Unformatted text preview: 03/05/2004 11:53 FAX 510 642 9454 .001 Math 53 Midterm #2, 11/13/03, 8:10 AM — 9:30 AM Hu'l'clnilzjs No calculators or notes are permitted. Each of the 6 questions is worth
10 points. Please write your solution to each of the 6 questions on a separate
sheet of paper with your name and your TA’s name on it. Please put a box
around the ﬁnal answer. To maximize credit, please show your work, and
use extra time to double check that you got the correct answer and didn’t
misunderstand the question. Good luck! General hint: if something is hard to calculate, try using something you’ve
learned in order to calculate it in a different and easier way. 1. Find the minimum and maximum values of the function f=(zv—1)2+(y—1)2
on the unit disc 2:2 + 3/2 S 1. 1 1
f j :1: cos (y4)dy dm.
0 932/3 1 M
f / ($2 + y2)2003dﬂ? dy
0 O 4. Find the area of the region enclosed by the curve 2. Calculate 3. Calculate 9:2 + my + y2 = 1.
Hint: use the substitution
ac=u+m/§, y=u—v\/§.
5. Let C' be a plane curve starting at (0,0) and ending at (1,1). Let
F = <$Z+y,y2+x>.
(a) Show that f0 F  dr has the same value for every C as above.
(b) Compute f0 F  dr for a curve C as above. 6. Calculate f0 Fdr, where C is the unit circle oriented counterclockwise,
and F is the following vector ﬁeld in the plane: F = (—y3 + sin(sin cc), 1:3 + sin(sin (9)). ...
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 Spring '08
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