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(a) Falsa (b) Falsa 8. (a) y ′ = 4 cos 4x (b) y ′ = −5 sin 5x (c) y ′ = 3e 3x (b) y ′ = (e) y ′ = 3t 2 cos t 3 (f) g ′ (t) = 2 2t + 1 (g) (h) 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 x ′ = e sin t cos t f ′ (x) = −e x sin e x (i) y ′ = 3(sin x + cos x) 2 (cos x − sin x) 3 (j) y ′ = 2 √ 3x + 1 (k) √ ( ) y ′ 2 x + 1 2 = 3(x + 1) 2 · 3 x − 1 (l) y ′ = 2 tan(sin θ) sec 2 (sin θ) cos θ (m) x ′ 2t + 3 = t 2 + 3t + 9 (n) f ′ (x) = e tan x sec 2 x (o) y ′ = − sin x cos(cos x) (p) g ′ (t) = 8t(t 2 + 3) 3 (q) f ′ (x) = −2x sin(x 2 + 3) (r) y ′ = 1 + ex 2 √ x + e x (s) y ′ = (ln(t4 ) + 4) 2 √ t ln(t 4 ) (t) y ′ = 3x2 cos(tan √ 1 + x 3 ) sec 2 √ 1 + x 3 (u) y ′ = e 3x (1 + 3x) 2 √ 1 + x 3 (v) y ′ = e x (cos 2x − 2 sin 2x) (w) y ′ = e −x (cos x − sin x) (x) y ′ = e −2t (3 cos 3t − 2 sin 3t) (y) f ′ 2 (x) = 2x + 1 − 2xe−x2 (z) g ′ 4e 2t (t) = (e 2 t + 1) 2 (a 1 ) y ′ −5 sin 5x sin 2x − 2 cos 5x cos 2x = sin 2 2x (b 1 ) f ′ (x) = 3(e −x + e x2 ) 2 (−e −x + 2xe x2 ) (c 1 ) y ′ = 3t 2 e −3t (1 − t) (d 1 ) y ′ = 3(sin 3x + cos 2x) 2 (3 cos 3x − 2 sin 2x) (e 1 ) y ′ 2x − e−x = 2 √ x 2 + e −x (f 1 ) y ′ 2x = ln(2x + 1) + (2x + 1) (g 1 ) y ′ = 6x[ln(x2 + 1)] 2 (h 1 ) y ′ = sec x x 2 + 1 (i 1 ) y ′ 2x + 8 = x 2 + 8x + 1 (j 1 ) f ′ (x) = 3 √ 6x + 2 (k 1 ) f ′ (x) = e 3x x 3 (4 + 3x) 9. (a) y ′ = 1 − y4 − 2xy 4xy 3 + x 2 − 3 (c) Verdadeira (d) Falsa (l 1 ) f ′ (x) = 4 sin 3 x cos x (m 1 ) f ′ (x) = 10 sec 2 2x (n 1 ) f ′ (x) = (5 − x 2 ) 2 [(6x 2 − 3)(5 − x 2 ) − 6x(2x 3 − 3x)] (o 1 ) f ′ 9 (x) = 2 √ (3x − 5) 3 (p 1 ) y ′ = e x2 +x+1 (2x + 1) (q 1 ) y ′ = 2 cos 2x cos x − sin 2x sin x (r 1 ) y ′ = 32(2x 2 − 4x + 1) 7 (x − 1) (s 1 ) q ′ = 1 − r √ 2r − r 2 (t 1 ) s ′ = 3π 2 cos ( 3πx 2 ) − 3π 2 ( ) 3πx sin 2 (u 1 ) h ′ (x) = tan(2 √ x) + √ x sec 2 (2 √ x) (v 1 ) r ′ = 2θ cos θ 2 cos 2θ − 2 sin θ 2 sin 2θ (w 1 ) y ′ = (4x + 3)3 (4x + 7) (x + 1) 4 (x 1 ) y ′ = e x sin e x − e cos x sin x (y 1 ) y ′ = 5 sec 5x (z 1 ) y ′ = −6x · cossec 2 (3x + 5) (a 2 ) y ′ = (2e2x + e 3x ) (e x + 1) 2 (b 2 ) y ′ = 6x(x 4 − 3x 2 + 5) 2 (2x 2 − 3) (c 2 ) y ′ = − sin(tan x) sec 2 x (d 2 ) y ′ = 3x + 5 √ 2x + 1(2x + 1) (e 2 ) y ′ = 2(2x2 + 1) √ x 2 + 1 (f 2 ) y ′ = ex (1 + x 2 − 2x) (1 + x 2 ) 2 (g 2 ) y ′ = 2e sin 2θ cos 2θ (h 2 ) y ′ = e mx (m cos nx − n sin nx) (i 2 ) y ′ = 1 2 √ x (cos √ x − √ x sin √ x) (j 2 ) y ′ = cotan 4x − 4x · cossec 4x (k 2 ) y ′ cos x − cos(x − sin x) = sin 2 (x − sin x) (l 2 ) y ′ = −5cotan 5x (m 2 ) y ′ 2 sec 2θ(tan 2θ − 1) = (1 + tan2θ) 2 (n 2 ) y ′ = e cx (c 2 sin x + sin x) (o 2 ) y ′ = 2 + x x (p 2 ) y ′ = 2x sec(1 + x 2 ) tan(1 + x 2 ) (q 2 ) y ′ 1 = − (x − 1) 2 (r 2 ) y ′ = − 1 2 √ x + 1 √ 6 √ x 3 (x + √ x) 4 (s 2 ) y ′ = cos√ x 4 √ x sin √ x (t 2 ) y ′ = (cotan x − sin x cos x) = cos3 x (u 2 ) y ′ = − (x2 + 1) 3 (x 2 + 56x + 9) (2x + 1) 4 (3x − 1) 6 (c) y ′ = (2x − y cos xy) x cos xy + 1 sin x (b) y ′ = y − 2x cos y 2 cos 2y − x 2 sin y − x (d) y ′ = ey 2 − xe y 6
(e) y ′ = −x y 10. (a) y ′ = 8 x ln(8) (b) y ′ = −3 cossec x ln(3)cossec x · cotan x (c) y ′ = (x + 2 + 1)x x2 + x x2 +1 ln x · 2x (d) y ′ = 7 x2 +2x ln(7)(2x + 2) (e) y ′ = 3 x ln x ln 3(ln x + 1) (f) y ′ 2 = (1 + 2x) ln 5 (g) y ′ = cos x x(ln(cos x) − x tan x) 11. (a) y ′ = 4 arcsin(2x) √ 1 − 4x 2 (b) y ′ = cos √ x 2 √ x(1 + sin 2 √ x) (f) y ′ = cos x + 3y2 − 2xy 3y 2 + x 2 − 6y (h) y ′ = sinh(x 2 ) + 2x 2 cosh(x 2 ) (i) y ′ 3 sinh 3x = cosh 3x (j) y ′ sinh(sinh(x)) cosh x = − cosh(sinh(x)) 2 (k) y ′ = π10 tan πx sec 2 πx ln 10 (l) y ′ = 2 tanh √ x − √ xsech 2√ x 2 tanh 2 √ x (c) y ′ = 1 2 √ x (d) y ′ = 1 x Coletânea de exercícios elaborada pelos professores: • Dra. Dayse Batistus; • Msc. Ana Munaretto; • Msc. Cristiane Pendeza; • Msc. Adriano Delfino; • Msc. Marieli Musial Tumelero Digitação: • 1 a versão: Acadêmico Bruno Brito. • Versão atual: Acadêmica Larissa Hagedorn Vieira. Referência Bibliográfica: ANTON, H., BIVENS, I. e DAVIS, S. Cálculo. vol. 1. Tradução: Claus I. Doering. 8 ed. Porto Alegre: Bookman, 2007. GUIDORIZZI, H. L. Um curso de cálculo, vol.1 e 2. 5 a ed. LTC Editora, Rio de Janeiro, RJ: 2002. LEITHOLD, L. O cálculo com geometria analítica. Vol.1. 3 a ed. São Paulo: Harbra, 1994. LIMA, J. D. Apostila de Cálculo I. UTFPR, Pato Branco, 2008. STEWART, James. Cálculo. Vol. 2. 6 a ed. São Paulo: Pioneira Thomson Learning, 2009. SWOKOWSKI, E. W. Cálculo com geometria analítica. Vol. 1. 2 a ed. São Paulo: Makron Books do Brasil,1994. THOMAS, G. B. Cálculo. Vol. 1. 10 a ed. São Paulo: Person, 2002. 7
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