isDepartmental Final Exam
Multiple-Choice Section
2 pts 1. The value of
is
a) –6 b) 0 c) infinite d) 1
e) 1/2 f) 6 g) 3 h) 2
2 pts 2. The value of
is
a) 1/2 b) 0 c) 2 d) e
e) 2/3 f) 1 g) –1 h) infinite
2 pts 3. Determine the value of the
limit 
.
a) 0 b) 1 c) e/3 d) e
e) 3 f) ln(3) g) infinite h) does not exist
2 pts 4. For x = 0, the derivative
of 
is
a) 
b) 
c) 
d) 
e) 
f) 
g) 
h) 
2 pts 5. The derivative of csc x is
a) cot x b) cot x csc
x c) 
d) 
e) csc x f) 
g) 
h) 
2 pts 6. Find 
,
a) 4 b) 1/4 c) 1 d) 2
e) 5/2 f) 0 g) 1/2 h) 16/3
2 pts 7. Compute the integral 
a) 3 b) 1 c) ln 2 d) ln 3
e) ln (3/2) f) 3 ln 2 g) ln 6 h) does not exist
2 pts 8. The definite integral 
is equal to
a) 1 b) 
c) 
d) 
e) 0 f) 
g) 
h) 
2 pts 9. Compute the integral 
.
a) 
b) 
c) 
d)
e) 
f) 
g) 
h) 
2 pts 10. Compute the integral 
a) 
b) 
c) 
d) 
e) 
f) 
g) 
h) 
i) 
3 pts 11. Only one of the following statements is false. Which one?
a) The derivative of a constant is always zero.
b) If 
exists, then ƒ(x) is always continuous at x = a.
c) If ƒ(x) is continuous on
[a,b], then 
.
d) If ƒ(x) is continuous on [a,b], then ƒ(x) always attains its maximum value on [a,b].
e) If ƒ(x) is continuous on
[a,b], then 
.
f) If 
,
then (c, ƒ(c)) is either a local maximum or a local minimum.
g) If ƒ(x) and g(x)
are continuous on the interval [a,b], and 
are constants, then
.
3 pts 12. Which answer best describes
the graph of the function 
?
a) One local minimum and two local maxima.
b) One local maximum and one local minimum and one inflection point.
c) One local maximum and two local minima.
d) Two local minima and one inflection point.
e) No local maxima or minima.
3 pts 13. If 
,
then which of the following is true?
a) 
b) 
c) 
d) 
e) 
f) 
g) 
h) 
Written Answer Section. Show your work.
6 pts 14. Determine whether each of the following limits exist. If it exists, compute it. If it does not exist, explain why.
a) 
b) 
9 pts 15. Find the derivatives of the functions:
a) 
b) sin(ln 2x)
c) 
7 pts 16. a) Give the definition of a function ƒ(x) being differentiable at a point x = a.
b) Compute the derivative of 
from this definition.
6 pts 17. A heap of rubbish in the shape of a cube is being compacted and remains in the shape of a cube throughout this process.
a) Given that the volume decreases at a rate of 2 cubic meters per minute, find the
rate of change of the length of an edge of the cube when the volume is exactly 27
cubic meters.
b) What is the rate of change of the surface area of the cube at that instant?
7 pts 18. A company derives an average net profit of $12 per customer if it services 50 customers or less. If it services over 50 customers, then the average profit is decreased by 6 cents for each customer over 50. What number of customers produces the greatest total net profit for the company?
7 pts 19. Graph the function ƒ(x) = 4– x on the interval 0 = x = 9. Indicate maxima, minima and inflection points.
7 pts 20. Find the derivative of
the function 
.
7 pts 21. a) Define the limit of
a function, 
.
b) Using this definition, prove that
.
7 pts 22. Let y be defined
implicitly as a function of x by the formula 
.
Find 
at the point (1,1) on
the curve.
8 pts 23. The sum
is the Riemann sum of a definite integral. Estimate this sum by determining and then evaluating this definite integral.
