MATH 253.504–506 An unsupported answerFinal Exam, v ersion A is a wrong answer!12/13/99This test has 2 pages, 13 problems, and 175 points.Conversion from spherical to Cartesian coordinates:x =ρ sin φ cos θy =ρ sin φ sin θz =ρ cos φdV =ρ2sin φdρdθdφ1. (10 pts.) Find the equation of the plane tangent to the surface xy2z3= 2 at the point (2, 1, 1).2. (10 pts.) Evaluate the line integralRCydx−xdy on the semicircle x2+ y2=4,y≥0, fromthe point (2, 0) to (−2, 0).3. (15 pts.) Reverse the order of integration to writeZ31Z−2x+86/xφ(x, y) dy dx as an iteratedintegral in the order dx dy. (You won’t be evaluating an integral in this problem.)4. (10 pts.) Write a parameterization for the part of the cylinder x2+ z2= 4, which lies betweenthe planes y = −1andx+2y+z= 8. Be sure to specify the parameter domain.5. (15 pts.) Using the divergence theorem, computeRRS(~F ·~n) dS,whereSisthesurfaceofthecylinder x2+y2≤ 1, 0 ≤ z ≤ 2, ~n is the outward pointing unit normal, and~F = hxy2,xz,x2zi.6. (20 pts.) Suppose that E is the region in space bounded below by the cone z =px2+ y2andabove by the sphere x2+ y2+ z2= 4. Write (but don’t evaluate)RRREx2dV in(a) spherical coordinates.(b) cylindrical coordinates.7. (15 pts.) Set up (but do not evaluate) an iterated integral, in the order dz dy dx, forRRREzdV,whereEis the region in space bounded below by z = x2+ y2and above by theplane 2x +4y−z=−4.8. (15 pts.) Set up (but do not evaluate) an iterated integral in u and v to find the flux of~F = h2x, −z, yi across the surface ~r(u, v)=hu2,uv,v2i,1≤u≤2, 1 ≤ v ≤ 3, using theupward pointing unit normal.9. Suppose that z = f(x, y), where x =5u+2v,y=3u+v,andthatfhas continuous secondpartials.(a) (5 pts.) Find∂z∂uin terms of u, v, and the partial derivatives of f .(b) (10 pts.) Find∂2z∂u∂vin terms of u, v, and the partial derivatives of f.10. (15 pts.) FindHCx3dx +(x3+y2)dy,whereCconsists of the line segment from (−1, 0) to(1, 0) followed by the parabolic arc y =1−x2from (1, 0) to (−1, 0) (see diagram).00.20.40.60.81–1–0.8 –0.4 0.2 0.4 0.6 0.8 1Figure for problem 10.11. (10 pts.) Determine the equation of the plane which contains the point (1, −1, 2) and the linex = −1+2t,y=1+t,z=1+3t.12. (10 pts.) Describe the domain and range of the function f (x, y)=√16 − 4x2−p9 − y2.13. (15 pts.) Determine the maximum and minimum values of f(x, y)=x2+2y2−x on the diskx2+ y2≤ 1, giving the points where they
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