Survey of Calculus Exam #3
Fall 2004
Name ______________________________
You must show all your work on this paper. Solutions without correct supporting work will not be accepted.
Please circle your answers. You may omit one problem by clearly writing “OMIT” by the problem. If you do
not omit a problem I will omit the last one for you. Each problem is worth 10 points. (Note: When you omit a
problem, you are omitting all parts of the problem.) You may work the problem you omit for up to 5 points
extra credit if you wish.
1. a. Approximate the area below the graph of f ( x)   x 2  4 x  1 between x=1 and x=3 using 4 rectangles.
b. Find the actual area under the curve in part a.
2. Evaluate the integrals.
5
a.  (  e  2 x )dx
x
3
b.
 (6 x
2
 4 x  1)dx
1
3. Evaluate the integrals.
8
1

a.   x  7  dx
2

b.
x e
2 x 3 1
dx
4. a. The population of the world is predicted to be P(t )  6e 0.01t billion, where t is the number of years after
the year 2000. Find the average population between the years 2000 and 2100.
b. Find the Gini index for the Lorenz curve L( x)  0.4 x  0.6 x3
5. a. Find the consumers’ surplus for the demand function d ( x)  1800  0.03x 2 and demand level x=200.
b. Find f xy for f ( x, y)  5x 3  2 x 2 y 3  3 y 4  7
6. Find the relative extrema and/or saddle points for the function f ( x, y)  x 2  y 2  2 x  6 y  14 .
7.
A parking lot, divided into two equal parts, is to be constructed against a building, as shown in the diagram.
Only 6000 feet of fence are to be used, and the side along the building needs no fence. What are the
dimensions of the largest area that can be enclosed and what is the largest area? You may solve using the
calculus method of your choice.
BUILDING
8. Use Lagrange multipliers to maximize f ( x, y)  y 2  x 2  5 subject to the constraint x  2 y  9 .
9. Evaluate the following limits without the use of a table or graphing calculator.
x 2  3x  2
a. lim
b. lim 3  x 2
2
x2
x 2
x 4
10. Find the slope of the tangent line of f ( x) 
11. Find the derivative
3x  1
at the point (1,2). Give an exact answer.
x  3x  4
2
dy
if y  2e4 x 4 x 2  1 . Do not simplify your answer.
dx