Lista de Calculo 1

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Universidade Tecnolo´gica Federal do Parana´
UTFPR — Campus Pato Branco
Exerc´ıcios de Derivadas de Func¸o˜es Reais de Varia´vel Real
1. Usando a definic¸a˜o de derivadas f ′(x) = lim
∆x→0
f(x+ ∆x)− f(x)
∆x
ou f ′(p) = lim
x→p
f(x)− f(p)
x− p ,
calcule a derivada das seguintes func¸o˜es nos pontos dados:
(a) f(x) = 2x2 − 3x+ 4, p = 2
(b) f(x) =
3
x2
, p = 1
(c) f(t) = 3
√
t, p = 8
(d) g(x) = cos x, p =
pi
2
(e) f(x) = 3 sinx, p = 2pi e 0
(f) v =
3√
t
− 2√t; p = 4
(g) f(x) = 5x− x2, f ′(−3) e f ′(0)
(h) f(x) = x+
9
x
, p = −3
2. Calcule a derivada das func¸o˜es abaixo usando as propriedades adequadas:
(a) f(x) = 16x3 − 4x2 + 3
(b) f(x) = −5x3 + 21x2 − 3x+ 4
(c) f(x) = 5
(d) f(t) = 2t− 1
(e) y = 8
(f) y = 2x+ 1
(g) y =
5
√
x2 − 4√x3 + x4
(h) y = x
4
5 − x 16
(i) f(x) = 10100
1000
(j) s(t) =
5t− 1
2t− 7
(k) f(x) =
3
x
+ 2
√
x− 1
4
√
x
(l) f(r) =
4
r2
+
5
r3
(m) f(x) = (2x2 − 1) · (1− 2x)
(n) y = (x2 − 3x4) · (x5 − 1)
(o) g(t) =
5t− 2
1 + t+ t2
(p) f(x) = (x2 + 3x+ 3) · (x+ 3)
(q) f(x) =
2x3
4x+ 2
(r) y = (x+ 2) · (x5 − 6x)
(s) g(x) =
x2 − 4
x+ 0, 5
(t) r = 2 · ( 1√
θ
+
√
θ)
(u) f(x) =
1
(x2 − 1) · (x2 + x+ 1)
(v) v = (1− t) · (1− t2)−1
(w) y =
√
x+
1
3
√
x4
3. Calcule a derivada das func¸o˜es trigonome´tricas abaixo usando as regras de derivac¸a˜o:
(a) f(x) = tan x =
sinx
cosx
(b) g(x) = sec x =
1
cosx
(c) f(x) =
√
x · (2 sinx+ x2)
(d) h(θ) =
pi
2
sin θ − cos θ
(e) y = x3 − 1
2
cosx
(f) y =
5
(2x)3
+ 2 sinx
(g) y =
3
x
+ 5 sinx
(h) y =
cotg x
1 + cotg x
4. Calcule a derivada das func¸o˜es exponenciais e logar´ıtmicas abaixo usando as regras de
derivac¸a˜o:
1
(a) f(x) =
ex
cosx
(b) y = ex · sinx
(c) f(x) = x2 · lnx
(d) f(x) = (x2 + 1) · ex
(e) y =
ex
2ex + 1
(f) y = xex − ex
(g) y = x2ex − xex
(h) y = 2ex
(i) y = e−t(t2 − 2t+ 2)
5. Usando a regra do quociente e do produto, ache
dy
dx
no ponto x = 1:
(a) y =
2x− 1
x+ 3
(b) y =
4x+ 1
x2 − 5
(c) y =
(
3x+ 2
x
)
· (x−5 + 1)
(d) y = (2x8 − x678) ·
(
x+ 1
x− 1
)
6. Resolva e determine se e´ verdadeiro ou falso, se g(x) = x5, enta˜o lim
x→2
g(x)− g(2)
x− 2 = 80.
7. Resolva e determine se e´ verdadeiro ou falso:
(a)
d
dx
(10x) = x10x−1
(b)
d
dx
(ln 10) =
1
10
(c)
d
dx
(tan2 x) =
d
dx
(sec2 x)
(d)
d
dx
|x2 + x| = |2x+ 1|
8. Derive utlizando as regras de derivac¸a˜o.
(a) y = sin 4x
(b) y = cos 5x
(c) y = e3x
(d) y =
√
x+ 1(2− x)5
(x+ 3)7
(e) y = sin t3
(f) g(t) = ln(2t+ 1)
(g) x = esin t
(h) f(x) = cos(ex)
(i) y = (sinx+ cosx)3
(j) y =
√
(3x+ 1)
(k) y = 3
√(
x− 1
x+ 1
)
(l) y = tan2(sin θ)
(m) x = ln(t2 + 3t+ 9)
(n) f(x) = etanx
(o) y = sin(cosx)
(p) g(t) = (t2 + 3)4
(q) f(x) = cos(x2 + 3)
(r) y =
√
(x+ ex)
(s) y =
√
t · ln(t4)
(t) y = sin(tan
√
1 + x3)
(u) y = x · e3x
(v) y = ex · cos 2x
(w) y = e−x · sinx
(x) y = e2t · sin 3t
(y) f(x) = e−x
2
+ ln(2x+ 1)
(z) g(t) =
et − e−t
et + e−t
(a1) y =
cos 5x
sin 2x
(b1) f(x) = (e
−x + ex
2
)3
(c1) y = t
3 · e−3t
(d1) y = (sin 3x+ cos 2x)
3
(e1) y =
√
x2 + e−x
(f1) y = x · ln(2x+ 1)
(g1) y = [ln(x
2 + 1)]
3
(h1) y = ln(secx+ tanx)
(i1) f(x) = ln(x
2 + 8x+ 1)
(j1) f(x) =
√
6x+ 2
(k1) f(x) = x
4 · e3x
(l1) f(x) = sin
4 x
2
(m1) f(x) = 5 tan 2x
(n1) f(x) = (2x
3 − 3x) · (5− x2)3
(o1) f(x) = − 3√
3x− 5
(p1) y = e
x2+x+1
(q1) y = sin 2x · cosx
(r1) y = (2x
2 − 4x+ 1)8
(s1) q =
√
2r − r2
(t1) s = sin
(
3pit
2
)
+ cos
(
3pit
2
)
(u1) h(x) = x tan(2
√
x) + 7
(v1) r = sin(θ
2) cos(2θ)
(w1) y = (4x+ 3)
4(x+ 1)−3
(x1) y = x tan
−1(4x)
(y1) y = e
cosx + cos(ex)
(z1) y = cotg (3x
2 + 5)
(a2) y =
ex
e−x + 1
(b2) y = (x
4 − 3x2 + 5)3
(c2) y = cos(tanx)
(d2) y =
3x− 2√
2x+ 1
(e2) y = 2x
√
x2 + 1
(f2) y =
ex
1 + x2
(g2) y = e
sin 2θ
(h2) y = e
mx cosnx
(i2) y =
√
x cos
√
x
(j2) y =
(x2 + 1)4
(2x+ 1)3(3x− 1)5
(k2) y =
1
sin(x− sin(x))
(l2) y = ln(cossec 5x)
(m2) y =
sec 2θ
1 + tan 2θ
(n2) y = e
cx(c sinx− cosx)
(o2) y = ln(x
2ex)
(p2) y = sec(1 + x
2)
(q2) y = (1− x−1)−1
(r2) y =
1
3
√
(x+
√
x)
(s2) y =
√
sin
√
x
(t2) y = ln(sinx)− 1
2
sin2 x
9. Derive utilizando a derivada impl´ıcita:
(a) xy4 + x2y = x+ 3y
(b) x2 cos y + sin 2y = xy
(c) sin(xy) = x2 − y
(d) y = xey − y − 1
(e) y2 + x2 = 1
(f) y3 + yx2 = sen x+ 3y2x
10. Encontre a derivada das seguintes func¸o˜es:
(a) y = 8x
(b) y = 3cossec (x)
(c) y = x(x
2+1)
(d) y = 7x
2+2x
(e) y = 3x lnx
(f) y = log5(1 + 2x)
(g) y = (cosx)x
(h) y = x sinhx2
(i) y = ln(cosh 3x)
(j) y = cosh−1(sinhx)
(k) y = 10tanpiθ
(l) y = x · tanh−1√x
11. Derive utilizando a derivada inversa:
(a) y = (arcsin 2x)2
(b) y = arctan(arcsin
√
x)
(c) y =
√
x
(d) y = ln x
3
Respostas
1. (a) f ′(2) = 5
(b) f ′(1) = −6
(c) f ′(8) =
1
12
(d) g′
(pi
2
)
= −1
(e) f ′(2pi) = 3
(f) v′(4) = −11
16
(g) f ′(−3) = 11 e f ′(0) = 5
(h) f ′(−3) = 0
2. (a) f ′(x) = 48x2 − 8x
(b) f ′(x) = −15x2 + 42x− 3
(c) f ′(x) = 0
(d) f ′(x) = 2
(e) y′ = 0
(f) y′ = 2
(g) y′ =
2
5
5
√
x3
− 3
4 4
√
x
+ 4x3
(h) f ′(x) =
4
5 5
√
x
− 1
6
6
√
x5
(i) f ′(x) = 0
(j) s′(t) =
−33
(2t− 7)2
(k) f ′(x) = − 3
x2
+
1√
x
+
1
8x
√
x
(l) f ′(r) =
−8r − 15
r4
(m) f ′(x) = 2(−6x2 + 2x+ 1)
(n) y′ = x(−27x7 + 7x5 + 12x2 − 2)
(o) g′(t) =
7− 5t2 + 4t
(1 + t+ t2)2
(p) f ′(x) = 3(x2 + 4x+ 4)
(q) f ′(x) =
x2(4x+ 3)
(2x+ 1)2
(r) y′ = 2(3x5 + 5x4 − 6x− 6)
(s) g′(x) =
x2 + x+ 4
(x+ 0, 5)2
(t) r′ =
1√
θ
− 1
θ
√
θ
(u) f ′(x) =
−4x3 − 3x2 + 1
((x2 − 1)(x2 + x+ 1))2
(v) v′ =
−t2 + 2t− 1
(1− t2)2
(w) y′ =
1
2
√
x
− 4
3x2 3
√
x
3. (a) f ′(x) = sec2 x
(b) g′(t) = tan t · sec t
(c) f ′(x) =
2 sinx+ x2
2
√
x
+ 2
√
x(cosx+ x)
(d) h′(θ) =
pi
2
cos θ + sin θ
(e) y′ = 3x2 +
1
2
sinx
(f) y′ = − 15
8x4
+ 2 cosx
(g) y′ = 5 cos x− 3
x2
(h) y′ = − 1
2 sinx cosx+ 1
4. (a) f ′(x) =
ex(sinx+ cosx)
cos2 x
(b) f ′(x) = ex(sinx+ cosx)
(c) f ′(x) = x(2 lnx+ 1)
(d) f ′(x) = ex(x2 + 2x+ 1)
(e) y′ =
ex
(2ex + 1)2
(f) y′ = ex · x
(g) y′ = ex(x2 + x− 1)
(h) y′ = 2ex
(i) y′ =
(−t2 + 4t− 4)
et
5. (a) y′(1) =
7
16
(b) y′(1) = −13
8
(c) y′(1) = −29
(d) Descont´ınua em x = 1
6. verdadeira
4
7. (a) Falsa
(b) Falsa
(c) Verdadeira
(d) Falsa
8. (a) y′ = 4 cos 4x
(b) y′ = −5 sin 5x
(c) y′ = 3e3x
(b) y′ = 1(2− x)
5
2
√
x + 1(x + 3)7
− 5
√
x + 1(2− x)4
(x + 3)7
− 7
√
x + 1(2− x)5
(x + 3)8
(e) y′ = 3t2 cos t3
(f) g′(t) =
2
2t + 1
(g) x′ = esin t cos t
(h) f ′(x) = −ex sin ex
(i) y′ = 3(sin x + cos x)2(cos x− sin x)
(j) y′ =
3
2
√
3x + 1
(k) y′ =
2
3(x + 1)2
· 3
√(
x + 1
x− 1
)2
(l) y′ = 2 tan(sin θ) sec2(sin θ) cos θ
(m) x′ =
2t + 3
t2 + 3t + 9
(n) f ′(x) = etan x sec2 x
(o) y′ = − sin x cos(cos x)
(p) g′(t) = 8t(t2 + 3)3
(q) f ′(x) = −2x sin(x2 + 3)
(r) y′ =
1 + ex
2
√
x + ex
(s) y′ =
(ln(t4) + 4)
2
√
t ln(t4)
(t) y′ =
3x2 cos(tan
√
1 + x3) sec2
√
1 + x3
2
√
1 + x3
(u) y′ = e3x(1 + 3x)
(v) y′ = ex(cos 2x− 2 sin 2x)
(w) y′ = e−x(cos x− sin x)
(x) y′ = e2t(3 cos 3t + 2 sin 3t)
(y) f ′(x) =
2
2x + 1
− 2xe−x2
(z) g′(t) =
4e2t
(e2t + 1)2
(a1) y
′ =
−5 sin 5x sin 2x− 2 cos 5x cos 2x
sin22x
(b1) f
′(x) = 3(e−x + ex
2
)2(−e−x + 2xex2)
(c1) y
′ = 3t2e−3t(1− t)
(d1) y
′ = 3(sin 3x + cos 2x)2(3 cos 3x− 2 sin 2x)
(e1) y
′ =
2x− e−x
2
√
x2 + e−x
(f1) y
′ = ln(2x + 1) +
2x
(2x + 1)
(g1) y
′ =
6x[ln(x2 + 1)]2
x2 + 1
(h1) y
′ = sec x
(i1) y
′ =
2x + 8
x2 + 8x + 1
(j1) f
′(x) =
3
√
6x + 2
(k1) f
′(x) = e3xx3(4 + 3x)
(l1) f
′(x) = 4 sin3 x cos x
(m1) f
′(x) = 10 sec2 2x
(n1) f
′(x) = (5− x2)2[(6x2 − 3)(5− x2)− 6x(2x3 − 3x)]
(o1) f
′(x) =
9
2
√
(3x− 5)3
(p1) y
′ = ex
2+x+1(2x + 1)
(q1) y
′ = 2 cos 2x cos x− sin 2x sin x
(r1) y
′ = 32(2x2 − 4x + 1)7(x− 1)
(s1) q
′ =
1− r√
2r − r2
(t1) s
′ =
3pi
2
cos
(
3pix
2
)
− 3pi
2
sin
(
3pix
2
)
(u1) h
′(x) = tan(2√x) +√x sec2(2√x)
(v1) r
′ = 2θ cos θ2 cos 2θ − 2 sin θ2 sin 2θ
(w1) y
′ =
(4x + 3)3(4x + 7)
(x + 1)4
(x1) y
′ = tg−1(4x) +
4x
1 + 16x2
(y1) y
′ = −sen x ecos x + ex sen (ex)
(z1) y
′ = −6x · cossec2(3x + 5)
(a2) y
′ =
(2e2x + e3x)
(ex + 1)2
(b2) y
′ = 6x(x4 − 3x2 + 5)2(2x2 − 3)
(c2) y
′ = − sin(tan x) sec2 x
(d2) y
′ =
3x + 5
√
2x + 1(2x + 1)
(e2) y
′ =
2(2x2 + 1)√
x2 + 1
(f2) y
′ =
ex(1 + x2 − 2x)
(1 + x2)2
(g2) y
′ = 2esin 2θ cos 2θ
(h2) y
′ = emx(m cosnx− n sinnx)
(i2) y
′ =
1
2
√
x
(cos
√
x−√x sin√x)
(j2) y
′ = cotan 4x− 4x · cossec 4x
(k2) y
′ =
cos x− cos(x− sin x)
sin2(x− sin x)
(l2) y
′ = −5cotan 5x
(m2) y
′ =
2 sec 2θ(tan 2θ − 1)
(1 + tan2θ)2
(n2) y
′ = ecx(c2 sin x + sin x)
(o2) y
′ =
2 + x
x
(p2) y
′ = 2x sec(1 + x2) tan(1 + x2)
(q2) y
′ = − 1
(x− 1)2
(r2) y
′ = − 1
6
2
√
x + 1
√
x 3
√
(x +
√
x)4
(s2) y
′ =
cos
√
x
4
√
x sin
√
x
(t2) y
′ = (cotan x− sin x cos x) = cos
3 x
sin x
(u2) y
′ = − (x
2 + 1)3(x2 + 56x + 9)
(2x + 1)4(3x− 1)6
9.
5
(a) y′ =
1− y4 − 2xy
4xy3 + x2 − 3
(b) y′ =
y − 2x cos y
2 cos 2y − x2 sin y − x
(c) y′ =
(2x− y cosxy)
x cosxy + 1
(d) y′ =
ey
2− xey
(e) y′ =
−x
y
(f) y′ =
cos x+ 3y2 − 2xy
3y2 + x2 − 6yx
10. (a) y′ = 8x ln(8)
(b) y′ = −3cossec x ln(3)cossec x · cotan x
(c) y′ = (x2 + 1)xx
2
+ x(x
2+1) lnx · 2x
(d) y′ = 7x
2+2x ln(7)(2x+ 2)
(e) y′ = 3x lnx ln 3(lnx+ 1)
(f) y′ =
2
(1 + 2x) ln 5
(g) y′ = cosx x(ln(cosx)− x tanx)
(h) y′ = sinh(x2) + 2x2 cosh(x2)
(i) y′ =
3 sinh 3x
cosh 3x
(j) y′ = −sinh(sinh(x)) coshx
cosh(sinh(x))2
(k) y′ = pi10tanpix sec2 pix ln 10
(l) y′ =
2 tanh
√
x−√xsech2√x
2 tanh2
√
x
11. (a) y′ =
4 arcsin(2x)√
1− 4x2
(b) y′ =
cos
√
x
2
√
x(1 + sin2
√
x)
(c) y′ =
1
2
√
x
(d) y′ =
1
x
Coletaˆnea de exerc´ıcios elaborada pelos professores:
• Dra. Dayse Batistus;
• Msc. Ana Munaretto;
• Msc. Cristiane Pendeza;
• Msc. Adriano Delfino;
• Msc. Marieli Musial Tumelero
Digitac¸a˜o:
• 1a versa˜o: Acadeˆmico Bruno Brito.
• Versa˜o atual: Acadeˆmica Larissa Hagedorn Vieira.
Refereˆncia Bibliogra´fica:
ANTON, H., BIVENS, I. e DAVIS, S. Ca´lculo. vol. 1. Traduc¸a˜o: Claus I. Doering. 8 ed.
Porto Alegre: Bookman, 2007.
GUIDORIZZI, H. L. Um curso de ca´lculo, vol.1 e 2. 5a ed. LTC Editora, Rio de Janeiro,
RJ: 2002.
LEITHOLD, L. O ca´lculo com geometria anal´ıtica. Vol.1. 3a ed. Sa˜o Paulo: Harbra,
1994.
LIMA, J. D. Apostila de Ca´lculo I. UTFPR, Pato Branco, 2008.
STEWART, James. Ca´lculo. Vol. 2. 6a ed. Sa˜o Paulo: Pioneira Thomson Learning, 2009.
SWOKOWSKI, E. W. Ca´lculo com geometria anal´ıtica. Vol. 1. 2a ed. Sa˜o Paulo: Makron
Books do Brasil,1994.
THOMAS, G. B. Ca´lculo. Vol. 1. 10aed. Sa˜o Paulo: Person, 2002.
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