Math 324E
1
Final Exam
Autumn 2005
(10 points)
Evaluate
y
1
xy dx + x2 dy,
2
C
where C is the curve given by r(t) = t, t2 1 for 1 t 1
x
2
and r(t) = 2 t, t + 4t 3 for 1 t 3. See the gure to
the right.
The mistakes in the original versions parametric equat
Math 324E
Your Name
Final Exam
Your Signature
Problem
Total Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
Total
Autumn 2005
50
Score
This exam is closed book. There is a page of formulas attached to the back of the test. Feel
free to tear it o to use as refe
Math 324E
1
(10 points)
Exam 2 Solutions
Autumn 2005
Evaluate
xey ds,
C
along the curve r( t ) = cos t, sin t, t from t = 0 to t =
2.
f ds =
f ( r( t ) r ( t ) dt. So then, r ( t ) = sin t, cos t, 1 , and so
First things rst.
C
C
r ( t ) = 2. The integral
Math 324E
Your Name
Exam 2
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Problem
Total Points
1
10
2
10
3
10
4
10
5
10
Total
Autumn 2005
50
Score
This exam is closed book. You may use one hand-written 3x5 card for notes, though if
you dont have it by now, its a little late to do anyth
Math 324E
1
(10 points)
Exam 1 Solutions
Autumn 2005
Consider the function
f ( x, y ) = x4 2 x2 + y2 + 2
(a) Find the critical points of this function and classify them.
First nd the rst derivatives and set them equal to zero. f x ( x, y ) = 4 x3 4 x =
4
Math 324E
Your Name
Exam 1
Your Signature
Problem
Total Points
1
10
2
10
3
10
4
10
5
10
Total
Autumn 2005
50
Score
This exam is closed book and closed notes.
Only non-graphing scientic calculators are allowed.
In order to receive credit, you must show