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Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 1
1. Partea intreaga si partea fractional a unui numar real
[ ] [ ] [ ] { } { } { } [ ]
Proprietati:
[ ] { }
[ ]
{ }
[ ]
[ ] [
[ ] [ ]
{ } { }
√ | |
2. Progresii
2.1 Progresii Aritmetice
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 2
2.2 Progresii geometrice
√
3. Functii
3.1 Imaginea unei functii
{ | }
{ | }
(
)
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 3
3.2 Compunerea functiilor
3.3 Functii pare, impare, periodice
3.4 Functii injective
a) [ ]
b) [ ]
c)
d)
Propietate: Orice functie monodtona (crescatoare sau descrescatoare) este injective.
3.5 Functii surjective
a)
b)
c)
d)
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 4
3.6 Functia bijectiva
a)
b)
c)
3.7 Functii inversabile
Proprietate:
3.8 Monotonia Functiei
1) [ ]
2) [ ]
3)
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 5
3.8 Functia de gradul I
{ }
(
)
{ }
3.8.1 Semnul functiei de gradul I
x -∞
∞
ax+b Semn contrar lui a 0 Acelasi semn cu a
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 6
3.9 Functia de gradul II
(
)
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 7
3.9.1 Semnul functiei de gradul II
1.
x -∞ ∞
Semnul lui a
2.
x -∞
∞
Semnul lui a 0 Semnul lui a
3.
x -∞ ∞
Semnul lui a 0 Semn contrar lui a 0 Semnul lui a
3.10 Relatiile lui Viete
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 8
3.11 Functia radical
√
{ }
[ ∞
: Functia radical este o functie crescatoare.
1.
2.
3.
4.
5. (
)
6.
7.
√
3.12 Rationalizarea
1. √
⁄
√
√
2. √ √
⁄
√ √
√ √
3.
√
⁄
√ √
4.
√ √
√
⁄
√
√
√ √
√
√ √ √
√
√
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 9
1. √
2. √ √
-conditiile se pun numai la radicali de ordin impar
√
√ | |
3.13 Functia exponential
∞ { }
1.
2.
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 10
3.14 Functia logaritmica
∞ { }
1.
2.
3.14.1 Proprietatile logaritmilor
Notam:
1.
2.
3.
4.
5.
6.
7. √
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 11
I.
II. 1)
2)
III.
III.
4. Numere Complexe
4.1 Forma algebrica a numerelor complexe:
| | √
4.2 Proprietati
MODUL CONJUGAT
| | | | | |
| | | | | |
|
| | |
| | (
)
| | | |
| |
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 12
4.3 Puterile lui i
4.4 Forma trigonometrica a numerelor complexe
| | √ [
4.5 Operatii
1. [ ]
2.
[ ]
3.
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 13
4.6 Radicinile de ordinal n a numerelor complexe z
√
(
)
5. Metode de numarare
5.1 Permutari
5.2 Aranjamente
5.3 Combinari
5.3.1 Cazuri particulare
1.
2.
3.
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 14
5.3.2 Proprietati
1.
2.
3.
4.
5.4 Binomul lui Newton
5.5 Probabilitati
| |
1.
2.
3. ⁄
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 15
6. Vectori in plan
Vectori
| |
6.1 Adunarea vectorilor
6.2 Inmultirea unui vector cu un scalar
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 16
6.3 Teorema medianei
( )
T
T
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 17
6.4 Vectori in plan XOY
| | | |
6.5 Produsul sclar a doi vectori
| | | |
| | | |
√
√
| | √
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 18
7. Geometrie analitica
7.1 Distanta dintre doua puncte
√
7.2 Panta dreptei AB
7.3 Coordonatele mijlocului unui segment
[ ]
7.4 Coordonatele centrului de greuta
7.5 Ecuatia generala a dreptei
7.6 Ecuatia dreptei cand cunoastem un punct si panta m
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 19
7.7 Ecuatie dreptei cand cuonoastem 2 puncte
7.8 Aria unui triunghi
| | |
|
7.9 Coliniaritatea a trei puncte
7.10 Distanta de la un punct la o dreapta
√
1.
2.
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 20
3. { }
7.11 Paralelismul si perpendicularitatea a doua drepte
8. Trigonometrie
x
0° 30° 45° 60° 90° 180° 270° 360°
√
√
√
√
√
√
√ √
8.1 Formule trigonometrice
1.
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 21
2.
3. (
)
(
)
(
)
(
)
8.1.1 Reducerea la cadranul I
1.
2.
3.
4.
5.
6.
8.1.2 Paritatea
1.
2.
3.
4.
8.1.3 Periodicitatea
1.
2.
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 22
8.1.4 Alte formule trigonometrice
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
[ ]
11.
[ ]
12.
[ ]
8.2 Semnul functiei trigonometrice
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 23
8.3 Aplicatii ale trigonometriei in geometrie
8.4 Teorema sinusului
8.5 Teorema cosinusului (Pitagora generalizata)
8.6 Aria unui triunghi
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 24
√
9. Matrici
(
) ( )
9.1 Adunarea
( )
(
) (
) (
)
9.2 Inmultirea cu un scalar
( )
9.3 Inmultire
(
) (
) (
)
9.4 Ridicarea la putere
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 25
(
) (
) (
)
9.5 Determinati de ordinul doi:
|
| |
|
9.6 Determinanti de ordinul trei
9.7 Proprietatile determinantilor
1.
2.
3. |
| |
|
4. |
| |
| |
|
9.8 Inversa unei matrici
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 26
9.8.1 La matrici de gradul doi
(
)
(
)
9.8.2 La matrici de gradul trei
(
)
(
) | |
9.9 Ecuatii matriciale
1. |
2. |
9.10 Sisteme
(
) (
)
Sistem compatibil determinat(o solutie metoda lui Cramer)
nedeterminat(o infinitate de solutii)
Incompatibil(S=Ø)
( ) → solutie daca verifica toate ecuatiile sistemului (1).
(1)
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 27
10. Analiza matematica
10.1 Limite de functii
10.1.1 Cazuri de nedeterminare
1. ∞
2.
3.
4. ∞
5.
6. ∞ ∞
7.
10.1.2 Calcule cu ∞
1. ∞ ∞
2. ∞ ∞
3. ∞ ∞
∞
4.
5. ∞ ∞ ∞
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 28
6. ∞ ∞ ∞
7. ∞ ∞ ∞
8. ∞ ∞ ∞
9. ∞ ∞ ∞
10.1.3 Limite laterale
10.1.4 Proprietati
[ ]
10.2 Teorema clestelui
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 29
10.3 Formule
∞
∞
(
)
∞
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 30
(
)
10.4 Derivabilitatea
10.5 Regula lui L’hospital
10.5.1 Cazuri
a) ∞
b) ∞ ∞
(
) (
)
c) ∞
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 31
10.6 Formule
10.6.1 Operatii derivati
(
)
10.6.2 Ecuatia tangentei in
10.7 Rolul derivatei de ordinul I
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 32
10.8 Rolul derivatei de ordinul II
11. Asimptotele
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 33
11.1 Asimptote orizontale
∞
11.2 Asimptote verticale
∞
∞
11.3 Asimptote oblice
∞
[ ]
∞
12. Derivate formule
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 34
(
)
(
)
(√ )
√
√
√
(
)
(√ )
√
√
√
12. Derivare inverse unei functii
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 35
12.2 Derivate de ordin superior
12.3 Teorema lui Fermat
12.4 Teorema lui Lagrange
[ ]
[ ]
13. Formule pentru integrale nedefinte
∫
∫
∫
| |
∫
∫
|
|
√
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 36
∫
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 37
∫
√
∫
√ ( √ )
∫
√ | √ |
∫
∫
∫
∫
∫ | |
∫ | |
∫
∫ ∫
∫
∫
√ √
∫
√ √
∫
| |
∫
∫
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 38
13.1 Metoda integrarii prin parti
[ ] sunt doua functii derivabile, cu derivate continue
atunci:
∫
| ∫
∫
| ∫
13.2 Schimbare de variabila
[ ]
[ ]
∫
∫
[ ] [ ]
∫
∫
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 39
13.3 Aplicatii ale integralei definite
[ ]
{ | }
( ) ∫
[ ]
∫| |
13.3.1 Aria cuprinsa intre doua curbe
∫
[ ]
∫| |
13.3.2 Volumul corpului de rotatie
∫
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 40
Continut
1. Partea intreaga si partea fractional a unui numar real .............................................................. 1
2. Progresii ........................................................................................................................................................... 1
2.1 Progresii Aritmetice ............................................................................................................................ 1
2.2 Progresii geometrice .......................................................................................................................... 2
3. Functii ................................................................................................................................................................ 2
3.1 Imaginea unei functii .......................................................................................................................... 2
3.2 Compunerea functiilor........................................................................................................................ 3
3.3 Functii pare, impare, periodice ...................................................................................................... 3
3.4 Functii injective ...................................................................................................................................... 3
3.5 Functii surjective ................................................................................................................................... 3
3.6 Functia bijectiva ..................................................................................................................................... 4
3.7 Functii inversabile ................................................................................................................................ 4
3.8 Monotonia Functiei .............................................................................................................................. 4
3.8 Functia de gradul I ................................................................................................................................ 5
3.8.1 Semnul functiei de gradul I ...................................................................................................... 5
3.9 Functia de gradul II .............................................................................................................................. 6
3.9.1 Semnul functiei de gradul II .................................................................................................... 7
3.10 Relatiile lui Viete ............................................................................................................................... 7
3.11 Functia radical .................................................................................................................................. 8
3.12 Rationalizarea .................................................................................................................................... 8
3.13 Functia exponential ......................................................................................................................... 9
3.14 Functia logaritmica ........................................................................................................................10
3.14.1 Proprietatile logaritmilor ...................................................................................................10
4. Numere Complexe ......................................................................................................................................11
4.1 Forma algebrica a numerelor complexe: ................................................................................11
4.2 Proprietati..............................................................................................................................................11
4.3 Puterile lui i ...........................................................................................................................................12
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 41
4.4 Forma trigonometrica a numerelor complexe .....................................................................12
4.5 Operatii ...................................................................................................................................................12
4.6 Radicinile de ordinal n a numerelor complexe z .................................................................13
5. Metode de numarare ................................................................................................................................13
5.1 Permutari ...............................................................................................................................................13
5.2 Aranjamente .........................................................................................................................................13
5.3 Combinari...............................................................................................................................................13
5.3.1 Cazuri particulare .......................................................................................................................13
5.3.2 Proprietati ......................................................................................................................................14
5.4 Binomul lui Newton ..........................................................................................................................14
5.5 Probabilitati ..........................................................................................................................................14
6. Vectori in plan ..............................................................................................................................................15
6.1 Adunarea vectorilor ..........................................................................................................................15
6.2 Inmultirea unui vector cu un scalar ..........................................................................................15
6.3 Teorema medianei .............................................................................................................................16
6.4 Vectori in plan XOY ............................................................................................................................17
6.5 Produsul sclar a doi vectori ...........................................................................................................17
7. Geometrie analitica ...................................................................................................................................18
7.1 Distanta dintre doua puncte .........................................................................................................18
7.2 Panta dreptei AB .................................................................................................................................18
7.3 Coordonatele mijlocului unui segment ....................................................................................18
7.4 Coordonatele centrului de greuta ..............................................................................................18
7.5 Ecuatia generala a dreptei .............................................................................................................18
7.6 Ecuatia dreptei cand cunoastem un punct si panta m ...............................18
7.7 Ecuatie dreptei cand cuonoastem 2 puncte ..........................................................................19
7.8 Aria unui triunghi ...............................................................................................................................19
7.9 Coliniaritatea a trei puncte ............................................................................................................19
7.10 Distanta de la un punct la o dreapta......................................................................................19
7.11 Paralelismul si perpendicularitatea a doua drepte ........................................................20
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 42
8. Trigonometrie ..............................................................................................................................................20
8.1 Formule trigonometrice ..................................................................................................................20
8.1.1 Reducerea la cadranul I ...........................................................................................................21
8.1.2 Paritatea ..........................................................................................................................................21
8.1.3 Periodicitatea ...............................................................................................................................21
8.1.4 Alte formule trigonometrice .................................................................................................22
8.2 Semnul functiei trigonometrice ..................................................................................................22
8.3 Aplicatii ale trigonometriei in geometrie ...............................................................................23
8.4 Teorema sinusului .............................................................................................................................23
8.5 Teorema cosinusului (Pitagora generalizata) ......................................................................23
8.6 Aria unui triunghi ...............................................................................................................................23
9. Matrici..............................................................................................................................................................24
9.1 Adunarea ................................................................................................................................................24
9.2 Inmultirea cu un scalar ....................................................................................................................24
9.3 Inmultire .................................................................................................................................................24
9.4 Ridicarea la putere.............................................................................................................................24
9.5 Determinati de ordinul doi: ...........................................................................................................25
9.6 Determinanti de ordinul trei ........................................................................................................25
9.7 Proprietatile determinantilor ......................................................................................................25
9.8 Inversa unei matrici ..........................................................................................................................25
9.8.1 La matrici de gradul doi ..........................................................................................................26
9.8.2 La matrici de gradul trei..........................................................................................................26
9.9 Ecuatii matriciale................................................................................................................................26
9.10 Sisteme ................................................................................................................................................26
10. Analiza matematica ...............................................................................................................................27
10.1 Limite de functii ..............................................................................................................................27
10.1.1 Cazuri de nedeterminare ....................................................................................................27
10.1.2 C u e u ∞ ..............................................................................................................................27
10.1.3 Limite laterale ..........................................................................................................................28
Memorator de Matematica cls a XII-a 2012
© 2012 Onodi Botond Page 43
10.1.4 Proprietati ..................................................................................................................................28
10.2 Teorema clestelui ...........................................................................................................................28
10.3 Formule ...............................................................................................................................................29
10.4 Derivabilitatea .................................................................................................................................30
10.5 Regu u L’h p t ......................................................................................................................30
10.5.1 Cazuri ...........................................................................................................................................30
10.6 Formule ...............................................................................................................................................31
10.6.1 Operatii derivati ......................................................................................................................31
10.6.2 Ecuatia tangentei in .......................................................................................31
10.7 Rolul derivatei de ordinul I ........................................................................................................31
10.8 Rolul derivatei de ordinul II ......................................................................................................32
11. Asimptotele ...............................................................................................................................................32
11.1 Asimptote orizontale ....................................................................................................................33
11.2 Asimptote verticale .......................................................................................................................33
11.3 Asimptote oblice .............................................................................................................................33
12. Derivate formule.....................................................................................................................................33
12. Derivare inverse unei functii .......................................................................................................34
12.2 Derivate de ordin superior ........................................................................................................35
12.3 Teorema lui Fermat .......................................................................................................................35
12.4 Teorema lui Lagrange ..................................................................................................................35
13. Formule pentru integrale nedefinte ..............................................................................................35
13.1 Metoda integrarii prin parti ......................................................................................................38
13.2 Schimbare de variabila ................................................................................................................38
13.3 Aplicatii ale integralei definite .................................................................................................39
13.3.1 Aria cuprinsa intre doua curbe ........................................................................................39
13.3.2 Volumul corpului de rotatie ....................................................................................................39
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