MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

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1 Chapter Practice Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the general solution to the eact differential equation. ) dy dt = cos t - e -t - 35t6 ) A) y = sin t - e-t - 35t + C B) y = sin t + e-t - 5t + C C) y = sin t - e-t - 5t + C D) y = -sin t + e-t - 5t6 + C Solve the initial value problem eplicitly. 2) dy d = 2e - cos and y = 8 when = 0 2) A) y = 2e - sin + 6 B) y = 2e+ - sin + 8 C) y = 2e + sin + 6 D) y = 2e - sin + 8 Solve the initial value problem using the Fundamental Theorem. Your answer will contain a definite integral. 3) du = 5 - sin and u = when = 0 3) d A) u = 5 - sin d B) u = 5 + cos t dt C) u = 5 - sin t dt + D) u = 5 - sin t dt 0 Match the differential equation with the appropriate slope field. 4) y = y + 2 A) 4)

2 B) C) D) 2

3 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the slope field to sketch the graph of the particular solution through the indicated point. (The slope field is shown in the window [-6, 6] by [-4, 4].) 5) dy d = -y; (-2, 2) 5) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Euler's method to solve the initial value problem. 6) dy = y - 4 and y = 4 when = d Use Euler's method with increments of Δ = 0. to approimate the value of y when =.3. A) 4.08 B) 4 C) 3.9 D) 4.2 6) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Verify that f(u) du f(u) d. ) f(u) = u and u = 4 ( > 0). ) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral using the given substitution. 8) cos (32) d, u = 32 A) 2 2 sin (3 2) + C B) sin(32) + C 8) C) 6 sin (3 2) + C D) u sin (u) + C 3

4 9) 2 sin 2 d, u = - 9) A) - + sin C B) sin + C C) sin 4 + C D) sin 4 + C Evaluate the integral. 0) d 52-6 A) - 0 (52-6) 2 + C B) (52-6) 2 + C 0) C) 0 ln C D) ln C ) sin (9-8) d ) A) -cos (9-8) + C B) 9 cos (9-8) + C C) 9 cos (9-8) + C D) - 9 cos (9-8) + C 2) cos(3θ + ) dθ sin2(3θ + ) 2) A) - 3 sin(3θ + ) + C B) sin(3θ + ) + C C) 3 sin(3θ + ) + C D) - cos(3θ + ) 3 sin(3θ + ) + C Use the given trig identity to set up a u-substitution and then evaluate the indefinite integral. d 3), csc 8 = sin28 sin 8 A) - cot 8 + C B) tan 8 + C 8 3) C) - 8 cot 8 + C D) - 8 cot 8 csc 8 + C 4

5 Evaluate the definite integral by making a u-substitution and integrating from u(a) to u(b). 0 4) 4t t2 5 dt A) B) C) D) ) 5) 5 d 2-5) A) ln 9 B) 2 ln 9 C) ln D) 2 ln 9 Evaluate the integral. 6) y ln 8y dy A) y 2 2 ln 8y - y C B) y 2 2 ln 8y + y C 6) C) y 2 2 ln 8y - y C D) 8 y ln 8y - 64 ln 8y + C Use parts and solve for the unknown integral. ) e-4 cos d A) e -4 (sin - 4 cos ) + C B) e -4 (sin + 4 cos ) + C ) C) e -4 ( cos - 4 sin ) 4 + C D) e -4 (4 sin - cos ) + C Use tabular integration to find the antiderivative. 8) 3 cos 6 d A) 6 3 cos sin 6-36 cos 6-26 sin 6 + C 8) B) 6 3 sin cos 6 - sin 6 - cos 6 + C C) 6 3 sin cos sin D) 6 3 sin cos 6-36 sin 6-26 cos 6 + C cos 6 + C 5

6 Solve the differential equation. 9) dy dθ = θ csc - (8θ), θ > 9) A) y = 8 θ cos 8θ - 64 sin 8θ + C B) y = - 8 θ cos 8θ + 8 sin 8θ + C C) y = - 8 θ cos 8θ + 64 sin 8θ + C D) y = - 8 θ cos 8θ - 64 θ sin 8θ + C Solve the problem. 20) Find the area between y = ln and the -ais from = to = 3. 20) A) 2 3 B) 3 ln 3-3 C) 3 ln 3 + (-2) D) ln 3 Evaluate the integral by using a substitution prior to integration by parts. 2) 2 e + 4 d 2) A) e + 4 [ ] + C B) e + 4 [ ] + C C) ( + 4) e C D) + 4 e C Use separation of variables to solve the initial value problem. 22) dy d = 3-3 3y + and y = 2 when = 0 22) A) 3 2 y 2 + y = B) 3 2 y 2 + y = C) 3 2 y 2 + y = D) 3 2 y 2 + y = Find the solution of the differential equation dy/dt = ky, k a constant, that satisfies the given conditions. 23) y(0) = 0, k = A) y = 0e-2.5t B) y = 0-2.5t C) y = 0t-2.5 D) y = -0e2.5t 23) Solve the problem. 24) A sample of 600 grams of radioactive substance decays according to the function A(t) = 600e -.034t, where t is the time in years. How much of the substance will be left in the sample after 0 years? Round your answer to the nearest whole gram. A) 42 grams B) 2 grams C) gram D) 0 grams 24) 25) A certain population is growing at a continuous rate so that the population doubles every 3 years. How long does it take for the population to triple? A) 9.5 years B) 2. years C) 20.6 years D) 2.2 years 25) 6

7 Find the eponential function y = y 0 ekt whose graph passes through the two given points. 26) y 26) (0, 550) (30, 98) A) y = 98e t B) y = 550e t C) y = 550e t D) y = 550e t Use Newton's Law of Cooling to solve the problem. 2) A cup of coffee with temperature 04 F is placed in a freezer with temperature 0 F. After 6 minutes, the temperature of the coffee is 65.9 F. When will its temperature be 39 F? Round your answer to the nearest minute. A) 2 minutes after being placed in the freezer B) 4 minutes after being placed in the freezer C) 3 minutes after being placed in the freezer D) 24 minutes after being placed in the freezer 2) Solve the problem. 28) The resisting force on a moving object such as a car coasting to a stop is proportional to its velocity and is thus equal to -kv for some positive constant k. Using the law Force = Mass Acceleration, the velocity of an object slowed by air resistance satisfies the differential equation m dv dt = -kv. Solving this equation gives v = v 0e-(k/m)t, where v 0 is the velocity of the object at time t = 0. A 60-kg cyclist on a 6-kg bicycle starts coasting on level ground at 9 m/sec. The value of k is about 3. kg/sec. How long will it take the cyclist's speed to drop to m/sec? A) 3.8 seconds B) 39.2 seconds C) 45.0 seconds D) 35.6 seconds 28)

8 Find the values of A and B that complete the partial fraction decomposition ) = A B - 3 A) C) B) - D) ) Evaluate the integral. 8 d 30) ) A) 8 ln C B) 8 tan C C) 2 ln C D) tan- 8 + C Solve the differential equation. 3) F'() = A) F() = ln ( - 2) 6 ( + 2) C B) F() = ln ( - 2)6 ( + 2)6 + C 3) C) F() = ln ( - 2)6 ( + 2) 6 + C D) F() = -2 ln + 2 tan C Solve the problem. 32) A wild animal preserve can support no more than 0 elephants. 30 elephants were known to be in the preserve in 980. Assume that the rate of growth of the population is dp dt = P(0 - P) where t is time in years. How long will it take for the elephant population to increase from 30 to 30? [First find a formula for the elephant population in terms of t.] A) 2.8 years B) 32.0 years C) 30. years D) 35.8 years 32) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 33) A population of rabbits is given by the formula P(t) = 200, + e-c t where t is the number of months after a few rabbits are released and the constant C is determined by an appropriate initial condition. Show that this function is the solution of a logistic differential equation. 33) 8

9 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 34) The table shows the population of a certain city for selected years between 950 and Years after 950 Population , , , , ,6 34) By finding a logistic regression equation to model the data, determine what the population of the city will approach in the long run. A) 254,80 B) 266,08 C) 2,96 D) 28, 5 9