MATH
221 Part II Problems from old Finals
1.
Find
the derivative of 
using the limit
definition of the derivative.
2.
A
cube is growing at a rate of 100 cubic inches per minute. How fast are the side
of the cube and the surface area of the cube growing when the volume is 1000 cubic
inches?
3.
Find
and classify the critical points of each of the following functions.
a)
b)
c)
4.
The
derivative, 
, of a function, f(x), is graphed below. Answer the following questions based on this
graph.
a)
On
what intervals is f(x) increasing?
b)
What
are the critical points of f(x)?
c)
On
what intervals is f(x) concave downwards?
d)
Where
are the points of inflection of f(x)?
e)
Where
does f(x) have a local minimum point (or points)?
5.
Let
and let R be the
region bounded by y = 0, x = ½, x = 2, and
the graph of f(x). Write the integral necessary
to find each of the following quantities and evaluate the integrals.
a)
Find
the area of R.
b)
Find
the volume of the solid that results when R is rotated about the x-axis.Find
the volume of the solid that results when R is rotated about the line x = -5.
c)
Find
the length of the graph of f(x) from x = ½ to x = 2.
6.
a) Solve the following inequality and express
your solution in interval notation.
b)
Give
examples (by formula) of functions f(x) and g(x) such that
i)
f(x)
has a removable discontinuity at x = 2 and f(2) = 0
ii)
g(x)
has a jump discontinuity at x = 1 and g(1) = 4.
7.
Find
the derivative of 
using the limit
definition of the derivative.
8.
A
rectangular box with no top is to be constructed so that its base is three
times as long as it is wide. It is to
have an volume of 18,000 cubic inches.
find the dimensions of such a box that yield the minimum surface area.
9.
Find
the intervals on which each of the following functions are increasing and the
intervals on which they are decreasing.
a)
b)
c)
10.
Let
R be the region bounded by the graphs of 
. Write the integrals
needed to find each of the following quantities and then evaluate the
integrals.
a)
The
area of R.
b)
The
volume of the solid that results when R is rotated about the x-axis.
c)
The
volume of the solid which results when R is rotated about the line x = -2.
11.
Let
.
a)
Find
the equation of the line tangent to the graph of f(x) at x = 2.
b)
Find
all points on the graph of f(x) at which the tangent lines are perpendicular to
the line y = 4x + 6.
c)
Verify
the Mean Value Theorem for f(x) over the interval [-4,-2].
12.
a) Find and classify the discontinuities of the
function 
.
b)
Find
and classify the discontinuities of the function 
.
c)
Give
an example by formula of a function with an essential discontinuity at x=2.
d)
13. Solve each of the following inequalities
and express your solutions in interval notation.
a)
b)
c)
14. A rectangular poster has a rectangular region of print with an area of 180 square inches. If the margins surrounding the printed area are 2” on the sides, 1” on the bottom, and 4” on the top, what is the minimum possible area of the poster?
15. a) Show that the function f(x), given below, is continuous at x = 0 and at x = 2.
b)
Let
. Find a value of a
such that f(x) has an essential
discontinuity at x = 3. Find a value of
a such that f(x) has a removable discontinuity
at x=3.
c) Give an example (by formula) of a function f(x) which has a jump discontinuity at x= 0 and a removable discontinuity at x = 2.
16. a) Let R be the region bounded by the
graphs of f(x) = x2+2 and
g(x) = 5x-x2. Find the volume
of the solid that results when R is rotated about the y-axis.
b)
Find
the length of the graph of f(x) = x3 from (1,1) to (3,27).
c)
Find
the area of the region bounded by the graphs of f(x) = x4 and
g(x)=5x2 – 4.
17. Find and classify the critical points of the following functions:
a)
b)
c)
18. a)
Find the equation of the line tangent to the graph of 
at the point
corresponding to x = 2.
b)
Let
f(x) = x3-6x. Find all
points on the graph of f(x) at which the tangent lines are perpendicular to the
line x+6y+2=0.
c)
At what points does the graph of 
not have a tangent
line?