Lista 8 Alessandra Verri

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Universidade Federal de Sa˜o Carlos-Departamento de Matema´tica
89109-Ca´lculo 1-Turma E: Lista 8
Prof(a) Alessandra Verri 11 de maio de 2017
Exerc´ıcio 1. Usando a regra da cadeia, derive a func¸o˜es:
(a) F (x) = (x3+4x)7 (b) g(x) =
√
x2 − 7x (c) h(t) =
(
t− 1
t
)3/2
(d) y = cos(a3+x3), a ∈ R
(e) y = e−mx, m ∈ R (f) G(x) = (3x− 2)10(5x2 − x + 1)12 (g) y =
√
x +
√
x
(h) y = (2x− 5)4(8x2 − 5)−3 (i) y = xe−x2 (j) F (y) =
(
y − 6
y + 7
)3
(k) f(z) =
1
5
√
2z − 1
(l) y = tg (cosx) (m) y = 5−1/x (n) sen3x + cos3 x (o) y = (1 + cos2 x)6
(p) y =
e3x
1 + ex
(q) y = ex cosx
Exerc´ıcio 2. Suponha que F (x) = f(g(x)) e g(3) = 6, g′(3) = 4, f ′(3) = 2 e f ′(6) = 7. Econtre
F ′(3).
Exerc´ıcio 3. Encontre dy/dx diferenciando implicitamente.
(a) x2 + y2 = 1 (b) x3 + x2y + 4y2 = 6 (c) x2y + xy2 = 3x (d)
y
x− y = x
2 + 1
(e)
√
xy = 1 + x2y (f) 4 cosx sen y = 1
Respostas: 1. (a) 7(x3 + 4x)6(3x2 + 4) (b) (2x− 7)/(2√x2 − 7x) (c) (3/2)(t− 1/t)1/2(1 + 1/t2)
(d) −3x2sen (a3 + x3) (e) −me−mx (f) 6(3x− 2)9(5x2 − x + 1)11(85x2 − 51x + 9)
(g)[1 + 1/(2
√
x)]/(2
√
x +
√
x) (h) 8(2x− 5)3(8x2 − 5)−4(−4x2 + 30x− 5) (i) e−x2(1− 2x2)
(j) 39(y − 6)2/(y + 7)4 (k) −(2/5)(2z − 1)−6/5 (l) −senx sec2(cosx)
(m) 5−1/x(ln 5)/x2 (n) 3 senx cosx(senx− cosx) (o) −12 cosxsenx(1 + cos2 x)5
(p) (3e3x + 2e4x)/(1 + ex)2 (q) (cosx− x senx)ex cosx
2. 28
3. (a) y′ = −x/y (b) y′ = −x(3x + 2y)/(x2 + 8y) (c) y′ = (3− 2xy − y2)/(x2 + 2xy)
(d) y′ = (y/x) + 2(x− y)2 (e) y′ = (4xy√xy − y)/(x− 2x2√xy) (f) y′ = tg x tg y