MATH 201 Final, Spring 2016

Spring 2015-2016 Date: May 7 | : “nal Exam almrs poe ee Canin Mat! 1 Sections 5 to 8 Duration: 2 hou™ a¥) mal 2 } + ss 4 » b : 2 en a eicanid ms csasnasaahiceinaca aaa Problem 1 (answer on pages 1 and 2 a 0 the woklet) Use the curl t f : ) aluate the es ‘ a at “ se gi " . AVE U LEe t to show that the differential form in the following integral Is exa t, then eva integral. (15 pts) (1 #0) @ia.. E + er ) / J; | (20 COS Y | yz)\d - (2) q sin yu) dy - (ry) a (8.0.9) Problem a (answer on page 3 of the hooklet) Find the maximum and minimum values of the function f(x,y, z) = xyz on the ellipsoid a 7 42° . 6. EE pts) ar > * Aw eenee Problem 3 (answer on page 4 of the booklet) — y Integrate f(x,y) — —2_ ( _ ee fe 64 e wele \ ; ies yey abled the curve C:; z= siny from the origin to the point (U,7)- (49 P™ \ A, Problem 4 (answer o shee B (@ er on pages 5 and 6 of the booklet) Let R be the region in the zxy-plane | r : : & the ry-plane bounded by the lines y = 1, y = — and 2 sy e. Use the g y rl ‘ Kp transformation z = 3u+v, y = u to rewrite || 2 3 dA(x,y) as an integral over an appropriate Rea region G in the wv-plane. Then evaluate the wv-integral over G. (20 pts) Problem 5 (answer on page 7 of the booklet) A mountain climber’s oxygen mask is leaking. If the mountain is represented by the surface : ae 2 : i 2 eine”. ae a. 5 — g* — 2y* and the climber is at (5,—35, 7), in what direction should the climber turn to descend most rapidly ? Find also the instantaneous rate of change of the climber’s altitude in that direction a — (at the point (5,—5)). (15 pts) Problem 6 (answer on page 8 of the booklet) Find the volume of the region bounded from below by the plane z = z, on the sides by the circular para- boloid y = 4(x?+2”) and the plane y = —2, and trom above by the parabolic cylinder z = y z. (20 pts Problem 7 (answer on pages 9, 10 and 11 9g; ee pole | Let D be the larger region cut from the sphere r + y” + 22 =4 by the plane z = J3. | 1. Set up and evaluate the iterated integral in spherical coordinates that gives the volume of D | using the order of integration dp dg dé. (26 pts) \/ 2. Set up the iterated integral in spherical ~~ that calculates the triple integral of a function NM f(z,y,z) on D using the order of integration dédp dé. (14 pts) | Set up the iterated integral in cylindrical coordinates that calculates the triple integral of function f(z,y,z) on D using the order of integration dé dz dr. (8 pts) Problem 8 (answer on pages 12 and 13 of the booklet) Which of the following series converge, and which diverge ‘ When possible, find the sum of the series. (7 pts each) ; oo ; | n oul / jvn 4 vole \\" ( ,)\" i) - (1 .. ) | ii) \ Jn In( hI ae") iii) \ t rj t (2 rt) aie - tomweil ya alter i} a Ti “ A f 6 n= nel i ney Nie