Departmental Final Exam
Fall Semester 1998
Multiple-Choice Section
Choose your answers for questions 1-10 from the following list. (Several questions may have the same answer.)
(a) -1 (b) 1 (c) -2 (d) 2 (e) -3
(f) 3 (g) 4 (h) 5 (i) 6 (j) None
Find the derivatives at the given points.
3 pts 1. If 
,
then 
3 pts 2. If 
,
then 
3 pts 3. If 
,
then 
3 pts 4. If 
,
then 
3 pts 5. If 
,
then 
3 pts 6. If 
,
then the second derivative 
Find the following integrals:
3 pts 7. 
3 pts 8. 
Find the following limits:
3 pts 9. 
3 pts 10. 
Written Answer Section. Show your work.
3 pts 11. a) If 
is given implicitly by 
then
give 
as a
function of 
and 
.
3 pts b) Find the derivative of 
.
3 pts c) Find the indefinite integral
6 pts 12. Find the volume generated
when the region bounded by 
and
is rotated about the 
-axis.
2 pts 13. a) Give the formal definition of
5 pts b) Use this definition to prove that
7 pts 14. Suppose that ƒ(x)
is twice differentiable and satisfies ƒ(0)=0, ƒ(2)=4, ƒ(3)=0, ƒ(4)=2. Use
the Mean Value Theorem to prove that 
for some point z in the interval (0,4).
8 pts 15. The cost of operating a
small aircraft is $400 per hour plus the cost of fuel. Fuel costs are 
dollars per hour, where s is the speed of the aircraft in kilometers
per hour. You must fly from here to Annville, Nebraska, which is 1,000
km from here. Find the speed at which you should fly to minimize
the cost of the flight.
(Hint: rate 
time = distance, i.e. st = 1000.) Justify your answer.
8 pts 16. Sketch the graph of a function
f(x) which is twice differentiable on the interval 
and which has the following properties:
• The second derivative 
is positive on the interval 
and negative on the interval 
.
• 
• The first derivative is zero only at x = -1.
Clearly indicate all extrema, asymptotes, and points of inflection.
7 pts 17. Find the centroid of the
region in the plane bounded by the curve 
and the line 
.
6 pts 18. Recall that two functions ƒ and g are said to be inverse functions if
.
Let ƒ and 
be differentiable and assume 
.
Show that if ƒ and g are inverse functions, then
where 
.
6 pts 19. A 13-foot ladder leans against a vertical wall. If the bottom of the ladder is being pulled away from the wall at a rate of 2 feet per second, how fast is the angle between the ladder and the ground changing at the instant the bottom of the ladder is 5 feet from the wall?
20. Let 
3 pts a) Determine a and b so that ƒ is differentiable at 0.
3 pts b) Using the values found in part (a), sketch the graph of ƒ.