Name SID Section Math 231 Final Examination December 15 2005 Instructions To receive full credit you must solve each problem on this exam fully and correctly Be sure that your answers are legible and complete Some questions have more than one part Check carefully to ensure you don t miss any parts Do not write on the line marked Score on the bottom of each page You MAY NOT use a calculator during this examination There are 12 questions for a total of 150 points Math 231 Final Exam December 15 2005 15points 1 Find all critical points of the function f x y Use the second derivative test to determine whether each point is a local maximum local minimum or saddle point 1 f x y xy 2 x2 4x 4 Page 2 of 13 Score Math 231 Final Exam December 15 2005 5points 2 a Find the function D f x y z for the length of the long diagonal in a box with dimensions x y z 5points b Find the linearization of this function at the point 20 10 20 Page 3 of 13 Score Math 231 Final Exam December 15 2005 15points 3 Using Lagrange multipliers find the maximum and the minimum values of f x y xy 2 subject to the constraint x2 y 2 1 No credit will be given if another method is used Page 4 of 13 Score Math 231 Final Exam December 15 2005 15points 4 Find an equation for the plane containing the following parallel lines L1 x 6t y 1 3t z 3t L2 x 1 2s y 4 s z s Page 5 of 13 Score Math 231 Final Exam December 15 2005 10points 5 Find the maximum rate of change of z f x y 4 2x2 xy 2y at the point 1 2 0 and the direction in which it occurs Page 6 of 13 Score Math 231 Final Exam December 15 2005 10points 6 For what value of t is the tangent vector to the curve r t 4t t2 perpendicular to h1 1i Page 7 of 13 Score Math 231 7 f x y x2 y 8points a Find fx 7points b Find fyx Final Exam x y December 15 2005 ey Page 8 of 13 Score Math 231 Final Exam December 15 2005 10points 8 The volume of a torus with major radius R and minor radius r is V R r 2 2 R r2 If dR dt 2 and dr dt 1 use the chain rule to find dV dt Page 9 of 13 when R 5 and r 3 Score Math 231 Final Exam December 15 2005 15points 9 If a t h6t 4 6t 2i is the acceleration of a particle v 1 h2 4 2i is the velocity at t 1 and r 0 h2 0 1i is the position at t 0 what is r 2 Page 10 of 13 Score Math 231 Final Exam December 15 2005 10 Which of the following are equal to 0 either the scalar 0 or the vector 0 for arbitrary vectors u and v Mark all answers that are zero 3points a u u 3points b u u 3points c u v u 3points d u v v u 3points e u v v u Page 11 of 13 Score Math 231 Final Exam December 15 2005 10points 11 Consider A x4 y 4 x y 0 0 x2 y 2 B x3 y x y 0 0 x4 y 4 lim lim Show your work in order to receive partial credit a A does not exist and B 0 b A 0 and B does not exist c Both A and B do not exist d Both A and B exist and are different from 0 Page 12 of 13 Score Math 231 Final Exam December 15 2005 10points 12 Which of the following is the equation for the tangent plane to the surface x2 4x y 2 2y z 2 at the point 1 1 2 Show your work in order to receive partial credit a 6x 4z 14 0 b 6x 4z 2 0 c x 3 z 2 4 3 0 d 6x 2z 4 0 Page 13 of 13 Score