Universidade Tecnológica Federal do Paraná - UTFPR
Universidade Tecnológica Federal do Paraná - UTFPR
Universidade Tecnológica Federal do Paraná - UTFPR
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<strong>Universidade</strong> <strong>Tecnológica</strong> <strong>Federal</strong> <strong>do</strong> <strong>Paraná</strong><br />
<strong>UTFPR</strong> — Campus Pato Branco<br />
Exercícios de Derivadas de Funções Reais de Variável Real<br />
1. Usan<strong>do</strong> a definição de derivadas f ′ f(x + ∆x) − f(x)<br />
(x) = lim<br />
ou f ′ f(x) − f(p)<br />
(p) = lim<br />
,<br />
∆x→0 ∆x<br />
x→p x − p<br />
calcule a derivada das seguintes funções nos pontos da<strong>do</strong>s:<br />
(a) f(x) = 2x 2 − 3x + 4, p = 2<br />
(b) f(x) = 3 x , p = 1<br />
2<br />
(c) f(t) = 3√ t, p = 8<br />
(d) g(x) = cos x, p = π 2<br />
(e) f(x) = 3 sin x, p = 2π e 0<br />
(f) v = √ 3 − 2 √ t; t = 4<br />
t<br />
(g) f(x) = 5x − x 2 , f ′ (−3) e f ′ (0)<br />
(h) f(x) = x + 9 x , x = −3<br />
2. Calcule a derivada das funções abaixo usan<strong>do</strong> as propriedades adequadas:<br />
(a) f(x) = 16x 3 − 4x 2 + 3<br />
(b) f(x) = −5x 3 + 21x 2 − 3x + 4<br />
(c) f(x) = 5<br />
(d) f(t) = 2t − 1<br />
(e) y = 8<br />
(f) y = 2x + 1<br />
(g) y = 5√ x 2 − 4√ x 3 + x 4<br />
(h) y = x 4 5 − x 1 6<br />
(i) f(x) = 10 1001000<br />
(j) s(t) = 5t − 1<br />
2t − 7<br />
(k) f(x) = 3 x + 2√ x − 1<br />
4 √ x<br />
(l) f(r) = 4 r 2 + 5 r 3<br />
(m) f(x) = (2x 2 − 1) · (1 − 2x)<br />
(n) y = (x 2 − 3x 4 ) · (x 5 − 1)<br />
(o) g(t) = 5t − 2<br />
1 + t + t 2<br />
(p) f(x) = (x 2 + 3x + 3) · (x + 3)<br />
(q) f(x) =<br />
2x3<br />
4x + 2<br />
(r) y = (x + 2) · (x 5 − 6x)<br />
(s) g(x) = x2 − 4<br />
x + 0, 5<br />
(t) r = 2 · ( 1 √<br />
θ<br />
+ √ θ)<br />
(u) f(x) =<br />
1<br />
(x 2 − 1) · (x 2 + x + 1)<br />
(v) v = (1 − t) · (1 − t 2 ) −1<br />
(w) y = √ x + 1<br />
3√<br />
x<br />
4<br />
3. Calcule a derivada das funções trigonométricas abaixo usan<strong>do</strong> as regras de derivação:<br />
(a) f(x) = tan x = sin x<br />
cos x<br />
(b) g(x) = sec x = 1<br />
cos x<br />
(c) f(x) = √ x · (2 sin x + x 2 )<br />
(e)<br />
(d) h(θ) = π sin θ − cos θ<br />
2<br />
y = x 3 − 1 2 cos x<br />
(f) y = 5<br />
(2x) 3 + 2 sin x<br />
(g) y = 3 x + 5 sin x<br />
(h) y =<br />
cotg x<br />
1 + cotg x<br />
4. Calcule a derivada das funções exponenciais e logarítmicas abaixo usan<strong>do</strong> as regras de<br />
derivação:<br />
1
(a) f(x) =<br />
ex<br />
cos x<br />
(b) y = e x · sin x<br />
(c) f(x) = x 2 · ln x<br />
(d) f(x) = (x 2 + 1) · e x<br />
(e) y =<br />
ex<br />
2e x + 1<br />
(f) y = xe x − e x<br />
(g) y = x 2 e x − xe x<br />
(h) y = 2e x<br />
5. Usan<strong>do</strong> a regra <strong>do</strong> quociente e <strong>do</strong> produto, ache dy<br />
dx<br />
(a) y = 2x − 1<br />
(c) y =<br />
x + 3<br />
(b) y = 4x + 1<br />
x 2 − 5<br />
(i) y = e −t (t 2 − 2t + 2)<br />
no ponto x = 1:<br />
( ) 3x + 2<br />
· (x −5 + 1)<br />
x<br />
( ) x + 1<br />
x − 1<br />
(d) y = (2x 8 − x 678 ) ·<br />
6. Resolva e determine se é verdadeiro ou falso, se g(x) = x 5 , então lim<br />
x→2<br />
g(x) − g(2)<br />
x − 2<br />
7. Resolva e determine se é verdadeiro ou falso:<br />
= 80.<br />
(a)<br />
(b)<br />
d<br />
dx (10x ) = x10 x−1<br />
d<br />
dx (ln 10) = 1 10<br />
(c)<br />
(d)<br />
d<br />
dx (tan2 x) = d<br />
dx (sec2 x)<br />
d<br />
dx |x2 + x| = |2x + 1|<br />
8. Derive utlizan<strong>do</strong> as regras de derivação.<br />
(a) y = sin 4x<br />
(b) y = cos 5x<br />
(c) y = e 3x<br />
√ x + 1(2 − x)<br />
5<br />
(d) y =<br />
(x + 3) 7<br />
(e) y = sin t 3<br />
(f) g(t) = ln(2t + 1)<br />
(g) x = e sin t<br />
(h) f(x) = cos(e x )<br />
(i) y = (sin x + cos x) 3<br />
(j) y = √ (3x + 1)<br />
√ (x ) − 1<br />
(k) y = 3 x + 1<br />
(l) y = tan 2 (sin θ)<br />
(m) x = ln(t 2 + 3t + 9)<br />
(n) f(x) = e tan x<br />
(o) y = sin(cos x)<br />
(p) g(t) = (t 2 + 3) 4<br />
(q) f(x) = cos(x 2 + 3)<br />
(r) y = √ (x + e x )<br />
(s) y = √ t · ln(t 4 )<br />
(t) y = sin(tan √ 1 + x 3 )<br />
(u) y = x · e 3x<br />
(v) y = e x · cos 2x<br />
(w) y = e −x · sin x<br />
(x) y = e 2t · sin 3t<br />
(y) f(x) = e −x2 + ln(2x + 1)<br />
(z) g(t) = et − e −t<br />
e t + e −t<br />
cos 5x<br />
(a 1 ) y =<br />
sin 2x<br />
(b 1 ) f(x) = (e −x + e x2 ) 3<br />
(c 1 ) y = t 3 · e −3t<br />
(d 1 ) y = (sin 3x + cos 2x) 3<br />
(e 1 ) y = √ x 2 + e −x<br />
(f 1 ) y = x · ln(2x + 1)<br />
(g 1 ) y = [ln(x 2 + 1)] 3<br />
(h 1 ) y = ln(sec x + tan x)<br />
(i 1 ) f(x) = ln(x 2 + 8x + 1)<br />
(j 1 ) f(x) = √ 6x + 2<br />
(k 1 ) f(x) = x 4 · e 3x<br />
(l 1 ) f(x) = sin 4 x<br />
2
(m 1 ) f(x) = 5 tan 2x<br />
(e 2 ) y = 2x √ x 2 + 1<br />
(n 1 ) f(x) = (2x 3 − 3x) · (5 − x 2 ) 3<br />
(f 2 ) y = ex<br />
3<br />
1 + x 2<br />
(o 1 ) f(x) = −√ 3x − 5<br />
sin 2θ<br />
(g 2 ) y = e<br />
(p 1 ) y = e x2 +x+1<br />
(h 2 ) y = e mx cos nx<br />
(q 1 ) y = sin 2x · cos x<br />
(i 2 ) y = √ x cos √ x<br />
(r 1 ) y = (2x 2 − 4x + 1) 8<br />
(s 1 ) q = √ (x 2 + 1) 4<br />
(j 2 ) y =<br />
2r − r 2<br />
(2x + 1) 3 (3x − 1) 5<br />
( ) ( )<br />
3πt 3πt<br />
1<br />
(t 1 ) s = sin + cos<br />
(k 2 ) y =<br />
2<br />
2<br />
sin(x − sin(x))<br />
(u 1 ) h(x) = x tan(2 √ x) + 7<br />
(l 2 ) y = ln(cossec 5x)<br />
(v 1 ) r = sin(θ 2 ) cos(2θ)<br />
sec 2θ<br />
(m 2 ) y =<br />
(w 1 ) y = (4x + 3) 4 (x + 1) −3<br />
1 + tan 2θ<br />
(n<br />
(x 1 ) y = x tan −1 2 ) y = e cx (c sin x − cos x)<br />
(4x)<br />
(o<br />
(y 1 ) y = e cos x + cos(e x 2 ) y = ln(x 2 e x )<br />
)<br />
(z 1 ) y = cotg (3x 2 (p<br />
+ 5)<br />
2 ) y = sec(1 + x 2 )<br />
(a 2 ) y = ex<br />
(q 2 ) y = (1 − x −1 ) −1<br />
e −x + 1<br />
1<br />
(r 2 ) y = √ √<br />
(b 2 ) y = (x 4 − 3x 2 + 5) 3<br />
3<br />
(x + x)<br />
(c 2 ) y = cos(tan x)<br />
(s 2 ) y = √ sin √ x<br />
(d 2 ) y = √ 3x − 2<br />
(t 2x + 1 2 ) y = ln(sin x) − 1 2 sin2 x<br />
9. Derive utilizan<strong>do</strong> a derivada implícita:<br />
(a) xy 4 + x 2 y = x + 3y<br />
(b) x 2 cos y + sin 2y = xy<br />
(c) sin(xy) = x 2 − y<br />
(d) y = xe y − y − 1<br />
(e) y 2 + x 2 = 1<br />
(f) y 3 + yx 2 = sen x + 3y 2 x<br />
10. Encontre a derivada das seguintes funções:<br />
(a) y = 8 x<br />
cossec (x)<br />
(b) y = 3<br />
(c) y = x (x2 +1)<br />
(d) y = 7 x2 +2x<br />
(e) y = 3 x ln x<br />
(f) y = log 5 (1 + 2x)<br />
(g) y = (cos x) x<br />
(h) y = x sinh x 2<br />
(i) y = ln(cosh 3x)<br />
(j) y = cosh −1 (sinh x)<br />
tan πθ<br />
(k) y = 10<br />
(l) y = x · tanh −1 √ x<br />
11. Derive utilizan<strong>do</strong> a derivada inversa:<br />
(a) y = (arcsin 2x) 2<br />
(b) y = arctan(arcsin √ x)<br />
(c) y = √ x<br />
(d) y = ln x<br />
3
Respostas<br />
1. (a) f ′ (2) = 5<br />
(b) f ′ (1) = −6<br />
(c) f ′ (8) = 1<br />
(<br />
12<br />
π<br />
)<br />
(d) g ′ = −1<br />
2<br />
2. (a) f ′ (x) = 48x 2 − 8x<br />
(b) f ′ (x) = −15x 2 + 42x − 3<br />
(c) f ′ (x) = 0<br />
(d) f ′ (x) = 2<br />
(e) y ′ = 0<br />
(f) y ′ = 2<br />
(g) y ′ = 2<br />
5 5√ x 3 − 3<br />
4 4√ x + 4x3<br />
(h) f ′ (x) = 4<br />
5 5√ x − 1<br />
6 6√ x 5<br />
(i) f ′ (x) = 0<br />
(j) s ′ (t) =<br />
−33<br />
(2t − 7) 2<br />
(k) f ′ (x) = − 3 x + √ 1 + 1<br />
2 x 8x √ x<br />
(l) f ′ −8r − 15<br />
(r) =<br />
r 4<br />
(m) f ′ (x) = 2(−6x 2 + 2x + 1)<br />
(e) f ′ (2π) = 3<br />
(f) v ′ (4) = − 11<br />
16<br />
(g) f ′ (−3) = 11 e f ′ (0) = 5<br />
(h) f ′ (−3) = 0<br />
(n) y ′ = x(−27x 7 + 7x 5 + 12x 2 − 2)<br />
(o) g ′ (t) = 7 − 5t2 + 4t<br />
(1 + t + t 2 ) 2<br />
(p) f ′ (x) = x 2 + 4x + 4<br />
(q) f ′ (x) = x2 (4x + 3)<br />
(2x + 1) 2<br />
(r) y ′ = 3x 5 + 5x 4 − 6x − 6<br />
(s) g ′ (x) = x2 + x + 4<br />
(x + 0, 5) 2<br />
(t) r ′ = 1 √<br />
θ<br />
− 1<br />
θ √ θ<br />
(u) f ′ (x) =<br />
(v) v ′ = t2 − 2t − 1<br />
(1 + t 2 ) 2<br />
−4x 3 − 3x 2 + 1<br />
((x 2 − 1)(x 2 + x + 1)) 2<br />
(w) y ′ = 1<br />
2 √ x − 4<br />
3x 2 3√ x<br />
3. (a) f ′ (x) = sec 2 x<br />
(b) g ′ (t) = tan t · sec t<br />
(c) f ′ (x) =<br />
2 sin x + x2<br />
2 √ x<br />
(e)<br />
(d) h ′ (θ) = π cos θ + sin θ<br />
2<br />
+ 2 √ x(cos x + x)<br />
y ′ = 3x 2 + 1 2 sin x<br />
(f) y ′ = − 15<br />
8x 4 + 2 cos x<br />
(g) y ′ = 5 cos x − 3 x 2<br />
(h) y ′ 1<br />
= −<br />
2 sin x cos x + 1<br />
4. (a) f ′ (x) = ex (sin x + cos x)<br />
(f) y ′ = e x · x<br />
cos 2 x<br />
(b) f ′ (x) = e x (sin x + cos x)<br />
(g) y ′ = e x (x 2 + x − 1)<br />
(c) f ′ (x) = x(2 ln x + 1)<br />
(d) f ′ (x) = e x (x 2 (h) y ′ = 2e x<br />
+ 2x + 1)<br />
(e) y ′ e x<br />
=<br />
(2e x + 1) 2 (i) y ′ = (−t2 + 4t − 4)<br />
e t<br />
5. (a) y ′ (1) = 7 16<br />
(b) y ′ (1) = − 13<br />
8<br />
6. Verdadeira<br />
4<br />
(c) y ′ (1) = −29<br />
(d) Descontínua em x = 1
7.<br />
5
(a) Falsa<br />
(b) Falsa<br />
8. (a) y ′ = 4 cos 4x<br />
(b) y ′ = −5 sin 5x<br />
(c) y ′ = 3e 3x<br />
(b) y ′ =<br />
(e) y ′ = 3t 2 cos t 3<br />
(f) g ′ (t) = 2<br />
2t + 1<br />
(g)<br />
(h)<br />
1(2 − x) 5<br />
2 √ x + 1(x + 3) 7 − 5√ x + 1(2 − x) 4<br />
(x + 3) 7 − 7√ x + 1(2 − x) 5<br />
(x + 3) 8<br />
x ′ = e sin t cos t<br />
f ′ (x) = −e x sin e x<br />
(i) y ′ = 3(sin x + cos x) 2 (cos x − sin x)<br />
3<br />
(j) y ′ =<br />
2 √ 3x + 1<br />
(k)<br />
√ ( )<br />
y ′ 2<br />
x + 1 2<br />
=<br />
3(x + 1) 2 · 3<br />
x − 1<br />
(l)<br />
y ′ = 2 tan(sin θ) sec 2 (sin θ) cos θ<br />
(m) x ′ 2t + 3<br />
=<br />
t 2 + 3t + 9<br />
(n)<br />
f ′ (x) = e tan x sec 2 x<br />
(o) y ′ = − sin x cos(cos x)<br />
(p) g ′ (t) = 8t(t 2 + 3) 3<br />
(q) f ′ (x) = −2x sin(x 2 + 3)<br />
(r) y ′ = 1 + ex<br />
2 √ x + e x<br />
(s) y ′ = (ln(t4 ) + 4)<br />
2 √ t ln(t 4 )<br />
(t) y ′ = 3x2 cos(tan √ 1 + x 3 ) sec 2 √ 1 + x 3<br />
(u) y ′ = e 3x (1 + 3x)<br />
2 √ 1 + x 3<br />
(v) y ′ = e x (cos 2x − 2 sin 2x)<br />
(w) y ′ = e −x (cos x − sin x)<br />
(x) y ′ = e −2t (3 cos 3t − 2 sin 3t)<br />
(y) f ′ 2<br />
(x) =<br />
2x + 1 − 2xe−x2<br />
(z) g ′ 4e 2t<br />
(t) =<br />
(e 2 t + 1) 2<br />
(a 1 ) y ′ −5 sin 5x sin 2x − 2 cos 5x cos 2x<br />
=<br />
sin 2 2x<br />
(b 1 ) f ′ (x) = 3(e −x + e x2 ) 2 (−e −x + 2xe x2 )<br />
(c 1 ) y ′ = 3t 2 e −3t (1 − t)<br />
(d 1 ) y ′ = 3(sin 3x + cos 2x) 2 (3 cos 3x − 2 sin 2x)<br />
(e 1 ) y ′ 2x − e−x<br />
=<br />
2 √ x 2 + e −x<br />
(f 1 ) y ′ 2x<br />
= ln(2x + 1) +<br />
(2x + 1)<br />
(g 1 ) y ′ = 6x[ln(x2 + 1)] 2<br />
(h 1 ) y ′ = sec x<br />
x 2 + 1<br />
(i 1 ) y ′ 2x + 8<br />
=<br />
x 2 + 8x + 1<br />
(j 1 ) f ′ (x) =<br />
3<br />
√ 6x + 2<br />
(k 1 ) f ′ (x) = e 3x x 3 (4 + 3x)<br />
9. (a) y ′ = 1 − y4 − 2xy<br />
4xy 3 + x 2 − 3<br />
(c) Verdadeira<br />
(d) Falsa<br />
(l 1 ) f ′ (x) = 4 sin 3 x cos x<br />
(m 1 ) f ′ (x) = 10 sec 2 2x<br />
(n 1 ) f ′ (x) = (5 − x 2 ) 2 [(6x 2 − 3)(5 − x 2 ) − 6x(2x 3 − 3x)]<br />
(o 1 ) f ′ 9<br />
(x) =<br />
2 √ (3x − 5) 3<br />
(p 1 ) y ′ = e x2 +x+1 (2x + 1)<br />
(q 1 ) y ′ = 2 cos 2x cos x − sin 2x sin x<br />
(r 1 ) y ′ = 32(2x 2 − 4x + 1) 7 (x − 1)<br />
(s 1 ) q ′ = 1 − r<br />
√<br />
2r − r 2<br />
(t 1 ) s ′ = 3π 2 cos ( 3πx<br />
2<br />
)<br />
− 3π 2<br />
( ) 3πx<br />
sin<br />
2<br />
(u 1 ) h ′ (x) = tan(2 √ x) + √ x sec 2 (2 √ x)<br />
(v 1 ) r ′ = 2θ cos θ 2 cos 2θ − 2 sin θ 2 sin 2θ<br />
(w 1 ) y ′ = (4x + 3)3 (4x + 7)<br />
(x + 1) 4<br />
(x 1 ) y ′ = e x sin e x − e cos x sin x<br />
(y 1 ) y ′ = 5 sec 5x<br />
(z 1 ) y ′ = −6x · cossec 2 (3x + 5)<br />
(a 2 ) y ′ = (2e2x + e 3x )<br />
(e x + 1) 2<br />
(b 2 ) y ′ = 6x(x 4 − 3x 2 + 5) 2 (2x 2 − 3)<br />
(c 2 ) y ′ = − sin(tan x) sec 2 x<br />
(d 2 ) y ′ =<br />
3x + 5<br />
√ 2x + 1(2x + 1)<br />
(e 2 ) y ′ = 2(2x2 + 1)<br />
√<br />
x 2 + 1<br />
(f 2 ) y ′ = ex (1 + x 2 − 2x)<br />
(1 + x 2 ) 2<br />
(g 2 ) y ′ = 2e sin 2θ cos 2θ<br />
(h 2 ) y ′ = e mx (m cos nx − n sin nx)<br />
(i 2 ) y ′ = 1<br />
2 √ x (cos √ x − √ x sin √ x)<br />
(j 2 ) y ′ = cotan 4x − 4x · cossec 4x<br />
(k 2 ) y ′ cos x − cos(x − sin x)<br />
=<br />
sin 2 (x − sin x)<br />
(l 2 ) y ′ = −5cotan 5x<br />
(m 2 ) y ′ 2 sec 2θ(tan 2θ − 1)<br />
=<br />
(1 + tan2θ) 2<br />
(n 2 ) y ′ = e cx (c 2 sin x + sin x)<br />
(o 2 ) y ′ = 2 + x<br />
x<br />
(p 2 ) y ′ = 2x sec(1 + x 2 ) tan(1 + x 2 )<br />
(q 2 ) y ′ 1<br />
= −<br />
(x − 1) 2<br />
(r 2 ) y ′ = − 1 2 √ x + 1<br />
√<br />
6<br />
√ x<br />
3<br />
(x + √ x) 4<br />
(s 2 ) y ′ = cos√ x<br />
4 √ x sin √ x<br />
(t 2 ) y ′ = (cotan x − sin x cos x) = cos3 x<br />
(u 2 ) y ′ = − (x2 + 1) 3 (x 2 + 56x + 9)<br />
(2x + 1) 4 (3x − 1) 6<br />
(c) y ′ =<br />
(2x − y cos xy)<br />
x cos xy + 1<br />
sin x<br />
(b) y ′ =<br />
y − 2x cos y<br />
2 cos 2y − x 2 sin y − x<br />
(d) y ′ =<br />
ey<br />
2 − xe y<br />
6
(e) y ′ = −x<br />
y<br />
10. (a) y ′ = 8 x ln(8)<br />
(b) y ′ = −3 cossec x ln(3)cossec x · cotan x<br />
(c) y ′ = (x + 2 + 1)x x2 + x x2 +1 ln x · 2x<br />
(d) y ′ = 7 x2 +2x ln(7)(2x + 2)<br />
(e) y ′ = 3 x ln x ln 3(ln x + 1)<br />
(f) y ′ 2<br />
=<br />
(1 + 2x) ln 5<br />
(g) y ′ = cos x x(ln(cos x) − x tan x)<br />
11. (a) y ′ = 4 arcsin(2x) √<br />
1 − 4x<br />
2<br />
(b) y ′ =<br />
cos √ x<br />
2 √ x(1 + sin 2 √ x)<br />
(f) y ′ = cos x + 3y2 − 2xy<br />
3y 2 + x 2 − 6y<br />
(h) y ′ = sinh(x 2 ) + 2x 2 cosh(x 2 )<br />
(i) y ′ 3 sinh 3x<br />
=<br />
cosh 3x<br />
(j) y ′ sinh(sinh(x)) cosh x<br />
= −<br />
cosh(sinh(x)) 2<br />
(k) y ′ = π10 tan πx sec 2 πx ln 10<br />
(l) y ′ = 2 tanh √ x − √ xsech 2√ x<br />
2 tanh 2 √ x<br />
(c) y ′ = 1<br />
2 √ x<br />
(d) y ′ = 1 x<br />
Coletânea de exercícios elaborada pelos professores:<br />
• Dra. Dayse Batistus;<br />
• Msc. Ana Munaretto;<br />
• Msc. Cristiane Pendeza;<br />
• Msc. Adriano Delfino;<br />
• Msc. Marieli Musial Tumelero<br />
Digitação:<br />
• 1 a versão: Acadêmico Bruno Brito.<br />
• Versão atual: Acadêmica Larissa Hage<strong>do</strong>rn Vieira.<br />
Referência Bibliográfica:<br />
ANTON, H., BIVENS, I. e DAVIS, S. Cálculo. vol. 1. Tradução: Claus I. Doering. 8 ed.<br />
Porto Alegre: Bookman, 2007.<br />
GUIDORIZZI, H. L. Um curso de cálculo, vol.1 e 2. 5 a ed. LTC Editora, Rio de Janeiro,<br />
RJ: 2002.<br />
LEITHOLD, L. O cálculo com geometria analítica. Vol.1. 3 a ed. São Paulo: Harbra,<br />
1994.<br />
LIMA, J. D. Apostila de Cálculo I. <strong>UTFPR</strong>, Pato Branco, 2008.<br />
STEWART, James. Cálculo. Vol. 2. 6 a ed. São Paulo: Pioneira Thomson Learning, 2009.<br />
SWOKOWSKI, E. W. Cálculo com geometria analítica. Vol. 1. 2 a ed. São Paulo: Makron<br />
Books <strong>do</strong> Brasil,1994.<br />
THOMAS, G. B. Cálculo. Vol. 1. 10 a ed. São Paulo: Person, 2002.<br />
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