MATH 221 Final Exam – Part I Name_____________
Instructions: Answer all 12 questions on Part I. You may not use a calculator on this
part. Each question is worth five
points.
Find
the indicated limits:
1. 
1.____________
2. 
2.____________
3. 
= 3.____________
4. 
4.____________
Find
for each of the
following:
5. 
5.____________
6. 
6.____________
7. 
7._____________
8. Find the slope of the tangent line to 
at x = 2. 8.____________
Evaluate
each of the following integrals:
9. 
9.____________
10. 
10.___________
11. 
11.____________
12. 
12._____________
MATH 221 Final Exam Part II Name____________
Instructions:
Answer any five (5) of the six questions on this part. Clearly indicate which questions you have
answered. Show all your work. Each
question is found on one page of the test. Each question is worth 18 points.
1. Find the derivative of 
using the limit
definition of the derivative.
2. A poster is designed with a central
rectangle of text surrounded by margins of 2" on each side, 2" on the
top, and 3" on the bottom. There is
to be 80 square inches of text. What are
the dimensions of the poster of smallest area that contains this text area as
its central rectangle?
3. Below is the graph of the derivative, 
, of a function, f(x).
Answer the questions below about the function, f(x).
a) 
On which intervals is
f(x) increasing?
b) What are the critical points
of f(x)?
c) Where does f(x) have a local
maximum point?
d) Where does f(x) have a local
minimum point?
e) Where are the inflection points
of f(x)?
f)
Find the intervals on which f(x) is concave upwards.
4. Find and classify the
discontinuities of the function f(x) defined below. Include all necessary limits.
.
5. a) Let 
. Find the equation of
the line tangent to the graph of f(x) at x = -1.
b) Let 
. Find all points on
the graph of f(x) at which the tangent lines are perpendicular to the line y =
4x-1. Find the equations of those
tangent lines.
c) Find the equations of the lines tangent to
f(x) = x2 + 1 that pass through (1,1).
6. a) Let R be the region bounded by f(x) = x3
+ 1 and g(x) = 4x + 1. Find the area of
the region R.
b)
Let R be the region bounded by 
and 
. For
each of the following write an integral that gives the desired quantity and
then evaluate the integral.
i) The volume of the solid that results when R
is rotated about the x-axis.
ii)
The volume of the solid that results when R is rotated about the y-axis.