Math 116 Practice for Exam 2

Transcription

1 Math 116 Practice for Exam 2 Generated November 11, 2016 Name: SOLUTIONS Instructor: Section Number: 1. This exam has 9 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 2. Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you hand in the exam. 3. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam. 4. Show an appropriate amount of work (including appropriate explanation) for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 5. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 5 note card. 6. If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 7. You must use the methods learned in this course to solve all problems. Semester Exam Problem Name Points Score Winter caffeine drip 9 Winter leaky cone 15 Fall currency 18 Winter Fall termites 8 Fall Fall Fall robot uprising 12 Winter Total 107 Recommended time (based on points): 96 minutes

2 Math 116 / Exam 2 (March 23, 2015) page 8 6. [9 points] An extremely sleepy graduate student is grading Math 116 exams. She has been drinking coffee all day, but it just is not enough. She hooks up a caffeine drip that delivers caffeine into her body at a constant rate of 170 mg/hr. The amount of caffeine in her body decays at a rate proportional to the current amount of caffeine in her body. The half-life of caffeine in her body is 6 hours. a. [4 points] Using the blank provided, write a differential equation which models the scenario described above. Use Q(t) for the amount of caffeine in the graduate student s body, measured in mg, t for hours after she hooked up the caffeine drip, and k > 0 for the constant of proportionality. The rate that the amount of caffeine in the graduate student s body is changing over time should be the rate that caffeine is entering their body minus the rate that caffeine is leaving their body. The rate that caffeine is entering the graduate students body is a constant 170 mg/hr. The rate that caffeine is leaving the graders body is proportional to the current amount, so it is kq mg/hr. Putting all this together gives us the equation written below. dq dt = 170 kq b. [5 points] Use the half-life of caffeine to determine the constant of proportionality. We know that the amount of caffeine in the graduate student s body decays exponentially with decay rate k. If C 0 is the initial amount of caffeine, then a half-life of 6 hours means that 1 2 C 0 = C 0 e k6. Solving for k gives us that k = 1 ( ) 1 6 ln. 2 Winter, 2015 Math 116 Exam 2 Problem 6 (caffeine drip) Solution

3 Math 116 / Exam 2 (March 20, 2013) page [15 points] A cone is filled with water up to a depth of H 0 m. At time t = 0, a valve at the bottom of the cone is opened. Water leaks out of the cone through the opened valve. Let H(t) be the depth of the water (in m) in the cone at time t (in hours). The function H(t) satisfies the differential equation dh dt = k H 3 2 a. [2 points] What must be the sign and units of k? The sign of k is negative, because the water is dripping out. Because dh/dt is in meters per hour and h 3/2 is in m 3/2, we know that k must have units m 5/2 per hour. b. [7 points] Find a formula for H(t). Your formula should include k and H 0 We use separation of variables. H 3/2 dh = kdt Since H 5/2 5/2 = kt+c 0 H 5/2 = 5 2 kt+c 1 H = we know that C 1 = H 5/2 0. Therefore, H(t) = ( ) 5 2/5 2 kt+c 1 H 0 = H(0) = C 2/5 1 ( ) 5 2/5 2 kt+h5/2 0. This problem continues on the next page. Winter, 2013 Math 116 Exam 2 Problem 7 (leaky cone) Solution

4 Math 116 / Exam 2 (March 20, 2013) page 11 Problem 7 continued c. [4 points] If the cone is filled with water up to a depth of 4 m at t = 0. What should the value of k be in order for the cone to be empty after an hour? Show all your work. We know H 0 = 4. Therefore, the equation is (5/2kt+32) 2/5 = H(t). Setting this equal to zero, we have ( ) 5 2/5 2 k +32 = 0 So we want to solve 5 k +32 = 0 k = d. [2 points] Does the differential equation satisfied by H have equilibrium solutions? If it does, find them. No Winter, 2013 Math 116 Exam 2 Problem 7 (leaky cone) Solution

5 Math 116 / Exam 2 (November 18, 2015) DO NOT WRITE YOUR NAME ON THIS PAGE page 7 7. [18 points] A certain small country called Merrimead has 25 million dollars in paper currency in circulation, and each day 50 thousand dollars comes into Merrimead s banks. The government decides to introduce new currency by having the banks replace the old bills with new ones whenever old currency comes into the banks. Assume that the new bills are equally distributed throughoutallpapercurrency. LetM = M(t)denotetheamountofnewcurrency,inthousands of dollars, in circulation at time t days after starting to replace the paper currency. a. [5 points] Write a differential equation involving M(t), including an appropriate initial condition. The concentration of old bills among all bills in circulation is M 25000, and 50 thousand dollars moves through the bank each day, so dm dt = Concentration Money per day = M , M(0) = 0. Now consider the differential equation B 2 +2B db dt = b. [4 points] Find all equilibrium solutions and classify their stability. If db dt = 0 then we see that B 2 = 2500, so the equilibrium solutions are B = ±50. Both equilibrium solutions are stable. Brightcrest, a second small country, also wants to replace all of their old paper bills as well, using a different strategy than Merrimead. The amount B(t), in millions of dollars, of new paper currency in circulation in Brightcrest at a time t years after starting to replace the paper currency is modeled by the differential equation for B above with initial condition B(0) = 0. c. [6 points] Find a formula for B(t). Using separation of variables, we have 2B 2500 B 2 db = dt. Using the substitution w = 2500 B 2,dw = 2BdB, we see that ln 2500 B 2 = t+c. Solving for B, we see B = 2500 Ae t. Since B(0) = 0, we see that A = 2500, so B(t) = e t. d. [3 points] Assuming that all of the old bills are replaced in the long run, how much time will pass after starting to replace the paper bills until the new currency accounts for 99% of all currency in Brightcrest? Since all of the bills are replaced in the long run, the total amount of money in circulation is lim B(t) = 50 million dollars. So the amount of time that passes until t the new bills account for 99% of all currency is the value of t so that So t = ln(1 (.99) 2 ) 3.92 years. (.99)(50) = e t. Fall, 2015 Math 116 Exam 2 Problem 7 (currency) Solution

6 Math 116 / Exam 2 (March 24th, 2014) page 2 1. [10 points] Consider the differential equation y = xy 1. a. [2 points] The slope field of y = xy 1 is shown below. On the graph, sketch a solution curve passing through the point (0, 0). y x b. [5 points] Starting with the initial condition y(0) = 0, use Euler s method with 3 steps to estimate y(3/2). Show your work for each step. y(1/2) y(0)+ 1 2 y (0) = 1 2 y(1) y(1/2)+ 1 2 y (1/2) 9/8 y(3/2) y(1)+ 1 2 y (1) 35/16. c. [3 points] Can you determine if your estimate of y(3/2) is an underestimate or overestimate? Circle your answer and explain your reasoning in one sentence. Underestimate Overestimate Not enough information y is concave down on the interval (0, ) therefore Euler s method will give an overestimate. Winter, 2014 Math 116 Exam 2 Problem 1 Solution

7 Math 116 / Exam 2 (November 13, 2013) page 9 7. [8 points] The population of termites P(t) (in thousands) in a tree grows at a rate f(p), in thousands of termites per day. A pesticide is applied to the tree to eliminate the termites. As a result, the population of termites P(t) satisfies dp dt = f(p) 3e 1 3 t, where t is measured in days since the pesticide is applied. a. [4 points] Use Euler s method with steps of t = 0.5 to estimate the amount of termites in the tree one day after the pesticide is applied. It is estimated that there are 2500 termites in the tree at the time the pesticide is applied (t = 0). The graph of f(p) is given below. Show all your computations. P(0) = 2.5 P(0.5) 2.5+(0.5)(3 3e 1 3 (0) ) = 2.5 P(1) 2.5+(0.5)(3 3e 1 3 (0.5) ) = t P(t) b. [2 points] The slope field of the differential equation satisfied by P(t) is shown below. Sketch the graph of P(t). c. [2 points] Is the estimate obtained in part (a) guaranteed to be an overestimate or an underestimate? Justify. Underestimate because P(t) is concave up. Fall, 2013 Math 116 Exam 2 Problem 7 (termites) Solution

8 Math 116 / Exam 2 (November 13, 2013) page 3 2. [12 points] a. [10 points] Suppose the function y(t) satisfies the differential equation dy dt the graph of g(y) is shown below: = g(y), where 1.(4 pts) Use inequalities to describe the regions in the y-t plane where the solution curves of the differential equation are strictly increasing. y(t) is increasing on 1 < y < 1 and 3 < y since dy dt intervals. = g(y) > 0 in these 2.(6 pts) Find all equilibrium solutions (if any) to the differential equation for y(t). Classify each one as stable or unstable. If the equation does not have equilibrium solutions, write none. y = 1 and y = 3 are unstable, y = 1 is stable. b. [2 points] Consider the differential equation dy dt = (2y +5t)t. Find all equilibrium solutions (if any) to the differential equation for y. If the equation does not have equilibrium solutions, write none. None Fall, 2013 Math 116 Exam 2 Problem 2 Solution

9 Math 116 / Exam 2 (November 18, 2015) DO NOT WRITE YOUR NAME ON THIS PAGE page 5 5. [10 points] The graph of a slope field corresponding to a differential equation is shown below. y x -1-2 a. [4 points] On the slope field, carefully sketch a solution curve passing through the point (-1,-1). See graph above. b. [2 points] The slope field pictured above is the slope field for one of the following differential equations. Which one? Circle your answer. You do not need to show your work. dy dx = cosxcos(2y) dy dx = cosxsin(2y) dy dx = sinxcos(2y) dy dx = sinxsin(2y) c. [4 points] Find two equilibrium solutions to the differential equation you circled. The equilibrium solutions of dy dx = cosxsin(2y) are the values of y such that sin(2y) = 0. Solving, we see that the equilibrium solutions are y = 0,± π 2,±π,... Fall, 2015 Math 116 Exam 2 Problem 5 Solution

10 Math 116 / Exam 2 (November 12, 2014) page 2 1. [12 points] Franklin, your robot, is on the local news. Let R(t) be the number of robots, in millions, that have joined the robot uprising t minutes after the start of the broadcast. After watching the news for a little bit, you find that R(t) obeys the differential equation: dr dt = f(r) for some function f(r). A graph of f(r) is shown below. f(r) R a. [3 points] If R(t) is the solution to the above differential equation with R(0) = 0, what is R(t)? Justify your answer. lim t If R = 0, f(r) = R (t) is positive, so R will increase as t increases. As R increases to 1, R (t) = f(r) goes to 0, so lim t R(t) = 1. b. [6 points] Find the equilibrium solutions to the above differential equation and classify them as stable or unstable. R = 2 Stable Unstable R = 1 Stable Unstable R = 1 Stable Unstable R = 3 Stable Unstable c. [3 points] Let R(t) be a solution to the given differential equation, with R(3) = 0.5. Is the graph of R(t) concave up, concave down, or neither at the point (3,0.5)? Justify your answer. d 2 R dt 2 = d dt f(r) = f (R) dr dt = f (R)f(R) At R = 0.5, f (0.5) < 0 and f(0.5) > 0 so d2 R dt 2 < 0. Therefore, the solution curve will be concave down. Fall, 2014 Math 116 Exam 2 Problem 1 (robot uprising) Solution

11 Math 116 / Exam 2 (March 19, 2012) page 6 5. [13 points] Consider the following differential equations A. y = 2x B. y = 5y 1 C. yy = 2 D. y = y x a. [6 points] Each of the following functions is a solution to one of the differential equations listed above. Indicate which differential equation with the corresponding letter (A,B,C or D) on the given line. 1. y = 1 5 +e5x B 2. y = x 2 +1 A 3. y = 2 x C b. [3 points] Each of the following slope fields belongs to one of the differential equations listed above. Indicate which differential equation on the given line. A D C c. [4 points] Find the equilibrium solutions of the differential equations given above (if any). Write the equation of the equilibrium solutions in the space provided below. If the equation does not have equilibrium solutions, write none. A. None B. y = 1 5 C. None D. y = 0 Winter, 2012 Math 116 Exam 2 Problem 5 Solution