Midterm Math - Math 211 Exam 1 The use of calculators to do...

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Unformatted text preview: Math 211, Exam 1 The use of calculators to do arithmetic, or to evaluate functions such as e“, is permitted. However, the use of calculators for any other purpose, such as graphing or integrating, is banned. If you use a fact which depends on a theorem, then you must say which theorem, and you must verify that the hypotheses of that theorem are satis— fied. Please make sure that your name, your professor’s name, and your section number are all indicated clearly on your paper, 1. (15 points) Find the exact solution for the following initial value problem. ~~2t 1'— 2 y'zw, withy(0)=1. y What is the interval of existence? 2. (10 points) Consider the differential equation yy’ 2 11/2 it (34‘. a) Is 3/(t) : 62‘ a solution? b) Is y(t) = 262‘ a solution? 3. (15 points) Consider the differential equation rp’ — %;r, = t4. 3.) Find the general solution. b) Find a particular solution satisfying 11(2) 2 10. 4. (10 points) Consider the initial value problem y’ = .112, y(0)=1~ What is the approximate value of 11(1) given by Euler’s method, with step size 0.5? 5. (‘20 points) Consider the autonomous equation 3," z (y + 1)(g2 4). 3. Find and classify the equilibrium points. b. Draw the phase line. c. On the same set of axes sketch the equilibrium solutions, and i. a solution satisfying 31(0) 2 3 ii. a solution satisfying y(0) : 0 iii. a solution satisfying 31(1) 2 0 iv. a solution satisfying 31(0) 2 rrrr 3 2 6. (15 points) Consider the differential equation 7/ = 112 — y _ ye ‘. a) Sl’row that gilt) : e” is a. solution. h) Find it solution of the form 3/ = constant. 0) Sketch the solutions from part a) and h) on the same set of axes. (1) Suppose that Mt) is a solution for which 0 _<_ 31(0) 3 1. Show that fit) -—r 0 as t ~+ 0C. 7. (15 points) Suppose that you’re currently paying $1200/ month to rent your 2—bedroom house. If you were to put that rent money toward a house mortgage instead, what price of house could you afford? Assmno a 30 year mortgage with a 7% interest rate, compounded continuously, and that your payments are made continuously, at a constant rate. Note: A “mortgage” just means a loan from the bank. You must pay interest on the loan at a rate of 7‘3), and the loan must be completely paid oil at the end of 30 years. r ...
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