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UNIVERSIDADE FEDERAL DE SERGIPE
DEPARTAMENTO DE MATEMA´TICA
Disciplina: Ca´lculo Diferencial e Integral I
SEGUNDA LISTA DE EXERCI´CIOS - GABARITO
1. (a) y = −x+ 5 (b) y = −x+ 2 (c) y = pi (d) y = x
2. (a) b = −4 (b) (1, 1); y = 3x− 2 (c) x = 0 (d) y = 12x− 15; y = 12x+ 17
(e) No ponto (−1,−2) (f) y = −x− 1; y = 11x− 25 (g) –
(h) Dois; (−2±√3, 1
2
(1∓√3)) (i) x = 0 (j) – (k) – (l) –
(m) (1,−1) (n) y = 3; y = 2
3
x− 5
3. (a) (i) h′(2) = 2 (ii) i′(2) = −22
25
(iii) j′(2) = 44
(b) 120
(c) 198
4. (1) f ′(x) = − 1
x4
+
1
x3
(2) f ′(x) = (3− 2x)(3x3 − 4) + (3x− x2)(9x2)
(3) f ′(x) =
−20x
(x2 − 1)2
(4) f ′(x) =
−20
(4x+ 1)6
(5) f ′(x) = −8(2
√
3x+ 1)(
√
3x2 + x−
√
11)−9
(6) f ′(x) =
3x2 + 1
2
√
x
3 3
√
(x3 +
√
x)2
(7) f ′(x) = 3
(x2 + x
1− 2x
)21 + 2x− 2x2
(1− 2x)2
(8) f ′(x) =
2x√
1− x4 arctg(x
2) + arcsen(x2)
2x
1 + x4
(9) f ′(x) = −−x
2 + 4x+ 4
(x2 + 4)2
cossec2
( x− 2
x2 + 4
)
(10) f ′(x) = sec2(7x− 1)
(11) f ′(x) =
1
4
sec(4x− 3)tg(4x− 3)
(12) f ′(x) =
−3xcossec(3x)cotg(3x)− cossec(3x)
x2
(13) f ′(x) = 3sen2(x) cos(x)
(14) f ′(x) = 10x cos(tg(5x2)) sec2(5x2)
(15) f ′(x) = 6cotg(6x)
(16) f ′(x) =
3√
e6x − 1
(17) f ′(x) = e
√
4−x2+5 lnx
( −x√
4− x2 +
5
x
)
(18) f ′(x) = −3e−3x(3cos(x) + sen(2x)) + e−3x(−3sen(x) + 2cos(2x))
(19) f ′(x) =
1
x(1 + (ln x)2)
(20) f ′(x) =
−(2x− 1)√
1− (x2 − x)2
(21) f ′(x) =
8x
4x2 + 1
+ 3x2sen(ex) + x3excos(ex)
(22) f ′(x) = x
1
x (
1− ln(x)
x2
)
(23) f ′(x) = xcos(x)
(
− sen(x) ln(x) + cos(x)
x
)
(24) f ′(x) = (sen(x))ln(x)
( ln(sen(x))
x
+ cotg(x) ln(x))
)
(25)
dy
dx
=
cosh(x)
senh(x) · ln 4
(26)
dy
dx
=
2x · ln 2 · log3(cosh(x))− 2x · senh(x)cosh(x)·ln 3
log23(cosh(x))
(27)
dy
dx
= sech2(5x + ex) · (5x · ln 5 + ex)
(28)
dy
dx
= −cossech(x)coth(x) · esech(2x) − 2cossech(x) · esech(2x) · sech(2x)tgh(2x)
(29)
dy
dx
=
3e3x(x2 + 1) ln(x2 + 1)− 2xe3x
(x2 + 1)(ln(x2 + 1))2
(30)
dy
dx
=
10arcsin(x) ln 10√
1− x2
(31)
dy
dx
=
2x− 1
(ln 4)[tan−1(x2 − x)][(x2 − x)2 + 1]
(32)
dy
dx
=
((1 + x2)13(e6 arctanx)
(cos(3x))10
)(26x+ 6
1 + x2
+ 30 tan(3x)
)
(33)
dy
dx
= 3 tanh(3x)
(34)
dy
dx
= cosh(x)esinhx + ex cosh(ex)
(35)
dy
dx
= ((cosx)sinx)(cosx ln(cosx)− tanx sinx)
(36)− cos(cos(cossec(cotg(pix))))sen(cossec(cotg(pix)))cossec(cotg(pix))cotg(cotg(pix))cossec2(pix)pi
5. (a) y′ =
3x2y − 4x3y2 − 2y3
2x4y − x3 + 6xy2 (b) y
′ =
y2 − yex/y
y2 − xex/y
(c) y′ =
4x
√
x+ y − 1
1− 4y√x+ y (d) y
′ =
cos(x)cos(y)− cos(x)
sen(x)sen(y)− sen(y)
6. (a) y′′ =
x2/3 + y2/3
3y1/3x4/3
(b) y′′ =
81x2 − 9y2
y3
7. −250 cos(2x)
8. 1
2
9. y′(0) = 0
10. (a) (1, 999)4 ≈ 15, 968 (b) 3√1001 ≈ 10, 003 (c) sec(0, 08) ≈ 1
11. (12 +
3pi
2
)cm2
12. (a) 270cm3 (b) 36cm2
13. 140 cm2/s
14. 6 cm/s
15.
9
4
m/s
16.
4
3pi
m/min
17. −0, 01 rad/s
18. 150
√
3 cm2/min