Calculus III
Test 2 Solutions
July 3, 2012
1. The following is a contour map for a function f ( x, y).
2.5
-5
-2.5
0
2.5
5
-2.5
(a) Estimate the value of f at the point (1, 1.5).
The point (1, 1.5) is approximately on the level curve labeled 1, so
f (1, 1
Calculus III
Test 3 Solutions
July 19, 2012
1
1. Carefully sketch the region of integration for the integral
0
3
3y
2
e x dx dy, then reverse
the order of integration. Do not evaluate the integral.
The limits of integration tell us that that the region is
Math 227
Carter
Test 1
Fall 2011
Instructions. Write your name on only the outside of the blue book. Do all your work and
write all of your solutions inside your blue book. Do not write on this sheet. Write neat
complete solutions to the problems written
Math 227
Carter
Test 2
Fall 2011
General instructions. Do all your work in your blue books. Write your solutions in your
blue book. Show all work. Write your name on only the outside of your blue book. Please
insert this sheet into your blue book as you l
Calculus III
Test 1 Solutions
June 13, 2012
1. Let C denote the curve parametrized by r(t) = t, t2 , t3 with 0 t 1, and let
F( x, y, z) = xy, yz, xz . Find
C
F ds.
We compute r (t) = 1, 2t, 3t2 and F(r(t) = t4 , t5 , t4 , so
F(r(t) r (t) = t4 + 2t6 + 3t6
Calculus III
Test 1 Solutions
June 16, 2010
1. Let C denote the portion of the graph of y = x2 + x 2 with 0 x 1, and let
F( x, y) = y x, x2 . Find
F ds.
C
r(t) = t, t2 + t 2 0 t 1
F(r(t) = t2 2, t2
r (t) = 1, 2t + 1
F(r(t) r (t) = t2 2, t2 1, 2t
Calculus III
Solutions to Practice Problems for Test 2
1. Consider the function f ( x, y) = x3 y3 2x2 y. Compute
f
f
and
and verify Clairauts
x
y
theorem for this function.
f
= 3x2 y3 4xy
x
f
= 3x3 y2 2x2
y
2 f
2 f
= 9x2 y2 4x =
xy
yx
2. Compute the gradi
INTRODUCTION
If S is a surface and f is a function dened at all points of S , then the integral of f over
S does the following (roughly speaking):
Chop S into small rectangular pieces, where the ith piece has area Si ;
For each piece Si of S , pick a po
Calculus III
Review Information for Test 2
1. Consider the function f ( x, y) = x3 y3 2x2 y. Compute
June 28, 2012
f
f
and
and verify Clairauts
x
y
theorem for this function.
2. Compute the gradient of the function in the previous problem and evaluate at
INTRODUCTION
If C is a curve and f is a function dened at all points of C , then the integral of f over C
does the following (roughly speaking):
Chop C into small pieces each of which has size s;
For each piece Ci of C , pick a point pi ; then f (pi ) i
Math 227
Carter Sample for test 3
By popular request, I am preparing a sample study guide. The rst question is familar,
but now more questions are attached. A problem such as this will be a substantial portion
of the test. The remaining integrals should b
Math 227
Carter
Test 1
Fall 2006
Do all your work in your blue books. Write your solutions in your blue book. Show all work.
Write your name on only the outside of your blue book. Write neatly, and use complete
sentences when appropriate. My hope is that
Math 227
Carter
Sample 1
Spring 2007
Sample Instructions. Do all your work in your blue books. Write your solutions in your blue
book. Show all work. Write your name on only the outside of your blue book. Write neatly,
and use complete sentences when appr
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Calculus III
Final Exam
Name:
This is an open-book take-home test, due in my mailbox in the math ofce (ILB 325)
by 12:00 noon, Wednesday, July 25. You may consult the textbook, your notes, and any
material posted on our course webpage. No other sources of