lista11_cal1

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Lista de Exerc´ıcios 11
1. Calcule a diferencial.
(a) y = x3
(b) y = x2 − 2x
(c) y = x
x+1
(d) y = 3
√
x
(e) y = xe−x
2. Use diferenciais para aproximar:
(a)
√
50
(b)
√
37, 5
(c)
√
16, 01,
(d) e0,01
(e) ln(0, 99)
3. Seja V (r) = 4pi
3
r3, r > 0.
(a) Calcule a diferencial.
(b) Interprete geometricamente ∆V −
dV .
4. Seja y = x2 + 3x.
(a) Calcule a diferencial.
(b) Interprete geometricamente ∆y −
dy.
5. Seja f : R −→ R deriva´vel. Prove que:
(a) Se f ′(x) = f(x) para todo x ∈ R,
enta˜o existe uma constante k tal
que f(x) = kex para todo x ∈ R.
(b) Se f ′(x) = αf(x) para todo x ∈ R,
enta˜o existe uma constante k tal
que f(x) = keαx para todo x ∈ R.
Sugesta˜o: use o exercicio 11, da
lista no9.
(c) Determine y = f(x), x ∈ R, tal
que
(i)
{
f ′(x) = 2f(x);
f(0) = 1
(ii)
{
f ′(x) = −2f(x);
f(0) = 1
Esboce o gra´fico de f .
6. Calcule as seguintes integrais in-
definidas:
(a)
∫
2cos xdx
(b)
∫
3
2cos2x
dx
(c)
∫
1 + tg2x
tg2x
dx
(d)
∫
tg x
sen 2x
dx
(e)
∫
1√
9− 9x2dx
(f)
∫
x
√
x dx
(g)
∫
x3 + 1
x2
dx
(h)
∫
1
1 + x2
dx
(i)
∫
tg2x dx
(j)
∫
cos2x dx
7. Verifique que
(a)
∫
sec x dx = ln | sec x+ tgx|+ C
(b)
∫
tgx dx = − ln | cos x|+ C
1
8. Sejam a 6= 0 e b constantes. Mostre que
(a)
∫
ax+ b
1 + x2
dx =
a
2
ln(1 + x2) +
b arc tg x+ C
(b)
∫
1
a2 + (x+ b)2
dx =
1
a
arc tg (
x+ b
a
) + C
9. Use a integrac¸a˜o por substuic¸a˜o para
calcular as seguintes integrais:
(a)
∫
x cos x2 dx
(b)
∫
(3x+ 1)19dx
(c)
∫
e3xdx
(d)
∫
xex
2
dx
(e)
∫
x
1 + x2
dx
(f)
∫
2x+ 3
x2 + 3x+ 1
dx
(g)
∫
x
1 + x4
dx
(h)
∫
ln x
x
dx
(i)
∫
x2
1 + x3
dx
(j)
∫
x2
(1 + x3)2
dx
(k)
∫
1
x ln x
dx
(l)
∫
esen xcos x dx
(m)
∫
x5
√
1− x2dx
10. Calcule as integrais indefinidas usando a
substituic¸a˜o indicada.
(a)
∫
cos3 xsen x dx; u = cos x
(b)
∫
1√
x
sen
√
xdx; u =
√
x
(c)
∫
3xdx√
4x2 + 5
; u = 4x2 + 5
(d)
∫
cotg x cossec2x dx; u = cotg x
(e)
∫
(1 + sen x)9cos x dx; u = 1 +
sen x
11. Use a integrac¸a˜o por partes para calcular
cada integral:
(a)
∫
xcos x dx
(b)
∫
xexdx
(c)
∫
x2ex dx
(d)
∫
arc tg x dx
(e)
∫
ln x dx
(f)
∫
x ln x dx
(g)
∫
x2 sen x dx
(h)
∫
ex cos xdx
(i)
∫
cos2x dx
(j)
∫
sec3xdx
Respostas
1. (a) dy = 3x2dx
(b) dy = (2x− 2)dx
(c) dy = 1
(x+1)2
dx
(d) dy = 1
3
3
√
x
2
(e) dy = (e−x − xe−x)dx
2. (a) 7, 0714285
(b) 6, 125
2
(c) 4, 00125
(d) 1, 01
(e) −0, 01
3. (a); (b)
4. (a); (b)
5. (a); (b); (c)
6. (a) 2sen x+ C
(b) 3
2
tg x+ C
(c) −cotg x+ C
(d) 1
2
tg x+ C
(e) 1
3
arc sen x+ C
(f) 2
5
√
x5 + C
(g) x
3
3
+ ln |x|+ C
(h) arc tg x+ C
(i) tg x− x+ C
(j) 1
2
x+ 1
4
sen 2x+ C
7. (a)
(b)
8. (a)
(b)
9. (a) 1
2
sen x2 + C
(b)
(3x+1)20
60
+ C
(c) e
3x
3
+ C
(d) 1
2
ex
2
+ C
(e) 1
2
ln(1 + x2) + C
(f) ln |x2 + 3x+ 1|+ C
(g) 1
2
arc tg x2 + C
(h)
(ln x)2
2
+ C
(i) 1
3
ln |1 + x3|+ C
(j) − 1
1+x2
+ C
(k) ln | ln x|+ C
(l) esen x + C
(m)
2(
√
1−x2)5
5
− (
√
1−x2)3
3
− (
√
1−x2)7
7
+ C
10. (a) − cos4x
4
+ C
(b) −2cos √x+ C
(c) 3
4
√
4x2 + 5 + C
(d) − 1
2
cotg2x+ C
(e) 1
10
(1 + sen x)10 + C
11. (a) xsen x+ cos x+ C
(b) xex − ex + C
(c) ex(x2 − 2x+ 2) + C
(d) x arc tgx− 1
2
ln(1 + x2) + C
(e) x ln x− x+ C
(f) x
2
2
ln x− x2
4
+ C
(g) −x2cos x+ 2x sen x+ 2cos x+ C
(h) 1
2
ex(sen x+ cos x) + C
(i) 1
2
(x+ senx cos x) + C
(j) 1
2
sec x tg x+ 1
2
ln |sec x+ tg x|+ C
3