Gabarito parcial Lista de exercícios 4

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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 parcial da 4a lista de exerc´ıcios (21/08/2017 a 01/09/2017)
1. f ′ na˜o esta´ definida nos pontos x = −1; 1; 3; pois nestes pontos a curva tem uma ”quina”.
2.
3. f ′(x) = −2x; f ′(−3) = 6, f ′(0) = 0, f ′(1) = −2.(a)
g′(t) = − 2
t3
; g′(−1) = 2, g′(2) = −1
4
, g′(
√
3 ) = − 2
33/2
.(b)
p′(θ) =
√
3
2
√
θ
; p′(1) =
√
3
2
, p′(3) =
1
2
, p′
(
2
3
)
=
3
23/2
.(c)
4. g′(1) = 2.(a)
(b)
5. (a)
Na˜o, pois os limites laterais sa˜o diferentes.(b)
6. f(0) = 1.(a)
(b)
7.
8. y − 1
p2
= − 2
p3
(x− p), assim para y = 0 temos x = 3p
2
.
9. x = 1 e x = −2.
10. Na˜o existe x ∈ R tal que f ′(x) = 4.
11. a = −3; b = 2 e c = 1.
12. y =
x
4
− 7
2
.
13. (c)
14.
15.
7(a) (b)
1
4
(c)
16.
d
dx
(uvw) = uv
dw
dx
+ uw
dv
dx
+ vw
du
dx
1
17. f ′(x3) = 6x6.
18.
19. f ′(x) = 0(a)
f ′(x) = 17(b)
f ′(s) = 3
√
3 s2 − 2√3 s(c)
(d)
g′(x) = 2(e)
(f)
F ′(x) =
6x(x2 + 3)2
(5x− 8)2 −
10(x2 + 3)3
(5x− 8)3(g)
f ′(x) =
1
4
√
1 +
√
1 + x
√
1 + x
(h)
f ′(x) = 1000 x249(i)
f ′(x) =
3
7
7
√
x4
− 101−x2 (ln 10) 2x(j)
(k)
(l)
g′(x) = (7x2 + 6x)6(3x− 1)3(378x2 + 100x− 42)(m)
H ′(z) = (−9z2 + 18z)(z3 − 3z2 + 1)−4(n)
(o)
f ′(x) = pix ln pi(p)
(q)
f ′(x) =
1
2
√
x+ 2
− 18x
2 + 12
(x3 + 2x)2
(r)
(s)
f ′(x) =
ex(x5 − 5x4 + 2x− 2)
(x5 + 2x)2
(t)
(u)
f ′(x) =
1
(x− 2)3/4√x+ 2(v)
g′(x) = 3x2ex
3
ln(3 +
√
x) + ex
3 1
2
√
x(3 +
√
x)
(w)
(x)
2
20. (a)
f ′(x) = − cossec2 x(b)
f ′(x) = −cossecx cotg x(c)
(d)
f ′(x) = −sen
2 x+ cossec2 x
(secx− cosx)2(e)
(f)
f ′(x) =
1− senx
x+ cosx
(g)
f ′(x) = 2f(x) + e2x
senx+ x cosx− e−x x
5 + 5x4 + 1
(x5 + 1)2
x senx+
e−x
(x5 + 1)
(h)
(i)
f ′(x) =
3x4 senx− x5 cosx− 2x2 cosx
x2 sen2 x
(j)
f ′(x) =
2x3 cosx− 6x2 senx
(x3 − senx)2(k)
(l)
f ′(x) =
x+ 4
x lnx
(m)
f ′(x) =
(
1
3
3
√
x2
+
1
2
√
x
)
ex cotg x+ ( 3
√
x+
√
x ) ex cotg x− ( 3√x+√x ) ex cossec2 x(n)
(o)
y′ = 2 tan x sec2 x etan
2 x(p)
y′ = −cossecx cotg x− cossec
2 x
cossecx+ cotg x
(q)
(r)
(s)
y′ = −4
3
(2t+ 5)−5/3 cos[(2t+ 5)−2/3](t)
21. (a) Na˜o e´ diferencia´vel.(b)
22. Sim.(a)
Sim.(b)
f ′(x) =
 2x sen
(
1
x
)
− cos
(
1
x
)
se x 6= 0
0 se x = 0.
(c)
Na˜o.(d)
3
23.
24. f ′(0) = 23.
25.
26.
T: y − 1 = 2(x− 2) e N: y − 1 = −1
2
(x− 2).(a)
T: y − 4 = 1
2
(x− 4) e N: y − 4 = −2(x− 4).(b)
T: y = 1 e N: x = 0.(c)
T: y = −pi(x− 1) e N: y = 1
pi
(x− 1).(d)
27. H ′(2) = 8.
28.
29.
(a) g′(x) = |x|+ x
2
|x|(b) f
′(x) =
|x|+ x
3x 3
√
(|x|+ x)2(c)
30. f ′(5) = lim
h→0
f(5 + h)− f(5)
h
na˜o existe, pois os limites laterais sa˜o diferentes.
4