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Chapter 3 97 - CHAPTER 3 REVIEW 247(b C(t 2 K(e_at —...

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Unformatted text preview: CHAPTER 3 REVIEW 247 (b) C(t) 2 K(e_at — e4") 2 C'(t) 2 K(e"‘”(—a,) — e‘bt(—b)) 2 K(—ae“at + lye—l") b 2 e(_a+b)t 2> In E 2 (I) ~— a)t 2> t 2 ln( /a) a, b — a (C) C’(t) 2 0 2> be‘bt 2 0.6“” 2> 85. 3(t) ll 32 m i. on. O ,‘Q E FF + 3 U C A H. V II CIJ \ A H- \_/ I] A{e‘c‘ [2w si11(wt + 6)] + cos(wt + 0005(1)} flare-rt [e sin(Wt + 6) + ecosw + 5)] 2 a(t) 2 t/(t) 2 2A{e_"t[w2 cos(wt + 6) — cw sin(wt + 6)] + [w sin(wt + 6) + ccos(wt + 6)](2ce‘a)} 2 —Ae_°t[w2 cos(wt + 6) — cwsin(wt + 6) — cwsin(wt + 6) — 02 cos(wt + 6)] 2 2Ae’d[(w2 — 62) cos(wt + 6) — 20w sin(wt + 6)] 2 Ae‘d[(c2 — 0.12) cos(wt + 6) + 2cm sin(wt + 6)] 86. (a) a: 2 b2 + 02t2 2> v(t) 2 :L" 2 [1/(2 vb? + czt2 )] 20215 2 02t/ b2 + 02752 2> Cz\/b2 + 02t2 — c2t (th/x/b2 + c2t2) i902 a“) : “'05) 2 b2 + C2152 _ (b2 + c2t2)3/2 (b) v(t) > 0 fort > 0. so the particle always moves in the positive direction. 87. (a) y 2 t3 — 12t+3 2 U03) 2 y’ 2 32:2 — 12 2 a(t) 2 v’(t) 2 6t (b) v(t) 2 3(t2 — 4) > 0 when t > 2. so it moves upward when t > 2 and downward when 0 g t < 2. (c) Distance upward 2 y(3) — 34(2) 2 26 — (—13) 2 7. Distance downward 2 y(0) — 31(2) 2 3 2 (213) 2 16. Total distance 2 7 +16 2 23. 88. (a) V 2 gmfih 2 dV/dh 2 7W2 [r constant] 1 3 (b) V 2 gmflh 2 dV/dr 2 §7rrh [h constant] 89. The linear density p is the rate of change of mass m with respect to length x. m 2 03(1 + fl) 2 a: + :63/2 2> p 2 dm/dm 2 1 +% at. so the linear density when m 2 4 is l + gx/KT 2 4 kg/m. 90. (a) C(x) 2 920 + 2x 2 0.029:2 + 000007953 2 C’(ac) 2 2 2 0.04m + 0.00021w2 (b) C'(100) 2 2 — 4 + 2.1 2 $0.10/unit, This value represents the rate at which costs are increasing as the hundredth unit is produced. and is the approximate cost of producing the 101 st unit. (c) The cost of producing the 101st item is C(101) 2 C(100) 2 99010107 — 990 2 $010107. slightly larger than C’(100). 91. Ifa: 2 edge length. then v 2 n3 2 dV/dt 2 3m2 dzc/dt 2 10 2 dm/dt 2 10/(3932) and S 2 6n:2 2 dS/dt 2 (12m) dm/dt 212x[10/(3x2)]2 40/33. When a: 2 30, dS/dt 2 £ 2 g cmZ/min. 92. Given dV/dt 2 2. find dh/dt when h 2 5. V 2 émfih and, from similar . r 3 71' 3h 2 3n t 1 ,—2— 2 _— 3 rianges h 10 V 3(10) h 100h.so , 2_fl_97rh2dh 9 @ 200 200 _8 / 1" dt 100 dt dt *9nh2 ‘97r(5)2 T971" ems when h 2 5. ...
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