MATH 202 Exam 1 with Solution, Spring 2008

! © Math 202 — Spring 2008 Differential Equations, sections 17-20 Quiz 1, March 7 — Duration: 1 hour 10 minutes JUST A MINUTE — PLEASE READ THE INSTRUCTIONS BELOW FIRST 1. Write your name, AUB ID number, and section number ON THE FRONT COVER OF YOUR AUB EXAMINATION BOOKLET. To remind you, the sections are as follows: Section 17 Section 18 Section 19 Section 20 Recitation W 8 Recitation Tu 2 Recitation Tu 3:30 Recitation Tu 5 Prof. Makdisi Mrs. Karam Mrs. Karam Mrs. Karam 2. You may work on the problems in ANY ORDER in your exam booklet, but please make it clear which problem you are solving on any given page. In particular, PLEASE INDICATE IF THE SOLUTION TO A PROBLEM IS CONTINUED ON A LATER PAGE. 3. Do as much of the exam as you can, and budget your time carefully. The problems are arranged APPROXIMATELY in increasing order of difficulty. Take a minute at the start of the exam to decide which problems to work on first and to plan strategy. 4, Explain your steps precisely and clearly to ensure full credit. Partial solutions will receive partial credit. Each problem is worth 12 points, except for problems 1 and 6, and there are 6 problems for a TOTAL 72 points. 5. If you cannot do a certain integral, just leave it as an integral in your solution for partial credit on the rest of the problem. Also do NOT bother to simplify fractions; if the answer to a problem is 3e?(17/6 + 5/11 —7/4), just leave it that way. 6. No calculators, books, or notes allowed. Turn off and put away any cell phones or beepers. GOOD LUCK! (Remember, each problem is worth 12 points, except for problems 1 and 6, for a total of 72) 1. (8 pts) Find the solution to the initial value problem dy/dx = 2rysin(x — x”), y({0) = 10. 2. Let C be the closed curve in the plane shown at right; part of C is the parabola y = x? — 2z from (0,0) to (2,0), and part is the line segment from (2,0) to (0,0). Let F be the vector field F= (xy, 0). a) (6 pts) Compute the work integral f, C F- dr by directly parametriz- ing (both parts of) C. Remember instruction 5 above. b) (3 pts) Set up but do not evaluate the double integral that is : equal to f, F . dF by Green’s theorem. \ c) (3 pts) Set up but do not evaluate the double integral that is equal to f., F. itds by Green’s theorem. 3. Consider the parametrized surface S given by f(u,w) = (u, ucosw, usinw), with O<u<1 and 0 < w < 27. a) (4 pts) Make a rough sketch of S (your picture should also show the «x, y, and z coordinate axes). 2 b) (8 pts) Find the surface area of S. 4, (12 pts) Let C be the curve lying on the surface z = y?+1, whose sides lie vertically above the triangle in the «cy-plane with x=0,y=0,2+y=1. (See diagram for C and its orientation). Use Stokes’ theorem to compute & i Patera [ 3azda+ydy +x" dz. nn) Cc Remember instruction 5! \ \ 5. a) (5 pts) Given that the vector field \ N . \ é “shadow of C F = (Qzy, a7 + ze¥, ye” +1) x " plane is conservative, find a potential function for F. Orr b) (3 pts) Use part (a) to compute [ F . T ds where C is the parametrized curve given by c F(t) = (sin(at), e+, 25 + £5), O<t<1. c) (4 pts) Show that the vector field G = (Qeyt+2,07 + ze¥*, ye¥* 4.1) is not conservative. 6. Let S be the hemisphere x? + y?+22=4, z>0. The flat “bottom” is NOT included. Orient S with a normal vector that points away from the origin. Let F be the vector field ras F = (2az, 0,1). a) (6 pts) Set up but do not compute the flux integral [[#-aa0 in terms of an integral in the ry-plane. 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