Gabarito parcial Lista de exercícios 5

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Universidade Federal de Santa Catarina
Centro de Cieˆncias F´ısicas e Matema´ticas
Departamento de Matema´tica
MTM3101 - Ca´lculo 1
Gabarito da 5a lista de exerc´ıcios (04/09/2017 a 15/09/2017)
1. (a)
f ′(x) = − 1
(1− x2)√1− 2x2(b)
g′(x) = (3 + tg x)x
[
ln(3 + tan x) +
x sec2 x
3 + tan x
]
(c)
y′ = (lnx)x
[
1
lnx
+ ln(lnx)
]
(d)
(e)
f ′(x) = 2x arctg(2x) +
2x2
1 + 4x2
(f)
f ′(x) = pi xpi−1 + xx
x
[
xx(lnx+ ln2 x) +
xx
x
]
(g)
f ′(x) = (5 + cos(5x))x
2
[
2x ln(5 + cos(5x))− 5x
2 sen(5x)
5 + cos(5x)
]
(h)
(i)
y′ = e3x
[
3 arcsen(2x) +
2√
1− 4x2
]
(j)
(k)
f ′(x)′ = ex xe
x
[
lnx+
1
x
]
(l)
2.
1
9
3.
4.
5. y − 1 = −(x− 1) ou tambe´m y + 1 = −(x+ 1).
1
6.
y′ =
4xy
√
xy − y
x− 2x2√xy(a) (b) y
′ =
2x(x− y)2 + y
x
(c)
y′ =
−2xy2 − sen y
2yx2 + x cos y
(d) y′ =
2e2x − cos(x+ 3y)
3 cos(x+ 3y)
(e) (f)
7.
2 cos(x+ y) sen(x+ y)
(
1 +
dy
dx
)
= 0(a) (b)
arctg x+
x
1 + x2
+ 2y
dy
dx
= 0(c) (d)
8. dy
dx
(0) = −1(a) (b)
dy
dx
(0) = 0(c) (d)
9. y′ = θ cos θ + sen θ e y′′ = −θ sen θ + 2 cos θ(a)
(b)
F ′(t) = −49
2
(1− 7t)5/2 e F ′′(t) = 1715
4
(1− 7t)3/2(c)
10. 6t2 − 6t− 12 ; 12t− 6(a) 6(b) ±18(c)
11. λ =
1 +
√
5
2
e λ =
1−√5
2
12.
13. f ′(1) = −4
7
14.
15.
16. f (2001)(0) = 4500.
17. − 6√
55
m/s
18. −4.000 galo˜es/min −2.000 galo˜es/min
19.
20. 1 rad/s
21.
22. 1, 03
23. dy = 0, 03 ∆y − dy = 0, 000301.
24. dV = 4pir2dr(a)
(b)
25.
2