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Unformatted text preview: 09/10/2003 WED 11:55 FAX 6434330 MOFFITT LIBRARY 001 EPR major:5 MATH 53 — FINAL EXAM L. Eva :45 Each problem counts 14 points. Be sure to show all your work. Good luck. Problem #1. Find the length of the curve
r(t) =< cost + tsint,sint — tcost >
for 0 S t S 10. Problem #2. Compute Ir“ >< rv for r(u,v) :< v,1w,u+v >. Problem #3. Let C denote the circle (m— 2)2 + (y~—3)2 : 4 in the try—plane, oriented
counterclockwise. Compute / (2m + y2)dx + (2333; + 32:)dy.
0 Problem #4. Find the Critical points of
ﬂay) = :64 ~ 2932 + 312 H 2, and classify each as a local maximum, a local minimum, or a saddle point. [LEas, where F =< 32,3; —I— 23, ey > and S is the boundary of the solid region E determined
by 0 S m2 + y2 S 1, O 3 z 5 1. Orient S by the outward pointing unit normal ﬁeld. Problem #5 . Calculate Problem #6. Use Lagrange multipliers to ﬁnd the point on the plane
a: 4» 2y + 32 : 14
that is closest to the origin. Problem #7. Let 5' denote that part of the surface 2 = 4 — (:62 + 112) lying in the
half—space z 2 O. Orient S by the upward pointing unit normal ﬁeld. Compute ffS(V><F)dS for the vector ﬁeld F =< as + z, w + y, m3 >. (OVER) 09/10/2003 WED 11:55 FAX 6434330 to MOFFITT LIBRARY 002 Problem #8. Assume b > a > 0. Calculate the value of the integral 00 —a:c —b:r:
6 '— e
/ _____.. a
D 33 6—1131? _ b=r b _
__E_ 31:13 rydy) in terms of a and b. (Hint: w Problem #9. Let u and v be two parametric curves in three dimensions that
satisfy u’ : v — u v’ = V + u
for all times t, where ’ = 34;. Show that u x v is constant in time. Problem #10. A point moves along the curve of intersection of the paraboloid
z = m2 + iyz and circular cylinder 3:2 + y2 = 25. We are given that a: z 3, y = 4 and :B’ = 4 at time t = 0,.where ’ = %. Calculate y’ and z’ at time t = 0. Problem #11. Suppose that f is a function of a single variable and that the
expression u=f(:t—ut) implicitly deﬁnes u as a function of :1: and it. Show that ﬁt +U'Um 0. Problem #12. Prove the Divergence Theorem in the special case that
F =< 0, 0, R > and E is a solid region lying between two graphs in the zﬁiirection: E = {(w,y,z) l (219)6‘ D1 Udall) S 2 3 H203, 31)} Draw a picture to illustrate the various terms in your calculation. Problem #13. Let S be a surface whose boundary is the positively oriented curve
0. Suppose also that f, g are real—valued functions. Show that f/S(foVg)dS:/C(ng)dr. Problem #14. Find the volume of the solid enclosed by the surface
($2 + y2 + z2)2 = 221,032 + $12) (Hints: Note that we must have z 2 O in this formula. You will need to use the
identity 0033 rt 2 cos 45(1 — sin2 ...
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