MATH 201 Final Sample, Summer 2013

a 4 Math 201-Final Exam (Summer 13) = B. Shayya and H. Yamani 0 =— 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. % - © Unjustified answers will receive little or no credit. : Problem 1 (answer on pages 1 and 2 of the booklet.) (i) (15 pts) Find the work done by F = ayzit+ (c+y+z)j+(Qet+ty+ z)k over the curve G r(t)=3ti+tj+2tk, O<t<1. (ii) (15 pts) Show that the differential form (1+ y?+ 2?) dx + 2ay dy + 2xz dz is exact and evaluate SoA + y? + 2) de + 2xy dy + 2xz dz, where C is the same as in part (i). (iii) (15 pts) Find the outward flux of the field F(z,y) = ryi+ y’j across the boundary of the region enclosed by the curves y = x? and y = « in the first quadrant. (Hint. Use Green’s theorem.) Problem 2 (answer on pages 3 and 4 of the booklet.) Let D be the region bounded below by the plane z = 0, above by the plane z = 2, and on the sides by the cone z = /z?+y?. (a) (9 pts) Set up (but do not evaluate) the iterated triple integral in Cartesian coordinates that gives the volume of D using the order of integration dz dy dz. (b) (12 pts) Set up (but do not evaluate) the iterated triple integrals in Cartesian coordinates that give the volume of D using the order of integration dz dz dy. (c) (12 pts) Set up (but do not evaluate) the iterated triple integrals in spherical coordinates that give the volume of D using the order of integration dpd¢ dé. (d) (12 pts) Set up (but do not evaluate) the iterated triple integrals in spherical coordinates that give the volume of D using the order of integration dd dp dé. Problem 3 (answer on pages 5 and 6 of the booklet.) (25 pts) Let R be the region in the first quadrant in the zy-plane bounded by the lines x = 0, y = 0, and z+y=1. Use the transformation r=u y=(1—u)v [f,evdd(e.y) as an integral over an appropriate region G in the uv-plane. Then evaluate the uv-integral over G. to rewrite Problem 4 (answer on pages 7, 8, and 9 of the booklet.) (i) (15 pts) Find the normal line and tangent plane of the surface x2 +y? — z = 0 at the point (1,2, 5). (ii) (15 pts) Find the absolute maximum and minimum values of the function f(z, y, z) = z—2y+5z on the sphere x? + y? + z* = 30. (iii) (15 pts) Consider the function f(x,y, z) = ysina + 2yz. Estimate f (1/15, 31/30, -1/15). Problem 5 (answer on pages 10, 11, and 12 of the booklet.) (a) (7 pts) Find a power series expansion for the function f(x) = (1 +2)~‘/? about the point < = 0. (b) (7 pts) Find a power series expansion for [(1 + z*)-1/2dz about the point x = 0. (c) (6 pts) Estimate fo?(1 + 24)~1/2dx with an error of magnitude less than 10-*. ? te, we wre = Ne yw \* ee rs ~ mS) 4 a oo J x ho nag S, s its ee