{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter62002solutions

# Chapter62002solutions - Chapter 6 Test x cos 2 x x Name 1...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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, ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online