Chapter62002solutions - Chapter 6 Test x December 10, 2002...

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Unformatted text preview: Chapter 6 Test x December 10, 2002 cos 2 x x Name 1. Evaluate x e2 dx. e2 u 2 x 2 cos 2 x 2 du dx 1 2 1 2 1 dx and cos v dv x C v e 2 dx 2 x, 2 eu cos 2 dv 1 2 2 x dx dx or C Now, use 1 2 dv or dx so eu du x sin v 2e2 sin 2 2. Evaluate 2x 0 18 1 3 dx. 1 18 2x 1 1 2 3 dx u 3 2x 1, du 2 dx or 1 2 du dx so 0 3 18 u3 du 9 1 u 3 du 9 2 u 2 3 1 9 2 1 9 1 or 1 9 2 3. 8 9 4 Solve the initial value problem. Support your answer by overlaying your solution on a slope field for the differential equation. dy 3 x2 2x 2 y0 1 dx 3 2 1 -2 -1 1 2 -1 dy dx dy y0 3 x2 2x 3 x2 2 2x 20 or 2 dx C 1 dy dx dx y x3 C 1 3 x2 x2 2x and 2x C y 2 dx and x3 x2 or 03 02 so 2x 1 3 2 1 -2 -1 1 2 -1 6 4. Evaluate 2 1 sin2 x dx. 6 1 sin2 x 6 dx 2 csc2 x dx cot x 6 2 3 0 3 2 5. Use separation of variables to solve the following differential equation : dy dx 4 y ln x x ye 1. dy dx y y y and 6. 1 2 1 2 4 y ln x x 4 ln x x y dx 1 2 dy dx 4 ln x x u ln x 2 u2 ln e C1 2 y so du 1 2 dy dx dx and ln x 2 4 ln x x dx or dy dy ln x y 2 and 1 1 x 1 dx 4 u du C1 2 ln x 4 e3 x cos x 2 2 2y2 and ye or C1 2 2 y2 1 C1 2 or C1 2 0 C1 2 2 1 so Evaluate x 2 dx. e3 x cos u du e3 x dx v dv Use integration by parts, with LIPET, so 2 sin cos x 2 x 2 v dv 2 cos sin x 2 x 2 x 2 x 2 3 e3 x dx x 2 dx dx 6 and e3 x sin x 2 dx integration by parts again e3 x cos u du e3 x 2 e3 x sin 3 e3 x dx x 2 6 x 2 x 2 dx and 2 e3 x sin 36 36 I x 2 x 2 37 I x 2 C 2 e3 x sin x 2 6 dx 2 e3 x cos or I x 2 e3 x cos x 2 12 e3 x cos x 2 6 e3 x cos x 2 dx dx 2 e3 x sin or I so e3 x sin dx x 2 x 2 e3 x sin 2 e3 x sin I 2 37 12 e3 x cos 12 e3 x cos x 2 3 e3 x cos e3 x sin x 2 12 37 e3 x cos 7. Evaluate e dx. 3 e x dx choose 3 w 3 1 x x3 or x w3 and dx 3 w2 dw so ew 3 w2 dw ew w2 dw Now, using tabular integration, u dv ——————————————— w2 ew 2w ew 2 ew 0 ew Now, multiply across and down, alternating signs 3 w2 ew 8. 2 w ew 2 ew 2 3 C or 3x3 e x 1 3 6x3 e x 3 6e x C Evaluate cot2 x sec x dx. cos2 x sin2 x and du csc x x2 ln x dx. 1 cos x so u cos x sin2 x 2 cot2 x sec x dx Now, 1 sin x 9. Evaluate x2 ln x dx u du ln x 1 x x2 ln x dx 1 3 x3 ln x 1 9 x3 1 3 C x3 ln x dx v dv u C sin x dx dx 1 u C or cos x dx C du Use integration by parts, with LIPET, so 1 3 x2 dx 1 3 x3 x dx and 1 3 x3 ln x 1 3 x2 dx or x3 10. The relative growth rate of the population of the state of South Dakota is 0.1 and its current population is P0 500000. When will the population reach 1 million people ? Memorize the exponential growth and decay equations, y so y 500000 e0.1 t and 1000000 500000 e0.1 t 2 e0.1 t ln 2 0.1 t or t 10 ln 2 years ky and or y y0 ek t 11. The population of students at Monta Vista High School is represented by dP 1 1 the logistic differential equation P P2 , where t is dt 2 6000 measured in years a Find k and the carrying capacity, M, for the population b The initial population is P 0 2000 students. Find a formula for the population in terms of t. a We want the form : dP dt k b 1 2 We want the form so P 1 P 1 3000 3000 2000 2000 dP dt k M 1 PM P, so 1 2 1 2 P and 1 6000 M P2 3000 P 3000 P P 3000 P so 6000 3000 M Ae 1t 2 kt where or P A M P0 3000 P0 e 1 1 2 e 1t 2 12. The temperature of a plate is 80 °C as it is taken out of the oven. The temperature of the room is 30 °C, and after 40 minutes the plate cools to 60 °C. How long will it take for the plate to cool to 50 °C ? Know the formula : T Ts T 60 when t 40 so 30 50 e 40 k T0 Ts e k t 60 30 80 40 k and here 30 e k 40 40 k ln 3 5 40 t T0 80, Ts or k 30 ln 3 5 and 3 5 ln 3 5 40 e t ln 3 5 or 2 5 40 ln so T 30 2 5 50 e ln so 50 30 50 e 40 ln 2 5 3 5 e 3 5 40 t or ln 3 5 t or t ln minutes 40 13. Mr. DeRuiter is riding his bike, and together he and the bike have a mass of 90 kilograms. His initial velocity is 10 m sec. The bike s motion obeys the equation v v0 e k m t with k 2 kg sec a Find about how far the bike will coast before reaching a complete stop b About how long will it take the bike s speed to drop to 5 m sec ? a Using the given velocity equation model, and for the position function, s v 10 e 45 10 e 90 1t 2t 10 e 45 1t 1t 1t dt 450 e 45 C and assuming that s 0 0, C 450 and st 450 e 45 450 According to our model, the bike will never come to a "complete stop", but we can check b set t 5 10 e 45 1 2 14. Use Euler s Method to numerically solve the initial value problem y x2 2 y, y1 2, on the interval 1 x 4 starting at x0 1 with dx 1. Know the formula for Euler s Method, yn 1 yn f xn , yn x so x0 1, x 1, and y0 1 and it will take three steps to reach x 2 y1 2 1 2 2 1 2 31 1 2, 1 and 2 y2 1 2 21 1 1 61 5 3, 5 and y3 5 32 2 5 1 5 11 4 4, 4 1t lim t 450 e 45 1 2 1t 1t 450 ln t 0 1 2 450 t 45 450 meters or e 45 or 45 ln seconds 45 ln 2 seconds 4, ...
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This note was uploaded on 08/29/2010 for the course MATH 44323 taught by Professor Anderson during the Spring '09 term at University of California, Berkeley.

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