1º Prova - Caderno_de_Exercícios_de_Cálculo_I

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1 
 
FUNÇÕES 
Determinação do Domínio de uma Função 
Caso 1) O domínio de toda função polinomial ( ) 11 0...n nn nf x a x a x a−−= + + + é R, isto é, 
não há restrições para os valores da variável x. 
Exercício 1) Determine o domínio das seguintes funções: 
(a) ( ) 5f x x= +
 
(b) ( ) 2 4 2f x x x= + − 
 (d) ( ) 5 43 5 2f x x x x= + −
 
(e) ( )
3
8 7 45 27
2 3
f x x x x= + −
 
Caso 2) O domínio de toda função da forma
( )
( )
f x
g x
 é dado por ( ){ }/ 0x g x∈ℜ ≠ , isto é, 
são todos os valores de x para os quais o denominador da função é diferente de zero. 
Exercício 2) Determine o domínio das seguintes funções: 
Bloco 1 
(a) ( ) 5
1
x
f x
x
+
=
−
 (b) ( ) 3 5
2 1
x
f x
x
+
=
−
 
(c) ( ) 7 5
3 8
x
f x
x
− +
=
− 
(d) ( ) 6 3
4 7
x
f x
x
− +
=
− 
(e) ( ) 5 8
4 9
x
f x
x
−
=
− − 
(f) ( ) 5 9
3 12
x
f x
x
−
=
− + 
 
Bloco 2 
(a) ( ) 9 5
3
7
2
x
f x
x
− +
=
−
 
(b) ( ) 5
2
5
7
x
f x
x
− +
=
−
 
(c) ( ) 7 4
5
1
2
x
f x
x
− +
=
−
 
(d) ( ) 3 4
3
5
8
x
f x
x
− +
=
−
 
(e) ( ) 6 7
7
2
5
x
f x
x
− +
=
−
 
(f) ( ) 8 2
9
11
4
x
f x
x
− +
=
−
 
Bloco 3 
 (a) ( ) 2
4 5
1
x
f x
x
+
=
−
 (b) ( ) 2
3 7
81 3
x
f x
x
− +
=
− +
 
(c) ( ) 2
5 2
3 27
x
f x
x
+
=
− 
(d) ( ) 2
3 5
64 4
x
f x
x
−
=
−
 
 (e) ( ) 2
2 6
121 11
x
f x
x
−
=
−
 (f) ( ) 2
3 5
144 12
x
f x
x
−
=
− + 
Bloco 4 
 (a) ( ) 2
5 4
2
x
f x
x x
+
=
− −
 (b) ( )
2
2
2 4 5
6
x x
f x
x x
− +
=
+ −
 
(c) ( ) 2
2 5
7 6
x
f x
x x
−
=
− − 
(d) ( )
2
2
5 3 4
9 20
x x
f x
x x
− +
=
+ +
 
 (e) ( ) 2
6 2
11 30
x
f x
x x
+
=
− + 
(f) ( )
2
2
7 3
3 28
x x
f x
x x
− +
=
− − 
2 
 
Caso 3) Para funções da forma ( )n f x seu domínio é dado por: 
• ( ){ }/ 0x f x∈ℜ ≥ se n for um número par. 
• ℜ se n for um número ímpar. 
(Exercício 3) Determine o domínio das seguintes funções: 
Bloco 1 
(a) ( ) 2f x x= − (b) ( ) 5 3 1f x x= − 
(c) ( ) 6 4 8f x x= − (d) ( ) 9 2 9f x x= − 
(e) ( ) 10 57
3
x
f x = −
 
(f) ( ) 9 2 9
3
x
f x = − 
(g) ( ) 11 21
3
x
f x = +
 
(h) ( ) 16 25
3
f x x= − 
Bloco 2 
 (a) ( ) 3 2 4f x x= − (b) ( ) 2 9f x x= − 
(c) ( ) 8 2 25f x x= − (d) ( ) 7 22 72f x x= −
 
(e) ( ) 5 2243 3f x x= − +
 
(f) ( ) 6 2200 2f x x= − 
(g) ( ) 24 2 18f x x x= − +
 
(h) ( ) 214 2 32f x x x= − + 
(i) ( ) 7 23 8f x x x= − +
 
(j) ( ) 212 3 27f x x x= − 
Bloco 3 
 (a) ( ) 2 2 8f x x x= + − (b) ( ) 24 6f x x x= − − 
(c) ( ) 6 2 8 15f x x x= − + (d) ( ) 8 2 12f x x x= − + +
 
(e) ( ) 10 2 2 35f x x x= − − +
 
(f) ( ) 212 3 18f x x x= − − + 
Caso 4) Para funções da forma
( )
( )n
g x
f x
 seu domínio é dado por: 
• ( ){ }/ 0x f x∈ℜ > se n for um número par. 
• ( ){ }/ 0x f x∈ℜ ≠ se n for um número ímpar. 
(Exercício 4) Determine o domínio das seguintes funções: 
Bloco 1 
 (a) ( ) 4
2
x
f x
x
+
=
−
 (b) ( )
7
4
4
x
f x
x
+
=
+
 
(c) ( )
4
6 3
4 8
x
f x
x
− +
=
−
 (d) ( )
6
6 3
8 2
x
f x
x
− +
=
−
 
(e) ( ) 5 3
3 1
x
f x
x
−
=
−
 (f) ( )
5
5 3
3 2
x
f x
x
−
=
+
 
(g) ( )
3
4 2
5 2
x
f x
x
−
=
−
 (h) ( )
8
3 5
2 1
x
f x
x
−
=
−
 
3 
 
Bloco 2 
(a) ( )
9
3 4
5
2
3
x
f x
x
− +
=
−
 
(b) ( ) 2 5
5
7
3
x
f x
x
− +
=
−
 
(c) ( )
11
2 5
7
3
5
x
f x
x
− +
=
−
 
(d) ( ) 3 4
2
9
3
x
f x
x
− +
=
−
 
(e) ( )
13
4 5
3
2
x
f x
x
− −
=
+
 
(f) ( ) 3 9
2
1
3
x
f x
x
− −
=
+
 
Bloco 3 
(a) ( )
3 2
2 1
3 12
x
f x
x
−
=
−
 (b) ( )
2
3 2
2 18
x
f x
x
+
=
−
 
(c) ( )
8 2
4 3
2 50
x
f x
x
+
=
−
 (d) ( )
7 2
5 2
2 32
x
f x
x
− +
=
− 
(e) ( )
5 2
6 9
162 2
x
f x
x
− −
=
− + 
(f) ( )
6 2
7 3
100
x
f x
x
−
=
− +
 
(g) ( )
2
2
3 2 1
2 8
x x
f x
x x
− +
=
+ −
 (h) ( )
2
8 2
2 3 4
12
x x
f x
x x
+ +
=
− + +
 
 (i) ( )
2
10 2
4 7 5
2 35
x x
f x
x x
+ −
=
− − + 
(j) ( )
2
212
5 9 2
3 18
x x
f x
x x
− −
=
− − +
 
 
Caso 5) Para funções da forma
( )
( )
n
g x
f x
 seu domínio é dado por: 
• 
( )
( )
/ 0
f x
x
g x
  
∈ℜ ≥ 
  
se n for um número par. 
• ( ){ }/ 0x f x∈ℜ ≠ se n for um número ímpar. 
(Exercício 5) Determine o domínio das seguintes funções: 
Bloco 1 
 (a) ( ) 4
2
x
f x
x
+
=
−
 (b) ( ) 3 4
3 9
x
f x
x
+
=
− +
 
(c) ( ) 4 8
6 3
x
f x
x
−
=
− +
 (d) ( ) 7 8 2
6 3
x
f x
x
−
=
− +
 
(e) ( ) 3 1
5 3
x
f x
x
−
=
−
 (f) ( ) 5 3 2
5 3
x
f x
x
+
=
−
 
 (g) ( ) 3 9
5
4
3
x
f x
x
− −
=
+
 
(h) ( ) 9 5 3
3
x
f x
x
−
=
−
 
(i) ( ) 5 7
3 9
x
f x
x
−
=
− 
(j) ( ) 2 7
3 4
x
f x
x
−
=
− + 
(l) ( ) 11
5
3
7
2 5
x
f x
x
−
=
− + 
(m) ( ) 13
2
3
5 8
x
f x
x
−
=
− −
 
 
4 
 
Bloco 2 
(a) ( ) 3 2
2 1
3 12
x
f x
x
−
=
−
 (b) ( ) 2
3 2
2 18
x
f x
x
+
=
−
 
(c) ( )
2
8
2 50
3 4
x
f x
x
−
=
+
 (d) ( )
2
7
2 32
4 7
x
f x
x
−
=
− + 
(e) ( ) 5 2
4 8
162 2
x
f x
x
−
=
− 
(f) ( )
2
6
100
7 14
x
f x
x
− +
=
−
 
 (g) ( )
2
4
2
2 18
3 9
x x
f x
x x
− +
=
− + 
(h) ( )
2
14
2
2 32
3 27
x x
f x
x x
− +
=
−
 
(i) ( )
2
2
2 8
12
x x
f x
x x
+ −
=
− + +
 (j) ( )
2
10
2
2 35
3 18
x x
f x
x x
− − +
=
− − +
 
 
Caso 6) Para funções da forma
( )
( )
n g x
f x
 seu domínio é dado por: 
• ( ) ( ){ }/ 0 e 0x g x f x∈ℜ ≥ ≠ se n for um número par. 
• ( ){ }/ 0x f x∈ℜ ≠ se n for um número ímpar. 
 
 
 
(Exercício 6) Determine o domínio das seguintes funções: 
Bloco 1 
 (a) ( ) 4
2
x
f x
x
+
=
−
 (b) ( )
3 6
7
x
f x
x
+
=
−
 
(c) ( ) 3 9
4
x
f x
x
−
=
−
 (d) ( )
13 3 4
2
x
f x
x
+
=
−
 
(e) ( ) 7 14
3 7
x
f x
x
−
=
−
 (f) ( )
3 5 10
3 8
x
f x
x
−
=
−
 
(g) ( )
5 2 8
4 9
x
f x
x
−
=
− 
(h) ( ) 9 3
3 2
x
f x
x
−
=
−
 
(i) ( )
7 14 7
2 6
x
f x
x
+
=
− 
(j) ( ) 4 7
3 9
x
f x
x
−
=
−
 
(l) ( )
3 12 3
4 2
x
f x
x
−
=
− 
(m) ( )
4 4 2
3 18
x
f x
x
−
=
−1 
 
LIMITE DE UMA FUNÇÃO 
 
Limite. Seja :f D ⊂ℜ→ℜ uma função dada. Se ( )f x se aproxima de 
um número L quando x se aproxima de um número c tanto pela esquerda 
como pela direita, dizemos que L é o limite de ( )f x quando x tende a c, o 
que em notação matemática é escrito como ( )lim
x c
f x L
→
= . 
Geometricamente, a igualdade ( )lim
x c
f x L
→
= 
significa que a ordenada do gráfico de
( )y f x= se aproxima de L quando x se 
aproxima de c, como mostra a figura ao lado. 
 
Exemplo. Vamos usar uma tabela para estimar o limite
1
1
lim
1x
x
x→
−
−
 . 
 
O gráfico de aparece na figura abaixo. O cálculo do limite mostra que a 
ordenada do gráfico da função ( )y f x= tende para L=0,5 quando x tende 
para 1. Assim escrevemos: 
1
1
lim 0,5
1x
x
x→
−
=
− 
É importante não esquecer que os 
limites descrevem o comportamento 
de uma função “perto” de um ponto, 
mas não necessariamente no próprio 
ponto. 
PROPRIEDADES DOS LIMITES 
Se ( )lim
x c
f x
→
 e ( )lim
x c
g x
→
 existem, então: 
• ( ) ( ) ( ) ( )lim lim lim
x c x c x c
f x g x f x g x
→ → →
± = ±   
• ( ) ( )lim lim
x c x c
k f x k f x
→ →
⋅ = ⋅   
• ( ) ( ) ( ) ( )lim lim lim
x c x c x c
f x g x f x g x
→ → →
⋅ = ⋅   
• 
( )
( )
( )
( )
( )
lim
lim , se lim 0
lim
x c
x c x c
x c
f xf x
g x
g x g x
→
→ →
→
 
= ≠ 
 
 
• ( ) ( )lim lim
pp
x c x c
f x f x
→ →
 =    
 
• lim
x c
k k
→
= 
Estas propriedades são demonstradas formalmente em cursos mais teóricos. 
Elas são importantes porque simplificam o cálculo dos limites de funções 
algébricas. 
EXERCÍCIOS 
1. Calcule os limites abaixo: 
(a) ( )
4
lim 5
x→
− = 
(b) 
2
6
lim
5x→
  = 
 
 
(c) 
8
3
lim 7
x→−
= 
(d) 
9
5
lim
3x→−
 
=  
 
 
 
2 
 
2. Calcule os limites abaixo: 
(a) ( )
3
lim 4 2
x
x
→
+ = 
(b) ( )2
4
lim 5 9 8
x
x x
→
− − = 
(c) ( )
2
lim 7 6
x
x
→−
− = 
(d) ( )3
2
lim 3 2 7
x
x x
→−
− + = 
(e) 
1
4
lim 5
3x
x
→−
 + = 
 
 
(f) ( )
7
lim 5 3
x
x
→
− + = 
(g) ( )2
4
lim 4 8 7
x
x x
→
− + = 
(h) ( )
4
lim 5 2
x
x
→−
− = 
(i) ( )2
2
lim 6 7 5
x
x x
→−
− − = 
(j) 
2
3
lim 2
4x
x
→−
 − = 
 
 
3. Calcule os limites abaixo: 
(a) ( )
1
lim 2 4 1
x
x x
→
− =   
(b) ( ) ( )
1
lim 2 7 2
x
x x
→−
− + + =   
(c) ( ) ( )
2
lim 5 3 3 9
x
x x
→−
− + − =   
(d) 
1
3 2
lim 1 1
2 3x
x x
→−
 −   + − =      
 
(e) 
0
7 5
lim 2 9
2 3x
x x
→
   − − =      
 
(f) ( )
1
lim 3 2 1
x
x x
→−
− =   
(g) ( )( )
1
lim 4 3 5 2
x
x x
→
− + − =   
(h) ( )( )
1
lim 3 5 6 2
x
x x
→−
− + − =   
(i) 
1
2 3
lim 1 2
3 2x
x x
→
 −   − + =      
 
(j) 
1
2 5 4
lim 4
3 2 3x
x x
→
   + − =      
 
 
3 
 
4. Calcule os limites abaixo: 
(a) ( )3
3
lim 2 4
x
x
→−
+ = 
(b) ( )6
1
lim 6 1
x
x
→
− − = 
(c) ( )2
1
lim 6 4
x
x
→−
− = 
(d) 
6
0
1
lim 5
2x
x
→
 − + = 
 
 
(e) 
5
3
5
lim 1
3x
x
→−
 − = 
 
 
(f) 
4
4
2
lim 5
3x
x
→−
 − = 
 
 
(g) 
2
3
4
lim 2
5x
x
→−
 − = 
 
 
(h) 
2
2
2
3
lim 1
5x
x
x
→−
 + − = 
 
 
(i) 
42
1
4
lim 5
5 3x
x x
→−
 
+ − = 
 
 
(j) 
36
0
5 5
lim 3
4 7x
x x
→
 
− − = 
 
 
5. Calcule os limites abaixo: 
(a) 
3
4 5
lim
5 1x
x
x→
−
=
−
 
(b) 
1
6 4
lim
3 2x
x
x→−
−
=
−
 
(c) 
1
3 9
lim
8 1x
x
x→
− +
=
−
 
(d) 
2
3 4
lim
8 1x
x
x→
+
=
−
 
(e) 
1
4
lim
2 1x
x
x→−
+
=
+
 
(f) 
5
2
lim
4x
x
x→
+
=
−
 
(g) 
2
31
4 6 3
lim
16 8 7x
x x
x x→
− +
=
+ −
 
(h) 
2
21
2 5 3
lim
6 7 2x
x x
x x→−
+ −
=
− +
 
(i) 
2
30
5 3 2
lim
19 6 2x
x x
x x→
− +
=
+ −
 
(j) 
2
21
5 9
lim
9 17 12x
x x
x x→
− +
=
+ −
 
4 
 
6. Calcule os limites abaixo: 
(a) 
2
1 1
2lim
2x
x
x→
−
=
−
 
(b) 
3
1 1
3lim
3x
x
x→
−
=
−
 
(c) 
5
1 1
5lim
5x
x
x→
−
=
−
 
(d) 
8
1 1
8lim
8x
x
x→
−
=
−
 
(e) 
9
1 1
9lim
9x
x
x→
−
=
−
 
(f) 
4
1 1
4lim
4x
x
x→
−
=
−
 
(g) 
6
1 1
6lim
6x
x
x→
−
=
−
 
(h) 
10
1 1
10lim
10x
x
x→
−
=
−
 
(i) 
13
1 1
13lim
13x
x
x→
−
=
−
 
(j) 
1 1
lim
x a
a x
x a→
−
=
−
 
7. Calcule os limites abaixo: 
(a) 
21
1
lim
1x
x
x→
−
=
−
 
(b) 
22
2
lim
4x
x
x→
−
=
−
 
(c) 
23
3
lim
9x
x
x→
−
=
−
 
(d) 
24
4
lim
16x
x
x→
−
=
−
 
(e) 
25
5
lim
25x
x
x→
−
=
−
 
(f) 
21
1
lim
1x
x
x→−
+
=
−
 
(g) 
22
2
lim
4x
x
x→−
+
=
−
 
(h) 
23
3
lim
9x
x
x→−
+
=
−
 
(i) 
25
5
lim
25x
x
x→
+
=
−
 
(j) 
26
6
lim
36x
x
x→−
+
=
−
 
 
5 
 
8. Calcule os limites abaixo: 
(a) 
2
2
4
lim
2x
x
x→−
−
=
+
 
(b) 
2
3
9
lim
3x
x
x→−
−
=
+
 
(c) 
2
4
16
lim
4x
x
x→
−
=
−
 
(d) 
2
1
1
lim
1x
x
x→
−
=
−
 
(e) 
2
4
16
lim
4x
x
x→−
−
=
+
 
(f) 
2
5
25
lim
5x
x
x→−
−
=
+
 
(g) 
2
6
36
lim
6x
x
x→−
−
=
+
 
(h) 
2
7
49
lim
7x
x
x→
−
=
−
 
(i) 
2
8
64
lim
8x
x
x→−
−
=
+
 
(j) 
2
9
81
lim
9x
x
x→
−
=
−
 
9. Calcule os limites abaixo: 
(a) 
25
5
lim
25x
x
x→
−
=
−
 
(b) 
24
4
lim
16x
x
x→
−
=
−
 
(c) 
26
6
lim
36x
x
x→−
+
=
−
 
(d) 
21
1
lim
1x
x
x→−
+
=
−
 
(e) 
27
7
lim
49x
x
x→
−
=
−
 
(f) 
28
8
lim
64x
x
x→−
+
=
−
 
(g) 
210
10
lim
100x
x
x→
−
=
−
 
(h) 
29
9
lim
81x
x
x→−
+
=
−
 
(i) 
211
11
lim
121x
x
x→
−
=
−
 
(j) 
212
12
lim
144x
x
x→
−
=
−
 
 
6 
 
10. Calcule os limites abaixo: 
(a) 
2
21
2
lim
2 3x
x x
x x→
+ −
=
+ −
 
(b) 
2
22
6
lim
2 8x
x x
x x→
+ −
=
+ −
 
(c) 
2
23
12
lim
6x
x x
x x→
− −
=
− −
 
(d) 
2
24
9 20
lim
10 24x
x x
x x→
− −
=
− +
 
(e) 
2
25
11 30
lim
7 10x
x x
x x→
− +
=
− +
 
(f) 
2
26
9 18
lim
8 12x
x x
x x→
− +
=
− +
 
11. Calcule os limites abaixo: 
(a) 
2
23
12
lim
6x
x x
x x→−
− −
=
+ −
 
(b) 
2
24
20
lim
3 4x
x x
x x→−
− −
=
+ −
 
(c) 
2
25
30
lim
2 15x
x x
x x→−
− −
=
+ −
 
(d) 
2
26
5 6
lim
4 12x
x x
x x→−
+ −
=
+ −
 
(e) 
2
27
5 14
lim
4 21x
x x
x x→−
+ −
=
+ −
 
(f) 
2
28
5 24
lim
6 16x
x x
x x→−
+ −
=+ −
 
 
12. Calcule os limites abaixo: 
(a) 
31
1
lim
1x
x
x→
−
=
−
 
(b) 
32
2
lim
8x
x
x→
−
=
−
 
(c) 
33
3
lim
27x
x
x→
−
=
−
 
(d) 
34
4
lim
64x
x
x→
−
=
−
 
(e) 
35
5
lim
125x
x
x→
−
=
−
 
(f) 
36
6
lim
216x
x
x→
−
=
−
 
 
13. Calcule os limites abaixo: 
(a) 
4
2
16
lim
2x
x
x→
−
=
−
 
(b) 
4
3
81
lim
3x
x
x→−
−
=
+
 
(c) 
4
4
256
lim
4x
x
x→
−
=
−
 
(d) 
4
5
625
lim
5x
x
x→−
−
=
+
 
(e) 
4
6
1296
lim
6x
x
x→
−
=
−
 
 
 
 
7 
 
14. Calcule os limites abaixo: 
(a) 
( )2
0
1 1
lim
x
x
x→
+ −
= 
(b) 
( )2
0
2 4
lim
x
x
x→
+ −
= 
(c) 
( )2
0
3 9
lim
x
x
x→
+ −
= 
(d) 
( )2
0
5 25
lim
x
x
x→
− −
= 
(e) 
( )2
0
6 36
lim
x
x
x→
− −
= 
(f) 
( )2
0
7 49
lim
x
x
x→
− −
= 
(g) 
( )2
0
9 81
lim
x
x
x→
− −
= 
(h) 
( )2
0
10 100
lim
x
x
x→
− −
= 
(i) 
( )2
0
11 121
lim
x
x
x→
− −
= 
(j) 
( )2
0
12 144
lim
x
x
x→
− −
= 
 
 
15. Calcule os limites abaixo: 
(a) 
1
1
lim
1x
x
x→
−
=
−
 
(b) 
4
4
lim
2x
x
x→
−
=
−
 
(c) 
9
9
lim
3x
x
x→
−
=
−
 
(d) 
16
16
lim
4x
x
x→
−
=
−
 
(e) 
25
25
lim
5x
x
x→
−
=
−
 
(f) 
36
36
lim
6x
x
x→
−
=
−
 
(g) 
49
49
lim
7x
x
x→
−
=
−
 
(h) 
81
81
lim
9x
x
x→
−
=
−
 
(i) 
100
100
lim
10x
x
x→
−
=
−
 
(j) 
121
121
lim
11x
x
x→
−
=
−
 
 
8 
 
16. Calcule os limites abaixo: 
 
(a) 
16
16
lim
4x
x
x→
−
=
−
 
(b) 
4
4
lim
2x
x
x→
−
=
−
 
(c) 
25
25
lim
5x
x
x→
−
=
−
 
(d) 
36
36
lim
6x
x
x→
−
=
−
 
(e) 
49
49
lim
7x
x
x→
−
=
−
 
(f) 
64
64
lim
8x
x
x→
−
=
−
 
(g) 
81
81
lim
9x
x
x→
−
=
−
 
(h) 
100
100
lim
10x
x
x→
−
=
−
 
(i) 
121
121
lim
11x
x
x→
−
=
−
 
(j) 
144
144
lim
12x
x
x→
−
=
−
 
17. Calcule os limites abaixo: 
 
(a) 
4
5 1
lim
4x
x
x→−
+ −
=
+
 
(b) 
6
15 3
lim
6x
x
x→−
+ −
=
+
 
(c) 
61
3 8
lim
61x
x
x→
+ −
=
−
 
(d) 
9
7 4
lim
9x
x
x→
+ −
=
−
 
(e) 
32
4 6
lim
32x
x
x→
+ −
=
−
 
 
18. Calcule os limites abaixo: 
(a) 
0
1 1
3 3lim
x
x
x→
−
+ = 
(b) 
0
1 1
4 4lim
x
x
x→
−
+ = 
(c) 
0
1 1
5 5lim
x
x
x→
−
+ = 
(d) 
0
1 1
6 6lim
x
x
x→
−
+ = 
(e) 
0
1 1
7 7lim
x
x
x→
−
+ = 
9 
 
19. Calcule os limites abaixo: 
(a) 
0
3 3
lim
x
x
x→
+ −
= 
(b) 
0
5 5
lim
x
x
x→
+ −
= 
(c) 
0
7 7
lim
x
x
x→
+ −
= 
(d) 
0
9 9
lim
x
x
x→
+ −
= 
(e) 
0
13 13
lim
x
x
x→
+ −
= 
(f) 
0
lim
2 2x
x
x→
=
+ −
 
(g) 
0
lim
4 2x
x
x→
=
+ −
 
(h) 
0
lim
6 6x
x
x→
=
+ −
 
(i) 
0
lim
10 10x
x
x→
=
+ −
 
(j) 
0
lim
8 8x
x
x→
=
+ −
 
20. Calcule os limites abaixo: 
(a) 
2
20
9 3
lim
x
x
x→
+ −
= 
(b) 
2
20
4 2
lim
x
x
x→
+ −
= 
(c) 
2
20
25 5
lim
x
x
x→
+ −
= 
(d) 
2
20
16 4
lim
x
x
x→
+ −
= 
(e) 
2
20
49 7
lim
x
x
x→
+ −
= 
(f) 
2
20
6 36
lim
x
x
x→
− +
= 
(g) 
2
20
9 81
lim
x
x
x→
− +
= 
(h) 
2
20
8 64
lim
x
x
x→
− +
= 
(i) 
2
20
10 100
lim
x
x
x→
− +
= 
(j) 
2
20
11 121
lim
x
x
x→
− +
= 
10 
 
21. Calcule os limites abaixo: 
( )( )3 32 23 3 3x a x a x ax a− = − + + 
(a) 
33
2
2
lim
2x
x
x→
−
=
−
 
(b) 
3 3
5
5
lim
5x
x
x→
−
=
−
 
(c) 
3 3
6
6
lim
6x
x
x→
−
=
−
 
(d) 
3 37
7
lim
7x
x
x→
−
=
−
 
(e) 
3 39
9
lim
9x
x
x→
−
=
−
 
(f) 
3 310
10
lim
10x
x
x→
−
=
−
 
22. Calcule os limites abaixo: 
(a) 
31
1
lim
1x
x
x→
−
=
−
 
(b) 
3
41
1
lim
1x
x
x→
−
=
−
 
(c) 
4
51
1
lim
1x
x
x→
−
=
−
 
(d) 
5
61
1
lim
1x
x
x→
−
=
−
 
(e) 
6
71
1
lim
1x
x
x→
−
=
−
 
 
23. Calcule os limites abaixo: 
(a) 
31
1
lim
1x
x
x→
−
=
−
 
(b) 
3
41
1
lim
1x
x
x→
−
=
−
 
(c) 
4
51
1
lim
1x
x
x→
−
=
−
 
(d) 
5
61
1
lim
1x
x
x→
−
=
−
 
(e) 
6
71
1
lim
1x
x
x→
−
=
−
 
24. Sabendo que
0
lim 1
x
senx
x→
= calcule os limites abaixo: 
(a) 
0
sen 2
lim
x
x
x→
= 
(b) 
0
sen 3
lim
x
x
x→
= 
(c) 
0
sen 4
lim
x
x
x→
= 
(d) 
0
sen 5
lim
x
x
x→
= 
(e) 
0
sen 6
lim
x
x
x→
= 
11 
 
(f) 
0
sen 2
lim
3x
x
x→
= 
(g) 
0
sen 4
lim
5x
x
x→
= 
(h) 
0
sen 6
lim
4x
x
x→
= 
(i) 
0
sen 7
lim
3x
x
x→
= 
(j) 
0
sen8
lim
9x
x
x→
= 
(k) 
0
sen 3
lim
sen 2x
x
x→
= 
(l) 
0
sen 5
lim
sen 4x
x
x→
= 
(m) 
0
sen 4
lim
sen 6x
x
x→
= 
(n) 
0
sen 3
lim
sen 7x
x
x→
= 
(o) 
0
sen 9
lim
sen8x
x
x→
= 
(p) 
0
tg
lim
x
x
x→
= 
(q) 
0
tg 2
lim
x
x
x→
= 
(r) 
0
tg5
lim
x
x
x→
= 
(s) 
0
tg 7
lim
x
x
x→
= 
(f) 
0
tg9
lim
x
x
x→
= 
(g) 
0
tg 2
lim
3x
x
x→
= 
(h) 
0
tg3
lim
4x
x
x→
= 
(i) 
0
tg5
lim
7x
x
x→
= 
(j) 
0
tg 7
lim
9x
x
x→
= 
 
12 
 
25. Sabendo que
0
1
lim ln
x
x
a
a
x→
−
= calcule os limites abaixo: 
(a) 
0
2 1
lim
x
x x→
−
= 
(b) 
0
4 1
lim
x
x x→
−
= 
(c) 
0
3 1
lim
5
x
x x→
−
= 
(d) 
0
6 1
lim
7
x
x x→
−
= 
(e) 
0
9 1
lim
3
x
x x→
−
= 
(f) 
0
2 3
lim
x x
x x→
−
= 
(g) 
0
3 4
lim
x x
x x→
−
= 
(h) 
0
5 7
lim
x x
x x→
−
= 
(i) 
0
6 8
lim
2
x x
x x→
−
= 
 
(j) 
0
5 3
lim
4
x x
x x→
−
= 
(k) 
0
9 4
lim
5
x x
x x→
−
= 
(l) 
0
7 3
lim
6
x x
x x→
−
= 
(m) 
2
3 9
lim
2
t
t t→
−
=
−
 
(n) 
2
4 16
lim
2
t
t t→
−
=
−
 
(o) 
3
3 27
lim
3
t
t t→
−
=
−
 
(p) 
3
2 8
lim
3
t
t t→
−
=
−
 
(q) 
2
5 25
lim
2
t
t t→
−
=
−
 
(r) 
2
6 36
lim
2
t
t t→
−
=
−
 
(s) 
3
4 64
lim
3
t
t t→
−
=
−
 
13 
 
26. Sabendo que
0
1
lim ln
x
x
a
a
x→
−
= calcule os limites abaixo: 
(a) 
2 3
0
lim
t t
t
e e
t→
−
= 
(b) 
4 5
0
lim
t t
t
e e
t→
−
= 
(c) 
2 5
0
lim
t t
t
e et→
−
= 
(d) 
3 6
0
lim
t t
t
e e
t→
−
= 
(e) 
2 4
0
lim
t t
t
e e
t
− −
→
−
= 
(f) 
3 7
0
lim
t t
t
e e
t
− −
→
−
= 
27. Sabendo que ( )
1
0
lim 1 x
x
x e
→
+ = calcule os limites abaixo: 
(a) 
2
1
lim 1
n
n n
+
→∞
 + = 
 
 
(b) 
4
1
lim 1
n
n n
+
→∞
 + = 
 
 
(c) 
3
1
lim 1
n
n n
+
→∞
 + = 
 
 
(d) 
5
1
lim 1
n
n n
+
→∞
 + = 
 
 
(e) 
7
1
lim 1
n
n n
+
→∞
 + = 
 
 
(f) 
2
lim 1
t
t t→∞
 + = 
 
 
(g) 
4
lim 1
t
t t→∞
 + = 
 
 
(h) 
5
lim 1
t
t t→∞
 + = 
 
 
(i) 
7
lim 1
t
t t→∞
 + = 
 
 
(j) 
9
lim 1
t
t t→∞
 + = 
 
 
 
 
14 
 
Limites Laterais, Limites Infinitos e 
Limites no Infinito 
 
1. Calcule ( ) ( )lim , lim
x a x a
f x f x
+ −→ →
 e ( )lim
x a
f x
→
caso existam: 
(a) 
 ( ) 5 ; 5f x x a= − = 
(b) ( ) 7 ; 7f x x a= − = 
(c) ( ) 9 ; 9f x x a= − = 
(d) ( ) 10 ; 10f x x a= − = 
(e) ( ) 38 ; 2f x x a= − = 
(f) ( ) 327 ; 3f x x a= − = 
 
 
2. Calcule os limites abaixo: 
( ) ( )
( )
( )
1
1
1
1
lim
lim
x
x
a f x
x
f x
f x
−
+
→
→
=
−
=
=
 
( ) ( )
( )
( )
3
3
2
3
lim
lim
x
x
b f x
x
f x
f x
−
+
→
→
=
−
=
=
 
( ) ( )
( )
( )
5
5
3
5
lim
lim
x
x
c f x
x
f x
f x
−
+
→
→
=
−
=
=
 
( ) ( )
( )
( )
7
7
4
7
lim
lim
x
x
d f x
x
f x
f x
−
+
→
→
=
−
=
=
 
( ) ( )
( )
( )
8
8
5
8
lim
lim
x
x
e f x
x
f x
f x
−
+
→
→
=
−
=
=
 
( ) ( )
( )
( )
2
2
3
2
lim
lim
x
x
f f x
x
f x
f x
−
+
→
→
−
=
−
=
=
 
( ) ( )
( )
( )
3
3
5
3
lim
lim
x
x
g f x
x
f x
f x
−
+
→
→
−
=
−
=
=
 
( ) ( )
( )
( )
4
4
7
4
lim
lim
x
x
h f x
x
f x
f x
−
+
→
→
−
=
−
=
=
 
15 
 
( ) ( )
( )
( )
5
5
8
5
lim
lim
x
x
i f x
x
f x
f x
−
+
→
→
−
=
−
=
=
 
( ) ( )
( )
( )
3
3
4
3
lim
lim
x
x
j f x
x
f x
f x
−
+
→
→
=
−
=
=
 
( ) ( )
( )
( )
10
10
6
10
lim
lim
x
x
l f x
x
f x
f x
−
+
→
→
=
−
=
=
 
 
6. Determine os limites: 
(a) 
3 4
lim
5 2x
x
x→+∞
−
=
+
 
(b) 
3 2
lim
6 3x
x
x→+∞
−
=
+
 
(c) 
9 7
lim
3x
x
x→−∞
− −
=
−
 
(d) 
2
2
5 2
lim
3 4x
x
x→+∞
−
=
+
 
(e) 
4
2 4
3 2
lim
8 5x
x x
x x→−∞
− −
=
+
 
7. Determine os limites: 
(a) 
2
3
5 3
lim
2 4x
x
x→+∞
−
=
+
 
(b) 
3
2 4
4 3
lim
7 5x
x x
x x→−∞
− −
=
+
 
(c) 
2
3
2 7
lim
4 3x
x
x→+∞
−
=
+
 
(d) 
5 2
5 7
3 5
lim
2 3x
x x
x x→−∞
− −
=
−
 
(e) 
3 4
9 5
3 4
lim
5 2x
x x
x x→+∞
−
=
−
 
8. Determine os limites: 
(a) 
6
2 3
3 3 9
lim
6 5 3x
x x
x x x→−∞
− − +
=
+ −
 
(b) 
5
3
10 7 9
lim
5 3 4x
x x
x x→+∞
− −
=
+ +
 
(c) 
7 2
3 5
5 3 2 1
lim
3 4 2x
x x x
x x x→−∞
− − + −
=
− + −
 
(d) 
8 3
4 5 3
4 3 2 7
lim
2 5 4 2x
x x x
x x x→+∞
− + −
=
− + −
 
(e) 
9 3 2
2 4
5 7 6 3
lim
3 2 5x
x x x
x x→−∞
− + −
=
− +
 
 
16 
 
9. Determine os limites: 
(a) 
2
4 3
lim
5 2x
x
x→+∞
+
=
−
 
(b) 
2
3 4
lim
2 5x
x
x→−∞
+
=
−
 
(c) 
2
5 4
lim
6 3x
x
x→+∞
−
=
+
 
(d) 
2
4 5
lim
3 6x
x
x→−∞
+
=
−
 
10. Determine os limites: 
(a) 
23 4
lim
2 5x
x
x→+∞
+
=
−
 
(b) 
25 2
lim
4 3x
x
x→−∞
−
=
+
 
(c) 
23 6
lim
4 5x
x
x→+∞
−
=
−
 
(d) 
26 3
lim
3 7x
x
x→−∞
−
=
+
 
 
 
 
 
11. Determine os limites: 
(a) ( )2lim 9 4 3
x
x x
→+∞
+ − = 
(b) ( )2lim 16 3 4
x
x x
→−∞
+ + = 
(c) ( )4 2lim 3 25 5
x
x x
→+∞
+ − = 
(d) ( )6 3lim 36 2 6
x
x x
→−∞
+ + = 
(e) ( )4 2lim 5 49 7
x
x x
→+∞
+ + = 
Continuidade 
1. Verifique se f é contínua no ponto dado: 
(a) ( )
2 4
, 2
2
x
f x a
x
−
= =
−
 
(b) ( )
2 4
, se 2
, 22
3, se 2
x
x
f x ax
x
 −
≠
= =−
 =
 
(c) ( )
2 9
, 3
3
x
f x a
x
−
= = −
+
 
(d) ( )
2 9
, se 3
3
6, se 3
x
x
f x x
x
 −
≠ −
= +
− = −
 
(e) ( )
2 16
, 4
4
x
f x a
x
−
= =
−
 
17 
 
(f) ( )
2 16
, se 4
4
8, se 4
x
x
f x x
x
 −
≠
= −
− =
 
(g) ( )
225
, 5
5
x
f x a
x
−
= = −
+
 
(h) ( )
225
, se 5
, 55
10, se 5
x
x
f x ax
x
 −
≠ −
= = −+
− = −
 
(i) ( )
236
, 6
6
x
f x a
x
−
= =
−
 
(j) ( )
236
, se 6
 , 66
12, se 6
x
x
f x ax
x
 −
≠
= =−
 =
 
2. Verifique se f é contínua no ponto dado: 
(a) ( ) 3 , 2
2
f x a
x
= = −
+
 
(b) ( ) 5 , 1
1
f x a
x
= =
−
 
(c) ( )
2 3 se 1
, 1
2 se 1
x x
f x a
x
+ ≠
= =
≠
 
(d) ( )
3 se 1
, 1
3 se 1
x x
f x a
x x
+ ≤
= =
− <
 
(e) ( )
2 9
3
, 33
4 3
x
se x
f x ax
se x
 −
≠
= =−
 =
 
(f) ( )
2 9
3
, 33
2 3
x
se x
f x ax
se x
 −
≠ −
= = −+
 = −
 
(g) ( )
1 4
, 4
0 4
se x
f x a
se x
≠
= =
=
 
(h) ( )
3 se 3
, 3
2 se 3
x x
f x a
x
 − ≠
= =
=
 
(i) ( )
5
5
, 55
1 5
x
se x
f x ax
se x
 −
≠
= =−
 =
 
3. Determine todos os números para os quais f é descontínua: 
(k) ( ) 2
3
6
f x
x x
=
+ −
 
(l) ( ) 2
5
4 12
f x
x x
=
− −
 
(m) ( ) 2
1
2
x
f x
x x
−
=
+ −
 
(n) ( ) 2
4
12
x
f x
x x
−
=
− −
 
(o) ( )
2
2
16
16
x
f x
x
−
=
−
 
 
4. Determine todos os números para os quais f é contínua: 
(a) ( ) 2
3 5
2 3
x
f x
x x
−
=
− −
 
(b) ( )
2 9
3
x
f x
x
−
=
−
 
18 
 
(c) ( ) 22 3f x x x= − + 
(d) ( )
2
1
1
x
f x
x
−
=
− 
 
(e) ( )
21
x
f x
x
=
−
 
(f) ( )
9
9
x
f x
x
+
=
+
 
(g) ( ) 2 1
x
f x
x
=
+
 
(h) ( ) 3 2
5
f x
x x
=
−
 
(i) ( )
( )( )2
4 7
3 2 8
x
f x
x x x
−
=
+ + −
 
(j) ( )
2 29 25
4
x x
f x
x
− −
=
−
 
(k) ( ) 9
6
x
f x
x
−
=
−
 
5. Mostre que f é contínua no intervalo dado: 
(a) ( ) [ ]4 ; 4,8f x x= − 
(b) ( ) ( )1 ; 1,3
1
f x
x
=
−
 
(c) ( ) ( )16 ; ,16f x x= − −∞ 
(d) ( ) ( )2
1
; 0,f x
x
= ∞ 
(e) ( ) ( )2 3 ; 2,
2
x
f x
x
+
= ∞
−
 
(f) ( ) ( )2 3 ; ,3f x x= − −∞ 
6. Explique por que a função é descontínua no número dado a. 
(a) ( ) ln 2 ; 2f x x a= − = 
(b) ( )
1
se 1
; 11
2 se 1
x
f x ax
x
 ≠
= =−
 =
 
(c) ( ) 2
se 0
; 0
se 0
xe x
f x a
x x <
= =
≥
 
(d) ( )
2
2
se 1
; 11
1 se 1
x x
x
f x ax
x
 −
≠
= =−
 =
 
 
(e) ( )
22 5 3
se 3
; 33
6 se 3
x x
x
f x ax
x
 − −
≠
= =−
 =
 
7. Mostre que f é contínua em toda a reta. 
(a) ( )
2 se 1
 s e 1
x x
f x
x x
 <
= 
≥
 
(b) ( )
sen se
4
cos se
4
x x
f x
x x
π
π
 <
= 
 ≥

 
8. Encontre os pontos nos quais f é descontínua. 
(a) ( )
( )
2
2
1 se 0
2 se 0 2
2 se 2
x x
f x x x
x x
 + ≤

= − < ≤

− >
 
 
19 
 
(b) ( )
1 se 1
1
 se 1 3
3 se 3
x x
f x x
x
x x
 + ≤


= < <

 − ≥
 
 
9. Para quais valores da constante c a função f é contínua em toda a reta? 
 
(a) ( )
2
3
2 se 2
- s e 2
cx x x
f x
x cx x
 + <
= 
≥
 
(b) ( )
2
2
3 se 3
4 - s e 3
cx x x
f x
x cx x
 − <
= 
≥
 
(c) ( )
3
2
4 7 se 1
2 -5 s e 1
cx x x
f x
x cx x
 − ≤
= 
>
 
 
10. Investigue a continuidade das funções abaixo nos pontos indicados. 
 
(a) ( )
sen
 se 0
em 0.
0 s e 2
x
x
f x xx
x
 ≠
= =
 =
 
(b) ( )
3
2
8
 se 2
em 2.4
3 s e 3
x
x
f x xx
x
 −
≠
= =−
 =
 
(c) ( )
2 1 se 0
0 s e 0
x sen x
f x x
x
   ≠  =  
 =
 
1 
 
DERIVADA 
 
Definição. A derivada de uma função f é a função denotada por f ′ , 
tal que seu valor em qualquer número x do domínio de f seja dado por
( ) ( ) ( )
0
lim
x
f x x f x
f x
x∆ →
+∆ −
′ =
∆
 se esse limite existir. 
Quando ( )y f x= , isto é, y é uma função de x, escrevemos também
dy
dx
para indicar a derivada de y com respeiro a x. 
 
1. Regras de Derivação 
 
1.1. Derivada de uma função constante 
Teorema. Se ( ) ,f x c c= ∈ℜ então 
( ) 0 , .f x x′ = ∀ ∈ℜ 
Exercício: Calcule ( )f x′ . 
(a) ( ) 2f x = 
(b) ( ) 3
5
f x
−
= 
(c) ( ) 3 7f x = 
 
1.2. Derivadas de nx 
 Teorema. Seja n um número racional. Então 
( ) ( ) 1n nf x x f x nx −′= ⇒ =
 
 Teorema. Se f for uma função, c uma constante e g a função 
definida por ( ) ( )g x c f x= ⋅ , então, se ( )f x′ existir, tem-se 
( ) ( )g x c f x′ ′= ⋅
. 
EXERCÍCIOS 
1. Calcule ( )f x′ . 
(a) ( ) 4f x x= 
(b) ( ) 6f x x= 
(c) ( ) 100f x x= 
(d) ( ) 2f x x= 
(e) ( )f x x= 
(f) ( ) 32f x x= 
(g) ( ) 74f x x= 
(h) ( ) 405f x x= 
(i) ( ) 206f x x= 
(j) ( ) 87f x x= 
 
2 
 
2. Calcule ( )f x′ . 
(a) ( )
3
1
f x
x
= 
(b) ( ) 5
1
f x
x
= 
(c) ( ) 7
1
f x
x
= 
(d) ( ) 4
1
f x
x
= 
(e) ( ) 1f x
x
= 
(f) ( )
2
8
f x
x
= 
(g) ( ) 4
3
f x
x
−
= 
(h) ( ) 8
6
f x
x
= 
(i) ( ) 5
2
f x
x
−
= 
(j) ( ) 9
2
f x
x
= 
 
2. Calcule ( )f x′ . 
(a) ( )f x x= 
(b) ( ) 4f x x= 
(c) ( ) 6f x x= 
(d) ( ) 8f x x= 
(e) ( ) 9f x x= 
(f) ( ) 32f x x= 
(g) ( ) 6 73f x x= 
(h) ( ) 8 54f x x= 
(i) ( ) 9 45f x x= 
(j) ( ) 5123f x x= 
3 
 
2. Calcule ( )f x′ . 
(a) ( )
3
1
f x
x
= 
(b) ( )
54
1
f x
x
= 
(c) ( )
6 7
1
f x
x
= 
(d) ( )
8 5
1
f x
x
= 
(e) ( )
9 4
1
f x
x
= 
(f) ( )
5
2
f x
x
= 
(g) ( )
74
3
f x
x
−
= 
(h) ( )
7 6
4
f x
x
= 
(i) ( )
6 5
5
f x
x
−
= 
(j) ( )
8 3
6
f x
x
= 
Teorema. Sejam f e g deriváveis em p e seja k uma constante. 
Então as funções f+g, kf e fg são deriváveis em p e têm-se: 
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) 
( ) 
( )
i f g p f p g p
ii kf p kf p
iii f g p f p g p f p g p
′ ′ ′+ = +
′ ′=
′ ′ ′⋅ = + ⋅
 
 
EXERCÍCIOS 
1. Calcule ( )f x′ . 
(a) ( ) 4 3 25 6 7 8 3f x x x x x= − + + − 
(b) ( ) 5 4 3 26 7 8 9 4 1f x x x x x x= − + − + − + 
(c) ( ) 6 4 5 37 8 9 10 11 2f x x x x x x= − + − + − + 
(d) ( ) 3 4 5 6 58 9 10 11 12 3f x x x x x x= − + − + − + 
(e) ( ) 2 3 4 5 69 8 7 6 5 4f x x x x x x x= − + − + − + 
(f) ( ) 5 2 3 43 6 5 8 7 2f x x x x x x= − + − + + − 
(g) ( ) 2 3 4 57 6 9 8 4f x x x x x x= − + − + − 
(h) ( ) 4 7 3 5 26 7 10 9 15f x x x x x x x= − + − + − + 
(i) ( ) 7 6 5 4 39 8 11 12 16 3f x x x x x x x= − + − + − 
(j) ( ) 9 8 7 6 5 48 9 6 7 4 5f x x x x x x x= − + − + − + 
 
4 
 
2. Calcule ( )f x′ . 
 
(a) ( ) 3 4 5 6 72 3 4 5 6 4 3
5 2 3 2 5 3 2
f x x x x x x x= − + − + − + + 
(b) ( ) 4 5 6 5 4 35 2 3 2 5 3 2
2 3 4 5 6 4 3
f x x x x x x x= − + − + − + + 
(c) ( ) 5 6 7 8 9 103 4 5 6 7 8 4
2 3 4 5 6 4 3
f x x x x x x x= − − − − − − − 
(d) ( ) 3 4 5 6 7 84 3 8 7 5 3 9
5 6 7 8 9 10 4
f x x x x x x x x= − + − + − + 
(e) ( ) 4 5 6 7 8 96 3 9 4 6 3 5
5 4 8 7 5 7 3
f x x x x x x x x= − + − + − + 
(f) ( ) 3 4 5 6 7
3 5 4 7 8 3 2
5 2 3 2 5 3 3
f x
x x x x x x
= − + − + − + + 
(g) ( ) 4 5 6 5 4 3
5 4 7 3 9 5 4
2 3 4 5 6 4 3
f x
x x x x x x
= − + − + − + + 
(h) ( ) 5 6 7 8 9 10
5 7 3 8 4 6 10
2 3 4 5 6 4 13
f x
x x x x x x
= − − − − − − − 
(i) ( ) 3 4 5 6 7 8
3 4 9 5 7 3 5 3
5 6 7 8 9 5 6 5
f x
x x x x x x x
= − + − + − + −
.
 
(j) ( ) 4 5 6 7 8 9
5 6 7 9 3 4 2 1
3 7 8 5 4 9 3 2
f x
x x x x x x x
= + − + − + − + 
3. Calcule ( )f x′ . 
(a) ( ) 33 2 34f x x x x= + − 
(b) ( ) 6 7 87 8 7f x x x x= − + 
(c) ( ) 8 7 65 4 32 3 4f x x x x= + − 
(d) ( ) 9 8 74 3 25 6 7f x x x x= − + 
(e) ( ) 7 9 55 7 38 9 10f x x x x= − + 
(f) ( )
33 2 34
2 3 4
f x
x x x
= + − 
(g) ( )
6 7 87 8 7
5 6 7
f x
x x x
= − + 
(h) ( )
8 7 65 4 3
3 4 5
2 3 4
f x
x x x
= + − 
(i) ( )
9 8 74 3 2
7 5 6
5 6 7
f x
x x x
= − + 
(j) ( )
9 8 75 7 3
3 4 5
7 6 5
f x
x x x
= − + 
5 
 
4. Calcule ( )f x′ . 
(a) ( ) ( ) ( )4 3 26 5 3 7 8f x x x x x= − ⋅ + − 
(b) ( ) ( ) ( )5 4 3 27 8 6 9 4f x x x x x x= − + − ⋅ − + 
(c) ( ) ( ) ( )6 4 5 38 9 7 2 3 4f x x x x x x= − + − ⋅ − + 
(d) ( ) ( ) ( )3 4 5 6 59 10 8 12 3 15f x x x x x x= − + − ⋅ − + 
(e) ( ) ( ) ( )2 3 4 5 68 7 9 5 4 3f x x x x x x x= − + − ⋅ − + 
(f) ( ) 3 4 5 62 3 4 5
5 2 3 2
f x x x x x
  = − + − +  
  
 
(g) ( ) 4 5 6 55 2 3 2
2 3 4 5
f x x x x x
  = − + − +  
  
 
(h) ( ) 5 6 7 82 3 4 5
3 4 5 6
f x x x x x
  = − − − −  
  
 
(i) ( ) 6 5 6 44 3 3 7
3 2 5 6
f x x x x x
  = − + − −  
  
 
(j) ( ) 8 6 7 55 4 4 8
4 3 7 9
f x x x x x
  = − + − −  
  
 
Teorema (regra do quociente). Se f e g forem deriváveis em p e se 
g(p) ≠ 0, então f/g será derivável em p e 
( ) ( ) ( ) ( ) ( )
( )
2
f p g p f p g pf
p
g g p
′ ′ ′− 
= 
     
EXERCÍCIOS 
1. Calcule ( )f x′ . 
(a) ( ) 1
2 3
f x
x
=
+
 
(b) ( ) 2
3 4
f x
x
=
−
 
(c) ( ) 3
3 4
f x
x
−
=
+
 
(d) ( ) 4
4 5
f x
x
=
−
 
(e) ( ) 5
5 6
f x
x
−
=
+
 
(f) ( ) 3
2
f x
x
=
− +
 
(g) ( ) 4
2 3
f x
x
=
−
 
(h) ( ) 2
5 7
f x
x
−
=
+
 
(i) ( ) 5
3 2
f x
x
−
=
−
 
(j) ( ) 6
7 1
f x
x
−
=
+
 
6 
 
2. Calcule ( )f x′ . 
(a) ( ) 3 2
4 6
x
f x
x
−
=
+
 
(b) ( ) 2 5
4 3
x
f x
x
+
=
−
 
(c) ( ) 4 6
5 7
x
f x
x
− −
=
+
 
(d) ( ) 5 3
3 2
x
f x
x
−
=
−
 
(e) ( ) 6 9
3 4
x
f x
x
− +
=
+(f) ( ) 4 1
3 5
x
f x
x
−
=
+
 
(g) ( ) 3 5
2 1
x
f x
x
−
=
−
 
(h) ( ) 6 4
7 5
x
f x
x
− −
=
+
 
(i) ( ) 3 5
4
x
f x
x
− −
=
−
 
(j) ( ) 9 6
4 3
x
f x
x
− +
=
+
 
 
3. Calcule ( )f x′ . 
(a) ( )
2 3 2
5 7
x x
f x
x
+ −
=
+
 
(b) ( )
23 5 4
3 4
x x
f x
x
− +
=
−
 
(c) ( )
25 4 6
7 5
x x
f x
x
− −
=
+
 
(d) ( )
22 3 5
2 3
x x
f x
x
+ −
=
−
 
(e) ( )
24 9 6
4 4
x x
f x
x
− +
=
+
 
(f) ( )
2
2
3 2 1
6 7 5
x x
f x
x x
+ −
=
− +
 
(g) ( )
2
2
4 3 5
5 4 3
x x
f x
x x
− +
=
+ −
 
(h) ( )
2
2
6 5 4
6 5 7
x x
f x
x x
+ −
=
− +
 
(i) ( )
2
2
5 2 3
4 3 2
x x
f x
x x
+ −
=
− +
 
(j) ( )
2
2
5 8 7
7 3 4
x x
f x
x x
− +
=
+ −
 
7 
 
Derivadas de algumas funções especiais 
( ) xf x e= ( ) xf x e′ = 
( ) lnf x x= ( ) 1f x
x
′ = 
( ) senf x x= ( ) cosf x x′ = 
( ) cosf x x= ( ) senf x x′ = − 
( ) tgf x x= ( ) 2secf x x′ = 
( ) secf x x= ( ) sec tgf x x x′ = 
 
EXERCÍCIOS 
1. Calcule ( )f x′ . 
(a) ( ) 23 xf x x e= ⋅ 
(b) ( ) 34 lnf x x x= ⋅ 
(c) ( ) 45 senf x x x= ⋅ 
(d) ( ) 56 cosf x x x= 
(e) ( ) lnxf x e x= 
(f) ( ) senxf x e x= 
(g) ( ) cosxf x e x= 
(h) ( ) ln senf x x x= 
(i) ( ) ln cosf x x x= 
(j) ( ) cos senf x x x= 
2. Calcule ( )f x′ . 
(a) 
( )
25
x
x
f x
e
=
 
(b) ( )
43
ln 1
x
f x
x
=
+
 
(c) ( )
54
sen 1
x
f x
x
=
−
 
(d) ( )
45
cos 1
x
f x
x
=
+
 
(e) ( )
ln
xe
f x
x x
=
−
 
(f) ( ) 1
sen
xe
f x
x x
−
=
+
 
(g) ( )
cos
xe x
f x
x x
+
=
−
 
(h) ( ) ln 2
sen
x
f x
x x
−
=
+
 
(i) ( ) ln
cos
x x
f x
x x
+
=
−
 
(j) ( ) 1 cos
1 sen
x
f x
x
−
=
+
 
8 
 
3. Seja ( ) xf x a= , onde 0, 1a a> ≠ . Sabendo que ( ) lnxf x a a′ =
calcule ( )f x′ nos casos abaixo. 
(a) ( ) 2xf x = 
(b) ( ) 3xf x = 
(c) ( ) 4xf x = 
(d) ( ) 5xf x = 
(e) ( ) 6xf x = 
(f) ( ) 3
2
x
f x
 =  
 
 
(g) ( ) 4
5
x
f x
 =  
 
 
(h) ( ) 5
3
x
f x
 =  
 
 
(i) ( ) 6
7
x
f x
 =  
 
 
(j) ( ) xf x e= 
4. Seja ( ) logaf x x= , onde 0, 1a a> ≠ . Sabendo que 
( ) 1
ln
f x
x a
′ = calcule ( )f x′ nos casos abaixo. 
(a) ( ) 2logf x x= 
(b) ( ) 3logf x x= 
(c) ( ) 4logf x x= 
(d) ( ) 5logf x x= 
(e) ( ) 6logf x x= 
(f) ( ) 7logf x x= 
(g) ( ) 8logf x x= 
(h) ( ) 9logf x x= 
(i) ( ) 10logf x x= 
(j) ( ) lnf x x= 
 
9 
 
{ { 
dy dx
dx dt
dy
dt
dy dy dx
y x t
dt dx dt
→ → ⇒ = ⋅
14243
5. Calcule ( )f x′ . 
(a) ( )
2
6
log
x
f x
x
= 
(b) ( )
4
7
log
x
f x
x
= 
(c) ( )
3
8
log
x
f x
x
= 
(d) ( )
6
9
log
x
f x
x
= 
(e) ( )
7
10
log
x
f x
x
= 
2. Regra da Cadeia 
 
Sejam y=f(x) e x=g(t) duas funções deriváveis, com Im(g)⊂D(f). 
Temos que a composta h(t)=f(g(t)) é derivável e vale a regra da 
cadeia ( ) ( )( ) ( ).h t f g t g t′ ′ ′= ⋅
 
Na notação de Leibniz, a regra da cadeia se escreve: 
dy dy dx
dt dx dt
= ⋅
 
Uma maneira de fixar a regra da cadeia é fazer o seguinte 
diagrama: 
 
 
 
 
EXERCÍCIOS 
 
1. Calcule ( )f x′ . 
(a) ( ) ( )422f x x= + 
(b) ( ) ( )163f x x= − 
(c) ( ) ( )1004f x x= + 
(d) ( ) ( )156f x x= + 
(e) ( ) ( )139f x x= − 
(f) ( ) ( )2423 4f x x x= − 
(g) ( ) ( )153 24 5f x x x= − − 
(h) ( ) ( )104 35 4f x x x= − 
(i) ( ) ( )55 47 8f x x x= − 
(j) ( ) ( )67 36 5f x x x= − 
 
10 
 
2. Calcule ( )f x′ . 
 
(a) ( )
( )5
1
7
f x
x
=
−
 
(b) ( )
( )7
3
8
f x
x
=
+
 
(c) ( )
( )4
5
9
f x
x
=
−
 
(d) ( )
( )6
2
10
f x
x
−
=
+
 
(e) ( )
( )7
4
14
f x
x
=
−
 
(f) ( )
( )4
3
2 7
f x
x
=
−
 
(g) ( )
( )6
4
3 5
f x
x
=
− +
 
(h) ( )
( )5
3
4 2
f x
x
=
−
 
(i) ( )
( )7
4
2 10
f x
x
−
=
− +
 
(j) ( )
( )9
5
2 4
f x
x
=
−
 
3. Calcule ( )f x′ . 
 
(a) ( ) 4 3 2f x x= − 
(b) ( ) 6 2 3f x x= + 
(c) ( ) 8 5 4f x x= − 
(d) ( ) 9 4 7f x x= + 
(e) ( ) 10 9 6f x x= − 
(f) ( ) ( )32 1f x x= + 
(g) ( ) ( )76 3 1f x x= − 
(h) ( ) ( )58 4 2f x x= + 
(i) ( ) ( )49 5 3f x x= − 
(j) ( ) ( )610 3 5f x x= − 
 
11 
 
4. Calcule ( )f x′ . 
 
(a) ( )
( )3
1
6 4
f x
x
=
−
 
(b) ( )
( )54
1
7 5
f x
x
=
+
 
(c) ( )
( )76
1
8 6
f x
x
=
−
 
(d) ( )
( )34
3
2 1
f x
x
=
−
 
(e) ( )
( )45
2
3 2
f x
x
−
=
+
 
5. Calcule ( )f x′ . 
(a) ( ) 2xf x e−= 
(b) ( ) 3xf x e= 
(c) ( ) 4xf x e−= 
(d) ( ) 6xf x e= 
(e) ( ) 8xf x e−= 
(f) ( )
4
3
xe
f x
−
= 
(g) ( )
4
5
xe
f x = 
(h) ( )
5
6
xe
f x
−
= 
(i) ( )
3
7
xe
f x = 
(j) ( )
8
4
xe
f x
−
= 
(k) ( ) 3 5xf x e= 
(l) ( ) 7 6xf x e−= 
(m) ( ) 8 5xf x e−= 
(n) ( ) 9 7xf x e−= 
(o) ( ) 10 8xf x e−= 
12 
 
6. Calcule ( )f x′ . 
 
(a) ( ) ( )ln 2 1f x x= + 
(b) ( ) ( )ln 3 2f x x= − 
(c) ( ) ( )ln 2 5f x x= − 
(d) ( ) ( )ln 5 7f x x= − 
(e) ( ) ( )ln 7 1f x x= − 
(f) ( ) ( )3ln 7 3f x x= − 
(g) ( ) ( )4ln 5 2f x x= − 
(h) ( ) ( )5ln 6 1f x x= − 
(i) ( ) ( )6ln 3 12f x x= − 
(j) ( ) ( )27ln 4 10f x x= − 
 
7. Calcule ( )f x′ . 
 
(a) ( ) ( )sen 3 2f x x= + 
(b) ( ) ( )cos 7 3f x x= − 
(c) ( ) ( )sen 5 3f x x= − 
(d) ( ) 6cos 2
5
x
f x
 = − 
 
 
(e) ( ) 5sen 3
2
x
f x
 = + 
 
 
 
8. Calcule ( )f x′ . 
 
(a) ( ) ( )sen xf x e= 
(b) ( ) ( )cos xf x e= 
(c) ( ) ( )sen lnf x x= 
(d) ( ) ( )cos lnf x x= 
(e) ( ) ( )sen cosf x x= 
13 
 
9. Calcule ( )f x′ . 
(a) ( ) sen xf x e= 
(b) ( ) cos xf x e= 
(c) ( ) nl xf x e= 
(d) ( ) ( )ln cosf x x= 
(e) ( ) ( )ln senf x x= 
(f) ( ) ( )cos lnf x x= 
(g) ( ) ( )sen lnf x x= 
 
10. Calcule ( )f x′ . 
(a) ( ) ( )2sen 2f x x= 
(b) ( ) ( )3cos 3f x x= 
(c) ( ) ( )4sen 5f x x= 
(d) ( ) 5 5cos
3
x
f x
 =  
 
 
(e) ( ) 6 2sen
3
x
f x
 =  
 
 
(f) ( ) 7 3cos
5
x
f x
 =  
 
 
(g) ( ) 8 3sen
7
x
f x
 =  
 
 
(h) ( ) 8 9cos
4
x
f x
 =  
 
 
(i) ( ) 6 3sen
2
x
f x
 =  
 
 
(j) ( ) 7 8cos
3
x
f x
 =  
 
 
11. Calcule ( )f x′ . 
(a) ( ) ( ) ( )2 42 34 7 2 1f x x x= + + 
(b) ( ) ( ) ( )3 223 5 3 1f x x x= + − 
(c) ( ) ( ) ( )2 12 2 24 1f x x x x −−= − + 
(d) ( ) ( ) ( )1 22 5 4 3f x x x− −= − + 
(e) ( ) ( ) ( )32 32 9 4 5f x x x x= − + − 
 
14 
 
12. Calcule ( )f x′ . 
(a) ( )
2
2 3
6 4
x
f x
x
− =  + 
 
(b) ( )
3
5 2
3 4
x
f x
x
+ =  − 
 
(c) ( )
4
6 4
7 5
x
f x
x
− − =  + 
 
(d) ( )
5
3 5
2 3
x
f x
x
− =  − 
 
(e) ( )
6
2 5
4
x
f x
x
− + =  + 
 
 
13. Calcule ( )f x′ . 
(a) ( )
2
3
3
4
x
f x
x
− =  + 
 
(b) ( )
3
4
4 1
3 2
x
f x
x
− =  − 
 
(c) ( )
4
5
1
5
x
f x
x
− − =  + 
 
(d) ( )5
7
5 3
4 1
x
f x
x
− =  − 
 
(e) ( )
6
8
3 4
2 5
x
f x
x
− + =  + 
 
14. Calcule ( )f x′ . 
(a) ( ) ( )ln 3 2f x x= + 
(b) ( ) ( )3 ln 4 3f x x= − 
(c) ( ) ( )4 ln 5 2f x x= + 
(d) ( ) ( )5 ln 6 3f x x= − − 
(e) ( ) ( )6 ln 4 1f x x= − + 
(f) ( ) ( )4 sen 7 3f x x= + 
(g) ( ) ( )5 cos 3 7f x x= − 
(h) ( ) ( )6 sen 3 7f x x= − 
(i) ( ) ( )7 cos 5 9f x x= − 
(j) ( ) ( )sen xf x e= 
 
 
15 
 
3. Derivação Implícita 
 
Seja ( ), 0F x y = uma equação nas variáveis x e y. 
Definição. A função ( )y f x= é definida implicitamente pela 
equação ( ), 0F x y = ,quando ( )( ), 0F x f x = . 
Em outras palavras, quando ( )y f x= satisfaz a equação
( ), 0F x y = . 
Exemplo. Seja a equação ( ), 0F x y = ,onde ( ) 3, 1F x y x y= + − ; 
a função
31y x= − é definida implicitamente pela equação
( ), 0F x y = , pois ( )( ) ( )3 3, 1 1 0F x f x x x= + − − = . 
 
Método do Cálculo da derivação implícita 
 
Dada uma equação que define y implicitamente com uma função 
derivável de x, calcula-se y′ do seguinte modo: Deriva-se ambos 
os lados da equação em relação a x, termo a termo. Ao fazê-lo, 
tenha em mente que y é uma função de x e use a regar da cadeia, 
quando necessário, para derivar as expressões nas quais figure y. O 
resultado será uma equação ode figura não somente x e y, mas 
também y′ . Expresse y′em função de x e y. Tal processo é 
chamado explicitar y’. 
 
 
 
EXERCÍCIOS 
 
1. Admitindo que a equação determine implicitamente uma função 
diferenciável ( )y f x= , calcule ( )f x′ : 
(a) 
2 2 4x y− = 
(b) 
3 2 4y x y x+ = + 
(c) 
2 2 3xy y+ = 
(d) 
5y y x+ = 
(e) 
2 24 5x y+ = 
(f) 
3xy y x+ = 
(g) 
2 2 2 0x y y+ + = 
(h) 
2 3 2x y xy+ = 
(i) 
2 28 10x y+ = 
(j) 
3 34 2x y x− = 5y y x+ = 
(k) 
3 2 32 1x x y y+ + = 
16 
 
(l) 
2 2 25 2 8x x y y+ + = 
(m) 
2 25 4 0x xy y− − = 
(n) 
4 2 2 34 3 2 0x x y xy x+ − + = 
(o) 
3 2 32 1 0x x y y− + − = 
(p) 
4 2 33 4 4 0y x x y+ − − = 
(q) 
2 23 4 4x y+ = 
(r) 
2 2 33 4 12x x y y− + = 
(s) 
4 33 4 5 1y y x x+ − = + 
(t) 
3 2 34 5 6 0xy x y x x− + − + = 
(u) 
4 3 3 2 4 33 5 4 2 0x y x y x y xy+ − + = 
(v) 
5 4 4 3 5 4 2 24 6 5 3 0x y x y x y x y+ − + = 
(w) 
4 5 3 4 5 6 3 25 7 9 11 0x y x y x y x y+ − + = 
(x) 
3 4 5 6 7 8 9 102 3 4 5 0x y x y x y x y+ − + = 
2. Admitindo que a equação determine implicitamente uma função 
diferenciável ( )y f x= , calcule ( )f x′ : 
(a) 7x y+ = 
(b) 4
x
y x
y
− = 
(c) 
23 3 4x xy y+ = 
(d) 3 4y y y x+ + = 
(e) 2xy x y+ = 
 
3. Admitindo que a equação determine implicitamente uma função 
diferenciável ( )y f x= , calcule ( )f x′ : 
(a) ( )cosy x y= − 
(b) ( )seny x y= + 
(c) ( )cos 2 3y x y= − 
(d) ( )sen 4 5y x y= + 
(e) ( )cos 7 9y x y= − 
17 
 
4. Admitindo que a equação determine implicitamente uma função 
diferenciável ( )y f x= , calcule ( )f x′ : 
 
(a) ( )ln 2 4x x y= + 
(b) ( )ln 3 5y x y= − 
(c) ( )ln 9 7x x y= − 
(d) ( )ln 3 2y x y= − + 
(e) ( )ln 5x x y= − + 
(f) 
( )5 3x y
y e
−= 
(g) 
( )3 4x y
x e
+= 
(h) 
( )6 4x y
y e
−= 
(i) 
( )3 7x y
x e
+= 
(j) 
( )5 3x y
y e
−= 
 
 
4. Equação da Reta tangente 
 
A função { }( )0: , domínio de F D x D f− →ℜ = ,definida por
( ) ( ) ( )0
0
f x f x
F x
x x
−
=
−
, representa, geometricamente, o 
coeficiente angular da reta secante ao gráfico de f passando 
pelos pontos ( )( ) ( )( )0 0, e ,x f x x f x . Logo, quando f é 
derivável no ponto 0x , a reta de coeficiente angular ( )0f x′ e 
passando pelo ponto ( )( )0 0,x f x é a reta tangente ao gráfico 
de f no ponto ( )( )0 0,x f x . Se f admite derivada no ponto 0x , 
então, a equação da reta tangente ao gráfico de f no ponto
( )( )0 0,x f x é: ( ) ( )( )0 0 0y f x f x x x′− = − . A equação da 
reta normal ao gráfico de f no ponto ( )( )0 0,x f x é: 
( )
( )
( ) ( )0 0 0
0
1
; 0.y f x x x f x
f x
−
′− = − ≠
′
 
 
 
 
 
 
 
18 
 
EXERCÍCIOS 
1. Determine a equação da reta tangente em ( )( ),p f p sendo 
dados: 
(a) 
( ) 4 e 1f x x p= =
 
(b) ( ) 2 e 2f x x p= = 
(c) ( ) 3 e 2f x x p−= = − 
(d) ( ) 5
1
e 3f x p
x
= = 
(e) ( ) 6 e 1f x x p= = 
(f) ( )
3
1
e 1f x p
x
= = − 
(g) ( ) 3 42 3 4 5 e 1
5 2 3 2
f x x x x p= − + − + = − 
(h) ( )
8 7 65 4 3
3 4 5
e 1
2 3 4
f x p
x x x
= + − = 
(i) ( ) ( ) ( )4 3 26 5 3 7 8f x x x x x= − ⋅ + − e 1p = 
(j) ( ) 5 6 7 8
7 5 9 3
2 3 4 5
f x
x x x x
   = − − − −  
   
e 1p = 
 
 
2. Determine a equação da reta tangente em ( )( ),p f p sendo dados: 
 
(a) 
 
( ) 23 e 1xf x x e p= ⋅ = 
(b) ( ) 34 ln ef x x x p e= ⋅ = 
(c) ( ) 45 sen e
2
f x x x p
π
= ⋅ = 
(d) ( ) 56 cos e
2
f x x x p
π
= = 
(e) ( ) ln e 1xf x e x p= = 
(f) ( ) sen exf x e x p π= = 
(g) ( ) cos exf x e x p π= = 
(h) ( ) ln sen ef x x x p e= = 
(i) ( ) ln cos ef x x x p π= = 
(j) ( ) cos senf x x x= e
4
p
π
= 
 
19 
 
3. Determine a equação da reta tangente em ( )( ),p f p sendo dados: 
(a) ( ) 1 e 0
4 3
f x p
x
= =
+
 
(b) ( ) 2 e 1
3 5
f x p
x
= =
−
 
(c) ( ) 3 e 0
2 4
f x p
x
−
= =
+
 
(d) ( ) 1 e 1
4 3
f x p
x
= = −
−
 
(e) ( ) 5 e 0
2 3
f x p
x
−
= =
+
 
(f) ( ) 3 2 e 1
4 6
x
f x p
x
−
= =
+
 
(g) ( ) 2 5 e 1
4 3
x
f x p
x
+
= = −
−
 
(h) ( ) 4 6 e 0
5 7
x
f x p
x
− −
= =
+
 
(i) ( ) 5 3 e 1
3 2
x
f x p
x
−
= =
−
 
(j) ( ) 6 9 e 1
3 4
x
f x p
x
− +
= = −
+
 
 
4. Determine a equação da reta tangente em ( )( ),p f p sendo dados: 
(a) ( )
23
e 0
x
x
f x p
e
= = 
(b) ( )
42
e 1
ln 1
x
f x p
x
= =
+
 
(c) ( )
54
e
sen 1
x
f x p
x
π= =
−
 
(d) ( )
23
e
cos 1
x
f x p
x
π= =
+
 
(e) ( ) e 1
ln
xe
f x p
x x
= =
−
 
(f) ( ) 1 e
sen 2
xe
f x p
x x
π−
= =
+
 
(g) ( ) e 0
cos
xe x
f x p
x x
+
= =
−
 
(h) ( ) ln e 1
sen
x
f x p
x x
= =
+
 
(i) ( ) ln e 1
cos
x x
f x p
x x
+
= =
−
 
(j) ( ) 1 cos e 1
1 sen
x
f x p
x
−
= =
+
 
 
20 
 
5. Taxa de Variação
 
A velocidade de uma partícula que se move ao longo do gráfico da 
função derivável u=u(t) no tempo t é v(t)=u’(t) e representa a razão 
do deslocamento por unidade de variação de tempo. u’(t) expressa a 
taxa de variação de u(t) por unidade de tempo 
( ) ( ) ( )
0
lim
h
u t h u t
u t
h→
+ −
′ = . 
( )y f x= ( )dy f x
dx
′=Se é uma função derivável, então é a taxa de 
variação de y em relação a x. 
 
EXERCÍCIOS 
 
1. Admita que todas as variáveis sejam funções de t: 
(a) Se 2 e 3
dx
A x
dt
= = quando 10x = determine 
dA
dt
. 
(b) Se 3 e 2
dz
S z
dt
= = − quando 3z = determine 
dS
dt
. 
(c) Se
3
25 e 4
dV
V p
dt
= − = − quando 40V = − determine 
dp
dt
. 
(d) Se
3
e 5
dP
P
w dt
= = quando 9p = determine 
dw
dt
. 
(e) Se 2 23 2 10 e 2
dx
x y y
dt
+ + = = quando 3 e 1x y= = − 
determine 
dy
dt
. 
(f) Se 3 22 4 10 e 3
dy
y x x
dt
− + = − = − quando 2e 1x y= − = 
determine 
dx
dt
. 
(g) Se 23 2 32 e 4
dy
x y x
dt
+ = − = − quando 2 e 3x y= = − 
determine 
dx
dt
. 
(h) Se 
2 2 4 44 e 5
dx
x y y
dt
− − = − = quando 3 e 2x y= − = 
determine 
dydt
 
2. Ao ser aquecida uma chapa circular de metal, seu diâmetro varia à 
razão de 0,01 cm/min. Determine a taxa à qual a área de uma das faces 
varia quando o diâmetro está em 30 cm. 
 
3. Um incêndio em um campo aberto se alastra em forma de círculo. O 
raio do círculo aumenta à razão de 1 m/min. Determine a taxa à qual a 
área incendiada está aumentando quando o raio é de 20 m. 
 
4. Gás está sendo bombeado para um balão esférico à razão de 0,1 
m³/min. Ache a taxa de variação do raio quando este é de 0,45 m. 
 
 
21 
 
5. Suponha que uma bola de neve (esférica) esteja se derretendo, com o 
raio decrescendo à razão constante, passando de 30 cm para 20 cm em 
45 minutos. Qual a variação do volume quando o raio está com 25 cm? 
 
6. Uma escada de 6 m de comprimento está apoiada em uma parede 
vertical. Se a base da escada começa a deslizar horizontalmente à razão 
de 1 m/s, com que velocidade o topo da escada percorre a parede, 
quando está a 2,5 m acima do solo? 
 
 
7. Uma luz está no alto de um poste de 5m. Um menino de 1,6 m se afasta 
do poste à razão de 1,2 m/s. A que taxa se move a ponta de sua sombra 
quando ele está a 6m do poste? A que taxa aumenta o comprimento de 
sua sombra? 
 
6. Estudo da Variação das Funções 
 
6.1. Intervalo de Crescimento e de Decrescimento – Pontos Críticos 
Teorema: Seja f contínua no intervalo I. 
- Se ( ) 0f x′ > para todo x interior a I, então f será estritamente 
crescente em I. 
- Se ( ) 0f x′ < para todo x interior a I, então f será estritamente 
decrescente em I. 
 
EXERCÍCIOS 
1. Determine os intervalos de crescimento e de decrescimento das 
funções abaixo: 
 
(a) ( ) 3 22 2f x x x x= − + + 
(b) ( ) 3 28 30 24 10f x x x x= + + + 
(c) ( ) 3 23 1f x x x= − + 
(d) ( ) 3 22 1f x x x x= + + + 
(e) ( ) 4 3 24 4 2f x x x x= − + − + 
(f) ( ) 23 2 5f x x x= + − 
(g) ( ) 3 22 1f x x x x= − + + 
(h) ( ) 3 210 25 50f x x x x= + + − 
(i) ( ) 4 33 4 6f x x x= − + 
(j) ( ) 2 48 2f x x x= − 
(k) ( ) 1f x x
x
= + 
(l) ( ) 2 1f x x
x
= + 
22 
 
(m) ( ) 2
1
f x x
x
= + 
(n) ( )
2
2 1
x
f x
x
=
−
 
(o) ( )
2
21 3
x x
f x
x
−
=
+
 
(p) ( )
21
x
f x
x
=
+
 
(q) ( )
2
21
x
f x
x
=
+
 
(r) ( ) 2 xf x e−= − 
(s) ( )
2xf x e−= 
(t) ( ) 2x xf x e e= − 
(u) ( ) xf x xe= 
(v) ( )
xe
f x
x
= 
(w) ( ) ln xf x
x
= 
(x) ( ) xf x x e= − 
 
 
6.2. Concavidade e Ponto de Inflexão 
Teorema: Seja f uma função que admite derivada até segunda 
ordem no intervalo aberto I: 
- Se ( ) 0f x′′ > em I, então f terá concavidade para cima em I. 
- Se ( ) 0f x′′ < em I, então f terá concavidade para baixo em I. 
EXERCÍCIOS 
1. Determine os intervalos em que o gráfico de f é côncavo para cima 
ou côncavo para baixo: 
(a) ( ) 3 22 2f x x x x= − + + 
(b) ( ) 3 28 30 24 10f x x x x= + + + 
(c) ( ) 3 23 1f x x x= − + 
(d) ( ) 3 22 1f x x x x= + + + 
(e) ( ) 4 3 24 4 2f x x x x= − + − + 
(f) ( ) 23 2 5f x x x= + − 
(g) ( ) 3 22 1f x x x x= − + + 
(h) ( ) 3 210 25 50f x x x x= + + − 
(i) ( ) 4 33 4 6f x x x= − + 
23 
 
(j) ( ) 2 48 2f x x x= − 
(k) ( ) 1f x x
x
= + 
(l) ( ) 2 1f x x
x
= + 
(m) ( ) 2
1
f x x
x
= + 
(n) ( )
2
2 1
x
f x
x
=
−
 
(o) ( )
2
21 3
x x
f x
x
−
=
+
 
(p) ( ) 21
x
f x
x
=
+
 
(q) ( )
2
21
x
f x
x
=
+
 
(r) ( ) 2 xf x e−= − 
(s) ( )
2xf x e−= 
(t) ( ) 2 x xf x e e= − 
(u) ( ) xf x xe= 
(v) ( )
xe
f x
x
= 
 6.3. Máximos e Mínimos 
 
Uma maneira de se determinar os pontos de máximo e de mínimo de uma 
função f é estudá-la com relação a crescimento e decrescimento. Sejam
a c b< < : 
- Se f for crescente em ]a,c[ e decrescente em [c,b[, então c será um ponto 
de máximo local de f. 
- Se f for decrescente em ]a,c[ e crescente em [c,b[, então c será um ponto 
de mínimo local de f. 
Exercícios 
1. Estude a função dada com relação a máximos e mínimos locais. 
(a) ( )
3 2
2 1
3 2
x x
f x x= + − + 
(b) ( )
3 23
2 2
3 2
x x
f x x
−
= − − + 
(c) ( )
3
2 15 12
3
x
f x x x= + − − 
(d) ( )
3
24 15 5
3
x
f x x x
−
= − − + 
(e) ( )
3 23
40 3
3 2
x x
f x x= + − + 
24 
 
(f) ( )
3
26 32 7
3
x
f x x x
−
= − − + 
(g) ( )
3 29
14 26
3 2
x x
f x x= − + + 
(h) ( )
3 213
42 18
3 2
x x
f x x
−
= − − + 
(i) ( )
4 3
2 1
4 3
x x
f x x= + − − 
(j) ( )
4 3 23
3
2 3 2
x x x
f x = − − + 
(k) ( )
4 3 25 13 3
7
2 3 2
x x x
f x = + − − 
(l) ( )
4
3 23 2 18 9
2
x
f x x x
−
= − + + 
(m) ( )
4
3 23 18 1
4
x
f x x x= + − + 
(n) ( )
4
3 23 16 36 11
2
x
f x x x= + + + 
(o) ( )
4
3 25 20 40 4
2
x
f x x x= − + + 
(p) ( )
4 3 221 168 315
14
4 3 2
x x x
f x = + + + 
2. Estude a função dada com relação a máximos e mínimos locais. 
(a) 
( ) 1f x x
x
= +
 
(b) ( ) 2 1f x x
x
= + 
(c) ( ) 2
1
f x x
x
= + 
(d) ( )
2
2 1
x
f x
x
=
−
 
(e) ( )
2
21 3
x x
f x
x
−
=
+
 
(f) ( )
21
x
f x
x
=
+
 
(g) ( )
2
21
x
f x
x
=
+
 
(h) ( )
2xf x e−= 
(i) ( ) 2x xf x e e= − 
(j) ( ) xf x xe= 
(k) ( )
xe
f x
x
= 
(l) ( ) ln xf x
x
= 
(m) ( ) xf x x e= − 
25 
 
Teorema. Sejam f uma função que admite derivada de 2ª. Ordem contínua 
no intervalo aberto I e p∈I. 
(a) ( ) ( )0 0 é ponto de mínimo local.f p e f p p′ ′′= > ⇒ 
(b) ( ) ( )0 0 é ponto de máximo local.f p e f p p′ ′′= < ⇒ Se 
( ) 0f p′′ = não se pode usar este teste. Em tal caso, deve-se usar o 
teste da derivada primeira. 
EXERCÍCIOS 
 
1. Determine os extremos locais de f usando o teste da derivada 
segunda quando aplicável: 
(a) ( ) 3 22 1f x x x x= − + + 
(b) ( ) 3 210 25 50f x x x x= + + − 
(c) ( ) 4 33 4 6f x x x= − + 
(d) ( ) 2 48 4f x x x= − 
(e) ( ) 6 42 6f x x x= − 
(f) ( ) 5 33 5f x x x= − 
(g) ( ) 4 34 10f x x x= − + 
(h) ( )
3 23
2 2
3 2
x x
f x x
−
= − − + 
(i) ( )
4 3
2 1
4 3
x x
f x x= + − − 
(j) ( )
4 3 25 13 3
7
2 3 2
x x x
f x = + − − 
 
1 
 
PRIMITIVA DE UMA FUNÇÃO 
Seja f uma função definida num intervalo I. Uma primitiva de f 
em I é uma função F definida em I, tal que
( ) ( )F .x f x x I′ = ∀ ∈ Sendo F(x) uma primitiva de f em I, 
então, para toda constante k, F(x)+k é, também, primitiva de f. 
Segue que as primitivas de f em I são funções da forma F(x)+k, 
com k constante. Diremos, então, que ( )Fy x k= + é a 
família das primitivas de f em I. A notação ( )f x dx∫ será 
usada para representar a família das primitivas de f:
 
( ) ( )Ff x dx x k= +∫ 
Na notação ( )f x dx∫ , a função f denomina-se integrando e 
referimo-nos a ( )f x dx∫ como a integral indefinida de f. 
 
EXERCÍCIOS 
 
1. Sabendo que 
1
, e 1
1
x
x dx
α
α α α
α
+
= ∈ℜ ≠ −
+∫ calcule: 
 
(a) x dx =∫ 
(b) 
2x dx =∫ 
(c) 
3x dx =∫ 
(d) 
4x dx =∫ 
(e) 
5x dx =∫ 
(f) 
10x dx =∫ 
(g) 
15x dx =∫ 
(h) 
20x dx =∫ 
(i) 
27x dx =∫ 
(j) 
37x dx =∫ 
(k) 
3
1
dx
x
=∫ 
(l) 
5
1
dx
x
=∫ 
(m) 
7
1
dx
x
=∫ 
(n) 
4
1
dx
x
=∫ 
(o) 
6
1
dx
x
=∫ 
(p) 
9
1
dx
x
=∫ 
(q) 
11
1
dx
x
=∫ 
(r) 
8
1
dx
x
=∫ 
(s) 
10
1
dx
x
=∫ 
(t) 
12
1
dx
x
=∫ 
2. Sabendo que 
1
1
x
x dx
α
α
α
+
=
+∫ 
, e 1αα∈ℜ ≠ − calcule: 
 
(a) x dx =∫ 
(b) 3 2x dx =∫ 
(c) 34 x dx =∫ 
(d) 5 4x dx =∫ 
(e) 7 5x dx =∫ 
(f) 5 2x dx =∫ 
(g) 3 5x dx =∫ 
2 
 
(h) 9 7x dx =∫ 
(i) 5 7x dx =∫ 
(j) 
3 2
1
dx
x
=∫ 
(k) 
34
1
dx
x
=∫ 
(l) 
5 4
1
dx
x
=∫ 
(m) 
7 5
1
dx
x
=∫ 
Obs. Mostra-se que: 
 •
1
ln , 0dx x k x
x
= + >∫ . 
 
1x xe dx e kα α
α
= +∫• 
 3. Calcule: 
 
 
xe dx∫(a) 
 
2xe dx∫(b) 
 
3xe dx∫(c) 
 
4xe dx∫(d) 
 
5xe dx∫(e) 
 
11
1
x
dx
e∫(f) 
 
12
1
x
dx
e∫(g) 
 
13
1
x
dx
e∫(h) 
 
15
1
x
dx
e∫(i) 
 
 
3 2x
e dx∫(j) 
(k) 
5 3x
e dx∫ 
(l) 
7 5
1
x
dx
e
=∫ 
Teorema: Admita que f e g são funções que 
possuem primitivas num intervalo I. Então: 
( ) ( )( ) i c f x dx c f x dx⋅ = ⋅∫ ∫
 
( ) ( ) ( ) ( )( ) ii f x g x dx f x dx g x dx± = ±  ∫ ∫ ∫
 
4.Calcule: 
 
(a) ( )3 22 2x x x dx− + +∫ 
(b) ( )3 28 30 24 10x x x dx+ + +∫ 
(c) ( )3 23 1x x dx− +∫ 
(d) ( )3 22 1x x x dx+ + +∫ 
(e) ( )4 3 24 4 2x x x dx− + − +∫ 
(f) ( )23 2 5x x dx+ −∫ 
(g) ( )3 22 1x x x dx− + +∫ 
(h) ( )3 210 25 50x x x dx+ + −∫ 
(i) ( )4 33 4 6x x dx− +∫ 
(j) ( )2 48 2x x dx−∫ 
(k) 
3
x dx
x
 + 
 ∫
 
(l) 
2 5x dx
x
 − 
 ∫
 
3 
 
(m) 
2
1
x dx
x
 + 
 ∫
 
(n) 
3 2
43 6
3 2
x x
x dx
 
+ − − 
 
∫ 
(o) 
4 3 25 13 3
7
2 3 2
x x x
dx
 
+ − − 
 
∫ 
(p) 
4
3 23 2 18 9
2
x
x x dx
 −
− + + 
 
∫ 
(q) 
3 211
30 13
3 2
x x
x dx
 −
+ − + 
 
∫ 
Obs. Mostra-se que: 
 • ( ) ( )1 cossen x dx x kα α
α
= − +∫ 
Exemplo: 
( ) ( )12 cos 2
2
sen x dx x k= − +∫ 
 ( ) ( )1cos x dx sen x kα α
α
= +∫• 
Exemplo: 
( ) ( )1cos 3 3
3
x dx sen x k= +∫ 
 4. Calcule: 
 sen x dx∫(a) 
 cos x dx∫(b) 
 ( )3sen x dx∫(c) 
 ( )cos 4x dx∫(d) 
 ( )5sen x dx∫(e) 
 ( )cos 6x dx∫(f) 
 ( )7sen x dx∫(g) 
 ( )cos 8x dx∫(h) 
 ( )9sen x dx∫(i) 
 ( )cos 10x dx∫(j) 
 5. Calcule: 
 ( )3 xx e dx+∫(a) 
 ( )2x senx dx+∫(b) 
 ( )3 cos x dx+∫(c) 
 ( )cos5 3x sen x dx+∫(d) 
 
2
x
sen dx
 
 
 ∫
(e) 
(f) cos
3
x
dx
 
 
 ∫
 
(g) 
3
5
x
sen dx
 
 
 ∫
 
 
5
3
x
sen dx
 
 
 ∫
(h) 
(i) 
7
cos
3
x
dx
 
 
 ∫
 
(j) 
5
6
x
sen dx
 
 
 ∫
 
 6. Calcule: 
 
(a) ( )24 8 1x x dx− +∫ 
(b) ( )29 4 3t t dt− +∫ 
(c) ( )3 22 3 7t t t dt− + −∫ 
(d) 
3 2
1 3
dz
z z
 − 
 ∫
 
(e) 
7 4
4 7
z dz
z z
 − + 
 ∫
 
(f) 
1
3 u du
u
 
+ 
 
∫ 
(g) 
3
2
1
5u du
u
 − + 
 ∫
 
(h) 
5 1
44 42 6 3v v v dv−
 
+ + 
 
∫ 
4 
 
(i) 
2
1
x dx
x
 − 
 ∫
 
(j) 
2
1
2x dx
x
 + 
 ∫
 
(k) ( )2 3x x dx+∫ 
(l) ( )2 3 1x x dx+∫ 
(m) ( )32 2x x dx− −∫ 
(n) ( )( )2 5 3 1x x dx− +∫ 
(o) ( )( )3 1 4 2x x dx− +∫ 
(p) ( )( )5 1 2 1x x dx− −∫ 
(q) 
3
8 5x
dx
x
− 
 
 
∫ 
(r) 
22 3x x
dx
x
 − +
 
 
∫ 
(s) 
2 1
1
x
dx
x
 −
 + 
∫ 
(t) 
3 1
1
x
dx
x
 −
 − 
∫ 
(u) 
( )22
6
3t
dx
t
+
∫ 
 
( )2
3
2t
dt
t
+
∫(v) 
 
3cos
4
u
du∫(w) 
 
sen
5
u
du
−
∫(x) 
(y) ( )cost t dt+∫ 
(z) ( )3 2 sent t dt−∫ 
 7. ( ) ,y y x x= ∈ℜDetermine a função , tal que: 
 ( )3 1e 0 2dy x y
dx
= − =(a) 
 ( )3 1e 1 1dy x x y
dx
= − + =(b) 
 ( )cos e 0 0dy x y
dx
= =(c) 
 ( ) ( )3 1e 0 1dy sen x y
dx
= − =(d) 
 ( )3 e 1 0
2
dy x
y
dx
= + − =(e) 
 ( )e 0 1xdy e y
dx
−= =(f) 
 ( )2
1
e 1 1
dy
y
dx x
= =(g) 
 
. 
( )13 e 1 2dy y
dx x
= + =(h) 
(i) ( )1 e 1 0dy x y
dx x
= + = 
(j) ( )2
1 1
e 1 1
dy
y
dx x x
= + = 
INTEGRAL DE RIEMANN - PROPRIEDADES 
DA INTEGRAL 
 
Teorema. Seja f,g funções integráveis em [a,b] e k uma 
constante. Então: 
( ) é integrável em [a,b] e 
[ ( ) ( )] ( ) ( ) .
b b b
a a a
i f g
f x g x dx f x dx g x dx
+
+ = +∫ ∫ ∫ 
(ii)
 
( ) é integrável em [a,b] e 
( ) ( ) .
b b
a a
ii kf
kf x dx k f x dx=∫ ∫ 
(iii)
 
( ) ( )Se 0 em [a,b], então ( ) 0.
b
a
iii f x f x dx≥ ≥∫ 
5 
 
( )Se c ]a,b[ e é integrável em [a,c] e em [c,b], 
então ( ) ( ) ( ) .
b c b
a a c
iv f
f x dx f x dx f x dx
∈
= +∫ ∫ ∫ 
 
Teorema Fundamental do Cálculo. Se f for integrável 
em [a,b] e se F for uma primitiva de f em [a,b], então 
( ) ( ) ( ) .
b
a
f x dx F b F a= −∫
 
EXERCÍCIOS 
 
1. Calcule: 
(a) 
3
1
3 dx
−
∫ 
(b) ( )
4
2
5 dx
−
−∫ 
(c) 
2
1
7
2
dx∫ 
(d) 
4
2
5
3
dx∫ 
(e) 
0
1
3 dx
−
∫ 
(f) ( )
1
0
3x dx+∫ 
(g) ( )
1
1
2 1x dx
−
+∫ 
(h) ( )
2
1
3 2x dx−∫ 
(i) ( )
1
0
4 1x dx+∫ 
(j) 
1
3
0
1
5
2
x dx
 − 
 ∫
 
(k) ( )
1
2
2
1x dx
−
−∫ 
(l) ( )
2
2
0
3 3x x dx+ −∫ 
(m) ( )
4
2
1
4 3x x dx− −∫ 
(n) ( )
3
2
2
5 6x x dx
−
+ −∫ 
(o) ( )
3
3
2
8 3 1z z dz
−
+ −∫ 
(p) ( )
0
3
1
2 3x x dx
−
− +∫ 
(q) ( )
1
7 3
1
x x x dx
−
+ +∫ 
(r) ( )
0
7
1
3x x dx− +∫ 
(s) 
2
5 4
1
4 5
dx
x x
 − 
 ∫
 
(t) 
2
3
3
1
1
x x dx
x
 + + 
 ∫
 
(u) ( )
2
2
0
3 1t t dt+ +∫ 
(v) ( )
3
2
0
2 3u u du− +∫ 
(w) ( )
2
2
1
3 1s s ds+ +∫ 
(x) ( )
1
2
0
1x dx+∫ 
(y) ( )
0
2
1
3x dx
−
−∫ 
(z) ( )
1
2
0
2 1x dx+∫ 
(aa) 
 
( )
1
2
1
3 2x dx
−
−∫ 
6 
 
(bb) ( )
1
2
2
0
5 1x dx−∫ 
(cc) 
 
( )
2
2
4 3
0
2z z dz−∫ 
(dd) 
3
1
1
1 dx
x
 + 
 ∫
 
(ee) 
 
1
2
2
1
x dx
x
−
−
 + 
 ∫
 
(ff) 
 
21
2
1
x dx
x
−
−
 − 
 ∫
 
(gg) 
 
3
2
1
1
5 dx
x
 + 
 ∫
 
(hh) 
2
3
1
1 x
dx
x
+ 
 
 ∫
 
(ii) 
 
4
1
1 x
dx
x
+ 
 
 
∫ 
(jj) 
 
2 2
4
1
1 t
dt
t
 +
 
 
∫ 
(kk) 
 
2 2
1
1 3x
dx
x
 +
 
 
∫ 
(ll) 
 
9
4
3t
dt
t
− 
 
 
∫ 
(mm) 
2
3
1
2 7t
dt
t
−
−
 −
 
 
∫ 
(nn) 
4
0
x dx∫ 
(oo) 
8
3
0
x dx∫ 
(pp) 
1
3
1
t dt
−
∫ 
(qq) 
1
8
0
x dx∫ 
(rr) 
 
2
3
1
1
dx
x
∫ 
(ss) 
 
( )
4
1
5x x dx+∫ 
(tt) 
 
( )
1
4
0
x x dx+∫ 
(uu) 
 
( )
8
3 2
8
2 s ds
−
+∫ 
(vv) 
 
( )
0
2 3
1
s s s ds+∫ 
 
2. Calcule: 
 
(a) 
6
0
sen x dx
π
∫ 
(b) 
6
0
cos x dx
π
∫ 
(c) 
4
0
sen x dx
π
∫ 
(d) 
4
0
cos x dx
π
∫ 
(e) 
3
0
sen x dx
π
∫ 
(f) 
3
0
cos x dx
π
∫ 
(g) 
2
0
sen x dx
π
∫ 
(h) 
2
0
cos x dx
π
∫ 
(i) 
0
sen x dx
π
∫ 
(j) 
0
cos x dx
π
∫ 
(k) 
3
2
0
sen x dx
π
∫(l) 
3
2
0
cos x dx
π
∫ 
(m) 
2
4
sen x dx
π
π
∫ 
7 
 
(n) 
2
cos x dx
π
π
∫ 
(o) 
2
sen x dx
π
π
∫ 
 
3. Calcule: 
 
(a) ( )
6
0
sen 2x dx
π
∫ 
(b) ( )
6
0
cos 2x dx
π
∫ 
(c) ( )
2
0
sen 3x dx
π
∫ 
(d) ( )
2
0
cos 4x dx
π
∫ 
(e) ( )
2
0
sen 4x dx
π
∫ 
(f) 
2
cos
3
x
dx
π
π
 
 
 ∫
 
(g) 
2
0
3sen
2
x
dx
π
 
 
 ∫
 
(h) 
0
cos
2
x
dx
π
 
 
 ∫
 
(i) 
0
2sen
3
x
dx
π
 
 
 ∫
 
(j) ( ) ( )
3
4
4sen 2 6cos 3x x dx
π
π
+  ∫ 
(k) 
1
2
0
xe dx∫ 
(l) 
0
3
1
xe dx
−
∫ 
(m) 
0
2
1
xe dx−
−
∫ 
(n) 
0
4
1
xe dx−∫ 
(o) 
1
7
0
1
x
dx
e
−
∫ 
(p) 
1
3
1
1
x
dx
e−
∫ 
 
INTEGRAL DE RIEMANN – MUDANÇA DE 
VARIÁVEL NA INTEGRAL 
 
Teorema. Seja f contínua num intervalo I e sejam a e b 
dois reais quaisquer em I. Seja : [ , ]g c d I→ , com g’ 
contínua em [c,d], tal que g(c)=a e g(d)=b. Nestas 
condições 
( )( ) ( ) ( ) .
b d
a c
f x dx f g u g u du′=∫ ∫
 
EXERCÍCIOS 
1. Calcule: 
(a) ( )
1
10
0
1x dx−∫ 
(b) ( )
0
5
1
2 1x dx
−
−∫ 
(c) ( )
2
6
1
2x dx−∫ 
(d) ( )
1
4
0
3 1x dx+∫ 
(e) ( )
0
3
1
2 5x dx
−
+∫ 
(f) ( )
1
11
0
2x dx−∫ 
(g) ( )
0
5
1
3 4x dx
−
−∫ 
(h) ( )
2
9
1
5 1x dx−∫ 
8 
 
(i) ( )
1
14
0
2 1x dx− −∫ 
(j) ( )
0
7
1
5 2x dx
−
− −∫ 
2. Calcule: 
(a) 
1
1
2
2 1x dx−∫ 
(b) 
1
0
3 1x dx+∫ 
(c) 
4
4
3
5 x dx
−
−∫ 
(d) 
2
3
1
3 2x dx−∫ 
(e) 
1
5
0
4 1x dx+∫ 
(f) 
2
1
3 1x dx−∫ 
(g) 
1
0
4 1x dx+∫ 
(h) 
4
4
3
3 x dx
−
−∫ 
(i) 
2
3
1
5 1x dx−∫ 
(j) 
1
5
0
4 1x dx+∫ 
3. Calcule: 
(a) 
( )
1
9
0
1
2
dx
x −∫
 
(b) 
( )
0
5
1
2
3 1
dx
x− −
∫ 
(c) 
( )
2
4
1
1
2
dx
x −∫
 
(d) 
( )
1
5
0
3
4 2
dx
x +∫
 
(e) 
( )
0
4
1
2
5 3
dx
x− +
∫ 
(f) 
( )
1
8
0
1
3
dx
x −∫
 
(g) 
( )
0
6
1
4
4 2
dx
x− −
∫ 
(h) 
( )
2
5
1
1
4
dx
x
−
−∫
 
(i) 
( )
1
7
0
2
3 4
dx
x +∫
 
(j) 
( )
0
3
1
3
4 2
dx
x−
−
− +∫
 
4. Calcule: 
(a) 
1
1
2
3
4 1
dx
x −∫
 
(b) 
1
0
1
2 1
dx
x +∫
 
(c) 
4
5
3
3
4
dx
x− −
∫ 
(d) 
2
3
1
1
7 2
dx
x −∫
 
(e) 
1
5
0
2
5 1
dx
x −∫
 
(f) 
2
1
8
3 1
dx
x −∫
 
9 
 
(g) 
1
0
3
2 1
dx
x −∫
 
(h) 
4
5
3
4
3
dx
x− −
∫ 
(i) 
2
3
1
1
2 7
dx
x
−
−∫
 
(j) 
1
5
0
1
4 1
dx
x −∫
 
5. Calcule: 
(a) 
2
1
3
2 1
dx
x −∫ 
(b) 
1
0
1
7 1
dx
x −∫ 
(c) 
4
3
2
3 4
dx
x−∫ 
(d) 
2
1
1
2 5
dx
x −∫ 
(e) 
1
0
2
3 1
dx
x −∫ 
(f) 
2
1
2
3 1
dx
x− −∫ 
(g) 
1
0
2
6 1
dx
x −∫ 
(h) 
4
3
3
1 2
dx
x−∫ 
(i) 
2
1
5
2 3
dx
x −∫ 
(j) 
1
0
4
5 2
dx
x −∫ 
 
 
6. Calcule: 
(a) 
1
2
0
3x x dx+∫ 
(b) 
1
2
0
1x x dx−∫ 
(c) 
1
2
0
1 2x x dx+∫ 
(d) 
1
2
0
2 3x x dx−∫ 
(e) 
1
2
0
5 7x x dx−∫ 
(f) 
1
2 3
0
1 3x x dx−∫ 
(g) 
1
2 3
0
2 1x x dx+∫ 
(h) 
1
2 3
0
5 2x x dx−∫ 
(i) 
1
3 4
0
7 1x x dx−∫ 
(j) 
1
5 6
0
4 1x x dx−∫ 
7. Calcule: 
(a) ( )
1
5
2
0
1x x dx+∫ 
(b) ( )
1
7
2
0
2 3x x dx−∫ 
(c) ( )
1
3
2
0
7 2x x dx−∫ 
(d) ( )
1
5
2 3
0
3 4x x dx−∫ 
10 
 
(e) ( )
1
6
3 4
0
3 1x x dx−∫ 
(f) ( )
1
7
4 5
0
9 2x x dx−∫ 
(g) ( )
1
8
5 6
0
7 1x x dx−∫ 
(h) ( )
1
4
6 7
0
4 6x x dx−∫ 
(i) ( )
1
4
7 6
0
2 3 1x x dx−∫ 
(j) ( )
1
2
8 9
0
3 2 3x x dx−∫ 
8. Calcule: 
(a) 
2
2
1
2
3 1
x
dx
x −∫ 
(b) 
1
2
0
5
6 3
x
dx
x −∫ 
(c) 
2
2
1
3
4 5
x
dx
x−∫ 
(d) 
2
2
1
5
7 2
x
dx
x −∫ 
(e) 
2
2
0
3
4 1
x
dx
x −∫ 
(f) 
2
2
1
4
3 5
x
dx
x
−
−∫ 
(g) 
2
2
1
7
4 7
x
dx
x
−
−∫ 
(h) 
1
2
0
6
3 5
x
dx
x
−
−∫ 
(i) 
1
2
0
6
8 1
x
dx
x −
∫ 
(j) 
2 2
3
1 1 2
x
dx
x−
∫ 
(k) 
1 2
3
0 4 1
x
dx
x −
∫ 
(l) 
1 2
3
0 2 1
x
dx
x −
∫ 
9. Calcule: 
(a) 
1
2
0
2x x dx+∫ 
(b) 
0
2
1
1x x dx
−
−∫ 
(c) 
1
2
0
3x x dx+∫ 
(d) 
0
2
1
5x x dx
−
−∫ 
(e) 
1
2
0
4 2x x dx+∫ 
(f) 
1
2
0
2 1x x dx+∫ 
(g) 
0
2
1
3 2x x dx
−
−∫ 
(h) 
1
2
0
4 1x x dx−∫ 
(i) 
0
2
1
2 4x x dx
−
−∫ 
(j) 
1
2
0
3 2 3x x dx− +∫ 
 
 
 
11 
 
TÉCNICAS DE PRIMITIVAÇÃO 
 
PRIMITIVAS IMEDIATAS. 
Seja α≠0 e c constantes reais. Das fórmulas de 
derivação já vistas seguem as seguintes de primitivação: 
c dx cx k= +∫ 
1x xe dx e kα α
α
= +∫ 
cos senx dx x k= +∫ 
sec ln sec tgx dx x x k= + +∫ 
2
sec tgx dx x k= +∫ 
2
1
arctg
1
dx x k
x
= +
+∫ 
( )
1
1
1
x
x dx k
α
α α
α
+
= + ≠
+∫ 
1
lndx x k
x
= +∫ 
sen cosx dx x k= − +∫ 
tg ln cosx dx x k= − +∫ 
sec tg secx x dx x k= +∫ 
2
1
arcsen
1
dx x k
x
= +
−
∫ 
Exercícios 
1. Lembrando que 
1
cos senx dx x kα α
α
= +∫ e 
1
sen cosx dx x kα α
α
= − +∫ , calcule: 
(a) cos5x dx∫ 
(b) sen 2x dx∫ 
(c) cos7x dx∫ 
(d) sen 3x dx∫ 
(e) 
1 1
cos 2
2 2
x dx
 − 
 ∫
 
(f) 
1
2 sen 2
3
x dx
 − 
 ∫
 
(g) 
1
cos3
5
x x dx
 + 
 ∫
 
(h) 
1
4sen3x dx
x
 − 
 ∫
 
(i) 
1 5
cos7
3 2
x dx
 + 
 ∫
 
(j) 
1
cos3 sen 4
2
x x dx
 + 
 ∫
 
(k) 
1 1
sen 2 cos3
3 2
x x dx
 + 
 ∫
 
(l) 
1 1
cos3 sen 7
3 7
x x dx
 − 
 ∫
 
(m) 3
1
sen3
3
x
e x dx
 + 
 ∫
 
(n) 
3
0
sen 2x dx
π
∫ 
(o) 
2
2
cos
2
x
dx
π
π−
 
 
 ∫
 
(p) ( )
3
0
sen 3 cos3x x dx
π
+∫ 
(q) 
2
0
1 1
cos2
2 2
x dx
π
 + 
 ∫
 
2. Sabendo que ( ) ( )2 1 1sen cos 2
2 2
x xβ β= − e que 
( ) ( )2 1 1cos cos 2
2 2
x xβ β= + , calcule: 
(a) 2cos x dx∫ 
(b) 2sen x dx∫ 
(c) 2cos 2x dx∫ 
(d) 2sen 2x dx∫ 
(e) 2cos 3x dx∫ 
(f) 2sen 4x dx∫ 
12 
 
(g) 2
4
cos
3
x
dx
 
 
 ∫
 
(h) 2
2
sen
5
x
dx
 
 
 ∫
 
(i) 2
3
cos
7
x
dx
 
 
 ∫
 
(j) 2
5
sen
3
x
dx
 
 
 ∫
 
3. Calcule: 
(a) 4cos x dx∫ 
(b) 4sen x dx∫ 
(c) ( )2sen cosx x dx+∫ 
(d) ( )2sen cosx x dx−∫ 
(e) ( )25 sen3x dx+∫ 
(f) ( )21 cos2x dx−∫ 
(g) ( )24 sen 2x dx−∫ 
(h) ( )23 cos5x dx+∫ 
(i) ( )26 sen3x dx−∫ 
(j) ( )27 cos 4x dx+∫ 
4. Sabendoque 
 ( ) ( )1sen cos sen sen
2
a b a b a b⋅ = + + −   , calcule: 
(a) ( )sen 2 cos3x x dx⋅∫ 
(b) ( )sen 4 cos3x x dx⋅∫ 
(c) ( )sen3 cos5x x dx⋅∫ 
(d) ( )sen6 cos4x x dx⋅∫ 
(e) ( )sen cos3x x dx⋅∫ 
(f) ( )sen3 cosx x dx⋅∫ 
5. Sabendo que 
 ( ) ( )1sen sen cos cos
2
a b a b a b⋅ = − − +   , calcule: 
(a) ( )sen3 sen5x x dx⋅∫ 
(b) ( )sen5 sen 4x x dx⋅∫ 
(c) ( )sen 4 sen6x x dx⋅∫ 
(d) ( )sen7 sen5x x dx⋅∫ 
(e) ( )sen 2 sen 4x x dx⋅∫ 
(f) ( )sen 4 senx x dx⋅∫ 
6. Sabendo que 
 ( ) ( )1cos cos cos cos
2
a b a b a b⋅ = + + −   , calcule: 
(a) ( )cos4 cos6x x dx⋅∫ 
(b) ( )cos6 cos5x x dx⋅∫ 
(c) ( )cos5 cos7x x dx⋅∫ 
(d) ( )cos8 cos6x x dx⋅∫ 
(e) ( )cos3 cos5x x dx⋅∫ 
(f) ( )cos5 cos2x x dx⋅∫ 
7. Lembrando que
2
1
arctg
1
dx x k
x
= +
+∫ calcule: 
(a) 
2
1
1 4
dx
x+∫ 
(b) 
2
1
1 9
dx
x+∫ 
(c) 
2
1
1 16
dx
x+∫ 
(d) 
2
2
1 25
dx
x+∫ 
(e) 
2
3
1 49
dx
x
−
+∫ 
13 
 
--------------------------- 
(f) 
2
1
1 3
dx
x+∫ 
(g) 
2
1
1 8
dx
x+∫ 
(h) 
2
1
1 15
dx
x+∫ 
(i) 
2
2
1 17
dx
x+∫ 
(j) 
2
3
1 19
dx
x
−
+∫ 
 --------------------- 
(k) 
2
2
3 4
dx
x+∫ 
(l) 
2
3
4 9
dx
x+∫ 
(m) 
2
5
6 16
dx
x+∫ 
(n) 
2
7
8 25
dx
x
−
+∫ 
(o) 
2
3
8 49
dx
x
−
+∫ 
-------------------- 
(p) 
2
7
5 3
dx
x+∫ 
(q) 
2
4
3 7
dx
x+∫ 
(r) 
2
2
5 7
dx
x+∫ 
(s) 
2
3
6 5
dx
x+∫ 
(t) 
2
5
2 11
dx
x
−
+∫ 
8. Lembrando que 
2
1
arcsen
1
dx x k
x
= +
−
∫ , 
calcule: 
(a) 
2
1
1 4
dx
x−
∫ 
(b) 
2
1
1 9
dx
x−
∫ 
(c) 
2
1
1 16
dx
x−
∫ 
(d) 
2
1
1 25
dx
x−
∫ 
(e) 
2
1
1 36
dx
x−
∫ 
------------------------ 
(f) 
2
2
1 49
dx
x−
∫ 
(g) 
2
3
1 81
dx
x−
∫ 
(h) 
2
4
1 64
dx
x−
∫ 
(i) 
2
6
1 121
dx
x−
∫ 
(j) 
2
7
1 144
dx
x−
∫ 
------------------ 
(k) 
2
3
4 9
dx
x−
∫ 
(l) 
2
4
9 16
dx
x
−
−
∫ 
(m) 
2
5
16 25
dx
x−
∫ 
(n) 
2
6
25 49
dx
x
−
−
∫ 
(o) 
2
7
64 81
dx
x−
∫ 
(p) 
2
5
9 4
dx
x−
∫ 
(q) 
2
5
16 9
dx
x
−
−
∫ 
(r) 
2
7
25 16
dx
x−
∫ 
(s) 
2
8
49 25
dx
x
−
−
∫ 
 
14 
 
INTEGRAÇÃO POR PARTES. 
Suponhamos f e g definidas e deriváveis num mesmo 
intervalo I. Temos: 
( ) ( ) ( ) ( ) ( ) ( )f x g x f x g x f x g x′ ′ ′⋅ = ⋅ + ⋅   
ou 
( ) ( ) ( ) ( ) ( ) ( )f x g x f x g x f x g x′ ′′⋅ = ⋅ − ⋅   
Supondo, então, que ( ) ( )f x g x′ ⋅ admita primitiva em I e 
observando que ( ) ( )f x g x⋅ é uma primitiva de
( ) ( )f x g x ′⋅   , então ( ) ( )f x g x′⋅ também admitirá 
primitiva em I e: 
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
f x g x dx f x g x dx f x g x dx
f x g x dx f x g x f x g x dx
′ ′′⋅ = ⋅ − ⋅  
′′⋅ = ⋅ − ⋅
∫ ∫ ∫
∫ ∫
 
que é a regra de integração por partes. 
Fazendo ( ) ( ) e u f x v g x= = teremos 
( ) ( ) e ddu f x v g x′ ′= = , 
o que nos permite escrever a regra de integração por 
partes de uma forma mais prática e fácil de ser 
lembrada: u dv u v v du= ⋅ −∫ ∫ . 
 
Exercícios 
1. Calcule: 
 
(a) 2xxe dx∫ 
(b) 3xxe dx∫ 
(c) 4
2
xx e dx∫ 
(d) 6
3
5
xx e dx∫ 
(e) 7
4
5
xx e dx∫ 
------------ 
(f) 2 3xx e dx∫ 
(g) 2 4xx e dx∫ 
(h) 
2
5
3
xx e dx∫ 
(i) 
2
63
4
xx e dx∫ 
(j) 
2
75
3
xx e dx∫ 
2. Calcule: 
 
(a) cosx x dx∫ 
(b) senx x dx∫ 
(c) cos 2x x dx∫ 
(d) sen3x x dx∫ 
(e) cos 4x x dx∫ 
(f) sen5x x dx∫ 
(g) 2 cosx x dx∫ 
(h) 2 senx x dx∫ 
3. Calcule: 
(a) cosxe x dx∫ 
(b) senxe x dx∫ 
(c) 2 cos3xe x dx∫ 
(d) 3 sen 2xe x dx∫ 
(e) 4 cos5xe x dx∫ 
4. Calcule: 
(a) ln x dx∫ 
(b) lnx x dx∫ 
(c) 2 lnx x dx∫ 
(d) lnx x dx∫ 
(e) 3 lnx x dx∫ 
15 
 
INTEGRAIS INDEFINIDAS DO TIPO 
( )
( )( )
P x
dx
x xα β− −∫ 
Teorema. Sejam , , e m nα β reais dados, comα β≠ . 
Então existem constantes A e B tais que: 
( )( )
( ) ( )2 2
A B
( ) 
A B
( ) 
mx n
a
x x x x
mx n
b
xx x
α β α β
αα α
+
= +
− − − −
+
= +
−− −
 
 
Exercícios 
 
1. Calcule: 
(a) 
2
3
3 2
x
dx
x x
+
− +∫ 
(b) 
2
2
2 3
x
dx
x x
+
− −∫ 
(c) 
2
5
2 8
x
dx
x x
−
− −∫ 
(d) 
2
7
3 4
x
dx
x x
+
− −∫ 
(e) 
2
1
3 10
x
dx
x x
−
+ −∫ 
(f) 
2
3 2
6 5
x
dx
x x
−
− +∫ 
(g) 
2
2 3
6 5
x
dx
x x
+
+ +∫ 
(h) 
2
4 6
6
x
dx
x x
−
− −∫ 
(i) 
2
5 1
7 12
x
dx
x x
−
+ +∫ 
(j) 
2
2 5
8 15
x
dx
x x
−
+ +∫ 
 
2. Calcule: 
Sugestão: Para calcular 
( )2
mx n
dx
x α
+
−
∫ é menos 
trabalhoso fazer a mudança de variável 
.u x α= − 
(a) 
( )2
1
2
x
dx
x
+
+
∫ 
(b) 
( )2
2 5
1
x
dx
x
+
+
∫ 
(c) 
( )
2
2
3
2
x
dx
x
+
−
∫ 
(d) 
( )
2
2
2 4
3
x
dx
x
+
+
∫ 
(e) 
( )
3
2
1
1
x
dx
x
+
−
∫ 
(f) 
( )2
7
3
x
dx
x
+
+
∫ 
(g) 
( )2
3 4
1
x
dx
x
+
−
∫ 
(h) 
( )
2
2
5 3
4
x
dx
x
+
−
∫ 
(i) 
( )
2
2
8 3
6
x
dx
x
−
+
∫ 
(j) 
( )
3
2
4 1
2
x
dx
x
+
−
∫