Chapter 3 23 - SECTION 3.4 DERIVATIVES OFTRiGONOMETRlC...

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Unformatted text preview: SECTION 3.4 DERIVATIVES OFTRiGONOMETRlC FUNCTIONS C 173 E”. _ i _ L "01% _ (10)(0.0821) _ 0.821 g 2 W121 [P(t)V’(t) + V(t)P’(t)] = 071% [(8)(—0.15)+(10)(0.10)] z —0.2436 K/min. 34. (a) If dP/dt = 0, the population is stable (it is constant). (b)£1di:=0 => flP—ro(1 P>P fl 1 P —> P_1 B => P=Pr< —£>. 33. PV 2 nRT => T 2 (PV). Using the Product Rule. we have Pr: 7‘0 PC PC 7'0 T0 If P. : 10,000.1‘0 = 5% z 0.05, and 0 = 4% = 0.04, then P : 10.000(1 — g) = 2000. (C) Iffi : 0.05. then P = 10.000(1 — g) = 0. There is no stable population. 35. (a) If the populations are stable, then the growth rates are neither positive nor negative; that is. dC’ dW dt 0 an dt (b) “The caribou go extinct” means that the population is zero. or mathematically, C = 0. (c) We have the equations % : aC — bCW and 31g : —cW + dCW. Let dC/dt : dW/dt : 0, a = 0.05. b = 0.001. C : 0.05. and d = 0.0001 to Obtain 0.05C * 0.001C'W 2 0 (l) and —0.05W + 0.0001C'W : 0 (2). Adding 10 times (2) to (1) eliminates the CW—terms and gives us 0.050 — 0.5W : 0 => C : 10W. Substituting C : 10W into (1) results in 0.05(10W) , 0.001(10W)W : 0 c» 0.5W — 0.01W2 : 0 4:» 50W — W2 : 0 4:» W(50 — W) : 0 4:) W = 0 or 50. Since G = 10W. 0 = 0 or 500. Thus. the population pairs (C, W) that lead to stable populations are (0, 0) and (500, 50). So it is possible for the two species to live in harmony. 3.4 Derivatives of Trigonometric Functions \ f(a:):$~3sin:1: => f'(3:):1~3cosx .f(:c):xsinx => f'(:c):x-Cosx—I—(sinx)~1:chosm—I—sinx .y:sinx+10tanm :> y’:cosa:+10sec2a: 1. 2 3 4.y:2csc:c+5cosm :> y’:~2cscmc0tm—5sinx 5. g(t) : t3 cost 2)» g'(t) = t3(~sint) + (cost) - 3252 : 3t2 cost — t3 sintor t2(3cost — tsint) 6.g(t)=4sect+tant :> g'(t):4secttant+sec2t 7 . I109): csc6l+egcot6 :> h'(9) = — cch cot 6 —I— 60 (~ csc2 I9) + (cot 6)€9 = — csc 0c0t 0 + 69 (cot 0 — csc2 9) 8. y : eu(cosu+ cu) :> y’ : e”(— sinu + c) + (cosu +0106" 2 eu(cosu ~ sinu + cu—I—c) 9. y 2 m y' _ (cosm)(1) — (2:)(— sinx) _ cosm + msinw cos 90 (cos ac)2 cos2 a: 10.y:1+Sinw : x+cosx y’ _ (m + cosm)(cosz) i (1 + sin 03)(1 ~ sinm) wcosw + cos2m — (1 — sin2 x) (ac + cos $)2 _ (a: —I— cos (5)2 _ at 60300 + cos2 90 ~ (cos2 3:) xcosx (m +cosx)2 — (w+cosw)2 ...
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