Math 116 Practice for Exam 2

Transcription

1 Math 116 Practice for Exam 2 Generated October 28, 2016 Name: Instructor: Section Number: 1. This exam has 6 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 drug 10 Fall cell growth 15 Winter wood 14 Fall Winter Adele 6 Winter Total 65 Recommended time (based on points): 66 minutes

2 Math 116 / Final (April 26, 2013) page 8 6. [10 points] At a hospital, a patient is given a drug intravenously at a constant rate of r mg/day as part of a new treatment. The patient s bo depletes the drug at a rate proportional to the amount of drug present in his bo at that time. Let M(t) be the amount of drug (in mg) in the patient s bo t days after the treatment started. The function M(t) satisfies the differential equation dm dt = r 1 M with M(0) = 0. 4 a. [7 points] Find a formula for M(t). Your answer should depend on r. b. [1 point] Find all the equilibrium solutions of the differential equation. c. [2 points] The treatment s goal is to stabilize in the long run the amount of drug in the patient at a level of 200 mg. At what rate r should the drug be administered? Winter, 2013 Math 116 Exam 3 Problem 6 (drug)

3 Math 116 / Exam 2 (November 14, 2012) page 4 3. [15 points] A model for cell growth states that the volume V(t) (in mm 3 ) of a cell at time t (in days) satisfies the differential equation dv dt = 2e t V. a. [2 points] Find the equilibrium solutions of this equation. b. [8 points] Solve the differential equation. The initial volume of the cell is V 0 mm 3. Your answer should contain V 0. c. [3 points] How long does it take a cell to double its initial size? d. [2 points] What happens to the value of the volume of the cell in the long run? Fall, 2012 Math 116 Exam 2 Problem 3 (cell growth)

4 Math 116 / Final (April 2011) page 7 5. [14 points] Years later, after being rescued from the island, you design a machine that will automatically feed wood into a fire at a constant rate of 500 pounds per day. At the same time, as it burns, the weight of the wood pile (in pounds) decreases at a rate (in pounds/day) proportional to the current weight with constant of proportionality 1 2. a. [3points] LetW(t)betheweightofthewoodpiletdaysafteryoustartthemachine. Write a differential equation satisfied by W(t). b. [4 points] Find all equilibrium solutions to the differential equation in part (a). For each equilibrium solution, determine whether it is stable or unstable, and give a practical interpretation of its stability in terms of the weight of the wood pile as t. c. [7 points] Solve the differential equation from part (a), assuming that the wood pile weighs 200 pounds when you start the machine. Winter, 2011 Math 116 Exam 3 Problem 5 (wood)

5 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). 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. dx = cosxcos(2y) dx = cosxsin(2y) dx = sinxcos(2y) dx = sinxsin(2y) c. [4 points] Find two equilibrium solutions to the differential equation you circled. Fall, 2015 Math 116 Exam 2 Problem 5

6 Math 116 / Exam 2 (March 21, 2016) DO NOT WRITE YOUR NAME ON THIS PAGE page 5 4. [5 points] Drake is running for president. Suppose F(t) is the fraction of the total population of the country who supports him t months after he announces he is running. Drake gains supporters at a stea rate of 2% of the total population of the country per month, but he also steadily loses 3% of his supporters per month. Write a differential equation that models F(t). 5. [6 points] Adele is also running for president. Suppose P(t), the total number of supporters she has in millions t days after she announces, is modeled by the differential equation with k > 0. dp dt = kp(100 P) a. [4 points] Find the equilibrium solutions to this differential equation and indicate stabilities for each. Make sure your answer is clear. b. [2 points] If Adele starts with one million supporters, what is the maximum number of supporters she can get in the long run? You do not need to show your work. Winter, 2016 Math 116 Exam 2 Problem 5 (Adele)

7 Math 116 / Exam 2 (March 23, 2015) page 8 8. [10 points] Consider the differential equation where F(y) is graphed below. F(y) 2 2 dt = F(y) y 4 a. [4 points] Identify all equilibrium solutions to the equation above. b. [4 points] Determine the stability of each equilibrium solution of the differential equation. c. [2 points] Suppose y(t) solves the differential equation above subject to the initial condition y(0) = 3. Compute lim t y(t). Write your answer in the blank provided. lim y(t) = t Winter, 2015 Math 116 Exam 2 Problem 8