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ANS.] 5 + 3 3 ( f ) max = 6
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ANSWERS TO SUPPLEMENTARY PROBLEMS 5 3 3 at , min = at ; 3 6 3 3 < x0 < , min = 5 (at x0 + ). (g) max = 5 at x0 , where sin x0 = and 5 2
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27.15 (b) (i) Amplitude = 5, period = 2 ; (ii) amplitude = 130, period= ; (iii) amplitude =
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6, period = 2 .
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2 3 3 9 2 3 27.16 (a) ; (b) . 27.17 A = . 27.18 y = x+ . 2 2 2 12 27 4 3 2 3 27.19 y = x+ . 27.20 n = 4. 27.21 (a) See Fig. A-20; (b) yes; (c) no. 3 18
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27.22 (a) 1; (b) 0; 27.23 (a) y = 1 1 y cos (xy) = . ; (b) y = 2 sin y x cos (xy) 2y 1 x
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1 27.24 (a) rad/sec; (b) 200 3 km/h. 8 27.25 (a) = ; (b) = 0. 2 27.26 (a) max = 1 at , min = 0 (at 0). 2 2 0.349; (b) 0.857. 9
27.27 Continuous, but not differentiable.
27.28 (a)
27.29 (a) cos x, sin x, cos x, sin x; (b) sin x. 27.30 (b) 0.8654740331. 27.31 3.141592410. 27.32 1.895494267.
28.8 See Fig. A-21. x 28.10 (a) sec2 ; (b) (sec x)(sec x tan x); (c) 2 cot x csc2 x; (d) 3 4 csc2 4x; (e) 6 sec2 3x tan 3x; 2 cot x csc x sec2 x ( f ) 3 cot (3x 5) csc (3x 5); (g) . ; (h) 3 2 x 3 tan2 x 28.11 (a) y = (d) y = y( y2 + 1) y sec2 (xy) csc2 x cot x 3 cos x = ; (b) y = ; (c) y = ; 2 tan ( y + 1) sec2 ( y + 1) x sec2 (xy) 1 x( y2 + 1) 1 sec2 y tan y 2 tan (x + y) sec2 (x + y) 2(1 + y) tan (x + y) . = 1 2(1 + y) tan (x + y) 1 2 tan (x + y) sec2 (x + y)
ANSWERS TO SUPPLEMENTARY PROBLEMS
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Fig. A-21
28.12 y = 4 x 3 28.15 320 rad/h = + 3. 28.13 y = 3 x 8 6 3 . 4
+ 4.
28.14 (a) 1; (b) 8; (c)
4 rad/sec 5 per second. 45
28.16 tan( 2 1 ) = 1.5, 2 1 56 . 28.19 y = 4x . 2( 2 + 4)
28.17 tan( 2 1 ) = 2, 2 1 63 .
28.18 tan( 2 1 ) = 3, 2 1 71 .
28.20 (a) Rel. max. at /4, rel. min. at 3 /4, vertical asymptote at /2, in ection points at 0 and ; (b) rel. max. at 2 /3, rel. min. at /3, vertical asymptote at /2, in ection points at 0 and . 28.21 On all intervals where it is de ned: , , 2 2 3 , , etc. 2 2
28.22 (a) 1.318116072; (b) 4.493409457; (c) 0.860333589. 28.23 sec2 4 = 2. 2 deg/sec.
28.24 40 rad/h =
29.7 (a) 5 x4 3 8 1 x 3 + x 2 + x + C; (b) 5x 2 x + C; (c) x 5/4 + C; (d) 3x 5/3 + C; (e) 3 + C; 2 3 2 5 x 2 2 3/2 1 1 1 2 x (3x 5) + C; (g) 2 + 4 + C = 4 (1 2x 2 ) + C; ( f ) x 5/2 x 3/2 + C = 5 3 15 2x 4x 4x 4 2 6 (h) x 5/2 x 3/2 + 2x 1/2 + C = x (9x 2 10x + 15) + C; (i) 3 cos x + 5 sin x + C; ( j) 7 tan x sec x + C; 5 3 15 2 3x 1 10 2 6 3 cot x + C; (l) 2 3 x 5/2 + C = 2x + C; (m) sin x + C; (n) tan x x + C; (o) x + x + 2x 2 + C. (k) x 5 5 10 3
ANS.]
ANSWERS TO SUPPLEMENTARY PROBLEMS
29.8 (a)
x 1 1 2 (7x + 4)3/2 + C; (b) 2(x 1)1/2 + C; (c) (3x 5)13 + C; (d) cos(3x 1) + C; (e) 2 tan + C; 21 39 3 2 1 1 ( f ) 2 sin x + C; (g) (4 2t 2 )8 + C; (h) (x 3 + 5)4/3 + C; 32 4 2 3 2 x + 1(x 2) + C; ( j) (x 1)5/3 + C; (i) (x + 1)3/2 2(x + 1)1/2 + C = 3 3 5 3 1 4 3 4 1 1 (k) (x + 1)7/3 (x 4 + 1)4/3 + C = (x + 1)4/3 (4x 4 3) + C; (l) 1 + 5x 2 + C; 4 7 4 112 5 2(ax + b)3/2 1 1 + C = csc 3x + C; (3ax 2b) + C; (n) (m) 3 sin 3x 3 15a2 2 1 2 1 (o) 2(1 x)3/2 (1 x) + (1 x)2 + C = (1 x)3/2 (15x 2 + 12x + 8) + C; 3 5 7 105 1 1 1 1 1 3 (3x 5)13 (39x + 5); (q) (4 7t 2 )8 + C; (r) sin2 + C; (s) tan 4 + C. (p) 1638 112 2 x 12 x
29.9 (a) t = 7 sec; (b) 1024 ft; (c) t = 15 sec; (d) 256 ft/sec. 29.10 (a) v = t 2 3t + 4; (b) x = 3t 2 t3 + 4t; (c) Never; (d) Never. 3 2
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