Name: Due Thursday, 7/5/2012 Math 392 Exam 1 Directions Answer all questions in the space provided and box your nal answers. If you need more room, you may continue your work on the other side of the page. Do not use a calculator or graphing device. Do not consult the internet. If you get stuck, you may consult Stewart's Calculus, your class notes, and your/my solutions to assigned exercises. You may collaborate with other students, but you may not copy each other's answers. If you do collaborate, please list the names of your collaborators clearly at the top of this page. Good luck! 2t 1. (a) (4 points) For the path parameterized by r(t) =ti + sintj +e k, compute parametric equations 2 for the tangent line at the point (; 0;e ). p 2 2 (b) (4 points) For the surfaceS given byz = x +y , compute parametric equations for the tangent plane to the surface at (3; 4; XXXXXXXXXX 2. Let F = (ax y +y + 1)i + (2x +bxy + 2)j be a vector eld, where a and b are constants. (a) (4 points) Find the values of a and b for which F is conservative. (b) (4 points) For these values of a and b, nd f(x;y) such that F =Of. Z (c) (4 points) Still using the values ofa andb from part (a), compute Fdr along the curveC such C t t that x =e cost, y =e sint, 0t. Page 23. (a) (3 points) Find the arc length of the curve parametrized by r(t) = 3 cos(t)i + 3 sin(t)k + 4tk, 0t 2. Z 3 2 2 (b) (5 points) For F = yx i +y j, nd F dr on the portion of the curve y = x from (0; 0) to C (1; 1). Page 3
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