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Unformatted text preview: Name: TA: Math 20D. Midterm Exam 1 April 24, 2009 Sec. No: PID: Sec. Time: Turn oﬀ and put away your cell phone. No calculators or any other electronic devices are allowed during this exam. You may use one page of notes, but no books or other assistance during this exam. Read each question carefully, and answer each question completely. Show all of your work; no credit will be given for unsupported answers. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clariﬁcation. # 1 2 3 4 Σ Points Score 8 6 6 6 26 1. A college graduate with no initial capital invests m dollars per year at an annual rate of return of r . Assume that investments are made continuously and that the return is compounded continuously. (a) (2 points) Write a diﬀerential equation and initial condition for the amount A(t) of the investment at time t (in years). (b) (4 points) Solve the initial value problem to ﬁnd amount A(t) of the investment at any time t. (c) (2 points) If r = 4%, determine m so that $1 million will be available for retirement in 40 years. 2. (6 points) Solve the initial value problem y ′ − tan(t) y = 4t3 sec(t), y (0) = 1 (Hint: Recall that tan(t) dt = − ln  cos(t) + C .) 3. Consider the autonomous diﬀerential equation dy = y (y − 2). dt (a) (2 points) Identify the equilibrium solutions. (b) (4 points) Using the axes below, sketch f (y ) = y (y − 2) versus y and determine which, if any, of the equilibrium solutions are asymptotically stable and which, if any, are unstable.
f(y) y 4. (a) (2 points) Verify that (3x +4y )+(4x +5y )y ′ = 0 is an exact diﬀerential equation. (b) (4 points) Find the general solution to the diﬀerential equation. ...
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This note was uploaded on 01/26/2010 for the course MATH 20D 20D taught by Professor Eggers,john during the Spring '09 term at UCSD.
 Spring '09
 Eggers,John

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