Download Math 116 Practice Problems for Exam 3: Differential Equations and more Exams Differential Equations in PDF only on Docsity!
Math 116 — Practice for Exam 3
Generated November 29, 2017
Name:
Instructor: Section Number:
- This exam has 4 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- You must use the methods learned in this course to solve all problems.
Semester Exam Problem Name Points Score Fall 2015 3 3 rumor 13 Winter 2015 2 6 caffeine drip 9 Winter 2012 2 5 13 Fall 2015 2 5 10 Total 45
Recommended time (based on points): 45 minutes
Math 116 / Final (December 17, 2015) DO NOT WRITE YOUR NAME ON THIS PAGE page 4
- [13 points] a. [4 points] The number of people R that have heard a rumor increases at a rate proportional to the product of the number of people that have heard the rumor and the number of people that haven’t yet heard the rumor. Write a differential equation for R which models the scenario described assuming that the total number of people is 1,000. Use k > 0 for the constant of proportionality.
dR dt
b. [4 points] For what values of A, B is y(t) = At cos t + Bt a solution to the differential equation ty′^ = y + t^2 sin t satisfying the initial condition y
( (^) π 2
= 2π? Be sure to show your work.
A =
B =
c. [5 points] Find the solution to the differential equation
e−x^ + y^2
dy dx = 0, with initial condition y(0) = 2.
y =
University of Michigan Department of Mathematics Fall, 2015 Math 116 Exam 3 Problem 3 (rumor)
Math 116 / Exam 2 (March 19, 2012) page 6
- [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.
- y = 15 + e^5 x
- y = x^2 + 1
- y = 2
x
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.
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. B. C. D.
University of Michigan Department of Mathematics Winter, 2012 Math 116 Exam 2 Problem 5
Math 116 / Exam 2 (November 18, 2015) DO NOT WRITE YOUR NAME ON THIS PAGE page 5
- [10 points] The graph of a slope field corresponding to a differential equation is shown below.
x
y
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 differ- ential equations. Which one? Circle your answer. You do not need to show your work.
dy dx = cos x cos(2y)
dy dx = sin x cos(2y)
dy dx = cos x sin(2y)
dy dx = sin x sin(2y)
c. [4 points] Find two equilibrium solutions to the differential equation you circled.
University of Michigan Department of Mathematics Fall, 2015 Math 116 Exam 2 Problem 5
