AP Calculus AB Exam Review Differential Equations and Mathematical Modelling MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the general solution to the exact differential equation. 1) du dx = 15x 14 sin (x15) A) u = x15 cos (x15) + C B) u = cos (x15) + C C)u = - cos (x15) + C D) u = - x15 cos (x15) + C Solve the initial value problem explicitly. 2) ds = t(18t - 8) and s = 3 when t = 1 A) s = 6t3-8t + 5 B) s = 18t3-8t2-7 C)s = 6t3-4t2 + 3 D) s = 6t3-4t2 + 1 Solve the initial value problem using the Fundamental Theorem. Your answer will contain a definite integral. 3) du = 3 - sin x and u = 9 when x = 0 dx x x A) u = 3 + cos t + 9 B) u = 3 - sin t + 9 0 0 9 x C)u = 3 - sin x dx D) u = 3 - sin t 0 9 1
Match the differential equation with the appropriate slope field. 4) y = (y + 3)(y - 3) A) B) C) D) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Obtain a slope field and add to its graphs of the solution curves passing through the given points. 5) y = -y with (0, 2) 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 = x - 2y and y = 2 when x = 2 dx Use Euler's method with increments of Δx = -0.1 to approximate the value of y when x = 1.7. A) 2.76 B) 2.55 C) 2.68 D) 2.88 2
Use separation of variables to solve the initial value problem. 7) dy dx = 5 - x 2 and y = -1 when x = 0 3y + 5 A) 3 2 y 2 + 5y = 5 2 x 3-2 B) 3 2 y 2 + 5y = 5 3 x 3-7 2 C) 3 2 y 2 + 5y = 5 3 x 3-1 D) 3 2 y 2 + 5y = 5x - x3-7 2 Find the solution of the differential equation dy/ = ky, k a constant, that satisfies the given conditions. 8) y(0) = 1301, k = - 1.5 A) y = -1301e1.5t B) y = 1301t-1.5 C)y = 1301-1.5t D) y = 1301e-1.5t Solve the problem. 9) How long would it take $5000 to grow to $15,000 at 6% compounded continuously? Round your answer to the nearest tenth of a year. A) 18.5 years B) 16.9 years C) 18.9 years D) 18.3 years 10) The decay equation for a radioactive substance is known to be y = y0e-0.054t, with t in days. About how long will it take for the amount of substance to decay to 79% of its original value? A) 80.9 days B) 1.9 days C)78.6 days D) 4.4 days 11) Suppose the amount of oil pumped from an oil well decreases at the continuous rate of 8% per year. When will the well's output fall to 25% of its present level? A) after 9.4 years B) after 16.6 years C) after 0.50 years D) after 17.3 years Find the exponential function y = y0ekt whose graph passes through the two given points. 12) y 800 600 400 200 (0, 300) (20, 178) 10 20 30 40 x A) y = 300e-0.043t B) y = 300e-0.026t C)y = 178e-0.026t D) y = 300e-0.032t 3
Use Newton's Law of Cooling to solve the problem. 13) A dish of lasagna is taken out of the oven into a kitchen that is 68 F. After 5 minutes, the temperature of the lasagna is 330.9 F. 16 minutes after being taken out of the oven its temperature is 255 F. What was the temperature of the lasagna when it was taken out of the oven? Round your answer to the nearest degree. A) 425 F B) 390 F C) 375 F D) 360 F Solve the problem. 14) A temperature probe is removed from a cup of coffee and placed in water whose temperature (Ts) is 6 C. The data in the table were collected over the next 30 seconds using a temperature probe. Time (sec) T( C) T - Ts ( C) 2 61.6 55.6 5 50.8 44.8 10 37.5 31.5 15 27.8 21.8 20 21.2 15.2 25 16.8 10.8 30 13.4 7.4 Use exponential regression to find a model for the (t, T) data. A) T = 70.25(0.9307t) B) T = 6 + 64.25(0.9307t) C)T = 66.05(0.9467t) D) T = 6 + 65.45(0.9217t) 15) 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 = -kv. Solving this equation gives v = v 0 e-(k/m)t, where v0 is the velocity of the object at time t = 0. A 63-kg cyclist on a 9-kg bicycle starts coasting on level ground at 7 m/sec. The value of k is about 3.9 kg/sec. How long will it take the cyclist's speed to drop to 2 m/sec? A) 90.2 seconds B) 20.2 seconds C) 23.1 seconds D) 78.9 seconds Find the values of A and B that complete the partial fraction decomposition. 5x + 5 16) x2-4x + 3 = A x - 3 + B A) 10 x - 3-5 B) - 10 x - 3 + 5 C) 5 x - 3-10 D) 20 x - 3-10 Evaluate the integral. 4x + 36 17) dx x2 + 8x + 12 A) ln (x + 2)3 (x + 6)7 + C B) ln (x + 2)7 (x + 6)3 + C C)ln (x + 2) 6 (x + 6)7 + C D) ln (x + 2)4 (x + 6)7 + C 4
Solve the differential equation. 3t + 26 18) G'(t) = t2 + 8t + 12 A) G(t) = ln (t + 2)3 (t + 6)5 + C B) G(t) = ln (t + 2)5 (t + 6)2 + C C)G(t) = ln (t + 2)2 (t + 6)5 + C D) G(t) = ln (t + 6)5 (t + 2)2 + C Solve the problem. 19) The growth of a population is described by the logistic differential equation dp = 0.007P(700 - P) where t is measured in years and P = 8 when t = 0. Solve the initial value problem using partial fractions. 700 A) P = 1 + 82.38e-4.9t C)P = 700 1 + 86.50e-0.49t B) P = D) P = 4.9 1 + 86.50e-0.49t 700 1 + 86.50e-4.9t 20) A wild animal preserve can support no more than 120 elephants. 33 elephants were known to be in the preserve in 1980. Assume that the rate of growth of the population is dp = 0.0007P(120 - P) where t is time in years. Find a formula for the elephant population in terms of t. 120 A) P = where t is the number of years since 1980 1 + 2.64e-0.084t B) P = C)P = D) P = 120 1-2.64e-0.84t 120 1 + 3.64e-0.084t 120 1 + 2.64e-0.84t where t is the number of years since 1980 where t is the number of years since 1980 where t is the number of years since 1980 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 21) A population of rabbits is given by the formula P(t) = 1 + e-c - 0.84t, 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. 5
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 22) The table shows the population of a certain city for selected years between 1950 and 2003. Years after 1950 Population 0 7891 20 103,087 30 191,064 40 241,552 50 265, 058 55 269,116 Use logistic regression to find a curve to model the data. 266,076.8 A) P = 1 + 23.128e-0.1215t C) P = 254,180.3 1 + 26.118e-0.1402t B) P = D) P = 278,715.3 1 + 25.311e-0.1374t 271,976.2 1 + 24.361e-0.1345t 6
Answer Key Testname: DIFFERENTIAL EQUATIONS 1) C 2) D 3) B 4) A 5) 6) A 7) B 8) D 9) D 10) D 11) D 12) B 13) C 14) B 15) C 16) A 17) B 18) B 19) D 20) A 7
Answer Key Testname: DIFFERENTIAL EQUATIONS 21) Let dp = 0.0007P( - P) dp Then P( - P) = 0.0007 1 P( - P) dp = 0.0007 A P + B - P dp = 0.0007 We know that A( - P) + B(P) = 1 Setting P = 0: A() + B(0) = 1, so A = 0.00083333 Setting P = : A(0) + B() = 1, so B = 0.00083333 0.00083333 P 1 P + 1 - P + 0.00083333 - P dp = 0.84 ln P - ln( - P) = 0.84t + C ln( - P) - ln P = -0.84t- C ln - P = -0.84t- C P P - 1 = e -0.84t - C P = 1 + e -C - 0.84t P = 22) D 1 + e-c - 0.84t So P(t) = 1 + e-c - 0.84t dp = 0.0007 where the constant C is determined by an appropriate initial condition. is the solution of the logistic differential equation dp = 0.0007P( - P). 8