Calculus I
Test File
Test #1
In #1 - #5, find the limit. If it does not exist, tell why not.
2
4
1
x - x -6
x - 5x + 3
1.)
2.)
3.)
lim x 1
lim x 3 2
lim
x

1
2x + 3
(x - 1 )2
x - 5x + 6
1
4.)
5.)
limx0 x csc(3x)
lim x 3
x-3
6.)
Use the definition of limit to prove that limx2 3x - 2 = 4.
7.)
For the following function, does the IVT guarantee the existence of a point between
-1 and 2 such that f(c) = 4?
f(x) = x2 - 3x + 4
8.)
Determine whether or not the function is continuous. If it is not, then for all points of
discontinuity, determine whether they are removable or non-removable and tell why.
x+3
if x < 0
x+1
.
2
if x = 0
f(x)=
.
3 sin x
if 0 < x  
x
.
1
if x > 
x -
For #9 and #10, find all vertical asymptotes. Give reasons why each is a vertical
asymptote and why any other candidates are not vertical asymptotes.
2
x -1
x + 3x - 10
f(x)= 2
9.)
10.)
f(x)=
x -1
x - 5x + 6
Test #2
In #1 - 5, find dy/dx. Show any necessary work. Simplify.
1.)
y = x2 - x + cos2x
2.)
y = (3 - 2x)234 3.)
y = (2x4 + 1)3(sec(5x) - 5)3
4
4.)
6.)
7.)
8.)
 8 x2 - 4 
y = 
5.)
y = sin(6x)

3
 1- 9 x 
Use the definition of derivative to find the derivative of f(x) = x2 - x.
A ball is thrown from the top of a 1000 foot tall tower with a downward velocity of
42 feet/second.
a.)
How long does it take the ball to hit the ground?
b.)
How fast is it going when it hits the ground?
Find dy/dx if sin(xy) = x.
9.)
10.)
11.)
Find d2y/dx2 if x2 + y2 = y. Do NOT leave dy/dx in the answer.
Find equations for any horizontal tangent lines to the curve y = 4x3 - 3x2 - 16x+8.
Explain the origins of the formula for the definition of the derivative. Use
complete sentences and proper grammar. Use figures if necessary for your
explanation to make sense.
Test #3
In #1-2, find the absolute extrema, if any exist, on the given interval.
1.)
f(x) = 2x3 + 3x2 - 12x + 1, [0, 2]
2.)
f(x) = cos2x - sin x, [0, 2π] Find exact values.
3.)
Use the first derivative test to find local extrema for : f(x) = (x - 1)/(x2 + 1)
4.)
On what intervals is f(x) = x3 - 2x + 6 strictly increasing? strictly decreasing?
x
5.)
Use the second derivative test to find local extrema for: f(x)= 2
x +1
2/3
6.)
Let f(x) = 2x - x - 3.
a.)
Does Rolle's Theorem apply on the interval [-1, 1]? If so, find the point, c,
that it guarantees. If not, why not?
b.)
Does the Mean Value Theorem apply on the interval [-1, 1]? If so, find the
point, c, that it guarantees. If not, why not?
7.)
Let f(x) = sin (πx) - x. Find exact values.
a.)
Does Rolle's Theorem apply on the interval [-1, 1]? If so, find the point, c,
that it guarantees. If not, why not?
b.)
Does the Mean Value Theorem apply on the interval [-1, 1]? If so, find the
point, c, that it guarantees. If not, why not?
8.)
Water is flowing out of a tank that has the shape of an inverted cone (the small
end is at the bottom). The top of the tank is 10 feet across and the tank is 10
feet high. If the height of the water is decreasing at a rate of 1 foot per hour,
what is the rate of change of the volume?
9.)
A 24 foot long ladder is leaning against a house. There is a 6-foot tall man
standing against the wall of the house under the ladder. He doesn't notice the
ladder is sliding down the wall. If the foot of the ladder is moving away from the
wall 3 feet/second, how fast is the top of the ladder coming down the wall when it
hits the man on the head?
10.) Suppose that the graph below is the graph of f'(x). Draw a possible graph of f(x).
Test #4
1.)
2.)
3.)
4.)
5.)
6.)
Find the following limits, if they exist. If they do not exit, tell why not.
1
sin x
3x + 1
a.) lim x  cos
b.) lim x 2
c.) lim x
x
x
x +1
1/2
2
Find both fixed points of the function f(x) = (cos x) - x . For each fixed point,
list your initial guess and the number of iterations necessary to get your answer.
Sketch graph of the following function. Find the following.
x
1 - x2
2 x3 - 6x

y= 2
,y = 2
, y" = 2
( x + 1 )2
( x + 1 )3
x +1
local max, local min, absolute max, absolute min, critical numbers, x-intercepts,
possible inflection pts, y-intercepts, horiz. asympt., vert. asympt., oblique
asympt., intervals where incr., intervals where decr., intervals where concave up
and down
Sketch graph of the following function. Find the following.
2
x
y=
x +1
y', y", local max, local min, absolute max, absolute min, critical numbers, xintercepts, possible inflection pts, y-intercepts, horiz. asympt., vert. asympt.,
oblique asympt., intervals where incr., intervals where decr., intervals where
concave up and down
A farmer wants to enclose a rectangular pasture along a straight river.
Additionally, he wants to put another fence line perpendicular to the river that will
bisect the pasture. If the farmer wants to enclose 6000 square feet what
dimensions of the pasture will require the least amount of fencing? Assume no
fencing is necessary along the river.
Find the minimum distance between the point (2, 0) and the curve f(x) = x 2.
Test #5
In #1 and #2, find the antiderivatives.
1.)
 (2x2 - x1/2 - 2x-2/3) dx
2.)
 sin(x/2) cos(x/2) dx
3.)
Evaluate the following definite integral. EXACT ANSWER!!

3
 sec x tan x dx

6
4.)
Solve the following initial value problem.
f"(x) = 2x2 - x, f'(0) = 3, f(0) = 2
5.)
Find the area under the given curve in the indicated interval.
f(x) = x3 + 2x2 - 3x, a = -3, b = 1.
6.)
Use the Second Fundamental Theorem of Calculus to find the derivative of g(x)
2
x
14
=  (2t + 1 ) dt .
0
7.)
An experiment involves finding the area to the right of the y-axis and under the
curve y = tan x as x increases. Find the instantaneous rate of change in the area
when x = π/3.
8.)
A box is supposed to be a cube with each side having length 5 inches. Find the
possible error in the volume if the error in measuring the sides may be as much
as 0.1 inches.
3
9.) Use the limit definition to find  ( x 2 - x) dx.
1
Test #6
In #1 and #2, find the antiderivatives.
1.)
 (3x - 1)21 dx
2.)
 sin4(2x) cos(2x) dx
In #3 and #4, evaluate the following definite integrals. Give exact answers and
simplify your answer!!

3.)
 sin x cos x dx
4.)
0
5.)
4
 x x - 1 dx
2
In Maple I did the following:
> int(1/x,x=-1..1);
1
 1
 dx


 x
-1
> evalf(");
Error, (in evalf/int) integrand has a pole in the interval
What is the problem?
In #6 - #7, for y as given find y.
6.)
y = ln(x2 + sin x)
8.)
9.)
10.)
7.)
y = (x2 + 1)3 (x + sin x)4
(2x - 7)3
5 ln x
. Do the following.
x
a.)
Graph f(x).
b.)
Find any critical numbers for f(x). Exact answers! If you can't find exact
answers, calculator approximations will be accepted but not for full credit.
c.)
Find all local extrema.
Suppose f(x) = 2x + 4x1/2, x > 0. Without finding f-1, find the derivative of f-1 at the
point where x = 6.
Use Simpson's rule, with n = 12, to approximate the area under the curve y
= tan x from x = 0 to x = 1. Give at least 5 places to the right of the decimal
point.
Let f(x) =
Test #7
In #1 - #4, given y find the derivative.
-1 2
-1
y = etan x
1.)
2.)
y = 2cos x
3.)
y = xln x
4.)
y = sin-1(e2x)
In #5 and #6, find the antiderivative.
3 x3 - 4 x 2 + 6x + 3
e
(x + 1) dx

dx
5.)
6.)
2
x - 2x + 2
In #7 and #8, evaluate the definite integral. Give EXACT answers only and simplify
your answers completely.
ln 2 1
2 dx
dx
7.) 
8.)  x
02
0 e2x - 1
x2 + 2x
9.)
10.)
x-2
x
- 2 Sin -1
is constant for 0  x  4.
2
2
C14 dating assumes that the carbon dioxide on earth today has the same
radioactive content as it did centuries ago. If this is true, then the amount of C 14
absorbed by a tree that grew several centuries ago should be the same as the
amount of C14 absorbed by a tree growing today. A piece of ancient charcoal
contains only 15% as much of the radioactive carbon as a piece of modern
charcoal. How long ago was the tree burned to make the ancient charcoal?
(The half-life of C14 is 5730 years.)
Show that the function f(x)= Sin -1
Final Exam
In #1 - #4, find dy/dx.
1.)
y = (Tan-1 e2x + x)4
2.)
y = ln
2
x (x + 1)( sin x - 2x)
(3x - 2 )3
4
3.)
5.)
6.)
-1
2
y = Cos (x + 3)
 8 x2 - 4 
4.) y = 

3
 1- 9 x 
What happens at an inflection point?
Suppose f is a function. Suppose that f'(2) = 0. Suppose further that if x < 2,
f'(x) > 0 and if x > 2, f'(x) < 0. What does this mean? Why?
7.)
Explain the origins of the formula for the definition of the derivative. Use
complete sentences and proper grammar. Use figures if necessary for your
explanation to make sense.
In #8 - #9, find the antiderivatives.
3 x 2 + 6x + 3

dx
8.)
 cos x sin2x dx
9.)
2
x +1
In #10 - 13, evaluate the definite integral. Give EXACT answers only.
2

e
4
1
dx

x x - 1 dx
10.)  x3 + x dx 11.)  sin x dx 12.) 
13.)
1 x 1 + ln 2 x 
-2
-
2
In #14 - #15, find the limit if it exists. If it does not exist, write "DNE" and tell why the
limit fails to exist.
lim sin 5x
ln x
14.) lim x 0+
15.)
x_0 3x
x
16.) Derive the formula for tanh-1x.
sin x
= 1 . Use your calculator to try to determine
17.) We know that lim x 0
x
sinh x
. Explain what method(s) you used.
lim x 0
x
18.) Find the absolute extrema, if they exist, for the function f(x) = x3 - x on the
interval [0, 1].
19.) A farmer wants to enclose a rectangular pasture along a straight river. If the
farmer wants to enclose 6000 square feet, what dimensions of the pasture will
require the least amount of fencing? Assume no fencing is necessary along the
river.
20.) There is a searchlight on a small island 1 mile from a straight shore. The light is
kept on a car that is traveling 40 miles per hour on a road along the shore. At
what rate is the angle the light makes with the shore changing when the car is
1320 feet from the point on the shore closest to the light?