MTH 234 Michigan State University Department of Mathematics Name PID Section No Problem Total 1 16 2 16 3 17 4 17 5 16 6 17 7 17 8 17 9 17 10 16 11 17 12 17 Total 200 Score 1 MTH 234 Practice Final Exam Fall 2010 Michigan State University Department of Mathematics PID Name Section No Signature Total Score 1 Check that you have pages 1 through 16 and that none are blank 2 Fill in the information at the top of the page 3 You will need a pen or pencil and this booklet for the exam Please clear everything else from your desk 4 The use of calculators cell phones or any other electronic device as an aid to writing this exam is strictly prohibited 5 The grading of this exam is based on your method Show all of your work There are problems however that will be graded right or wrong If you need additional space use the backs of the exam pages 6 If you present different answers the worst answer will be graded 7 Box your answers MTH 234 1 Practice Final Exam Fall 2010 16 points a Find a unit vector in the opposite direction of v h1 2 3i b Find the scalar projection of w h1 1 2i onto v c Find the vector projection of w onto v 2 MTH 234 2 Practice Final Exam Fall 2010 3 16 points Find the equation of the plane that contains the lines r1 t h1 2 3it and r2 t h1 1 0i h1 2 3it MTH 234 3 Practice Final Exam Fall 2010 4 17 points A particle moves along the curve r t hsin 2t2 t3 cos 2t2 i for t 0 a Find the velocity v t and acceleration a t functions of the particle b Find the arc length function for the curve r t measured from the point where t 0 in the direction of increasing t MTH 234 4 Practice Final Exam Fall 2010 17 points a Find and sketch the domain of the function f x t ln t x2 b Determine whether the function f above is solution of the wave equation ftt fxx 0 5 MTH 234 5 Practice Final Exam Fall 2010 16 points a Find the tangent plane approximation of f x y sin 2x 5y at the point 5 2 b Use the linear approximation computed above to approximate the value of f 4 8 2 1 6 MTH 234 6 Practice Final Exam Fall 2010 7 17 points Find the absolute maximum and absolute minimum of the function f x y x 3y 2 2xy in the triangle formed by the lines y 0 x 1 and y x 2 MTH 234 7 Practice Final Exam Fall 2010 17 points a Sketch the region of integration D whose area is given by the double integral Z Z Z Z 2 3 dA D x 2 dy dx 0 3 x 2 b Compute the double integral given in a c Change the order of integration in the integral given in a 8 MTH 234 8 Practice Final Exam Fall 2010 17 points Transform to polar coordinates and then evaluate the integral Z 1Z 0 ln x2 y 2 1 dx dy I 1 1 y 2 9 MTH 234 9 Practice Final Exam Fall 2010 10 17 points Find the component z of the centroid for a wire lying along the the curve given 3 2 by r t ht cos t t sin t 2 2 3 t i for t 0 1 MTH 234 10 Practice Final Exam Fall 2010 11 16 points Use the Green Theorem area formula to find the area of the region enclosed by the curve r t hcos2 t sin2 t i for t 0 2 16 4 23 MTH 234 11 Practice Final Exam Fall 2010 12 17 points Find the flux of F outward through the surface S where F h y x x2 i and S x2 y 2 a2 z 0 h x2 y 2 6 a2 z h MTH 234 12 Practice Final Exam Fall 2010 13 17 points Find the outward flux of the field F hx2 2xy 3xzi across the boundary of the region D x2 y 2 z 2 6 4 x 0 y 0 z 0