lista calc1 2 2012

Universidade Veiga de Almeida
Ciclo B´asico das Engenharias
C´alculo Diferencial e Integral I
2a Lista de Exerc´ıcios
Prof(a): Andreia Nogueira
Conte´
udo: Regras de deriva¸c˜ ao, Derivada das fun¸co
˜es trigonom´etricas, Regra da Cadeia, Equa¸ca
˜o da reta tangente

1. Derive cada fun¸c˜ ao abaixo.
(a) f (x) = (x2 + 2x + 1).(tg(x)).
(b) f (x) = x1/2 .sen(x) + x1/3 .
(c) f (x) =

x.sec(x)
.
x2 + 2x + 3

(d) f (x) =

cosec(x)
.
x2 + 1

(e) f (x) = cosec(x).cotg(x).
(f) f (x) = sec(x).tg(x).
(g) f (x) = x3 sen(x) − 5cos(x).
(h) f (x) =

cotg(x)
.
1 + cosec(x)

(i) f (x) =

sen(x) + cos(x)
.
sen(x) − cos(x)

2. Derive cada fun¸c˜ ao dada utilizando as regras b´asicas de deriva¸c˜ao e a regra da cadeia.
(a) f (x) = (x3 + 2x)37 .
(b) f (x) = (x4 + 2x3 + 3)1/2 .
(c) f (x) =

7 x − x −2

3

.

(d) y = (5x + 8)13 (x3 + 7x)12 .
(e) f (x) =

x−5
2x + 1

3

.

(f) y = sen(x2 + 3x).
(g) f (x) = 4cos(x5 ).
(h) f (x) = tg(x2 + 1).
(i) f (x) = (sen(2x))(x3 + 2x)2/3 .
(j) f (x) = (1 + sec(5x))1/2 .
(l) f (x) = cosec(x2 + 1)1/2 .

1

2
(m) Desafio: f (x) = [4 + (3x)1/2 ]1/2 .
(n) Desafio: f (x) = cos3 (sen(2x)).

3. Seja f (x) = 2x2 − x. Determine o ponto no gr´afico de f onde a tangente ´e paralela `a reta 3x − y − 4 = 0.
Determine tamb´em a equa¸c˜ ao da reta tangente a esse ponto e esquematize o gr´afico.

4. Determine a equa¸c˜ ao da reta tangente ao gr´afico de y = xcos(3x) no ponto x = π.
Gabarito
2

1. (a) f (x) = (2x + 2)tg(x) + (x + 2x + 1)sec2 (x).
1
1
(b) f (x) = x−1/2 .sen(x) + x1/2 .cos(x) + x−2/3 .
2
3
2
(sec(x) + xsec(x)tg(x))(x + 2x + 3) − (xsec(x))(2x + 2)
.
(c) f (x) =
(x2 + 2x + 3)2
(d) f (x) =

(−cosec(x)cotg(x))(x2 + 1) − 2xcosec(x)
.
(x2 + 1)2

(e) f (x) = −cosec3 (x) − cosec(x)cotg 2 (x).
(f) y = sec(x)tg 2 (x) + sec3 (x).
(g) f (x) = 3x2 sen(x) + x3 cos(x) + 5sen(x).
(h) f (x) =

−cosec(x)
.
1 + cosec(x)

(i) f (x) =

−2
.
(sen(x) − cos(x))2

2. (a) f (x) = 37(x3 + 2x)36 (3x2 + 2).
1
(b) f (x) = (x4 + 2x3 + 3)−1/2 (4x3 + 6x2 ).
2
7
(c) f (x) = −2 x − x −3