Math 128 Midterm 2 Spring 2009

Transcription

1 Name: ID: Discussion Section: This exam consists of 16 questions: 14 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS: Read each problem carefully and answer the question as written. You may use a non-graphing calculator and a standard sized (no larger than 4 6 ) index card worth of notes for the exam, but you may use no other aids. Record your answer to the multiple choice questions on the accompanying answer card. Show your work on the written problems and write clearly the ease with which your answer can be read will be a factor in your grade. 1. Solve the IVP dy dt = t2 y, y 0 =5 (a) 2 3 t 3 5 (b) 5 e t3 / 3 (c) (d) (e) (f) 2 3 t / t 3 5 e 2t t 3 5 (g) 25 t 2 Page 1 of 18

2 2. Find the integrating factor used to solve the following linear differential equation: t y ' 2 y=ln t, t > 0. (a) t (b) t 2 (c) (d) (e) 1 t 1 t 2 e t (f) (g) (h) (i) e t e 2t e t2 ln t Page 2 of 18

3 3. Find all equilibrium solutions to the differential equation y' 4 y= y 2 5. (a) 0 (b) ± 5 (c) 4,± 5 (d) -4 (e) 4, 5 (f) 1, 4 (g) -1, 4 (h) -1, 5 (i) -4, 5 (j) There are no equilibrium solutions. Page 3 of 18

4 4. At time t=0, a certain tank contains 50 kg of salt dissolved in 1000 L of water. Brine with a concentration of 0.6 kg/l of salt is pumped into the tank at a rate of 5 L/sec. The brine in the tank is kept uniform by mixing and brine is pumped out of the tank at a rate of 5 L/sec. Let y = f(t) represent the amount of salt in the tank at time t. For which of the following differential equations is f(t) a solution? (a) y' =2.75 (b) y' =3 (c) (d) (e) y' =3 y y' =0.6 y 5 t 200 y' =5 e (f) y ' =3 y 200 (g) (h) (i) y' =0.6 y 50 y' =0.6 y 200 y' =0.6 y 50 y (j) y' = 600 y Page 4 of 18

5 5. The population of a fish pond satisfies the logistic equation y' = ,000 y 1000 y where y = f(t) is the number of fish in the population at time t, with t in months. What is the maximal rate of growth the population can ever experience? (a) 2 fish/month (b) 5 fish/month (c) 7.5 fish/month (d) 10 fish/month (e) 15 fish/month (f) 20 fish/month (g) 25 fish/month (h) 50 fish/month Page 5 of 18

6 6. Use Euler's method with n=4 to estimate f(2) where f(t) is a solution to the Initial Value Problem y' = y t 2, y 0 =1. (Round to 1 decimal place.) (a) 2.3 (b) 2.4 (c) 2.5 (d) 2.6 (e) 2.7 (f) 2.8 (g) 2.9 (h) 3.0 (i) 3.1 (j) 3.2 Page 6 of 18

7 7. You start with a lump-sum of $100,000 in a savings account which yields 5% compounded continuously. Starting immediately, you begin continuously withdrawing money at a rate of t dollars per year. Which of the following IVP's would you solve to find the amount of money in the account at time t? (a) y' = ( t), y(0) = 2,000 (b) y' = 0.05 y, y(0) = 98,000 (c) y' = e^{0.05 t} ( t), y(0) = 100,000 (d) y' = e^{0.05 t} + ( t), y(0) = 0 (e) y' = 0.05 y + ( t), y(0) = 100,000 (f) y' = 0.05(y t), y(0) = 100,000 (g) y' = 0.05 y ( t), y(0) = 100,000 (h) y' = e^{0.05 t} ( t), y(0) = 2,000 Page 7 of 18

8 8. Beer is leaking out of a large vat at a rate proportional to the square root of the amount of beer left in the vat, with k = 0.5. Suppose there is 400 L of beer in the vat at time t=0 hours. Then the amount of beer in the vat after t hours is the solution to the Initial Value Problem: y' = 0.5 y, y 0 =400 (Note the negative sign!) How long does it take for all of the beer to leak out of the vat? (a) 2 hours (b) 4 hours (c) 20 hours (d) 25 hours (e) 40 hours (f) 80 hours (g) 200 hours (h) 400 hours (i) The vat never empties out. Page 8 of 18

9 9. Suppose weekly sales of Duff Beer in Shelbyville (in hundreds of dollars) are governed by the formula S x, y =12 x y where x represents Duff's billboard weekly advertising and y represents Duff's radio advertising (each in hundreds of dollars). If Duff spends $1,000 on billboards and $1,600 on radio advertising in a given week, what are its sales that week? (Be careful about your units!) (a) $5,000 (b) $7,500 (c) $12,000 (d) $14,400 (e) $16,000 (f) $22,100 (g) $28,900 (h) $36,000 (i) $42,000 (j) $48,000 Page 9 of 18

10 10. Find the equation of the level curve of f x, y =2 x 2 y which contains the point (1, 1). (a) y= 1 x 2 (b) y= 1 x (c) y= 2 x (d) y=1 (e) (f) y=x y=2 x (g) y=x 2 (h) y=2 x 2 (i) (j) 2 3 y=e x3 1 y=ln x 1 Page 10 of 18

11 11. Let f x, y =ln x 2 x y. Find f 1, 2. x (a) 0 (b) 1 3 (c) ln 2 (d) 1 (e) ln 3 (f) (g) 2 (h) e 4 3 (i) 5 ln 2 Page 11 of 18

12 12. The function f x, y = y 2 x y 3 y 2 x 3 has one critical point. What is it? (a) (-2, 3) (b) (-2, 0) (c) (-1, 0) (d) (-1, 1) (e) (0, 3) (f) (1, -2) (g) (1, 1) (h) (1, 2) (i) (2, -1) (j) (2, 3) Page 12 of 18

13 13. Which of the following best describes the behavior of the function f x, y =sin x sin y at the point 2, 2? (a) f(x,y) has a local minimum at, 2 2 (b) f(x,y) has a local maximum at, 2 2 (c) f(x,y) has a saddle point at 2, 2 (d) f(x,y) may have a local minimum, local maximum, or saddle at 2, 2 but there is insufficient information to tell which (e) f(x,y) does not have a local minimum, local maximum, or saddle at 2, 2 Page 13 of 18

14 14. Suppose that f 10,15 =12, f x 10,15 =1.5, f y 10, 15 = 2. Use this information to estimate f 10.2,15 : (a) 10 (b) 11.6 (c) 12 (d) 12.3 (e) 12.5 (f) 13.5 (g) 14 (h) 15 Page 14 of 18

15 Name: ID: Discussion Section: Written Problem Instructions: Answer below. Show your work. Write clearly. You may receive partial credit for partially worked-out answers. 15. (15 Points) You have a room temperature (25 degree C) beer that you wish to cool quickly, so you put it in your ( -5 degree C) freezer. Assume that the the temperature of the beer follows Newton's Law of Cooling. (a) (5 Points) Suppose that your beer is cooling at a rate of 0.5 degrees/minute when it is 20 degrees. Find the value of the constant of proportionality k in Newton's Law of Cooling. (b) (5 Points) Find a formula for the temperature of the beer at time t. Page 15 of 18

16 (c) (5 Points) How long does it take your beer to reach 5 degrees C? Page 16 of 18

17 Name: ID: Discussion Section: Written Problem Instructions: Answer below. Show your work. Write clearly. You may receive partial credit for partially worked-out answers. 16. (15 Points) Consider the Cobb-Douglas production function f x, y =240 x 1 /3 y 2 /3 where x represents units of labor and y represents units of capital (a) (5 Points) Find an equation for the production isoquant containing (64, 8) (i.e., the curve containing all levels of inputs having the same production level). (b) (5 Points) Calculate the marginal productivity of labor and the marginal productivity of capital when x=64 and y = 8 Page 17 of 18

18 (c) (5 Points) Estimate how much additional production we would have if we increased capital to 8.5 units. Page 18 of 18