MATH 201 Exam 2, Fall 2016

Math 201—Final Exam (Summer 17) B. Shayya and H. Yamani e Please write your section number on your booklet. e Please answer each problem on the indicated page(s) of the booklet. Any part of your answer not written on the indicated page(s) will not be graded. e Unjustified answers will receive little or no credit. Problem 1 (answer on pages 1 and 2 of the booklet.) (i) (12 pts) Find the interval of convergence of the power series }>°° , In (1 + 1\(¢ — 8)". (ii) (12 pts) Estimate the error if cos(x) is approximated by 1— 5 + 2 in the integral Jo cos(x*) dx. = ce (12 pts) The estimate JI +2 21+ £ is used when z is small. Estimate the error when x| < 0.01. Problem 2 (answer on pages 3 and 4 of the booklet.) (i) (12 pts) The derivative of f(x,y) at the point (1,2) in the direction of i+j is 2/2 and in the direction of —2j is —3. What is the derivative of f in the direction of —i — 2j? (ii) (12 pts) Consider the function f(x,y) = xcosy — ye*. Find the plane tangent to the surface z= f(x,y) at the point (0, 0,0). (iii) (12 pts) Let f(x,y) = xcosy — ye* be as in part (ii). Estimate f(—0.3, 0.4). Problem 3 (answer on pages 5 and 6 of the booklet.) (15 pts) Find the absolute maximum and minimum values of the function f(x,y, z) = x? + 2yz — 8z + 600 on the closed ball e+y?+2 <16. Problem 4 (answer on pages 7 and 8 of the booklet.) (15 pts) Use triple integrals to find the volume of the tetrahedron cut from the first octant by the plane 6z + 3y + 2z =6. Problem 5 (answer on pages 9 and 10 of the booklet.) Consider the integral a l= i i i r dz dr dé 0 Jo J3r o Cartesian coordinates with the order of integration dz dx dy. You do not (i) (13 pts) Convert J t need to evaluate the integral. (ii) (13 pts) Convert I to spherical coor need to evaluate the integral. (iii) (13 pts) Convert / to spherical coor need to evaluate the integral. dinates with the order of integration do dé dé. You do not dinates with the order of integration dp dp d@. You do not Problem 6 (answer on pages 11 and 12 of the booklet.) (y+4)/2. 93 (O0 — y)e?2-¥ dady 2 (24 pts) Use the transformation x = u + 2 y =v to evaluate So Suje by first writing it as an integral over a region G in the uv-plane. Problem 7 (answer on pages 13 and 14 of the booklet.) (t) = (#2 -1)j+ 2tk, -1<t <1. (15 pts) A wire of density (x,y, z) = Vy + 2 lies along the curve r Find the curve’s center of mass.