
i
≥0,i=
1,n.
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaakiabgwMiZkaaicdacaGGSaGaamyAaiabg2da9maanaaabaGaaGymaiaacYcacaWGUbGaaiOlaaaaaaa@4042@
egalitate când
a
i
=aj,∀i,j=
1,n.
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaakiabg2da9iaadggacaWGQbGaaiilaiabgcGiIiaadMgacaGGSaGaamOAaiabg2da9maanaaabaGaaGymaiaacYcacaWGUbGaaiOlaaaaaaa@430C@
30. Inegalitatea lui Cauchy-Buniakowsky-Schwartz.
(
a
1
2
+...+
a
n
2
)(
b
1
2
+...+
b
n
2
)≥
(
a
1
b
1
+...+
a
n
b
n
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5B5B@
∀
a
i
,
b
i
∈R.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamyyamaaBaaaleaacaWGPbaabeaakiaacYcacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaeyicI4SaamOuaiaac6caaaa@3E95@
31. Inegalitatea mediilor generalizate:
"="⇔
ai
bi
=
aj
bj
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOiaiabg2da9iaackcacqGHuhY2daWcaaqaaiaadggacaWGPbaabaGaamOyaiaadMgaaaGaeyypa0ZaaSaaaeaacaWGHbGaamOAaaqaaiaadkgacaWGQbaaaiaac6caaaa@43CD@
(
a
1
α
+...+
a
n
α
n
)
1
α
≥
(
a
1
β
+...+
a
n
β
n
)
1
β
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWcaaqaaiaadggadaqhaaWcbaGaaGymaaqaaiabeg7aHbaakiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadggadaqhaaWcbaGaamOBaaqaaiabeg7aHbaaaOqaaiaad6gaaaaacaGLOaGaayzkaaWaaWbaaSqabeaadaWcaaqaaiaaigdaaeaacqaHXoqyaaaaaOGaeyyzIm7aaeWaaeaadaWcaaqaaiaadggadaqhaaWcbaGaaGymaaqaaiabek7aIbaakiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadggadaqhaaWcbaGaamOBaaqaaiabek7aIbaaaOqaaiaad6gaaaaacaGLOaGaayzkaaWaaWbaaSqabeaadaWcaaqaaiaaigdaaeaacqaHYoGyaaaaaaaa@580F@
,∀
a
i
,
b
i
∈
R
+
,α≥β,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiilaiabgcGiIiaadggadaWgaaWcbaGaamyAaaqabaGccaGGSaGaamOyamaaBaaaleaacaWGPbaabeaakiabgIGiolaadkfadaWgaaWcbaGaey4kaScabeaakiaacYcacqaHXoqycqGHLjYScqaHYoGycaGGSaaaaa@4611@
α,β∈R.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaiilaiabek7aIjabgIGiolaadkfacaGGUaaaaa@3CF0@
⇓
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey40H8naaa@3852@
32.
(
a
1
2
+...+
a
n
2
n
)
1
2
≥
a
1
+...+
a
n
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWcaaqaaiaadggadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkcaWGHbWaa0baaSqaaiaad6gaaeaacaaIYaaaaaGcbaGaamOBaaaaaiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaaikdaaaaaaOGaeyyzIm7aaSaaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaaiOlaiaac6cacaGGUaGaey4kaSIaamyyamaaBaaaleaacaWGUbaabeaaaOqaaiaad6gaaaaaaa@4E00@
33.Inegalitatea lui
Bernoulli:
(
1+a
)
n
≥1+na,a≥−1,∀n∈N.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIXaGaey4kaSIaamyyaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaaaakiabgwMiZkaaigdacqGHRaWkcaWGUbGaamyyaiaacYcacaWGHbGaeyyzImRaeyOeI0IaaGymaiaacYcacqGHaiIicaWGUbGaeyicI4SaamOtaiaac6caaaa@4AE5@
3.Multimi. Operatii cu multimi.
1. Asociativitatea reuniunii si a intersectiei:
A
(BC)=(AB)C A
(BC)=(AB)C
2. Comutativitatea reuniunii si a intersectiei:
AB=BA A
B=BA
3. Idempotenta reuniunii si intersectiei:
A
A=A A
A=A
4. A
Ø=A A
Ø=Ø
5. Distributivitatea reuniunii fata de
intersectie:
A
(B
C)=(AB)(AC)
6. Distributivitatea intersectiei fata de
reuniune:
A(BC)=(AB)(AC)
7. A,B
E, 
(AB)=A
B (A
B)=
A
B
8. A
E, 
(A)=A
9. A\B=
(AB)
10. A\(BC)=(A\B)\C
A\(B
C)=(A\B)(A\C)
(AB)\C=(A\C)(B\C)
(AB)\C=A(B\C)=(A\C)B
11. A×(B
C)=(A×B)(A×C)
A×(BC)=(A×B)(A×C)
A×(B\C)=(A×B)\ (A×C)
A×B≠B×A
A
B
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
(
x)
(x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
A=>x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
B)
A
B
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
(
x)((x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
A) 
(x
B))
x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
AB
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
(x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
A) 
(x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
B)
x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
A
B
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
(x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
A) 
(x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
B)
x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
C EA
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
(x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
E)
(x
A)
x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
A\B
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
(x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
A) (x
B)
12. Relatiile lui de Morgan
1.
ך (p
q)=ךp 
ךq, ך(pq)=
ךp 
ךq .
2.
p(q
r)
= (p
q)
(p
r),
p(qr)=(p
q)
(p
r).
3.
ךp
p=A, ךp 
p = F.
4.
p
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3850@
q 
ךp q.
5.
p
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
q (p
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3850@
q)
(q
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3850@
p) (ךp 
q) (ך q
p).
6.
p A = p , p 
A=A
7.
p q = q p , p q = q p
8.
ך(ךp)=p
9.
p ךp =F , p
ךp =A
10.
(p q) r = p (q r)
(p 
q) r = p (q r)
11.
p 
F = p
p 
F = F
4. Progresii
1. Siruri
Se cunosc deja sirul numerelor
naturale 0,1,2,3,4,…….,sirul numerelor
pare 2,4,6,…… Din observatiile directe asupra acestor siruri, un sir de numere reale este dat in forma
a
1
,
a
2
,
a
3
,.....
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadggadaWgaaWcbaGaaG4maaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caaaa@4105@
unde
a
1
,
a
2
,
a
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadggadaWgaaWcbaGaaG4maaqabaaaaa@3CD1@
sunt termenii sirului iar indicii 1,2,3,
reprezinta pozitia pe care ii ocupa termenii in sir.
Definitie: Se numeste sir de numere reale o functie f:
N*→R , definita prin f(n)=a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
Notam
(
a
n
)
n∈N*
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobGaaiOkaaqabaaaaa@3DAF@
sirul de termen general , a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
Observatie: Numerotarea termenilor unui sir se mai
poate face incepând cu zero:
a
0
,
a
1
,
a
2
,.....
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIWaaabeaakiaacYcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadggadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caaaa@4102@
a
i
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaaaaa@37F3@
, i
≥
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImlaaa@37B9@
1 se numeste termenul de rang i.
Un sir poate fi definit prin :
a) descrierea
elementelor multimii de termeni. 2,4,6,8,……..
b) cu
ajutorul unei formule a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
=2n
c)
printr-o relatie de recurenta.
a
n+1
=
a
n
+2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaey4kaSIaaGOmaaaa@3E52@
Un sir constant este un sir in care toti termenii sirului
sunt constanti : 5,5,5,5,…..
Doua siruri
(
a
n
)
n
,
(
b
n
)
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaaqabaGccaGGPaWaaSbaaSqaaiaad6gaaeqaaOGaaiilaiaacIcacaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaaiykamaaBaaaleaacaWGUbaabeaaaaa@3FBC@
sunt egale
daca
a
n
=
b
n
,∀n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabg2da9iaadkgadaWgaaWcbaGaamOBaaqabaGccaGGSaGaeyiaIiIaamOBaiabgIGiolaad6eaaaa@3FE2@
Orice sir are o infinitate de termeni.
2.
Progresii aritmetice
Definitie: Se numeste progresie
aritmetica un sir in care diferenta oricaror doi termeni consecutivi este
un numar constant r, numit ratia progresiei
aritmetice.
1. Relatia de
recurenta intre doi termeni consecutivi:
a
n+1
=
a
n
+r,∀n≥1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaey4kaSIaamOCaiaacYcacqGHaiIicaWGUbGaeyyzImRaaGymaaaa@4381@
2.
a1,a2, … an-1, an, an+1
sunt termenii unei progresii aritmetice
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
a
n
=
a
n−1
+
a
n+1
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabg2da9maalaaabaGaamyyamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaOqaaiaaikdaaaaaaa@4219@
3. Termenul
general este dat de :
a
n
=
a
1
+(
n−1
)r
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabg2da9iaadggadaWgaaWcbaGaaGymaaqabaGccqGHRaWkdaqadaqaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaaGaamOCaaaa@40DC@
4. Suma oricaror doi termeni egal departati de
extremi este egal cu suma termenilor
extremi :
a
k
+
a
n−k+1
=
a
1
+
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGRbaabeaakiabgUcaRiaadggadaWgaaWcbaGaamOBaiabgkHiTiaadUgacqGHRaWkcaaIXaaabeaakiabg2da9iaadggadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaad6gaaeqaaaaa@442E@
5. Suma primilor n termeni :
S
n
=
(
a
1
+
a
n
)⋅n
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWGUbaabeaakiabg2da9maalaaabaWaaeWaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaaiabgwSixlaad6gaaeaacaaIYaaaaaaa@4354@
6. Sirul termenilor unei progresii
aritmetice:
a
1
,
a
1
+r,
a
1
+2r,
a
1
+3r
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamOCaiaacYcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaaGOmaiaadkhacaGGSaGaamyyamaaBaaaleaacaaIXaaabeaakiabgUcaRiaaiodacaWGYbaaaa@4663@
,…….
a
m
−
a
n
=(
m−n
)r
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGTbaabeaakiabgkHiTiaadggadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaqadaqaaiaad2gacqGHsislcaWGUbaacaGLOaGaayzkaaGaamOCaaaa@4155@
7. Trei numere x1, x2,
x3 se scriu in progresie aritmetica de forma :
x1
= u
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@37A0@
v x2
= u x3 = u + v
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
u,v
∈ℜ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaeyihHimaaa@3902@
.
8. Patru numere x1, x2,
x3, x4 se scriu in
progresie aritmetica astfel:
x1 = u
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@37A0@
3v, x2
= u
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@37A0@
v , x3
= u + v , x4 = u + 3v,
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
u,v
∈ℜ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaeyihHimaaa@3902@
.
9. Daca
÷
a
i
⇒
a
k
a
k+1
〈
a
k+1
a
k+2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey49aGRaamyyamaaBaaaleaacaWGPbaabeaakiabgkDiEpaalaaabaGaamyyamaaBaaaleaacaWGRbaabeaaaOqaaiaadggadaWgaaWcbaGaam4AaiabgUcaRiaaigdaaeqaaaaakiabgMYiHpaalaaabaGaamyyamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaaakeaacaWGHbWaaSbaaSqaaiaadUgacqGHRaWkcaaIYaaabeaaaaaaaa@4B6C@
4.
Progresii geometrice
Definitie :
Se numeste progresie geometrica un sir
in care raportul oricaror doi
termeni consecutivi este un numar
constant q, numit ratia progresiei geometrice.
1. Relatia de
recurenta :
b
n+1
=
b
n
⋅q,∀n≥1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaeyyXICTaamyCaiaacYcacqGHaiIicaWGUbGaeyyzImRaaGymaaaa@44EA@
2. b1,b2, … bn-1, bn, bn+1
sunt termenii unei progresii
geometrice cu termeni pozitivi
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
b
n
=
b
n−1
⋅
b
n+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGUbaabeaakiabg2da9maakaaabaGaamOyamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHflY1caWGIbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaeqaaaaa@42BE@
3. Termenul general este dat de :
b
n
=
b
1
⋅
q
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGUbaabeaakiabg2da9iaadkgadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWGXbWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGymaaaaaaa@40E9@
4. Produsul oricaror doi termeni
egal departati de extremi este egal cu produsul extremilor
b
k
⋅
b
n−k+1
=
b
1
⋅
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGRbaabeaakiabgwSixlaadkgadaWgaaWcbaGaamOBaiabgkHiTiaadUgacqGHRaWkcaaIXaaabeaakiabg2da9iaadkgadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWGIbWaaSbaaSqaaiaad6gaaeqaaaaa@4702@
5. Suma primilor n termeni ai unei
progresii geometrice :
S
n
=
b
1
⋅
1−
q
n
1−q
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWGUbaabeaakiabg2da9iaadkgadaWgaaWcbaGaaGymaaqabaGccqGHflY1daWcaaqaaiaaigdacqGHsislcaWGXbWaaWbaaSqabeaacaWGUbaaaaGcbaGaaGymaiabgkHiTiaadghaaaaaaa@4392@
6. Sirul termenilor unei
progresii geometrice :
b
1
,
b
1
⋅q,
b
1
⋅
q
2
,...
b
1
⋅
q
n
,....
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaamyCaiaacYcacaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaamyCamaaCaaaleqabaGaaGOmaaaakiaacYcacaGGUaGaaiOlaiaac6cacaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaamyCamaaCaaaleqabaGaamOBaaaakiaacYcacaGGUaGaaiOlaiaac6cacaGGUaaaaa@50CE@
7. Trei numere x1, x2,
x3 se scriu in progresie geometrica de forma :
x1 =
u
v
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG1baabaGaamODaaaaaaa@37F8@
x2 = u x3 =
u⋅v
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabgwSixlaadAhaaaa@3A32@
,
∀u,v∈
R
*
+
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamyDaiaacYcacaWG2bGaeyicI4SaamOuamaaCaaaleqabaGaaiOkaaaakmaaBaaaleaacqGHRaWkaeqaaaaa@3DB6@
8. Patru numere x1, x2,
x3, x4 se scriu in progresie geometrica astfel :
x1 =
u
v
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG1baabaGaamODamaaCaaaleqabaGaaG4maaaaaaaaaa@38E2@
x2 =
u
v
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG1baabaGaamODaaaaaaa@37F8@
x3 =
u⋅v
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabgwSixlaadAhaaaa@3A32@
x4 =
u⋅
v
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabgwSixlaadAhadaahaaWcbeqaaiaaiodaaaaaaa@3B1C@
∀u,v∈
R
*
+
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamyDaiaacYcacaWG2bGaeyicI4SaamOuamaaCaaaleqabaGaaiOkaaaakmaaBaaaleaacqGHRaWkaeqaaaaa@3DB6@
5. Functii
I. Fie
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jzaaaa@3820@
: A→B.
1) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbyaqaaaaaaaaaWdbiaa=jzaaaa@3821@
este injectiva,daca
∀
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=bGiIaaa@37D6@
x,y
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
A, x≠ y=>
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jzaaaa@3820@
(x) ≠
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(y).
2) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este injectiva,daca din
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(x)=
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(y) =>x=y.
3) Functia f este injectiva, daca orice
paralela la axa 0x intersecteaza graficul
functiei in cel mult un punct.
II.
1)Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbyaqaaaaaaaaaWdbiaa=jzaaaa@3821@
este surjectiva, daca
∀
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=bGiIaaa@37D6@
y
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
B, exista cel putin un punct x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
A, a.i.
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(x)=y.
2) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jzaaaa@3820@
este surjectiva, daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jzaaaa@3820@
(A) =B.
3) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jzaaaa@3820@
este surjectiva, daca orice paralela la axa
0x, dusa printr-un punct al lui B, intersecteaza graficul functiei in cel putin
un punct.
III.
1) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacceqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CC@
este bijectiva
daca este injectiva si surjectiva.
2) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este bijectiva daca pentru orice y
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
B exista un singur x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
A a.i.
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(x) =y (ecuatia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(x)=y,are o singura solutie,pentru orice y din B)
3) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este bijectiva daca orice paralela la axa 0x,
dusa printr-un punct al lui B, intersecteaza graficul functiei intr-un punct si
numai unul.
IV.
1A: A→A prin 1A(x) =x,
∀
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=bGiIaaa@37D6@
x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
A.
1) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacceqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CC@
: A→B este inversabila
, daca exista o functie g:B→A astfel incât g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
= 1A si
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
o g =1B, functia g este inversa
functiei
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
si se noteaza cu
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jzaaaa@3820@
-1.
2)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(x) = y <=>
x=
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1(y)
3)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este bijectiva <=>
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este inversabila.
V. Fie
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
:A→B si g: B→C,
doua functii.
1)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
si g sunt injective, atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este injectiva.
2)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
si g sunt surjective,atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este surjectiva.
3)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
si g sunt bijective, atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este bijectiva.
4)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
si g sunt (strict) crescatoare,atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este (strict) crescatoare.
5)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
si g sunt (strict) descrescatoare, atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este (strict) descrescatoare.
6)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
si g sunt monotone, de monotonii
diferite,atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este descrescatoare.
7)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este periodica, atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este periodica.
8)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este para, atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este para.
9)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
si g sunt impare, atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este impara,
10)
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este impara si g para, atunci g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este para.
VI. Fie
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
: A→ B si g:B→C,
doua functii.
Daca g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este injectiva, atunci
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este injectiva.
Daca g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este surjectiva, atunci g este surjectiva.
Daca
g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este
bijectiva, atunci
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este injectiva si g surjectiva.
Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
,g: A → B iar h: B→ C
bijectiva si h o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
= h
o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
, atunci
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
= g.
VII. Fie
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
: A→B si X,Y multimi
oarecare.
Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este bijectiva, daca si numai daca oricare ar
fi functiile
u,v:
X→A,din
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
o u
=
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
o v, rezulta u=v.
Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este surjectiva, daca si numai daca oricare ar
fi functiile u,v :B→Y, din u o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
= v
o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
, rezulta u=v
VIII.
1)Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
:A→B este strict monotona,atunci
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este injectiva.
2) Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
: R→R este periodic si monotona, atunci
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este constanta.
3) Daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
: R→R este bijectiva si impara,atunci
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1
este impara.
4) Fie A finita si
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
:A→A. Atunci
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este injectiva <=> este surjectiva.
IX. Fie
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
: E → F, atunci
1)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
injectiva <=> (
∃
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=nGiKaaa@37DB@
) g : F →E (surjectiva) a.i. g o
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
=1E.
2)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
surjectiva <=>(
∃
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=nGiKaaa@37DB@
) g : E→F
(injectiva) a.i.
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
o g =1F
3)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
bijectiva <=> inversabila.
X. Fie
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
: E → F.
1)Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este injectiva daca si numai daca (
∀
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=bGiIaaa@37D6@
) A,B
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
E
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A ∩ B) =
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A) ∩ (B).
2) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este surjectiva daca si numai daca (
∀
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=bGiIaaa@37D6@
) B
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
F exista A
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
E, astfel incât
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A)=B.
3) Functia
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
este injectiva daca
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A
—
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=rbiaaa@37C1@
B)=
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A)
—
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=rbiaaa@37C1@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(B),
∀
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=bGiIaaa@37D6@
A, B
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
E.
XI. Fie
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
: E → F si A
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
E, B
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
E, atunci
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A) ={y
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
F ***TRANSLATION ERROR***
∃
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=nGiKaaa@37DB@
x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
A a.i.
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(x)=y}
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (B)
= {x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
E
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=zo+9caa@3A2D@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(x)
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=HGiodaa@388A@
B}.
1.Fie
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
: E→ F si A,B
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
E, atunci
a) A
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
B =>
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A)
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(B),
b)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A
∪
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=PIiidaa@38A6@
B)=
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A)
∪
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=PIiidaa@38A6@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(B),
c)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A ∩ B)
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A) ∩
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(B),
d)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A)
—
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=rbiaaa@37C1@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(B)
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
(A
—
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=rbiaaa@37C1@
B).
2.Fie
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
: E → F si A,B
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
F atunci
a) A
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
B =>
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (A)
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1
(B),
b)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (A)
∪
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=PIiidaa@38A6@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (B)
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jOimdaa@3902@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
--1 (A
∪
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=PIiidaa@38A6@
B),
c)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (A)
∩
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (B)
=
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 ( A
∩ B),
d)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (A)
—
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=rbiaaa@37C1@
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (B)
=
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (A
—
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=rbiaaa@37C1@
B),
e)
ƒ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbyaqaaaaaaaaaWdbiab=jzaMcaa@38CB@
-1 (F)
= E.
Functia de gradul al doilea
Forma
canonica a functiei f:R→R,
f(x)=a
x
2
+bx+c,
a,b,c∈R,a≠0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaadggacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyaiaadIhacqGHRaWkcaWGJbGaaiilamaavacabeWcbeqaaaqdbaaaaOGaamyyaiaacYcacaWGIbGaaiilaiaadogacqGHiiIZcaWGsbGaaiilaiaadggacqGHGjsUcaaIWaaaaa@4D37@
este
f(x)=a
(
x+
b
2a
)
2
−
Δ
4a
,∀x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaadggadaqadaqaaiaadIhacqGHRaWkdaWcaaqaaiaadkgaaeaacaaIYaGaamyyaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHsisldaWcaaqaaiabfs5aebqaaiaaisdacaWGHbaaaiaacYcacqGHaiIicaWG4bGaeyicI4SaamOuaaaa@4AF3@
;
Graficul
functiei este o parabola de vârf
V(
−
b
2a
,−
Δ
4a
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabmaabaGaeyOeI0YaaSaaaeaacaWGIbaabaGaaGOmaiaadggaaaGaaiilaiabgkHiTmaalaaabaGaeuiLdqeabaGaaGinaiaadggaaaaacaGLOaGaayzkaaaaaa@4094@
,
unde
Δ=
b
2
−4ac
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeyypa0JaamOyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaisdacaWGHbGaam4yaaaa@3DB2@
a〉0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgQYiXlaaicdaaaa@395D@
f este convexa;
Δ〈0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeyykJeUaaGimaaaa@39CC@
;
x1,x2
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
C
f(x) >0,
∀x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaadkfaaaa@3A1B@
;
V(
−
b
2a
,−
Δ
4a
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabmaabaGaeyOeI0YaaSaaaeaacaWGIbaabaGaaGOmaiaadggaaaGaaiilaiabgkHiTmaalaaabaGaeuiLdqeabaGaaGinaiaadggaaaaacaGLOaGaayzkaaaaaa@4094@
- punct de minim;
Δ=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeyypa0JaaGimaaaa@3919@
,
x1=x2
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
R 
f(x)
≥
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImlaaa@37B9@
0,
∀x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaadkfaaaa@3A1B@
;
f(x)=0
⇔x=−
b
2a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaamiEaiabg2da9iabgkHiTmaalaaabaGaamOyaaqaaiaaikdacaWGHbaaaaaa@3DD8@
Δ〉0,
x
1
≠
x
2
∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeyOkJeVaaGimaiaacYcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyiyIKRaamiEamaaBaaaleaacaaIYaaabeaakiabgIGiolaadkfaaaa@428C@
f(x)
≥
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImlaaa@37B9@
0,
∀x∈(−∞,
x
1
]∪[
x
2
,+∞)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaacIcacqGHsislcqGHEisPcaGGSaGaamiEamaaBaaaleaacaaIXaaabeaakiaac2facqGHQicYcaGGBbGaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacqGHRaWkcqGHEisPcaGGPaaaaa@47EB@
;
f(x)<0,
∀x∈(
x
1
,
x
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@3F2A@
Pentru
x∈(
−∞,−
b
2a
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiopaabmaabaGaeyOeI0IaeyOhIuQaaiilaiabgkHiTmaalaaabaGaamOyaaqaaiaaikdacaWGHbaaaaGaayjkaiaawMcaaaaa@4091@
functia este strict descrescatoare;
Pentru
x∈[−
b
2a
,+∞),
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaacUfacqGHsisldaWcaaqaaiaadkgaaeaacaaIYaGaamyyaaaacaGGSaGaey4kaSIaeyOhIuQaaiykaiaacYcaaaa@4139@
functia este strict crescatoare
a<0 functia este concava
Δ〈0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeyykJeUaaGimaaaa@39CC@
;
x1,x2
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
C
f(x) <0,
∀x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaadkfaaaa@3A1B@
;
V(
−
b
2a
,−
Δ
4a
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabmaabaGaeyOeI0YaaSaaaeaacaWGIbaabaGaaGOmaiaadggaaaGaaiilaiabgkHiTmaalaaabaGaeuiLdqeabaGaaGinaiaadggaaaaacaGLOaGaayzkaaaaaa@4094@
- punct de maxim
Δ=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeyypa0JaaGimaaaa@3919@
,
x1=x2
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
R
f(x)
≤
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyizImkaaa@37A8@
0,
∀x∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaadkfaaaa@3A1B@
;
f(x)=0
⇔x=−
b
2a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaamiEaiabg2da9iabgkHiTmaalaaabaGaamOyaaqaaiaaikdacaWGHbaaaaaa@3DD8@
Δ〉0,
x
1
≠
x
2
∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeyOkJeVaaGimaiaacYcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyiyIKRaamiEamaaBaaaleaacaaIYaaabeaakiabgIGiolaadkfaaaa@428C@
f(x)
≥
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImlaaa@37B9@
0,
∀x∈[
x
1
,
x
2
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaacUfacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGDbaaaa@3F91@
;
f(x)<0,
∀x∈(−∞,
x
1
)∪(
x
2
,+∞)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaacIcacqGHsislcqGHEisPcaGGSaGaamiEamaaBaaaleaacaaIXaaabeaakiaacMcacqGHQicYcaGGOaGaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacqGHRaWkcqGHEisPcaGGPaaaaa@4784@
Pentru
x∈(
−∞,−
b
2a
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiopaabmaabaGaeyOeI0IaeyOhIuQaaiilaiabgkHiTmaalaaabaGaamOyaaqaaiaaikdacaWGHbaaaaGaayjkaiaawMcaaaaa@4091@
functia este strict crescatoare;
Pentru
x∈[−
b
2a
,+∞),
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaacUfacqGHsisldaWcaaqaaiaadkgaaeaacaaIYaGaamyyaaaacaGGSaGaey4kaSIaeyOhIuQaaiykaiaacYcaaaa@4139@
functia este strict descrescatoare.
6. NUMERE COMPLEXE
1. NUMERE COMPLEXE SUB FORMA ALGEBRICA
C={
z|
z=a+ib,
a,b∈R,
i
2
=−1
}
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabg2da9maacmaabaGaamOEamaaeeaabaGaamOEaiabg2da9iaadggacqGHRaWkcaWGPbGaamOyaiaacYcadaWfGaqaaaWcbeqaaaaakiaadggacaGGSaGaamOyaiabgIGiolaadkfacaGGSaWaaCbiaeaaaSqabeaaaaGccaWGPbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaeyOeI0IaaGymaaGaay5bSdaacaGL7bGaayzFaaaaaa@4D94@
-
multimea numerelor complexe.
z=a+ib=Re z+Im z
OPERATII CU NUMERE COMPLEXE
Fie
z
1
=a+ib,
z
2
=c+id
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBaaaleaacaaIXaaabeaakiabg2da9iaadggacqGHRaWkcaWGPbGaamOyaiaacYcadaqfGaqabSqabeaaa0qaaaaakiaadQhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaWGJbGaey4kaSIaamyAaiaadsgaaaa@442E@
.
Atunci:
1.
z
1
=
z
2
⇔a=c
si
b=d
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBaaaleaacaaIXaaabeaakiabg2da9iaadQhadaWgaaWcbaGaaGOmaaqabaGccqGHuhY2caWGHbGaeyypa0Jaam4yamaavacabeWcbeqaaaqdbaaaaOGaam4CaiaadMgadaqfGaqabSqabeaaa0qaaaaakiaadkgacqGH9aqpcaWGKbaaaa@4586@
.
2.
z
1
+
z
2
=(a+c)+i(b+d).
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadQhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaGGOaGaamyyaiabgUcaRiaadogacaGGPaGaey4kaSIaamyAaiaacIcacaWGIbGaey4kaSIaamizaiaacMcacaGGUaaaaa@4652@
3.
z
1
⋅
z
2
=(a⋅c−b⋅d)+i(a⋅d+b⋅c).
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBaaaleaacaaIXaaabeaakiabgwSixlaadQhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaGGOaGaamyyaiabgwSixlaadogacqGHsislcaWGIbGaeyyXICTaamizaiaacMcacqGHRaWkcaWGPbGaaiikaiaadggacqGHflY1caWGKbGaey4kaSIaamOyaiabgwSixlaadogacaGGPaGaaiOlaaaa@548B@
4.
z
1
¯
=a−ib,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWG6bWaaSbaaSqaaiaaigdaaeqaaaaakiabg2da9iaadggacqGHsislcaWGPbGaamOyaiaacYcaaaa@3D52@
conjugatul lui
z
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBaaaleaacaaIXaaabeaaaaa@37D9@
5.
z
1
z
2
=
a⋅c+b⋅d
c
2
+
d
2
+i
b⋅c−a⋅d
c
2
+
d
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG6bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOEamaaBaaaleaacaaIYaaabeaaaaGccqGH9aqpdaWcaaqaaiaadggacqGHflY1caWGJbGaey4kaSIaamOyaiabgwSixlaadsgaaeaacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamizamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkcaWGPbWaaSaaaeaacaWGIbGaeyyXICTaam4yaiabgkHiTiaadggacqGHflY1caWGKbaabaGaam4yamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadsgadaahaaWcbeqaaiaaikdaaaaaaaaa@5835@
6.
1
z
1
=
a
a
2
+
b
2
−i
b
a
2
+
b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOEamaaBaaaleaacaaIXaaabeaaaaGccqGH9aqpdaWcaaqaaiaadggaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaaaaGccqGHsislcaWGPbWaaSaaaeaacaWGIbaabaGaamyyamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaaikdaaaaaaaaa@469C@
.
PUTERILE LUI i
1.
i
4k
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCaaaleqabaGaaGinaiaadUgaaaGccqGH9aqpcaaIXaaaaa@3A87@
;
2.
i
4k+1
=i
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCaaaleqabaGaaGinaiaadUgacqGHRaWkcaaIXaaaaOGaeyypa0JaamyAaaaa@3C57@
;
3.
i
4k+2
=−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCaaaleqabaGaaGinaiaadUgacqGHRaWkcaaIYaaaaOGaeyypa0JaeyOeI0IaaGymaaaa@3D12@
;
4.
i
4k+3
=−i
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCaaaleqabaGaaGinaiaadUgacqGHRaWkcaaIZaaaaOGaeyypa0JaeyOeI0IaamyAaaaa@3D46@
;
5.
i
−n
=
1
i
n
,
i
−1
=
1
i
=−i
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCaaaleqabaGaeyOeI0IaamOBaaaakiabg2da9maalaaabaGaaGymaaqaaiaadMgadaahaaWcbeqaaiaad6gaaaaaaOGaaiilaiaadMgadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGPbaaaiabg2da9iabgkHiTiaadMgaaaa@45FE@
;
6.
i
−n
=
(−i)
n
=
(−1)
n
⋅
i
n
={
i
n
,n
par
−
i
n
,n
impar
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCaaaleqabaGaeyOeI0IaamOBaaaakiabg2da9iaacIcacqGHsislcaWGPbGaaiykamaaCaaaleqabaGaamOBaaaakiabg2da9iaacIcacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaamOBaaaakiabgwSixlaadMgadaahaaWcbeqaaiaad6gaaaGccqGH9aqpdaGabaabaeqabaGaamyAamaaCaaaleqabaGaamOBaaaakiaacYcacaWGUbWaaubiaeqaleqabaaaneaaaaGccaWGWbGaamyyaiaadkhaaeaacqGHsislcaWGPbWaaWbaaSqabeaacaWGUbaaaOGaaiilaiaad6gadaqfGaqabSqabeaaa0qaaaaakiaadMgacaWGTbGaamiCaiaadggacaWGYbaaaiaawUhaaaaa@5ABD@
PROPRIETATILE MODULULUI
| z |=
a
2
+
b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG6baacaGLhWUaayjcSdGaeyypa0ZaaOaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaaaeqaaaaa@3FB5@
- modulul nr. complexe
1.
| z |≥0,| z |=0⇔z=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG6baacaGLhWUaayjcSdGaeyyzImRaaGimaiaacYcadaabdaqaaiaadQhaaiaawEa7caGLiWoacqGH9aqpcaaIWaGaeyi1HSTaamOEaiabg2da9iaaicdaaaa@4840@
2.
z⋅
z
¯
=
| z |
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabgwSixpaanaaabaGaamOEaaaacqGH9aqpdaabdaqaaiaadQhaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaaaaa@405C@
3.
| z |=|
z
¯
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG6baacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaadaqdaaqaaiaadQhaaaaacaGLhWUaayjcSdaaaa@3F4C@
4.
|
z
1
⋅
z
2
|=|
z
1
|⋅|
z
2
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG6bWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaamOEamaaBaaaleaacaaIYaaabeaaaOGaay5bSlaawIa7aiabg2da9maaemaabaGaamOEamaaBaaaleaacaaIXaaabeaaaOGaay5bSlaawIa7aiabgwSixpaaemaabaGaamOEamaaBaaaleaacaaIYaaabeaaaOGaay5bSlaawIa7aaaa@4CB5@
5.
|
z
1
z
2
|=
|
z
1
|
|
z
2
|
,
z
2
≠0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWcaaqaaiaadQhadaWgaaWcbaGaaGymaaqabaaakeaacaWG6bWaaSbaaSqaaiaaikdaaeqaaaaaaOGaay5bSlaawIa7aiabg2da9maalaaabaWaaqWaaeaacaWG6bWaaSbaaSqaaiaaigdaaeqaaaGccaGLhWUaayjcSdaabaWaaqWaaeaacaWG6bWaaSbaaSqaaiaaikdaaeqaaaGccaGLhWUaayjcSdaaaiaacYcacaWG6bWaaSbaaSqaaiaaikdaaeqaaOGaeyiyIKRaaGimaaaa@4D63@
6.
|
|
z
1
|−|
z
2
|
|≤|
z
1
±
z
2
|≤|
z
1
|+|
z
2
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaabdaqaaiaadQhadaWgaaWcbaGaaGymaaqabaaakiaawEa7caGLiWoacqGHsisldaabdaqaaiaadQhadaWgaaWcbaGaaGOmaaqabaaakiaawEa7caGLiWoaaiaawEa7caGLiWoacqGHKjYOdaabdaqaaiaadQhadaWgaaWcbaGaaGymaaqabaGccqGHXcqScaWG6bWaaSbaaSqaaiaaikdaaeqaaaGccaGLhWUaayjcSdGaeyizIm6aaqWaaeaacaWG6bWaaSbaaSqaaiaaigdaaeqaaaGccaGLhWUaayjcSdGaey4kaSYaaqWaaeaacaWG6bWaaSbaaSqaaiaaikdaaeqaaaGccaGLhWUaayjcSdaaaa@5B89@
7.
|
z
n
|=
| z |
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG6bWaaWbaaSqabeaacaWGUbaaaaGccaGLhWUaayjcSdGaeyypa0ZaaqWaaeaacaWG6baacaGLhWUaayjcSdWaaWbaaSqabeaacaWGUbaaaaaa@4185@
8.
z∈C;
z∈R⇔Imz=0⇔z=
z
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabgIGiolaadoeacaGG7aWaaubiaeqaleqabaaaneaaaaGccaWG6bGaeyicI4SaamOuaiabgsDiBlGacMeacaGGTbGaamOEaiabg2da9iaaicdacqGHuhY2caWG6bGaeyypa0Zaa0aaaeaacaWG6baaaaaa@4A03@
ECUATII:
z
2
=a+ib⇒
z
1,2
=±
a+ib
⇒
z
1,2
=±[
a+
a
2
+
b
2
2
±i
−a+
a
2
+
b
2
2
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6675@
‚
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=Pbiaaa@37C7@
+’ daca b
pozitiv;
‚
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=Pbiaaa@37C7@
-
‚
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=Pbiaaa@37C7@
daca b negativ
a
x
2
+bx+c=0⇒
x
1,2
=
−b±
b
2
−4ac
2a
daca
Δ=
b
2
−4ac≥0
sau
x
1,2
=
−b±i
−Δ
2a
daca
Δ〈0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHbGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadkgacaWG4bGaey4kaSIaam4yaiabg2da9iaaicdacqGHshI3caWG4bWaaSbaaSqaaiaaigdacaGGSaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiabgkHiTiaadkgacqGHXcqSdaGcaaqaaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI0aGaamyyaiaadogaaSqabaaakeaacaaIYaGaamyyaaaaaeaacaWGKbGaamyyaiaadogacaWGHbWaaCbiaeaaaSqabeaaaaGccqqHuoarcqGH9aqpcaWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGinaiaadggacaWGJbGaeyyzImRaaGimamaaxacabaaaleqabaaaaOGaam4CaiaadggacaWG1baabaWaaubiaeqaleqabaaaneaaaaGcdaqfGaqabSqabeaadaqfGaqabWqabeaadaqfGaqabeqabaWaaubiaeqabeqaaaqaaaaaaeaaaaaaoeaaaaaaneaaaaGcdaqfGaqabSqabeaaa0qaaaaakmaavacabeWcbeqaaaqdbaaaaOWaaubeaeqaleaaaeqaneaaaaGccaWG4bWaaSbaaSqaaiaaigdacaGGSaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiabgkHiTiaadkgacqGHXcqScaWGPbWaaOaaaeaacqGHsislcqqHuoaraSqabaaakeaacaaIYaGaamyyaaaadaWfGaqaaaWcbeqaaaaakiaadsgacaWGHbGaam4yaiaadggadaqfGaqabSqabeaaa0qaaaaakiabfs5aejabgMYiHlaaicdaaaaa@7A2B@
NUMERE COMPLEXE SUB FORMA GEOMETRICA
Forma trigonometrica
a numerelor complexe:
z=
ρ(cosϕ+isinϕ)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdiNaaiikaiGacogacaGGVbGaai4Caiabew9aMjabgUcaRiaadMgaciGGZbGaaiyAaiaac6gacqaHvpGzcaGGPaaaaa@4417@
,
ϕ=arctg
b
a
+kπ,
k={
0,
(a,b)∈I
1,
(a,b)∈II,III
2,
(a,b)∈IV
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@639B@
ρ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdihaaa@37B3@
=
| z |=
a
2
+
b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG6baacaGLhWUaayjcSdGaeyypa0ZaaOaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaaaeqaaaaa@3FB5@
se numeste raza polara a lui z
Fie z1=
ρ
1
(cos
ϕ
1
+isin
ϕ
1
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaSbaaSqaaiaaigdaaeqaaOGaaiikaiGacogacaGGVbGaai4Caiabew9aMnaaBaaaleaacaaIXaaabeaakiabgUcaRiaadMgaciGGZbGaaiyAaiaac6gacqaHvpGzdaWgaaWcbaGaaGymaaqabaGccaGGPaaaaa@46EA@
si z2=
ρ
2
(cos
ϕ
2
+isin
ϕ
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaSbaaSqaaiaaikdaaeqaaOGaaiikaiGacogacaGGVbGaai4Caiabew9aMnaaBaaaleaacaaIYaaabeaakiabgUcaRiaadMgaciGGZbGaaiyAaiaac6gacqaHvpGzdaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@46ED@
;
z1=z2
ρ
1
=
ρ
2
,si
exista
k∈Z
a.i
ϕ
1
=
ϕ
2
+kπ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeqyWdi3aaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadohacaWGPbWaaCbiaeaaaSqabeaaaaGccaWGLbGaamiEaiaadMgacaWGZbGaamiDaiaadggadaWfGaqaaaWcbeqaaaaakiaadUgacqGHiiIZcaWGAbWaaCbiaeaaaSqabeaaaaGccaWGHbGaaiOlaiaadMgadaWfGaqaaaWcbeqaaaaakiabew9aMnaaBaaaleaacaaIXaaabeaakiabg2da9iabew9aMnaaBaaaleaacaaIYaaabeaakiabgUcaRiaadUgacqaHapaCaaa@55CB@
z
1
⋅
z
2
=
ρ
1
⋅
ρ
2
[cos(
ϕ
1
+
ϕ
2
)+isin(
ϕ
1
+
ϕ
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBaaaleaacaaIXaaabeaakiabgwSixlaadQhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqaHbpGCdaWgaaWcbaGaaGymaaqabaGccqGHflY1cqaHbpGCdaWgaaWcbaGaaGOmaaqabaGccaGGBbGaci4yaiaac+gacaGGZbGaaiikaiabew9aMnaaBaaaleaacaaIXaaabeaakiabgUcaRiabew9aMnaaBaaaleaacaaIYaaabeaakiaacMcacqGHRaWkcaWGPbGaci4CaiaacMgacaGGUbGaaiikaiabew9aMnaaBaaaleaacaaIXaaabeaakiabgUcaRiabew9aMnaaBaaaleaacaaIYaaabeaakiaacMcaaaa@5C87@
z
1
¯
=
ρ
1
(cos
ϕ
1
−isin
ϕ
1
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWG6bWaaSbaaSqaaiaaigdaaeqaaaaakiabg2da9iabeg8aYnaaBaaaleaacaaIXaaabeaakiaacIcaciGGJbGaai4BaiaacohacqaHvpGzdaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGPbGaci4CaiaacMgacaGGUbGaeqy1dy2aaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@49FC@
1
z
1
=
1
ρ
1
[cos(−
ϕ
1
)+isin(−
ϕ
1
)]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOEamaaBaaaleaacaaIXaaabeaaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacqaHbpGCdaWgaaWcbaGaaGymaaqabaaaaOGaai4waiGacogacaGGVbGaai4CaiaacIcacqGHsislcqaHvpGzdaWgaaWcbaGaaGymaaqabaGccaGGPaGaey4kaSIaamyAaiGacohacaGGPbGaaiOBaiaacIcacqGHsislcqaHvpGzdaWgaaWcbaGaaGymaaqabaGccaGGPaGaaiyxaaaa@5069@
z
2
z
1
=
ρ
2
ρ
1
[
cos(
ϕ
2
−
ϕ
1
)+isin(
ϕ
2
−
ϕ
1
)
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG6bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamOEamaaBaaaleaacaaIXaaabeaaaaGccqGH9aqpdaWcaaqaaiabeg8aYnaaBaaaleaacaaIYaaabeaaaOqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaaaGcdaWadaqaaiGacogacaGGVbGaai4CaiaacIcacqaHvpGzdaWgaaWcbaGaaGOmaaqabaGccqGHsislcqaHvpGzdaWgaaWcbaGaaGymaaqabaGccaGGPaGaey4kaSIaamyAaiGacohacaGGPbGaaiOBaiaacIcacqaHvpGzdaWgaaWcbaGaaGOmaaqabaGccqGHsislcqaHvpGzdaWgaaWcbaGaaGymaaqabaGccaGGPaaacaGLBbGaayzxaaaaaa@593C@
z
1
n
=
ρ
1
n
(cosn
ϕ
1
+isinn
ϕ
1
),n∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBaaaleaacaaIXaaabeaakmaaCaaaleqabaGaamOBaaaakiabg2da9iabeg8aYnaaBaaaleaacaaIXaaabeaakmaaCaaaleqabaGaamOBaaaakiaacIcaciGGJbGaai4BaiaacohacaWGUbGaeqy1dy2aaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyAaiGacohacaGGPbGaaiOBaiaad6gacqaHvpGzdaWgaaWcbaGaaGymaaqabaGccaGGPaGaaiilaiaad6gacqGHiiIZcaWGsbaaaa@5218@
z
1
n
=
ρ
1
n
(cos
ϕ
1
+2kπ
n
+isin
ϕ
1
+2kπ
n
),k∈
0,n−1
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWG6bWaaSbaaSqaaiaaigdaaeqaaaqaaiaad6gaaaGccqGH9aqpdaGcbaqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaaeaacaWGUbaaaOGaaiikaiGacogacaGGVbGaai4CamaalaaabaGaeqy1dy2aaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaaGOmaiaadUgacqaHapaCaeaacaWGUbaaaiabgUcaRiaadMgaciGGZbGaaiyAaiaac6gadaWcaaqaaiabew9aMnaaBaaaleaacaaIXaaabeaakiabgUcaRiaaikdacaWGRbGaeqiWdahabaGaamOBaaaacaGGPaGaaiilaiaadUgacqGHiiIZdaqdaaqaaiaaicdacaGGSaGaamOBaiabgkHiTiaaigdaaaaaaa@5DBC@
7. FUNCTIA EXPONENTIALA
Def. f: R→ (0,∞), f(x)=
a
x
,a〉0,a≠1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCaaaleqabaGaamiEaaaakiaacYcacaWGHbGaeyOkJeVaaGimaiaacYcacaWGHbGaeyiyIKRaaGymaaaa@403F@
Daca a
〉1⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOkJeVaaGymaiabgkDiEdaa@3AD5@
f este strict crescatoare
x
1
〈
x
2
⇒
a
x
1
〈
a
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabgMYiHlaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHshI3caWGHbWaaWbaaSqabeaacaWG4bWaaSbaaWqaaiaaigdaaeqaaaaakiabgMYiHlaadggadaahaaWcbeqaaiaadIhadaWgaaadbaGaaGOmaaqabaaaaaaa@459A@
Daca a
∈(
0,1
)⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI48aaeWaaeaacaaIWaGaaiilaiaaigdaaiaawIcacaGLPaaacqGHshI3aaa@3D82@
f este strict descrescatoare
x
1
〈
x
2
⇒
a
x
1
〉
a
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabgMYiHlaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHshI3caWGHbWaaWbaaSqabeaacaWG4bWaaSbaaWqaaiaaigdaaeqaaaaakiabgQYiXlaadggadaahaaWcbeqaaiaadIhadaWgaaadbaGaaGOmaaqabaaaaaaa@45AB@
Proprietati:
Fie a,b
∈(
0,∞
),a,b≠1,x,y∈R⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI48aaeWaaeaacaaIWaGaaiilaiabg6HiLcGaayjkaiaawMcaaiaacYcacaWGHbGaaiilaiaadkgacqGHGjsUcaaIXaGaaiilaiaadIhacaGGSaGaamyEaiabgIGiolaadkfacqGHshI3aaa@499D@
a
x
⋅
a
y
=
a
x+y
(
a⋅b
)
x
=
a
x
⋅
a
y
(
a
x
)
y
=
a
x⋅y
a
x
a
y
=
a
x−y
,a≠0
(
a
b
)
x
=
a
x
b
x
,b≠0
a
0
=1
a
−x
=
1
a
x
,a≠0
pentru
a〈0,nu
se
defineste
a
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9E7E@
Tipuri
de ecuatii:
1. a
f(x)
=b,a〉0,a≠1,b〉0⇒f(x)=
log
a
b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabeaacaWGMbGaaiikaiaadIhacaGGPaaaaOGaeyypa0JaamOyaiaacYcacaWGHbGaeyOkJeVaaGimaiaacYcacaWGHbGaeyiyIKRaaGymaiaacYcacaWGIbGaeyOkJeVaaGimaiabgkDiElaadAgacaGGOaGaamiEaiaacMcacqGH9aqpciGGSbGaai4BaiaacEgadaWgaaWcbaGaamyyaaqabaGccaWGIbaaaa@531C@
2. a
f(x)
=
a
g(x)
,a〉0,a≠1⇒f(x)=g(x)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabeaacaWGMbGaaiikaiaadIhacaGGPaaaaOGaeyypa0JaamyyamaaCaaaleqabaGaam4zaiaacIcacaWG4bGaaiykaaaakiaacYcacaWGHbGaeyOkJeVaaGimaiaacYcacaWGHbGaeyiyIKRaaGymaiabgkDiElaadAgacaGGOaGaamiEaiaacMcacqGH9aqpcaWGNbGaaiikaiaadIhacaGGPaaaaa@50E8@
3. a
f(x)
=
b
g(x)
,a,b〉0,a,b≠1⇒f(x)=g(x)⋅
log
a
b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabeaacaWGMbGaaiikaiaadIhacaGGPaaaaOGaeyypa0JaamOyamaaCaaaleqabaGaam4zaiaacIcacaWG4bGaaiykaaaakiaacYcacaWGHbGaaiilaiaadkgacqGHQms8caaIWaGaaiilaiaadggacaGGSaGaamOyaiabgcMi5kaaigdacqGHshI3caWGMbGaaiikaiaadIhacaGGPaGaeyypa0Jaam4zaiaacIcacaWG4bGaaiykaiabgwSixlGacYgacaGGVbGaai4zamaaBaaaleaacaWGHbaabeaakiaadkgaaaa@5B34@
4. ecuatii exponentiale reductibile la ecuatii algebrice printr-o
substitutie.
5. ecuatii ce se rezolva utilizând monotonia functiei
exponentiale.
Inecuatii
a>1,
a
f(x)
≤
a
g(x)
⇒f(x)≤g(x)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCaaaleqabaGaamOzaiaacIcacaWG4bGaaiykaaaakiabgsMiJkaadggadaahaaWcbeqaaiaadEgacaGGOaGaamiEaiaacMcaaaGccqGHshI3caWGMbGaaiikaiaadIhacaGGPaGaeyizImQaam4zaiaacIcacaWG4bGaaiykaaaa@4AFA@
a
∈(0,1)
a
f(x)
≤
a
g(x)
⇒f(x)≥g(x)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaaiikaiaaicdacaGGSaGaaGymaiaacMcadaqfGaqabSqabeaaa0qaaaaakiaadggadaahaaWcbeqaaiaadAgacaGGOaGaamiEaiaacMcaaaGccqGHKjYOcaWGHbWaaWbaaSqabeaacaWGNbGaaiikaiaadIhacaGGPaaaaOGaeyO0H4TaamOzaiaacIcacaWG4bGaaiykaiabgwMiZkaadEgacaGGOaGaamiEaiaacMcaaaa@506D@
FUNCTIA LOGARITMICA
Def: f:(0,∞)
→R, f(x)=
log
a
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamiEaaaa@3ADC@
,
a〉0,a≠1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgQYiXlaaicdacaGGSaGaamyyaiabgcMi5kaaigdaaaa@3D75@
,x>0
Daca a
〉1⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOkJeVaaGymaiabgkDiEdaa@3AD5@
f este strict crescatoare
x
1
〈
x
2
⇒
log
a
x
1
〈
log
a
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabgMYiHlaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHshI3ciGGSbGaai4BaiaacEgadaWgaaWcbaGaamyyaaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyykJeUaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamiEamaaBaaaleaacaaIYaaabeaaaaa@4B4A@
Daca a
∈(
0,1
)⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI48aaeWaaeaacaaIWaGaaiilaiaaigdaaiaawIcacaGLPaaacqGHshI3aaa@3D82@
f este strict descrescatoare
x
1
〈
x
2
⇒
log
a
x
1
〉
log
a
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabgMYiHlaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHshI3ciGGSbGaai4BaiaacEgadaWgaaWcbaGaamyyaaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOkJeVaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamiEamaaBaaaleaacaaIYaaabeaaaaa@4B5B@
Proprietati:
Fie a,b
c∈(
0,∞
),a,b,c≠1,x,y∈(0,∞),m∈R⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgIGiopaabmaabaGaaGimaiaacYcacqGHEisPaiaawIcacaGLPaaacaGGSaGaamyyaiaacYcacaWGIbGaaiilaiaadogacqGHGjsUcaaIXaGaaiilaiaadIhacaGGSaGaamyEaiabgIGiolaacIcacaaIWaGaaiilaiabg6HiLkaacMcacaGGSaGaamyBaiabgIGiolaadkfacqGHshI3aaa@5377@
a
y
=x〉0⇒y=
log
a
x
log
a
x⋅y=
log
a
x+
log
a
y
log
a
x
y
=
log
a
x−
log
a
y
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6B90@
log
a
a
m
=m,
log
a
b
m
=m
log
a
b
log
a
b=
log
c
b
log
c
a
,
1
log
a
b
=
log
b
a
a
log
b
c
=
c
log
b
a
,
x=
a
log
a
x
log
a
1=0,
log
a
a=1.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8CA5@
Tipuri de ecuatii:
1.
log
f(x)
g(x)=b,f,g〉0,f≠1⇒g(x)=f
(x)
b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadAgacaGGOaGaamiEaiaacMcaaeqaaOGaam4zaiaacIcacaWG4bGaaiykaiabg2da9iaadkgacaGGSaGaamOzaiaacYcacaWGNbGaeyOkJeVaaGimaiaacYcacaWGMbGaeyiyIKRaaGymaiabgkDiElaadEgacaGGOaGaamiEaiaacMcacqGH9aqpcaWGMbGaaiikaiaadIhacaGGPaWaaWbaaSqabeaacaWGIbaaaaaa@563B@
2.
log
a
f(x)=
log
a
g(x)⇒f(x)=g(x)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iGacYgacaGGVbGaai4zamaaBaaaleaacaWGHbaabeaakiaadEgacaGGOaGaamiEaiaacMcacqGHshI3caWGMbGaaiikaiaadIhacaGGPaGaeyypa0Jaam4zaiaacIcacaWG4bGaaiykaaaa@4F3A@
3.
log
a
f(x)=
log
b
g(x)⇒f(x)=
a
log
b
g(x)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iGacYgacaGGVbGaai4zamaaBaaaleaacaWGIbaabeaakiaadEgacaGGOaGaamiEaiaacMcacqGHshI3caWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamyyamaaCaaaleqabaGaciiBaiaac+gacaGGNbWaaSbaaWqaaiaadkgaaeqaaSGaam4zaiaacIcacaWG4bGaaiykaaaaaaa@543D@
4.
ecuatii logaritmice reductibile la ecuatii algebrice printr-o substitutie.
5.
ecuatii ce se rezolva utilizând
monotonia functiei logaritmice.
Inecuatii
a>1,
log
a
f(x)≤
log
a
g(x)⇒f(x)≤g(x)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamOzaiaacIcacaWG4bGaaiykaiabgsMiJkGacYgacaGGVbGaai4zamaaBaaaleaacaWGHbaabeaakiaadEgacaGGOaGaamiEaiaacMcacqGHshI3caWGMbGaaiikaiaadIhacaGGPaGaeyizImQaam4zaiaacIcacaWG4bGaaiykaaaa@5098@
a
∈(0,1)
log
a
f(x)≤
log
a
g(x)⇒f(x)≥g(x)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaaiikaiaaicdacaGGSaGaaGymaiaacMcaciGGSbGaai4BaiaacEgadaWgaaWcbaGaamyyaaqabaGccaWGMbGaaiikaiaadIhacaGGPaGaeyizImQaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaam4zaiaacIcacaWG4bGaaiykaiabgkDiElaadAgacaGGOaGaamiEaiaacMcacqGHLjYScaWGNbGaaiikaiaadIhacaGGPaaaaa@55AB@
8.
BINOMUL LUI NEWTON
In 1664 Isaac Newton (1643-1727) a gasit urmatoarea
formula pentru dezvoltarea binomului (a+b)n. Desi formula era cunoscuta inca din
antichitate de catre matematicianul arab Omar
Khayyam (1040-1123), Newton a
extins-o si pentru coeficienti rationali.
TEOREMA: Pentru orice numar natural n si
a si b numere reale exista relatia:
(
a+b
)
n
=
C
n
0
a
n
+
C
n
1
a
n−1
⋅b+
C
n
2
a
n−2
⋅
b
2
+..........+
C
n
k
a
n−k
⋅
b
k
+.....+
C
n
n
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7613@
(1)
Numerele
C
n
0
,
C
n
1
,....,
C
n
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaaGimaaaakiaacYcacaWGdbWaaSbaaSqaaiaad6gaaeqaaOWaaWbaaSqabeaacaaIXaaaaOGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaGGSaGaam4qamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaamOBaaaaaaa@43A1@
se numesc coeficientii binomiali ai dezvoltarii;
Este necesar sa se
faca distinctie intre coeficientul unui termen al dezvoltarii si coeficientul
binomial al acelui termen.
Exemplu: (a+2b)4= a4
+ 4a 3 .2b+…..
Coeficientul celui de-al
doilea termen este 8 iar coeficientul
binomial este C41 =4;
Pentru (a-b)n
avem urmatoarea forma a binomului lui Newton:
(
a−b
)
n
=
C
n
0
a
n
−
C
n
1
a
n−1
⋅b+
C
n
2
a
n−2
⋅
b
2
−...........+
(−1)
k
C
n
k
a
n−k
⋅
b
k
+.....+
(−1)
n
C
n
n
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7F39@
(1’)
Proprietati:
1. Numarul termenilor dezvoltarii
binomului (a+b)n este n+1;
Daca n=2k
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3850@
coeficientul binomial al termenului din mijloc
al dezvoltarii este Cnk si
este cel mai mare.
Daca n=2k+1
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3850@
Cnk si Cnk+1
sunt egali si sunt cei mai mari;
Cno<Cn1<……<Cnk
>Cnk+1>…..>Cnn daca
n este par, n=2k
Cno<Cn1<……<Cnk
=Cnk+1>…..>Cnn daca
n este impar, n=2k+1.
2. Coeficientii
binomiali din dezvoltare, egal departati de termenii extremi ai dezvoltarii sunt egali intre ei.
C
n
k
=
C
n
n−k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaam4Aaaaakiabg2da9iaadoeadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaad6gacqGHsislcaWGRbaaaaaa@3EFF@
(2)
3. Termenul de
rang k+1 al dezvoltarii (sau termenul general al dezvoltarii) este
T
k+1
=
C
n
k
a
n−k
⋅
b
k
,
k=0,1,2,....,n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaad6gaaeqaaOWaaWbaaSqabeaacaWGRbaaaOGaamyyamaaCaaaleqabaGaamOBaiabgkHiTiaadUgaaaGccqGHflY1caWGIbWaaWbaaSqabeaacaWGRbaaaOGaaiilamaavacabeWcbeqaaaqdbaaaaOGaam4Aaiabg2da9iaaicdacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaGGSaGaamOBaaaa@51A4@
(3)
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3850@
Formula binomului lui Newton scrisa restrâns
are forma:
(
a+b
)
n
=
∑
k=0
n
C
n
k
a
n−k
b
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbGaey4kaSIaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaaaakiabg2da9maaqahabaGaam4qamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaam4AaaaakiaadggadaahaaWcbeqaaiaad6gacqGHsislcaWGRbaaaOGaamOyamaaCaaaleqabaGaam4AaaaaaeaacaWGRbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoaaaa@4B3E@
.
(4)
4. Relatia de recurenta intre
termenii succesivi ai dezvoltarii este urmatoarea:
T
k+2
T
k+1
=
n−k
k+1
⋅
b
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGubWaaSbaaSqaaiaadUgacqGHRaWkcaaIYaaabeaaaOqaaiaadsfadaWgaaWcbaGaam4AaiabgUcaRiaaigdaaeqaaaaakiabg2da9maalaaabaGaamOBaiabgkHiTiaadUgaaeaacaWGRbGaey4kaSIaaGymaaaacqGHflY1daWcaaqaaiaadkgaaeaacaWGHbaaaaaa@47D6@
(5)
5. Pentru a=b=1 se obtine
C
n
0
+
C
n
1
+
C
n
2
+.............+
C
n
n
=
(1+1)
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaaGimaaaakiabgUcaRiaadoeadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaaigdaaaGccqGHRaWkcaWGdbWaaSbaaSqaaiaad6gaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiabgUcaRiaadoeadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaad6gaaaGccqGH9aqpcaGGOaGaaGymaiabgUcaRiaaigdacaGGPaWaaWbaaSqabeaacaWGUbaaaaaa@5420@
(6)
ceea ce inseamna ca numarul tuturor submultimilor unei multimi
cu n elemente este 2n .
9. Vectori si operatii cu vectori
Definitie:
Se numeste segment orientat, o pereche ordonata de puncte din plan;
Se numeste vector, multimea tuturor
segmentelor orientate care au aceeasi directie, aceeasi lungime si acelasi sens
cu ale unui segment orientat.
Observatii:
Orice vector
AB
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGbbGaamOqaaGaay51Gaaaaa@3934@
se caracterizeaza prin:
-
modul(lungime,norma), dat de lungimea
segmentului AB;
-
directie, data de dreapta AB sau orice dreapta
paralela cu aceasta;
-
sens, indicat printr-o sageata de la originea
A la extremitatea B.
Notatii:
AB
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGbbGaamOqaaGaay51Gaaaaa@3934@
vectorul cu originea A si extremitatea B;
|
AB
|
→
=
(x−
x
0
)
2
+
(y−
y
0
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaadaabdaqaaiaadgeacaWGcbaacaGLhWUaayjcSdaacaGLxdcacqGH9aqpdaGcaaqaaiaacIcacaWG4bGaeyOeI0IaamiEamaaBaaaleaacaaIWaaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOaGaamyEaiabgkHiTiaadMhadaWgaaWcbaGaaGimaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@4A8C@
- modulul vectorului
AB
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGbbGaamOqaaGaay51Gaaaaa@3934@
unde A(x0,y0), B(x.y).
Definitie:
Se numesc vectori egali, vectorii care au
aceeasi directie, acelasi sens si acelasi modul. Doi vectori se numesc opusi
daca au aceeasi directie, acelasi modul si sensuri contrare:
-
AB
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGbbGaamOqaaGaay51Gaaaaa@3934@
=
BA
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGcbGaamyqaaGaay51Gaaaaa@3934@
.
Adunarea vectorilor se poate face dupa regula
triunghiului sau dupa regula paralelogramului:
λ⋅
v
→
=
0
→
⇔λ=0
sau
v
→
=
0
→
,∀λ∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyyXIC9aa8HaaeaacaWG2baacaGLxdcacqGH9aqpdaWhcaqaaiaaicdaaiaawEniaiabgsDiBlabeU7aSjabg2da9iaaicdadaWfGaqaaaWcbeqaaaaakiaadohacaWGHbGaamyDamaaxacabaaaleqabaaaaOWaa8HaaeaacaWG2baacaGLxdcacqGH9aqpdaWhcaqaaiaaicdaaiaawEniaiaacYcacqGHaiIicqaH7oaBcqGHiiIZcaWGsbaaaa@5514@
Daca
λ≠0,
v
→
≠
0
→
⇒|
λ⋅
v
→
|=| λ |⋅|
v
→
|,λ⋅
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiaadggacaWGJbGaamyyamaaxacabaaaleqabaaaaOGaeq4UdWMaeyiyIKRaaGimaiaacYcadaWhcaqaaiaadAhaaiaawEniaiabgcMi5oaaFiaabaGaaGimaaGaay51GaGaeyO0H49aaqWaaeaacqaH7oaBcqGHflY1daWhcaqaaiaadAhaaiaawEniaaGaay5bSlaawIa7aiabg2da9maaemaabaGaeq4UdWgacaGLhWUaayjcSdGaeyyXIC9aaqWaaeaadaWhcaqaaiaadAhaaiaawEniaaGaay5bSlaawIa7aiaacYcacqaH7oaBcqGHflY1daWhcaqaaiaadAhaaiaawEniaaaa@670C@
are directia si sensul vectorului
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcaaaa@38A2@
daca
λ〉0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyOkJeVaaGimaaaa@3A2B@
si sens opus lui
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcaaaa@38A2@
daca
λ〈0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyykJeUaaGimaaaa@3A1A@
.
Definitie:
Doi vectori se numesc coliniari daca cel putin unul este nul sau daca amândoi sunt nenuli
si au aceeasi directie. In caz contrar se numesc necoliniari.
vectori coliniari vectori
necoliniari
Teorema:
Fie
u
→
≠
0
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG1baacaGLxdcacqGHGjsUdaWhcaqaaiaaicdaaiaawEniaaaa@3CD6@
si
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcaaaa@38A2@
un vector oarecare.
Vectorii
u
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG1baacaGLxdcaaaa@38A1@
si
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcaaaa@38A2@
sunt coliniari
⇔∃λ∈R
a.i.
v
→
=λ⋅
u
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaey4aIqIaeq4UdWMaeyicI4SaamOuamaaxacabaaaleqabaaaaOGaamyyaiaac6cacaWGPbGaaiOlamaaxacabaaaleqabaaaaOWaa8HaaeaacaWG2baacaGLxdcacqGH9aqpcqaH7oaBcqGHflY1daWhcaqaaiaadwhaaiaawEniaaaa@4B72@
.
Punctele A, B, C sunt coliniare
⇔
AB
→
si
AC
→
sunt
coliniari⇔
∃λ∈Ra.i.
AB
→
=λ⋅
AC
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aa8HaaeaacaWGbbGaamOqaaGaay51GaWaaCbiaeaaaSqabeaaaaGccaWGZbGaamyAamaaxacabaaaleqabaaaaOWaa8HaaeaacaWGbbGaam4qaaGaay51GaWaaCbiaeaaaSqabeaaaaGccaWGZbGaamyDaiaad6gacaWG0bWaaCbiaeaacaWGJbGaam4BaiaadYgacaWGPbGaamOBaiaadMgacaWGHbGaamOCaiaadMgacqGHuhY2aSqabeaaaaGccqGHdicjcqaH7oaBcqGHiiIZcaWGsbGaamyyaiaac6cacaWGPbGaaiOlamaaFiaabaGaamyqaiaadkeaaiaawEniaiabg2da9iabeU7aSjabgwSixpaaFiaabaGaamyqaiaadoeaaiaawEniaaaa@6448@
.
AB| |CD⇔
AB
→
si
CD
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadkeadaabdaqaaaGaay5bSlaawIa7aiaadoeacaWGebGaeyi1HS9aa8HaaeaacaWGbbGaamOqaaGaay51GaWaaCbiaeaaaSqabeaaaaGccaWGZbGaamyAamaaxacabaaaleqabaaaaOWaa8HaaeaacaWGdbGaamiraaGaay51Gaaaaa@47A1@
sunt coliniari;
Daca
u
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG1baacaGLxdcaaaa@38A1@
si
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcaaaa@38A2@
sunt vectori necoliniari atunci
∃x,y∈R
a.i.
x⋅
u
→
+y⋅
v
→
=
0
→
⇔x=y=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqIaamiEaiaacYcacaWG5bGaeyicI4SaamOuamaaxacabaaaleqabaaaaOGaamyyaiaac6cacaWGPbGaaiOlamaaxacabaaaleqabaaaaOGaamiEaiabgwSixpaaFiaabaGaamyDaaGaay51GaGaey4kaSIaamyEaiabgwSixpaaFiaabaGaamODaaGaay51GaGaeyypa0Zaa8HaaeaacaaIWaaacaGLxdcacqGHuhY2caWG4bGaeyypa0JaamyEaiabg2da9iaaicdaaaa@570B@
.
Teorema: Fie
a
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGHbaacaGLxdcaaaa@388D@
si
b
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGIbaacaGLxdcaaaa@388E@
doi vectori necoliniari. Oricare ar fi
vectorul
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcaaaa@38A2@
,
exista
α,β∈R(unice)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaiilaiabek7aIjabgIGiolaadkfacaGGOaGaamyDaiaad6gacaWGPbGaam4yaiaadwgacaGGPaaaaa@4244@
astfel incât
v
→
=α⋅
a
→
+β⋅
b
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcacqGH9aqpcqaHXoqycqGHflY1daWhcaqaaiaadggaaiaawEniaiabgUcaRiabek7aIjabgwSixpaaFiaabaGaamOyaaGaay51Gaaaaa@4793@
.
Vectorii
a
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGHbaacaGLxdcaaaa@388D@
si
b
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGIbaacaGLxdcaaaa@388E@
formeaza o baza.
α,β
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaiilaiabek7aIbaa@39E3@
se numesc coordonatele vectorului
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcaaaa@38A2@
in baza
(
a
→
,
b
→
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWhcaqaaiaadggaaiaawEniaiaacYcadaWhcaqaaiaadkgaaiaawEniaaGaayjkaiaawMcaaaaa@3D61@
.
Definitie:
Fie XOY un reper
cartezian. Consideram punctele A(1,0), B(0,1).
Vectorii
i
→
=
OA
→
si
j
→
=
OB
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGPbaacaGLxdcacqGH9aqpdaWhcaqaaiaad+eacaWGbbaacaGLxdcadaWfGaqaaaWcbeqaaaaakiaadohacaWGPbWaaCbiaeaadaWhcaqaaiaadQgaaiaawEniaaWcbeqaaaaakiabg2da9maaFiaabaGaam4taiaadkeaaiaawEniaaaa@466D@
se numesc versorii axelor de coordonate. Ei
au modulul egal cu 1, directiile axelor si sensurile semiaxelor pozitive cu OX si
OY.
Baza
(
i
→
,
j
→
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWhcaqaaiaadMgaaiaawEniaiaacYcadaWhcaqaaiaadQgaaiaawEniaaGaayjkaiaawMcaaaaa@3D71@
se numeste baza ortonormata.
v
→
=
A'B'
→
+
A''B''
→
=x⋅
i
→
+y⋅
j
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcacqGH9aqpdaWhcaqaaiaadgeacaGGNaGaamOqaiaacEcaaiaawEniaiabgUcaRmaaFiaabaGaamyqaiaacEcacaGGNaGaamOqaiaacEcacaGGNaaacaGLxdcacqGH9aqpcaWG4bGaeyyXIC9aa8HaaeaacaWGPbaacaGLxdcacqGHRaWkcaWG5bGaeyyXIC9aa8HaaeaacaWGQbaacaGLxdcaaaa@52CA@
x=xB-
xA, y=yB- yA
v
→
=p
r
OX
v
→
⋅
i
→
+p
r
OY
v
→
⋅
j
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcacqGH9aqpcaWGWbGaamOCamaaBaaaleaacaWGpbGaamiwaaqabaGcdaWhcaqaaiaadAhaaiaawEniaiabgwSixpaaFiaabaGaamyAaaGaay51GaGaey4kaSIaamiCaiaadkhadaWgaaWcbaGaam4taiaadMfaaeqaaOWaa8HaaeaacaWG2baacaGLxdcacqGHflY1daWhcaqaaiaadQgaaiaawEniaaaa@5168@
|
AB
→
|=
(
x
B
−
x
A
)
2
+
(
y
B
−
y
A
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWhcaqaaiaadgeacaWGcbaacaGLxdcaaiaawEa7caGLiWoacqGH9aqpdaGcaaqaaiaacIcacaWG4bWaaSbaaSqaaiaadkeaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGbbaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOaGaamyEamaaBaaaleaacaWGcbaabeaakiabgkHiTiaadMhadaWgaaWcbaGaamyqaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@4C9E@
Teorema:
Fie
u
→
(x,y),
v
→
(x',y')
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG1baacaGLxdcacaGGOaGaamiEaiaacYcacaWG5bGaaiykaiaacYcadaWhcaqaaiaadAhaaiaawEniaiaacIcacaWG4bGaai4jaiaacYcacaWG5bGaai4jaiaacMcaaaa@455E@
.
Atunci:
1)
u
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG1baacaGLxdcaaaa@38A1@
+
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2baacaGLxdcaaaa@38A2@
are coordonatele (x+x’.y+y’);
2)
∀λ∈R,λ⋅
v
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaeq4UdWMaeyicI4SaamOuaiaacYcacqaH7oaBcqGHflY1daWhcaqaaiaadAhaaiaawEniaaaa@422F@
are coordonatele (
λ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@37A7@
x’,
λ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@37A7@
y’);
3)
u
→
(x,y),
v
→
(x',y')
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG1baacaGLxdcacaGGOaGaamiEaiaacYcacaWG5bGaaiykaiaacYcadaWhcaqaaiaadAhaaiaawEniaiaacIcacaWG4bGaai4jaiaacYcacaWG5bGaai4jaiaacMcaaaa@455E@
sunt coliniari
⇔
x
x'
=
y
y'
=k,x',y'≠0.⇔xy'−x'y=0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aaSaaaeaacaWG4baabaGaamiEaiaacEcaaaGaeyypa0ZaaSaaaeaacaWG5baabaGaamyEaiaacEcaaaGaeyypa0Jaam4AaiaacYcacaWG4bGaai4jaiaacYcacaWG5bGaai4jaiabgcMi5kaaicdacaGGUaGaeyi1HSTaamiEaiaadMhacaGGNaGaeyOeI0IaamiEaiaacEcacaWG5bGaeyypa0JaaGimaiaac6caaaa@53A2@
4) Produsul scalar a doi vectori nenuli.
u
→
⋅
v
→
=|
u
→
|⋅|
v
→
|⋅cosα
unde
α=m(
u
→
,
v
→
)
,α∈[0,π].
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG1baacaGLxdcacqGHflY1daWhcaqaaiaadAhaaiaawEniaiabg2da9maaemaabaWaa8HaaeaacaWG1baacaGLxdcaaiaawEa7caGLiWoacqGHflY1daabdaqaamaaFiaabaGaamODaaGaay51GaaacaGLhWUaayjcSdGaeyyXICTaci4yaiaac+gacaGGZbGaeqySde2aaCbiaeaaaSqabeaaaaGccaWG1bGaamOBaiaadsgacaWGLbWaaCbiaeaacqaHXoqycqGH9aqpcaWGTbGaaiikamaaFiaabaGaamyDaaGaay51GaGaaiilamaaFiaabaGaamODaaGaay51GaGaaiykaaWcbeqaaaaakiaacYcacqaHXoqycqGHiiIZcaGGBbGaaGimaiaacYcacqaHapaCcaGGDbGaaiOlaaaa@6C16@
cosα=
x⋅x'+y⋅y'
x
2
+
y
2
⋅
(x')
2
+
(y')
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaeqySdeMaeyypa0ZaaSaaaeaacaWG4bGaeyyXICTaamiEaiaacEcacqGHRaWkcaWG5bGaeyyXICTaamyEaiaacEcaaeaadaGcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG5bWaaWbaaSqabeaacaaIYaaaaaqabaGccqGHflY1daGcaaqaaiaacIcacaWG4bGaai4jaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOaGaamyEaiaacEcacaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaaaaaaa@562B@
α∈[0,
π
2
]⇒
u
→
⋅
v
→
≥0;α∈(
π
2
,π]⇒
u
→
⋅
v
→
〈0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4Saai4waiaaicdacaGGSaWaaSaaaeaacqaHapaCaeaacaaIYaaaaiaac2facqGHshI3daWhcaqaaiaadwhaaiaawEniaiabgwSixpaaFiaabaGaamODaaGaay51GaGaeyyzImRaaGimaiaacUdacqaHXoqycqGHiiIZcaGGOaWaaSaaaeaacqaHapaCaeaacaaIYaaaaiaacYcacqaHapaCcaGGDbGaeyO0H49aa8HaaeaacaWG1baacaGLxdcacqGHflY1daWhcaqaaiaadAhaaiaawEniaiabgMYiHlaaicdaaaa@6229@
Fie
u
→
(x,y),
v
→
(x',y')
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG1baacaGLxdcacaGGOaGaamiEaiaacYcacaWG5bGaaiykaiaacYcadaWhcaqaaiaadAhaaiaawEniaiaacIcacaWG4bGaai4jaiaacYcacaWG5bGaai4jaiaacMcaaaa@455E@
nenuli. Atunci:
u
→
⋅
v
→
=0⇔
u
→
⊥
v
→
⇔x⋅x'+y⋅y'=0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG1baacaGLxdcacqGHflY1daWhcaqaaiaadAhaaiaawEniaiabg2da9iaaicdacqGHuhY2daWhcaqaaiaadwhaaiaawEniaiabgwQiEnaaFiaabaGaamODaaGaay51GaGaeyi1HSTaamiEaiabgwSixlaadIhacaGGNaGaey4kaSIaamyEaiabgwSixlaadMhacaGGNaGaeyypa0JaaGimaiaac6caaaa@5854@
u
→
⋅
u
→
=
|
u
→
|
2
≥0,∀
u
→
.
u
→
⋅
u
→
=0⇔
u
→
=0.
i
→
⋅
i
→
=
j
→
⋅
j
→
=1;
i
→
⋅
j
→
=0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7AA4@
Vectori de pozitie. Daca
r
A
→
,
r
B
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGYbWaaSbaaSqaaiaadgeaaeqaaaGccaGLxdcacaGGSaWaa8HaaeaacaWGYbWaaSbaaSqaaiaadkeaaeqaaaGccaGLxdcaaaa@3DF2@
sunt vectori de pozitie, atunci:
AB
→
=
r
B
→
−
r
A
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGbbGaamOqaaGaay51GaGaeyypa0Zaa8HaaeaacaWGYbWaaSbaaSqaaiaadkeaaeqaaaGccaGLxdcacqGHsisldaWhcaqaaiaadkhadaWgaaWcbaGaamyqaaqabaaakiaawEniaaaa@4276@
10.
Functii trigonometrice
Semnul
functiilor trigonometrice:
Sin:
[
−
π
2
,
π
2
]→[
−1,1
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaaaiaawUfacaGLDbaacqGHsgIRdaWadaqaaiabgkHiTiaaigdacaGGSaGaaGymaaGaay5waiaaw2faaaaa@4586@
arcsin:[-1,1]→
[
−
π
2
,
π
2
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaaaiaawUfacaGLDbaaaaa@3E94@
Cos:
[
0,π
]→[
−1,1
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaIWaGaaiilaiabec8aWbGaay5waiaaw2faaiabgkziUoaadmaabaGaeyOeI0IaaGymaiaacYcacaaIXaaacaGLBbGaayzxaaaaaa@41FE@
arccos:[-1,1]
→[
0,π
]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH46aamWaaeaacaaIWaGaaiilaiabec8aWbGaay5waiaaw2faaaaa@3CF9@
Tg:
(
−
π
2
,
π
2
)→R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaaaiaawIcacaGLPaaacqGHsgIRcaWGsbaaaa@40EF@
arctg:R→
(
−
π
2
,
π
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaaaiaawIcacaGLPaaaaaa@3E2B@
Reducerea
la un unghi ascutit
Fie u
∈(0,
π
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaaiikaiaaicdacaGGSaWaaSaaaeaacqaHapaCaeaacaaIYaaaaiaacMcaaaa@3CC3@
Notam sgn f= semnul functiei f; cof = cofunctia
lui f
sin(
k
π
2
±u
)={
sgnf(k
π
2
±u)⋅sinu,k=par
sgnf(k
π
2
±u)⋅cosu,k=impar
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@72D5@
Analog pentru celelalte;
In general,
f(k
π
2
±u)={
sgnf(k
π
2
±u)⋅f(u),k=par
sgnf(k
π
2
±u)⋅cof(u),k=impar
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7171@
Ecuatii
trigonometrice
Fie x un unghi, a un
numar real si k
∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaamOwaaaa@3856@
.
sinx=a,| a |≤1⇒x=
(−1)
k
arcsina+kπ,dacă
a∈[0,1]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaamiEaiabg2da9iaadggacaGGSaWaaqWaaeaacaWGHbaacaGLhWUaayjcSdGaeyizImQaaGymaiabgkDiElaadIhacqGH9aqpcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaadUgaaaGcciGGHbGaaiOCaiaacogacaGGZbGaaiyAaiaac6gacaWGHbGaey4kaSIaam4Aaiabec8aWjaacYcacaWGKbGaamyyaiaadogacaWGdeWaaCbiaeaaaSqabeaaaaGccaWGHbGaeyicI4Saai4waiaaicdacaGGSaGaaGymaiaac2faaaa@6005@
=
(−1)
k+1
arcsin| a |+kπ,dacă
a∈[−1,0]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbGaey4kaSIaaGymaaaakiGacggacaGGYbGaai4yaiaacohacaGGPbGaaiOBamaaemaabaGaamyyaaGaay5bSlaawIa7aiabgUcaRiaadUgacqaHapaCcaGGSaGaamizaiaadggacaWGJbGaam4abmaaxacabaaaleqabaaaaOGaamyyaiabgIGiolaacUfacqGHsislcaaIXaGaaiilaiaaicdacaGGDbaaaa@5468@
cosx=a,| a |≤1⇒x=±arccosa+2kπ,dacă
a∈[0,1]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaamiEaiabg2da9iaadggacaGGSaWaaqWaaeaacaWGHbaacaGLhWUaayjcSdGaeyizImQaaGymaiabgkDiElaadIhacqGH9aqpcqGHXcqSciGGHbGaaiOCaiaacogacaGGJbGaai4BaiaacohacaWGHbGaey4kaSIaaGOmaiaadUgacqaHapaCcaGGSaGaamizaiaadggacaWGJbGaam4abmaaxacabaaaleqabaaaaOGaamyyaiabgIGiolaacUfacaaIWaGaaiilaiaaigdacaGGDbaaaa@5E7D@
=
±arccosa+(2k+1)π,dacă
a∈[−1,0]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyySaeRaciyyaiaackhacaGGJbGaai4yaiaac+gacaGGZbGaamyyaiabgUcaRiaacIcacaaIYaGaam4AaiabgUcaRiaaigdacaGGPaGaeqiWdaNaaiilaiaadsgacaWGHbGaam4yaiaadoqadaWfGaqaaaWcbeqaaaaakiaadggacqGHiiIZcaGGBbGaeyOeI0IaaGymaiaacYcacaaIWaGaaiyxaaaa@511C@
tgx=a,a∈R⇒x=arctga+kπ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadEgacaWG4bGaeyypa0JaamyyaiaacYcacaWGHbGaeyicI4SaamOuaiabgkDiElaadIhacqGH9aqpcaWGHbGaamOCaiaadogacaWG0bGaam4zaiaadggacqGHRaWkcaWGRbGaeqiWdahaaa@4C31@
arcsin(sinx)=a⇒x=
(−1)
k
a+kπ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaackhacaGGJbGaai4CaiaacMgacaGGUbGaaiikaiGacohacaGGPbGaaiOBaiaadIhacaGGPaGaeyypa0JaamyyaiabgkDiElaadIhacqGH9aqpcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaadUgaaaGccaWGHbGaey4kaSIaam4Aaiabec8aWbaa@4FA4@
arccos(cosx)=a⇒x=±a+2kπ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaackhacaGGJbGaai4yaiaac+gacaGGZbGaaiikaiGacogacaGGVbGaai4CaiaadIhacaGGPaGaeyypa0JaamyyaiabgkDiElaadIhacqGH9aqpcqGHXcqScaWGHbGaey4kaSIaaGOmaiaadUgacqaHapaCaaa@4E1C@
arctg(tgx)=a⇒x=a+kπ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadkhacaWGJbGaamiDaiaadEgacaGGOaGaamiDaiaadEgacaWG4bGaaiykaiabg2da9iaadggacqGHshI3caWG4bGaeyypa0JaamyyaiabgUcaRiaadUgacqaHapaCaaa@4999@
sinf(x)=sing(x)⇒f(x)=
(−1)
k
g(x)+kπ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iGacohacaGGPbGaaiOBaiaadEgacaGGOaGaamiEaiaacMcacqGHshI3caWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbaaaOGaam4zaiaacIcacaWG4bGaaiykaiabgUcaRiaadUgacqaHapaCaaa@54C9@
sinf(x)=sing(x)⇒f(x)=
(−1)
k
g(x)+kπ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iGacohacaGGPbGaaiOBaiaadEgacaGGOaGaamiEaiaacMcacqGHshI3caWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbaaaOGaam4zaiaacIcacaWG4bGaaiykaiabgUcaRiaadUgacqaHapaCaaa@54C9@
tgf(x)=tgg(x)⇒f(x)=g(x)+kπ,k∈Z
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadEgacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamiDaiaadEgacaWGNbGaaiikaiaadIhacaGGPaGaeyO0H4TaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaadEgacaGGOaGaamiEaiaacMcacqGHRaWkcaWGRbGaeqiWdaNaaiilaiaadUgacqGHiiIZcaWGAbaaaa@52BE@
Ecuatii
trigonometrice reductibile la ecuatii care contin aceeasi functie a aceluiasi
unghi;
Ecuatii omogene in
sin x si cos x de forma: asin x+bcos x=0; asin2 x+bsin x .cos x+
ccos2 x=0
Ecuatii
trigonometrice care se rezolva prin descompuneri in factori;
Ecuatii simetrice in
sin x si cos x;
Ecuatii de forma:
asinx+bcosx+c=0
|:
a⇒sinx+tgϕcosx=−
c
a
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkcaWGIbGaci4yaiaac+gacaGGZbGaamiEaiabgUcaRiaadogacqGH9aqpcaaIWaWaaCbiaeaadaabbaqaaiaacQdaaiaawEa7aaWcbeqaaaaakiaadggacqGHshI3ciGGZbGaaiyAaiaac6gacaWG4bGaey4kaSIaamiDaiaadEgacqaHvpGzciGGJbGaai4BaiaacohacaWG4bGaeyypa0JaeyOeI0YaaSaaaeaacaWGJbaabaGaamyyaaaacqGHshI3aaa@5C1B@
x+ϕ=
(−1)
k
arcsin(−
c
a
cosϕ)+kπ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgUcaRiabew9aMjabg2da9iaacIcacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaam4AaaaakiGacggacaGGYbGaai4yaiaacohacaGGPbGaaiOBaiaacIcacqGHsisldaWcaaqaaiaadogaaeaacaWGHbaaaiGacogacaGGVbGaai4Caiabew9aMjaacMcacqGHRaWkcaWGRbGaeqiWdahaaa@50B0@
|
asinx+bcosx
|≤
a
2
+
b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbGaci4CaiaacMgacaGGUbGaamiEaiabgUcaRiaadkgaciGGJbGaai4BaiaacohacaWG4baacaGLhWUaayjcSdGaeyizIm6aaOaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaaaeqaaaaa@49B9@
Observatie importanta: Prin ridicarea la putere a unei ecuatii
trigonometrice pot aparea solutii straine iar prin impartirea unei ecuatii
trigonometrice se pot pierde solutii;
FORMULE
TRIGONOMETRICE
1.
sin
2
α+
cos
2
α=1⇒cosα=±
1−
sin
2
α
;
sinα=±
1−
cos
2
α
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHXoqycqGHRaWkciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccqaHXoqycqGH9aqpcaaIXaGaeyO0H4Taci4yaiaac+gacaGGZbGaeqySdeMaeyypa0JaeyySae7aaOaaaeaacaaIXaGaeyOeI0Iaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeqySdegaleqaaOGaai4oamaavacabeWcbeqaaaqdbaaaaaGcbaGaci4CaiaacMgacaGGUbGaeqySdeMaeyypa0JaeyySae7aaOaaaeaacaaIXaGaeyOeI0Iaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySdegaleqaaaaaaa@6418@
α∈R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaamOuaaaa@39ED@
2.
tgα=±
sinα
1−
sin
2
α
=±
1−
cos
2
α
cosα
⇒
t
g
2
α+1=
1
cos
2
α
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6797@
3.
cosα=±
1
1+t
g
2
α
;
sinα=±
tgα
1+t
g
2
α
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaeqySdeMaeyypa0JaeyySae7aaSaaaeaacaaIXaaabaWaaOaaaeaacaaIXaGaey4kaSIaamiDaiaadEgadaahaaWcbeqaaiaaikdaaaGccqaHXoqyaSqabaaaaOGaai4oamaavacabeWcbeqaaaqdbaaaaOGaci4CaiaacMgacaGGUbGaeqySdeMaeyypa0JaeyySae7aaSaaaeaacaWG0bGaam4zaiabeg7aHbqaamaakaaabaGaaGymaiabgUcaRiaadshacaWGNbWaaWbaaSqabeaacaaIYaaaaOGaeqySdegaleqaaaaakiaacUdadaqfGaqabSqabeaaa0qaaaaaaaa@57C9@
4.
cos(α+β)=cosαcosβ−sinαsinβ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiikaiabeg7aHjabgUcaRiabek7aIjaacMcacqGH9aqpciGGJbGaai4BaiaacohacqaHXoqyciGGJbGaai4BaiaacohacqaHYoGycqGHsislciGGZbGaaiyAaiaac6gacqaHXoqyciGGZbGaaiyAaiaac6gacqaHYoGyaaa@520A@
;
5.
cos(α−β)=cosαcosβ+sinαsinβ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiikaiabeg7aHjabgkHiTiabek7aIjaacMcacqGH9aqpciGGJbGaai4BaiaacohacqaHXoqyciGGJbGaai4BaiaacohacqaHYoGycqGHRaWkciGGZbGaaiyAaiaac6gacqaHXoqyciGGZbGaaiyAaiaac6gacqaHYoGyaaa@520A@
;
6.
sin(α+β)=sinαcosβ+sinβcosα
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaiikaiabeg7aHjabgUcaRiabek7aIjaacMcacqGH9aqpciGGZbGaaiyAaiaac6gacqaHXoqyciGGJbGaai4BaiaacohacqaHYoGycqGHRaWkciGGZbGaaiyAaiaac6gacqaHYoGyciGGJbGaai4BaiaacohacqaHXoqyaaa@5204@
;
7.
sin(α−β)=sinαcosβ−sinβcosα
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaiikaiabeg7aHjabgkHiTiabek7aIjaacMcacqGH9aqpciGGZbGaaiyAaiaac6gacqaHXoqyciGGJbGaai4BaiaacohacqaHYoGycqGHsislciGGZbGaaiyAaiaac6gacqaHYoGyciGGJbGaai4BaiaacohacqaHXoqyaaa@521A@
;
8.
tg(α+β)=
tgα+tgβ
1−tgα⋅tgβ
;
tg(α−β)=
tgα−tgβ
1+tgα⋅tgβ
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadEgacaGGOaGaeqySdeMaey4kaSIaeqOSdiMaaiykaiabg2da9maalaaabaGaamiDaiaadEgacqaHXoqycqGHRaWkcaWG0bGaam4zaiabek7aIbqaaiaaigdacqGHsislcaWG0bGaam4zaiabeg7aHjabgwSixlaadshacaWGNbGaeqOSdigaaiaacUdadaqfGaqabSqabeaaa0qaaaaakiaadshacaWGNbGaaiikaiabeg7aHjabgkHiTiabek7aIjaacMcacqGH9aqpdaWcaaqaaiaadshacaWGNbGaeqySdeMaeyOeI0IaamiDaiaadEgacqaHYoGyaeaacaaIXaGaey4kaSIaamiDaiaadEgacqaHXoqycqGHflY1caWG0bGaam4zaiabek7aIbaacaGG7aWaaubiaeqaleqabaaaneaaaaaaaa@6EEE@
9.
ctg(α+β)=
ctgα⋅ctgβ−1
ctgα+ctgβ
;
ctg(α−β)=
ctgα⋅ctgβ+1
ctgα−ctgβ
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@77FE@
10.
sin2α=2sinαcosα;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaGOmaiabeg7aHjabg2da9iaaikdaciGGZbGaaiyAaiaac6gacqaHXoqyciGGJbGaai4BaiaacohacqaHXoqycaGG7aWaaubiaeqaleqabaaaneaaaaaaaa@46E6@
11.
cos2α=
cos
2
α−
sin
2
α=2
cos
2
α−1=1−2
sin
2
α
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaGOmaiabeg7aHjabg2da9iGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHjabgkHiTiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeg7aHjabg2da9iaaikdaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccqaHXoqycqGHsislcaaIXaGaeyypa0JaaGymaiabgkHiTiaaikdaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHXoqyaaa@5986@
12.
cos
2
α=
1+cos2α
2
;
sin
2
α=
1−cos2α
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaeyypa0ZaaSaaaeaacaaIXaGaey4kaSIaci4yaiaac+gacaGGZbGaaGOmaiabeg7aHbqaaiaaikdaaaGaai4oamaavacabeWcbeqaaaqdbaaaaOGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaeyypa0ZaaSaaaeaacaaIXaGaeyOeI0Iaci4yaiaac+gacaGGZbGaaGOmaiabeg7aHbqaaiaaikdaaaaaaa@5326@
;
13.
cos
α
2
=±
1+cosα
2
;sin
α
2
=±
1−cosα
2
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbWaaSaaaeaacqaHXoqyaeaacaaIYaaaaiabg2da9iabgglaXoaakaaabaWaaSaaaeaacaaIXaGaey4kaSIaci4yaiaac+gacaGGZbGaeqySdegabaGaaGOmaaaaaSqabaGccaGG7aGaci4CaiaacMgacaGGUbWaaSaaaeaacqaHXoqyaeaacaaIYaaaaiabg2da9iabgglaXoaakaaabaWaaSaaaeaacaaIXaGaeyOeI0Iaci4yaiaac+gacaGGZbGaeqySdegabaGaaGOmaaaaaSqabaGccaGG7aaaaa@55E5@
14.
tg
α
2
=±
1−cosα
1+cosα
;
ctg
α
2
=±
1+cosα
1−cosα
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadEgadaWcaaqaaiabeg7aHbqaaiaaikdaaaGaeyypa0JaeyySae7aaOaaaeaadaWcaaqaaiaaigdacqGHsislciGGJbGaai4BaiaacohacqaHXoqyaeaacaaIXaGaey4kaSIaci4yaiaac+gacaGGZbGaeqySdegaaaWcbeaakiaacUdadaqfGaqabSqabeaaa0qaaaaakiaadogacaWG0bGaam4zamaalaaabaGaeqySdegabaGaaGOmaaaacqGH9aqpcqGHXcqSdaGcaaqaamaalaaabaGaaGymaiabgUcaRiGacogacaGGVbGaai4Caiabeg7aHbqaaiaaigdacqGHsislciGGJbGaai4BaiaacohacqaHXoqyaaaaleqaaaaa@5F34@
15.
tg2α=
2tgα
1−t
g
2
α
;
ctg2α=
ct
g
2
α−1
2ctgα
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadEgacaaIYaGaeqySdeMaeyypa0ZaaSaaaeaacaaIYaGaamiDaiaadEgacqaHXoqyaeaacaaIXaGaeyOeI0IaamiDaiaadEgadaahaaWcbeqaaiaaikdaaaGccqaHXoqyaaGaai4oamaavacabeWcbeqaaaqdbaaaaOGaam4yaiaadshacaWGNbGaaGOmaiabeg7aHjabg2da9maalaaabaGaam4yaiaadshacaWGNbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaeyOeI0IaaGymaaqaaiaaikdacaWGJbGaamiDaiaadEgacqaHXoqyaaGaai4oaaaa@59F3@
16.
tgα=
2tg
α
2
1−t
g
2
α
2
;
ctgα=
1−t
g
2
α
2
2tg
α
2
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadEgacqaHXoqycqGH9aqpdaWcaaqaaiaaikdacaWG0bGaam4zamaalaaabaGaeqySdegabaGaaGOmaaaaaeaacaaIXaGaeyOeI0IaamiDaiaadEgadaahaaWcbeqaaiaaikdaaaGcdaWcaaqaaiabeg7aHbqaaiaaikdaaaaaaiaacUdadaqfGaqabSqabeaaa0qaaaaakiaadogacaWG0bGaam4zaiabeg7aHjabg2da9maalaaabaGaaGymaiabgkHiTiaadshacaWGNbWaaWbaaSqabeaacaaIYaaaaOWaaSaaaeaacqaHXoqyaeaacaaIYaaaaaqaaiaaikdacaWG0bGaam4zamaalaaabaGaeqySdegabaGaaGOmaaaaaaGaai4oaaaa@59DB@
17.
sin3α=3sinα−4
sin
3
α;
tg3α=
3tgα−t
g
3
α
1−3t
g
2
α
cos3α=4
cos
3
α−3cosα;
ctg3α=
ct
g
3
α−3ctgα
3ct
g
2
α−1
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8FE2@
18.
tg
α
2
=
sinα
1+cosα
=
1−cosα
sinα
=
1
ctg
α
2
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadEgadaWcaaqaaiabeg7aHbqaaiaaikdaaaGaeyypa0ZaaSaaaeaaciGGZbGaaiyAaiaac6gacqaHXoqyaeaacaaIXaGaey4kaSIaci4yaiaac+gacaGGZbGaeqySdegaaiabg2da9maalaaabaGaaGymaiabgkHiTiGacogacaGGVbGaai4Caiabeg7aHbqaaiGacohacaGGPbGaaiOBaiabeg7aHbaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGJbGaamiDaiaadEgadaWcaaqaaiabeg7aHbqaaiaaikdaaaaaaiaacUdaaaa@594E@
19.
sinα=
2tg
α
2
1+t
g
2
α
2
;
cosα=
1−t
g
2
α
2
1+t
g
2
α
2
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaeqySdeMaeyypa0ZaaSaaaeaacaaIYaGaamiDaiaadEgadaWcaaqaaiabeg7aHbqaaiaaikdaaaaabaGaaGymaiabgUcaRiaadshacaWGNbWaaWbaaSqabeaacaaIYaaaaOWaaSaaaeaacqaHXoqyaeaacaaIYaaaaaaacaGG7aWaaubiaeqaleqabaaaneaaaaGcciGGJbGaai4BaiaacohacqaHXoqycqGH9aqpdaWcaaqaaiaaigdacqGHsislcaWG0bGaam4zamaaCaaaleqabaGaaGOmaaaakmaalaaabaGaeqySdegabaGaaGOmaaaaaeaacaaIXaGaey4kaSIaamiDaiaadEgadaahaaWcbeqaaiaaikdaaaGcdaWcaaqaaiabeg7aHbqaaiaaikdaaaaaaiaacUdaaaa@5C9D@
sina+sinb=2sin
a+b
2
⋅cos
a−b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaamyyaiabgUcaRiGacohacaGGPbGaaiOBaiaadkgacqGH9aqpcaaIYaGaci4CaiaacMgacaGGUbWaaSaaaeaacaWGHbGaey4kaSIaamOyaaqaaiaaikdaaaGaeyyXICTaci4yaiaac+gacaGGZbWaaSaaaeaacaWGHbGaeyOeI0IaamOyaaqaaiaaikdaaaaaaa@4F0A@
sina−sinb=2sin
a−b
2
⋅cos
a+b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaamyyaiabgkHiTiGacohacaGGPbGaaiOBaiaadkgacqGH9aqpcaaIYaGaci4CaiaacMgacaGGUbWaaSaaaeaacaWGHbGaeyOeI0IaamOyaaqaaiaaikdaaaGaeyyXICTaci4yaiaac+gacaGGZbWaaSaaaeaacaWGHbGaey4kaSIaamOyaaqaaiaaikdaaaaaaa@4F15@
cosa−cosb=−2sin
a+b
2
⋅sin
a−b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaamyyaiabgkHiTiGacogacaGGVbGaai4CaiaadkgacqGH9aqpcqGHsislcaaIYaGaci4CaiaacMgacaGGUbWaaSaaaeaacaWGHbGaey4kaSIaamOyaaqaaiaaikdaaaGaeyyXICTaci4CaiaacMgacaGGUbWaaSaaaeaacaWGHbGaeyOeI0IaamOyaaqaaiaaikdaaaaaaa@4FFD@
cosa+cosb=2sin
a−b
2
⋅cos
a+b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaamyyaiabgUcaRiGacogacaGGVbGaai4CaiaadkgacqGH9aqpcaaIYaGaci4CaiaacMgacaGGUbWaaSaaaeaacaWGHbGaeyOeI0IaamOyaaqaaiaaikdaaaGaeyyXICTaci4yaiaac+gacaGGZbWaaSaaaeaacaWGHbGaey4kaSIaamOyaaqaaiaaikdaaaaaaa@4F00@
tga−tgb=
sin(a−b)
cosa⋅cosb
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadEgacaWGHbGaeyOeI0IaamiDaiaadEgacaWGIbGaeyypa0ZaaSaaaeaaciGGZbGaaiyAaiaac6gacaGGOaGaamyyaiabgkHiTiaadkgacaGGPaaabaGaci4yaiaac+gacaGGZbGaamyyaiabgwSixlGacogacaGGVbGaai4Caiaadkgaaaaaaa@4E35@
ctga+ctgb=
sin(a+b)
sina⋅sinb
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaadshacaWGNbGaamyyaiabgUcaRiaadogacaWG0bGaam4zaiaadkgacqGH9aqpdaWcaaqaaiGacohacaGGPbGaaiOBaiaacIcacaWGHbGaey4kaSIaamOyaiaacMcaaeaaciGGZbGaaiyAaiaac6gacaWGHbGaeyyXICTaci4CaiaacMgacaGGUbGaamOyaaaaaaa@4FF9@
ctga−ctgb=
sin(b−a)
sina⋅sinb
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaadshacaWGNbGaamyyaiabgkHiTiaadogacaWG0bGaam4zaiaadkgacqGH9aqpdaWcaaqaaiGacohacaGGPbGaaiOBaiaacIcacaWGIbGaeyOeI0IaamyyaiaacMcaaeaaciGGZbGaaiyAaiaac6gacaWGHbGaeyyXICTaci4CaiaacMgacaGGUbGaamOyaaaaaaa@500F@
cosa⋅cosb=
cos(a+b)+cos(a−b)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaamyyaiabgwSixlGacogacaGGVbGaai4CaiaadkgacqGH9aqpdaWcaaqaaiGacogacaGGVbGaai4CaiaacIcacaWGHbGaey4kaSIaamOyaiaacMcacqGHRaWkciGGJbGaai4BaiaacohacaGGOaGaamyyaiabgkHiTiaadkgacaGGPaaabaGaaGOmaaaaaaa@5025@
arcsinx+arcsiny=arcsin(x
1−
y
2
+y
1−
x
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaackhacaGGJbGaai4CaiaacMgacaGGUbGaamiEaiabgUcaRiGacggacaGGYbGaai4yaiaacohacaGGPbGaaiOBaiaadMhacqGH9aqpciGGHbGaaiOCaiaacogacaGGZbGaaiyAaiaac6gacaGGOaGaamiEamaakaaabaGaaGymaiabgkHiTiaadMhadaahaaWcbeqaaiaaikdaaaaabeaakiabgUcaRiaadMhadaGcaaqaaiaaigdacqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaaqabaGccaGGPaaaaa@562B@
arcsin x+arccos x=
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
arctg x +arcctg x=
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
arctg
x+arctg
1
x
=
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiEaaaacqGH9aqpdaWcaaqaaiabec8aWbqaaiaaikdaaaaaaa@3B4A@
arccos(-x)=
π
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdahaaa@37B0@
-arccos x
11. ECUATIILE
DREPTEI IN PLAN
1. Ecuatia carteziana generala a dreptei:
ax+by+c=0 (d)
Punctul M(x0,y0)
∈d⇔a⋅
x
0
+b⋅
y
0
+c=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaamizaiabgsDiBlaadggacqGHflY1caWG4bWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaamOyaiabgwSixlaadMhadaqhaaWcbaGaaGimaaqaaaaakiabgUcaRiaadogacqGH9aqpcaaIWaaaaa@4965@
2. Ecuatia dreptei determinata
de punctele A(x1,y1), B(x2,y2):
y−
y
1
y
2
−
y
1
=
x−
x
1
x
2
−
x
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG5bGaeyOeI0IaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiaadMhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG5bWaaSbaaSqaaiaaigdaaeqaaaaakiabg2da9maalaaabaGaamiEaiabgkHiTiaadIhadaWgaaWcbaGaaGymaaqabaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaaIXaaabeaaaaaaaa@4857@
3. Ecuatia dreptei determinata
de un punct M(x0,y0) si o directie data( are panta m)
y-y0=m(x-x0)
4. Ecuatia explicita a dreptei
(ecuatia normala):
y=mx+n, unde
m=tgϕ=
y
2
−
y
1
x
2
−
x
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9iaadshacaWGNbGaeqy1dyMaeyypa0ZaaSaaaeaacaWG5bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaaaaa@463A@
este panta dreptei si n este ordonata la
origine.
5. Ecuatia dreptei prin taieturi:
x
a
+
y
b
=1,
a,b≠0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG4baabaGaamyyaaaacqGHRaWkdaWcaaqaaiaadMhaaeaacaWGIbaaaiabg2da9iaaigdacaGGSaWaaCbiaeaaaSqabeaaaaGccaWGHbGaaiilaiaadkgacqGHGjsUcaaIWaGaaiOlaaaa@4331@
6. Fie (d): y=mx+n si (d’):
y=m’x+n’
Dreptele d si d’ sunt paralele
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
m=m’si n
≠
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKlaaa@37BA@
n’.
Dreptele d si d’ coincid
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
m=m’si n=n’.
Dreptele d si d’ sunt perpendiculare
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
mm’= -1.
Tangenta
unghiului
ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BB@
a celor doua drepte este
tgϕ=|
m−m'
1+m⋅m'
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadEgacqaHvpGzcqGH9aqpdaabdaqaamaalaaabaGaamyBaiabgkHiTiaad2gacaGGNaaabaGaaGymaiabgUcaRiaad2gacqGHflY1caWGTbGaai4jaaaaaiaawEa7caGLiWoaaaa@47CA@
7. Fie d: ax+by+c=0 si d’: a’x+b’y+c’=0
cu a’,b’,c’
≠0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKRaaGimaiaac6caaaa@3926@
si
θ=m(〈d,d')
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaeyypa0JaamyBaiaacIcacqGHPms4caWGKbGaaiilaiaadsgacaGGNaGaaiykaaaa@3FE0@
Dreptele d si d’ sunt paralele
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
a
a'
=
b
b'
≠
c
c'
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamyyaiaacEcaaaGaeyypa0ZaaSaaaeaacaWGIbaabaGaamOyaiaacEcaaaGaeyiyIK7aaSaaaeaacaWGJbaabaGaam4yaiaacEcaaaaaaa@405B@
Dreptele d si d’ coincid
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
a
a'
=
b
b'
=
c
c'
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamyyaiaacEcaaaGaeyypa0ZaaSaaaeaacaWGIbaabaGaamOyaiaacEcaaaGaeyypa0ZaaSaaaeaacaWGJbaabaGaam4yaiaacEcaaaaaaa@3F9A@
Dreptele d si d’ sunt concurente
⇔
a
a'
≠
b
b'
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aaSaaaeaacaWGHbaabaGaamyyaiaacEcaaaGaeyiyIK7aaSaaaeaacaWGIbaabaGaamOyaiaacEcaaaGaeyi1HSnaaa@4182@
ab’-ba’
≠0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKRaaGimaiaac6caaaa@3926@
cosθ=
v
→
⋅
v
'
→
|
v
|
→
⋅
|
v
'
|
→
=
a
⋅a
'
+b⋅b
'
a
2
+
b
2
⋅
a
'
2
+b
'
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6D2A@
unde
v
→
(−
b
,
a
),
v
'
→
(−b
'
,a
'
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWG2bWaaSbaaSqaaaqabaaakiaawEniaiaacIcacqGHsislcaWGIbWaaSbaaSqaaaqabaGccaGGSaGaamyyamaaBaaaleaaaeqaaOGaaiykaiaacYcadaWhcaqaaiaadAhacaGGNaWaaSbaaSqaaaqabaaakiaawEniaiaacIcacqGHsislcaWGIbGaai4jamaaBaaaleaaaeqaaOGaaiilaiaadggacaGGNaWaaSbaaSqaaaqabaGccaGGPaaaaa@48CC@
sunt vectorii directori ai dreptelor d si d’.
Dreptele d si d’ sunt perpendiculare,
d⊥d'⇔a⋅a
'
+b⋅b
'
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgwQiEjaadsgacaGGNaGaeyi1HSTaamyyaiabgwSixlaadggacaGGNaWaaSbaaSqaaaqabaGccqGHRaWkcaWGIbGaeyyXICTaamOyaiaacEcadaWgaaWcbaaabeaakiabg2da9iaaicdaaaa@490F@
8. Fie punctele A(x1,y1),
B(x2,y2), C(x3,y3), D(x4,y4)
in plan.
Dreptele AB si CD sunt paralele, AB|| CD
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
∃α∈R*,a.î
AB
→
=α
CD
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqIaeqySdeMaeyicI4SaamOuaiaacQcacaGGSaGaamyyaiaac6cacaWGUdWaa8HaaeaacaWGbbGaamOqaaGaay51GaGaeyypa0JaeqySde2aa8HaaeaacaWGdbGaamiraaGaay51Gaaaaa@4856@
sau
mAB=mCD.
Dreptele AB si CD sunt perpendiculare,
AB⊥CD⇔
AB
→
⋅
CD
→
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadkeacqGHLkIxcaWGdbGaamiraiabgsDiBpaaFiaabaGaamyqaiaadkeaaiaawEniaiabgwSixpaaFiaabaGaam4qaiaadseaaiaawEniaiabg2da9iaaicdaaaa@47AE@
Conditia ca punctele A(x1,y1),
B(x2,y2), C(x3,y3) sa fie coliniare
este:
y
3
−
y
1
y
2
−
y
1
=
x
3
−
x
1
x
2
−
x
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG5bWaaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiaadMhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG5bWaaSbaaSqaaiaaigdaaeqaaaaakiabg2da9maalaaabaGaamiEamaaBaaaleaacaaIZaaabeaakiabgkHiTiaadIhadaWgaaWcbaGaaGymaaqabaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaaIXaaabeaaaaaaaa@4A3D@
9. Distanta dintre punctele A(x1,y1) si B(x2,y2) este
AB=
(
x
2
−
x
1
)
2
+
(
y
2
−
y
1
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadkeacqGH9aqpdaGcaaqaamaabmaabaGaamiEamaaBaaaleaacaaIYaaabeaakiabgkHiTiaadIhadaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaaiaadMhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG5bWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@47FC@
Distanta de la un punct M0(x0,y0)
la o dreapta h de ecuatie (h): ax+by+c=0 este data de:
d(
M
0
,h)=
|
a
x
0
+b
y
0
+c
|
a
2
+
b
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaacIcacaWGnbWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiaadIgacaGGPaGaeyypa0ZaaSaaaeaadaabdaqaaiaadggacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaamOyaiaadMhadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaWGJbaacaGLhWUaayjcSdaabaWaaOaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaaaeqaaaaaaaa@4CBB@
.
12. CONICE
1.CERCUL
Definitie: Locul
geometric al tuturor punctelor din plan egal departate de un punct fix, numit
centru se numeste cerc.
C(O,r)={M(x,y)|OM=r}
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaacIcacaWGpbGaaiilaiaadkhacaGGPaGaeyypa0Jaai4Eaiaad2eacaGGOaGaamiEaiaacYcacaWG5bGaaiykaiaacYhacaWGpbGaamytaiabg2da9iaadkhacaGG9baaaa@470E@
1.
Ecuatia generala a cercului
A(x
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
+ y
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
) + Bx + Cy + D =
0
2.
Ecuatia cercului de centru: O(a, b) respectiv O(0, 0) si raza
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=5biaaa@37AB@
r”
(x - a)
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
+ (y + b)
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
= r
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
; x
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
+ y
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
= r
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
3. Ecuatia
cercului de diametru A(x1;y1),
B(x2; y2)
(x - x1)(x
- x2) + ( y- y1)(y - y2) = 0
4.
Ecuatia tangentei dupa o directie
O(0,0) : y = mx ± r
1+m²
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaqGXaGaey4kaSIaaeyBaiaabklaaSqabaaaaa@39C9@
O(a,b) : y-b = m(x-a) ± r
1+m²
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaqGXaGaey4kaSIaaeyBaiaabklaaSqabaaaaa@39C9@
5.
Ecuatia tangentei in punctul M(x0, y0)
(x· x0) +
(y ·y0) = r
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
respectiv
(x - a)(x0 -
a) + (y - b)(y0 - b) = r
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
6.
Ecuatia normala a cercului
x
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
+ y
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
+ 2mx + 2ny + p
= 0 cu
O(-m; -n) si r
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
= m
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
+ n
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
- p
7.
Ecuatia tangentei in punctul M(x0,y0)
x · x0 + y
· y0 + m(x + x0) + n(y + y0) + p = 0
8.
Distanta de la centrul cercului O(a, b) la dreapta de ecuatie
y = mx + n este
d(0,d) =
|ma−b+n|
m²+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaGG8bGaamyBaiaadggacqGHsislcaWGIbGaey4kaSIaamOBaiaacYhaaeaadaGcaaqaaiaad2gacaGGYcGaey4kaSIaaGymaaWcbeaaaaaaaa@4164@
sau (
d=
|ax0+by0+c|
a²+b²
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg2da9maalaaabaGaaiiFaiaadggacaWG4bWccaaIWaGccqGHRaWkcaWGIbGaamyEaSGaaGimaiabgUcaROGaam4yaiaacYhaaeaadaGcaaqaaiaadggacaGGYcGaey4kaSIaamOyaiaacklaaSqabaaaaaaa@473A@
)
9.
Ecuatiile tangentelor din punctul exterior M(x0, y0)
I.
Se scrie ecuatia 4 si se pune conditia ca M sa apartina cercului de ecuatie 4.
II.
y - y0 = m(x - x0)
x
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
+ y
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
= r
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
,
Δ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqeaaa@3759@
=0
2. ELIPSA
Definitie: Locul geometric al punctelor din plan care au suma distantelor la doua
puncte fixe, constanta, se numeste elipsa.
F,F’- focare, FF’ distanta focala
E=
{
M(x,y)|
MF+MF'=2a
}
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGnbGaaiikaiaadIhacaGGSaGaamyEaiaacMcadaabbaqaaiaad2eacaWGgbGaey4kaSIaamytaiaadAeacaGGNaGaeyypa0JaaGOmaiaadggaaiaawEa7aaGaay5Eaiaaw2haaaaa@45FD@
MF,MF’- raze focale
1. Ecuatia elipsei
x²
a²
+
y²
b²
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG4bGaaiOSaaqaaiaadggacaGGYcaaaiabgUcaRmaalaaabaGaamyEaiaacklaaeaacaWGIbGaaiOSaaaacqGH9aqpcaaIXaaaaa@4156@
, b
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jlaaaa@383F@
= a
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jlaaaa@383F@
- c
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jlaaaa@383F@
2. Ecuatia tangentei la elipsa
y = mx ±
a²m²+b²
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGHbGaaiOSaiaad2gacaGGYcGaey4kaSIaamOyaiaacklaaSqabaaaaa@3D51@
3. Ecuatia tangentei in punctul M(x0, y0) la elipsa
x⋅x0
a²
+
y⋅y0
b²
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG4bGaeyyXICTaamiEaSGaaGimaaGcbaGaamyyaiaacklaaaGaey4kaSYaaSaaaeaacaWG5bGaeyyXICTaamyEaSGaaeimaaGcbaGaamOyaiaacklaaaGaeyypa0JaaGymaaaa@4710@
,
m=−
b²
a²
⋅
x0
y0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9iabgkHiTmaalaaabaGaamOyaiaacklaaeaacaWGHbGaaiOSaaaacqGHflY1daWcaaqaaiaadIhaliaaicdaaOqaaiaadMhaliaaicdaaaaaaa@430A@
4. Ecuatiile tangentelor dintr-un punct exterior M(x0, y0)
la elipsa
VAR I Se scrie ecuatia 2 si se pune
conditia ca M sa apartina elipsei de ecuatie 2 de unde rezulta m
VAR II Se rezolva sistemul y
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
y0 = m(x-x0)
,
cu conditia
Δ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqeaaa@3759@
= 0
3. HIPERBOLA
Definitie: Locul geometric al punctelor din plan a caror diferenta la doua puncte
fixe este constanta, se numeste hiperbola
H: = { M(x,y) | |MF
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@37A0@
MF’| = 2a }
y = ±
b
a
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGIbaabaGaamyyaaaacaWG4baaaa@38CD@
--ecuatia asimptotelor
1.
Ecuatia hiperbolei
x²
a²
−
y²
b²
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG4bGaaiOSaaqaaiaadggacaGGYcaaaiabgkHiTmaalaaabaGaamyEaiaacklaaeaacaWGIbGaaiOSaaaacqGH9aqpcaaIXaaaaa@4161@
, b
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
= c
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
- a
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
;
Daca a = b => hiperbola echilaterala
2.Ecuatia
tangentei la hiperbola
y = mx ±
a²m²−b²
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGHbGaaiOSaiaad2gacaGGYcGaeyOeI0IaamOyaiaacklaaSqabaaaaa@3D5C@
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F2@
3. Ecuatia tangentei in punctul M(x0,
y0)
x⋅x0
a²
−
y⋅y0
b²
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG4bGaeyyXICTaamiEaSGaaGimaaGcbaGaamyyaiaacklaaaGaeyOeI0YaaSaaaeaacaWG5bGaeyyXICTaamyEaSGaaGimaaGcbaGaamOyaiaacklaaaGaeyypa0JaaGymaaaa@4722@
,
m=
b²
a²
⋅
x0
y0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9maalaaabaGaamOyaiaacklaaeaacaWGHbGaaiOSaaaacqGHflY1daWcaaqaaiaadIhaliaaicdaaOqaaiaadMhaliaaicdaaaaaaa@421D@
4.
Ecuatiile tangentelor dintr-un punct exterior M(x0, y0)
VAR I. Se scrie ecuatia 2 si se pune conditia ca M sa apartina
hiperbolei de ecuatie 2, de unde rezulta m.
VAR II. Se rezolva sistemul
y - y0 = m(x - x0)
x²
a²
−
y²
b²
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG4bGaaiOSaaqaaiaadggacaGGYcaaaiabgkHiTmaalaaabaGaamyEaiaacklaaeaacaWGIbGaaiOSaaaacqGH9aqpcaaIXaaaaa@4161@
, cu
Δ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqeaaa@3759@
= 0
4. PARABOLA
Definitie: Locul geometric al punctelor egal departate
de un punct fix, (numit focar) si o dreapta fixa (numita directoare), se numeste
parabola.
P: = { M(x, y) | MF =
MN }
(d): x =
−
p
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaSaaaeaacaWGWbaabaGaaGOmaaaaaaa@38A1@
( locul geometric al punctelor din plan de
unde putem duce tangente la o parabola).
1.
Ecuatia parabolei
y
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
= 2px
2.
Ecuatia tangentei la parabola
y = mx +
P
2m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGqbaabaGaaGOmaiaad2gaaaaaaa@3886@
3.
Ecuatia tangentei in M (x0, y0)
y·y0 = p(x
+ x0)
4.
Ecuatia tangentelor dintr-un punct exterior M(x0, y0)
VAR I. Se scrie ecuatia 2 si se pune conditia ca M
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
(ecuatia 2) => m
VAR II.
Se rezolva sistemul
y - y0 =
m(x - x0)
y
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqaaaaaaaaaWdbiaa=jlaaaa@381F@
= 2px cu
Δ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqeaaa@3759@
= 0
13. ALGEBRA LINIARA
1. MATRICE.
Adunarea matricelor
(
a
b
c
d
)+(
x
y
z
t
)=(
a+x
b+y
c+z
d+t
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGacaaabaGaamyyaaqaaiaadkgaaeaacaWGJbaabaGaamizaaaaaiaawIcacaGLPaaacqGHRaWkdaqadaqaauaabeqaciaaaeaacaWG4baabaGaamyEaaqaaiaadQhaaeaacaWG0baaaaGaayjkaiaawMcaaiabg2da9maabmaabaqbaeqabiGaaaqaaiaadggacqGHRaWkcaWG4baabaGaamOyaiabgUcaRiaadMhaaeaacaWGJbGaey4kaSIaamOEaaqaaiaadsgacqGHRaWkcaWG0baaaaGaayjkaiaawMcaaaaa@4F50@
a⋅(
x
y
z
t
)=(
a⋅x
a⋅y
a⋅z
a⋅t
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgwSixpaabmaabaqbaeqabiGaaaqaaiaadIhaaeaacaWG5baabaGaamOEaaqaaiaadshaaaaacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaafaqabeGacaaabaGaamyyaiabgwSixlaadIhaaeaacaWGHbGaeyyXICTaamyEaaqaaiaadggacqGHflY1caWG6baabaGaamyyaiabgwSixlaadshaaaaacaGLOaGaayzkaaaaaa@5201@
Inmultirea
matricelor
(
a
b
c
d
)⋅(
x
y
z
t
)=(
a⋅x+b⋅z
a⋅y+b⋅t
c⋅x+d⋅z
c⋅y+d⋅t
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGacaaabaGaamyyaaqaaiaadkgaaeaacaWGJbaabaGaamizaaaaaiaawIcacaGLPaaacqGHflY1daqadaqaauaabeqaciaaaeaacaWG4baabaGaamyEaaqaaiaadQhaaeaacaWG0baaaaGaayjkaiaawMcaaiabg2da9maabmaabaqbaeqabiGaaaqaaiaadggacqGHflY1caWG4bGaey4kaSIaamOyaiabgwSixlaadQhaaeaacaWGHbGaeyyXICTaamyEaiabgUcaRiaadkgacqGHflY1caWG0baabaGaam4yaiabgwSixlaadIhacqGHRaWkcaWGKbGaeyyXICTaamOEaaqaaiaadogacqGHflY1caWG5bGaey4kaSIaamizaiabgwSixlaadshaaaaacaGLOaGaayzkaaaaaa@6A99@
Transpusa unei matrice
(
a
b
c
d
)
T
=(
a
c
b
d
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGacaaabaGaamyyaaqaaiaadkgaaeaacaWGJbaabaGaamizaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaadsfaaaGccqGH9aqpdaqadaqaauaabeqaciaaaeaacaWGHbaabaGaam4yaaqaaiaadkgaaeaacaWGKbaaaaGaayjkaiaawMcaaaaa@4277@
2.
DETERMINANTI.
|
a
b
c
d
|=a⋅d−b⋅c
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaafaqabeGacaaabaGaamyyaaqaaiaadkgaaeaacaWGJbaabaGaamizaaaaaiaawEa7caGLiWoacqGH9aqpcaWGHbGaeyyXICTaamizaiabgkHiTiaadkgacqGHflY1caWGJbaaaa@46E8@
;
|
a
b
c
d
e
f
g
h
i
|=a⋅e⋅i+d⋅h⋅c+g⋅b⋅f−c⋅e⋅g−f⋅h⋅a−i⋅b⋅d
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaafaqabeWadaaabaGaamyyaaqaaiaadkgaaeaacaWGJbaabaGaamizaaqaaiaadwgaaeaacaWGMbaabaGaam4zaaqaaiaadIgaaeaacaWGPbaaaaGaay5bSlaawIa7aiabg2da9iaadggacqGHflY1caWGLbGaeyyXICTaamyAaiabgUcaRiaadsgacqGHflY1caWGObGaeyyXICTaam4yaiabgUcaRiaadEgacqGHflY1caWGIbGaeyyXICTaamOzaiabgkHiTiaadogacqGHflY1caWGLbGaeyyXICTaam4zaiabgkHiTiaadAgacqGHflY1caWGObGaeyyXICTaamyyaiabgkHiTiaadMgacqGHflY1caWGIbGaeyyXICTaamizaaaa@72E3@
Proprietati:
1. Determinantul unei matrice este egal cu
determinantul matricei transpuse;
2. Daca toate elementele unei linii (sau
coloane) dintr-o matrice sunt nule, atunci determinantul matricei este nul;
3. Daca intr-o matrice schimbam doua linii(sau
coloane) intre ele obtinem o matrice care are determinantul egal cu opusul
determinantului matricei initiale.
4. Daca o matrice are doua linii (sau coloane)
identice atunci determinantul sau este nul;
5. Daca toate elementele unei linii(sau
coloane) ale unei matrice sunt inmultite cu un element a, obtinem o matrice al
carei determinant este egal cu a inmultit cu determinantul matricei initiale.
6. Daca elementele a doua linii(sau coloane)
ale unei matrice sunt proportionale atunci determinantul matricei este nul;
7. Daca la o matrice patratica A de ordin n
presupunem ca elementele unei linii i sunt de forma
a
ij
=
a
ij
'
+
a
ij
''
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakmaaCaaaleqabaGaai4jaaaakiabgUcaRiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaWbaaSqabeaacaGGNaGaai4jaaaaaaa@432B@
atunci det A = det A’ +det A’’;
8. Daca o linie (sau coloana) a unei matrice patratice este o combinatie
liniara de celelate linii(sau coloane) atunci determinantul matricei este nul.
9. Daca la o linie (sau coloana) a matricei A adunam elementele altei
linii (sau coloane) inmultite cu acelasi element se obtine o matrice al carei
determinant este egal cu determinantul matricei initiale;
10. Determinantul Vandermonde:
|
1
1
1
a
b
c
a
2
b
2
c
2
|=(b−a)(c−a)(c−b)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaafaqabeWadaaabaGaaGymaaqaaiaaigdaaeaacaaIXaaabaGaamyyaaqaaiaadkgaaeaacaWGJbaabaGaamyyamaaCaaaleqabaGaaGOmaaaaaOqaaiaadkgadaahaaWcbeqaaiaaikdaaaaakeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaaaOGaay5bSlaawIa7aiabg2da9iaacIcacaWGIbGaeyOeI0IaamyyaiaacMcacaGGOaGaam4yaiabgkHiTiaadggacaGGPaGaaiikaiaadogacqGHsislcaWGIbGaaiykaaaa@50E2@
;
11. Daca intr-un determinant toate elementele
de deasupra diagonalei principale sau de dedesubtul ei sunt egale cu zero,
atunci determinantul este egal cu
a⋅c⋅f
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgwSixlaadogacqGHflY1caWGMbaaaa@3D40@
;
|
a
0
0
b
c
0
d
e
f
|=a⋅c⋅f
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaafaqabeWadaaabaGaamyyaaqaaiaaicdaaeaacaaIWaaabaGaamOyaaqaaiaadogaaeaacaaIWaaabaGaamizaaqaaiaadwgaaeaacaWGMbaaaaGaay5bSlaawIa7aiabg2da9iaadggacqGHflY1caWGJbGaeyyXICTaamOzaaaa@4920@
12. Factor comun
|
a⋅x
a⋅y
a⋅z
b⋅m
b⋅n
b⋅p
u
v
r
|=a⋅b⋅|
x
y
z
m
n
p
u
v
r
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaafaqabeWadaaabaGaamyyaiabgwSixlaadIhaaeaacaWGHbGaeyyXICTaamyEaaqaaiaadggacqGHflY1caWG6baabaGaamOyaiabgwSixlaad2gaaeaacaWGIbGaeyyXICTaamOBaaqaaiaadkgacqGHflY1caWGWbaabaGaamyDaaqaaiaadAhaaeaacaWGYbaaaaGaay5bSlaawIa7aiabg2da9iaadggacqGHflY1caWGIbGaeyyXIC9aaqWaaeaafaqabeWadaaabaGaamiEaaqaaiaadMhaaeaacaWG6baabaGaamyBaaqaaiaad6gaaeaacaWGWbaabaGaamyDaaqaaiaadAhaaeaacaWGYbaaaaGaay5bSlaawIa7aaaa@686F@
3. Rangul unei matrice
Fie A
∈
M
m,n
(C)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaamytamaaBaaaleaacaWGTbGaaiilaiaad6gaaeqaaOGaaiikaiaadoeacaGGPaaaaa@3D35@
,
r
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
N,
1≤r≤min(m,n)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgsMiJkaadkhacqGHKjYOciGGTbGaaiyAaiaac6gacaGGOaGaamyBaiaacYcacaWGUbGaaiykaaaa@41CF@
.
Definitie: Se numeste minor de ordinul r
al matricei A, determinantul format cu
elementele matricei A situate la intersectia celor r linii si r coloane.
Definitie: Fie A
≠
O
m,n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKRaam4tamaaBaaaleaacaWGTbGaaiilaiaad6gaaeqaaaaa@3B4F@
o matrice . Numarul natural r este rangul
matricei A
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
exista un minor de ordinul r al lui A, nenul
iar toti minorii de ordin mai mare decât r+1 (daca exista) sunt nuli.
Teorema: Matricea A are rangul r
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
exista un
minor de ordin r al lui A iar toti minorii de ordin r+1 sunt zero.
Teorema: Fie A
∈
M
m,n
(C),B∈
M
n,s
(C)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaamytamaaBaaaleaacaWGTbGaaiilaiaad6gaaeqaaOGaaiikaiaadoeacaGGPaGaaiilaiaadkeacqGHiiIZcaWGnbWaaSbaaSqaaiaad6gacaGGSaGaam4CaaqabaGccaGGOaGaam4qaiaacMcaaaa@45F4@
.
Atunci orice minor de ordinul k ,
1≤k≤min(m,s)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgsMiJkaadUgacqGHKjYOciGGTbGaaiyAaiaac6gacaGGOaGaamyBaiaacYcacaWGZbGaaiykaaaa@41CD@
al lui AB se poate scrie ca o combinatie
liniara de minorii de ordinul k al lui A (sau B).
Teorema: Rangul produsului a doua
matrice este mai mic sau egal cu rangul fiecarei matrice.
Definitie:
∈
M
n
(C)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaamytamaaBaaaleaacaWGUbaabeaakiaacIcacaWGdbGaaiykaaaa@3B93@
.
A este inversabila
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
det A
≠
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKlaaa@37BA@
0.( A este nesingulara).
Teorema: Inversa unei matrice daca exista este unica.
Observatii: 1) det
(A·B) =det A· det B.
2)
A
−1
=
1
detA
⋅A*
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaaleqabaGaeyOeI0IaaGymaaaakiabg2da9maalaaabaGaaGymaaqaaiGacsgacaGGLbGaaiiDaiaadgeaaaGaeyyXICTaamyqaiaacQcaaaa@41B8@
(
A→
A
τ
→A*=
(
(−1)
i+j
d
ij
)
i,j
→
A
−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgkziUkaadgeadaahaaWcbeqaaiabes8a0baakiabgkziUkaadgeacaGGQaGaeyypa0JaaiikaiaacIcacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaamyAaiabgUcaRiaadQgaaaGccaWGKbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcadaWgaaWcbaGaamyAaiaacYcacaWGQbaabeaakiabgkziUkaadgeadaahaaWcbeqaaiabgkHiTiaaigdaaaaaaa@5166@
)
3) A-1
∈
M
n
(Z)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaamytamaaBaaaleaacaWGUbaabeaakiaacIcacaWGAbGaaiykaaaa@3BAA@
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
det A =
±1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyySaeRaaGymaaaa@389C@
.
Stabilirea rangului unei
matrice:
Se ia determinantul de ordinul k-1 si
se bordeaza cu o linie (respectiv cu o coloana). Daca noul determinant este nul
rezulta ca ultima linie(respectiv coloana )este combinatie liniara de celelalte
linii (respectiv coloane).
Teorema: Un determinant este nul
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
una din coloanele (respectiv linii) este o
combinatie liniara de celelalte coloane(respectiv linii).
Teorema: Rangul r al unei matrice A este egal cu numarul maxim de
coloane(respectiv linii) care se pot alege dintre coloanele (respectiv liniile)
lui A astfel incât nici una dintre ele sa nu fie combinatie liniara a
celorlalte.
4. Sisteme de ecuatii
liniare
Forma generala a
unui sistem de m ecuatii cu n necunoscute este:
(1
{
a
11
x
1
+
a
12
x
2
+...........+
a
1n
x
n
=
b
1
.............................................
a
m1
x
1
+
a
m2
x
2
+..........+
a
mn
x
n
=
b
m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8CC1@
sau 
b
i
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGPbaabeaaaaa@37F4@
Unde A (aij)
1≤i≤m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgsMiJkaadMgacqGHKjYOcaWGTbaaaa@3BF8@
,
1≤j≤n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgsMiJkaadQgacqGHKjYOcaWGUbaaaa@3BFA@
- matricea coeficientilor necunoscutelor.
Matricea
A
¯
=(
a
11
...
a
1n
b
1
...
a
m1
....
a
mn
b
m
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWGbbaaaiabg2da9maabmaabaqbaeqabmWaaaqaaiaadggadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaaiOlaiaac6cacaGGUaaabaqbaeqabeGaaaqaaiaadggadaWgaaWcbaGaaGymaiaad6gaaeqaaaGcbaGaamOyamaaBaaaleaacaaIXaaabeaaaaaakeaaaeaacaGGUaGaaiOlaiaac6caaeaafaqabeqacaaabaaabaaaaaqaaiaadggadaWgaaWcbaGaamyBaiaaigdaaeqaaaGcbaGaaiOlaiaac6cacaGGUaGaaiOlaaqaauaabeqabiaaaeaacaWGHbWaaSbaaSqaaiaad2gacaWGUbaabeaaaOqaaiaadkgadaWgaaWcbaGaamyBaaqabaaaaaaaaOGaayjkaiaawMcaaaaa@4F98@
se numeste matricea extinsa a sistemului.
Definitie: Un sistem de numere
α
1
,
α
2
,.......
α
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiabeg7aHnaaBaaaleaacaaIYaaabeaakiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaeqySde2aaSbaaSqaaiaad6gaaeqaaaaa@4410@
se numeste solutie a sistemului (1)
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
∑
j=1
n
a
ij
α
j
=
b
i
,i=
1,m
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabeg7aHnaaBaaaleaacaWGQbaabeaaaeaacaWGQbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoakiabg2da9iaadkgadaWgaaWcbaGaamyAaaqabaGccaGGSaGaamyAaiabg2da9maanaaabaGaaGymaiaacYcacaWGTbaaaaaa@49AD@
.
Definitie:
- Un sistem se numeste incompatibil
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
nu are solutie;
- Un sistem se numeste compatibil
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
are cel putin o solutie;
- Un sistem se numeste compatibil determinat
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
are o singura
solutie;
- Un sistem se numeste compatibil nedeterminat
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
are o infinitate de solutii;
Rezolvarea matriceala a unui sistem
Fie A,
B∈
M
n
(C)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiabgIGiolaad2eadaWgaaWcbaGaamOBaaqabaGccaGGOaGaam4qaiaacMcaaaa@3C5A@
.
A
−1
|A⋅X=B⇒X=
A
−1
⋅B⇒
X
j
=
1
detA
⋅
∑
i=1
n
a
ij
⋅
b
i
,
j=1,n
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6822@
.
Rezolvarea sistemelor prin metoda lui Cramer:
Teorema lui Cramer: Daca
det A
not
¯
¯
Δ≠0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWqaaeaacaWGUbGaam4BaiaadshaaaWaaCbiaeaaaSqabeaaaaGccqqHuoarcqGHGjsUcaaIWaaaaa@3D1E@
,
atunci sistemul AX=B are o solutie unica
Xi=
Δ
i
Δ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqqHuoardaWgaaWcbaGaamyAaaqabaaakeaacqqHuoaraaaaaa@39F3@
.
Teorema lui Kronecker- Capelli: Un sistem de ecuatii liniare este compatibil
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
rangul matricei sistemului este egal cu rangul
matricei extinse.
Teorema lui Rouche: Un
sistem de ecuatii liniare este
compatibil
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
toti minorii caracteristici sunt nuli.
Notam cu m-numarul
de ecuatii;
n- numarul de necunoscute;
r -rangul matricei coeficientilor.
|
I
|
m=n=r
|
Sistem compatibil determinat
|
Δ≠0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeyiyIKRaaGimaaaa@39DA@
|
|
II
|
m=r
〈n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeUaamOBaaaa@389F@
|
Sistem compatibil nedeterminat
|
Minorul principal este nenul
|
|
III
|
n=r
〈m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeUaamyBaaaa@389E@
|
Sistem compatibil determinat sau
|
Daca toti minorii caracteristici sunt nuli
|
|
Sistem incompatibil
|
Exista cel putin un minor caracteristic
nenul
|
|
IV
|
r〈n,r〈m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabgMYiHlaad6gacaGGSaGaamOCaiabgMYiHlaad2gaaaa@3DE8@
|
Sistem compatibil nedeterminat sau
|
Daca toti minorii caracteristici sunt nuli
|
|
Sistem incompatibil
|
Exista cel putin un minor caracteristic
nenul
|
Teorema: Un sistem liniar si omogen admite numai solutia banala
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
Δ≠0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeyiyIKRaaGimaaaa@39DA@
14. SIRURI DE NUMERE REALE
1. Vecinatati. Puncte de acumulare.
Definitia 1 : Se numeste sir , o functie f : N → R
definita prin f(n) = a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
.
Notam
(
a
n
)
n∈N
:
a
0
,
a
1
,
a
2
,.............sau
¯
a
1
,
a
2
,
a
3
,...........
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@608F@
Orice sir are o infinitate de termeni;
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
este termenul general al sirului
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
.
Definitia 2 : Doua siruri
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
,
(
b
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D02@
sunt egale
⇔
a
n
=
b
n
,∀n≥k∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaamyyamaaBaaaleaacaWGUbaabeaakiabg2da9iaadkgadaWgaaWcbaGaamOBaaqabaGccaGGSaGaeyiaIiIaamOBaiabgwMiZkaadUgacqGHiiIZcaWGobaaaa@44F4@
Definitia 3: Fie a
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
R. Se numeste vecinatate a punctului a
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
R, o multime V pentru care
∃
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqcaaa@36C8@
ε >0 si un interval deschis centrat in a
de forma (a- ε , a+ ε)
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOGIWmaaa@37EF@
V.
Definitia 4: Fie D
⊆
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa@37F4@
R. Un punct α
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
R
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWGsbaaaaaa@36DB@
se numeste punct de acumulare pentru D daca in
orice vecinatate a lui α exista cel putin un punct din D-
{ α }
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacqaHXoqyaiaawUhacaGL9baaaaa@39C3@
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
V ∩(D-
{ α }
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacqaHXoqyaiaawUhacaGL9baaaaa@39C3@
)
≠
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKlaaa@37BA@
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbwaqaaaaaaaaaWdbiabl2==Ubaa@39DE@
. Un punct x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
D care nu e punct de acumulare se numeste
punct izolat.
2. Siruri
convergente
Definitia 5 : Un sir
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
este convergent catre un numar a
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
R
¯
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacaWGsbaaaaaa@36DB@
daca in orice vecinatate a lui a se afla toti
termenii sirului cu exceptia unui numar finit si scriem a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→
n→∞
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4akaSqabeaacaWGUbGaeyOKH4QaeyOhIukakiaawkziaiaadggaaaa@3CC6@
sau
lim
a
n
=a
n→∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGSbGaaiyAaiaac2gacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0Jaamyyaaqaaiaad6gacqGHsgIRcqGHEisPaaaa@4116@
a se numeste limita sirului .
Teorema 1: Daca un sir e convergent , atunci limita sa
este unica.
Teorema 2: Fie
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
un sir de numere reale. Atunci:
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
este monoton crescator
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
≤
a
n+1
,∀n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyizImQaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGSaGaeyiaIiIaamOBaiabgIGiolaad6eaaaa@401E@
sau
a
n+1
−
a
n
≥0,sau
a
n+1
a
n
≥1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyyzImRaaGimaiaacYcacaWGZbGaamyyaiaadwhadaWcaaqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaaaaGccqGHLjYScaaIXaaaaa@4AEF@
;
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
este stict crescator
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
〈
a
n+1
,∀n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeUaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGSaGaeyiaIiIaamOBaiabgIGiolaad6eaaaa@4022@
sau
a
n+1
−
a
n
〉0,sau
a
n+1
a
n
〉1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyOkJeVaaGimaiaacYcacaWGZbGaamyyaiaadwhadaWcaaqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaaaaGccqGHQms8caaIXaaaaa@4AF7@
;
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
este monoton descrescator
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
≥
a
n+1
,∀n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImRaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGSaGaeyiaIiIaamOBaiabgIGiolaad6eaaaa@402F@
sau
a
n+1
−
a
n
≤0,sau
a
n+1
a
n
≤1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyizImQaaGimaiaacYcacaWGZbGaamyyaiaadwhadaWcaaqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaaaaGccqGHKjYOcaaIXaaaaa@4ACD@
;
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
este strict descrescator
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
〉
a
n+1
,∀n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOkJeVaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGSaGaeyiaIiIaamOBaiabgIGiolaad6eaaaa@4033@
sau
a
n+1
−
a
n
〈0,sau
a
n+1
a
n
〈1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyykJeUaaGimaiaacYcacaWGZbGaamyyaiaadwhadaWcaaqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaaaaGccqGHPms4caaIXaaaaa@4AD5@
.
Definitia 6. Un sir
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
este marginit
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@384F@
∃
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqcaaa@36C8@
M
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
R astfel incât
|
a
n
|≤M
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLhWUaayjcSdGaeyizImQaamytaaaa@3DAB@
sau
∃α,β∈R
astfel
încât
α≤
a
n
≤β
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqIaeqySdeMaaiilaiabek7aIjabgIGiolaadkfadaqfGaqabSqabeaaa0qaaaaakiaadggacaWGZbGaamiDaiaadAgacaWGLbGaamiBamaavacabeWcbeqaaaqdbaaaaOGaamO7aiaad6gacaWGJbGaamO4aiaadshadaqfGaqabSqabeaaa0qaaaaakiabeg7aHjabgsMiJkaadggadaWgaaWcbaGaamOBaaqabaGccqGHKjYOcqaHYoGyaaa@5237@
.
Teorema 3: Teorema lui Weierstrass: Orice sir monoton si marginit este
convergent.
Definitia 7: Daca un sir are limita finita
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3850@
sirul este convergent.
Daca un sir are limita
infinita
+∞
sau
−∞
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaeyOhIu6aaubiaeqaleqabaaaneaaaaGccaWGZbGaamyyaiaadwhadaqfGaqabSqabeaaa0qaaaaakiabgkHiTiabg6HiLoaavacabeWcbeqaaaqdbaaaaOGaeyO0H4naaa@40F9@
sirul este divergent.
Teorema 4: Orice sir convergent are limita finita si
este marginit dar nu neaparat monoton.
Teorema 5: Lema lui Cesaro:
Orice sir marginit are cel putin un subsir convergent.
Definitia 8: Un sir e divergent fie daca nu are limita,
fie daca are o limita sau daca admite doua subsiruri care au limite diferite.
OBS: Orice sir crescator are limita finita sau infinita.
Teorema 6: Daca
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
R
+
*
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBaaaleaacqGHRaWkaeqaaOWaaWbaaSqabeaacaGGQaWaaSbaaWqaaaqabaaaaaaa@38EA@
este un sir strict crescator si nemarginit
atunci
lim
a
n
=+∞
⇒lim
1
a
n
=0
n→∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGSbGaaiyAaiaac2gacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0Jaey4kaSIaeyOhIu6aaubiaeqaleqabaaaneaaaaGccqGHshI3ciGGSbGaaiyAaiaac2gadaWcaaqaaiaaigdaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9iaaicdaaeaacaWGUbGaeyOKH4QaeyOhIukaaaa@4CAA@
.
Un sir descrescator cu termenii pozitivi este marginit de primul termen si de
0.
3. Operatii cu siruri care au
limita
Teorema 7: Fie
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
,
(
b
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D02@
siruri care au limita: a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→
n→∞
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4akaSqabeaacaWGUbGaeyOKH4QaeyOhIukakiaawkziaiaadggaaaa@3CC6@
,
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→
n→∞
b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4akaSqabeaacaWGUbGaeyOKH4QaeyOhIukakiaawkziaiaadkgaaaa@3CC7@
.
Daca operatiile
a+b,ab
a
b
,
a
b
au
sens
atunci
şirurile
a
n
+
b
n
,
a
n
−
b
n
,α⋅
a
n
,
a
n
⋅
b
n
,
a
n
b
n
,
a
n
b
n
au
limită
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@78B8@
.
lim(
a
n
+
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgUcaRiaadkgadaWgaaWcbaGaamOBaaqabaaaaa@3AEA@
)= lim
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaaaaa@37F8@
+lim
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGUbaabeaaaaa@37F9@
; lim(
a
n
⋅
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgwSixlaadkgadaWgaaWcbaGaamOBaaqabaaaaa@3C52@
)=lim
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaaaaa@37F8@
.lim
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGUbaabeaaaaa@37F9@
;
n
→∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4QaeyOhIukaaa@3951@
n
→∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4QaeyOhIukaaa@3951@
n
→∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4QaeyOhIukaaa@3951@
lim(
α⋅
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyyXICTaamyyamaaBaaaleaacaWGUbaabeaaaaa@3BE1@
)=α·lim
a
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaaaaa@37F8@
; lim
a
n
b
n
=
lim
a
n
lim
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOyamaaBaaaleaacaWGUbaabeaaaaGccqGH9aqpdaWcaaqaaiGacYgacaGGPbGaaiyBaiaadggadaWgaaWcbaGaamOBaaqabaaakeaaciGGSbGaaiyAaiaac2gacaWGIbWaaSbaaSqaaiaad6gaaeqaaaaaaaa@44ED@
lim
a
n
b
n
=
(lim
a
n
)
lim
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaamOyamaaBaaameaacaWGUbaabeaaaaGccqGH9aqpcaGGOaGaciiBaiaacMgacaGGTbGaamyyamaaBaaaleaacaWGUbaabeaakiaacMcadaahaaWcbeqaaiGacYgacaGGPbGaaiyBaiaadkgadaWgaaadbaGaamOBaaqabaaaaaaa@4682@
lim(
log
a
a
n
)=
log
a
(
lim
a
n
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaacMgacaGGTbWaaeWaaeaaciGGSbGaai4BaiaacEgadaWgaaWcbaGaamyyaaqabaGccaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOWaaeWaaeaaciGGSbGaaiyAaiaac2gacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaaa@4BA1@
lim
a
n
k
=
lim
a
n
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaacMgacaGGTbWaaOqaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaqaaiaadUgaaaGccqGH9aqpdaGcbaqaaiGacYgacaGGPbGaaiyBaiaadggadaWgaaWcbaGaamOBaaqabaaabaGaam4Aaaaaaaa@42AD@
Prin conventie s-a stabilit: ∞+∞=∞ ;
a+∞=∞,a
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4maaa@3777@
R; a+(-∞)=-∞; -∞+(-∞)=-∞; a·∞=∞ ,a>0;
a·∞=-∞,a<0; ∞·(-∞)=-∞;
-∞·(-∞)=∞;
∞
∞
=∞;
∞
−∞
=
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIu6aaWbaaSqabeaacqGHEisPaaGccqGH9aqpcqGHEisPcaGG7aGaeyOhIu6aaWbaaSqabeaacqGHsislcqGHEisPaaGccqGH9aqpaaa@414E@
0;
0
∞
=0;
∞
a
={
∞,dacă
a〉0
0,dacă
a〈0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimamaaCaaaleqabaGaeyOhIukaaOGaeyypa0JaaGimaiaacUdadaqfGaqabSqabeaaa0qaaaaakiabg6HiLoaaCaaaleqabaGaamyyaaaakiabg2da9maaceaaeaqabeaacqGHEisPcaGGSaGaamizaiaadggacaWGJbGaam4abmaaxacabaaaleqabaaaaOGaamyyaiabgQYiXlaaicdaaeaacaaIWaGaaiilaiaadsgacaWGHbGaam4yaiaadoqadaWfGaqaaaWcbeqaaaaakiaadggacqGHPms4caaIWaaaaiaawUhaaaaa@515D@
Nu au sens operatiile: ∞-∞, 0·(±∞);
±∞
±∞
,
1
∞
,
1
−∞
,
∞
0
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHXcqScqGHEisPaeaacqGHXcqScqGHEisPaaGaaiilamaavacabeWcbeqaaaqdbaaaaOGaaGymamaaCaaaleqabaGaeyOhIukaaOGaaiilamaavacabeWcbeqaaaqdbaaaaOGaaGymamaaCaaaleqabaGaeyOeI0IaeyOhIukaaOGaaiilamaavacabeWcbeqaaaqdbaaaaOGaeyOhIu6aaWbaaSqabeaacaaIWaaaaOGaaiOlaaaa@48B8@
Teorema 8: Daca
|
a
n
−a
|≤
b
n
şi
b
n
→0⇒
a
n
→
n→∞
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamyyaaGaay5bSlaawIa7aiabgsMiJkaadkgadaWgaaWcbaGaamOBaaqabaGcdaqfGaqabSqabeaaa0qaaaaakiaad+vacaWGPbWaaubiaeqaleqabaaaneaaaaGccaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaeyOKH4QaaGimaiabgkDiElaadggadaWgaaWcbaGaamOBaaqabaGcdaGdOaWcbeqaaiaad6gacqGHsgIRcqGHEisPaOGaayPKHaGaamyyaaaa@5345@
Daca
a
n
≥
b
n
şi
b
n
→∞⇒
a
n
→
n→∞
∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgwMiZkaadkgadaWgaaWcbaGaamOBaaqabaGcdaqfGaqabSqabeaaa0qaaaaakiaad+vacaWGPbWaaubiaeqaleqabaaaneaaaaGccaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaeyOKH4QaeyOhIuQaeyO0H4TaamyyamaaBaaaleaacaWGUbaabeaakmaaoGcaleqabaGaamOBaiabgkziUkabg6HiLcGccaGLsgcacqGHEisPaaa@4FA3@
Daca
a
n
≤
b
n
şi
b
n
→−∞⇒
a
n
→
n→∞
−∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgsMiJkaadkgadaWgaaWcbaGaamOBaaqabaGcdaqfGaqabSqabeaaa0qaaaaakiaad+vacaWGPbWaaubiaeqaleqabaaaneaaaaGccaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaeyOKH4QaeyOeI0IaeyOhIuQaeyO0H4TaamyyamaaBaaaleaacaWGUbaabeaakmaaoGcaleqabaGaamOBaiabgkziUkabg6HiLcGccaGLsgcacqGHsislcqGHEisPaaa@516C@
Daca
a
n
→
n→∞
a⇒
|
a
n
|
→
n→∞
| a |
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakmaaoGcaleqabaGaamOBaiabgkziUkabg6HiLcGccaGLsgcacaWGHbGaeyO0H49aaubiaeqaleqabaaaneaaaaGcdaabdaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakiaawEa7caGLiWoadaGdOaWcbeqaaiaad6gacqGHsgIRcqGHEisPaOGaayPKHaWaaqWaaeaacaWGHbaacaGLhWUaayjcSdaaaa@50B8@
.
Daca
a
n
→
n→∞
0⇒
|
a
n
|
→
n→∞
| 0 |
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakmaaoGcaleqabaGaamOBaiabgkziUkabg6HiLcGccaGLsgcacaaIWaGaeyO0H49aaubiaeqaleqabaaaneaaaaGcdaabdaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakiaawEa7caGLiWoadaGdOaWcbeqaaiaad6gacqGHsgIRcqGHEisPaOGaayPKHaWaaqWaaeaacaaIWaaacaGLhWUaayjcSdaaaa@5060@
.
Teorema 9: Daca sirul
(
a
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D01@
este convergent la zero, iar
(
b
n
)
n∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqaaiaad6gacqGHiiIZcaWGobaabeaaaaa@3D02@
este un sir marginit, atunci sirul produs
a
n
⋅
b
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgwSixlaadkgadaWgaaWcbaGaamOBaaqabaaaaa@3C52@
este convergent la zero.
4.
Limitele unor siruri tip
lim
︸
n→∞
q
n
={
0,dacă
q∈(−1,1)
1,dacă
q=1
∞,dacă
q〉1
nu
există,dacă
q≤−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaGbaaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaOGaayjo+dGaamyCamaaCaaaleqabaGaamOBaaaakiabg2da9maaceaaeaqabeaacaaIWaGaaiilaiaadsgacaWGHbGaam4yaiaadoqadaWfGaqaaaWcbeqaaaaakiaadghacqGHiiIZcaGGOaGaeyOeI0IaaGymaiaacYcacaaIXaGaaiykaaqaaiaaigdacaGGSaGaamizaiaadggacaWGJbGaam4abmaaxacabaaaleqabaaaaOGaamyCaiabg2da9iaaigdaaeaacqGHEisPcaGGSaGaamizaiaadggacaWGJbGaam4abmaaxacabaaaleqabaaaaOGaamyCaiabgQYiXlaaigdaaeaacaWGUbGaamyDamaaxacabaaaleqabaaaaOGaamyzaiaadIhacaWGPbGaam4CaiaadshacaWGdeGaaiilaiaadsgacaWGHbGaam4yaiaadoqadaWfGaqaaaWcbeqaaaaakiaadghacqGHKjYOcqGHsislcaaIXaaaaiaawUhaaaaa@7049@
lim
︸
n→∞
(
a
0
n
p
+
a
1
n
p−1
+....+
a
p
)={
∞,
a
0
〉0
−∞,
a
0
〈0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6208@
lim
︸
n→∞
a
0
⋅
n
p
+
a
1
⋅
n
p−1
+.......+
a
p
b
0
⋅
n
q
+
b
1
⋅
n
q−1
+.....+
b
q
={
0,dacă
p〈q
a
0
b
0
,dacă
p=q
∞,dacă
p〉q
şi
a
0
b
0
〉0
−∞,dacă
p〉q
şi
a
0
b
0
〈0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A827@
lim
(
1+
1
n
)
n
=e≈2,71......
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaad6gaaaGccqGH9aqpcaWGLbGaeyisISRaaGOmaiaacYcacaaI3aGaaGymaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaaaa@46B6@
lim
(
1+
1
x
n
)
x
n
=e
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamiEamaaBaaaleaacaWGUbaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaadIhadaWgaaadbaGaamOBaaqabaaaaOGaeyypa0Jaamyzaaaa@404E@
n→∞ x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→∞
lim
(
1+
x
n
)
1
x
n
=e
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIXaGaey4kaSIaamiEamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaWaaSaaaeaacaaIXaaabaGaamiEamaaBaaameaacaWGUbaabeaaaaaaaOGaeyypa0Jaamyzaaaa@404E@
lim
sin
x
n
x
n
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaciGGZbGaaiyAaiaac6gacaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaaaaGccqGH9aqpcaaIXaaaaa@3EE8@
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→0 x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→0
lim
arcsin
x
n
x
n
=
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaciGGHbGaaiOCaiaacogacaGGZbGaaiyAaiaac6gacaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaaaaGccqGH9aqpcaaIXaWaaSbaaSqaaaqabaaaaa@41D6@
lim
tg
x
n
x
n
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG0bGaam4zaiaadIhadaWgaaWcbaGaamOBaaqabaaakeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9iaaigdaaaa@3DF5@
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→0 x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→0
lim
arctg
x
n
x
n
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbGaamOCaiaadogacaWG0bGaam4zaiaadIhadaWgaaWcbaGaamOBaaqabaaakeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9iaaigdaaaa@40BA@
lim
ln(1+
x
n)
x
n
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaciGGSbGaaiOBaiaacIcacaaIXaGaey4kaSIaamiEamaaBaaaleaacaWGUbGaaiykaaqabaaakeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9iaaigdaaaa@40EA@
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→0 x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→0
lim
a
x
n
−1
x
n
=lna
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbWaaWbaaSqabeaacaWG4bWaaSbaaWqaaiaad6gaaeqaaaaakiabgkHiTiaaigdaaeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9iGacYgacaGGUbGaamyyaaaa@40DB@
lim
(
1+
x
n
)
r
−1
x
n
=r
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaadaqadaqaaiaaigdacqGHRaWkcaWG4bWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWGYbaaaOGaeyOeI0IaaGymaaqaaiaadIhadaWgaaWcbaGaamOBaaqabaaaaOGaeyypa0JaamOCaaaa@4248@
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→0 x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→0
lim
e
x
n
x
n
p
=∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGLbWaaWbaaSqabeaacaWG4bWaaSbaaWqaaiaad6gaaeqaaaaaaOqaaiaadIhadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaadchaaaaaaOGaeyypa0JaeyOhIukaaa@3F0A@
lim
ln
x
n
x
n
p
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaciGGSbGaaiOBaiaadIhadaWgaaWcbaGaamOBaaqabaaakeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaOWaaWbaaSqabeaacaWGWbaaaaaakiabg2da9iaaicdaaaa@3F1F@
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→∞ x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaad6gaaeqaaaaa@3712@
→∞
15. LIMITE DE FUNCTII
Definitie: O functie f:D
⊆R→R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0SaamOuaiabgkziUkaadkfaaaa@3B8F@
are limita laterala la stânga ( respectiv la
dreapta) in punctul de acumulare
x
0
⇔
există
l
s
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaakiabgsDiBpaavacabeWcbeqaaaqdbaaaaOGaamyzaiaadIhacaWGPbGaam4CaiaadshacaWGdeWaaCbiaeaaaSqabeaaaaGccaWGSbWaaSbaaSqaaiaadohaaeqaaOGaeyicI4maaa@43E1@
R (respectiv
l
d
∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBaaaleaacaWGKbaabeaakiabgIGiodaa@3987@
R) a. i. lim f(x)=
l
s
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBaaaleaacaWGZbaabeaaaaa@3808@
,
(respectiv lim f(x) =
l
d
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBaaaleaacaWGKbaabeaaaaa@37F9@
).
x→
x
0
x〈
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4bGaeyOKH4QaamiEamaaBaaaleaacaaIWaaabeaaaOqaaiaadIhacqGHPms4caWG4bWaaSbaaSqaaiaaicdaaeqaaaaaaa@3F6A@
x→
x
0
x〉
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4bGaeyOKH4QaamiEamaaBaaaleaacaaIWaaabeaaaOqaaiaadIhacqGHQms8caWG4bWaaSbaaSqaaiaaicdaaeqaaaaaaa@3F7B@
Definitie: Fie f:D
⊆R→R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0SaamOuaiabgkziUkaadkfaaaa@3B8F@
,
x
0
∈D
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaakiabgIGiolaadseaaaa@3A2D@
un punct de acumulare. Functia f are limita in
x
0
⇔
l
s
(
x
0
)=
l
d
(
x
0
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaakiabgsDiBlaadYgadaWgaaWcbaGaam4CaaqabaGccaGGOaGaamiEamaaBaaaleaacaaIWaaabeaakiaacMcacqGH9aqpcaWGSbWaaSbaaSqaaiaadsgaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa@45FD@
Proprietati:
1. Daca lim f(x) exista,
atunci aceasta limita este unica.
x→
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgkziUkaadIhadaWgaaWcbaGaaGimaaqabaaaaa@3AC0@
2. Daca lim f(x) =l atunci
lim|
f(x)
|=| l |.
x→
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGSbGaaiyAaiaac2gadaabdaqaaiaadAgacaGGOaGaamiEaiaacMcaaiaawEa7caGLiWoacqGH9aqpdaabdaqaaiaadYgaaiaawEa7caGLiWoacaGGUaaabaGaamiEaiabgkziUkaadIhadaWgaaWcbaGaaGimaaqabaaaaaa@49C5@
x→
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgkziUkaadIhadaWgaaWcbaGaaGimaaqabaaaaa@3AC0@
Reciproc nu.
3. Daca
lim|
f(x)
|=0
⇒
limf(x)=0
x→
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGSbGaaiyAaiaac2gadaabdaqaaiaadAgacaGGOaGaamiEaiaacMcaaiaawEa7caGLiWoacqGH9aqpcaaIWaWaaubiaeqaleqabaaaneaaaaGccqGHshI3daqfGaqabSqabeaaa0qaaaaakiGacYgacaGGPbGaaiyBaiaadAgacaGGOaGaamiEaiaacMcacqGH9aqpcaaIWaaabaGaamiEaiabgkziUkaadIhadaWgaaWcbaGaaGimaaqabaaaaaa@50A8@
4. Fie f,g:D
⊆R→R
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0SaamOuaiabgkziUkaadkfaaaa@3B8F@
,
∃
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqcaaa@36C8@
U o vecinatate a lui
x
0
∈D
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaakiabgIGiolaadseaaaa@3A2D@
astfel incât f(x)
≤
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyizImkaaa@37A8@
g(x)
∀x∈D∩U−{
x
0
}
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamiEaiabgIGiolaadseacqWIPisscaWGvbGaeyOeI0YaaiWaaeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaaGccaGL7bGaayzFaaaaaa@4117@
si daca exista
limf(x),limg(x)
⇒
x→
x
0
,x→
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGSbGaaiyAaiaac2gacaWGMbGaaiikaiaadIhacaGGPaGaaiilaiGacYgacaGGPbGaaiyBaiaadEgacaGGOaGaamiEaiaacMcadaqfGaqabSqabeaaa0qaaaaakiabgkDiEdqaaiaadIhacqGHsgIRcaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiaadIhacqGHsgIRcaWG4bWaaSbaaSqaaiaaicdaaeqaaaaaaa@4FDE@
limf(x)
〈
limg(x)
x→
x
0
x→
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGSbGaaiyAaiaac2gacaWGMbGaaiikaiaadIhacaGGPaWaaCbiaeaaaSqabeaaaaGccqGHPms4daWfGaqaaaWcbeqaaaaakiGacYgacaGGPbGaaiyBaiaadEgacaGGOaGaamiEaiaacMcadaqfGaqabSqabeaaa0qaaaaaaOqaaiaadIhacqGHsgIRcaWG4bWaaSbaaSqaaiaaicdaaeqaaOWaaubiaeqaleqabaaaneaaaaGccaWG4bGaeyOKH4QaamiEamaaBaaaleaacaaIWaaabeaaaaaa@4EE0@
5. Daca
f(x)≤g(x)≤h(x)
∀x∈D∩U−{
x
0
}
şi
∃
limf(x)=limh(x)=l⇒∃
limg(x)=l.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6DAA@
x→x0 x→x0 x→x0
6.
Daca
|
f(x)−l
|≤g(x)
∀
x∈D∩U−{
x
0
}
şi
limg(x)=0⇒limf(x)=l
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6433@
7.
Dacă
limf(x)=0
şi
∃M〉0
a.î.|
g(x)
|≤M
⇒limf(x)⋅g(x)=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaaeaacaWGebGaamyyaiaadogacaWGdeWaaubiaeqaleqabaaaneaaaaGcciGGSbGaaiyAaiaac2gacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaaGimamaaxacabaaaleqabaaaaOGaam4xbiaadMgadaWfGaqaaaWcbeqaaaaakiabgoGiKiaad2eacqGHQms8caaIWaWaaCbiaeaaaSqabeaaaaGccaWGHbGaaiOlaiaad6oacaGGUaWaaqWaaeaacaWGNbGaaiikaiaadIhacaGGPaaacaGLhWUaayjcSdGaeyizImQaamytaaqaaiabgkDiElGacYgacaGGPbGaaiyBaiaadAgacaGGOaGaamiEaiaacMcacqGHflY1caWGNbGaaiikaiaadIhacaGGPaGaeyypa0JaaGimaaaaaa@64A5@
.
8.
Dacă
f(x)≥g(x)
şi
limg(x)=+∞
⇒limf(x)=+∞.
Dacă
f(x)≤g(x)
şi
limg(x=−∞
⇒limf(x)=−∞.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7DD4@
OPERATII CU FUNCTII
Dacă
există
limf(x)=
l
1
,limg(x)=
l
2
şi
au
sens
operatiile
l
1
+
l
2
,
l
1
−
l
2
,
l
1
⋅
l
2
,
l
1
l
2
,
l
1
l
2
,
l
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGebGaamyyaiaadogacaWGdeWaaCbiaeaaaSqabeaaaaGccaWGLbGaamiEaiaadMgacaWGZbGaamiDaiaadoqadaWfGaqaaaWcbeqaaaaakiGacYgacaGGPbGaaiyBaiaadAgacaGGOaGaamiEaiaacMcacqGH9aqpcaWGSbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiGacYgacaGGPbGaaiyBaiaadEgacaGGOaGaamiEaiaacMcacqGH9aqpcaWGSbWaaSbaaSqaaiaaikdaaeqaaOWaaCbiaeaaaSqabeaaaaGccaWGFfGaamyAamaaxacabaaaleqabaaaaOGaamyyaiaadwhadaWfGaqaaaWcbeqaaaaaaOqaaiaadohacaWGLbGaamOBaiaadohadaWfGaqaaaWcbeqaaaaakiaad+gacaWGWbGaamyzaiaadkhacaWGHbGaamiDaiaadMgacaWGPbGaamiBaiaadwgadaqfGaqabSqabeaaa0qaaaaakiaadYgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGSbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadYgadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGSbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadYgadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWGSbWaaSbaaSqaaiaaikdaaeqaaOGaaiilamaalaaabaGaamiBamaaBaaaleaacaaIXaaabeaaaOqaaiaadYgadaWgaaWcbaGaaGOmaaqabaaaaOGaaiilaiaadYgadaWgaaWcbaGaaGymaaqabaGcdaahaaWcbeqaaiaadYgadaWgaaadbaGaaGOmaaqabaaaaOGaaiilamaakaaabaGaamiBamaaBaaaleaacaaIXaaabeaaaeqaaaaaaa@80C7@
atunci:
1. lim(f(x)
±
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyySaelaaa@37E1@
g(x))=
l
1
±
l
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBaaaleaacaaIXaaabeaakiabgglaXkaadYgadaWgaaWcbaGaaGOmaaqabaaaaa@3B9C@
.
2. limf(x)g(x)=
l
1
⋅
l
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBaaaleaacaaIXaaabeaakiabgwSixlaadYgadaWgaaWcbaGaaGOmaaqabaaaaa@3BF8@
3.lim
f(x)
g(x)
=
l
1
l
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGMbGaaiikaiaadIhacaGGPaaabaGaam4zaiaacIcacaWG4bGaaiykaaaacqGH9aqpdaWcaaqaaiaadYgadaWgaaWcbaGaaGymaaqabaaakeaacaWGSbWaaSbaaSqaaiaaikdaaeqaaaaaaaa@4157@
4.lim
f
(x)
g(x)
=
l
1
l
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykamaaCaaaleqabaGaam4zaiaacIcacaWG4bGaaiykaaaakiabg2da9iaadYgadaWgaaWcbaGaaGymaaqabaGcdaahaaWcbeqaaiaadYgadaWgaaadbaGaaGOmaaqabaaaaaaa@419C@
5.lim
f(x)
=
l
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGMbGaaiikaiaadIhacaGGPaaaleqaaOGaeyypa0ZaaOaaaeaacaWGSbWaaSbaaSqaaiaaigdaaeqaaaqabaaaaa@3C47@
P(X)=a0xn + a1xn-1
+ ……………..+an ,a0
≠
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKlaaa@37BA@
0
lim
x
→
±∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyySaeRaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaaaa@3F44@
P(x)=
a
0
(±∞)
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacIcacaWG4bGaaiykaiabg2da9iaadggadaWgaaWcbaGaaGimaaqabaGccaGGOaGaeyySaeRaeyOhIuQaaiykamaaCaaaleqabaGaamOBaaaaaaa@41D2@
|
|
|
0,
|
daca
|
q
∈(
−1,1
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI48aaeWaaeaacqGHsislcaaIXaGaaiilaiaaigdaaiaawIcacaGLPaaaaaa@3C13@
|
|
lim
x
→
∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaaaa@3D56@
qx =
|
|
1,
|
daca
|
q=1
|
|
|
|
∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIukaaa@3764@
,
|
daca
|
q>1
|
|
|
|
nu
exista,
|
daca
|
q
≤−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyizImQaeyOeI0IaaGymaaaa@3950@
|
|
|
|
0,
|
daca
|
q
∈(
−1,1
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI48aaeWaaeaacqGHsislcaaIXaGaaiilaiaaigdaaiaawIcacaGLPaaaaaa@3C13@
|
|
lim
x
→
∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaaaa@3D56@
qx =
|
|
1,
|
daca
|
q=1
|
|
|
|
∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIukaaa@3764@
,
|
daca
|
q>1
|
|
|
|
nu exista,
|
daca
|
q
≤−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyizImQaeyOeI0IaaGymaaaa@3950@
|
|
lim
x
→
π
2
tgx=−∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgkHiTiabg6HiLcaa@44BE@
|
|
x>
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
|
lim
x
→
π
2
tgx=+∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgUcaRiabg6HiLcaa@44B3@
|
|
X<
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
|
|
|
0,
|
daca
|
q
∈(
−1,1
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI48aaeWaaeaacqGHsislcaaIXaGaaiilaiaaigdaaiaawIcacaGLPaaaaaa@3C13@
|
|
lim
x
→
∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaaaa@3D56@
qx =
|
|
1,
|
daca
|
q=1
|
|
|
|
∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIukaaa@3764@
,
|
daca
|
q>1
|
|
|
|
nu
exista,
|
daca
|
q
≤−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyizImQaeyOeI0IaaGymaaaa@3950@
|
|
lim
x
→
π
2
tgx=−∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgkHiTiabg6HiLcaa@44BE@
|
|
x>
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
|
lim
x
→
π
2
tgx=+∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgUcaRiabg6HiLcaa@44B3@
|
|
X<
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
|
lim
x
→
π
2
tgx=−∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgkHiTiabg6HiLcaa@44BE@
|
|
x>
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
|
lim
x
→
π
2
tgx=+∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgUcaRiabg6HiLcaa@44B3@
|
|
X<
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
|
lim
x
→
π
2
tgx=−∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgkHiTiabg6HiLcaa@44BE@
|
|
x>
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
|
lim
x
→
π
2
tgx=+∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgUcaRiabg6HiLcaa@44B3@
|
|
X<
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
|
lim
x
→
π
2
tgx=−∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgkHiTiabg6HiLcaa@44BE@
|
|
x>
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
|
lim
x
→
π
2
tgx=+∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiaadshacaWGNbGaamiEaiabg2da9iabgUcaRiabg6HiLcaa@44B3@
|
|
X<
π
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@387C@
|
lim
︸
x→∞
a
0
⋅
x
p
+
a
1
⋅
x
p−1
+.......+
a
p
b
0
⋅
x
q
+
b
1
⋅
x
q−1
+.....+
b
q
={
0,dacă
p〈q
a
0
b
0
,dacă
p=q
∞,dacă
p〉q
şi
a
0
b
0
〉0
−∞,dacă
p〉q
şi
a
0
b
0
〈0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A859@
a>1
lim
x
→
∞
a
x
=∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOGaamyyamaaCaaaleqabaGaamiEaaaakiabg2da9iabg6HiLcaa@41F1@
lim
x
→
−∞
a
x
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOeI0IaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOGaamyyamaaCaaaleqabaGaamiEaaaakiabg2da9iaaicdaaaa@4227@
a
∈(0,1)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaaiikaiaaicdacaGGSaGaaGymaiaacMcaaaa@3AF5@
lim
x
→
∞
a
x
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOGaamyyamaaCaaaleqabaGaamiEaaaakiabg2da9iaaicdaaaa@413A@
lim
x
→
−∞
a
x
=∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOeI0IaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOGaamyyamaaCaaaleqabaGaamiEaaaakiabg2da9iabg6HiLcaa@42DE@
a>1
lim
x
→
∞
log
a
x=∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamiEaiabg2da9iabg6HiLcaa@44C0@
lim
x
→
0
log
a
x=−∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiGacYgacaGGVbGaai4zamaaBaaaleaacaWGHbaabeaakiaadIhacqGH9aqpcqGHsislcqGHEisPaaa@44F6@
a
∈(0,1)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaaiikaiaaicdacaGGSaGaaGymaiaacMcaaaa@3AF5@
lim
x
→
∞
log
a
x=−∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadggaaeqaaOGaamiEaiabg2da9iabgkHiTiabg6HiLcaa@45AD@
lim
x
→
0
log
a
x=∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakiGacYgacaGGVbGaai4zamaaBaaaleaacaWGHbaabeaakiaadIhacqGH9aqpcqGHEisPaaa@4409@
lim
x
→
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaaaaa@3C9F@
sinx
x
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaciGGZbGaaiyAaiaac6gacaWG4baabaGaamiEaaaacqGH9aqpcaaIXaaaaa@3C96@
lim
u(
x
)
→
0
sinu(
x
)
u(
x
)
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakmaalaaabaGaci4CaiaacMgacaGGUbGaamyDamaabmaabaGaamiEaaGaayjkaiaawMcaaaqaaiaadwhadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaGaeyypa0JaaGymaaaa@4AD5@
lim
x
→
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaaaaa@3C9F@
tgx
x
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG0bGaam4zaiaadIhaaeaacaWG4baaaiabg2da9iaaigdaaaa@3BA3@
lim
u(
x
)
→
0
tgu(
x
)
u(
x
)
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakmaalaaabaGaamiDaiaadEgacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaaabaGaamyDamaabmaabaGaamiEaaGaayjkaiaawMcaaaaacqGH9aqpcaaIXaaaaa@49E2@
lim
x
→
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaaaaa@3C9F@
arcsinx
x
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaciGGHbGaaiOCaiaacogacaGGZbGaaiyAaiaac6gacaWG4baabaGaamiEaaaacqGH9aqpcaaIXaaaaa@3F58@
lim
u(
x
)
→
0
arcsinu(
x
)
u(
x
)
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakmaalaaabaGaciyyaiaackhacaGGJbGaai4CaiaacMgacaGGUbGaamyDamaabmaabaGaamiEaaGaayjkaiaawMcaaaqaaiaadwhadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaGaeyypa0JaaGymaaaa@4D97@
lim
x
→
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaaaaa@3C9F@
arctgx
x
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbGaamOCaiaadogacaWG0bGaam4zaiaadIhaaeaacaWG4baaaiabg2da9iaaigdaaaa@3E68@
lim
u(
x
)
→
0
arctgu(
x
)
u(
x
)
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakmaalaaabaGaamyyaiaadkhacaWGJbGaamiDaiaadEgacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaaabaGaamyDamaabmaabaGaamiEaaGaayjkaiaawMcaaaaacqGH9aqpcaaIXaaaaa@4CA7@
lim
x
→
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaaaaa@3C9F@
(
1+x
)
1
x
=e
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIXaGaey4kaSIaamiEaaGaayjkaiaawMcaamaaCaaaleqabaWaaSaaaeaacaaIXaaabaGaamiEaaaaaaGccqGH9aqpcaWGLbaaaa@3E05@
lim
u(
x
)
→
0
(
1+u(
x
)
)
1
u(
x
)
=e
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakmaabmaabaGaaGymaiabgUcaRiaadwhadaqadaqaaiaadIhaaiaawIcacaGLPaaaaiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaadwhadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaaaaOGaeyypa0Jaamyzaaaa@4C44@
lim
x
→
∞
(
1+
1
x
)
x
=e
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOWaaeWaaeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamiEaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaadIhaaaGccqGH9aqpcaWGLbaaaa@4572@
lim
u(
x
)
→
∞
(
1+
1
u(
x
)
)
u(
x
)
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOWaaeWaaeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaamyDamaabmaabaGaamiEaaGaayjkaiaawMcaaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaadwhadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaGccqGH9aqpcaaIWaaaaa@4CCB@
lim
x
→
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaaaaa@3C9F@
ln(
1+x
)
x
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaciGGSbGaaiOBamaabmaabaGaaGymaiabgUcaRiaadIhaaiaawIcacaGLPaaaaeaacaWG4baaaiabg2da9iaaigdaaaa@3EC8@
lim
u(
x
)
→
0
ln(
1+u(
x
)
)
u(
x
)
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakmaalaaabaGaciiBaiaac6gadaqadaqaaiaaigdacqGHRaWkcaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaaacaGLOaGaayzkaaaabaGaamyDamaabmaabaGaamiEaaGaayjkaiaawMcaaaaacqGH9aqpcaaIXaaaaa@4D07@
lim
x
→
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaaaaa@3C9F@
a
x
−1
x
=lna
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbWaaWbaaSqabeaacaWG4baaaOGaeyOeI0IaaGymaaqaaiaadIhaaaGaeyypa0JaciiBaiaac6gacaWGHbaaaa@3E92@
lim
u(
x
)
→
0
a
u(x)
−1
u(
x
)
=lna
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakmaalaaabaGaamyyamaaCaaaleqabaGaamyDaiaacIcacaWG4bGaaiykaaaakmaaCaaaleqabaaaaOGaeyOeI0IaaGymaaqaaiaadwhadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaGaeyypa0JaciiBaiaac6gacaWGHbaaaa@4CD8@
lim
x
→
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaaaaa@3C9F@
(
1+x
)
r
−1
x
=r
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaadaqadaqaaiaaigdacqGHRaWkcaWG4baacaGLOaGaayzkaaWaaWbaaSqabeaacaWGYbaaaOGaeyOeI0IaaGymaaqaaiaadIhaaaGaeyypa0JaamOCaaaa@3FF6@
lim
u(
x
)
→
0
(
1+u(
x
)
)
r
−1
u(
x
)
=r
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaaGimaaqab0qaaiGacYgacaGGPbGaaiyBaaaakmaalaaabaWaaeWaaeaacaaIXaGaey4kaSIaamyDamaabmaabaGaamiEaaGaayjkaiaawMcaaaGaayjkaiaawMcaamaaCaaaleqabaGaamOCaaaakiabgkHiTiaaigdaaeaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaaaaiabg2da9iaadkhaaaa@4E35@
lim
x
→
∞
x
k
a
x
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOWaaSaaaeaacaWG4bWaaWbaaSqabeaacaWGRbaaaaGcbaGaamyyamaaCaaaleqabaGaamiEaaaaaaGccqGH9aqpcaaIWaaaaa@436E@
lim
u(
x
)
→
∞
u
(
x
)
k
a
u(
x
)
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOWaaSaaaeaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaaWbaaSqabeaacaWGRbaaaaGcbaGaamyyamaaCaaaleqabaGaamyDamaabmaabaGaamiEaaGaayjkaiaawMcaaaaaaaGccqGH9aqpcaaIWaaaaa@4AF7@
lim
x
→
∞
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG4bWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaaaa@3D56@
lnx
x
k
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaciGGSbGaaiOBaiaadIhaaeaacaWG4bWaaWbaaSqabeaacaWGRbaaaaaakiabg2da9iaaicdaaaa@3CC8@
lim
u(
x
)
→
∞
lnu(
x
)
u
(
x
)
k
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqaleaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaa4akaWqabeaaaSGaayPKHaGaeyOhIukabeqdbaGaciiBaiaacMgacaGGTbaaaOWaaSaaaeaaciGGSbGaaiOBaiaadwhadaqadaqaaiaadIhaaiaawIcacaGLPaaaaeaacaWG1bWaaeWaaeaacaWG4baacaGLOaGaayzkaaWaaWbaaSqabeaacaWGRbaaaaaakiabg2da9iaaicdaaaa@4BBE@
16. FUNCTII CONTINUE
DEFINITIE. O functie f : D
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jOimdaa@38E4@
R → R
se numeste continua in punctul de acumulare x0
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
D
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=rDiBdaa@3944@
oricare ar fi vecinatatea V a lui f(x0) , exista o vecinatate U
a lui x0,
astfel incât pentru orice
x
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
U ∩ D
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jDiEdaa@3945@
f(x)
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
V.
DEFINITIE. f : D
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jOimdaa@38E4@
R → R este
continua in x0
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
D
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=rDiBdaa@3944@
f are limita in x0 si lim f(x) = f(x0)
sau ls (x0 )
= ld (x0 ) = f(x0).
x0 se numeste
punct de continuitate.
Daca functia nu este continua in x0
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jDiEdaa@3945@
f.se
numeste discontinua in x0 si x0
se numeste punct de discontinuitate. Acesta
poate fi:
- punct de discontinuitate de prima speta daca ls (x0 ), ld
(x0 ) finite, dar ≠ f(x0);
- punct de discontinuitate de a doua speta daca cel putin
o limita laterala e infinita sau nu exista.
DEFINITIE. f este continua pe o multime ( interval)
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbwaqaaaaaaaaaWdbiab=rDiBdaa@3942@
este continua in fiecare punct a multimii (
intervalului).
•
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbwaqaaaaaaaaaWdbiab=jci3caa@386B@
Functiile elementare sunt continue pe
domeniile lor de definitie.
Exemple
de functii elementare: functia
constanta c, functia identica x, functia polinomiala f(x) = a0xn + a1xn-1
+ .......an , functia rationala f(x)/g(x), functia radical
f(x)
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWGMbGaaiikaiaadIhacaGGPaaaleaacaWGUbaaaaaa@3A42@
, functia
logaritmica log f(x), functia putere xa, functia exponentiala ax, functiile trigonometrice sin x, cos x, tg x, ctg x.
PRELUNGIREA PRIN CONTINUITATE A
UNEI FUNCTII INTR-UN PUNCT DE ACUMULARE
DEFINITIE. Fie f : D
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jOimdaa@38E4@
R → R.
Daca f are limita
l
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
R in punctul de acumulare x0
∉
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbwaqaaaaaaaaaWdbiab=LGipdaa@386C@
D
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jDiEdaa@3945@
f: D
∪
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=PIiidaa@3888@
{ x0} →R, f(x) =
{
f(x),x∈D
l,x=
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaadAgacaGGOaGaamiEaiaacMcacaGGSaGaamiEaiabgIGiolaadseaaeaacaWGSbGaaiilaiaadIhacqGH9aqpcaWG4bWaaSbaaSqaaiaaicdaaeqaaaaakiaawUhaaaaa@43E0@
este o functie continua in x0 si se numeste
prelungirea prin continuitate a lui f in
x0.
OPERATII CU FUNCTII CONTINUE
T1. Daca f,g:D→R sunt continue in x0
( respectiv pe D) atunci f+g, αf, f
•
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jci3caa@386D@
g,f/g, fg,
f
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGMbaaleqaaaaa@36F9@
sunt continue in x0 ( respectiv pe D); α
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
R, g ≠ 0.
T2. Daca f:D→R e continua in x0
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
D (
respectiv pe D)
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbwaqaaaaaaaaaWdbiab=jDiEdaa@3943@
|
f(x)
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbGaaiikaiaadIhacaGGPaaacaGLhWUaayjcSdaaaa@3C56@
e continua in
x0
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
( respectiv pe D).
Reciproca nu e valabila.
T3. Fie f:D→R continua in in x0
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
A si g:B →A continua in x0
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
B, atunci g
•
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jci3caa@386D@
f e continua in x0
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
A.
|
lim f( g (x) = f( lim g(x))
x→x0 x→x0
|
Orice functie continua comuta cu
limita.
PROPRIETATILE FUNCTIILOR
CONTINUE PE UN INTERVAL
LEMA. Daca f este o functie
continua pe un interval [ a,b] si daca are valori de semne contrare la extremitatile
intervalului
( f(a)
•
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jci3caa@386D@
( f(b) <0 ) atunci exista cel putin un punct c
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386A@
( a,b)
astfel incât f(c) =
0.
•
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbwaqaaaaaaaaaWdbiab=jci3caa@386B@
Daca f este
strict monotona pe [ a,b]
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbwaqaaaaaaaaaWdbiab=jDiEdaa@3943@
ecuatia
f(x) = 0 are cel mult o radacina in
intervalul ( a, b).
f este strict
monotona
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbwaqaaaaaaaaaWdbiab=rDiBdaa@3942@
f: I →J - continua
f(I) =J - surjectiva
f -
injectiva
Orice functie continua pe un interval compact este marginita si isi
atinge marginile.
STABILIREA SEMNULUI UNEI FUNCTII
PROP. O functie continua pe un interval, care nu se anuleaza pe acest
interval pastreaza semn constant pe el.
DEFINITIE. Fie f : I
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jOimdaa@38E4@
R → R (
I = interval) f are proprietatea lui Darboux.
⇔
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=rDiBdaa@3944@
∀
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=bGiIaaa@37B8@
a,b
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
I cu a < b si
∀
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=bGiIaaa@37B8@
λ
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
( f(a), f(b)) sau λ
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
( f(b), f(a))
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jDiEdaa@3945@
∃
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=nGiKaaa@37BD@
c
∈
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=HGiodaa@386C@
( a,b), a.i. f(c) = λ.
TEOREMA. Orice functie continua
pe un
interval are P.D.
Daca f :I → R are P.D. atunci
⇒
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jDiEdaa@3945@
f( I) e interval.
( Reciproca e in general falsa).
CONTINUITATEA FUNCTIILOR
INVERSE
T1. Fie f : I
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jOimdaa@38E4@
R → R o
functie monotona a.i.
f( I) e interval. Atunci f este continua.
T2. Orice functie continua si injectiva pe un
interval este strict monotona pe
acest interval.
T3. Fie f : I → R, I, J
⊂
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciqcLbwaqaaaaaaaaaWdbiab=jOimdaa@38E4@
R intervale.
Daca f e bijectiva si continua
atunci inversa sa
f-1 e continua si strict monotona.
17. DERIVATE
FUNCTIA DERIVATA
C 0
x 1
xn nxn-1
xa axa-1
ax a
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabeaacaWG4baaaaaa@371D@
lna
ex e
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabeaacaWG4baaaaaa@371D@
1
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiEaaaaaaa@37BB@
-
1
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiEamaaCaaaleqabaGaaGOmaaaaaaaaaa@38A4@
1
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiEamaaCaaaleqabaGaamOBaaaaaaaaaa@38DB@
-
n
x
n+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGUbaabaGaamiEamaaCaaaleqabaGaamOBaiabgUcaRiaaigdaaaaaaaaa@3AB0@
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWG4baaleqaaaaa@370B@
1
2
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmamaakaaabaGaamiEaaWcbeaaaaaaaa@3892@
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacaWG4baaleaacaWGUbaaaaaa@37FE@
1
n
x
n−1
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOBamaakeaabaGaamiEamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaaabaGaamOBaaaaaaaaaa@3C79@
sin x
cosx
cos x
-sinx
tg x
1
cos
2
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaamiEaaaaaaa@3B81@
ctg x -
1
sin
2
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaamiEaaaaaaa@3B86@
arcsin x
1
1−
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIXaGaeyOeI0IaamiEamaaCaaaleqabaGaaGOmaaaaaeqaaaaaaaa@3A5C@
arccos x -
1
1−
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIXaGaeyOeI0IaamiEamaaCaaaleqabaGaaGOmaaaaaeqaaaaaaaa@3A5C@
arctg x
1
1+
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGymaiabgUcaRiaadIhadaahaaWcbeqaaiaaikdaaaaaaaaa@3A41@
arcctg x -
1
1+
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGymaiabgUcaRiaadIhadaahaaWcbeqaaiaaikdaaaaaaaaa@3A41@
lnx
1
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiEaaaaaaa@37BB@
log
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaadggaaeqaaaaa@3705@
x
1
xlna
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiEaiGacYgacaGGUbGaamyyaaaaaaa@3A85@
(uv)’ = v. uv-1.u’ + uv.v’.lnu
f(x)=
ax+b
cx+d
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbGaamiEaiabgUcaRiaadkgaaeaacaWGJbGaamiEaiabgUcaRiaadsgaaaaaaa@3D5F@
f’(x)=
|
a
b
c
d
|
(cx+d)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaadaabdaqaauaabeqaciaaaeaacaWGHbaabaGaamOyaaqaaiaadogaaeaacaWGKbaaaaGaay5bSlaawIa7aaqaaiaacIcacaWGJbGaamiEaiabgUcaRiaadsgacaGGPaWaaWbaaSqabeaacaaIYaaaaaaaaaa@42C5@
REGULI DE DERIVARE
(f.g)’=f’g+fg’
(
χf
)
'
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqaHhpWycaWGMbaacaGLOaGaayzkaaWaaWbaaSqabeaacaGGNaaaaaaa@3AF6@
=
χf'
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaamOzaiaacEcaaaa@3940@
(
f
g
)
'
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWcaaqaaiaadAgaaeaacaWGNbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaai4jaaaaaaa@3A3B@
=
f
'
g−f
g
'
g
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGMbWaaWbaaSqabeaacaGGNaaaaOGaam4zaiabgkHiTiaadAgacaWGNbWaaWbaaSqabeaacaGGNaaaaaGcbaGaam4zamaaCaaaleqabaGaaGOmaaaaaaaaaa@3E37@
(
f
−1
)
'
(
f(
x
0
)
)=
1
f
'
(
x
0
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGMbWaaWbaaSqabeaacqGHsislcaaIXaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaGGNaaaaOWaaeWaaeaacaWGMbGaaiikaiaadIhadaWgaaWcbaGaaGimaaqabaGccaGGPaaacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOzamaaCaaaleqabaGaai4jaaaakiaacIcacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaaiykaaaaaaa@47C6@
18. STUDIUL FUNCTIILOR
CU AJUTORUL DERIVATELOR
Proprietati generale ale functiilor derivabile .
1.Punctele
de extrem ale unei functii.
Fie Ι un interval si f:Ι
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4kaaa@37E0@
R.
Definitie. Se numeste
punct de maxim (respectiv de minim)(local) al functiei
f
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@36DE@
,
un punct
a∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiodaa@385D@
Ι
pentru care exista o vecinatate V a lui
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36D9@
astfel incât
f(
x
)≤f(
a
)(
respectiv.f(
x
)
)≥f(
a
)∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabgsMiJkaadAgadaqadaqaaiaadggaaiaawIcacaGLPaaadaqadaqaaiaadkhacaWGLbGaam4CaiaadchacaWGLbGaam4yaiaadshacaWGPbGaamODaiaac6cacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaaacaGLOaGaayzkaaGaeyyzImRaamOzamaabmaabaGaamyyaaGaayjkaiaawMcaaiabgcGiIaaa@5291@
x∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiodaa@3874@
V.
•
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOiGClaaa@3778@
Un punct de maxim sau de minim se numeste
punct de extrem.
•
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOiGClaaa@3778@
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36D9@
se numeste punct de maxim(respectiv de minim)
global daca
f(
x
)≤f(
a
)(
resp.f(
x
)≥f(
a
)
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabgsMiJkaadAgadaqadaqaaiaadggaaiaawIcacaGLPaaadaqadaqaaiaadkhacaWGLbGaam4CaiaadchacaGGUaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabgwMiZkaadAgadaqadaqaaiaadggaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@4D0D@
.
∀
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIicaaa@36C3@
x∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiodaa@3874@
Ι.
Obs.1.O functie poate
avea intr-un interval mai multe puncte de extrem.(vezi desenul).
Obs.2.O functie poate
avea intr-un punct
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36D9@
un maxim (local), fara a avea in
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36D9@
cea mai mare valoare din interval.(vezi
desenul
f(
a
)<f(
c
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamyyaaGaayjkaiaawMcaaiabgYda8iaadAgadaqadaqaaiaadogaaiaawIcacaGLPaaaaaa@3DAD@
).
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F2@
-puncte de maxim
-puncte de minim
TEOREMA LUI FERMAT
Daca
f
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@36DE@
este o functie derivabila pe un interval Ι si
x
0
∈
I
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaakiabgIGiopaaxacabaGaamysaaWcbeqaaiaaicdaaaaaaa@3B35@
un punct de extrem,atunci
f
'
(
x
0
)=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCaaaleqabaGaai4jaaaakmaabmaabaGaamiEamaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaaa@3CF6@
.
Interpretare geometrica:
•
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOiGClaaa@3778@
Deoarece
f
'
(
x
0
)=0⇒
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCaaaleqabaGaai4jaaaakmaabmaabaGaamiEamaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiabg2da9iaaicdacqGHshI3aaa@3F53@
tangenta la grafic in punctul
(
x
0
,f(
x
0
)
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiaadAgadaqadaqaaiaadIhadaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E7A@
este paralela cu OX.
Obs.1. Teorema este
adevarata si daca functia este derivabila numai in punctele de extrem.
Obs.2. Conditia ca
punctul de extrem
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaaaaa@37D6@
sa fie interior intervalului este esentiala.
(daca ar fi o
extremitate a intervalului I atunci s-ar putea ca
f
'
(
x
0
)≠0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCaaaleqabaGaai4jaaaakmaabmaabaGaamiEamaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiabgcMi5kaaicdaaaa@3DB7@
). Ex.
f(
x
)=x.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaadIhacaGGUaaaaa@3C19@
Obs.3. Reciproca T. lui FERMAT nu este adevarata.(se pot gasi functii
astfel incât
f
'
(
x
0
)=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCaaaleqabaGaai4jaaaakmaabmaabaGaamiEamaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaaa@3CF6@
dar
x
0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaaaaa@37D6@
sa nu fie punct de extrem).
•
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOiGClaaa@3778@
Solutiile ecuatiei
f
'
(
x
)=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCaaaleqabaGaai4jaaaakmaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaicdaaaa@3C06@
se numesc puncte critice . Punctele de extrem
se gasesc printre acestea.
•
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOiGClaaa@3778@
Teorema lui Fermat da conditii suficiente (dar
nu si necesare) pentru ca derivata intr-un punct sa fie nula.
O alta teorema care da
conditii suficiente pentru ca derivata sa se anuleze este :
TEOREMA
LUI ROLLE.
Fie
f:
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdaaaa@379C@
I
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4kaaa@37E0@
R,
a,b∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4maaa@39F4@
I,
a<b.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgYda8iaadkgacaGGUaaaaa@3976@
Daca:
1.
f
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@36DE@
este continua pe
[
a,b
];
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWGHbGaaiilaiaadkgaaiaawUfacaGLDbaacaGG7aaaaa@3B21@
2.
f
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@36DE@
este derivabila pe
(
a,b
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbGaaiilaiaadkgaaiaawIcacaGLPaaaaaa@39F9@
;
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4oaaaa@36B2@
3.
f(
a
)=f(
b
),
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamyyaaGaayjkaiaawMcaaiabg2da9iaadAgadaqadaqaaiaadkgaaiaawIcacaGLPaaacaGGSaaaaa@3E5E@
atunci
∃
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqcaaa@36C8@
cel putin un punct
c∈(
a,b
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgIGiopaabmaabaGaamyyaiaacYcacaWGIbaacaGLOaGaayzkaaaaaa@3C65@
a.i
f
'
(
c
)=0.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCaaaleqabaGaai4jaaaakmaabmaabaGaam4yaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaaaaa@3CA3@
INTEPRETAREA
GEOMETRICA
Daca functia
f
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@36DE@
are valori egale la extremitatile unui interval
[
a,b
],
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWGHbGaaiilaiaadkgaaiaawUfacaGLDbaacaGGSaaaaa@3B12@
atunci exista cel putin un punct in care
tangenta este paralela cu axa
ox
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4BaiaadIhaaaa@37E4@
.
Consecinta 1. Intre doua radacini ale unei functii derivabile se afla cel putin o radacina
a derivatei.
Consecinta 2. Intre doua radacini consecutive ale derivatei se afla cel mult o radacina
a functiei.
TEOREMA LUI LAGRANGE (sau a cresterilor finite)
Fie
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaadMgacaWGLbaaaa@3896@
f:
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdaaaa@379C@
I
→
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4kaaa@37E0@
R,I (interval,
a,b∈
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4maaa@39F4@
I,
a<b.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgYda8iaadkgacaGGUaaaaa@3976@
Daca:
1.
f
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@36DE@
este continua pe
2.
este
derivabila pe
atunci exista cel putin un punct 
a.i sa avem
INTERPRETAREA
GEOMETRICA
Daca graficul functiei
admite tangenta in fiecare punct(cu exceptia eventual,a
extremitatilor) exista cel putin un punct de pe grafic(care nu coincide cu
extremitatile), in care tangenta este paralela cu coarda care uneste extremitatile.
tangenta la grafic in
M are coeficientul. unghiular 
dar
Obs.1. Daca 
Teorema lui Rolle.
Consecinta 1. Daca o functie are derivata nula pe un interval,atunci ea este
constanta pe acest interval.
Daca o functie are derivata nula pe o reuniune disjuncta de
intervale proprietate nu mai ramâne adevarata in general. Expl.
Consecinta 2. Daca si 
sunt doua functii derivabile pe un interval I si daca au
derivatele egale 
atunci ele difera
printr-o constanta. 
Daca si sunt definite pe o reuniune disjuncta de intervale, proprietatea
e falsa in general. Expl. 
Consecinta 3.
Daca 
pe I 
e strict crescatoare pe I.
Daca 
pe I e strict descrescatoare I.
Consecinta 4. 
Daca 
. 
are derivata in 
si 
Daca 
e derivabila in
Consecinta 5.Daca 
pe I 
pastreaza semn constant pe I.
ETAPELE REPREZENTARII
GRAFICULUI UNEI FUNCTII
1. Domeniul de definitie;
2. Intersectia graficului cu
axele de coordonate :
Intersectia cu
axa Ox contine puncte de forma{x,0},unde x este o radacina a ecuatiei f(x)=0 {daca exista}.
Intersectia cu
axa Oy este un punct de forma {0,f{0}} {daca punctul 0 apartine domeniului de
definitie}
3. Studiul continuitatii functiei pe domeniul de definitie :
Daca functia este
definita pe R se studiaza limita functiei la 
iar
daca este definita pe un interval se studiaza limita la capetele intervalului.
4.Studiul primei derivate :
a. Calculul lui
f’.
b. Rezolvarea
ecuatiei f’(x)=0.Radacinile acestei ecuatii vor fi eventuale puncte de maxim
sau de minim ale functiei ;
c. Stabilirea
intervalelor pe care semnul lui f este constant. Acestea reprezinta intervalele
de monotonie pentru f.
5.Studiul derivatei a doua :
a.Se calculeaza
f’’
b.Se rezolva
ecuatia f’’(x)=0. Radacinile acestei ecuatii vor fi eventuale puncte de
inflexiune ale graficului
c.Determinarea
intervalelor pe care semnul lui f este constant. Astfel,pe intervalele pe care
f’’>0 functia este convexa si pe cele pe care f’’<0, functia eate concava.
6.Asimptote :
a. Asimptotele
orizontale sunt drepte de forma y=a, unde
a= 
daca cel putin una din aceste limite are sens si exista in R.
b) Asimptotele
verticale sunt drepte de forma x=x0, daca exista cel putin o limita
laterala a functiei in x0,
infinita.
c) Asimptotele
oblice sunt drepte de forma y=mx+n, unde 
, analog si pentru -∞.
7. Tabelul de variatie;
8. Trasarea graficului.
19. PRIMITIVE
Primitive.
Proprietati.
Fie I un
interval din R.
Definitia 1. Fie
f: I → R. Se spune ca f admite primitive pe I daca 
F : I →R astfel incât
a) F
este derivabila pe I;
b) F’(x) =f(x), 
x ε I.
F se numeste primitiva lui f. ( I poate fi si o reuniune finita disjuncta de intervale).
Teorema 1.1 Fie f : I → R. Daca 
sunt doua primitive ale functiei f, atunci exista o constanta
c 
R astfel incât 
xI.
Demonstratie : Daca 
sunt primitive atunci sunt derivabile 
x ε I
, x ε I. 
, c= constanta
OBS 1. Fiind data o primitiva 
a unei functii, atunci orice primitiva F a lui f are forma F = 
+ c , c= constanta
f admite o infinitate
de primitive.
OBS 2. Teorema nu mai ramâne adevarata daca I este o
reuniune disjuncta de intervale Expl: f:
R-
, f(x) = x
²
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F =
, G= 
F, G sunt primitive ale lui f dar F-G nu e constanta . Contradictie cu T 1.1
OBS 3. Orice functie care admite primitive are Proprietatea lui Darboux.
Se stie ca derivata oricarei functii are Proprietatea lui Darboux ,
rezulta ca f are Proprietatea lui
Darboux.
F’ =f.
OBS 4. Daca I este interval si f(I) 
nu este interval atunci f nu admite primitive.
Daca presupunem ca f admite primitive atunci din OBS 3 rezulta ca f are
P lui Darboux, rezulta f(I) este interval ceea ce este o contradictie.
OBS 5. Orice functie continua definita pe un interval admite primitive.
Definitia 2. Fie f: I →R o functie
care admite primitive. Multimea tuturor primitivelor lui f se numeste integrala nedefinita a functiei f si se noteaza prin simbolul 
dx. Operatia de calculare a primitivelor unei functii(care
admite primitive ) se numeste integrare.
Simbolul 
a fost propus pentru prima data
de Leibniz, in 1675.
Fie F(I)= 
Pe aceasta multime se
introduc operatiile :
(f+g)(x) =f(x)+ g(x) ,
(αf)(x)=α.f(x)
,α constanta
C= 
dx =
.
Teorema 1.2 Daca f,g:I→ R sunt functii care admit primitive si α R, α ≠0, atunci functiile
f+g, αf admit
de asemenea primitive si au loc relatiile:
∫(f+g) =∫f +∫g, ∫αf=α∫f, α≠0, ∫f =∫f +C
Formula de integrare prin parti.
Teorema 1.1 Daca
f,g:R→R sunt functii derivabile cu derivatele continue, atunci functiile fg, f’g,
fg’ admit primitive si are loc relatia:
f(x)g’(x)dx =f(x)g(x)- f’(x)g(x)dx
Formula schimbarii
de variabila
(sau metoda substitutiei).
Teorema: Fie I,J intervale din R si 
1)
este derivabila pe I;
2) f admite primitive. (Fie F o primitiva a sa.)
Atunci functia (f o)’ admite primitive, iar functia F o este o primitiva a lui
(f o)’ adica:
5. Integrarea functiilor trigonometrice
Calculul integralelor trigonometrice se poate face fie folosind formula integrarii prin parti, fie metoda substitutiei. In acest caz se pot face substitutiile:
1. Daca functia este impara in sin x,
R(-sin x,cos x)=-R(sin x,cos x) atunci cos x=t.
2. Daca functia este impara in cos x,
R(sin x,-cos x)=-R(sin x,cos x) atunci sin x=t.
3. Daca functia este para in raport cu ambele variabile R(-sin x,-cos
x) atunci tg x=t.
4. Daca o functie nu se incadreaza in cazurile 1,2,3,atunci se
utilizeaza substitutiile universale:
5. Se mai pot folosi si
alte formule trigonometrice:
sin 2x=2sin x .cos x, 
Integrarea functiilor rationale
Definitie: O functie f:I→R , I interval, se numeste rationala
daca R(x)= 
unde f,g sunt functii polinomiale.
Daca grad f 
grad g, atunci se efectueaza impartirea lui f la g f=gq+r, 0
grad r<grad g si deci 
PRIMITIVELE FUNCTIILOR
CONTINUE SIMPLE
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10.
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
Bibliografie:
- Arno Kahane. Complemente de
matematica, Editura Tehnica, Bucuresti,
1958.
-
C. Nastasescu,C. Nita, Gh. Rizescui:”Matematica-Manual
pentru clasa a IX-a”, E.D.P., Bucuresti, 1982.
-
C. Nastasescu, C Nita, I. Stanescu:
Matematica-Manual pentru clasa a X-a-Algebra”, E.D.P., Bucuresti,1984.
-
E. Beju, I. Beju:”Compendiu de
matematica”, editura Stiintifica si Enciclopedica, Bucuresti, 1996.
-
E. Rogai,”Tabele si formule
matematice”,Editura tehnica,1983.
-
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@
Mica
enciclopedie matematica”, Editura tehnica, Bucuresti,1980.
-
Luminita Curtui,” Memorator de
Matematica-Algebra, pentru clasele 9-12”, Editura Booklet,2006.
Probleme propuse si rezolvate
1.Sa se
determine numerele intregi a si b astfel incât 
Rezolvare:
Ridicam la puterea a doua expresia data: 
Din egalarea termenilor asemenea intre ei rezulta : ab=2 si 2a2+3b2=14
rezulta: a=1 si b=2.
2.Daca 
=7, sa se calculeze a4 + 
.
Rezolvare:
Ridicam la puterea a doua relatia data: ()2=49, a2+
=51 procedând analog se obtine 
.
3.Aflati
X din X.3 2008 = (3 2008
–
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1) : (1+ 
)
Rezolvare:
, dupa formula
4. Sa
se calculeze: 
unde 
Rezolvare:
5. Stiind
ca 
sa se calculeze partea
intreaga a numarului 
Rezolvare:
6.Se da
numarul x = 
Sa se
arate ca x
²
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= 4
Sa se
calculeze (X+2)2007
Rezolvare:
a)
x =
b. x = 
2007 =
0
7. Daca
, sa se calculeze 
.
Rezolvare:
8.Sa se
calculeze suma
S = 
.
Rezolvare:
S=
Am adaugat
si am scazut 1.
9.Calculati:
Rezolvare:
10.Determinati 
astfel incât 
Rezolvare
11. Sa se rezolve ecuatia: (2x-4)(2x-3)(2x+1)(2x+2)=-6
Rezolvare:
Ecuatia data este echivalenta cu:
(2x-4)(2x+2)(2x-3)(2x+1)=-6(4x2
–
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4x-8) (4x2
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
4x-3)=-6
Notam 4x2
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
4x-8=t
t(t-5)=-6 t2-5t+6=0 t1=2 si t2=3
4x2
–
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4x-8=2x1,2=
4x2
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
4x-8=3x3,4=
.
12 . Se da ecuatia:
x
²
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+ 18x + 1 = 0. Se cere sa se calculeze
, unde x1, x2 sunt solutiile ecuatiei .
Rezolvare :
Fie A = . Se ridica la puterea
a treia
A
³
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= x1
+ x2 + 3 
· A
Cum x1
+ x2=
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18 x1+x2=1 (Relatiile lui Viete)
A
³
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- 3A + 18=
0 ; Solutia reala a acestei ecuatii
este A = -3 ; restul nu sunt reale
A
³
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+ 3A
²
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=jlaaaa@383F@
-3A
²
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-9A+6A+18=0
A
²
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(A+3)
–
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3A (A+3)+6(A+3)=o
(A+3)(A
²
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-3A+6)=0
A=-3
13.
Doua drepte perpendiculare intre ele in punctul M(3;4) intersecteaza axa OY in
punctual A si OX in punctual B.
a) sa se scrie ecuatia dreptei AB
b) sa se arate ca diagonalele patrulaterului AOBM sunt perpendiculare ,unde 0
este originea sistemului.
Rezolvare :
Scriem ecuatiile dreptelor AM si MB
cum AM 
Aflam coordonatele lui A:
-
din (1) când 
Aflam coordonatele lui B:
-
din (2) când 
Fie P(x,y) mijlocul lui AB 
panta dreptei AB este 
Panta dreptei OM este evident 
A
M(3,4)
O B
14.
Se dau punctele A (2,6), B(-4,3),
C(6,-2). Se cere:
a) perimetrul triunghiului ABC si natura sa ;
b)
coordonatele centrului de greutate;
c) ecuatia
dreptei BC;
d) ecuatia
medianei AM si lungimea sa;
e) ecuatia
inaltimii din A pe BC si lungimea sa ;
f) ecuatia
dreptei care trece prin A si face un unghi de 300 cu axa OX;
g) ecuatia
dreptei care trece prin A si este paralela cu BC;
h) ecuatia
bisectoarei din A si lungimea ei
i) aria triunghiului ABC.
Rezolvare:
a) Aplicând formula
distantei pentru cele trei laturi ale triunghiului 
obtinem:
AB = 
, BC = 
,AC = 
;
Se verifica cu
reciproca teoremei lui Pitagora ca
triunghiul este dreptunghic cu unghiul de 900 in vârful A.
b) Coordonatele
centrului de greutate sunt date de formula: 
;
c) Ecuatia dreptei
BC se scrie folosind formula: 
5x+10y-10=0 x+2y-2=0
(forma generala a dreptei )sau 
(forma normala);
d) Coordonatele
mijlocului segmentului BC sunt : M
ecuatia medianei este:
11x-2y-10 =0; Pentru calculul lungimii medianei AM se poate
folosi faptul ca intr-un triunghi dreptunghic mediana corespunzatoare
ipotenuzei este jumatate din ipotenuza:
AM = 
, altfel se poate aplica formula distantei.
e) Fie AD inaltimea
din A AD si BC sunt
perpendiculare ceea ce inseamna ca produsul pantelor este egal cu -1. Cum panta
dreptei BC este 
panta lui AD este 2. Ramâne
sa scriem ecuatia dreptei care trece prin A si are panta 2 :
y-6=2(x-2)
2x-y+2=0 este ecuatia inaltimii
din A;
Pentru calculul inaltimii
(intr-un triunghi dreptunghic) este convenabil sa aplicam formula:
AD = 
;
Altfel, trebuia
rezolvat sistemul format din ecuatiile dreptelor BC si AD pentru a determina
coordonatele lui D.
f) y-6=
(x-2); Am aplicat
formula y-y0=m(x-x0)
in conditiile in care panta este tg300
g) y-6= (x-2) unde este panta dreptei BC
.
h) Fie AE
bisectoarea unghiului A.
Din teorema
bisectoarei k= 
k= 
.Folosindu-ne de raportul
in care un punct imparte un segment
rezulta coordonatele lui E
. Atunci ecuatia
bisectoarei este:
21x-7y=0. Pentru a calcula lungimea bisectoarei ne
putem folosi si de formula 
care este utilizata
de obicei când se cunoaste masura unghiului a carei bisectoare se
calculeaza. AE =
.
i) Aria
triunghiului dreptunghic ABC este data de formula A = 
.
Se va insista pe
faptul ca daca triunghiul nu ar fi fost dreptunghic ar fi trebuit sa se calculeze distanta de la A la dreapta BC adica tocmai lungimea inaltimii
iar aceasta s-ar putea face mai simplu folosind formula :
Distanta de la un punct M0(x0,y0)
la o dreapta h de ecuatie (h): ax+by+c=0 este data de:
.
15. Sa se rezolve ecuatia :
Rezolvare :
Ecuatia data este echivalenta cu :
Ridicam la puterea 
Din monotonia functiei 
care e strict crescatoare ecuatia are solutie unica 
16 . Sa
se rezolve ecuatia:
2x x
x x 3 3
2007
–
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2006 = 3(2006 + 2006 ) + 1
Rezolvare:
Ecuatia data este
echivalenta cu:
x
x 3 3
2007 = (2006 + 1)
. Ridicam la puterea 1/3 =>
x x
3 3
2007 = 2006 +1
=>
x x
3 3
2007
–
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2006 =1
(*)
Din monotonia functiei
f(x) = (1+ a)x
–
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ax care e strict crescatoare => ecuatia (*) are solutie unica: x = 3
17. Sa se determine numarul de cifre din care este
compus numarul 72007.
Rezolvare:
102
< abc <103 ; p = 3
______
103
< abcd < 104 ; p =
4
(*) 10p-1 ≤ N < 10p ,
unde p reprezinta numarul de cifre
ale lui N.
Din (*) =>
lg 10p-1 ≤lg N <lg 10p => p-1 ≤ lg N <p .
Pentru N = 72007 => lg N = 2007 lg 7
�
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1696 de cifre.
18. Sa se arate ca matricea A = 
e inversabila ,
unde :
2006 ori de 1
Rezolvare :
A e inversabila 
ultima cifra a numarului det A 
Probleme - sinteze
I. NUMERE REALE. APLICATII.
1. Sa se calculeze:
a) 
.
b) 
2. Daca
a=2006.2007, aratati ca 
3. Sa
se calculeze numarul 
4.
Comparati numerele: 
5. Daca
6. Aratati ca numarul 
e patrat perfect.
7. Sa
se arate ca expresia 
8. Sa se aduca la o forma mai simpla expresia: 
9. Care numar este mai mare: 
.
10*. Sa
se arate ca: a) 
11. Sa
se arate ca: 
.
12.
Stabiliti valoarea de adevar a propozitiei:
13. Sa
se afle x stiind ca 
14. Sa
se afle numerele intregi x pentru care 
15. Sa
se verifice egalitatile: 
16. Sa
se ordoneze crescator numerele: 
.
17. Sa
se rationalizeze numitorii fractiilor:
. 
; 
; d) 
; e) 
.
18. Sa
se determine radacina patrata a numarului
a= 
19. Sa se determine cel mai mare numar natural n
cu proprietatea:
.
20.
Fie a,b,c numere rationale astfel incât ab+ac+bc=1. Sa se demonstreze ca: 
.
21. Sa
se demonstreze ca 
nu este un numar rational.
II. PROGRESII ARITMETICE
1. Sa se scrie primii
patru termeni ai progresiei aritmetice 
daca :
a) 
=-3 ; r=5 b) =7 ;r=2 c) = 1,3 ; r= 0,3
2. Sa se gaseasca primii doi
termeni ai progresiei aritmetice :
a) 
b) 
3. Sa se calculeze primii
cinci termeni ai sirului cu termenul general 
a) =3n+1 ; b) = 3 + (-1)
c) = n
4. Fie o progresie aritmetica
. Daca se dau doi termeni ai progresiei sa se afle ceilalti :
5. Fie o progresie aritmetica.
Se dau :
se cere a
b) 
se cere a
c) 
se cere
d) 
se cere
6. Sa se gaseasca primul
termen si ratia unei progresii aritmetice daca :
7. Sirul 
este dat prin formula termenului general.
a) x
=2n-5 ; b) x=10-7n. Sa se arate ca
e o progresie aritmetica. Sa se afle primul termen si ratia.
8. 
. Sa se afle S
daca : 
9.Cunoscând Sn sa se gasesca :
a) primii cinci termeni ai progresiei aritmetice
daca Sn =5n
+3n ; Sn =3 n ; Sn = 
.
b) = ?, r= ? daca Sn = 2 n +3n ;
10. Este progresie
aritmetica un sir pentru care :
a) Sn = n-2n ; b) Sn=
7n-1 ; c) Sn = -4 n+11.
11. , S10 = 100, S30 =900 . Sa se calculeze S50.
12. Determina x R astfel incât urmatoarele
numere sa fie in progresie aritmetica.
a) x-3, 9, x+3 ; b) 
c) 
13. Sa se rezolve ecuatiile :
a) 1+7+13+….+x =280 ;
b) 1+3+5+…..+x = 169 ;
c) (x+1)+(x+4)+(x+7)+…..+(x+28) = 155 ;
d) (x+1)+(x+3)+(x+5)+ ……..+(x+25) = 338 ;
e)
x+(x+5)+(x+10)+………+(x+100) = 2100.
14. Sa se arate ca urmatoarele
numere sunt in progresie aritmetica :
a) (a+b)
²
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, a
²
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+b
²
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, (a-b)
²
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;
b) 
;
c) 
15. Sa se arate ca daca
numerele 
sunt in progresie
aritmetica atunci numerele 
sunt in progresie
aritmetica.
16. Fie o progresie
aritmetica.
Sa se arate ca : 
.
17. Fie ecuatia ax
²
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+bx+c =0
cu solutiile x1,x2. Daca numerele a,b,c sunt in progresie aritmetica atunci
exista relatia : 2(x1+x2)+x1.x2
+1 = 0
18. Sa se
demonstreze : a) 
b) 
c) 
III. PROGRESII GEOMETRICE
1. Sa se scrie primii
cinci termeni ai progresiei geometrice
(b)daca :
a) 
b) 
c) 
d) 
e) 
2. Sa se gaseasca primii
doi termeni ai progresiei geometrice (b) :
a) 
b) 
3. Daca se cunosc doi
termeni ai progresiei geometrice (b)
a) 
, sa se gaseasca 
b) 
,…………….
.
4. Sa se scrie formula
termenulei al n-lea al progresiei geomertice date prin :
a) 
b) 
c) 
d) 
5. Este progresie
geometrica un sir pentru care suma primilor n termeni este :
a) Sn = n
²
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-1 ; b) Sn = 
;
c) Sn = 
6. Sa se determine x a.i. numerele urmatoare sa fie in progresie
geometrica :
a) a+x, b+x, c+x ; b)
;
c) 
;
7. Sa se gaseasca primul termen b1 si ratia q a progresiei geometrice (b)daca :
a) 
b) 
c) 
8.Sa se calculeze
sumele :
a) 
b) 
c) 
d) 
e) 1+11+111+1111+………111111…1 (de n ori 1)
f) 3+33+333+……..33333…..3
g) 7+77+777+…..7777…7(de n ori 7)
h) 
9. Sa se rezolve ecuatiile :
a) 
b) 
IV. LOGARITMI
1. Sa se logaritmeze expresiile in baza a : a) E=a2 
.
b) E=
.
c) E=
2. Sa se determine expresia E stiind ca : lg E=2 lga-
lgb-3 lg3.
3. Sa se arate ca log26+log62>2.
4. Sa se calculeze expresiile:
a) 
b) 
c)
E=log225-log2
.
d) 
e) 
f) 
g) 
5. Sa se arate ca expresia: E=
este independenta de valorile strict mai mari ca 1 ale variabilelor
x,z,y.
6. Sa se calculeze expresiile: a) E= 
.
b) E= 
7.Sa se calculeze suma: 
8. Sa se arate ca daca a,b,c sunt in progresie geometrica atunci are loc
egalitatea:
9. Sa se arate ca daca x, y, z
sunt in progresie geometrica atunci 
sunt in progresie aritmetica.
PRIMITIVE
1. Sa se calculeze primitivele
urmatoarelor functii.
1. ∫(3x
2.
∫ x(x-1)(x-2)dx
3. ∫ 
4.
∫ 
5. 
6. 
7. ∫ x
8.
9. ∫( e
10. ∫ (x
11. 
12.
13. ∫ 
14. ∫ 
15. ∫ 
16. ∫ 
17. 
18.
19. 
20.
21. 
2..Sa se calculeze primitivele urmatoarelor functii
compuse.
1. 
2.
3. 
4. 
5. 
6.
7. 
8. 
9.
10. 
11. 
12. 
13. 
14. 
3. Sa se calculeze primitivele
urmatoare utilizând metoda integrarii prin parti:
1. 
2. 
3. 
4. 
5. 
6.
7. 
8. 
9.
10.
11.
12.
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 
41.
42.
43. 
44.
45.
46. 
47.
3. Sa se calculeze integralele prin metoda substitutiei
1. 
2. 
3. 
4.
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16.
17. 
18. 
19. 
20.
21. 
22. 
23. 
24. 
25. 
26.
27. 
28. 
29. 
30.
31.
32. 
33. 
34.
35.
36
.
37.
38. 
4. Sa se calculeze primitivele urmatoarelor functii
trigonometrice:
1.
2. 
3.
4. 
5. 
6. 
7. 
8. 
9. 
10.
11.
12. 
13.
14.
15.
16.
17.
18. 
19.
20. 
5.Sa se calculeze primitivele urmatoarelor functii rationale:
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10.
11.
12.
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
ISTORICUL NOTIUNILOR MATEMATICE
Sec. 18
i.e.n. mesopotamienii creeaza
primele tabele de inmultire;
sec. 6 i.e.n. este cunoscuta asemanarea triunghiurilor de catre
Thales;
Sec. 5 i.e.n.
pitagorienii introduc notiunile de
numar prim, numar compus, numere relativ prime, numere prime perfecte;
Sec. 4 i.e.n.
Aristotel
(384-322 i.e.n) filozof grec a introdus notiunile de perimetru, teorema,
silogism.
Sec. 3 i.e.n.
Matematicianul grec Euclid(330-275 i.e.n ) cel care a intemeiat celebra scoala din Alexandria (in 323 i.e.n) a introdus notiunile
de semidreapta, tangenta la o curba, puterea unui punct fata de un cerc sau
sfera, sau denumirile de paralelogram, poliedru, prisma, tetraedru. A enuntat
teorema catetei si a inaltimii pentru un triunghi dreptunghic si a demonstrat
concurenta mediatoarelor unui triunghi;
Apolonius
din Perga(262-200 i.e.n), unul din cei mai mari geometri ai antichitatii introduce
pentru prima data denumirile pentru conice, de elipsa, hiperbola, parabola si
notiunile de focare, normale si defineste omotetia si inversiunea si da o aproximare exacta a lui 
cu patru zecimale.
este data aria triunghiului in functie de
laturi sau in functie de raza cercului inscris si semiperimetru;
Eratostene
din Cyrene(275-195 i.e.n) introduce metoda de determinare a tuturor
numerelor prime mai mici decât un numar dat, metoda cunoscuta sub numele de
„
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Ciurul lui Eratostene”
in prima carte din
„
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Elementele” lui
Euclid este cunoscuta teorema impartirii cu
rest si
„
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algoritmul lui
Euclid” pentru aflarea c.m.m.d.c. a doua numere intregi
85-168 matematicianul
grec Ptolemeu prezinta in cartea sa
„
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Almagest”, pe
lânga vaste cunostinte de astronomie si trigonometrie si diviziunea cercului in
360 de parti congruente si exprimarea acestora in fractii sexagesimale.
Sec. 3 s-a dat formularea teoremei celor trei perpendiculare de catre Pappos; acesta a mai dat si definitia
conicelor precum si teorema despre volumul corpurilor de rotatie
Sec. 7
sunt cunoscute regulile de trei directa si
inversa de catre Bragmagupta, matematician indian;
Arhimede(287-212
i.e.n) precursor al calculului integral, a determinat aria si volumul elipsoidului de rotatie si ale hiperboloidului de rotatie
cu pânze.
1202- Leonardo Fibonacci (1170-1240) matematician
italian introduce notatia pentru fractia ordinara;
1228- Fibonacci introduce denumirea pentru numarul zero, precum si sistemul
de numeratie zecimal. Tot prin opera sa
„
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Liber abaci”
sunt introduse pentru data in Europa numerele negative, fiind interpretate ca
datorii;
1150- este descrisa extragerea radacinii patrate si a celei cubice in cartea
„
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Lilavati” a matematicianului indian Bhaskara(1114-1185), tot el prezinta si
operatiile de inmultire si impartire cu numere negative;
1515- rezolvarea ecuatiilor de gradul al treilea cu o necunoscuta de catre
Scipio del Fero, iar mai târziu de Niccolo
Tartaglia in 1530, si pe acelea de gradul al patrulea de Ludovico Ferrari in
1545. Acestea au fost facute cunoscute abia in 1545 de catre Girolamo
Cardano(1502-1576) in lucrarile sale, desi promisese autorilor lor sa nu le
divulge;
1591-matematicianul francez Francois
Viete(1540-1603) introduce formulele cunoscute sub numele de relatiile lui
Viete;
1614- inventarea logaritmilor naturali de catre John Neper(1550-1617);
1637- este introdusa notiunea de variabila de catre Rene Descartes(1596-1650), cel care a introdus literele alfabetului
latin pentru notatii si a folosit coordonatele carteziene (definite dupa numele
sau), reducând problemele de geometrie la probleme de algebra;
1640- este introdusa denumirea pentru cicloida de catre Galileo Galilei (1564-1642);
1654- inceputul crearii teoriei probabilitatilor datorat corespondentei
dintre Pierre Fermat(1601-1665) si Blaise
Pascal(1623-1662) si dezvoltarea combinatoricii odata cu aparitia lucrarii
lui Pascal,
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@
Combinationes”;
1656- matematicianul englez John
Wallis(1616-1703) introduce simbolul 
cu notatiile 
si a denumirilor de interpolare respectiv mantisa
1670- este determinat semnul sinusului si desenata sinusoida respectiv
secantoida de catre John Wallis);
1678- este data teorema lui Ceva de catre Ceva Giovani(1648-1734);
1679- in
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@
Varia opera
mathematica” aparuta postum, a lui Pierre Fermat(1601-1665), a fost data
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@
Marea teorema a
lui Fermat”, reguli de integrare, definitia derivatei.
1692-
este scris primul manual de calcul integral de catre matematicianul elvetian Jean Bernoulli(1667-1748)” Lectiones
mathematicae de methodo integralium aliisque”, tiparit abia in 1742 si de
asemenea a mai scris un manual de calcul diferential, descoperit abia in 1920.
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@
Regula lui
l’Hospital” este data de catre Jean Bernoulli lui Guillaume de l’Hospital pe
care acesta o publica in 1696;
1690- este propusa denumirea de integrala de catre Jacques Bernoulli(1654-1705)
1692- sunt descoperite proprietatile spiralei logaritmice (Jacques
Bernoulli)
1694- este descoperita curba numita lemniscata, caracterizata de inegalitatea
(1+x)n
1+nx (Jacques
Bernoulli);
1696-1697- introducerea calculului variational, punerea problemei izoperimetrelor
de catre Jean Bernoulli.
1705- este data
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@
Legea numerelor
mari” de catre Jacques Bernoulli;
1711- realizarea dezvoltarii in serie a functiilor ex, sinx,
cosx,arcsinx, de catre matematicianul englez Isaac Newton(1642-1727) cel care a pus bazele calculului diferential
si integral concomitent cu Gottfried
Leibniz(1646-1716);
1729- este demonstrata existenta radacinilor complexe in numar par a unei
ecuatii algebrice cu coeficienti reali de catre Mac Laurin Colin(1698-1746;
1731- utilizarea sistemului de axe perpendiculare pentru a determina pozitia
unui obiect in functie de cele trei coordonate;
1733- crearea trigonometriei sferoidale de catre Alexis Clairaut(1713-1765);
1735- Matematicianul elvetian Leonhard
Euler(1707-1783) introduce si calculeaza constanta e=
=0,577215..., n→∞;
1739- introducerea conceptului de integrala curbilinie de catre Alexis
Clairaut;
1746- relatia lui Stewart este demonstrata de Mathew Stewart dupa ce in prealabil
ea ii fusese comunicata de catre Robert
Simson in 1735;
1747
este
enuntata problema celor trei corpuri de catre Clairaut;
introducerea
metodei multiplicatorilor nedeterminati in studiul sistemelor de ecuatii diferentiale de catre Jean Le Rond D’Alembert(1717-1783);
1750- Gabriel Cramer da o regula de rezolvare a
sistemelor cunoscuta sub denumirea de metoda lui Cramer;
1755- sunt puse bazele calculului variational de catre
Lagrange(1736-1813) concomitent cu
Euler,
1765- inceputul crearii geometriei descriptive de catre Gaspard Monge(1746-1818);
1766- crearea mecanicii analitice de catre Joseph Lagrange(1736-1813) cu enuntarea principiului vitezelor
virtuale si a ecuatiilor Lagrange;
1767- demonstrarea irationalitatii lui
de catre Heinrich
Lambert(1728-1777);
1768- demonstrarea existenta
factorului integrant la ecuatiile diferentiale de ordinul intâi de catre D’Alembert;
1771- a fost data ecuatia planului normal si formula distantei dintre doua
puncte din spatiu de catre matematicianul francez G. Monge;
1775- introducerea notiunilor de solutie generala si solutie particulara in
teoria ecuatiilor diferentiale de catre Leonhard
Euler; acesta a introdus si functia 
- indicatorul lui Euler, precum si notatiile e, i, f(x)si a
creat teoria fractiilor continue;
1780- au fost introduse liniile de curbura ale suprafetelor(G. Monge);
sunt descoperite functiile automorfe de
matematicianul francez Henri Poincare(1854-1912);
1785- a fost data ecuatia planului tangent(G. Monge);
1796- este data
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@
Teorema lui
Fourier” de determinare a numarului radacinilor reale cuprinse intr-un interval,
de catre Joseph Fourier(1768-183);
1797- este data formula cresterilor finite, cunoscuta sub denumirea de
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@
teorema lui
Lagrange”;
1798- au fost considerate cosinusurile directoare ale unei drepte(G. Monge);
este
introdus simbolul [.], pentru partea intreaga de catre Arien Marie Legendre
(1752-1833);
1807-, 1822 sunt date seriile Fourier care au contribuit la crearea teoriei
analitice a caldurii.
1812- este introdusa seria hipergeometrica de catre Carl Friedrich
Gauss(1777-1855) matematician german, cel care a demonstrat teorema fundamentala
a algebrei;
1816-1835- Augustin Cauchy(1789-1857), fondatorul analizei matematice moderne, a enuntat
criteriul de convergenta al seriilor, criteriu care-i poarta numele, a dat
primele teoreme de existenta din teoria ecuatiilor diferentiale si al ecuatiilor
cu derivate partiale, a introdus notiunile de afix, modul al unui numar
complex, numere conjugate, transpozitie;
1820- introducerea notiunii de raport anarmonic de catre Chasles Michel(1793-1880), fondatorul geometriei proiective alaturi
de matematicianul francez Jean Poncelet;
1822
introducerea functiilor Bessel de catre Friedrich Bessel;
este introdusa notatia pentru integrala
definita 
, de catre Fourier.;
este propusa denumirea de reprezentare
conforma de catre Gauss;
cercul lui Euler sau cercul celor noua puncte
este considerat pentru prima data de catre Charles Brianchon , Jean Poncelet si
Karl Feuerbach, atribuinduse din greseala numele lui Euler acestei teoreme;
1823-1831-
inceputul crearii primei geometrii neeuclidiene de catre Janos
Bolyai(1802-1860) concomitent
si independent de cea a lui Lobacevski.
1824-
este data denumirea de geometrie neeuclidiana
de catre Gauss;
Niels
Abel(1802-1829) demonstreaza
imposibilitatea rezolvarii cu ajutorul
radicalilor, a ecuatiilor algebrice de grad mai mare decât patru;
1825- Abel introduce integralele ce-i poarta numele;
1827- este creata teoria functiilor eliptice de catre Abel;
1828
sunt
introduse formele fundamentale ale suprafetelor si curburii totala a unei suprafete(curbura Gauss) de catre Gauss;
demonstrarea
teoremei lui Fermat pentru n=5 de catre matematicianul german Dirichlet
(1805-1859);
1830- este propusa denumirea de grup cu intelesul
actual de catre matematicianul francez
Evariste Galois(1811-1832);
1831- definitivarea calculului cu numere complexe de catre Gauss ;
1834- introducerea notiunii de factor de discontinuitate, referitor la integralele
1837-
introducerea notatiilor pentru limite laterale de catre Dirichlet si a functiei
care ii poarta numele, functia
Dirichlet;
W. Hamilton
introduce termenul de asociativitate a unei legi de compozitie;
1839- introducerea notiunii de integrale multiple(Dirichlet);
1840- este data o forma a eliminantului a doua ecuatii algebrice de catre James Sylvester(1814-1897), matematician englez;
1841- descoperirea invariantilor de catre matematicianul irlandez George Bole
(1815-1864);
introducerea
notiunilor de margine inferioara si superioara ale unei functii, de convergenta
uniforma de catre Weierstrass(1815-1897);
1843- descoperirea cuaternionilor de catre William Hamilton (1805-1865);
1845-
„
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@
Teorema limita
centrala” este data de matematicianul rus Pafnuti
Cebâsev;
1846- Legea numerelor mari
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
Cebâsev;
introducerea variabilei complexe in teoria numerelor imaginare de catre
D’Alembert;
1847
este
introdus calculul logic de George Boole,
creatorul algebrei booleene;
este introdusa notiunea de ideal de catre Ernest Kummel(1810-1893);
1851- sunt
introduse notiunile de rang si signatura a unei forme patratice si sunt propuse
notiunile de matrice si jacobian(J. Sylvester);
introducerea sufrafetelor riemann de catre matematicianul german
Bernhard Riemann(1826-1866), lui datorându-se studiul integralei definite.
1852- introducerea segmentelor orientate 
de catre Chasles Michael(1793-188) care a formulat si
proprietatile axei radicale a doua cercuri precum si a conicelor si
cuadricelor.
1853- Kronecker(1823-1891) introduce notatia 
;
1854- este introdusa notiunea de oscilatie intr-un punct de catre Riemann
care creeaza o noua geometrie neeuclidiana, numita geometria sferica;
1858- crearea calculului matriceal de catre Arthur Cayley(1821-1895) matematician englez ;
1871 Dedekind introduce notiunile de corp si modul ceeace in limbajul actual exprima
notiunile de subcorp si Z-submodul ale lui C. Tot el introduce multimea intregilor
unui corp de numere algebrice, definind si
idealele acestei multimi si demonstreaza teorema fundamentala de descompunere
unica a oricarui ideal in produs de ideale prime;
1872-
introducerea structurilor de subinel si modul
de catre Dirichlet;
introducerea numerelor rationale prin taieturi
de catre Dedekind;
1873- Charles Hermite(1822-1901)
demonstreaza transcendenta numarului e=
1874- este data denumirea de subgrup de catre Sophus Lie(1842-1899);
1874-1897- crearea teoriei multimilor de catre Georg Cantor(1845-1918). El a introdus notiunile de multime deschisa,
multime inchisa, multime densa, multime bine ordonata, multime numarabila,
punct de acumulare, punct izolat, produs cartezian, reuniune, intersectie.
1878- rezolvarea problemei celor patru culori pentru colorarea hartilor de catre Cayley;
1880-sunt descoperite functiile automorfe de matematicianul francez Henri
Poincare(1854-1912);
1882- Ferdinand Lindemann(1852-1939) a demonstrat trascendenta numarului =3,141592......; (un
numar se numeste transcedent daca nu este solutia niciunei ecuatii algebrice cu
coeficienti rationali); tot el demonstreaza imposibilitatea cvadraturii
cercului cu rigla si compasul;
1893-
H. Weber, asociaza conceptului de corp, sensul de astazi, ca o structura cu
o lege de grup aditiv si o inmultire asociativa, distributiva si in care orice
element e inversabil;
1897- introducerea
denumirii de inel de catre Hilbert(1862-1943);
1899 -axiomatizarea geometriei de catre
David Hilbert;
1900- introducerea axiomatica a numerelor intregi(D.Hilbert);
1905-
este introdusa notiunea de distanta intre doua multimi inchise de catre
matematicianul român Dimitrie Pompeiu(1873-1954);
1910- este introdusa denumirea de functionala
de catre Jacques Hadamard
(1865-1963), unul din fondatorii analizei functionale;
1912 -este descoperita notiunea de derivata areolara(Pompeiu)
1927-s-a stabilit formula Onicescu referitoare
la geodezice data de Octav Onicescu(1892-1983);
1928 -este introdusa functia
areolar-conjugata de catre matematicianul român Miron Nicolescu(1903-1975);
1933
-introducerea functiilor convexe de ordin superior de catre Tiberiu Popoviciu(1906-1975);
1936 -Matematicianul
român Gheorghe Mihoc(1906-1981) da o
metoda cunoscuta sub numele de metoda Schulz-Mihoc, de determinare a legilor
limita ale unui lant Markov;
1941 -teorema
lui Moisil referitoare la geodezicele unui spatiu riemannian este introdusa de Grigore Moisil(1906-1973);
1944
-este introdusa in domeniul algebrei moderne notiunea de signatura de catre
matematicianul român Dan Barbilian(1895-1961);
1950 -este introdusa notiunea de 
- derivata de catre Dan Barbilian;
1996 -celebra conjectura a lui Fermat
este demonstrata de catre Andrew
Wiles de la institutul Isaac Newton din Cambridge.
2000 -este determinat cel mai mare numar prim 26972593-1, având
doua milioane de cifre, obtinut cu ajutorul a 20 de mii de calculatoare puse in
retea;
BIBLIOGRAFIE.
1: N. Mihaileanu- Istoria
matematicii,vol.1,vol2.,Editura Stiintifica si enciclopedica; Bucuresti,1974/
1981;
2: Vasile Bobancu-
Caleidoscop matematic, Editura Niculesu;
3. Neculai Stanciu,
100 de probleme rezolvate. Editura Rafet;
4. Mica
enciclopedie matematica, Editura Tehnica, Bucuresti
Cuprins
Aplicatii ale numerelor complexe in geometrie.............5
Sinteze matematice
Multimea numerelor
reale...........................................37
Inegalitati....................................................................42
Multimi. Operatii cu multimi.....................................
45
Progresii......................................................................47
Functii.........................................................................50
Numere complexe.......................................................56
Functia exponentiala si logaritmica............................59
Binomul lui
Newton....................................................63
Vectori si operatii cu vectori.....................................
.65
Functii trigonometrice.................................................69
Formule
trigonometrice...............................................72
Ecuatiile dreptei in plan..............................................75
Conice..........................................................................77
Algebra liniara..............................................................82
Siruri de numere
reale..................................................88
Limite de siruri.............................................................93
Functii continue...........................................................98
Derivate.......................................................................101
Studiul functiilor cu ajutorul
derivatelor.....................103
Primitive......................................................................109
Probleme propuse si rezolvate....................................117
Probleme.sinteze.........................................................128
Istoricul notiunilor
matematice...................................143
