memorator de matematica 2015

Memorator de Matematica cls a XII-a 2012

© 2012 Onodi Botond

1. Partea intreaga si partea fractional a unui numar real

[ ] [ ] [ ] { } { } { } [ ]

Proprietati:

[ ] { }

[ ]

{ }

[ ]

[ ] [

[ ] [ ]

{ } { }

√ | |

2. Progresii

2.1 Progresii Aritmetice



2.2 Progresii geometrice

√

3. Functii

3.1 Imaginea unei functii

{ | }

{ | }

(

)



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)



3.6 Functia bijectiva

a)

b)

c)

3.7 Functii inversabile

Proprietate:

3.8 Monotonia Functiei

1) [ ]

2) [ ]

3)



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



3.9 Functia de gradul II

(

)



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



3.11 Functia radical

√

{ }

[ ∞

: Functia radical este o functie crescatoare.

1.

2.

3.

4.

5. (

)

6.

7.

√

3.12 Rationalizarea

1. √

⁄

√

√

2. √ √

⁄

√ √

√ √

3.

√

⁄

√ √

4.

√ √

√

⁄

√

√

√ √

√

√ √ √

√

√



1. √

2. √ √

-conditiile se pun numai la radicali de ordin impar

√

√ | |

3.13 Functia exponential

∞ { }

1.

2.



3.14 Functia logaritmica

∞ { }

1.

2.

3.14.1 Proprietatile logaritmilor

Notam:

1.

2.

3.

4.

5.

6.

7. √



I.

II. 1)

2)

III.

III.

4. Numere Complexe

4.1 Forma algebrica a numerelor complexe:

| | √

4.2 Proprietati

MODUL CONJUGAT

| | | | | |

| | | | | |

|

| | |

| | (

)

| | | |

| |



4.3 Puterile lui i

4.4 Forma trigonometrica a numerelor complexe

| | √ [

4.5 Operatii

1. [ ]

2.

[ ]

3.



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.



5.3.2 Proprietati

1.

2.

3.

4.

5.4 Binomul lui Newton

5.5 Probabilitati

| |

1.

2.

3. ⁄



6. Vectori in plan

Vectori

| |

6.1 Adunarea vectorilor

6.2 Inmultirea unui vector cu un scalar



6.3 Teorema medianei

( )

T

T



6.4 Vectori in plan XOY

| | | |

6.5 Produsul sclar a doi vectori

| | | |

| | | |

√

√

| | √



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



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.



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.



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.



8.1.4 Alte formule trigonometrice

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

[ ]

11.

[ ]

12.

[ ]

8.2 Semnul functiei trigonometrice



8.3 Aplicatii ale trigonometriei in geometrie

8.4 Teorema sinusului

8.5 Teorema cosinusului (Pitagora generalizata)

8.6 Aria unui triunghi



√

9. Matrici

(

) ( )

9.1 Adunarea

( )

(

) (

) (

)

9.2 Inmultirea cu un scalar

( )

9.3 Inmultire

(

) (

) (

)

9.4 Ridicarea la putere



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)



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. ∞ ∞ ∞



6. ∞ ∞ ∞

7. ∞ ∞ ∞

8. ∞ ∞ ∞

9. ∞ ∞ ∞

10.1.3 Limite laterale

10.1.4 Proprietati

[ ]

10.2 Teorema clestelui



10.3 Formule

∞

∞

(

)

∞



(

)

10.4 Derivabilitatea

10.5 Regula lui L’hospital

10.5.1 Cazuri

a) ∞

b) ∞ ∞

(

) (

)

c) ∞



10.6 Formule

10.6.1 Operatii derivati

(

)

10.6.2 Ecuatia tangentei in

10.7 Rolul derivatei de ordinul I



10.8 Rolul derivatei de ordinul II

11. Asimptotele



11.1 Asimptote orizontale

∞

11.2 Asimptote verticale

∞

∞

11.3 Asimptote oblice

∞

[ ]

∞

12. Derivate formule



(

)

(

)

(√ )

√

√

√

(

)

(√ )

√

√

√

12. Derivare inverse unei functii



12.2 Derivate de ordin superior

12.3 Teorema lui Fermat

12.4 Teorema lui Lagrange

[ ]

[ ]

13. Formule pentru integrale nedefinte

∫

∫

∫

| |

∫

∫

|

|

√



∫



∫

√

∫

√ ( √ )

∫

√ | √ |

∫

∫

∫

∫

∫ | |

∫ | |

∫

∫ ∫

∫

∫

√ √

∫

√ √

∫

| |

∫

∫



13.1 Metoda integrarii prin parti

[ ] sunt doua functii derivabile, cu derivate continue

atunci:

∫

| ∫

∫

| ∫

13.2 Schimbare de variabila

[ ]

[ ]

∫

∫

[ ] [ ]

∫

∫



13.3 Aplicatii ale integralei definite

[ ]

{ | }

( ) ∫

[ ]

∫| |

13.3.1 Aria cuprinsa intre doua curbe

∫

[ ]

∫| |

13.3.2 Volumul corpului de rotatie

∫



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



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



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



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

memorator de matematica 2015

Documents