Memorator de Matematica 2015

Memorator de Matematica cls a XII-a 2012

1. Partea Partea intr eaga aga si si par tea tea fr acti acti onal a un ui num ar r eal

                                                                                         √  ||      

Proprietati:       

2. Progresii

                                                              2.1

Progresii Aritmetice

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© 2012 Onodi Botond

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         √                                                       

2.2

Progresii geometrice

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3. Functii

                    |   {} }                  3.1

Imaginea unei functii


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Memorator de Matematica cls a XII-a 2012 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)


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3.6 Functia bijectiva

a) b) c)

                                                                                    [        ]

3.7 Functii inversabile

Proprietate:

3.8 Monotonia Functiei

1)

[                        ]               

2) 3)


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                                                       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

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3.9 Functia de gradul II

                                         


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         

3.9.1 Semnul functiei de gradul II

1.



x

- ∞

                 

∞

Semnul lui a

2.

x



- ∞

Semnul lui a

3.

x

- ∞

Semnul lui a

∞

0



0

Semnul lui a



Semn contrar lui a 0

∞

Semnul lui a

3.10 Relatiile lui Viete


       

Page 7

Memorator de Matematica cls a XII-a 2012 3.11 Functia radical

     √         ∞                   √ 

: Functia radical este o functie crescatoare.

1. 2. 3. 4. 5. 6. 7.

3.12 Rationalizarea

1.

2.

3.

√   √  √   √ √   √ √  √ √   √    √  √    √ √ √   √ √ 

√ √ √  

    √               4.


Page 8


                   √   || 1.

2.

-conditiile se pun numai la radicali de ordin impar

3.13 Functia exponential

  ∞      

1. 2.

       


Page 9

Memorator de Matematica cls a XII-a 2012 3.14 Functia logaritmica

  ∞        

1. 2.

                                           √     

3.14.1 Notam:

1. 2. 3. 4. 5. 6.

7.

Proprietatile logaritmilor


Page 10


                      I. II.

1) 2)

III.

III.

4. Num ere Compl exe

4.1

Forma algebrica a numerelor complexe:

4.2

Proprietati

    ̅     ||                 MODUL

 | |  ||  ||  | |  ||  ||   ||||  ||  ||  ̅  || © 2012 Onodi Botond

CONJUGAT

                 ̅   Page 11

Memorator de Matematica cls a XII-a 2012 4.3

Puterile lui i

4.4

Forma trigonometrica a numerelor complexe

4.5

Operatii

                        ||                                                                   1. 2. 3.


Page 12


Radicinile de ordinal n a numerelor complexe z

5.

M etode de nu mar are

5.1

Permutari

5.2

Aranjamente

5.3

Combinari

       √              

                 

5.3.1 Cazuri particulare 1.

2. 3.


Page 13


                                                                         | |                                       ⁄                          5.3.2 Proprietati

1. 2. 3. 4.

5.4

Binomul lui Newton

5.5

Probabilitati

1.

2. 3.


Page 14

Memorator de Matematica cls a XII-a 2012 6.

Vectori in plan

Vectori

    ⃗   6.1

Adunarea vectorilor

    ⃗ ⃗ ⃗     ⃗ ⃗ ⃗   ⃗ ⃗   ⃗ ⃗ 6.2

Inmultirea unui vector cu un scalar

                 © 2012 Onodi Botond

Page 15


Teorema medianei

 ( ⃗ ⃗) ⃗    

T

⃗ ⃗

         ⃗ ⃗     ⃗        T


Page 16


Vectori in plan XOY

 ||  | |  ⃗ ⃗         ⃗         ⃗     ⃗    ⃗ ⃗         ⃗ ⃗     ⃗ ⃗     6.5

Produsul sclar a doi vectori

⃗ ⃗   |⃗|  |⃗|   |⃗⃗|⃗|⃗|      ⃗ ⃗      ||        © 2012 Onodi Botond

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Geometr ie analiti ca

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

              

                            © 2012 Onodi Botond

si panta m

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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.


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                                    3.

7.11 Paralelismul si perpendicularitatea a doua drepte

8.

Trigonometrie

0° x

      8.1

1.

    

  √  √  √

30°

 √  √   

45°

 √   √ √ 

60°

    

90°

180°

    

    

270°

360°

    

Formule trigonometrice

        


Page 20


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.


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                                                           

8.1.4 Alte formule trigonometrice 1.

2. 3. 4.

5. 6. 7. 8. 9.

10. 11. 12.

8.2

Semnul functiei trigonometrice


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8.3

Aplicatii ale trigonometriei in geometrie

8.4

Teorema sinusului

8.5

Teorema cosinusului (Pitagora generalizata)

8.6

Aria unui triunghi

  

            

                              

                  © 2012 Onodi Botond

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        9.

Matrici

            ()      ( )              ()                                  9.1

Adunarea

9.2

Inmultirea cu un scalar

9.3

Inmultire

9.4

Ridicarea la putere


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                            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


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                                                          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

(1)

Sistem

compatibil

 

 

determinat(o solutie metoda lui Cramer)

nedeterminat(o infinitate de solutii) Incompatibil(S=Ø)

    

(

) → solutie daca verifica toate ecuatiile sistemului (1).


Page 26


                                                  10.

Anal iza matemati ca

10.1 Limite de functii

∞ 

10.1.1 1.

2. 3. 4. 5.

Cazuri de nedeterminare

  ∞ 

∞∞    ∞∞ ∞∞ ∞ ∞   ∞∞∞ 6. 7.

10.1.2

Calcule cu ∞

1. 2. 3.

4.

∞

5.


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∞∞∞ ∞∞ ∞ ∞∞ ∞ ∞∞ ∞                                                    6. 7. 8. 9.

10.1.3

Limite laterale

10.1.4

Proprietati

10.2 Teorema clestelui

             © 2012 Onodi Botond

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Memorator de Matematica cls a XII-a 2012 10.3 Formule

                            ∞     ∞                         ∞                                           © 2012 Onodi Botond

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                                            10.4 Derivabilitatea

10.5 Regula lui L’hospital

       ∞        ∞∞             ∞         10.5.1 a)

Cazuri

b)

c)


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10.6 Formule

                                                                                                                                            10.6.1

Operatii derivati

10.6.2

Ecuatia tangentei in

10.7 Rolul derivatei de ordinul I


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10.8 Rolul derivatei de ordinul II

                                                                                                           11.

Asimptotele

               © 2012 Onodi Botond

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11.1 Asimptote orizontale

       ∞  11.2 Asimptote verticale

   ∞          ∞       11.3 Asimptote oblice

    ∞      [  ]                         ∞     



12.

Deri vate f ormu le

         © 2012 Onodi Botond

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                         (√  ) √                         √    √         12.

            ( √ )  √                                  √    √         

Derivare inverse unei functii

 


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    12.2 Derivate de ordin superior

              12.3 Teorema lui Fermat

                                                       12.4 Teorema lui Lagrange

13.

F ormu le pentr u in tegrale nedef in te

 ∫   ∫     ∫  ||   ∫        ∫       © 2012 Onodi Botond

                           √ 

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    ∫       


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   ∫ √     ∫ √      ∫ √        ∫  ∫  ∫     ∫     ∫||   ∫||   ∫    ∫∫  ∫      ∫               ∫ √  ∫    | |   ∫    ∫  © 2012 Onodi Botond

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Memorator de Matematica cls a XII-a 2012 13.1 Metoda integrarii prin parti

       ∫  | ∫                          ∫   | ∫   sunt doua functii derivabile, cu derivate continue

atunci:

13.2 Schimbare de variabila

                             ∫     ∫              [ ]   ∫   ∫          


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Memorator de Matematica cls a XII-a 2012 13.3 Aplicatii ale integralei definite

          {   |   }   ()∫             ∫  13.3.1

Aria cuprinsa intre doua curbe

   ∫           ∫          ∫   13.3.2 Volumul corpului de rotatie


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Memorator de Matematica cls a XII-a 2012 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


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Memorator de Matematica cls a XII-a 2012 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


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Memorator de Matematica cls a XII-a 2012 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 10.

Sisteme ................................................................................................................................................26

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


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Memorator de Matematica 2015

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