prerequisitos calculo

8/19/2019 Prerequisitos Calculo

1/15

Profa. Lena Bizelli

Pré-requisitos para o Cálculo

C. C , ,

C.

Frações

A C , ,

. ,

.

Regra no 1A , , C:

.

,

00

5

=

5

0

.

Regra no 2:

1.

,

5

4

4

5

7 1

7

1 x − 1

1−


2/15

Profa. Lena Bizelli

Regra no 3: Multiplicação de Frações

A , . A, , , ,

.

,

2 5 2 5 10 5

3 4 3 4 12 6

⋅⋅ = = =

⋅

a b ab

c d cd ⋅ =

Regra no 4: Divisão de Frações

A , , ,

.

,

(simplifica10 5 10 4 40

3 4 3r a expr

5 15

8

3

essão)÷ = ⋅ =

=

.

2

1

1010 5 4 8

3 4 3 5 3

/ /÷ = ⋅ =

/

Regra no 5: Adição e Subtração de Frações

A , ,

.

,

2 5 2 5 7

3 3 3 3

±± = =

A, , , :

a b a b

c c c

±± =


3/15

Profa. Lena Bizelli

A .

A, , , , . E,

.

C , :

( )0 e 0a b

c d c d

± ≠ ≠

, , ,

. , ,

, 2 4

3 5± ?

, (

) , , .

3 5 15 , , :

2 4 2 5 4 3 2 5 4 3 2 5 4 3 2 5 4 3

3 5 3 5 5 3 3 5 5 3 3 5 3 5 3 5

⋅ ⋅ ⋅ ⋅ ⋅ ± ⋅

± = ⋅ ± ⋅ = + = ± =⋅ ⋅ ⋅ ⋅ ⋅

A, a b

c d ± . ,

2, 3, 4 5. A,

2 4

3 5± , ,

a b

c d ± .

A, :

a b ad bc

c d cd

±± =

:


4/15

Profa. Lena Bizelli

Regra no 6: Simplificação de Frações

C,

. ,

.

Como Cancelar?

,

4 3

2 2 ( 0)

x y x

x z ≠

( ),

2 3

2 .

y

z

, :

4 3

2 2

y x x x x y y y

x x z z x z

⋅ ⋅ ⋅ ⋅ ⋅ ⋅=

⋅ ⋅ ⋅

A :

x x x y y y

x x z z

⋅ ⋅ ⋅ ⋅ ⋅ ⋅/ /

⋅ ⋅ ⋅/ /

2 3

2ou .

x x y y y x y

z z z

⋅ ⋅ ⋅ ⋅

⋅

( :

)

3 2 3 oua v w xy−

, ( , ).

A . , 4 3

2 2

x y

z

( ) xy p−

( )

( )

4 3

2 2.

y p y

y p z

−

−


5/15

Profa. Lena Bizelli

, , ( ) xy p−

:

( )2 3

2 .

y p y

z

−

Quando Cancelar?

A , .

, :

( ) ( )

( )

22 3 3

4 2

a b xy p y c d

ab z xy p

− +

−

,

( ).

, , , ( ) y p− :

( ) ( )3

2

a xy p y c d

bz

− +

A,

,

.

Módulo ou Valor AbsolutoA , 2 ,

, 2.

2 . E, 2 2= . C

2 , , 2.

2 2− . E, 2 2− = .


6/15

Profa. Lena Bizelli

D , ,

u . A,

5 5,= 5 5,− = 0 0.=

:

se 0

se 0

u uu

u u

≥=

−


7/15

Profa. Lena Bizelli

2 3 coma a . . ,

2 3 2 3 2 32 5 7 yz x yz x yz + =

7 57 5 2 5 7 2

5 7; ;

x x y

y

a a aa a a a a

a a a

− − − −= = = = =

( )

( ) ( )

52 2 5 10

e y

x x ya a a a a⋅ ⋅= = = ( )

( ) ( )3 3 3 3 e x x x xabc a b c abc a b c= = ( )

3 3

3 e

x x

x

a a a a

b bb b

= =

( )

( )2 2 2a b a b+ ≠ +

. A , :

( ) ( )( )2 2 2 2 22a b a b a b a ab ba b a ab b+ = + + = + + + = + + , , ( )

2 2 2a b a b+ = +

:

( )2 2

2 2

4 3 7 49

4 3 16 9 25

+ = =

+ = + =

Radiciação, , . E,

.

, ,

1 215 23 3 52 , , x x x x x= = = .


8/15

Profa. Lena Bizelli

Propriedades

2 2

4 2 pois 2 4 e 16 4 pois 4 16= = = =

A 16 (4 4) 16 . , 4 16.

642 4 6, , ... e assim por diante x x x x x x= = =

C :

) D 36.

)

D 36.E , .E ) 6,6, ) 6.

3 53 5

, ... e assim por diante x x x x= =

0 0 e 1 1= = ( , )

.

3 3 3, , n n n x y x y x y x y x y x y⋅ = ⋅ ⋅ = ⋅ ⋅ = ⋅

( )3

33

, , 0n

nn

x x x x x x y

y y y y y y= = = ≠

3 5 3 5 15 e m n m n x x x x x⋅ ⋅= = =

( )

mn mn x x=

2 2 x y x y+ = + .

, ,2 2 x y x y+ ≠ + .


9/15

Profa. Lena Bizelli

Simplificando Raízes

A :

1) C 400 12600ou ?

500 100 5 100 5 10 5= ⋅ = ⋅ =

12600 2 2 2 3 3 5 5 7= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 12.600

2 2 2 3 3 5 5 7= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

2 3 5 2 7= ⋅ ⋅ ⋅ ,

30 14=

2) , . ,

5

2 :

5 5 2 5 2

22 2 2= ⋅ =

Logarítmos .

,2

33 9 log 9 2= ⇔ = ( 3 9 2)

E , .

1) A loga b

1 ( )0 e 1a a> ≠ .

2) , 10, , ,log100 2= 10log 100 2.=

3) ( ≈ 2,72 ) loge , ,

ln5 log 5.e


10/15

Profa. Lena Bizelli

Propriedades

log 1 0

a

=

log 1a a =

( )log log loga a ab c b c⋅ = +

log log loga a a

bb c

c= −

log logca ab c b= ⋅

log ca b ( )log c

a b . L

log ca b . ,

( ) ( ) ( ) ( )3 33

5 5 5 5 5log 5 log 125 3 3 log 5 mas log 5 1 1 3 log 5 3= = = ⋅ = = ≠ ⋅ =

loglog

log

ca

c

bb

a=

( ,

)

loga b

a b=

FatoraçãoF .

C,

:

5 10 xy yz + a a b b y x y+ + +

A ,

C.

Casos de Fatoração

1) Fator Comum

A 3 4 2 5 4 35 10 15 x y x y x y z + + 2 35 y , ,

, , :

( )2 3 2 2

5 2 3 x y xy y x z + +

.


11/15

Profa. Lena Bizelli

2) Agrupamento

A ( )a a b b x y x y+ + +

:

( ) ( )a a b b y x y+ + + .

( ) ( )a b x y x y+ + + .

( ) ( )a b x y+ ⋅ + ( + ) .

, .

3) Diferença de Quadrados

:

( ) ( )( )2 2a b a b a b− = + − (*)

[ ] [ ]2 2

−

(*) . ,

( ) ( )2 229 16 3 4 x x− = −

, a 3 x= b 4= (*), ,

,

( ) ( ) ( )( )2 229 16 3 4 3 4 3 4 x x x x− = − = + −

, ( )2 2a b ,− , , ( )2 2a +b , .

4) Trinômio Quadrado Perfeito

2 2a 2ab+b±

:

( )2 2a e b . ( )2ab .


12/15

Profa. Lena Bizelli

,

( ) ( ) ( )2 2 22 6 9 2 3 3 3 x x x x x+ + = + ⋅ ⋅ + = +

( ) ( ) ( )2 2 224 4 1 2 2 2 1 1 2 1 x x x x x− + = − ⋅ ⋅ + = −

5) Trinômio do segundo grau

2x S P x+ +

D = a b+ = a b⋅ :

( )( )2x S P a b x x x+ + = + +

,

( ) ( )2x 5 6 2 3 x x x+ + = + +

( ) ( )2x 2 8 4 2 x x x+ − = + −

6) Soma e Diferença de cubos

( ) ( )( )3 3 2 2a b a b a ab b+ = + − +

( ) ( )( )3 3 2 2a -b a-b a +ab b= +

Trabalhando com Equações Quadráticas

, 2a b c 0 x x+ + = ,

, , a 0≠ .

:

Modo 1: Fatoração

2 5 6 x x− = :

2 5 6 0 x x− − =

( )( )6 1 0 x x− + =

( ) ( )6 0 e 1 0 x x− = + =

E, : 6 e 1. x x= = −


13/15

Profa. Lena Bizelli

Modo 2: A fórmula quadrática

, , 2a b c 0, x x+ + =

:

2 b b 4ac

2a x

− ± −=

Modo 3: Completando o quadradoC

.

, 23 24 27, x x= + ,

:23 24 27 x x− = 2

2 8 9 x x− = 2

2 8 16 9 16 x x− + = + ,

( 8 4 (4)2 16)

( )2

4 25 x − = (

3 4 )

( )2

4 25

4 5

x

x

− =

− = ±

,

.

4 5

9 ou 1

x

x x

= ±

⇒ = = −


14/15


15/15

Profa. Lena Bizelli

5 2 5 25 2

x x

x x

+ +/= = +/

2 2

2 2

5 2 5 2 5 2

11 1

x x x x x

x x x x x

+ + + + +/= =

++ + + +/

, ,

, .

prerequisitos calculo

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