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MAT397-2003Spring - MAT 397 Spring 2003 FINAL EXAM Name...

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Unformatted text preview: MAT 397, Spring 2003 FINAL EXAM Name: M (Please print) Instructor: Cox, Mori (9:35 or 12:50), Siagiova(8:30 or 12:50) (Circle one) INSTRUCTIONS 0 There are a total of 10 problems. It is your res ponsibility to make sure that all 10 are present. 0 A scientific graphics calculator may be used on this final. culator, such as a TI-92, may not be used. All differentiat solving, etc. must be done by hand and written dow However, a symbolic cal- ion, integration, equation 11 on the exam. 0 Show all your work. Minimal credit will be given for answers without supporting work. 0 Please simplify your answers when appropriate. 1. Find an equation for the plane containing the points P = (1,2,1), Q = (4,3,3), R = (2, 3, 2). 2. Find the point on the plane a: + 2y + z = 18 that is also on the line that contains the points (1,2,1) and (3,-1,2). 3. Find the directional derivative of the function f(x, 3/, z) = xyey‘ at the point P(2, 1,0) in the direction of the vector v = 2i —j + 2k. 4. A particle’s position at time t is given by r(t) = cos(2t)i + 3tj — sin(2t)k. Find the distance the particle travels between t = 0 and t = 3'? 5. Find the equation of the plane tangent to the surface 1‘2 + 2y2 + 172 = 4 at the point (1,1,1). 6. Find all critical points of the function f (x, y) = my — y2 — x3. Determine for each such point if it is a local maximum, local minimum or saddle point. 7. Use Lagrange multipliers to find the minimum value of f(z, y, 2) = 4x —— 2y— 32 subject to 2z2+y2+z=0. 8. Write the following iterated integral as an equivalent iterated integral with the order of integration reversed. Do NOT evaluate the resulting integral. 2 9—12 2 2 f / er +v dydx 0 5 9. Evaluate ff/ de where S is the solid in the first octant bounded by the parabolic 3 cylinder 2 = 4 — y2 and the planes 2 = 0, y = 3:, and x = O. 10. Find the surface area of that portion of the paraboloid z = 1 — x2 — 3/2 that lies above the my plane. ...
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