MATH 201 Final, Summer 2017

Taynani Math 201-Final Exa: end H. Yemen! m (Summer 17) B. Shayy: © Please write your section number © Please ans ch pre not written on the indic. © Unjustified answers wi 1 your booklet. ar answer t. Any part of you Problem 1 (answer on pages 1 and 2 of the booklet ) () (12 pts) Find the interval of convergence of the power series S92, In (1 + 2)(@— 8)". (ii) (12 pts) Estimate the error if cos(2" r 2 , —— + in the integral Jj cos(x”)dx. ie ope amie Se small. Estimate the error when |z] < 0.01. ?) is approximated by 1 — 2 1+ = is use 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) = cosy — 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 — yet 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(z, y, z) = 2? + 2yz — 8z + 600 on the closed ball 22 +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 62 + 3y + 2z =6. Problem 5 (answer on pages 9 and 10 of the booklet.) Consider the integral 2m pl p4—r? = [ if : r dzdr d0 0 Jo Jar (i) (13 pts) Convert J to Cartesian coordinates with the order of integration dzdxdy. You do not need to evaluate the integral. (ii) (13 pts) Convert J to spherical coordinates with the order of integration dod¢d@. You do not need to evaluate the integral. (iii) (13 pts) Convert J to spherical coordinates with the order of integration do dp dé. You do not need to evaluate the integral. Problem 6 (answer on pages 11 and 12 of the booklet.) aa (24 pts) Use the transformation z = u+ 4, y =v to evaluate fp OM Me y(2x — ye? dady 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.) (15 pts) A wire of density d(x, y, z) = /y +2 lies along the curve r(t) = (t? -1)j+2tk, -1<¢ <a Find the curve’s center of mass.