Math 53 Final, 5/18/07, 12:30 PM – 3:30 PMNo calculators or notes. Each question is worth 10 points. Please writeyour solution to each of the 10 questions on a separate sheet with your name,SID#, and GSI on it. (If you are removing an incomplete for professor X,write “Incomplete/X” on each page. For Math 49, do questions 6-10 only.)To get full credit for a question, you must obtain the correct answer, put abox around it, and show correct work/justification. Please do not leave theexam between 3:00 and 3:30. Good luck!1. Find the point on the sphere x2+ y2+ z2= 1 that minimizes thefunctionf(x, y, z) = (x −2)2+ (y − 2)2+ (z − 1)2.2. Find the volume of the region consisting of all points that are insidethe sphere x2+y2+z2= 4, above the plane z = 0, and below the planez = x.3. Let R be the triangle with vertices (0, 0), (1, 0), and (1/2, 1/2). Eval-uate the integralZZRex+yx + ydA.Hint: Use the change of variables u = x + y, v = x −y.4. A particle moves along the intersection of the surfacesx2+ y2+ 2z2= 4, z = xy.Let h(x(t), y(t), z(t)i denote the location of the particle at time t. Sup-pose that hx(0), y(0), z(0)i = h1, 1, 1i and x0(0) = 1. Calculate y0(0)and z0(0).5. Suppose f is a function on R2satisfying the following conditions on itsdirectional derivatives:DD1√2,1√2Ef(x, y) =√2 x, DD1√2,−1√2Ef(x, y) =√2 y.(a) Find fx(x, y) and fy(x, y).(b) Assuming also that f(0, 0) = 0, find f (x, y).6. Let S be the triangle with vertices (0, 0, 0), (1, 0, 1), and (1, 1, 2), ori-ented upward. Calculate the surface integralZZSh3, 4, 5i · dS.7. Consider the vector fieldF =px2+ y2+ z2hx, y, zi.(a) There is a constant c such that div F = cpx2+ y2+ z2. Find c.(b) Compute the outward flux of the vector field F through the bound-ary of the solid region x2+ y2+ z2≤ 1, z ≥px2+ y2.8. CalculateRCF ·dr, where C is the space curve r(t) = hcos t, 0, sin ti for0 ≤ t ≤ 2π, andF =sin(x3) + z3, sin(y3), sin(z3) − x3.Hint: Use Stokes’ Theorem.9. Let S1be the hemisphere x2+ y2+ z2= 1, z ≥ 0, oriented upward. LetF =x + y2+ z2, x2− y + z2, x2+ y2.(a) Use the Divergence Theorem to show that there is an orientedsurface S2in the x, y plane such thatRRS1F · dS =RRS2F · dS.(b) Use part (a) to calculateRRS1F · dS.10. Let C be the semicircle x2+ y2= 1, y ≥ 0, oriented counterclockwise.Calculate the line integralZC(−y + cos x) dx + (x + sin y) dy.Hint: Part of this integral can be evaluated directly from the definition,and the rest using the Fundamental Theorem of Line