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Unformatted text preview: Your Name: Circle your TA’s name: James Hunter Deigo Galindo Nansen Petrosyan Boian Popunkiov
Mathematics 234, Fall 2005 Lecture 2 (Wilson)
Final Exam December 20, 2005 Write your answers to the ten problems in the spaces provided. If you must continue an
answer somewhere other than immediately after the problem statement, be sure (a) to
tell where to look for the answer, and (b) to label the answer wherever it winds up. In
any case, draw a box around your ﬁnal answer to each problem or part of a problem. Wherever applicable, leave your answers in exact forms (using %, x/S, cos(0.6), and similar
numbers) rather than using decimal approximations. If you use a calculator to evaluate
your answer be sure to show what you were evaluating! There is a problem on the back of this sheet: Be sure not to skip over it by accident!
There is scratch paper at the end of the exam. You may refer to notes you have brought on index cards or notebook paper, as announced
. l l l l l . BE SURE TO SHOW YOUR WORK, AND EXPLAIN WHAT YOU DID. YOU MAY
RECEIVE REDUCED OR ZERO CREDIT FOR UNSUBSTANTIATED ANSWERS. (“I did it on my calculator” and “I used a formula from the book” (without more details)
are not suﬂicient substantiation...) Problem Points Score )_ 20
20
20
20
20
20
20
20
20
20 ©OONQOTH>OJM }_
CD TOTAL 200 Problem 1 (20 points)
One of the following two vector ﬁelds is conservative and the other is not: F1(1L“7y)= (3 + 2xy)?+ (x2 — 3y2)f Elm y) = (a: — y>z+ (x — 2); (a) Which vector ﬁeld i_s conservative? Which one is not conservative? Show work that leads to your conclusion: You should actually show that one is conserva—
tive and show that the other is not7 directly. Don’t just show one is conservative, or one
is not, and use the fact that there is one of each to decide about the other! —> (b) For the vector ﬁeld F that you found to be conservative, use the fact that it is conservative trnevaluate —> FdF C where C' is the curve Ht) 2 6t sin(t)7+ 6t cos(t)jwith O S t 3 7T. Problem 2 (20 points) For f(x,y) 2 6200—2 cos(x — g) : (a) Find an equation for the tangent plane to the graph of z = f (x, y) at the point where
96 = 1 and y = 2. (b) Find an approximate value for f(1.1, 1.9).
Your answer rnust visibly use calculus! Don’t just use a calculator to work out the function at that point. Problem 3 (20 points) Use an integral to ﬁnd the volume of the region that is
(a) under the paraboloid Z = x2 + y2, (b) inside the cylinder 362 + 342 = 2x, and (C) above the plane z = 0. (You must both set up the integral and evaluate it. It may be easier to do the evaluation using polar coordinates. l Problem 4 (20 points) Set up as an iterated integral but do not evaluate (my — ysin(2)) V Where 5' is the region in Space that is
(a) under the parabolic Sheet 2’ = 4 — x2,
(b) above the coordinate plane 2 = 0, (C) Where y 2 0, and (d) above the plane 2’ = y — 2. Problem 5 (20 points)
Find all local maximum points local minimum points, and saddle points for ﬂay) =$4—4$y+y4+2 Be sure to identify each point that you list as to maximum7 minimum, or saddle point. You do
not need to give the values of the function at the points. Problem 6 (20 points) Let f(x, y, 2) = $231 + (x — y) cos(7r Z). (a) Find the gradient of f at the point (1, 2, 1). (b) In What direction is the directional derivative of f at (1, 2, 1) the largest?
What is the derivative in that direction at that point? (c) What is the derivative of f at (1, 2, 1) in the direction of the vector v 2 2?+ f— 21?? Problem 7 (20 points) Use Stokes’ theorem to evaluate f F  T d5 where
C % (1) F (96, y, 2) = 97— xf+ 3E. (ii) C' is the Circle Where the plane 2x + 2g + 22' = 0 intersects the sphere x2 + y2 + 22 = 9. Problem 8 (20 points) Use Green’s Theorem to evaluate the integral / $2y dye — 3y2 dy C Where C' is the Circle x2 + y2 = 1, oriented counter—Clockwise. Page 10 Problem 9 (20 points)
For motion along the curve given pararnetrically by F(t) = t27+ gt3f+ tkj: (a) Find the velocity 17(25) at the point Where t = 1 (b) Find the acceleration 5(25) at the point where t = 1 (c) Find the unit tangent vector T (t) at the point Where t = 1 : (d) Find the principal unit normal vector N (t) at the point Where t = 1 (e) Find the curvature Mt) at the point Where t = 1 Page 11 Problem 10 (20 points) $9
L t = .
e f($7 y) 132 + yg
h h
hm 96,
(00 y)—>(0 0) f( 34) does not exist. Hint: There are lines other than the coordinate axes. M FRAZZ RR RRRRRRRRR Rf RRRRR R RRRR RRRRRR R N3; RR R R“ RRRRRRRRR RR RRR \
§§§R§R§§§ 5 RRR§§R§§§§ §§§R§§§RR éww § §§§§§§§ RR§§R§R§ W§§§§§k g WWR
‘ f m; RRRR ERR RRRRRRRRRR EMWRW% RRRRRRRRR R RRRR RRR:RE \mm‘ggﬁﬁ} RS RR M...» RR \WjR RR WW§R§ MM” ””‘ngwww é Scratch Paper ...
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 Multivariable Calculus

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