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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) First Midterm Exam October 4, 2004 Write your answers to the eight 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 this exam. If you need more scratch paper, please
ask for it. You may refer to notes you have brought on an index card, 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
1 10
2 20
3 12
4 12
5 12
6 12
7 10
8 l2 page 2 Problem 1 (10 points)
In class we saw pictures of the helix (corkscrew) given by the position vector m) : sin(t)i’+ tj+ cos(t)lZ. Calculate the arclength of one turn of this helix7 from t = 0 to t : 271 page 3
Problem 2 (20 points) Let Ht) : 3sin(t)?+ 3cos(t)f+ 4t]; describe the motion of an object along a curve in
space. Find as functions oft: The velocity 17(25)
The acceleration (3(25) The unit tangent vector ) ) ) d) The principal unit normal vector 1W1?) ) The curvature Mt) ) The tangential (scalar) component of acceleration CLT
) The normal (scalar) component of acceleration aN Be sure to label each answer! (You may have additional space at the bottom of the opposite
page, Problem 1, to use for this problem.) page 4 (answer)
(answer)
(answer)
(answer) 12 points) ( For each of the four descriptions7 ﬁll in A or B or C or D to indicate which graph below it corresponds to.
sin(gz§), any value of 6 (ii) 7“ = 1 + cos(6), any value of z (i) z = y sin(x)
(iii) z:3—Jc2—y Problem 3 (1V) p . N “.3. a» a? V a?» \\ . we.“ Maﬁa? ___._.__.______. \
my“, y§§§§ l
xfwweae§a§§§ . N f ( §§§§§§§§§x§§ X.
/ 2181.3 page 5 Problem 4 (12 points) For ﬁx, y) : sin(a:2 —— 2y)7 ﬁnd: (a) (b) g
(c) <d> 332530
(e) 88:81; 82f page 6 Problem 5 (12 points) (a) Evaluate 11m x2 c0s(2y).
(mam)
(b) Show that lim L does not exist. ($7y)—>(0,0) V $2 + y2 page 7 Problem 6 (12 points)
Let ﬁx, y) : x2 69.
Let P be the point (17 0). (a) Find the gradient Vf at the point P. (b) Find the directional derivative of f at P in the direction of an arbitrary vector 17 = 12121—1227. (c) Find the directional derivative of f at P in the direction from (1, 0) to (4, 4). (d) In what direction is the directional derivative of f at P largest? What is the directional
derivative in that direction? page 8 Problem 7 (10 points)
Suppose w : sin(xy) + xsin(y)7 where x = u2 + U2 and y = 2U + v — 2. Using the chain rule: (a) Find page 9 Problem 8 (12 points) Suppose you are walking over some hills. The north—
south direction is measured as y and the east—west direc—
tion as :13. The altitude in feet is given by my > 1’2 y2 : l — 1 ' —
ﬁx, y) 000 00 Sin (3000 100 200. If you are at the point where 36 : 100 and y : —100, and you start walking toward the center
where x = 0 and y = 0, will you begin by walking uphill, downhill7 or horizontally? Be sure to show how you determine your answer. page 10 ‘ §. M ~ K wmﬁmwwwmw
§§§§§3§§§§ ‘ mmawmimm ‘ * aim mm w} wﬁkh
\\\\\\\« \ m > Jung ‘
mm  ' i§ w ﬁ‘ﬁﬁ‘fﬁx REE} mmm W $3 am} 1% am & Rm mm} m mg» wmg
mmmwmmm mm m w E}? @iﬁ‘iﬁ? Wmﬂ/Mﬂmﬂ I Scratch Paper (not a. command!) ...
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This note was uploaded on 05/15/2008 for the course MATH 234 taught by Professor Dickey during the Fall '08 term at University of Wisconsin Colleges Online.
 Fall '08
 DICKEY
 Multivariable Calculus

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