6aListaIBM_2010

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6a
¯
Lista de Ca´lculo Diferencial e Integral I
IBM / 2010
Exerc´ıcio 1. Calcule f ′(x):
a)f(x) = 3x2 + 5 b)f(x) = x3 + x2 + 1 c)f(x) = 3x2 − 2x2 + 4
d)f(x) = 3x+
√
x e)f(x) = 5 + 3x−2 f)f(x) = 2 3
√
x
g)f(x) = 3x+
1
x
h)f(x) =
4
x
+
5
x2
i)f(x) =
2x3
3
+
x2
4
j)f(x) = 3
√
x+
√
x k)f(x) = 2x+
1
x
+
1
x2
l)f(x) = 6x3 + 3
√
x
Exerc´ıcio 2. Seja f(x) = x3 + 3x2 + 1.
a) Estude o sinal de f ′(x).
b) Calcule lim
x→+∞ f(x) e limx→−∞ f(x).
Exerc´ıcio 3. Calcule f ′(x):
a)f(x) =
x
x2 + 1
b)f(x) =
x2 − 1
x+ 1
c)f(x) =
3x2 + 3
5x− 3
d)f(x) =
√
x
x+ 1
e)f(x) = 5x+
x
x− 1 f)f(x) =
√
x+
3
x3 + 2
Exerc´ıcio 4.Seja g(x) =
x
x2 + 1
.
a) Determine os pontos do gra´fico de g em que as retas tangentes, nestes pontos, sejam
paralelas ao eixo x.
b) Estude o sinal de g′(x).
c) Calcule lim
x→+∞ g(x) e limx→−∞ g(x).
Exerc´ıcio 5. Calcule f ′(x):
a)f(x) = 3x2 + 5cosx b)f(x) =
cosx
x2 + 1
c)f(x) = xsenx
d)f(x) = x2tgx e)f(x) =
x+ 1
tgx
f)f(x) =
3
senx+ cosx
g)f(x) =
secx
3x+ 2
h)f(x) = cosx+ (x2 + 1)senx i)f(x) =
√
xsecx
j)f(x) =
x+ 1
xsenx
k)f(x) = x2 + 3xtgx l)f(x) =
x+ senx
x− cosx
Exerc´ıcio 6. Calcule f ′(x):
a)f(x) = sen4x b)f(x) = cos5x c)f(x) = senx3
d)f(x) = (senx+ cosx)3 e)f(x) =
√
3x+ 1 f)f(x) = 3
√
x− 1
x+ 1
g)f(x) = sen(cosx) h)f(x) = (x2 + 3)4 i)f(x) = cos(x2 + 3)
j)f(x) = tg3x k)f(x) = sec3x l)f(x) =
sen(x2)
sen2x
1
Exerc´ıcio 7. Calcule f ′(x):
1)f(x) =
x+ 1
x− 1 2)f(x) =
2x3 + 1
x+ 2
3)f(x) =
4x− x4
x3 + 2
4)f(x) = xsen(
√
x5 − x2) 5)f(x) =
3
√
x2cosx
(x4 + tg2x+ 1)2
6)f(x) =
√
xtg2(x)
7)f(x) =
√
x+ cossec(x)
x3 + 3x2
8)f(x) = sec
√
x2 + 1 9)f(x) =
x2tg(x)
sec(x)
10)f(x) = xsen(x)cos(x) 11)f(x) =
(x+ 2)4
x4 + 16
12)f(x) =
1
sen(x− sen(x))
13)f(x) =
2x
3
√
x+
√
x
14)f(x) = cotg(3x2 + 5) 15)f(x) =
x2
sen(x)cos(x)
16)f(x) =
x3sec(x)tg(x)
(x2 + 1)cos(x)
17)f(x) =
cos(x)cotg(x)
sec(x)− cos(x) 18)f(x) =
2cos(x)
x2 + x+ 1
19)f(x) = arcsen(x2) 20)f(x) = (arcsen(x))2 21)f(x) = (1− x2)arctg(x)
22)f(x) = xarccos(x)−
√
1− x2 23)f(x) = cos(ln(x)) 24)f(x) = ln√x
25)f(x) = ln(e−x + xe−x) 26)f(x) =
ln(x)
(1− x) 27)f(x) = e
√
x
28)f(x) = sen(ex) 29)f(x) = arcsen(ex) 30)f(x) = ln(arctg(x))
31)f(x) = e1/x
2
+
1
x2
32)f(x) = ln(ex + 1) 33)f(x) = ecos(x
3)
2