6aListaIBMGabarito_2010

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Gabarito da 6a
¯
Lista de Ca´lculo Diferencial e Integral I
IBM / 2010
Exerc´ıcio 1.
a)f ′(x) = 6x b)f ′(x) = 3x2 + 2x c)f ′(x) = 2x
d)f ′(x) = 3 +
1
2
√
x
e)f ′(x) =
−6
x3
f)f ′(x) =
2
3
3
√
x2
g)f ′(x) = 3− 1
x2
h)f ′(x) = − 4
x2
− 10
x3
i)f ′(x) = 2x2 +
x
2
j)f ′(x) =
1
3
3
√
x2
+
1
2
√
x
k)f ′(x) = 2− 1
x2
− 2
x3
l)f ′(x) = 18x2 +
1
3
3
√
x2
Exerc´ıcio 2. a) f ′(x) > 0 se x ∈ (−∞,−2)∪ (0,∞), f ′(x) < 0 se x ∈ (−2, 0).
b) lim
x→+∞
f(x) =∞ e lim
x→−∞
f(x) = −∞.
Exerc´ıcio 3.
a)f ′(x) =
−x2 + 1
(x2 + 1)2
b)f ′(x) = 1 c)f ′(x) =
15x2 − 18x− 15
(5x− 3)2
d)f ′(x) =
−x+ 1
2
√
x(x+ 1)2
e)f ′(x) = 5− 1
(x− 1)2 f)f
′(x) =
1
2
√
x
− 9x
2
(x3 + 2)2
Exerc´ıcio 4. a) ±1.
b) g′(x) < 0, se |x| > 1 e g′(x) > 0, se |x| < 1.
c) lim
x→+∞
g(x) = 0 e lim
x→−∞
g(x) = 0.
Exerc´ıcio 5.
a)f ′(x) = 6x− 5senx b)f ′(x) = −senx(x
2 + 1)− 2xcosx
(x2 + 1)2
c)f ′(x) = senx+ xcosx d)f ′(x) = 2xtgx+ x2sec2x
e)f ′(x) =
tgx+ sec2x(x+ 1)
tg2x
f)f ′(x) =
3(senx− cosx)
(senx+ cosx)2
g)f ′(x) =
secxtgx(3x+ 2)− 3secx
(3x+ 2)2
h)f ′(x) = −senx+ 2xsenx+ (x2 + 1)cosx
i)f ′(x) ==
secx+ 2xsecxtgx
2
√
x
j)f ′(x) ==
−senx− cosx(x2 + x)
x2sen2x
k)f ′(x) = 2x+ 3tgx+ 3xsec2x l)f ′(x) =
−1 + cosx(x− 1)− senx(x+ 1)
(x− cosx)2
1
Exerc´ıcio 6. Calcule f ′(x):
a)f ′(x) = 4cos4x b)f ′(x) = −5sen5x
c)f ′(x) = 3x2cosx3 d)f ′(x) = 3(senx+ cosx)2(cosx− senx)
e)f ′(x) =
3
2
√
3x+ 1
f)f ′(x) =
1
3
3
√
(
x+ 1
x− 1)
2
2
(x+ 1)2
g)f ′(x) = −cos(cosx)senx h)f ′(x) = 8x(x2 + 3)3
i)f ′(x) = −2xsen(x2 + 3) j)f ′(x) = 3sec2(3x)
k)f ′(x) = 3sec(3x)tg(3x) l)f ′(x) =
2xsen2xcos2x− sen(2x)sen(x2)
sen4x
Exerc´ıcio 7. 1)f ′(x) =
−2
(x− 1)2
2)f ′(x) =
4x3 + 12x2 − 1
(x+ 2)2
3)f ′(x) =
−x6 − 16x3 + 8
(x3 + 2)2
4)f ′(x) = sen(
√
x5 − x2) + x(5
2
√
x3 − 2x)cos(
√
x5 − x2)
5)f ′(x) =
(x4 + tg2x+ 1)(2/3cos x− xsen x)− 4xcos x(2x3 + tg xsec2x)
3
√
x(x4 + tg2x+ 1)3
6)f ′(x) =
tg2x+ 2xtgxsec2x
2
√
xtg2x
7)f ′(x) =
(1− 2√xcossec xcotg x)(x3 + 3x2)− (2x+ 2√xcossec x)(3x2 + 6x)
2
√
x(x3 + 3x2)2
8)f ′(x) =
xsec
√
x2 + 1tg
√
x2 + 1√
x2 + 1
9)f ′(x) =
2xtg x+ x2sec2x− x2tg2x
sec x
10)f ′(x) =
sen(2x)
2
+ xcos(2x)
11)f ′(x) =
(x+ 2)3(64− 8x3)
(x4 + 16)2
12)f ′(x) =
(cosx− 1)cos(x− senx)
(sen(x− sen x))2
13)f ′(x) =
4
3
3
√
x+
√
x
( 3
√
x+
√
x)2
14)f ′(x) = −6xcosse2(3x2 + 5)
15)f ′(x) =
xsen(2x)− x2cos(2x)
(sen xcos x)2
16)f ′(x) =
(x4 + 3x2)tg x+ (2x5 + 2x3)tg2x+ (x5 + x3)sec2x
(x2 + 1)2cos2x
2
17)f ′(x) = −3cotg2(x)− 3cotg4(x)
18)f ′(x) =
−2sen x(x2 + x+ 1)− 2cos x(2x+ 1)
(x2 + x+ 1)2
19)f ′(x) =
2x√
1− x4
20)f ′(x) =
2arcsen x√
1− x2
21)f ′(x) = −2xarctg x+ 1− x
2
1 + x2
22)f ′(x) = arccos x
23)f ′(x) =
−sen(ln(x)
x
24)f ′(x) =
1
2x
25)f ′(x) =
−x
1 + x
26)f ′(x) =
1− x+ xln x
x(1− x)2
27)f ′(x) =
e
√
x
2
√
x
28)f ′(x) = excos(ex)
29)f ′(x) =
ex√
1− e2x
30)f ′(x) =
1
(1 + x2)arctg x
31)f ′(x) =
−2
x3
(e1/x
2
+ 1)
32)f ′(x) =
ex
ex + 1
33)f ′(x) = −3x2sen(x3)ecos(x3)
3