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Unformatted text preview: , AS‘sacllx’ MATH 234 (A. Assadi) — Spring 2008  MIDTERM I
Thursday, March 6, 2008 ' NAME (print): Instructions:
1. Write your name on each page. 2. Circle the name of your TA: WU TONEJ C 3. Closed book. Closed notes. Calculators allowed. No laptops.
4. 2 index cards of size 3” X 5” with formulas allowed. 5. Answer your questions on the exam paper. There are some extra pages in the
back if you need them. ’ 6. Show all your work. Partial credit is given only if your work is clear.
7. Enclose your ﬁnal answers clearly in a rectangle. 8. Time allowed: 70 minutes for your work, plus5 minutes for checking your
. answers. ’ Problem 1
Problem 4  Problem 6 20
Problem 7 20 NAME (print):
Problem 1. A particle is moving downwards along the helix
r(t) 2 3 cos(2t)i + 3 sin(2t)j — tk. (a) Compute the velocity V and the acceleration a of the particle. (b) How far does the particle travel along its path from t = 0 to t = 27f? (c) Find the unit tangent vector T and the unit normal vector N and compute the
angle between Tand N. ‘ (d) Compute the curvature a and the radius of the curvature. What can you say
about K)? ' NAME (print): Problem 2. Find and graph the osculating circle (Le. the circle of curvature) for
the parabola P:33=1—y2 at the point of intersection of P with the xaxis. NAME (print): Problem 3. If capacitors of Cl, 02 and C3 microfarads are connected in series to
make a C—microfarad capacitor, the value of C can be found from the equation i_1+1+1
0—0102 03' Find the value of 815; when 01 = 8, 02 = 12 and C3 = 24 microfarads. NAME (print): Problem 4. Find the extreme. values of f (:c,y) : x2 + 3y2 + 2y on the unit circle
51:2 + 3/2 = 1. NAME (print): M Problem 5. Find the equation of the tangent plane and the normal line for the
surface 5' given by x2 — 33,2 — 2522 = —11
at the point p = (1, ~52, 0). 2'1} NAME (print): Problem 6. Find all critical points for f(a:,y) = 2:134 — 43:2 +y’2 +23; and indicate Whether each point is a local maximum, local minimum or a saddle point. NAME (print): Problem 7. Let _
f($,y) = :62 +y2 ~ 4 (a) Sketch the level sets f(a:,y) = k for k = —4, ~3, O and 5. (b) Sketch the graph of z =, f(x, (c) Find a unit vector V, in the direction in which f (as, y) decreases most rapidly
at the point p = (1, —2\/§). What is the rate of change in this direction? Sketch the vector V at the point p on the graph in part (a). What do you
observe? ’ ...
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This note was uploaded on 09/15/2009 for the course MATH 234 taught by Professor Dickey during the Spring '08 term at Wisconsin.
 Spring '08
 DICKEY
 Multivariable Calculus

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