RESPOSTAS DOS EXERCÍCIOS ÍMPARES

Capítulo 14
RESPOSTAS DOS EXERCÍCIOS
ÍMPARES
14.1 Introdução
√
1. (a) [0, +∞) (b) [1, +∞) (c) (−∞, 1] (d) R (e) 0 (f) 2 ± 2 e 0 (g) −7/2 (h) 0 e 4
√
√
√
√
√
√
√
√
√
3. (a) 2 41 (b) 3 17 (c) 3 5 (d) 97 (e) 2 (f) 18 + 2 π 2 (g) 337 (h) 265 (i) 4 (j) 3 − 3
7. Pontos situados sobre a reta x + y = 4 ou {(x, y) ∈ R2 / x + y = 4}.
√
√
√
9. 10 11. 2 x + 3 y − 4 = 0 15. (a) −3 2 < k < 3 2 (b) k = ±3 2
19. (a) a = 8, b = −24 (b) não existe b ∈ R (c) a = 5, b = 2 (d) a = −13, b = 2
√
√
23. 1 + 2, 1 − 2, -1, 2/3
25. (a) ∅ (b) ∅ (c) [−2, 8/7] (d) (−∞, −1/2] (e) (−∞, −4] (f) (−2, 1)
27.
6
40
3
30
2
6
4
4
2
-2
1
-1
1
20
2
10
2
-2
1
-1
-2
20
-4
-4
(a)
2
-2
(b)
-6
-2
40
60
80
100
-1
120
4
(c)
-10
(d)
-20
-2
-3
29. (0, 4]
31. x < 2.79955 e x > 357.2
14.2 Funções
1. (a) x2 (b)
4πx3
3
(c) πx3 (d)
10πx3
3


−2x − 4 se x < −4
3. f (x) = 4 se − 4 ≤ x < 0


2x + 4 se x ≥ 0
419
2
3
CAPÍTULO 14. RESPOSTAS DOS EXERCÍCIOS ÍMPARES
420
12
10
8
6
4
2
-8
-6
-4
-2
0
2
4
6
5. Dom(f ) = R − {−7/2}; f (1/x) =
1−x
2+7x ;
1
(e) 2
7. (a) x + a (b) x2 − ax + a2 (c) x + a + 1 (d) − ax
√
√
3
a+1− 3 1+x
a−x
2
2
2x+7
x−1
(f) − ax+a
2 x2
(f (x))−1 =
(g) x2 + ax + a2 + 1 (h) − x
2 +ax+a2
a3 x3
+a )
(j) − (x+a)(x
(i)
a4 x4
9. Não; Dom(f ) = R − {0}
11. (a) x2 + 2x + 2; 2x − x2 − 2; 2x3 + 4x; x22x
+2
13. a = 3; b = − 43 ou a = −3 e b = 32
√
2 +4
2
, x 6= 1 (f) −2x − 1
(d) −4x2 + 18x − 17 (e) x−1
15. (a) 3x + 7 (b) x2 + 2 (c) xx2 +1
√
√
√
√
17. x + 3n + 3
19. (a) [1, +∞) (b) (1, +∞) (c) R − {0} (d) (− 3, − 2) ∪ (0, 2) ∪ ( 3, +∞)
21. Sim 22. Se f (x) = a x 29. f (x) = −6x + 8 e g(x) = 2x2 − 7x + 4 31. 3/11 33. f (x) = 16−x
3
35. D(t) = −2500; V (5) = 12500 reais
37. 35000 reais 39. y = −1.6x + 416, 6
41. 93804.23 reais 43. 10% 45. (a) 45.67−p
0.0023p (b) 1551
14.3 Funções na Economia
1. (a) 0 ≤ p ≤ 1 (b) p ≥ 1 (c) p > 4 (d) p ≥ 0 (e) p ≥ 0 (f) [0, 1] ∪ [3, +∞) (g) p ≥ 1 (h) 0 < p ≤ 2
(i) p > 9 (j) p ≥ 0
0.8x
3. − 3x
50 + 1000 5. (a) 0.53 u.m.; 2 u.m. ; 7.2 u. m. (b) 99.9% (c) CM (x) = 100−x2
0.5
0.4
0.3
0.2
0.1
0
2
4
6
8
10
p
√
7. (b) p = 1 − x2 ; x =
1 − p2 9. (a) x = 40±26.46
, x = 11.07; x = 2.25 (b) Terálucro:
6
2 < x < 11; não terá x > 11 11. (a) C(x) = 25x + 120 (b) R(x) = 35x (c) L(x) = 10x − 120 (d)
262
13. x = 0.95p + 105.5 15. 25000 [1.0625]20 ∼
= 84046.33 reais 17. 1061.36; 1346.84; 1814.02
7
∼
∼
∼
19. (a) = 1990 (b) = 3556865 (c) = 2.67362 × 10
21. (a)V (10) ∼
= 283942.1 u.m. (b)V (15) ∼
=
a−y −1/k
11
25. W (6) = 2048 × 10 mg.
478458.94 u.m. 23. t = ln( ab )
380
16x
55
x
x
29. (a) 166x
131 + 393 (b) 221 + 221 31. (a) y = 0.969 × (3.031) (b) y = 0.964 × (0.247)
14.4 Limites e Continuidade
√
1
(h) 9 (i) 1 (j) 0 (k) 2
1. (a) -5 (b) 1 (c) 2 (d) 2 (e) 2 (f) 4 (g) − 1000
3. (a) 4 (b) 15/11 (c) 2√(d) −∞ (e)-1/a (f) -1 (g) 2 (h) 2t (i) 2 (j) -6 (k) 1/3 (l) 5/6 (m) 1/4 (n) -1/56
(o) 3 (p) 0 (q) 0 (r) 1/ 2a (s) 1/9 (t) 0
5. (a) 0 (b) 3 (c) 1/3 (d) 0 (e) 1/3 (f) -1/2 (g) 0 (h) 0 (i) 0 (j) 1 (k) 0 (l) 0 (m) 1 (n) 0 (o) 0 (p) 0 (q) 0
(r) 1 (s) 0 (t) 3
7. (a) -13/6 (b) 11/6 (c) 1/3 (d) 12 (e) 1/12 (f) 4 (g) -4 (h) -1/6 (i) -85/4 (j) 0
9. (a) +∞
14.5. APLICAÇÕES DE LIMITES E CONTINUIDADE
421
13. (a) -1 (b) 6 (c) 1 (d) 2 (e) 1 (f) 5
11. (a) sim (b) não (c) sim (d) sim
13. (a) -1 (b) 6 (c) 1 (d) 2 (e) 1 (f) 5 15. (a), (c) e (d) Sim (b) Não
14.5 Aplicações de Limites e Continuidade
1. (a) 400/19, 100/9, 16/3, 4, -4/15 A capacidade de produção diminui com o tempo até 100 u.
m.; a partir daí tende a aumentar.
(b) 4 (c) +∞
100
80
60
40
(d)
20
0
50
100
150
200
5. (a) contínua (b) A variação do consumo não é sensível em torno de 20 m3
3. (a) contínua
de água.
80
60
40
(c)
20
0
10
20
30
40
7. Limite pela direita: 1, limite pela esquerda: 3.68
5
4
3
2
(c)
1
0
5
10
15
8.
14
12
10
8
6
(a)
4
2
0
5
10
15
(b) termina
14.6 Derivada
1. (a) y = 10 − 6x (b) y = −1 − 2x (c) y = −4 + 5x (d)
py = 2(−3 + x) (e) y = −x
(f) y = (3 + x)/4 (g) y = 2(−3 + x) (h) y = (1 + 2x)/ (3) (i) y = 2(−1 + x) (j) y = (3 + x)/3
(k) y = (3 − x)/4
3. x0 = −1, x0 = −2 e x0 = 2/3
5. (a) 1 + 6x + 15x2 + 24x3 + 25x4 + 18x5 + 7x6 (b) 3(3 + 5x2 )(x + x4 + x6 )2
(c) (−10+4x+9x2 +6x3 )/(1+3x)2 (d) (x(−2+9x−12x2 −x3 +38x4 −21x5 −8x6 +5x7 ))/(−3+x2 )2
7. (a) 5−1+x ln(5) (b) 21−2x 5−2x (−1 + 10x )(1 + 10x )(1 + 102x )ln(10)
(c) 2/(xlog(5)) (d) (1 + log(x/4))/log(4) (e) 1/(x + x2 ) (f) ln(10) (g) 1/(xlog(x))
√ √
9. (a) −x2 /y 2 (b) (−3x2 − 2xy)/(x2 + 2y) (c) − y/ x (d) 1/(−1 + ey ) (e) (1 + x + y 2 )/(y(−2 +
3xy + 3y 3 )) (f) −y/x (g) (x − 2x3 + 2xy 2 )/(−y − 2x2 y + 2y 3 ) (h) y/x (i) (−e2x+y − 2x)/x
CAPÍTULO 14. RESPOSTAS DOS EXERCÍCIOS ÍMPARES
422
11. (a) 0 (b) 72 (c)√−9x/(3 − x2 )5/2 (d) 24/(−1 + x)5 (e) 8e1+2x (f) 6/x4 (g) ex (7 + x)
13. (a) (6 + x)/(2 3) (b) 1 − 2x (c) (2 + x)/2 (d) x (e) x + ln(5) (f) −1 + 21x
17. l(x) = −4490 + 303x, 4751.5
19. 150√
√
√ √
√
√
√
23. (a) 3 + x/(2 3) − x2 /(24 3), 3 + x/(2 3) − x2 /(24 3) + x3 /(144 3)
(b) 1 − 2x + 2x2 , 1 − 2x + 2x2 − (4x3 )/3 (c) 1 + x/3 − x2 /9, 1 + x/3 − x2 /9 + (5x3 )/81
(d) x, x − x3 (e) x − x2 /2 + (8x3 )/15, ln(5) + x − x2 /2 + (8x3 )/15
(f) −1 + 21x − 189x2 , −1 + 21x − 189x2 + 973x3
14.7 Aplicações da Derivada
1. (a) sem pontos críticos (b) 3/2 (c) 1 (d) -1 (e) 0 (f) sem pontos críticos (g) −3, 0 (h) sem pontos
críticos (i) 0
3. (a) 3/7 min (b) 2 max (c) -7 max, 1 min (d) 0 min (e) 2/9 max
(f) (g)
(h) -2 min, 2 max (i) não possui pontos extremos
(j) -2 max, -4/5 min
√
√
7. (a) −b/2a (b) Se b2 − 3ac > 0, (−b + b2 − 3ac)/3a mínimo relativo e (−b − b2 − 3ac)/3a
máximo relativo
√
√
3
11. raio 8/ 3 4π e altura 256/π 16π 2
13. 13.5 reais
15. x = 3
17. 6.07 cm
14.8 A Derivada em Economia
1. CM g(x) = 4x − 1000/x2
3. (a) CM g(x) = −3 x2 + 200 x+ 1; C(0) = 1, C(10) = 1701, C(100)
= −9999
√
3
2
2
(c) L(x) = −4 + x − 100 x + −3/2 + 1/2 6401 + 4 x − 320 x x
√
1
′
2
2
2
L (x) = 2√6401+4 x2 −320 x (6x − 400x − 3) 6401 + 4 x − 320 x + +6401 + 8 x − 480 x
5. (a) x = 3.33 (b) x = 0 e x = 3.33
1400
1200
1000
800
600
(c)
400
200
0
10
20
30
40
50
60
7. 14.71 reais
9. (a) CM g(x) = 2x+5, CM e(x) =√x+5+30/x, CM g(0) = 5, CM g(50) = 105,
CM e(50) = 55.6 e CM e(0) indefinido
(b) x = 30
11. x ∼
13. (a) CM g(x) = 3x2 − 20x + 40, CM g(0) = 40, CM g(100) =
= 359 (b) 90250 reais
28040
(b) R(x) = 200x − 10x2 , RM g(x) = 200 − 20x (c) L(x) = 160x − x3 (d) x ∼
=7
15. (a) 2010 (b) 10%
17. 13.5 u. m.
19. (a) εC(x) = (3x3 + 40x2 − x)/(x3 + 20x2 − x + 4) (b) εC(100) > 1 (c) εC(1000) > 1
14.9 Integração Indefinida
√
n
1. (a) x4 /4 + 4x3 /3 + 3x2 /2 + c (c) n xn−1 /(n − 1) (e) −1/2 x2 + 2/5 x5/2 + 2/3 x3/2 (g)
√
x
2
eax
9/2 143 − 234 x + 99 x2 (i) 10
2
2
−3
−1
1287 x
ln(10) (k) 5 a (m) 2/15 x 15 + x (o) 1/2 x +1/3 x −2 x
14.10. INTEGRAIS INDEFINIDAS E ECONOMIA
423
3. (a) ln√(ex ) − ln (ex + 1)
√
√
√
(−2 + x) (c) − ln (x −
1) + 2 x + ln (−1
+ x) − ln (1 + x)
(b) 2/3 x + 1p
p
p
√
√
√
√
(d) − 16
1 + 3 xx2/3 − 8/5 1 + 3 x 3 x + 16
1+ 3x
5 + 6/5
5
x2
2
4
5. 1/2 2 − 2 x + x e
2
ln(5))5x
√ x +4 5 (c) 1/3 ln x3 + 3 x2 + 4 (d) 1/2 x2 + ln (x) − ln x2 + 1
(b)
−1/3
7. (a) (−1+x
2
2
(ln(5))
(x +4)
(e) x − (x + 1)−1 − 2 ln (x + 1) (f) −1/16 ln (2 x + 7) + 1/16 ln (2 x − 1)
14.10 Integrais Indefinidas e Economia
1. 436 u. m.
√
3. [(0.9 + 0.2 x) x + 10] bilhões de u. m.
5. x = 20
14.11 Integrais Impróprias
√
1. (a) 2 (b) ln(2) (c) 1/2 (d) 1 (e) divergente (f) −1/2ln(5) (g) divergente (h) 1/2 (i) π/2 (j) ln( 2)
(k) 1/4 (l) divergente (m) 1/8 (n) divergente (o) 1/ln(2)
3.p(a) 4 (b) 2(e2 − 1)/e2 (c) 7(ln(2))5/7 /5 (d) 0 (e) 5/3 (f) divergente (g) 51/7 (h) divergente (i)
2 ln(2) (j) divergente
14.12 Integrais Definidas e Economia
3. a = 1/18
5. (a) P (8 ≤ x) = 0.6 (b) E = 10, V = 33.3 (c) P (x = 10) =
7. (a) P (x < 10) = 1 − e−5/11 (b) P (30 < x < 60) = e−15/11 − e−30/11
(c) P (x > t) = 0.1, então t ∼
= 51 minutos.
9. (a) α = 1/16, P (x ≤ 20) = 0.7135 (b) P (5 ≤ x ≤ 25) = 0.7316
R 10
10
f (x) dx = 0