F97_Second_Midterm-G.Bergman - F(r,y =<*2y5 o*by">...

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I0/o4/20OL THU 16:42 FAX 6494830 George M. Bergman 120 Latimer UOFFITT LIBRARY Fall 1997, Math 53M Second Midterm Exam 4 Oct., 1997 8:10-9:30AM 1. (30 points, 10 points apiece) Find the following. A correct answer will give full credit whether or not you show your computations. An incorrect answer, given with computations that are corect except fbr a minor error, will give partial cred.it. (a) The locations of all critical points of the function *2y - lc - y. (You are not asked to give the values of the function at these points.) 12 rZ rZ (b) J.=r lr=oJ,=rryz dxdY dz' (c) Ic xydx+Lnxdy, where C isthecurve x=et, y=e-t,0st<I. 2- (L6 points) show that if / is a continuous function defined on [0,1], then Il=oli=,, x f(v) dv dx = *ll=o(y - yz) f(y) dy. (Suggestion: First change the order of integration.)
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Unformatted text preview: F(r,y) = <*2y5, o*by"> on the plane is conscrvativc. (b) (6points) Findafunction / suchthar V.f =F, where F isthevecorfieldyou found in part (a). (c) (6 points) Find (by any method) J, f (r y). dr, where F is the vector field asked forin (a), and C isthepath -r= sin f, Jl= 1 + sin?t, -tt/ZK t < n/2. 4. (20 points) (a) (12 point^s) Express the integral t=rli=u, $/yz)sin(nx./y)dydx as an integral in new variables u and u, related to .r and y by the equations r = rr, ! = u/u. (You may take for granted that the above equations give a mapping that is one-to-one on its domain.) (b) (8 points) Evaluate the integral of (a) by any method. 5. (16 points) Find the maximum and minimum values of the function *2 * ,y * y2 on the circle *2 * yZ = 1, and list all points of the circle at which these values occur....
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This note was uploaded on 10/31/2009 for the course STAT 131A taught by Professor Isber during the Spring '08 term at Berkeley.

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