Calculus I Exam #2
Fall 2005
Name ________________________________
You must show all your work on this paper. Solutions without correct supporting work will not be accepted. All problems
must be worked manually.
1. Locate the absolute extrema of
f ( x)  4 x 5  5x 4 on 0,2.
2. Determine if Rolle’s Theorem can be applied to
the theorem.
(4 points)
3. Given the function
f ( x)  x 3 ( x  4) , find:
(4 points)
f ( x)  x 2  5 x  4 on [1,4]. If so, find the values of c guaranteed by
(8 points)
a. Interval(s) where the function is increasing: ______________________________
b. Relative extrema: ________________________________ (Make sure you indicate if it is a max or min.)
c. Interval(s) where the function is concave down: ______________________________________
d. Inflection Points: ___________________________
4. Evaluate:
(Make sure you show your work.)
(4 points each)
6  x2
a. lim
= ______________________
x  5  x
5. Find the differential dy given
6. Evaluate:
a.
 4 x
y  ln 4  x 2 .
x
b.
lim
b.
  x  sec
x 
3x 2  1
= ________________________
(4 points)
(4 points each)
3

 3 x 5  4e x dx
4
2

x  dx

7. A rectangular page is to contain 30 square inches of print. The margins on each side are 1 inch. Find the dimensions
of the page such that the least amount of paper is used.
(8 points)
8. A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate
of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.
(8 points)
9. On the moon, the acceleration due to gravity is -1.6 meters per second per second. A stone is dropped from a cliff on
the moon and hits the surface of the moon 20 seconds later. How far did it fall? What was its velocity at impact?
(8 points)
10. The graph of y  f (x) is given below. Sketch a graph of y  f (x) on the same set of axes.
The graph of y  f (x) is given below. Sketch a graph of y  f (x) on the same set of axes.
11. Solve the differential equation:
f ( x)  sin x,
f (0)  1,
f (0)  6
(6 points)
(4 points)
(4 points)
12. Find the derivative of f ( x ) 
4
using the definition .
x
13. Write the equation of the tangent line to the graph of
14. Find the derivative:
a.
y  e 4 x  ln(cos 3x)
b.
y
x 1
15. Evaluate manually:
a.
lim
b.
lim
x 3
 0
3 x
x2  9
cos  tan 

f ( x)  x 2  x  1 at the point where x  2 . (4 points)
Do not simplify your answers.
x3
(4 points each)
(6 points)
(4 points each)