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Unformatted text preview: page 1 Your Name: Circle your TA’s name: Jesse Holzer Evan (Alec) Johnson Asher Kach Liming Lin
Mathematics 234, Fall 2004 Lecture 3 (Wilson)
Final Exam December 22, 2004 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, be sure to make clear what is your ﬁnal answer to each problem. Wherever applicable, leave your answers in exact forms (using g, x/g, 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
in class and on the class website. 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 sufﬁcient substantiation...) Problem Points Score }_ 20
20
20
20
20
20
20
20
20
20 ©OO\I®O‘IH>OOL\D }_
O TOTAL 200 page 2 Problem 1 (20 points)
For the helical (“corkscrew”) motion ﬁt) 2 2 cos(t)?— 2 sin(t)f+ Btk: (a) Find the velocity 27(t) as a function of t. (b) Find the acceleration 07(t) as a function of t. (c) Find the unit tangent vector (d) Find the principal unit normal vector N (e) Find the curvature Mt). page 3 Problem 2 (20 points) 2xy2 x2+y4 The function f(x, y) 2 does not have a limit as (x, y) —> (O, 0). Show that this is true. page 4 Problem 3 (20 points)
Let ﬁx, y) = xgy + 6” sin y. (a) What is the gradient V f as a function of x and y? (b) At the point (1, 0), in What direction U is the directional derivative D1; f largest? In What
direction is the directional derivative smallest? Express the directions as speciﬁc unit vectors. There should not be any :3 or y in your
answers to this part of the problem. (c) What is the value of the directional derivative at (1,0), in the direction making the
directional derivative largest? page 5 Problem 4 (20 points)
Set up but do not evaluate an iterated integral for the integral of f(x, y, z) = 3:32 — 22 cos(xy) over the region in space which is:
Inside the cylinder :32 + z2 = 4 and between the planes y = 0 and x + y = 3. page 6 Problem 5 (20 points) 2 «47062
For x \/$2+ 2d dx:
0 0 y y 9 Evaluate the integral by converting to polar coordinates and evaluating the resulting polar
integral. Possibly useful formulas:
sin(26) = 2 sin(6) C0s(6)
cos(26) = cos2(9) — sin2(6) page 7 Problem 6 (20 points)
If :3, y, and z satisfy 23 — my + yz + 3/3 = 2: (a) Find (C) Find 3—2 at the point (1,1,1).
x page 8 Problem 7 (20 points)
Let f(x,y,z) = my +y + ,2.
Let C be the curve 770%) 2 277+ tf—i— (2 — 2t)k for 0 < t < 1. Evaluate the line integral / f (:3, y, z) d8.
C page 9 Problem 8 (20 points)
One of the following two vector ﬁelds is conservative and the other is not. —» F1(x, y, z) = (2x — 3)?— 217+ (cos z)k;
F2(x, y, z) 2 (egg cos y + yzW—l— (5m — 6‘” sin y)f+ (my + (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 not, 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, evaluate /F(x,y,z)d8
C where C is any path leading from (0, 0, 0) to (—1, g, 2). Hint: It is almost surely easier to do this using a potential function! page 10 Problem 9 (20 points) Evaluate % —y2 dm + my dy
0 around the square in the gay—plane With vertices (0, 0), (1, 0), (1, 1), and (0, 1), in that order. page 11 Problem 10 (20 points) Evaluate % 13 . ids Where
C 13(m,y,z) = yT—i— xzf+ $2]; and C is the boundary of the triangle in the plane :3 + y + z = 1 with vertices (1, 0, 0), (0, 1, 0), and
(0, 0, 1), traversed counterclockwise as viewed from above. Hint: Stokes’ Theorem could help a lot here. . . page 12 $§§§§§§§E§¥ﬁi¥ i § mm mm wg‘ mama ﬁsmm
§ \ ma mgm km} 3%
v x ~51} m: m \ mi §§i§§§§$~ *\ m § § «aka 1‘
it: Tm Fgggfﬁ, \ \\ Wu“? 33% mm: Emmi '
g; mm WEE \ 2%me is? RN‘NSRK uwguwwwww ‘ xx I «. \\ *WWW\~ Nﬁﬁiﬁ XS Coggright § ﬁ§ﬁ3 Uniteﬁ ¥emture Sgﬁdicmte, 3&2. Scratch Paper ...
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 Fall '08
 DICKEY
 Multivariable Calculus, Dot Product, Johnson Asher Kach Liming Lin

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