Math 111
Sample Exam 3
Read each problem carefully. Show your work clearly. No calculators.
1. The radius of a circular oil slick expands at the rate of 2 m/min. How fast is the area of
the oil slick increasing when the radius is 25m?
2. The side s of a square carpet is measured at 6m. Estimate the maximum error in the area
A of the carpet if s is accurate to within 2 cm.
3. For y  ( x  x 2 )1/ 3 on [-1, 2] find all local and global extrema.
1
on [1/2, 2]? Find
x
the relevant value of c. Can you do the same on [-2, 2]? Why or why not?
4. Why does the Mean Value Theorem apply to the function f ( x )  x 
5. Sketch y  xe x finding all relevant “interesting” points and regions of increase,
decrease, concave up and concave down.
6. Evaluate the limits
2
ln x
x  x1/ 2
a) lim
b) lim
x 1
8  x  3 x1/ 2
c) lim xsin x
x 0 
x 2  3x  2
 3
d) lim x ln 1  
x 
x

7. An architect wishes to enclose a rectangular garden with area 1000 square meters on one
side with a brick wall costing $90/m and on the other 3 sides by a metal fence costing
$30/m. What dimensions minimize the cost?
8. Evaluate
2
 ( x  1)
3
dx ,
 (x
2
7
dx , and  ( x 2  3sin 4 x)dx
 1)