7aListaIBMGabarito_2010

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Gabarito da 7a
¯
Lista de Ca´lculo Diferencial e Integral I
IBM / 2010
Exerc´ıcio 1. a) y′ =
1
x
, y′′ =
−1
x2
b) y′ =
1
xln10
,y′′ =
−1
x2ln10
c) y′ =
2e2x
1 + e2x
, y′′ =
4e2x
(1 + e2x)2
d)y′ = e3x+1
[
3ln(x2 − 2) + 2x
x2 − 2
]
, y′′ = e3x+1
[
9ln(x2 − 2) + 12x
x2 − 2 −
2x2 + 4
(x2 − 2)2
]
.
Exerc´ıcio 2.
a)f ′(x) = (2x+ 1)5(x4 − 3)6
[
10
2x+ 1
+
24x3
x4 + 3
]
b)f ′(x) = − x
x4 − 1
4
√
x2 + 1
x2 − 1
c)f ′(x) = xsenx
[
cosxlnx+
senx
x
]
d)f ′(x) = (ln x)x
[
ln(ln x) +
1
ln x
]
e)f ′(x) = xe
sen(2x)
esen(2x)
[
2cos(2x)ln x+
1
x
]
f)f ′(x) =
√
xe x2(x2 + 1)10
(4x4 + 45x2 + 1)
2x3 + 2x
Exerc´ıcio 3.f ′(x) =
(−1)n+1(n− 1)!
(x− 1)n , n ≥ 1.
Exerc´ıcio 4. g′(0) =
1
2
.
Exerc´ıcio 5.
a)y′ = −1 b)y′ = 2y
3y2(x+ y)2 + 2x
c)y′ =
(4x2 + 3x− 1)(8x+ 3)
4y(y2 − 9)3
d)y′ =
−3x2 − 2xy + 2y2
x2 − 4xy + 3y2 e)y
′ =
2x− ycos(xy)
xcos(xy) + 1
f)y′ =
−y
x
Exerc´ıcio 6.a) y = x b) y = −9/13x+ 40/13.
Exerc´ıcio 7. a) y′′ =
2
(1− x)3 b) y
′′ = 2− 6
x4
.
Exerc´ıcio 8. h′(e) =
1
e+ 1
.
1