A N S W E R S T O

O D D - N U M B E R E D

P R O B L E M S

CHA PTER 1 (c) 2 x — 3y = 1 2 ; (d) y = —4; Section 1.5, p. 28

(e) x = 1; (f) x + 3y + 2 = 0; (g) x + 1- (a) 42; (b) 17; (c) - 3 ; (d) 32;
Section 1.2, p. 8
2y = 11; (h) 3y — 2x = 17; (i) x + (e) 5a 2 + 30a + 42; (f) 125r2 - 3.
I • (a) Rational; (b) integer, rational;
2y = 9; (j) x + y = 1. 3. 5. 5. 2x + h.
(c) integer, rational; (d) rational;
(e) integer, rational; (f) irrational; , (a)^ _ + ^ 1;(b)^ +
x(x + h )‘
(g) integer, rational; (h) irrational;
(i) rational; (j) rational. X = 1 ;( c)£ + i = , ; ( d)£ + 9- /( 1 ) = 0 ,/(2 ) = 2 ,/( 3 ) = 10,
3. 1 1 . 5. 7 7 3. /(0 ) = - 2 , / ( - 1 ) = - 1 0 , / ( - 2 ) =
7. 5 - x. 9 ■x 2 + 10. -3 0 .
II • 3x2 — 1. T — *' 13. (a) x > 0; (b) x < 0; (c) all x;
13. (a) jc < 0 and x > 1 ; (b) —2 < " • ( f. -¾ . (d) x < - 2 , x > 2; (e) all x except 2,
x < 1; (c) x < —7 and x > 3; 13. F = jC + 32 or C = f(F - 32). - 2 ; (f) all x; (g) x < - 2 , x > 1;
(d) —| < x < 1; (e) —3 < x < \ \ (h) x < - 2, x > 1; (i) —3 ^ x < 1;
Section 1.4, p. 22
(f) all x. ( j) x ^ 0, x > 2.
I • (a) (x — 4 ) 2 + (_y — 6 ) 2 = 9;
15. (a) x > 0; (b) - 2 < x < 0 and 15. /( 0 ) = 0 ,/(1 ) does not exist,
(b) (x + 3 ) 2 + ( y - I ) 2 = 5;
x > 2; (c) x < —1 and x > 3; (d) x < /( 2 ) = 2 ,/(3 ) = 1 ,/( /( 3 ) ) = 3. In the
(c) (x + 5)2 + ( j + 9 ) 2 = 49;
- 1 , 0 < x < 1, and x > 3. last part, it is tacitly understood that x
(d) (x - I ) 2 + (y + 6 ) 2 = 2 ;
17 • a = b. is restricted to those values for which
(e) (x — a)2 + y2 = a2 or x 2 + y 2 =
19- (a) Vertical; (b) horizontal; /( /( x ) ) exists: that is, x ^ 1.
2ax\ (f) x 2 + (y — a)2 = a 2 or x 2 +
(c) horizontal; (d) vertical; 17. /( 0 ) = 1,/(1) does not exist,
y 2 — 2 ay.
(e) horizontal; (f) vertical; (g) vertical; /( 2 ) = - 1 , / ( / ( 2 ) ) = i / ( /( / ( 2 ) ) ) = 2.
3- (a) Circle, center (2, 2) and radius
(h) horizontal. I 9 -/ (x i)/ (x 2) = f i x X + x2).
2 \/ 2 ; (b) point (9, 7); (c) circle, center 21 • No; it is true if and only if ad +
21 • (a) 5 V 2 ; (b) V l3 ; (c) V 8 9 ;
( —4, —5) and radius 1 ; (d) circle, b = be + d.
(d) |a - b\ V 2 .
center ( —f , 4) and radius 3; (e) empty; 23. (a) a = 4, b = —5, c = 3.
27. Center ( —2, f), radius j V 113.
(f) point (7 V 2 , —y V 2 ); (g) circle, 25. y = —x + V 3 — x2 and y = —x —
29- ( - 1 , - 1 ) .
center ( 8 , —3) and radius 1 1 .
31 ■Symmetric with respect to the V 3 — x2.
5- D istinct real roots, b2 — 4ac > 0;
straight line through the origin that 27. A = jx V 16 — x 2.
equal real roots, b2 — 4ac = 0 ; no real
bisects the first and third quadrants.
roots, b2 — 4ac < 0 . 29. A = x V 4 a 2 - x 2.
33. { V 2 h.
7 ■y = ± 2 V 2 x + 4. c2
31- (a) Yes, A = - — ; (b) yes, A =
Section 1.3, p. 14 9- (a) y 2 = —1 2 x; (b) x 2 = 4y; 1 2 / \ 477
7 6 P , (c) no. _______
1 - (a) - f ; (b) f; (c) j; (d) - 1 ; (e) 0 ; (c) y 2 = 8 x; (d) 3x 2 = —4y; (e) y 2 +
33. y = 27rr2V a 2 — r 2, A = 27rr2 +
(f) 1 0 . 1 2 x + 1 2 = 0 ; (f) x 2 — 6 x — 8 y +
4 7 tt V a 2 — r 2 ; V = ^ 7 r (4 < 3 2/* — /z3) ,
5- (a) Yes; (b) no; (c) no; (d) yes. 17 = 0.
7- (a) y = —4x + 5; (b) 3x + l y = 2; I I ■ 2 0 ft. A = Y7r(4a2 - h2) + 7 r/zV 4a2 — /z2.

856
 ANSWERS TO ODD-NUMBERED PROBLEMS 857

35. A = 277/-2 + — .
r
37. (a) Largest area = 625 ft2, both
sides - 25 ft; (b) A - 10(k — 2x2 -
1250 —
2(x - 25)2, largest area=
1250 ft2, sides = 25 and 50 ft.

Section 1.6, p. 37
L (a) i

2 4

5. (a)

3. (a)

S e c t io n 1.7, p . 4 5
2_5tt
'■ « > ; (b)7~ (c ) ~
3
(d)
858 ANSWERS TO ODD-NUMBERED PROBLEMS

3. (a) - 4 ; (b) |V 3 ; (c) - { V 3 ; (d) odd; (e) neither; (f) odd;

(g) neither; (h) neither.
(d) { V 2 ; (e) (f) -\V ~2.
55. (a) E ven; (b) even; (c) odd.
7. (a) sin y ; (b) sin 0; (c) - s i n -y ; 57. y = 275(jc — 1)(jc — 2) —

V 3 ( jc - l ) ( j c - 3 ) + y ( j c - 2 ) ( j c - 3 ) .
(d) - s in y ; (e) cos 0 ; (f) cos y ;
59. (a)
(g) -cos y ; (h) sin y ; (i) cos y .

11. sin 15° = { V 2 - V 3; cos 15° =

7 V 2 + V 3.

13. (a) cos y = y V 2; (b) cos ^ =

~V 2. 67. \h2.
2
277 V3 5 TT (b)
15. (a) sin — = (b) cos — = CHA PTER 2

1 , / r , . . YI TT 1
, / / /M /i
- 2 -1 1 2 Section 2.2, p. 57
~2 sm — = r 1. (a) 4jc + y + 4 = 0 ; (b) 8jc - y =
19. |V 2 ( V 3 + 1). 16; (c) 8 x - y = 16.
(c) 5. (a) 2jco - 4 ; (b) 2jc0 — 2; (c) 4jc0 ;
Additional Problems, p. 47 (d) 2x0.
9. No, to both questions. I. 8jc + y + 7 = 0.
15. (ji - yi)x + (jc2 “ *i)y = *iy\ ~ I I . y = 4x + 1, y = — 4x + 2 5 .
x\yz.
Section 2.3, p. 62
19. (a) [ b , a b ~ b i ' 3. — 1 6 jc. 5. -72.
7. — 1 0 + 3 0 jc. 9. 6y + 7.
/un 1 a b2 + c2 — ab 1 1. 3500 - 1 4 *. 13. IOjc + 25.
( b ) | r --------& — 15. — 3 2 jc -4 0 . 17. (0, 6).
a + b c 19. (3 ,0 ). 21. (10,100).
(C)
3 ’ 3 23. 5 - 3 jc2. 25. 6jc2 - 6jc + 6.
23. (a) x — l y + 5 = 0, Ix + y — 1
27. 1 + 29.
15 = 0; (b) x = (1 ± V2)y. xL (jc+ I)2'
25. \b\ < 2 V T 0 . -2 —2x
31. 33.
27. (a) (x — f a ) 2 + y2 = ja 2; v-3 '
(jc2 + 1)2‘
(b) (x2 + y 2 ) 2 = 2a2(x2 - y2). —2(jc2 + 1) 1
35. 37.
31. Ix + y = 1 0 and x - y + 2 = 0 . Ot2 - l)2 ' 2Vx 1
33. y = - 2 x + 2; (0, 2) and (y, f).
39. (b) Area = 2.
35. x 2 + y 2 — 2 xy - 4x - Ay + 4 =
43. g'(0) = 0; y = 3.
0.
45. (1, 1).
37. The line is x = 2pm.
41. No. 43. g(x) = x 3. Section 2.4, p. 67
45. V = \A r - — 7 r r3 I . v = 6 1- 1 2 ; (a) t = 2 , (b) t > 2.

3. v = 4 t + 28; (a) t = - 7 ,
47. V = y 7ra
r z — a‘ (b) t > - 7 .
-b 5. v = 141; (a) t = 0, (b) t > 0.
49. a — ,/3 =
ad — b e ' H ad — b e ' 7. v = S t - 24; (a) t = 3, (b) t > 3.
-c a 9. (a) 7 seconds; (b) 48 ft/s;
y = -,8 =
ad — b e ' ad - be' (c) 176 ft/s; (d) 224 ft/s.
5 1. (jc — l)(x — 2) • • • (x —n); I I . (a) 1 2 ; (b) 6 ; (c) 18.

jc" + 1; jc”. 13. 1 0 seconds.

53. (a) Odd; (b) even; (c) even; 15. (a) 3200 gal/min; (b) 2400 gal/min.
 ANSWERS TO ODD-NUMBERED PROBLEMS 859

17. dr/dt decreases as r increases. 35. -5 . 37. j . 10x3 — 3 6 x 2 + 4 2 x

23.
39. 4. 41. 3a/2. (5 x - 7 ) 2
Section 2.5, p. 73
43. 1. 45. 0. 288x10 - 360x5
1. 15. 3. - 5 .
47. Does not exist. 25.
5. 3. 7. - 3 . (24x5 - 5)2 '
49. Does not exist.
9. 4. 11. 5. —x 2 — 2x + 1
51. 3. 53. 0. 27.
13. 0 15. j . x 2(x I)2
55. 0. 57. 1.
17. (a) 6; (b) 4; (c) - 2 ; (d) 0; (e) does -4 2x4 - 2
59. lim /(x ), lim /( x ) , and lim /(x ) 29.
not exist; (f) \ . x—>0+ x—>0— x —>0 v2 •
19. (a) 5; (b) j ; (c) 0; (d) 1; (e) 1; do not exist. —(x + 30)
(f) y; (g) I- 6 1 . Because there are rationals as 33.
23. See Fig. 2.20. close as we please to every irrational, 35. (3, 2) and ( - 3 , - 2 ) .
25. (a) Lim it = 1; (b) x = 0.4. and irrationals as close as we please to 37. Two; ( - 3 ± V 5 )/2 .
every rational. 39. 4x + 5y = 13, 5x - 4y = 6.
Section 2.6, p. 79
63. Slope = 0.693. 41. 2y = x + 2.
I. (a) None; (b) 1, - 1 ; (c) 1; (d) all
x < 0; (e) all x < 0; (f) none; (g) 3, 43. x — 3y = 2, 3x + y = 6.
CHAPTER 3
—4; (h) none. [Remember that a 45. Area = 1.
function is automatically discontinuous S ectio n 3.1, p. 87 47. (0, 2), (± 1, 1).
at every point not in its dom ain; thus, I. (a) 54x8; (b) 0; (c) - 6 0 x 3;
1/x is discontinuous at x = 0 even (d) 1500x"(x40° + 1); (e) 2x - 6; Section 3.3, p. 97
though it is a continuous function.] (f) x 4 + x 3 + x2 + x + 1; (g) 4x3 +
1 10
3. f. 5. 3. 3x2 + 2x + 1; (h) 5x4 - 40x3 + ■ (2 - 5x)3 '
7. f. 9. 1. 120x2 — 160x + 80; (i) 12x(x10 + 3. 6(x + 2)(x2 + 4x — I)2.
15. (a) \ at x = 7t/6; (b) at x = x 4 — x — 1); (j) 18x2 — 6x + 4. 5. (13 - 8x)(5 - x)2(4 + x)4.
7r/4; ( c ) 1 at x = tt/2 . 3. (a) v = - 6 + 6t, a = 6; (b) v = -1 2
- 9 + 18r2, a = 36 1\ (c) v = 18/ - 12, 7. 9. -3 6 (1 - 6x)5.
17. (a) Yes; (b) yes. (3x + 1)5 ‘
19. (a) Yes, at x = 0; (b) yes, at x = 0; a = 18. 12(x3 - I)3
(c) no; (d) yes, at x = 0. 5. y = 7x - 10. 11
21. (a) No; (b) yes, at x = 0. 7. (1, - 2 ) and ( - j , - § ) . 13. 16(2x + 1)3[1 + (2x + I)4].
23. No maximum, minimum = 1. I I . a = 1, 3. —5(x5 - 5)(x5 - I)3
25. M aximum = 1, no m inim um . > 3 .i-A c- a . x 26
27. Maximum = - 3 , m inim um = - 8 . 2 a 4a
17. 4(x5 - 3x)3 • (5x4 - 3).
29. No maximum, minimum = 2. 15. (a) 3ac < b2\ (b) 3ac = b2\ 19. 6(x + x 2 - 2 x 5)5 ■(1 + 2x - 10x4).
Additional Problems, p. 81 (c) 3ac > b2. 4x
17. y — a3 = 3a2(x — a); all a ^ 0. 21 ------- -------
1. b = - 6 . ~ ' ( 1 2 - x 2)3 ’
5. (b) Drop the perpendicular from P 19. y = 12x — 16 and y = 3x + 2. 23. 7(x2 + 3x - 5)6 • (2x + 3).
to a point A on the axis o f the 23. ( - 1, -2 ). 25. —6(3x2 - 5x + 2 )“ 7 • (6x - 5).
parabola. Draw the circle w hose center Section 3.2, p. 91 27. 4(5x + 3)3(4x - 3)6(55x + 6).
is the vertex V and which passes 1. 2x. 2x(x2 + 9)
through A. Let B be the second point 3. 15x4 + 57x2 + 6. (9 - x 2)3 •
at which this circle intersects the axis, 5. 18x2 + 2x — 1. 31. 2 (2 x - 3 )7( 3 x 2 - x + 2 ) 9( 8 4 x 2 -
and draw the line PB. This line will be 7. 72x5 + 20x4 + 6x + 1. 1 0 8 x + 3 1 ).
tangent to the parabola at P. -2 Z } ( 2 t ~ l ) 2(t2 ~ 2t - 9)
7. (a) x = 0; (b) x = ± 2 ; (c) x = f ; (x — I)2 (,t2 + 3)3
(d) differentiable at all points. 4x3 + 12x2 - 1 72
11. 35
13. m = 2a, b — —a2. (5 - 4r)4 '
(x + 2)2
15. When t = f ; 8 ft/s. 37. 5(2x2 + 5x - 3)4(4x + 5).
3 - 6x2 —4x
19. Does not exist. 13. 15.
(1 + 2x2)2’ ' (1 + x 2)2'
21. - 5 . 39. 20(31 ^ 41. 5Ox - 55.
23. Does not exist. —4x —3 (1 - 2 x )5
17. ------ —— 19 ------- -—
25. Does not exist. (x2 - l ) 2' (2x - 3)2' 43. y = 16x — 15.
27. 2. 29. 2. —x 2 + 2x + 3
21. 45. (a) 3u2 (b) 4(2u - 1)
31. - 3 . 33. j. ( x 2 + 2x + l ) 2 ' dx dx
860 ANSWERS TO ODD-NUMBERED PROBLEMS

. -) _ dii 29. (a) x + 4;y = 7; (b) x + 2y = 4; , , 18x4 - 24x3 - 9 ^ — 10(x + 3)

(c) Au(u2 — 2 ) — .
dx (c) x + 3y = 0; (d) x + 3y = 19. ( g ) ------ ( x - 1 ) 2 ; (h) T x - 2 ? •
33. (a) 2x3/2; (b) 2x5/2. 23. (a) (x + 2)(x + 3) + (x + 1)(x + 3) +
Section 3.4, p. 101
COS X
1. 5 cos (5x — 2). 35. (V 2 , V 4 ). 37. (x + l)(x + 2); (b) (x3 + 3x2)(x4 + 4) X
3y 2 + 2 y ' (2 x + 2) + (x2 + 2x)(x4 + 4) X
3. —cos (cos x) • sin x.
5. 3 sin 2 x • cos x. (3x2 + 6x) + (x2 + 2 x )(x 3 + 3x2) X
y cos V x
39. 41. (4x3).
1. 9. 3x2 cos y — x 2 V x
25. (0, 10V5), (± 3 , V 5 ).
1+ cos X
tan _Vx_ sec 2 V x 27. (2, - 2 ) and ( - 1 0 , f).
1 1 . 5 sec 2 5x. 43.
29. (a) -6 (1 + 2x)2(4 - 5x)5(15x + 1);
13. sec 2 (sin x) • cos x.
(b) 10x(x2 + l)9(x2 - l ) 14(5x2 + 1);
15. 8 x(l + tan 2 x 2)tan x 2 • sec 2 x 2.
5 sin 2x —2x (2 x 2 - 19)
17. 15(cos 3x + sin 5x). 45.
V — 5 cos 2 x
(c) -; (d) —3x 6(3 - 2x)2
19. -4 5 ( 5 x - 3 ) 2 sin 3(5x - 3)3. 6 (16 + x2)4
COS X 3 3 sec 2 (3x — I ) - 1/2
X (4x — 9)(32x2 — 96x + 63).
21 23. 47. -
( 1 — sin x ) 2 1 + cos 3x ’ 2(3x —- n3/2
1) 31. (a) y = (x4 + 1)3; (b) ^ =
—x sin x — cos x 2(x6 + I)6.
25. 33. (a) 3 sin (1 — 3x);
Section 3.6, p. 110
1
1 . (a) 8 , 0 , 0 , 0 ; (b) 16x - 1 1 , 16, 0 , (b) —7x6 cos (1 - x 7);
27. 3x 2 sin — 2 cos —7 .
—7
0; (c) 24x2 + 14x - 1, 48x + 14, 48, (c) sin (cos x) • sin x;
Xz X
0; (d) 4x 3 — 39x 2 + lOx + 3, 12x 2 — (d) sin [sin (cos x)] • cos (cos x) • sin x;
29. 6 sin x cos x (2 — cos 2 x)2.
78x + 10, 24x - 78, 24; (e) f x 3/2, (e) —4 cos3 x • sin x;
3 1 . cos (tan x) • sec 2 x. l l r \/2 1 5 -1 / 2 _ i i —3/2
4A , 8A ? 16a (f) 90x( 1 - 3x2)2 cos4 (1 - 3x2)3 •
35. 2m r + —— or 2mr — 7 , « an 3. (a) n\{\ — x)_(”+l);
6 6 sin (1 - 3x2)3; (g) ------—---- ;
integer. (b) ( —l) nn! 3"(1 + 3 x )-(" + l>; 1 — sin x
37. (a) The particle starts at s = A (c) ( —l) ”+ 1 n!(l + x ) - (”+1\
(h) 15 sin4 3x • cos 3x; (i) —4x3 sin x 4;
when t = 0, moves to s = —A when
(/ 7 — 1)anx n 2 (j) 15 sin x • cos4 x • (1 — cos5 x)2;
t = 7 r/k, and moves back to s = A 5. -
(k) —3 sec2 ( 1 - 3x);
when t = 2n/k. This oscillatory motion (1) —24x2 tan3 ( 1 - 2x3) •
continues with period 2ir/k. (b) v = 7. (a) t = j , s = 0, v = 12; (b) t = 4, sec2 (1 — 2y3); (m) —sin (tan x) • sec2 x;
—Ak sin kt. (c) v = 0 when x = ± A s = 32, v = 6; (c) t = 1 ,5 = 6, v — (n) —cos [cos (tan x)] • sin (tan x) •
and |f| has its largest value A k when -3 . sec2 x; (0 ) 20x4 tan3 x 5 • sec2 x5.
s = 0 . (d) a = —Ak2 cos kt = —k2s. 9. 3, j.
13. (a) sin x, cosx; (b) —s in x ,—cos x; 35. 2/7 7r ± y , /1 an integer.
Section 3.5, p. 107 (c) sin x, cos x; (d) - c o s x, sin x.
1
3x 2 1 39. (a) y —>0 as x —> ±°«;
3.
Ay2 ' 1 - ly 6’ A d d itio n al Problem s, p. I l l
I. ( - 1 , 10) and (3, -2 2 ) . (c) 2x sin — - cos —; (d) 0.
3x2 - Ay x x
5.
3y 2 + 4x ‘
7.
S 3. (1 ,2 ) and ( —1, —2); the smallest
6X+ 5
2
slope = 1, at (0, 0). 41. (a) r;■I'M
(b) 2
9. II. 3(3x2 + 5x — 1)2/3’ 5x3/5’
5. Slope = 4x3 — 4x; x = 0, ± 1 ;
3x 2 ‘
—1 < x < 0, x > 1. x l/2
±3 - 2 (C) 1, —4 ; (d) - ^ _ ^3/2y/3-
13. 15. 7. a = 1, b = 1, c = 0.
(1 + x ) 2 ‘
9. a = 1, b = 0, c = —1. 43. (a) y = lOx — 19; (b) x — Ay +
±9x 13. a = I, b = —2, c = 2, d = — 1. 9 = 0; (c) 12x - 13y = 11; (d) y =
17.
17. (6, 9), ( - 2 , 1), ( - 4 , 4). —2x - 15.
2 V 3 6 - 9x2
7/4 —4x —4(x + 1) 47. (a) -2 (1 + 3x)_5/3;
19. f5Jx - 1'5. 21.
•9. (a) ; (b)
6 - (x2 - I)2 (x - I)3 5
23. 4x_3/5.
5* (b ) ~ 4(x
77T +7 1)5/2’ (°) ~ ^ _6/5;
3(xJ - 16) . x(4 — x 3) —2 x 2 — 6x — 11
25. (c) (x3
, 3 +, 2)2 ’i (d) (x2 + x — 4)2 (d) f x 3/2; (e) - } x ~ 3/2 + f x - 5/2-
4x 3 x3 + 8 '
(f) 20(x2 + 1)(x2 + 4 ) 1/2.
2 7 _I (* + 2 ) 1/2 x 2(3 - x 2) - 2 53. (a) —20 cos x + x sin x;
2 (x - 1) 5/2 ( } (1 - X 2)2 ,( f ) (1 + x ) 2 ’ (b) —362 sin 3x.
 ANSWERS TO ODD-NUMBERED PROBLEMS 861

CHAPTER 4 13. 27. max - y-§-V3 at x = —; min

6
S e c tio n 4 .1 , p . 1 1 9 —|V 3 at jc =
6
29. 125.

Section 4.2, p. 122

15.

7.

9.

13.

23. (0, 3), ( tt-, - 1 ) , (2 tt, 3) and

,2 3 n /4 3x
( l ^ “ I) . ( l ^ “ l)-
862 ANSWERS TO ODD-NUMBERED PROBLEMS

Section 4.3, p. 129 13. 5000. 15. 345.

1. i 7. alb. 17. ( a ) p = 110 - yjjc; (b) $55.
9. $8.50. 1 1 . 2 p.m .; 30 mi. 19. $160. 21. $16.
4, 4. 15. 108. 23. $573.33; jc = 20.
4 by 8 in. 19. 1. 25. 636.
24 in. 23. V 3 .
3 Additional Problems, p. 156
JO . 27. 1.
(a2/3 + b 2/3)y2 (convince yourself
that this and the preceding problem are
essentially the same).

= 4 /S i , _
5
17. (2, 0). 19. a = 3. 33. \(B + / / ) 2.
23. (a) a > 0; (b) a < 0.
Section 4.4, p. 137
I. 1. 3. f R.
5. 4 by 4 in. 7. \ .
9. (1, 1).
I I . (a) f mi; (b) 1 h and 44 min;
(c) 8 min longer.
13. 1 5 /V T o mi/h.
19. 1. 21. a = 2.
23. x = y V 2 a. 25. (a) 0; (b) 1.
27. (2, 4). 29. Amax = |V 3 .
31. The spider should w alk straight to
the m idpoint of a side not containing S
or B, then straight on to B.

Section 4.5, p. 142

I. (a) 12077 ft2/s; (b) 2 4 0 tt ft2/s.
3. 2/77 ft/m in.
5. 4 ft/s at each of the stated
moments.
7. 3 ft/s. 9. 4y ft/s.
13. 52 mi/h.
15. y lb /in 2 per min.

17. (a) — in/min; (b) ---- ----- in/min.

^ 2V 277
19. - j— in/s.
577
21. 2/(V 2 - 1) = 7.69 h. 9.
23. 4 0 ft/min.

Section 4.6, p. 146

27. (a) PI x = tt, CU 0 < x < tt,
3. 0.618034. 5. 2.154435.
CD TT < X < 277; (b) ? \ x = ~ , 7. 1.305407 ft.
4 4
II. 0.918247 and 2.863580.
CU 0 < x < - j , CD < 13. (a) 1.236123; (b) 0.876726.
4 4 4 4
3 tt 3 tt 57t ^ 5 tt Section 4.7, p. 155
x < CU —— < x < — , CD ——
4 4 4 4 I. $43/unit; $43.03.
3. $80/unit; $80.14.
< x < - - , CU
4 4 <x< 2tt; 5. $193.89; x = 89.
(c) PI X = TT, CD 0 < X < TT, C U TT < 7. $0.85; jc = 232.
x < 27r. 9. $0.31; jc = 93. 11. 200.
ANSWERS TO ODD-NUMBERED PROBLEMS 863

79. A square w ith side g(a + b —

V a 2 — ab + b2).
81. V 2.
83. 3 by 6 by 12 inches.
bs
85. a - m, if this num ber
Vr2 -
is positive.
87. x2 + y2 = 32.
89. (3, 3).
91. (5, 0) and ( - 5 , 0).
99. (a) 12 ft/s; (b) 3 ft/s.
101. l/7rft/m in. 103. At least 9 ft.
ax
107. ~ r = _______ in/s.
dt V x2 + r2
109. 0.32 lb/min.
111. 14477 m 3/m in.
I 15. Decreasing 1 in2/min.

117. When t = R()^ ~

a \fa — bV b
I 19. (a) 3.316625; (b) 1.903778;
(c) 2.087798.
123. 1.856636 in approximately.
125. $42. 127. 30.

C H A PTER 5

Section 5.2, p. 169

1. (63;t8 - 15x4) dx.
(2x - 3x3) dx
3.
Vl
(2 - x) dx
5.
V 4x - x 2

7. (2 x~ m + 2 x ~ 4/5 - 1 7 ) dx.
q (15x2 + 8x) dx
2V3x + 2
37. (a) Point of inflection at x = 1;
(b) points of inflection at x = 1,2; 1 1 -dx? =
(c) points of inflection at x = —2, 0, 1,
—15(3¾2 - 2u + l)x 2(x3 + 2)4
2, 3.
39. a = - 3 . 41. j V 3 . («2 — w)2
43. jc = 21, y = 35. 13. AA = 27tt A r + 77 Ar2 and dA =
45. 18 = 16 + 2. 47. 5, 5, 5. 277r dr = 2 tjt Ar, since dr = Ar.
51. j V 3 a, \b. 59. 2 in. Imagine that the thin circular ring is
63. 4000 knives at a price of $18 cut across and u nrolled— with slight
apiece. distortions— into a long thin rectangle
65. 20 days. with length 277r and width Ar.
67. (a) 120 ft; (b) 312 ft. 15. 12, 12.048064.
69. i 71. V 2 . 17. 16.167, 16.166236.
73. 77/4. 75. 4. 19. 26.75, 26.749612.
77. 4 / tt. 21. 6.019, 6.018462.
864 ANSWERS TO ODD-NUMBERED PROBLEMS

35
23. 0.849,0.848048. 13. 3 \ /ry = x V x - 3. 9. 4 8 V 2 /5 . II. 3 •
25. 1.037 mi. 27. 125.66 ft. 21 26
13. • 15. 3•
Section 5.5, p. 187 2
26
Section 5.3, p. 177 I . 8 s; velocity = - 1 2 8 ft/s and 17. 33. 19. 3•
1. yx 2 + X + c. 4 2
speed = 128 ft/s. 2 1. 9- 23. 3'
3. yx 3 + yx 4 + yx 5 + c. 3. v= - 3 2 1 + 128; 5 = —16r2 + 13
25. 1. 27. 3•
5. 2 V x + c. 7. yx 7/4 + c. 128/. 5 —2 1
29. 48« • 31. 6-
9. f x 2/3 + c. 5. l o V I o s. 7. 40 ft/s.
1 1 . f x 3/2 — 2 V x + c.
33. a 4 /4. 35. b2!6.
9. 96 ft/s. 1 1
13. 6 V x + yx 3 /2 + c. 37. 30- 39. 2 •
II. i’02/64 ft; 96 ft/s.
15. yX3 + 7 X6 + C. 41. 3.
A dditional Problem s, p. 188
17. Ix 4- j x 2 + c.
1. fx 5 — ^x4 + lOx + c. Section 6.7, p. 216
19 f x 6 + 3x 2 — 5x + c.
1. (a) f ; ( b ) f ; (c) 19; (d) f -
2 1 . —2 x - 2 + c. 3. }x3 — f x 2 + x - 4 V x + c.
128
3. 5 • 13. Tra212.
23. f x 3/2 - 4 x 7 /2 — 3x_l + c. 5. {x4 + f x 3 + -jx2 + c.
25. 9x1/3 — 16x1/4 + c. 7. 17x3 — 27x4 + c. Additional Problems, p. 217
27. 2 \T x — iy x 10/3 + c. 9. 6x —f x 3/2 —jx 2 + c. 7. (a) f ; (b) f ; (c) 12; (d) 5V 5/3;
29. x 4 - 4x 2 + 17x + c. —f(2 - 3x ) 3 / 2 + c.
(e) 1
31. i x 10/3 + ^ x 7 /3 + 3x 4/3 + c. ~ ^(5 x + 2 ) 1 6 5 + c. 9. ( a ) 4 ( 2 V 2 - 1); (b) f; (c) 3; (d)
33. f x 3/2 - f x 7 /2 + ^fx l l / 2 + c. 5 V 1 + X2 + c. 4r^ 9r
II- (a) (b)
35. yx 500 + c. } V 2 x 3 — 1 + c. 1 + x 4 ’ vw 1 + x 2 '
37. j(3 + 4x ) 3 /2 + c. |( x 2 — 2 x + 3 ) 2/ 3 + c. 3x 2 5 ¾9
(c) ,..... :;; w
(d) ^
.--------
39. - \( 2 x - 3 )“ ' + c. —j \ / 2 — x 2 + c. V3x3 + 7 V I + JO
41. - { V 5 - 4 x 2 + c.
+ - I + c.
43. f ( l + V x ) 5 /4 + c. x, CHA PTER 7
45. y(l + x 2 ) 3 /2 + c.
y(x + 1 ) 7/3 + C.
Section 7.2, p. 224
47. {(7 - x ) ~ 6 + c. y(l + X)7/3 - |(1 + X)4/3 + C.
i- £ 3. 1
49. —y(2 — x 2 ) 3 /2 + c. -n-(x3 + x + 32)11/2 + c.
5. f. 7. ff V 2 .
JL (x 3 _ , )4 /3 ( 4 x 3 + 3) + c
51. 2 V x 3 - 5 + c. 9. 4. 36.
53. -no(10x + 10)11 + c. Ix2 13. 2 1 15.
(a) y = 320
55. -jy(3x2 + 4)5/2 + c. 3x 2 + 13 ’ 17. - 4 V 2 . 19. 3 *
57. 4 \ / 3 x 3 — x + 2 + c. (b) 3 V y - 4 = (x - 1) 3 / 2 - 2. 21. f . 23. (a) j,; (b) j.
59. | x 15 + c. 61. y = x 3 + 2. 35. x 2 — y 2 = c. 25. f .
63. (a) y sin 2x + c; (b) —j cos 5x + 39. (a) 25 s, 1200 ft/s; (b) 25'\ f l = 27. 2 ( V b - 1) —> 0 0 as b —>0 0 .
c; (c) 2 sin 2x — 3 cos 5x + c; 35 s, 8 0 0 V 2 = 1120 ft/s. 29. (a) 2 (V 2 - 1); (b) y (3 V 3 -
(d) - y cos 2x + j sin 5x + c. 41. 30 m/s.
65. (b) yx — y sin 2x + c. 43. (a) 44 ft; (b) 680 ft. V 2 - 3);
/, (c) 32 + ~2 V 2 - 2.
67. (a) y sin 5 x + c; (b) —y cos 6 x + 45. A bout 1.86 mi. About 0.36 in.
4 V 2 - 5
c; (c) sin (sin x) + c. 47. A bout 179,427 mi/s. 33. a 2 = (0.21895)a2.
69. (a) j sin 2 x + c; (b) —y cos 2 x + c.
They differ by a constant. CHA PTER 6
Section 7.3, p. 229
Section 6.3, p. 196 I. (a) 8 7 7 ; (b) 1 6 7 r / 1 5 ; (c) 3 7 7 / 5 ;
Section 5.4, p. 181
1. (a) 55; (b) 62; (c) 206; (d) 0; (e) 0; (d) 277/3; (e) 877/3; (f) 1677«3/105.
1. y = 2 x 3 + 2 x 2 — 5x + c.
3. y — 6 x 4 + 6 x 3 — 4x 2 + 3x + c. (f) 1500; (g) 7. 5. 477. 7. y a3.
9. -y-a 3. 1 1. 27T2a2b.
5. 3y2 — 4y 3/2 = 3 x 2 + 4x 3/2 + c. 5 (a) (” ~ (b) 0 ¾~ \)n(2n ~ 1 )
1 1 , 2 13. { V 3 « 3. 15. 9V 2/2.
/. y = ------- b —x z + c. 1)n
x 2 in 17. lj-a3. 19.
tx 2 in3. 5 7
(c) [H
9 - y — ~z ■ t - 2 1 . (a) The volumes are A(x) dx
Section 6.6, p. 212 [H °
1. 9. 3. f . and B(x) dx, which are equal
2 x 2 - 31
11. v =
33 - 2 x 2 ' 5. 12. 7. because A{x) = B(x) for every x.
 ANSWERS TO ODD-NUMBERED PROBLEMS 865

2 3 . t 7 ra 2h. 25. y = a x 4. 27. (a) 277a3; (b) j77a3; (c) \ \ tto3. (b) I ln (3x2 + 2) + c; (c) fx 2 +
29. a2h. 3 1 . j a 2h. 2 ln x + c; (d) x + ln x + c;
Section 7.4, p. 235 33. 12877/3. 35. 1 177/15. (e) x - ln (x + 1) + c;
3. 12877-/5. 5. 486tt/5. 37. 13577/2. 39. 277. (f) y ln (x2 + 1) + c;
7. lir ib — a). 9. - j - (2 — V 2 ). 43. t - 45. £ (g) ln (3 - 2x2) + c;
47. if. 49. f^ . (h) ln x(x — 1) + c; (i) {(In x)2 + c;
11. (a) 8 7 7 ; (b) 25677/15. (j) ln (ln x) + c; (k) 2 ln (V x + 1 ) +
51 . AB = 3½. 53. 9077.
13. 7 7 /2 ^/6 . 15. 8 7 7 / 9 . c; (1) ln (ex + e~ x) + c.
55. I6 8 7 7 . 57. 77a2/2.
17. By a factor of 1.71, 9. No max., no pt. of infl., min.
59. 7 ft-lb. 61. 2325 ft-lb.
approximately. (3, 9 — 18 ln 3) = (3, -1 1 ).
63. 9 4 5 - 2 13 ft-lb.
Section 7.5, p. 240 65. i f aB. 11. 77 ln 4. 15. min.
1. j f ( lO v T ) - 1). 67. -¾ c, where c is the constant of
proportionality. 17. 1/V e.
c
J.
11 8
Q .
6• 69. 12577W/6. 7 1. f • 83ttw ft-lb. 2x
7. 12. 19. (a) y ( 1 + ^ 2
6x - 2 r
73. 10 tons. 75. j ton.
2x
Section 7.6, p. 244 77. 187.5 tons. 79. 60 tons. (b) “ •
5\x2 + 3 x + 5
1. 25377/20. 3. 1277. 81. 4096 lb.
1
21. (a) x x x x + (ln x)(l + ln x)
5. ^ - ( l O V l O - 1). x
C HA PTER 8 1 — ln x
(b) V x . Max = V e .
7. - y ^ ( 2 V 2 - 1)/72. Section 8.2, p. 263
1. (a) log4 16 = 2; (b) log 3 8 1 = 4 ;
). - f 77a r. 1 1. ^0, — <3j. (c) log81 9 = 0.5; (d) log32 16 = j. Section 8.5, p. 282
3. (a) 4; (b) 6; (c) - 4 ; (d) f. I. (a) 2520; (b) 13.8.
2
13. (a) y = — a; (b) area - 4772a£>. 5. (a) a = 32; (b) a = (c) a = 6; 3. 20.1. 5. 95.8 percent.
77
(d) a = 49. 7. When t = 6. Can you solve this
Section 7.7, p. 249 9. (a) 7; (b) acidic pH < 7, basic problem w ithout calculation, by
I . 6 4 ft-lb. 3. 6000 ft-lb. pH > 7. merely thinking about it?
5. 550 ft-lb. 7. 50,000 ft-lb. 9. x = 10(j)'. 11. 2 more hours.
Section 8.3, p. 269 13. (a) About 3330 years (1380 B.C.);
I I . (b) 250 ft-lb. 13. 5 ft-lb.
1. j( e x — e~ x). 3. (x 2 + 2x)ex. (b) about 3850 years (1900 B.C.);
15. GMm/2a. 17. mgRh/(R + h).
5. e e'ex. 7. xe™. (c) about 10,510 years; (d) about 7010
19. 24077VV ft-lb. 21. ^77a3w(h + a).
9. 4x2<?2x. 11. j e 3x + c. years.
23. About 118,500.
1 3 . 5 ex/5 + c. 1 5 . 2 ex3 + c. 15. x —>A if A < B\ x —» B if A > B.
25. m\ = \rti2.
17. (a) Max. pt. (0, 1), no min. pt.,
Section 7.8, p. 254 Section 8.6, p. 287
pts. of infl. |± { V 2 , -^ = j; (b) no
I. 150 tons. 3. 125 tons. 1 .X
5. f ton. 7. I tons.
3. x =
9. 30077 1b. max. pt., min. pt. ( - 3 , —— ), pt. of xo + (xi - x0)e~
I I . 10-^ tons; 11 ft.
5. s = — (1 — e ct).
13. 300V 2 w or approximately infl. ( - 6 , ). c
13.2 tons. 7. When v < 1, the resisting force in
19. j( e b — 0 ~b)•
the second case becomes very sm all.
Additional Problems, p. 254 23. Area = 1 — e b —> 1 as b
I1. ±6 . -, 128 9. About 53.4 lb.
J . 15 . 25. (a) e; (b) e\ (c) e\ (d) e2\ (e) \T e.
29. 8%.
5. 36. 7. Additional Problems, p. 288
9. 18. 11. f. Section 8.4, p. 276 1. ~ x e ^ x~xl/ V 1 —x2.
13. 64. 15. f . I . (a) 2; (b) 3; (c) 1/x; (d) 1/x; (e) —x; 3. (2x - 2)ex2- 2x+l.
17. (f) 1/x; (g) x; (h) 3x; (i) 0; (j) j; (k) j; eVx 1 ---
5. - — + —V P .
19. (a) 56 tt/ 15; (b) 56 tt/15; ( c) 32 tt/3; (1) 0; (m) x 3y2; (n) 8; (0 ) 2e3; (p) x 2e*.
2V~x 2
(d) 4877/5; (e) 77a3/15. - (a) — ------
+ 2x)
3. — ; n(b)\ — ---------.
+x>’)
7. - y e -3' + c. 9 - e \/x +
21. (a) 51277/15; (b) 12877/3. x(3y - 1) x(l-xy)
23. V - 25. y77(fc5 - a 5). 5. (a) } ln (3x + 1) + c; 11. 2 V e x + 1 + c.
866 ANSWERS TO ODD-NUMBERED PROBLEMS

77 ^ A 13. sin 4 6 = (4 sin 6 - 8 sin3 6) cos 6. 11. sec2 jc e tanx.

13. (1/a, e). 17. ^ r( e 6 - 1).
15. (a) j ( V 6 - V 2); 13. —j cot 6 jc + c.

- - (w y(3x + 1} n-.'i y(2 x 2 + !) (b) y V 72 — V 3 . Show that these 15. —j cot 2x + c.

23' (a) W - D ’ (b) 4 1 - 3 , ) ' numbers are equal. 17. j tan5 jc + c.
17. (a) 0, 2 7 7 ; (b) 7 7 / 2 , 377/2; (c) 7 7 . 19. —y CSC 7JC + C.
25. (a) y ln (1 + 2jc) + c; (b) - y ln
,r, / x ^ 77 27771 77 2777X 21. y. 23. y(7r - 2).
(1 ~ 3 x ) + c; ( c ) y l n 2; (d) y ln 10; 19. (a) 6 = - + — — or — + — —
4 3 12 3 25. 2. 27. 4 V 3 ; no.
(e) ln 3; (f) 2 V ln jc + c; (g) y ln 7;
for all integers rr, (b) 6 = (1 + 2n)5-rr 29. 3.277 mi/s. 31. (c) 66°.
(h) - y ln (1 — x 2) + c; ( i ) l ( l n 3)2;
r „ . / ■> „ 777 , 27771
( j ) j ln (3jc2 - 3jc + 7 ) + c; ( k ) ln for all integers n; (c) 6 = -I---- — Section 9.5, p. 318
(ex + 1) + c; (1) ln (jc + 1)(jc + 2 ) + c; 1177 27m 1. y V 3 , —y V 3 , —V 3 , |V 3 , - 2 .
or -I-----— for all integers n.
(m) y (ln jc)3 + c; (n) j (ln x)2 + c; 3. (a) 7 7 ; (b) 77/2; (c) 0.123; (d) 0.8;
(o) y [ln (ln x)]2 + c; (p) - y (ln x )2 + c. 35. For a p ro o f by geometry, use the (e) 0.96; (f) 77/7; (g) 7 7 / 6 ; (h) 7 7 /4 .
fact that the vertex opposite the fixed 5. 1/(25 + jc2 ).
29- T2a~ + 74 ln 2; a = V ln 2 side must lie on a circle of w hich this 1
side is a chord.
7. 9. sin 1 jc.
31. (a) (ln 10)10*; (b) (ln 3)3*; V x ( jc + 1)
(c) (ln 7 7 ) 7 7 *; (d) (3 ln 7)73*; 37. f V 2 ft.
(e) (ln 6)(2x - 2)6*2-2*; 11. (sin -1 jc) 2 . 13. ---------.
Section 9.2, p. 304 5 + 3 cos jc
ln 5 :Vx
(f) I. 3 cos (3jc — 2). 15. / .
77 6 17. \ sin-1 2jc + c.
2V x, 3. 48 cos 16jc. 5. 2jc cos jc2 . 19. / .
77 8 21. y sin-1 fjc + c.
33. Max. at x = ; pts. of infl. at 7. 15(cos 3jc — sin 5jc).
ln 5 23. j ta n -1 jx + c.
9. jc cos jc + sin x.
2±V2 25 . -77/12. 27. 77/4.
II. f sin 1 2 jc. 13. cos5 jc.
ln 5 29 .(a) sin-1 f; (b) { rad/s.
15. 3e2x cos 3jc + 2 e 2* sin 3jc.
31. The formula is invalid, because
35. (a) (In*)* + ln (ln jc) 17. - t a n x. 21. 45°.
ln x 23. 5. the integrand 1 /V l — jc2 is
n n vAln* discontinuous at the point jc = 1 in the
(b) (2 ln jc)jcln *“ 1; (c) ----- — X 31. 7 7 / 3 , tangent = 3.
JC 33. 1. 35. 1. interval of integration.
33 . 4 7 7 ¾ 2.
[1 + ln (ln jc)]; (d) — (1 + y ln jc); 37. 39. 1.
V jc 41. 1. 43. - 1 . Section 9.6, p. 323
Y^/x
(e) v-2/3 ( 1 + y ln jc) . Section 9.3, p. 309
1. (a) jc = 5V2 sin \ t ~
I . — 7 cos 5jc + c.
37. In the year 3524, approximately. 3. | cos (1 — 9jc) + c. A = 5V 2 , T = 277; (b) jc =
39. In 4 more hours. 5. sin2 jc + c or - c o s 2 x + c or 2 tt '
41. 73.12°F. 2 sin ^3f + - y -j, A = 2, T
—\ cos 2x + c.
47. 17 more days. 7. j sin4 2jc + c. 9. \ sin8 \ x + c.
(c) jc = V 2 sin (t + J, A = V 2 ,
49. v 2 = £ ( 1 - e~2cs)\ v -> as I I. - 2 cos Vjc + c.
C y c
S —» 0 0 . 13. \ sin (sin 2 jc) + c. 77
T = 2 7 7 ; (d) jc = 4 sin 21 — —
5 1. When t = 4.86 min; when t = ~6J’
15. \ sec (^2x 3 + c. A = 4, T = 77.
21.50 min.
53. About 1.39 h. 17. —In (cos jc ) + c. , , V l4 5 . 77
19. sin ( jc 2 + x) + c. i~ ; 7 '
C H A PT ER 9 21. f. 23. V 2 - 1.
Section 9.1, p. 299 25. f. 27. 3. 5. T = 277\/ — = 89 min.
V g
I. (a) 77/12; (b) 7 tt/ 12; (c) 2 tt/3; 29. y772. 7. A bout 39 in.
(d) 577/12; (e) 5 7 7 / 6 ; (f) 377/4;
(g) 577/4; (h) 7 7 7 / 6 ; (i) 777/2; (j) 577. Section 9.4, p. 311 Section 9.7, p. 329
3. 6 = 2 radians. 1. 8jc sec2 4jc2 . 1. (a) h (b) 1; (c) t
5. A = 25 cot \d . 3. 2 tan (sin jc ) • sec2 (sin jc) • cos jc.
13. 3jc2 cosh x 3.
I . H = L tan 6. 5. 0. 15. 6 csch 6x. 17. 0.
I I . sin 36 = 3 sin 6 - 4 sin3 8, 7. 24 csc ( —6 jc ) cot ( ~ 6 jc) . 19. j cosh (5jc - 3) + c.
cos 36 = 4 cos3 6 — 3 cos 6. 9. - V c s c 2 jc cot 2 jc. 2 1 . 2 V 2 sinh {jc + c.
 ANSWERS TO ODD-NUMBERED PROBLEMS 867

2 3 . x — tanh x + c. (b) y tan2 x + ln (cos x), y tan4 x —

107. —J— tan 1 V 5 x + c.
V5 } tan2 x — ln (cos x), y tan6 x —
29. -y- sinh 1.
aL \ tan4 x + y tan2 x + ln (cos x).
109. y sin 1 | x + c. 27. (a) 772/2; (b) 7 7 ; (c) (477 — 772)/8;
Additional Problems, p. 330 (d) 3t72/16.
l l l . | tan-1 x 4 + c.
I. —9 cos (1 - 9x). 29. y[sec x tan x + ln (sec x + tan x)].
113. 36 ft from the point on the road
3. —2 sin x cos x = —sin 2x.
closest to the billboard. Section 10.4, p. 348
5. —10 sin 5x c o s 5 x = - 5 sin lOx
115. rad/s. _j x V q2 —x^
7. —6 sin 6x.
117. T = 2 tt/ V 2 w = 0.56 s.
9. —x 2 sin x + 2x cos x.
119. A = 5 , / = I / 7 7 .
I I . x cos x 1 -j £ _________ X

13. (sin x)[sin (cos x)]. 2a3 tan a 2a2{a2 + x 2) '

15. —(cos x)[sin (sin x)]. CH A PTER 10 5. - y V 9 - x 2(x2 + 18).
17. cos x. 25. 0.
27. 1. 29. 2. Add a constant o f integration to the
7. - in ----------
31. f . 33. 2. answer for each indefinite integral in a \ a + V a 2 + x2
35. 77/4. 37. y sin 3x + c. this chapter.
9. ln (x + V x 2 — a2).
39. - 2 sin (1 - jx ) + c.
4 1 . jg sin6 3x + c. Section 10.2, p. 339 11. yxVa 2 + x 2 +
43. —| cos 3x + y cos3 3x + c. I. -y (3 — 2x)3/2.
ya2 ln (x + V a 2 + x2).
45. j sin x 3 + c. 3. y l n [ l + (In x)2].
5. —2 cos 2x. 1 , a + x
47. y cos (cos 2x) + c. 13. — l n -------- .
7. y ln [sin (3x - 1)]. 2a a —x
49. —j csc 4x + c.
9. y(x2 + 1)3/2. 11 l5ep5x. 15. V a 2 + x 2 -
1 13. —y cot (3x + 2).
51. + c.
2(3 + 2 cos x) 15. 2. 17. 2 V l - cos x. a + V a 2 + x2
a ln
53. —I V 7 - sin 5x + c. 19. e1™ '*. 21. y sec 5x.
55. j . 57. i 23. y(ln x)2. 25. ln 2. V x 2 — a2
59. 2 V 2 . 61. 2177. 17. ln (x + V x 2 — a2) —
27. sin 1 ex. 29. y sin3 x. A,

—csc2 2x 31. ln (1 + ex). 33. —y ln (cos 3x).

67. 12 sec2 3x 69. 19. 7 - [a 4 sin - 1 — +
V c o t 2x 35. 4 V x 2 + 1. 37. tan-1 ex. 8 a
71. 4 sec2 x tan x. 39. \( e x + I)7. 4 1. y tan 5x. V a 2 — x 2(2 a 3 — a2x)].
73. —10 cot 5x csc2 5x. 43. - y csc 2x. 45. 4 (V 2 - 1).
47. f. 21. - V 4 - x 2. 23. - ta n " 1 - .
1 1
75. t a n ---------secz2 1
—. a a
x x x 49. (a) n = 3, ye*4; (b) n = 2, y sin x 3;
25. - y ( 9 - x 2)3/2.
(c) n = - 1 , y(ln x)2; (d) n = —
77.
V s e c V x + ta n V x 2 tan V x. 27. V 9 + x 2. 31. 2TT2ba2.
4V x 33. 3 - V 2 + ln (1 + y V 2 ).
79. sec2 x sec2 (tan x). Section 10.3, p.. 344 Section 10.5, p. 350
81. - 3 csc yx + c. 1. yx —
- y
I sin 2x.
I. sin - 1 (x — 1). 3. tan - 1 (x + 2).
83. —y cot 3x + c. 3. ^ x + y sin 2x + sin 4x — 5. - V 2 x — x 2 + 2 sin - 1 (x — 1).
85. —\ csc4 x + c. sin3 2x.
87. —y cot4 x + c. 5. —y cos3 x + y cos5 x. 7. -y sin - 1 ^ - 6 V 6 x —x 2 -
89. 477/3. 91. 300 km/h. 7. sin x — y sin3 x.
93. (a) -77/3; (b) 77/3; (c) - 7 7 ; 9. y sin372 x — y sin7/2 x. y(x — 3 ) V 6 x — x 2.
(d) 0.7; (e) 0.7; (f) —1; (g) tj/3. 11- ~ 16 sin 12x. 9. y ln (x 2 + 2 x + 5) +
1 _ x4 13. i » _ i (x + 1
95. 97. 3 ta n -
V 25 - x2 1 + x 10' 15. y sec7 x — f sec5 x + y sec3 x.
17. - c o t x — x. 19. —y cot 4
4x.
1 1 + X II. ln (x - 1 + V x 2 - 2x - 8).
99. 101. 1
21. —y cot 2x y1 csc 2x.
xV x2 - 1 1 + x 2' 13. y In (2x + 1 + V 4 x 2 + Ax + 17).
23. y sin 3x.
\/x 2 —1 25. (a) tan x — x, y tan3 x — tan x + . 1 5 . ---------,
103. -------------. 105. / .
77 2
y tan5 x — y tan3 x + tan x — x; 4 V x 2 - 2x - 3
868 ANSWERS TO ODD-NUMBERED PROBLEMS

Section 10.6, p. 356 Section 10.8, p. 368 57. - i( x + i r 4 + f(x + I) - 5 -

1. (a) x + 1 H--------- - , jx2 + x + I. - V l - X 2. l ( x + l ) - 6 + j ( x + I ) - 7.

x—1 3. j sin3 x — f sin5 x + y sin7 x. 5 9 . j tan5 x — y tan3 x + tan x — x.
ln (x — 1); (b) yx2 —f x + jf ~ 5. 2 V l + ln x + 61. - 2 ln (x + 2) + 3 ln (x + 3).

6 3 . x ln V 2 x — 1 — yx —
13 - ~gX
, gXJ 1 2 +■ JfX
4 — ln
V l + ln x — 1
\ ln (2 x -
3x + 2 V l + ln x + 1 65. sin -1 jc1- V T 7
In (3x + 2); (c) x ----- 5----- -, 7. - 2 V x cos Vx + 2 sin V x. 67. T7?T sin 20x.
xz + 1 9. —cos x. 69. y[ln (sin x)]2.
> In (jc2
-1 r ^2 — — + 1); (d) 1 + 11. {(x2 — x sin 2x — y cos 2x). 71 7 e sc 3 2x — jo csc5 2x.
jc + 2 ’
13. ex - ln (ex + 1). 73. x ln (2x + x 2) — 2x + 2 ln (x + 2).
jc + ln (jc + 2); (e) 1 -----~---- - , 15. f(x - 1)8/3 + f(x - 1)5/3 + 75. fx 4/3 - -^x 11/6.
jcz + 1
f(* - 1)2/3. 77. x ln (1 + x 2) — 2x + 2 tan-1 x.
jc— 2 tan-1 jc. 17. 2 V x tan 1 V x — ln (1 + x). 79. x tan x — y*2 + In (cos x).
3. 3 ln (jc — 3) + 4 ln (jc + 2). 19. 3x + 11 ln (x — 2). 8 1. y sec7 x.
5. 5 ln (jc - 7) — 3 ln jc.
2 - V4 -x 2 8 3. y(2x2 sin -1 x — sin-1 x +
7. 2 ln jc — 4 ln (jc + 8) +
21. 2 ln + V 4 - x 2.
3 ln (jc - 3). x V 1 —x 2).
9. 3 ln jc + 2 ln (jc + 13) — j tan 1 x 4 87. ln (tan x).
23. - } e ~ x3(x3 + 1). 2 e vS
ln (jc - 3). 91. 5(ln x )2.
2 2 V x — 2 — 4 tan-1 (y V x — 2 ).
I 1. —In (jc + 1) — 3 ln x. 25. In (x + 3) +
x+ 1 x + 3' —cot X.
13. 2 ln x + y ln (x2 + 2x + 2) — { (s in -1 3x + 3 x V 1 — 9 x 2).
1 _i x2 + 1
6 tan-1 (jc + 1). 27. — tan ln (x 2 + 5x + 6).
6
15. jc + 2 In (jc — 1 )- + x 2 — 3'
2 jc — 2 29. ln x V x 2 — 1 — V x 2 — 1 + 101. — In
x2 + 1
| ln (jc2 + 1 ) + 2 tan-1 jc.
tan-1 V x 2 — 1.
103 _ In (x — 1) — 7 In (x 2 +
I 7. yx2 - 2jc + 4 ln (jc + 2).
31 . - t1 cos
- - xv3. , 3. 1 f 2x + 1
19. yx2 — 2 jc + 5 ln (x + 2). 6 Ix 2 + 4 x + 1) + ------ tan 1 / ----------
V3 V V3
21 1
5 \ sin 6+ 4 35. In (x - 1) - 105. \ se '4 v — y sec 2x.
x —1 2 (x — 1 ) 2

„ 1 1 ( ex — 2 107. cos x — j cos3 x.

37. 7 tan6 x + j tan4 x. 109. f(x - 1)5/3 + |(jc - 1)2/3.
23’ 4 l n b T 2 / -
39. - I n (1 + V l - x2).
29. x =
*0
4 1. -^(x — j sin 4x + j sin3 2x). 1 1 1 . — (ax + b) ln (ax + b) —x.
x0 + (1 - x0)e kr a
43. 2 V x - 1 -
Section 10.7, p. 362 113. sin ex.
Vx - 1
I. yx2 In x - } x 2. 4 tan" 115. f x 3/2(3 ln x — 2).
3. yx2 tan-1 x — yx + y tan-1 x. 117. tan x — cot x.
45. x ln (x2 + 3) — 2x +
5. ye*(sin x —cos x). x —1
1 19. 7 ln
7. yxV 1 — x2 + y sin-1 x. 2 V 3 tan-1 — . x + 3 x + 3’
V3
12 1. (x 1) ln (1 Vx) — yx — V x.
e5x 123. y(x - I)2 ta n " 1 (x - 1) -
j V 1 - x 2. 47. -r— (3 sin 3x + 5 cos 3x).
34 y(x — 1) + (x — y) tan-1 (x — 1) —
I I. yx sin (3x — 2) + y cos (3x — 2).
1 fx 2 - 3
13. x tan x + ln (cos x). 49. — ln 1 ln [(x - I)2 + 1],
x2 + 1
15. x ln ( a2 + x 2) — 2 x + 2a tan-1 —. 125. y(xV 1 —x2 - sin-1 x).
51. y ln (x5 + 5x + 3).
17. y(ln x)2. 19. 77( 77- 2). 53. - 4 ln (V x + 1 + 2 ) + Section 10.9, p. 374
23. (b) x(ln x)5 - 5x(ln x)4 + 1. (a) 0.643; (b) 0.656.
6 ln ( V x + 1 + 3).
20x(ln x)3 - 60x(ln x )2 + 120x ln x — 3. 2.2845. 5. 0.881.
120x. 55. tan-1 ( V 3 cos x). 7. 3.14156.
25. 2 t7 [V 2 + ln ( V 2 + 1)]. V3 9. About 23,630 yd2.
 ANSWERS TO ODD-NUMBERED PROBLEMS 869

Additional Problems, p. 375 V a 2 + x2 5 _, (x + 2

115. ln (x + V a 2 + x2) —
1. f(3x + 5)3/2. 3. In (1 + 3x2). 2 ( 2
5. —f sin (1 — 5x).
161. fx 3 tan 1 x - jx 2 + { ln (1 + x2).
7. 2 sec V x. 9. ta n -1 x 2. 117. - 3 a 4 yT ^ fl2 ~ * 2 (2 x 2 + a2).
I I . } ln (sin 4x). 13. —1/ln x. 163. 7 x[cos (ln x) + sin (ln x)].
15. In (tan x). 1 _ I A. A 165. x 3 sin x + 3x 2 cos x — 6 x sin x —
119. — tan 1 ----- 2 2 .
2 a a 2 (a1 + x z) 6cos x.
2 / 3x — 5
1 7 . cos ^ -
V x2 —9 x ln x
121. ------------ . 123. 167. - + ln x - ln (x + 1 ).
19. —2 csc x 3. 21. tan- 1 (lnx). 9x x + 1
V l - 9x 2
2 3 -------- . 169. 7 V 1 + x 2 (x 2 - 2).
' 3(3x + 5) 1 , ( 3 + V 9 + 4x2
125. - l n ( --------- - ---------
25. — 7 ln (3 — 2x). e
17 1. —5----- TT (a sin bx — b cos bx).
27. y sin (1 + x 3). c r + bz
127. —/
x • -1 *
— sin 1 —.
29. — 7 cot (x2 + 1). 173. —x e ~ x — e~x.
V ^ 7 2 a
31. tan-1 (sin x). 175. —{x3^ -2 * — \ x 2e~ 2x — \x e ~ 2x —
33. 7 In (sin 2x). 35. {(tan - 1 x)2.
129. ln (x + V a 2 + x 2) — \ e ~ 2x.
37. 7 In (2x + 1). V a 2+ x2 177.277-. 179. a 1'2.
39. 3ex/3. 4 1 . tan (sin x).
43. j sin - 1 5x. 45. tan - 1 (sec x). 131. f x V x 2 — a2 +
183. +
47. {(In x)3. 7a 2 ln (x + V x 2 — a2). 6 12
49. —In (1 + cos x). x 6 ln x _ x 6
53. —cos (ln x). 133. sin - 1 + 4
51. 36 216*

57. 7 tan - 1 e2x. 135. i V 2 ta n - ' I ^ 5 . (b) {[sec x tan x + ln (sec x +

tan x)].
59. |( 2 + x4)372. 6 1 . \n (ex + x).
63. —4/Ve*. 65. —cot x. 137. _ L Si n - i / 3 x ^ 1
V7 C H A P TE R 11
67. —7 ln (cos x 2). V3
69. 7 In (1 + x 2). Section 11.2, p. 391
L „ 3 x 2—2
139. 7 sin 1 (x — 1) — 2 v 2 x —x 2 —
71. 73. tan x + sec x.
f(x — l ) V 2x — x 2. 1 . (it).
75. |(1 + x5/3)3/2.
77. etanx.
5. (0, fa ). 7. ( f , f).
79. - f ( l + cos x)5.
81. sin (tan x). 83. 77 6/ .
85. i 87. 9- ( 3 (4 - tt) a’ 3(4 - 7 T)al
89. j x — jo sin lOx. 1 1 . (fa , fa )
91. yx + jg sin 14x. 145. 3 V x 2 + 4x + 8 + ln (x + 2 +
4 a + ab + b2 , . 2
9 3 . —j cos 3 x + f cos 5 x — 7 cos 7 x. V x 2 + 4x + 8). 13. —-------------- --------- ; this —>— a as
95. { sin Ax — J2 sin 3 4x.
377 a + b TT
5x - 3
97. csc x - 7 csc 3 x. 147. b —» a.
99. f sin 8 /5 x. 4 V x 2 + 2x — 3
15. (a) On the axis, a distance \h from
1 0 1 . { tan 5 x + f tan 3 x + tan x. 149. 19 ln (x — 4) - 3 ln (x + 3). the center of the base; (b) on the axis,
103. 7 sec9 x — 7 sec 7 x. 151. 3 ln (2x + 1) - 5 ln (2x - 1). a distance f a from the center of the
105. cot4 x + 7 cot 2 x + ln (sin x). 153. 5 ln x + ln (x + 4) — base.
107. 7 tan 3x — 7 cot 3x — 3 ln (x — 3).
7 ln (csc 6 x + cot 6 x). 155. —2 ln x + 3 ln (x + 3) — Section 11.3, p. 393
3 ln (x — 3).
V3 1 3
157. 2 ln x + — — — j -
3 . (a) y-77 a 2; (b) 6V277a2.
a tan -1 _
5 ln (x + 1). 5. 777 a 3; 6 V 377 a 2.
1 13. —iy (a 2 “ x 2)3/2( 3 x 2 + 2 a 2). 159. - I n x + ln (x 2 + 4x + 8) - 7. (a) 77 r2h\ (b) 777 r2h.
870 ANSWERS TO ODD-NUMBERED PROBLEMS

Section 11.4, p. 396 3. (a) m = 7 ; (b) m = 0. 11. All terms must be zero from some
I. {M a2. 3. \M h 2. 5. 16, 34, 34, 14, 2 percent. point on.
5. \M a l. 7. {M a2. 13. (a) 0.6000 . . . ; (b) 1 . 6 6 6 . . . ;
Additional Problems, p. 424
9. ± M a 2. (c) 1 .0 8 0 0 0 .. . ; (d) 1 .1 2 5 0 0 0 ...;
1 I
1. 2- 3. 44. (e) 1.0384615384615 . . . .
II. (a) { V i a = 0.707a; 5. 12. 7. i
(b) ^ v f ( ) a s 0.548a; 9 - -75.
V. 11. 0 0 . Section 13.4, p. 449
13. 6 . 15. 3. 3. (a) \x\ < 1, ax/( 1 — x 2)\ (b) |jc| > 1,
(c) jVTOa = 0.632a.
17. i 19. 0 0 . M{x — 1); (c) |l 4- x\ > 1, 1 + x (also,
Additional Problems, p. 396 if x = 0 the sum is 0 ); (d) e ~ x < x < e,
2 1 . 0 . 23.
3. (a) ({, |) ; (b) (0, f); (c) (1, f); ln x/(\ — ln x).
25. 0. 27. 2J.
(d) (I, f ); (e) (0, f ) ; (f) (jf, ff); 5. * < 0 .
29. j . 31. 9.
33. 3. 35. Section 13.5, p. 454
® (t ^ t -°) 37. f . 39. I. (a) D; (b) C; (c) C; (d) C; (e) C;
5. Snabc', 877(a + £>)c.
41 (f) D; (g) C; (h) C.
7. iM a2. H1- — 16'
43. 0. No; instead, it emphasizes the 3. D. 5. D.
7. D. 9. D.
c h a p t e r 12 logical point that L’Hospital’s rule
II. D. 13. C.
makes a definite statement only when
Section 12.2, p. 403 15. C. 17. D.
the limit on the right exists.
1. 3. 3. £ . 19. C.
49. 0 . 51. 0 .
5 . I6 . 7/ . _ I9 .
j 2 1 . C if/? > 1, D if p < 1.
53. 0 . 55. 0 .
9. j . 11. - 6 . 23. C.
57. 0 . 59. 0 .
13. 3. 15. 4. 1 25. C i f p > 1, D i f p < 1.
61. P- 63. 3-
17. 19. I/ 77. 29. ln 2.
65. 0 . 67. 0 .
21. 16. 23. i 69. 6'
1
71 1. Section 13.6, p. 460
25. 6. 73. 1 75. 1. 1. C. 3. D.
27. /(0 ) = {a2(sin 6 — sin 0 cos 6), 77. 1. 79. 1. 5. C. 7. C.
g(6) = {a2(6 - sin 6 cos 6); limit = f.
81. 1. 83. —oo. Section 13.7, p. 464
Section 12.3, p. 408 85. 1. 87. 1. 1. C. 3. D.
1. - 3 . 3. 1. 89. 1. 91. 1. 5. C. 7. C.
5. 3. 7. 0. 93. e2. 95. e4. 9. D. ll.C.
9. 2. 1 1 . 1 . 97. e \ 99. 1. 13. D. 17. D.
13. 0. 15. 0. 1 0 1 . l/(3e6). 103. 1. 19. C. 21. D.
1
17. 0. 19. 2. 105 • 2- 107. 77/4. 23. C.
2 1 . i 1
23. 1. 109 . 77/8. 111. 3-
25. 1. 27. 1. 113,. 2. 1 15. Diverges. Section 13.8, p. 469
29. 1. 31. 1. I. CC. 3. D.
117,. Diverges. 119. 3.
33. e*. 35. 1. 5. AC. 7. AC.
37. 1. 39. I/V e . 9. D. 1 1 . CC.

41. eP. C H A PT ER 13 13. CC. 15. AC.

Section 13.2, p. 437 17. AC. 19. AC.
Section 12.4, p. 413
1. (a) D; (b) C, 0; (c) C, 0; (d) C, 0; 21. CC. 23. D.
1. l / ( l e 6). 3. f.
(e) D; (f) C, 1; (g) C, 0; (h) C, {■, 25. CC.
5. 1 — cos 1. 7. 1.
(i) C, 0; (j) D; (k) C, 0; (1) C, 0; 27. (a) F; (b) T; (c) F; (d) F; (e) T;
9. 0. 11. ln V 3.
(m) D; (n) C, 0; (0 ) C, 77; (p) C, (f) F.
13. V 2 (ln 4 — 4).
15. 1. 5. (a) a/2; (b) 4 a 3. Additional Problems, p. 470
17. Converges if p < 1, diverges if 9. A decreasing sequence of positive I. (a) 0 ; (b) | ; (c) 0 ; (d) 1 .
p> 1 .
numbers converges. 5. (a) i ; (b) i ; (c)
19. (a) 77/5; (b) 77. Section 13.3, p. 444
fan _ fin
7. xn = ----------- , where A and B are
Section 12.5, p. 423 5. (a) C; (b) D; (c) D; (d) C; (e) D; V5
(f) C; (g) D; (h) D. the positive and negative roots o f
1 . k = —, k = —, no k.
2 77 9. 40 mi. x2 - x - I =0.
 ANSW ERS TO ODD-NUMBERED PROBLEMS 871

1 + V l + 4a Section 15.3, p. 541

aQ l + * 2 + ^2!7 + TT
3! + T7
4! + 1. (a) x2/25 + y 2/21 = 1; (b) x 2/36 +
"■ --------- 2---------'
y 2/ 52 = 1; (c) Ax2/9 + y2/4 = 1;
23. |x| > V I a0ex2’, (b) y = a0 - (a0 - 1)* +
jc(1 — xn) nxn+x (d) j:2/16 + y2/ l = 1; (e) x 2/21 +
(ap 1) , _ (ap 1) , y 2/36 = 1; (f) 24*2/2500 + yVlOO = 1.
Sn~ (1 -x ) 2 \-x'
2! 3 ! 3. (a) (0, 0), (0, ± 5 ), (0, ±4), * = f;
31. (a) - I n 2; (b) 1.
1 + ( a0 - 1) X (b) (0, 0), (± 2 , 0), ( ± V 3 , 0), e =
35. (a) C; (b) D; (c) C; (d) D; (e) C;
(f) D; (g) C; (h) D; (i) C; (j) D; (k) C; V 3 /2 ; (c) ( - 2 , 1), ( - 2 , 1 ± V 2 ),
x2 x3
(1) C; (m) C; (n) C; (o) C; (p) D; ( - 2 , 2) and ( - 2 , 0), e = V 2/2;
' - x + v - v +
(q) C; (r) C; (s) C; (t) D; (u) C; (v) C. (d) (1, 0), (2, 0) and (0, 0), (1 ±
51. C. 53. Inconclusive. 1 + (ao - l)e~ x. { V 3 , 0), e = V 3 /2 ; (e) (2, - 1 ) ,
55. D. 57. C. 5. y = (5, - 1 ) and ( - 1 , - 1 ) , (2 ± V 5 , - 1 ) ,
59. D. .2 e = V 5 /3 ; (f) (0, 2), (± 2 V 2 , 2),
ai
X 1!2! 2!3! 3!4! (± 2 , 2), e = V 2 /2 .
C H A PTE R 14 7. (a) jTrab2', (b) jTra2b.
Section 14.2, p. 489 “i I („ - l) b i!: converSes for
b2
1. ( - 4 , 4). 3. R = 0. ,3 .
all jc.
5. [ - 1 , 1). 7. [ - 1 , 1],
9. R = 0. 11. ( - V 3 , V 3 ). Section 14.7, p. 519 IV. is. ________
13. [ - 1 , 1]. 15. ( - i l l - 21 . y = mx ± v b2 + a 2n
I . X + X2 + yJC3 — -^jJC5 — ^ X 6 +
17. [ - 1 , 1). 19. [ - 1 , 1]. 23. r\ + r2.
3 1 . x + j x 3 + -jyjt5 + ji jx 7 + • • •
21. (2, 6). 2 3 . / ? = °°.
R = tt/2. Section 15.4, p. 549
25. R = 0. 27. (0, 2e).
35. ( a ) i ( b ) | . 37. 272. I. (± 2 , 0), (± V T 3 , 0), 2 y = ±3jc,
29. (a) R = 1; (b) R = oo.
e = V T3/2, x = ± 4 V T 3 .
Additional Problems, p. 523
Section 14.3, p. 494 3. (0, ±2), (0, ± V T 3 ), 3y = ±2jc,
1. (a) °o; (b) oo; (c) e.
I. (a) 2 ( —1)”+1 nxn~x, |*| < 1; e = V I3 /2 , y = ± 4 / V l 3 .
5. ( - 1 , 1).
(b) 2 ( - 1)" in + 2)2(” + 1}- xn, \x\ < 1. 5. (0, ±2), (0, ± 2 V 5 ) , 2y = ± x,
. . f* tan 1 1
7. (a) --------- dt\ e = V 5 , y = ± 2 /V S .
1 1+ * ex - 1 Jo t
3. (a) - l „ _ ; ( b ) / W = — 7. (0, ± 1), (0, ± V 2 ) , y = ±jc,
(b) (1 + x) ln (1 + jc) — jc;
if x + 0 ,/ ( 0 ) = 1; e = V 2, V l y = ± x.
9. y 2/9 - x 2/\6 = 1.
x x - i M i - * 4);
(C) ; (d) I I . jc2/9 — y2/36 = 1.
(1 - X)2 (1 - x 2f
13. jc2/36 - y 2/2S = 1.
jc + 4 jc2 + jc3 ... 4 — 3jc
Section 14.4, p. 503 (e) — t( ;— » (f ) 15. jc2/36 - y2/45 = 1.
\-x) (1 - X ) 2' 17. Hyperbola with center (1, - 2 ) and
15(a)* - F l ! + T ^ ! 11. f 2(x) = 2x1(1 - x ) 3. horizontal principal axis.
(b) * + JX4 - j^x1 + • • • ; 15. (a) - j ; (b) 0; (c) 19. Two straight lines 5 ())+ 1) =
± 6 ( jc + 2).
(c) * - -fiX5 + j; x 9 - • • • .
17. 3.14085; 3.14159. C H A P T E R 15 31. ( —~ ? V b 2 + d 2, —~ t
\ d d
Section 14.5, p. 509 Section 15.2, p. 534
1. (a) Circle, center ( 1, 3) and radius Section 15.6, p. 557
1. (a) 9; (b) 13. 3 .1 .6 4 8 7 2 .
5; (b) empty set; (c) point (5, - 1 ) ; 1. 6 = 45°, jc'2 / 4 + y'2 = i 5 ellipse.
5. 0.978148. 7. 0.848048.
r3 r5 r7 (d) circle, center (8, —6) and radius 2; 3. 9 = 30°, / 2 / 2 - *'2/2 = l,
9. sinjc = j c - - + — - — . (e) point ( - 3 , 7); (f) empty set. hyperbola.
3. (a) ( - 2 , - 1 ) , ( - 2 , 0 ) , 3 ) = - 2 ; 5. 9 = 45°, jc'2 = 4 V 2 / , parabola.
13. 1.0833, 1.0981, 1.0985, 1.0986.
(b) (3, 1), (5, 1 ) , * = 1; (c) ( - 2 , 5), I . 9 = 45°, /2 /2 + /2 / 4 = 1, ellipse.
15. 0.000006.
( - 2 , 1),)) = 9; (d) ( - 2 , 1 ) ,( - 5 , 1), 9. 9 = 60°, x '2/3 + / 2/11 = 1, ellipse.
Section 14.6, p. 513 x = 1; (e) ( - 1 , 2 ) , ( - l , f ) , y = i I I . 9 = sin-1 1/V To, jc'2 + 3 /2 = 1,
1. (a) y = 5. b2y = 4hx(b — jc). ellipse.
872 ANSWERS TO ODD-NUMBERED PROBLEMS

Additional Problems, p. 558 Section 16.5, p. 582 Section 17.3, p. 605

I. 4p. I. |77<32. 5. / .
77 4 1. (a) V !0 , 13i - 34j, 4i - 13j;
9. 177-(2 + V T 7) ft3. (b) V 5 3 , —36i - 4j, —39i + 14j;
7. 77 + 3V 3. 9 . W £ - V 5
(c) 6, lOi - 33j, —2i - 33j; (d) V 34 ,
C H A PTE R 16 —i + 55j, 20i + 22j.
Additional Problems, p. 583
Section 16.1, p. 563 5. (a) (b) ± M L p i
I. (a) tan 0 = 4; (b) r2 = 36/(4 + 5 X
1. (a) (V 2 , V 2 ); (b) (2, - 2 V 3 ) ; sin2 0); (c) r = 2 cos 0 — 4 sin 0;
(c) (0, 0); (d) ({ V 3, {); (e) (0, - 2 ) ; (d) r = 3/(2 cos 0 — 5 sin 0); (e) r = (c) ± (5i ; (d) ± (24l2~ ? j) .
V 74
(f) (—2 V 2 , 2 V 2 ); (g) ( - 3 , 0); 4 cot 0 csc 0; ( f ) r = 1 + 4 sin 0;
A B
(g) r = 6 sin 20/(sin3 0 + cos3 0). 7. ± 2(12i + 5j).
(h) ( - 3 V 2 , 3 V 2 ); (i) (1, 0); ( j) (0, 0); 9‘ |A| + |B|
(k) (1, V 3 ); (1) (5, 12); (m) ( - 2 V 3 , 2); 3. (a) (V 2 a , / ); (b) the origin and
77 4 13. 0 = 45°.
(n) (0, 3).
2 - V2 3 t7\ / 2 + V 2
3. (1, 0), (1, 277-/5), (1, 477/5), a, — Section 17.4, p. 610
2 4 )’ \ 2 "’ 4
(1, 677/5), (1, 877/5). I . The line through the head of A
7. (x — 2)2 + ( y — 2)2 = 8; circle (c) (a/2, ± / ); (d) (3 V 2 , 77/4),
77 6 w hich is parallel to B.
with center (2, 2) and radius 2 V 2 . (3"\/2, 377/4); (e) the origin and 5. 2ri + j, 2i, V 4 /2 + 1.
9. (a) Line y = 2; (b) line x = 4; ( i 277/3), ( i 477/3); (f) ( ± a , 7 7 / 6 ), 7. i + (312 - 3)j, 6rj,
(c) line y -- —3; (d) line x = —2. ( ± a , —7 7 / 6 ); (g) the origin and
V l + 9(t2 - I)2.
(Sa/5, s i n ' 1 3/5); (h) (4, -77/3);
Section 16.2, p. 567 9. sec2 ?i + sec t tan rj, 2 sec2 t tan t X
(i) ( ± 2 , 7 7 / 2 ); (j) (2, ±77/3), ( —1, 7 7 );
5. (a) r = 5 sec 0; (b) r = —3 csc 0; i + (sec3 t + sec t tan2 r)j,
(k) the origin and (f, ± 7 7 / 3 );
(c) r = 3; (d) r 2 = 9 sec 20; |sec r|V 2 sec2 1 — 1.
(e) r = tan 0 sec 0; (f) r2 = 2 csc 20; ■ ,/2 + V2 77\
(1) the origin and I ----- ------ a, — I, I I. R = { a t 2j + \ 0t + R().
sir>2
sin- 0
(g) r
cos 0 cos 2 0 ' 2 - V2 577 Section 17.5, p. 615
2 cos 0 a, ]; (m) the origin and I. (a) —2/(1 + 4x)3/2; (b) cosx;
(h) r =
tan2 6 — 1 ' (c) 2x3/(2x4 - 2 x 2 + 1)3/2;
( ± a , 77/4), ( ± a , 377/4).
9. (1, 277/3) and (1, 4 t7/3); - j . (d) - \\/2 e ~ '- , (e) - 4 t 2/ ( 4 t A + 1)3/2.
15. Larger angle = 277/3.
1 I. r = a sin 20. 3. (a) 1; (b) ^ 5 5/46 1/2; (c) none.
17. ja [ (4 + 4t72)3/2 - 8],
13. x 3 = y 2(2a — x). 5. If (as usual) 5 increases on the
2 1. V 3 - J77.
15. (x — a )(x 2 + y2) = b2x 2. circle in the counterclockwise
25. j a 2(3 V ^ - 7 7 ).
direction, then k as calculated from (4)
Section 16.3, p. 573
has the wrong sign on the upper half­
1. 9 = r2 + 16 — 8r cos (0 — / ).
77 6
C H A P T E R 17 circle, because s increases in the
3. r = 10 cos 0.
direction of decreasing x. Change the
5. r = 4 V 3 cos ( 0 - 77/3). Section 17.1, p. 590 sign o f this result to get 1/a on both
7. r = a ( l + cos 0). 1. (a) x + y = 2; (b) 2x — y = —4. halves of the circle.
9. (a) V 5 a , V 3 a\ (b) { V 5 a, { V 3 a. 3. x + y = 3. 7. ( ~ { l n 2, } V 2 ), f V 3 .
I I. r = ¢7 7 /( 1 + e sin 0). 5. x — 1 = ( y — 3)2. 9. f a at 0 = 7 7 / 4 .
13. e = i \5.e = i 7. x 2 — j 2 = 1. 9. y = 1 — 2x2. I I . 4 a sin {0.
I I . No; the second is part o f the first.
17. ± V e 2 - 1.
13. (c) 45°. Section 17.6, p. 619
ep
19. (a) 15. x = a cos 0 + aO sin 0, y = 3. ( —aco sin iot)i + (aw cos cot)j,
a sin 0 — aO cos 0. ( —a(o2 cos cot)i + ( —aco2 sin cot)j, aco,
Tty_
21. (a) x = y cot 0, aco2.
2a' Section 17.2, p. 599
5. ( —e l sin t + e' cos t)i + (e1 cos t +
(b) r - —— 0 csc 0. . V 2ay - y 2 e 1 sin r)j, ( —2el sin r)i + (2e‘ cos t)j,
77
I. a sin 1 ------ ----- — =
a V 2 e r, V 2 e r, V i e '.
Section 16.4, p. 579 V 2 ay — y 2 + x. 41 . 2 t2 - 2 . 4(1 - t2) .
c 32 2 1 "I--- — --~ j , “ T ' . ..-,1 +
5. ■ 7. 6a. t2 + 1 t2 + \ (t2 + I)2
11. (a) 4 7 7 a 2; (b) 4772a2. I I . x = 2b cos 0 + b cos 20, y = 8 1
-j, 2, 0, 4/(f2 + 1).
17. V 2 . 2b sin 0 — b sin 20; -ya. ( t2 + 1) 2
 ANSWERS TO ODD-NUMBERED PROBLEMS 873

9. an = 0 when t = 0, an = - 1 when Section 18.4, p. 651 13. p = 4. 15. p = 6 cos ¢).

t = 77/2. I. (a) T; (b) F; (c) T; (d) T; (e) F; 17. p2 sin2 (f) + p cos 4> = A.
1 1. v > 30V 2. (f) T; (g) F; (h) F.
3. They are parallel.
Section 17.7, p. 626 C H A P T E R 19
5. (a) x = 2 + /, y = — 1 + At, z =
/ 4 tt2V «3 \ —3 — 2/; (b) x = 2 + 3/, y = —1 — t, Section 19.1, p. 669
1. Since M = ^ 2 j ’ determine

z = - 3 + 6/; (c) x = 2 + 2/, y = I . The entire plane except the line

the ratio a2IT 2 for any particular
- 1 - 2 t , z = - 3 + 51. y = 2jc.
planet ( for instance, the earth) and
7. (a) jc = 2 + 31, y = —3/, z = 3 — 2/; 3. The first and third quadrants,
proceed with the arithmetic.
(b) jc = 4 + At, y = 2, z = ~ 1• including the axes.
A dditional Problem s, p. 627 9 .(1 ,1 ,1 ) . 11. 6. 5. The part of the plane above the line
I. 3a2/2. 13. 2jc — v — z + 2 = 0. y = 3x.
3. (a) 57T2a 3; (b) ™ira2. 17. 8 / V 2 T. 19.(9,0,0). 7. All o f jcyz-space except the origin.
o, 862 y] x - 2 _ y + 1 = _z_ 9. The solid sphere jc2 + y 2 + z2 < 16.
5. 8b + ----- .
a 17 -2 —7 ’ I I . All o f jcyz-space for which z > 0
I I . Smallest radius = 9 /(7 ^ 2 8 ), at 23. The second plane. except the planes z = 77, 377,. . . .
t = iV f. 25. Ax + 3y + Az + 2 = 0. 25. Away from the origin.
29. (a) ^ (b) 0. 27. In the positive x, negative y,
13. Approximately 5.94 X 1027 g.
positive z direction.
Section 18.5, p. 656
C H A PT E R 18 I. Parabolic cylinder. Section 19.2, p. 674
Section 18.1, p. 635 5. Plane. I. 2, 3.
1. Faces: x = 1, y = 4, z = 5. Edges: 9. jc2 + (z — a )2 = a2 . 3. - 6 y 2/(3x + l)2, Ay/(3x + 1).
x = 1, y = 4; y = 4, z = 5; x = 1, I I. (a) z = e ~ <
'x2+y2)\ (b) x 2 + z 2 = 5. 2x sin y, x 2 cos y.
z = 5. e " 2?2. 7. tan 2 y + 3y sec2 3x, 2 x sec2 2 y +
3. 64. 13. (a) y = x 2 + z2; (b) 9(x2 + z2) + tan 3x.
5. (a) The yz-plane and the xz-plane Ay2 = 36; (c) z = 4 — jc2 — y 2\ (d) jc = 9. —3 sin (3x —y), sin (3x — y).
taken together; (b) all three coordinate y 2 + z2. I I . ex sin y, ex cos y.
planes taken together. 15. (jc - 2z)2 + ( y - 3 z )2 - 13. 2ey/x, ey ln x2.
7. (0 ,4 ,0 ) . 6( jc — 2z) = 0. 15. 2 x y 5z7, 5x2y4z7, 7x2y 5z6.
9. x 2 + y2 + (z ~ I ) 2 = 49 or x 2 + 17. l n A £ , - £
Section 18.6, p. 660 z y z
y 2 + z2 = 14z.
I. Ellipsoid. 19. (a) y = 3, z = 8x + 1; (b) x = 2,
I 1. (a) The sphere with center
3. Circular paraboloid. z = 6y - 1.
( —1, 3, 5) and radius 3; (b) the point
5. Hyperboloid of two sheets. 3 1. f ( x , y) = 3jcy2 - x 2 cos y + 2 y.
(5, - 1 , 3); (c) the empty set; (d) the
7. Hyperbolic paraboloid.
point ( —1, 7, 3); (e) the sphere with
9. Hyperboloid of two sheets. Section 19.3, p. 678
center (2, —3, 0) and radius
I I. Ellipsoid. 1. Z - 25 = 20(x — 1) + 40( y — 2).
15. f .
13. Hyperbolic paraboloid. 3. z = 4x + 5y.
17. j(7 i + 5j + 4k). 15. (6, - 2 , 2), (3, 4, - 2 ) . 5. z - 1 = 6(x - 3) - 8 (y - 2).
19. |(A + B + C + D). 7. z — 1 = y.
17. (a) A (k) = Trab ^1 - j>
Section 18.2, p. 639 9. lOx + 13y + 13z = 75.
3. (a) 60°; (b) 45°; (c) 90°. (b) jTrabc. 13. xox/a2 + yoy/b2 + zoz/c2 = 1.
7. No. 13. 2i — k; 23. Hyperbolic paraboloid. 15. 60°. 17. The origin.
19. Two; 45° and 135°. 19. h (l + a )/V a .
Section 18.7, p. 663
21. c(z\ ~ Z2 )•
1. (a) (2 V 2 , u/4, 1); Section 19.5, p. 685
Section 18.3, p. 646 (b) (2, -77-/3, 7); (c) (2 V 3 , tt/6, 2); I. (a) 8i + 4j + 2k; (b) 2i; (c) f(i +
I . (a) 14i + 7j; (b) 3i - 3j; (c) 2i - (d) (3 V 5 , ta n “ ‘ 2, 5). 2j - 2k); (d) } ( - i + 2j + fk ).
14j - 22k; (d) k. 3. (a) (2 V 2 , 77-/6, tt/4); (b) ( 2 V 2 , 3. (a) 3, - j ; (b) V 3 , i + j + k;
3. 2 V 6 . 5 7 7 / 6 , —77/4); (c) (2, 7 7 / 4 , 7 7 / 4 ); (c) 2 V T 9 , i + 3j + 3k; (d) 3<?2,
5. Assuming that their tails coincide, (d) (2,77/6, -77/2). i + 2j + 2k.
all three vectors lie in a plane. 5. r2 + z2 = 16. 7. r2 = z2. 5. ±(i + j — 2 k )/V 6 .
I I . (b) 11/VT07. 9. r = 2 sin 6. 11. r = 3. 7. 56/V 2 T ; i + j + 2k; 4 /V 6 .
874 ANSWERS TO ODD-NUMBERED PROBLEMS

Section 19.6, p. 691 sin 2 a

•5. 1 ' f* / ( x , y) dy dx. 37. jM a 2 ( 1 — ~ 2 a
1. 0.
3. - 9 (t2 + 9)/(t2 - 9)2. in x
5. At3 + Atu2, Au3 + 4f2w. n { f ( x , y) dy dx. Section 20.5, p. 735
1. £ . 3. 2abc/TT.
Section 19.7, p. 695 I9 ' /0 io 2x3 d y d x = j. 5. 24. 7. 4 tt.
I. (3, 5), a minimum. 9. § .
3. (1, 2), a minimum; (—1, 2), a 2 L 1' {02 ( 5 - 2 x - y ) d x d y = 5.
saddle point. "■ io U y f ( X ' y ' z) dz-
C2 f4 x -x * „
5. (—1, -2 ) and (-2 , 8), both saddle
points.
23- L L 2 ^ * = 1 - 13. a 4 / 8 .
fVl - y2 - ;
25. jabc. 27. 4.
7. (0, 0), a saddle point; (—1, -1 ), a - /:-Vl ->2 -
maximum. dx dy dz.
/ ( x , y, z)
13. 2, 4, 6. 15. 1/(21abc). Section 20.2, p., 722 17. 4. 19.
17. j. 19. V l4 . 1. f. 3. a 2. 21. j77abc. 23.
21 . x/6 + y/6 + zJ3 = 1. 5. 1 - e~ a. 7/. -3 . 27. fM a2.
Q 625
23. 3V3/2. y. 12. 11.“ .
1 40 , 3 Section 20.6, p. 738
27. Base of rectangle = (2 —V3)P; 13. f . 15. j77.
1. 77/2 .
height of rectangle = {(3 —V3)P; 17. \( b 3 - a3). 3. x = y = 0, z = ih .
height of triangle = j(2 \/3 - 3)P
Section 20.3, p. 726 5. \M (2 a 2 + 3^2).
Section 19.8, p. 701 1. M = a3; x = y = ija. 7. joM(a2 + Ah2).
1 / 128 —_ 20 — a
1. {. 3. M = x = y = 0. 9. x = y = 0, z. = ja .
3. Comer in first quadrant is 11. f(8 - 3V 3)77a3.
5 . M = f a 3; 3c = -^ 7ra, y = 0.
(V 2 , j V 2 ) . 13. 77/32. 15. a 3/3.
7. M = 77; X = (7T2 — 4)/77, y = 77/8. 17. f a 3(377 - 4).
5. 2r = h. 7. f, j.
9. yMa2. 11. zM a2.
9. x2/3 + y2/12 + e /2 1 = 1. 19. \M a 2. 21. jM a 2.
II. jr. 13. a3. 13. iM a 2.
23. j77a3(l —cos a).
17. (a) \d \N a 2 + b2 + c2. Section 20.4, p. 730
Section 20.7, p. 743
I . 7 7 CL2. 3. i(477 - 3 V 3 )a 2.
Section 19.9, p. 707 1. 5- f 77a5( f —cos a + j cos3 a)
5. a 2. 7. j( 3 V 3 - 77)a2.
7. W = C[X + C2- yMa2[f —j cos a (1 + cos a)].
9. 7 ( 8 + 77)a2.
I I . j ( 9 V 3 - 277)a2. 3. 2772a 3. 5. l l a A ~ b*
Section 19.10, p. 713 a 3 — b3
1. 3x/y. 13. ij(9 V 3 + 877). c877an+5
3. (1 —sin y)/(x cos y — 1). 7. f a a 3.
15. |77a4. 3(n + 5 )'
5. (ye*? - 2 y 2)/(Axy — xe*y). f i n il f3 13. |77a3.
7. 3z/(z ~ 1), 2z/(l - z). 17. \ z r d r dd.
J - tt/2 JO
q xyz cos xz + y sin xz r 7r/4 ftan 0 sec 0 is- «■ "=£ r r ** *■®
1 - x 2y cos xz
x sin xz
' J 0 Jo
z r dr d6.
p2 sin cp dO d(/> dp\
f n l4 n
1 — x 2y cos xz 21. z r d r dd. (b) M = 477 f j f?f(p) dp.
Jo Jo

11. 23. (a) y77[a3 — (a2 — b2)3n}\ 19. 7 7 GmSa sin 2 a.

19
(b) j77(a2 - b2)312 = { 77h3. 21. 2irGm 8.
25. x = fa , y = 0. Section 20.8, p. 747
C H A P T E R 20
27. x = 0, y = a. I. 3VT4. 3. 77a2V 3 .
Section 20.1, p. 717 5 IT

1. The triangle bounded by x = 0, 5. 277 V6. 7. a 2(77 — 2).

29. j77a3.
y = \ , y = x. 9. | t7(5V 5 - 1).
3 1. -j77<24b; x = y = 0 , z = \a 2b.
3. 4 5. 98. II. jira 2( 5 V 5 - 1).
7. 7 7 / 8 . 9. -r. / 1 - ln 2 \ 15. ^ [ 3 V l 0 + ln (3 + VlO)].
35. M a2
11. 77. 13. t In 2. I In 2 j’ 1 7 . 2 a 2. 1 9 . |( 2 0 - 3 t 7 ) .
 ANSWERS TO ODD-NUMBERED PROBLEMS 875

C H A PT E R 21 Section 21.4, p. 778 Section A.14, p. 832

1. (a) 0; (b) 0; (c) 2; (d) ez sin x; 3. C if X =£ |/C77, D if X = f&77.
Section 21.1, p. 757
8. (e) Hr.
1. (a) - 2 ; (b) - 3» (C) - 4 . c y1277-(25. A P P E N D IX B
3. 47Tdbc. 5.
3. 0.
7. (a) f; (b) § ; (c) t ; <:d) # . Section B .l, p. 848
7. 2 l y - + f \ r ) .
2 , , , 9!. 22!. , , 52!
9. (a) 0; (b) 0; (c) 3'
11. (a) 36; (b) 18. (a) 4 ! ’ ^ 16!’ (C) 5!47! ’
11. f for all paths.
13. 20VT 15. 47ra5. 3. 720; 120. 5. 2880.
13. 7 7 for both paths.
105 17. y for both integrals. 7. 9 • 9! = 3,265,920.
15. 2 • 9. 140,400,000. 11. 120; 60; 325.
19. -1277.
19. 0 along all paths. 13. 20,160. 15. 2 (3 !3!) = 72.
21. 0. Section 21.5, p. 783 17. 210. 19. 378.
5. 77. 7. - 1 . 21. 4200. 23. 8820.
Section 21.2, p. 763
9. 477. 11. 1877. 25. 211,680.
9. 0. 11. 6.
13. -877. 15. 0. 13
13. e. 15. 12. 27. 4 ‘ = 5148; 13 12
,5
APPENDIX A 3 7 4 4 .'
Section 21.3, p. 769
l 29. 286. 31. 84.
1. 12• 3. 2 ln 4 - Section A.9, p. 815 n
3 33.
5. 2- 7. 1. 5. 2 V x - 2 ln (1 + V x ). 2
1 3
9. 6' 11. 10' 7. 2 V x — 3 V x + b V x — 35. (a) 349,440x"; (b) -4 8 9 ,8 8 8 x 4;
1 6 ln (Vx + 1). (c) —2002a wb21.
13,• 2• 15. 3 77(22.
3 2 19. 2. 9. 2 V x — 2 tan -1 V x. Section B.2, p. 854
17 . g77^.
11. fx 3/4 — a V x + 4 tan-1 Vx. 1. (a) n(n + 1); (b) n(4n + 1); (c) 4«2;
21 , j3a 1.
2 23. xy3.
13. 2 V x + 2 - 2 tan-1 V x + 2. (d) 3n2; (e) { n(5n + 1).
25 . - x2 + y2-
27.. x sin y + y cos; x. Section A .12, p. 824 V- (a) n n — 2; (b) l — v2"*'
2n
31 . 277. 3. (a) k ^ 3; (b) all k\ (c) all k. n > 0.