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UW-Madison MATH 320 - Math 320 Take-Home Exam

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FW Math 320 Take-Home Exam Thur Sept 26, 2002This is a TAKE-HOME EXAM. You must work on this alone, by yourself, with noone’s help besides your own. You can use the book (Edwards and Penney), your classand homework notes but no other reference material. So any information obtained from theInternet in any way is also forbidden, except of course copies of this exam. You may useyour calculator. You can work on this for as long as you want but must state how long ittook you and turn it in by the due date and time: 1:20 pm, Monday Sept 30, 2002.YOUR NAME:TIME for completion:1. Solvex(x + 1)y0= sin x − xy, y(0) = 1.Your answer may contain integral(s) but all constants and limits of integration must bespecified. Estimate y(π).2. Make a beautiful hand-sketch (no computer plot!) of solution curves of x2y0+ y = 1.What can you say about limx→0y(x)? about limx→+∞y(x)? about the solution with initialcondition y(0) = y06= 0?Extra credit: same questions but for x2y0+ y = x.3. An initially empty cylindrical tank of constant cross-section A is filled with water at theconstant rate k. The water drains from a hole of cross-section a at the bottom of the tankunder the action of gravity. Assume Torricelli’s law for the water velocity at the hole.(a) Draw a sketch of this problem, identifying all relevant variables.(b) Derive the differential equation governing the evolution of the water depth h as afunction of time t.(c) Determine the water depth as t → ∞, explaining briefly your answer (assume the tankis deep enough that it does not overflow).(d) Find a definite integral expression relating h and t. Integrate.3. Edwards and Penney 2.1.20.4. Edwards and Penney 2.3.20.5. The mass meof an electron is much less than the mass Mpof a proton, so as a firstapproximation we can assume that the proton does not move during the interaction of anelectron and a proton (just like we neglect the effect of a space shuttle on the earth). Assumethat the interaction force between the proton and the electron is F (r) ≡ a/r2− b/r4wherer is the distance from the center of the proton to the center of the electron, a and b arepositive constants and the force is attractive if F (r) > 0, repulsive if F (r) < 0.(a) Derive the equation that governs the radial velocity v ≡ dr/dt of the electron, where tis time.(b) What is the equilibrium position of the electron?(c) Is that equilibrium stable or unstable?(d) What is the escape velocity for an electron located at the equilibrium position?6. Use elimination to answer Edwards and Penney 3.2.28. Make sure you clearly identifywhat operations you are performing on the equations to eliminate variables (for instance bywriting E1 - 4 E2 ≡ · · ·,


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UW-Madison MATH 320 - Math 320 Take-Home Exam

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