Calculus I
Study Guide
MA3310 Calculus I
Study Guide
EXERCISE 1.1 (1.5 HOURS)
Assessment Preparation Checklist:
To prepare for this assessment, you should:

Read Chapters 1 and 2, pp. 32–56, of your textbook, Calculus. Chapter 1 focuses on functions
used in calculus, and Chapter 2 focuses on limits.

Review the lesson for this module that explains various types of functions and how to work with
limits.
Title: Review of Functions and Limits
1. Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformation to
sketch the graph of the function p(x) = 3 sin (2x – π/3). Use a graphing utility only to check your
work.
2. Let f  x  
x2  5x  6
x2  2 x
a. Calculate lim f ( x), lim f ( x), lim f ( x), and lim f ( x) .
x 0
x 0
x 2
x 2
b. Does the graph of f have any vertical asymptotes? Explain.
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module1_Exercise.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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03/17/2014
MA3310 Calculus I
Study Guide
LAB 1.1 (1.5 HOURS)
Assessment Preparation Checklist:
To prepare for this lab, you should:

Read Chapters 1 and 2, pp. 32–56, of your textbook, Calculus. Chapter 1 focuses on functions
used in calculus, and Chapter 2 focuses on limits and continuity.

Review the lesson for this module that explains various types of functions and how to work with
limits.
Title: Functions and Limits
Solve the following problems, providing detailed steps wherever required.
1. If f(±10) = 10 and g(±10) = −10, evaluate f(g(10)) and g(f(−10)).
2. A stone is thrown vertically upward from the ground at a speed of 40 m/s at time t = 0. Its
distance d (in m) above the ground (neglecting air resistance) is approximated by the function
f(t) = 40t–5t².
Determine an appropriate domain of the function. Identify the independent and dependent
variables.
3. Let g(x) =
x
and h(x) = x³ − 4x² + 6. Calculate the function g(h(x)).
6x
4. If Q(x) = x² − 5x + 1, find
Q(4  h)  Q  4
h
.
5. Determine whether the graph of the following function has symmetry about the x-axis, the
y-axis, or the origin. Check your work by graphing.
f(x) = x5 − x³ − 5
6. Use transformations to explain how the graph of f can be found by using the graph of y =
x.
f(x) = ± x  4
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03/17/2014
MA3310 Calculus I
Study Guide
7. Find the slope of the line in the figure. If the slope of the line is undefined, so state. Then write an
equation of the given line. Express y as a function of x.
8. Give the piecewise function f defined for the graph. Give the domain and range.
9. Use transformations of f(x) =
x to graph the following function.
g x  3 x  5  2
10. Find the formula for a function, y = f(t), which describes the distance traveled by a vehicle
traveling at a constant rate of 7 mi/hr for t hours. Graph the function and give a domain for the
function, given that the vehicle can run for 6 hours at a stretch.
11. A unit circle centered at the origin, and an angle, α, with the positive x-axis as the initial side.
The terminal side of angle, α, is the line between the origin and any point (x, y) on the circle.
Define the relationship between x and y, the point coordinates, and the sine and the cosine
function values of angle, α.

12. Find the exact value of sec   .
4
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03/17/2014
MA3310 Calculus I
Study Guide
13. Solve the equation 2 sinx 2  0 . Give a general formula for all the solutions by using angle(s)
in the interval [0,2π] and adding multiples of some integer k.
14. Find the exact value of each of the remaining trigonometry functions of θ.
cos   
12
, θ in quadrant II
13
15. Graph the following function over a one-period interval.
y 
1


 sin 4  x  
5
6

16. What is the slope of the secant line between the points (a, f(a)) and (b, f(b)) on the graph of f?
What is the slope of the line tangent to the graph of f at (a, f(a))?
17. A rock is dropped off the edge of a cliff and its distance s (in feet) from the top of the cliff after t
seconds is s(t) = 16t2. Assume the distance from the top of the cliff to the water below is 1936ft.
a. When will the rock strike the water?
b. Make a table of average velocities and approximate the velocity at which the rock strikes
the water.
Time Interval
10,11 10.5,11 10.9,11 10.99,11 10.999,11
Average Velocity
 ft / sec 
18. Use the graph of h in the given figure to identify the following values, if they exist.
a. h(5)
b.
lim h  x 
x 5
c. h(7)
d.
e.
lim h  x 
x 7
lim h  x 
x 8
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03/17/2014
MA3310 Calculus I
19. Let g(t)=
t  100
t  10
Study Guide
.
a. Complete the following table by computing values of g(t) for the given values of t.
t
g(t)
99.900
99.990
99.999
100.100
100.010
100.001
b. Make a conjecture about the value of lim
x 100
t  100
t  10
.
20. Use the graph to find the following limits and function values.
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03/17/2014
MA3310 Calculus I
Study Guide
lim f  x 
a.
x 0
lim f  x 
b.
x 0
lim f  x 
c.
x 0
d. f(0)
21. Does lim
x 3
x 2  7x  12
 lim  x  4  ? Explain why or why not.
x 3
x 3
22. Suppose p and q are polynomial functions. If lim
x 0
p x
q x
=10 and q(0) =4, find p(0).
23. Compute the following limit assuming lim f  x  =7. State the limit law(s) used to justify the
x 3
computation.
lim 3f  x  
x 3
24. Let lim f  x  =7 and lim g  x  =3. Use the limit rules to find the following limit.
x 9
lim
x 9
f x  g x
x 9
4g  x 
25. Find the given limit.
lim (-x2+3x-6)
x 1
26. Find the given limit.
lim
h 0
7
7h  1  1
27. Use the following function to answer questions (a) and (b) below.
4  x, x  4
f x  
 x  1, x  4
a. Find lim f (x) and lim f (x) .
x  4
x  4
b. Does lim f (x) exist? If so, what is it? If not, why not?
x 4
 x if x  0
.
 x if x  0
28. Show that lim x = 0 by first evaluating lim x and lim x . Recall that x  


x 0
x 0
x 0
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03/17/2014
MA3310 Calculus I
Study Guide
29. Evaluate the following limit.
7
lim 21  x 
3
x 6
30. Using the following function determine a value of the constant a for which lim g  x  exists and
x 1
state the value of the limit, if possible.
 x2  5x if x  1

g x  
3

ax  5 if x  1
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03/17/2014
MA3310 Calculus I
Study Guide
Submission Requirements:
Submit your answers in a Word document. Name the document
MA3310_StudentName_Module1_Lab.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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03/17/2014
MA3310 Calculus I
Study Guide
READ AND BEGIN PROJECT (0.5 HOURS)
Refer to “Project: Problem-Solving Skills” for a detailed description of the project.
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03/17/2014
MA3310 Calculus I
Study Guide
EXERCISE 2.1 (2.5 HOURS)
Assessment Preparation Checklist:
To prepare for this assessment, you should:

Read Chapter 2, pp. 56–92, and Chapter 3, pp. 99–126, of your textbook, Calculus. Chapter 2
readings focus on limits and continuity. Chapter 3 readings focus on derivatives and rules of
differentiation.

Review the lesson for this module that explains infinite limits, continuity, the basics of
derivatives, and rules of differentiation.
Title: Infinite Limits and Derivatives
Solve the following problems, providing detailed steps wherever required.
1. Calculate the limit analytically.
5x  5h  5x
, where x is constant.
h
h 0
lim
2. Assume the function g satisfies the inequality 1≤ g(x) ≤ sin2 x + 1 for x near 0. Use the Squeeze
Theorem to find lim g(x).
x 0
3. Evaluate the following limit or state that it does not exist.
lim
r 
1
cos r  1
4. Determine whether the following function is continuous at x = a using the continuity checklist to
justify your answers.
 x2  16

g(x)   x  4

9

if x  4
if x  4
;a  4
5. Use the Intermediate Value Theorem to show that the equation x5 + 7x + 5 = 0 has a solution in
the interval (–1, 0).
Find a solution to x5 + 7x + 5 = 0 in (–1, 0) using a root finder.
6. Find the derivative of the function f(s) =
s
.
4
1
2
7. For the equation y = x4 + x; a = 2:
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03/17/2014
MA3310 Calculus I
Study Guide
a. Find an equation of the tangent line at x = a.
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
8. Find f’(x), f’’(x), and f(3)(x) for the following functions:
a. f(x) = 3x12 + 4x3
1
b. f(x) =8 𝑥 4 − 3𝑥 2 + 1
9. The position of a small rocket that is launched vertically upward is given by s(t) = –5t2 + 40t +
100, for 0 ≤ t ≤ 10, where t is measured in seconds and s is measured in meters above the
ground.
a. Find the rate of change in the position (instantaneous velocity) of the rocket, for 0 ≤ t ≤ 10.
b. At what time is the instantaneous velocity zero?
c. At what time does the instantaneous velocity have the greatest magnitude, for 0 ≤ t ≤ 10.
10. In 2008 the new social networking and microblogging service Twitter increases its number of
unique visitors from 0.5 million to more than 4.5 million. A fit to the visitor data over several
years using a quadratic polynomial gives V(t) = 0.0173t2 + 0.1736t + 0.5, where V is measured in
millions of visitors and t is measured in months, with t = 0 corresponding to January 1, 2008.
a. Compute V’(t). What units are associated with the derivative and what does it measure?
b. At what time during 2008 (on the interval [0, 12]) was the growth rate the greatest? What
was the growth rate at that time?
c. At what time during 2008 was the growth rate the least? What was the growth rate at that
time?
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module2_Exercise1.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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[12]
03/17/2014
MA3310 Calculus I
Study Guide
EXERCISE 2.2 (2.5 HOURS)
Assessment Preparation Checklist:
To prepare for this assessment, you should:

Read Chapter 2, pp. 56–92, and Chapter 3, pp. 99–126, of your textbook, Calculus. Chapter 2
readings focus on limits and continuity. Chapter 3 readings focus on derivatives and rules of
differentiation.

Review the lesson for this module, which explains infinite limits, continuity, the basics of
derivatives, and rules of differentiation.
Title: Infinite Limits and Rules of Differentiation
Solve the following problems, providing detailed steps wherever required.
1. Evaluate the following limit
2. For the function f ( x) 

100 sin4 x3

5 
x
x   
x2
lim

.


x 2  4x  3
,
x 1
a. Evaluate lim f (x) and lim f (x) , and then identify any horizontal asymptotes.
x 
x 
b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate lim f (x) and
x  a
lim f (x) .
x  a
3. Use theorem of polynomial and rational functions to determine the intervals on which the
x 2  4x  3
function s( x) 
is continuous.
x2 1
4. Determine the interval on which the function g(x)  x 4  1 is continuous. Be sure to
consider right- and left-continuity at the endpoints.
5. For the function f ( x)  3x  5 x  1; P(1,7) ,
2
a. Using the formula mtan  lim
x a
f (x )  f (a)
, find the slope of the line tangent to the graph
xa
of f at P.
b. Determine an equation of the tangent line at P.
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MA3310 Calculus I
6. For the function f ( x) 
Study Guide
1
; P (0,1) ,
2x  1
a. Using the formula mtan  lim
h 0
f (a  h)  f (a)
, find the slope of the line tangent to the
h
graph of f at P.
b. Determine an equation of the tangent line at P.
7. Find the derivative of the function g ( x)  6 x  2 x( x  3x ) using derivatives of product
10
3
method.
x3  4x 2  x
8. Find the derivative of the function f ( x) 
using derivatives of quotients method.
x2
9. For y 
2x 2
;a  1,
3x  1
a. Find an equation to the line tangent to the given curve at a.
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
10. Compute the derivative of the function h( x) 
( x  1)( 2 x 2  1)
.
( x 3  1)
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module2_Exercise2.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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03/17/2014
MA3310 Calculus I
Study Guide
LAB 2.1 (2.0 HOURS)
Assessment Preparation Checklist:
To prepare for this lab, you should:

Read Chapter 2, pp. 56–92, and Chapter 3, pp. 99–126, of your textbook, Calculus. Chapter 2
readings focus on limits and continuity. Chapter 3 readings focus on derivatives and rules of
differentiation.

Review the lesson for this module that explains infinite limits, continuity, the basics of
derivatives, and rules of differentiation.
Title: Problem Solving—Infinite Limits and Derivatives
Solve the following problems, providing detailed steps wherever required.
1. Use the graph of the function to determine the limit lim f  x  , if exists.
x 3
2. The graph of f shown below has vertical asymptotes at x = –4 and x = 5. Find the following limits.
Use ∞ or –∞ when appropriate.
a.
b.
c.
lim f  x 
x 4
lim f  x 
x 4
lim f  x 
x 4
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03/17/2014
MA3310 Calculus I
d.
e.
f.
Study Guide
lim f  x 
x 5
lim f  x 
x 5
lim f  x 
x 5
3. Find the limit.
lim
x  4
4
x4
4. Evaluate the limit, using ∞ or –∞ when appropriate, or state that it does not exist.
5x2  5
x 1 x  1
lim
5. Find all vertical asymptotes, x = a, of the function f  x  
x 3  9 x 2  14 x
x2  7x
. For each value of a,
evaluate lim f  x  , lim f  x  , and lim f  x  . Use ∞ or –∞ when appropriate.
x  a
x  a
x a
6. Find the limit of the rational function h  x  
16 x 3
13x 3  15x 2  11x
when:
a. X → ∞
b. X → –∞
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03/17/2014
MA3310 Calculus I
Study Guide
7. Evaluate lim f  x  and lim f  x  for the function f  x  
x 
x 
x3  8
4 x 3  64 x 6  4
. Use ∞ or –∞
where appropriate. Then give the horizontal asymptote(s) of f (if any).
8. If a function f represents a system that varies in time, the existence of lim f t  means that the
t 
system reaches a steady state (or equilibrium). For the system of the population of a culture of
tumor cells given by p  t  
3400t
, determine if a state exists and give the steady-state value.
t 2
9. What is the domain of f  x   1  x2 and where is f continuous?
10. State whether the indicated function is continuous at 4.
 t 3  64

if t  4
r t    t  4
 52
if t  4

11. Suppose f(x) is defined as shown below.
a. Use the continuity checklist to show that f is not continuous at 0.
b. Is f continuous from the left or right at 0?
c. State the interval(s) of continuity.
 x 3  3x  3 if x  0
f x  
3
if x  0
 3x
12. Determine the interval(s) on which the following function is continuous. Be sure to consider
right- and left-continuity at the endpoints.
f  x   3x2  24
13. Evaluate the following limit.
sin2 x  5 sin x  4
sin x  1
x  3
2
lim
14. Use the Intermediate Value Theorem to verify that the following equation has three solutions
on the interval (0, 1). Use a graphing utility to find the approximate roots.
140x3 – 139x2 + 38x – 3 = 0
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03/17/2014
MA3310 Calculus I
Study Guide
15. Check whether the function f  x  
x 2  11x  28
has a removable discontinuity at the point
x 5
a = 5.
16. Suppose x lies in the interval (4, 6) with x ≠ 5. Find the smallest positive value of δ such that the
inequality 0 < |x – 5| < δ is true for all possible values of x.
17. For the function f (x) = 4x + 7, P( –3, –5):
a. Use the definition mtan  lim
f  x   f  a
xa
x a
to find the slope of the line tangent to the graph
of f at P.
b. Determine an equation of the tangent line at P.
c. Plot the graph of f and the tangent line P.
18. For the function f (x) = 7x + 4, P (0,4):
a. Use the definition mtan  lim
f  a  h   f  a
h
h 0
to find the slope of the line tangent to the
graph of f at P.
b. Determine an equation of the tangent line at P.
19. Let f  x  
 1
1
and point p   1,  .
4  3x
 7
a. Use the following definitions of the slope of the tangent line at x = a to find the slope of the
line tangent to the graph of f at P.
mtan  lim
h 0
f  a  h  f a 
h
b. Determine an equation of the tangent line at P.
20. For the function and point f  x  
2
x
,a 
1
9
a. Find f’(a).
b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of
a.
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03/17/2014
MA3310 Calculus I
Study Guide
21. Use the graph of g in the figure to do the following.
a. Find the values of x in (–1, 7) at which g is not continuous.
b. Find the values of x in (–1, 7) at which g is not differentiable.
c. Sketch a graph of g’.
22. Find the derivative of the following function.
g  t   12 t
23. Find the derivative of the following function.
f  x   9x 3  4x
24. Find the derivative of the following function.
f  x   14x 3  25x 
1
4
25. Find the derivative of the following function by first expanding the expression.
f(x) = (6x + 5) (5x2 + 7)
26. Find f’ (x) for the function.
f x 
2 x 6  x5
12x
27. Find the derivative of the following function by first simplifying the expression.
g x 
x2  81
x 9
28. For what point on the curve of y = 4x2 + 2x is the slope of a tangent line equal to –30?
29. Find f’(x), f”(x), and f(3)(x) for the following function.
f(x) = 7x3 + 2x2 + 8x
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MA3310 Calculus I
Study Guide
30. Starting with a full tank of gas, the distance traveled by a car is related to the amount of gas
consumed by the function D (g) = 0.05g2 + 38g, where D is measured in miles and g in gallons.
a. Compute dD/dg. What units are associated with the derivative and what does it measure?
b. Find dD/dg for g = 0, 4, and 8 gal (include units). What does your answer say about the gas
mileage for this car?
c. What is the range of this car if it has a 12-gal tank?
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module2_Lab.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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[20]
03/17/2014
MA3310 Calculus I
Study Guide
QUIZ 1 (1.0 HOUR)
Assessment Preparation Checklist:
To prepare for this quiz, you should:

Read Chapters 1 and 2, pp. 32–56, from your textbook, Calculus.

Review the lesson for Module 1 that explains fundamental concepts of functions and limits.
Title: Quiz 1
Attempt the quiz.
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[21]
03/17/2014
MA3310 Calculus I
Study Guide
EXERCISE 3.1 (2.0 HOURS)
Assessment Preparation Checklist:
To prepare for this assessment, you should:

Read Chapter 3, pp. 129–169, and Chapter 4, pp. 177–225 and pp. 240–248, of your textbook,
Calculus. Chapter 3 readings focus on the chain rule and implicit differentiation. Readings from
Chapter 4 focus on applications of derivatives.

Review the lesson for this module that explains derivatives in detail and how derivatives can
help solve trigonometric problems and antiderivatives.
Title: More on Derivatives
Solve the following problems, providing detailed steps wherever required.
1. Suppose a stone thrown vertically upward from the edge of a cliff on mars (where the
acceleration due to gravity is only about 12 ft/s2) with an initial velocity of 64 ft/s from a height
of 192 ft above the ground. The height s of the stone above the ground after t seconds is given
by s = –6t2 + 64t + 192.
a. Determine the velocity v of the stone after t seconds.
b. When does the stone reach its highest point?
c. What is the height of the stone at the highest point?
d. When does the stone strike the ground?
e. With what velocity does the stone strike the ground?
2. Use the chain rule to evaluate
dy
.
dx
y = (5x2 + 11x)20
3. Use the chain rule to evaluate
dy
.
dx
y  sin x
4. Use the chain rule to evaluate
dy
.
dx
y = ((x + 2) (3x3 + 3x))4
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[22]
03/17/2014
MA3310 Calculus I
5. Find
d2y
dx 2
Study Guide
for the following function.
y = x cos x2
6. Find
d2y
dx 2
for the following function.
y  3x3  4x  1
7. Use the following graphs to identify the points on the interval [a, b] at which local and absolute
extreme values occur.
8. For the given function f  x  
1 3 1
x  x;  1, 3 :
8
2
a. Find the critical points of the function on the domain or on the given interval.
b. Use a graphing utility to determine whether each critical point corresponds to a local
maximum, or neither.
9. For the given function f(x) = sinx cosx; [0, 2π]:
a. Find the critical points of the following functions on the domain or on the given interval.
b. Use a graphing utility to determine whether each critical point corresponds to a local
maximum, or neither.
10. For the given function f  x   x 2  x 2 ;   2, 2  :


a. Find the critical points of f on the given interval.
b. Determine the absolute extreme values of f on the given interval (if they exist).
c. Use the graphing utility to confirm your conclusions.
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[23]
03/17/2014
MA3310 Calculus I
Study Guide
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[24]
03/17/2014
MA3310 Calculus I
Study Guide
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module3_Exercise.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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All Rights Reserved.
[25]
03/17/2014
MA3310 Calculus I
Study Guide
LAB 3.1 (2.0 HOURS)
Assessment Preparation Checklist:
To prepare for this lab, you should:

Read Chapter 3, pp. 129–169, and Chapter 4, pp. 177–225 and pp. 240–248, of your textbook,
Calculus. Chapter 3 readings focus on the chain rule and implicit differentiation. Readings from
Chapter 4 focus on applications of derivatives.

Review the lesson for this module that explains derivatives in details and how derivatives can
help solve trigonometric problems and antiderivatives.
Title: Problem Solving—More on Derivatives
Solve the following problems, providing detailed steps wherever required.
1. Find the derivative of the following function.
g(w) = (w4 + 4)(w4 – 1)
2. Where does cos x have a horizontal tangent line? Where does –sin x have a value of zero?
Explain the connection between these two observations.
3. Evaluate the limit.
lim
x 0
4. Find
sin 8 x
tan x
dr
for r = tanθsecθ.
d
5. Find dr/d θ.
r = tan(6 – 4θ)
6. Use the Chain Rule to find the derivative of the following function.
y= (3x7 – 5x5 + 2)24
7. For the function y4 = 16x; (16, 4):
a. Use implicit differentiation to find the derivative
dy
.
dx
b. Find the slope of the curve at the given point.
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[26]
03/17/2014
MA3310 Calculus I
Study Guide
8. Determine from the given graph whether the function has any absolute extreme value on [a, b].
9. A stone is launched vertically upward from a cliff 384 ft above the ground at a speed of 32 ft/s.
Its height above the ground t seconds after the launch is given by s = –16t2 + 32t + 384 for 0≤t≤6.
When does the stone reach its maximum height?
10. The following figure gives the graph of the derivative of a continuous function f that passes
through the origin. Sketch a graph of f on the same set of axes.
11. Find the intervals on which f is increasing and decreasing. Superimpose the graphs of f and f’ to
verify your work.
f(x) = x3 + 2x
12. Find the intervals on which f is increasing and decreasing. Superimpose the graphs of f and f’ to
verify your work.
f(x) = 9 + x – 2x2
13. Can the graph of a polynomial have vertical or horizontal asymptotes? Explain.
14. Sketch the following curve, indicating all relative extreme points and inflection points.
y = x3 – 12x + 1
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[27]
03/17/2014
MA3310 Calculus I
Study Guide
15. Summarize the pertinent information obtained by applying the graphing strategy and sketch the
graph of f  x  
4x
2
x  36
.
16. Sketch a continuous function f on some interval that has the properties described below. The
function f satisfies f’’(–2) = 3, f’(–1) = –2, and f’(6) = 2.
17. The sides of a square increase in length at a rate of 4 m/s.
a. At what rate is the area of the square changing when the sides are 20 m long?
b. At what rate is the area of the square changing when the sides are 30 m long?
c. Draw a graph of how the rate of change of the area varies with the side length.
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module3_Lab.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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[28]
03/17/2014
MA3310 Calculus I
Study Guide
QUIZ 2 (1.0 HOUR)
Assessment Preparation Checklist:
To prepare for this quiz, you should:

Read Chapter 2, pp. 56–92, and Chapter 3, pp. 99–126, of your textbook, Calculus.

Review the lesson for Module 2.
Title: Quiz 2
Attempt the quiz.
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[29]
03/17/2014
MA3310 Calculus I
Study Guide
EXERCISE 4.1 (3.0 HOURS)
Assessment Preparation Checklist:
To prepare for this assessment, you should:

Read Chapter 5 of your textbook, Calculus. This chapter introduces you to the integral and the
fundamental theorems of calculus.

Review the lesson for this module that focuses on the basics of integration.
Title: Fundamental Theorems of Calculus
Solve the following problems, providing detailed steps wherever required.
1. Evaluate the following integral.
 xsinx
2
cos8 x 2dx
2. The figure shows the areas of regions bounded by the graph of f and the x-axis.
Evaluate the following integrals.
c
a.
a f  x dx
b.
b f  x dx
c.
2 f  x dx
d.
4 f  x dx
e.
3 f  x dx
f.
2 f  x dx
d
b
c
d
a
b
a
d
b
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[30]
03/17/2014
MA3310 Calculus I
Study Guide
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module4_Exercise.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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[31]
03/17/2014
MA3310 Calculus I
Study Guide
LAB 4.1 (2.0 HOURS)
Assessment Preparation Checklist:
To prepare for this lab, you should:

Read Chapter 5 of your textbook, Calculus. This chapter introduces you to the integral and the
fundamental theorems of calculus.

Review the lesson for this module that focuses on the basics of integration.
Title: Fundamentals of Calculus and Integration by Substitution
Solve the following problems, providing detailed steps wherever required.
1. The graph of f is shown in the figure below. Let A  x  
 f t dt and F  x    f t dt be
x
x
2
4
two area functions for f.
Evaluate the following area functions.
a) A (–2)
b) F (8)
2. Evaluate the integral
 x
1
1
3
2
c) A (4)
d) A (8)
 3dx using the fundamental theorem of calculus.
3. Find the net area of the region bounded by y = 6cosx and the x-axis between x = –π/2 and x = π.
4. Find the area of the region R bounded by the graph of f(x) = x (x + 2) (x – 3) and the x-axis on the
interval [–2, 3]. Graph f and the region R.
5. Simplify the following expression.


d x
10t 2  t dt

0
dx
6. Is x38 an even or odd function? Is cos(x7) an even or odd function?
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[32]
03/17/2014
MA3310 Calculus I
Study Guide
7. Use symmetry to evaluate the following integral.
 9  x  x
1
 x3 dx
2
1
8. Use symmetry to evaluate the following integral. Draw a figure to interpret your result.

    sin x  dx
2

2
9. The elevation of a path is given by f (x) = x3 – 5x2 + 6, where x measures horizontal distances.
Draw a graph of the elevation function and find its average value for 0 ≤ x ≤ 4.
10. Use symmetry to evaluate the following integral.
3

2
4csc x cot xdx
4
11. On which derivative rule is the Substitution Rule based?
12. If the change of variables u = x2 – 5 is used to evaluate the definite integral
 f  x dx , what are
6
4
the new limits of integration?
13. Find the antiderivative of the following function by trial and error. Check your answer by
differentiation.
f (x)= (x + 2)15
14. Use the substitution u = x2 + 8 to find the following indefinite integral. Check your answer by
differentiation.
 2x  x
2
 8 dx
9
15. Use the substitution u = x2 + 4 to find the following indefinite integral. Check your answer by
differentiation.
 2x  x
2
 4  dx
15
16. Determine the integral by making the appropriate substitution.
 2x  x
2
3
 6  dx
4
17. Use a change of variables to find the following indefinite integral. Check your work by
differentiation.
x
5
 x 4   5 x 4  4 x3 dx
9
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[33]
03/17/2014
MA3310 Calculus I
Study Guide
18. Use a change of variables to evaluate the following definite integral.

12 x11 1 x12  dx
11
1
0
19. Use a change of variables to evaluate the following integral.
  csc  7 w cot  7 w dw
20. Use a change of variables to evaluate the following integral.
 sin x sec
8
xdx
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module4_Lab.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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[34]
03/17/2014
MA3310 Calculus I
Study Guide
QUIZ 3 (1.0 HOUR)
Assessment Preparation Checklist:
To prepare for this quiz, you should:

Read Chapter 3, pp. 129–169, and Chapter 4, pp. 177–225 and pp. 240–248, of your textbook,
Calculus.

Review the lesson for Module 3.
Title: Quiz 3
Attempt the quiz.
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[35]
03/17/2014
MA3310 Calculus I
Study Guide
PROJECT (2.0 HOURS)
Assessment Preparation Checklist:
To prepare for this project, you should:

Read Chapter 1 and Chapter 2, pp. 32–85, of your textbook, Calculus.

Review the lesson for this module, which explains various types of functions and how to work
with limits.
Title: Problem-solving Skills
Submit the project for evaluation to your instructor. Refer to “Project: Problem-Solving Skills”
for a detailed description of the project.
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[36]
03/17/2014
MA3310 Calculus I
Study Guide
EXERCISE 5.1 (3.0 HOURS)
Assessment Preparation Checklist:
To prepare for this assessment, you should:

Read Chapter 6 of your textbook, Calculus. This chapter focuses on physical applications of
integration.

Review the lesson for this module that explains how integration can be used to calculate areas
and volumes of various geometric shapes.
Title: Physical Applications of Integration
Solve the following problems, providing detailed steps wherever required.
1. Starting at the same point on a straight road, Anna and Benny begin running with velocities (in
mi/hr) given by vA (t) = 2t + 1 and vB (t) = 4 – t, respectively.
a. Graph the velocity functions, for 0 ≤ t ≤ 4.
b. If the runners run for 1 hr, who runs farther? Interpret your conclusion geometrically using
the graph in part (a).
c. If the runners run for 6 mi, who wins the race? Interpret your conclusion geometrically using
the graph in part (a).
2. Find the area of the region in the first quadrant bounded by the curve x 
y  1.
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module5_Exercise1.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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[37]
03/17/2014
MA3310 Calculus I
Study Guide
EXERCISE 5.2 (3.0 HOURS)
Assessment Preparation Checklist:
To prepare for this assessment, you should:

Read Chapter 6 of your textbook, Calculus. This chapter focuses on physical applications of
integration.

Review the lesson for this module that explains how integration can be used to calculate areas
and volumes of various geometric shapes.
Title: Evaluation of Volume by Integration
Solve the following problems, providing detailed steps wherever required.
1. The region bounded by the curves y = –x2 + 2x +2 and y = 2x2 – 4x +2 is revolved about the x-axis.
What is the volume of the solid that is generated?
2. The region bounded by the curves y = x + 1, y = 12/x, and y = 1 is revolved about the x-axis. What
is the volume of the solid that is generated?
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module5_Exercise2.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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All Rights Reserved.
[38]
03/17/2014
MA3310 Calculus I
Study Guide
LAB 5.1 (2.0 HOURS)
Assessment Preparation Checklist:
To prepare for this lab, you should:

Read Chapter 6 of your textbook, Calculus. This chapter focuses on physical applications of
integration.

Review the lesson for this module, which explains how integration can be used to calculate
areas and volumes of various geometric shapes.
Title: Applications of Integration
Solve the following problems, providing detailed steps wherever required.
1. Find the arc length of the curve below on the given interval by integrating with respect to x.
y = 4x + 2; [0, 3] (Use Calculus)
2. Find the length of the curve y = y 
3 5/3 3 1/3
x
 x
 9 for 1 ≤ x ≤ 27.
5
4
3. For y = cos 3x on [0, π]:
a. Write and simplify the integral that gives the arc length on the curve on the given integral.
b. If necessary, use a calculator to evaluate the approximate integral.
4. What differentiable functions have an arc length on the integral [a, b] given by the following
integrals? Note that the answers are not unique. Give all functions that satisfy the conditions.
b
a.

1  49 x 8 dx
a
b
b.

1  144 cos2 (3x )dx
a
5. Find the mass of the thin bar with the given density function.
p(x)  1  sin x; for

3
 x 
3
4
6. Find the mass of the following thin bar with the given density function.
p 5
x
; for 0  x  8
8
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[39]
03/17/2014
MA3310 Calculus I
Study Guide
7. A spring on a horizontal surface can be stretched and held 0.8 m from its equilibrium position
with a force of 64 N.
a. How much work is done in stretching the spring 1.5 m from its equilibrium position?
b. How much work is done in compressing the spring 2.5 m from its equilibrium position?
8. A water tank, shown below, is shaped like an inverted cone with the height 4 m and radius 0.5
m.
a. If the tank is full of water, how much work is required to pump the water to the level of the
top of the tank and out of the tank? Use 1000kg/m3 for the density of water and 9.8 m/s2 for
the acceleration due to gravity.
b. Is it true that it takes half as much work to pump the water out of the tank when it is filled
to half its depth as when it is full? Explain.
9. The figure below shows the shape and dimensions of a small dam.
Assuming the water level is at the top of the dam, find the total force on the face of the dam.
Use 1000 kg/m3 for the density of water and 9.8 m/s2 for the acceleration due to gravity.
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[40]
03/17/2014
MA3310 Calculus I
Study Guide
10. A large building shaped like a box is 35 m high with a face that is 70 m wide. A strong wind blows
directly at the face of the building, exerting a pressure of 180 N/m2 at the ground and increasing
with height according to P(y) = 180 + 2y, where y is the height above the ground. Calculate the
total force on the building, which is a measure of the resistance that must be included in the
design of the building.
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module5_Lab.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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All Rights Reserved.
[41]
03/17/2014
MA3310 Calculus I
Study Guide
QUIZ 4 (1.0 HOUR)
Assessment Preparation Checklist:
To prepare for this quiz, you should:

Read Chapter 5 of your textbook, Calculus.

Review the lesson for Module 4.
Title: Quiz 4
Attempt the quiz.
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[42]
03/17/2014
MA3310 Calculus I
Study Guide
LAB 6.1 (2.0 HOURS)
Assessment Preparation Checklist:
To prepare for this lab, you should:

Read Chapter 7 of your textbook, Calculus. This chapter focuses on the natural logarithmic and
exponential functions.

Review the lesson for this module that explains logarithmic and exponential functions and
exponential models.
Title: Integration of Trigonometric, Logarithmic, and Exponential Functions
Solve the following problems, providing detailed steps wherever required.
1. Express the inverse of f(x) = 9x – 2 in the form y = f–1(x).
2. Using the graph of f shown below, find three intervals on which f is one-to-one, making each
interval as large as possible.
3. Find the inverse function and graph both f(x) and f–1(x) on the same set of axes. Check your work
by looking for the required symmetry in the graph.
f(x) = 5x – 1
4. Find the derivative of the inverse of the following function at the specified point on the graph of
the inverse function. You do not need to find f-1.
f(x) = 2x – 9; (–5, 2)
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[43]
03/17/2014
MA3310 Calculus I
Study Guide
5. Find the derivative of the inverse of the following function at the specified point on the graph of
the inverse function. You do not need to find f-1.
f(x) = 5x – 1; (–6, –1)
6. Given the function f, find the slope of the line tangent to the graph of f–1 at the specified point
on the graph of f–1.
f  x   2x ;  4, 8 
7. If the slope of the curve y = f–1(x) at (2, 5) is
3
, find f’(5).
4
8. What is the slope of the line tangent to the graph of y = tan–1(–3x) at x = 1?
9. Find the exact value, in radians, of the expression.

3
sin1  
 2 


10. Evaluate the expression.
cos (cos–1(–0.5))
11. Use a right triangle to write the following expression as an algebraic expression. Assume that x is
positive and that the given inverse trigonometric function is defined for the expression in x.
cos (sin–16x)
12. Use a right triangle to write the following expression as an algebraic expression. Assume that x is
positive and that the given inverse trigonometric function is defined for the expression in x.
cos (sin–12x)
13. Use a right triangle to write the following expression as an algebraic expression. Assume that x is
positive and that the given inverse trigonometric function.
7

cos  sin1 
x

14. Find the exact value of the expression.
2 3 
sec1 
 3 


15. Find
dy
.
dx
y = sin–1(7x5)
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[44]
03/17/2014
MA3310 Calculus I
Study Guide
16. Find the derivative of y = sec–1(8s4 + 3) with respect to s.
17. Determine the indefinite integral
18. Evaluate the integral 
6 3
0

19ds
144  s2
8
49  x2
dx. Check the answer by differentiation.
.
19. Evaluate the following integral.
3 3
0
dx
2
9  x2
20. A boat is towed toward a dock by a cable attached to a winch that stands 5 ft above the water
level (see figure). Let θ be the angle of elevation of the winch and let L be the length of the cable
as the boat is towed toward the dock. Complete parts (a) to (c) below.
a. What is the rate of change of θ with respect to L,
d
?
dL
b. Compute
d
when L = 50, 20, and 6 ft.
dL
c. Find lim
d
and explain what is happening as the last foot of cable is reeled in (note that
dL
L 5
the boat is at the dock when L = 5). What is the value of limit?
21. Find the derivative of y with respect to x.
y 
x5
x5
ln x 
5
25
22. Evaluate the following integral. Include absolute values only when needed.
dx
 5x  4
23. Find the derivative of the following function.
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[45]
03/17/2014
MA3310 Calculus I
f x 
Study Guide
e3 x
e
x
5
24. Find the following integral.
e x  e x
 e x  e x dx
25. Use logarithmic differentiation to find the derivative,
dy
, of y 
d
  6 sin 7  .
26. Estimate the value of the following limit by creating a table of function values for h = 0.01,
0.001, and 0.0001, and h = –0.01, –0.001, and –0.0001.
lim
ln 1  h
h 0
h
27. Compute the following derivative using the method of your choice.
d  8x8 
e

dx 

28. Solve the equation for x.
logx 64 
3
4
29. Solve the equation for x.
logx 27 
3
4
30. Find the derivative of the following function.
y = 3·7x
31. Iodine-123 is a radioactive isotope used in medicine to test the function of the thyroid gland. Its
half-life is 13.1 hours. A 475-microcurie dose of isotope A is administered to a patient. The
t
 1 13.1
quantity, Q, left in the body after t hours is given by the equation Q  475  
.
2
Answer parts (a) through (d).
a. Determine the time it takes for the level of iodine-123 to drop to 25 microCurie.
b. Determine the rate of change of the quantity of Iodine-123 at 12 hours.
c. Determine the rate of change of the quantity of Iodine-123 at 1 day.
d. Determine the rate of change of the quantity of Iodine-123 at 2 days.
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MA3310 Calculus I
Study Guide
1
32. Evaluate the integral  9x dx .
1
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MA3310 Calculus I
33. Evaluate the integral
Study Guide
2
1 1  ln x  x
x
dx.
34. Devise the exponential growth function that fits the given data, then answer the accompanying
question. Be sure to identify the reference point (t = 0) and units of time.
The population of a town with a 2008 population 0f 73,000 grows at a rate of 2.5% per year. In
what year will the population double its initial value (to 146,000)?
35. Use the exponential growth model, A = A0ekt, to find the time it takes a population to multiply by
eleven (to grow from A0 to 11A0).
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module6_Lab.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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MA3310 Calculus I
Study Guide
FINAL EXAM (2.0 HOURS)
Assessment Preparation Checklist:
To prepare for this exam, you should:

Read Chapters 1–7 of your textbook, Calculus.

Review the lessons of Modules 1–6.
Title: Final Exam
Attempt the final exam.
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MA3310 Calculus I
Study Guide
PROJECT: PROBLEM-SOLVING SKILLS
Project Introduction:
Use the ITT Tech Virtual Library to research additional information about the topic and present both the
research and the computation work in the final unit of the course in the form of a presentation using
Microsoft PowerPoint.
Course Learning Objectives Tested:
1. Define and apply the concepts of functions and limits to introductory problems.
2. Differentiate algebraic and transcendental functions using the basic derivative rules.
3. Solve application problems involving derivatives.
4. Integrate algebraic and transcendental functions using the basic integration rules.
5. Solve application problems involving integrals.
6. Create a presentation of a problem solution.
7. Use the ITT Tech Virtual Library to research calculus topics as assigned.
PROJECT SUBMISSION PLAN
Description/Requirements of Project Part
Assessment Preparation Checklist:
Evaluation Criteria
Your submission will be evaluated
To prepare for this project, you should:
against the following criteria:


Read Chapter 1 and Chapter 2, pp. 32–85, of your

Did you include appropriate
textbook, Calculus.
steps, graphs, or rationales
Review the lesson for this module, which explains
to determine the answers to
various types of functions and how to work with
questions wherever
limits.
required?

Title: Problem-solving Skills
Using the ITT Tech Virtual Library, research additional
Did you correctly answer
each question?
information about a topic from the list that follows and
present both the research and the computation work at the
end in the form of a presentation using PowerPoint.
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MA3310 Calculus I
Study Guide
Description/Requirements of Project Part
Evaluation Criteria
Topic List

Numerical differentiation

Elasticity in economics
Submission Requirements

Submit the Word document to your instructor.
Ensure that you have correctly named the
document as MA3310_StudentName_Project.docx,
replacing StudentName with your name.

Submit the PowerPoint to your instructor. Ensure
that you have correctly named the document as
MA3310_StudentName_Project.docx, replacing
StudentName with your name.

Font: Arial, 12 point

Spacing: Double spaced

Length: Five to six pages

Citation Style: APA
Due: Module 4
Grading Weight: 10%
(End of Study Guide)
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MA3310 Calculus I
Study Guide
APPENDIX A: HANDOUTS AND WORKSHEETS
PROJECT TOPIC LIST
Topic: Numerical differentiation
Topics and skills: Derivatives, calculator
While the rules of differentiation allow us to compute the derivative of just about any function, there
are practical situations in which these rules cannot be used. For example, in some applications, a
relationship between two variables may be given as a set of data points, but not as a formula. In
situations like this, the rate of change of one variable with respect to the other (that is, the derivative)
might be needed, but the rules do not apply to sets of data. This project focuses on methods for
approximating the derivative of a function at a particular point.
Backward and Forward Difference Formulas
f (a  h)  f (a)
implies that
h
h 0
Assuming the limit exists, the definition of the derivative f (a)  lim
f (a) 
f (a  h)  f (a)
h
(1)
for h near 0. If h > 0, then (1) is referred to as a forward difference formula and if h < 0, (1) is a
backward difference formula. The geometry of these formulas is shown in Figure 1.
1. Why do you think (1) is called the forward difference formula if h > 0 and a backward difference
formula if h < 0?
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MA3310 Calculus I
2. Let f (x) =
Study Guide
x
a. Find the exact value of f '(4).
b. By equation (1), f (4) 
f (4  h)  f (4)

h
4h 2
. Therefore we estimate f '(4) by
h
4h 2
for values of h near 0. Complete columns 2 and 5 of table given below1
h
calculating
and describe
h
4h 2
behaves as h approaches 0.
h
4h 2
h
Error
h
0.1
–0.1
0.01
–0.01
0.001
–0.001
0.0001
–0.0001
4h 2
h
Error
Table 1
3. The accuracy of an approximation is given by
Error = |exact value – approximate value|.
Use the exact value of f’(4) in part (a) to complete columns 3 and 6 in Table 1. Describe the
behavior of the errors as h approaches 0.
Centered Difference Formulas
Another formula that is used to approximate the derivative of a function at a point is the centered
f (a)  lim
difference formula (CDF)
4. Again consider f (x) =
h 0
f (a  h)  f (a  h)
2h
(2)
x.
a. Graph f near the point (4, 2) and let h = ½ in the centered difference formula. Show the line
whose slope is computed by the centered difference formula and explain why the formula
approximates f '(4).
b. Use the centered difference formula to approximate f '(4) by completing Table 2.
h
Approximation
Error
0.1
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MA3310 Calculus I
Study Guide
h
Approximation
Error
0.01
0.001
0.0001
Table 2
5. Use the CDF (2) and a table similar to Table 2 to find a good approximation to f'(0) for
f(x) = (1 + x)–1.
6. Use the CDF (2) and a table similar to Table 2 to find a good approximation to f'(π/6) for
f(x) = sin x.
7. Table 3 gives the distance f(t) fallen by a smokejumper t seconds after she opens her chute.
a. Use the forward difference formula (1) with h = 0.5 to estimate the velocity of the skydiver
at t = 2 s.
b. Repeat part (a) using the centered difference formula (2).
t (seconds)
f(t) (feet)
0
0
0.5
4
1.0
15
1.5
33
2.0
55
2.5
81
3.0
109
3.5
138
4.0
169
Table 2
Computer Rounding Error
Using difference approximations to approximate derivatives with a computer or calculator is prone to
rounding errors. These errors occur when a calculator rounds a number before using it in an arithmetic
calculation. Such rounding may lead to remarkably inaccurate results.
8. Consider the function f (x) = x10.
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MA3310 Calculus I
Study Guide
a. Use analytical methods to find the exact value of f'(1).
b. Use the forward difference formula to approximate f'(1) using values of h = 10–2, 10–3, and
10–4. What do you observe?
c. Compute approximations to f'(1) using h = 10–n for n = 5, 6, 7, …, 15 What do you observe?
In Step 8c, you should find that for small enough values of h, the approximations to f'(1)
eventually are 0, which is clearly a bad estimate. Here is why this error occurs. Suppose h =
10–14. The calculator rounds f(1 + 10–14) to 1 and therefore the forward difference formula
becomes
f (1  1014 )  f (1)
14
10
, which is estimated to equal 1  1 or 0.
1014
d. The remedy to rounding errors in this situation is to use small—but not too small—values of
h. Based on the approximations computed in parts (b) and (c), what is a good approximation
to f'(1)?
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MA3310 Calculus I
Study Guide
Topic: Elasticity in economics
Topics and skills: Derivatives
Economists apply the term elasticity to supply, demand, income, capital, labor, and many other variables
in systems with input and output. In a few words, elasticity describes how changes in the input to a
system are related to changes in the output. And because elasticity involves change, it also involves
derivatives. In this project we investigate the idea of elasticity as it applies to demand functions. It’s a
common experience that as the price of an item increases, the number of sales of that item generally
decreases. This relationship is expressed in a demand function.
1. Suppose a gas station has the linear demand function D(p) = 1200 – 200p (Figure 1). According
to this function, how many gallons of gas can the gas station owners expect to sell per month if
the price is set at $4 per gallon?
2. Evaluate D’(p) and show that the demand function is decreasing. Explain why demand functions
are usually decreasing functions.
3. Suppose the price of a gallon of gasoline (Steps 1 and 2) increases from $3.50 to $4.00 per
gallon; call this change Δp. What is the resulting change in the number of gallons sold, call it ΔD?
(Note that the change is a decrease, so it should be negative.)
4. Now express the answer to Step 3 in terms of percentages: What is the percent change in price,
Δp/p, when it increases from $3.50 to $4.00 per gallon? What is the resulting percent change in
the number of gallons sold, ΔD/D? (Note that the percent change is negative.)
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MA3310 Calculus I
Study Guide
5. The elasticity in the demand is the ratio of the percent change in demand to the percent change
in price; that is, E 
D / D
. Compute the elasticity for the changes in Steps 3 and 4 (it should
p / p
be negative).
6. The elasticity is simplified by considering small changes in p and D. In this case we use the
definition of the derivative and write
E  lim
p 0
D / D
D  p  dD p
 lim
   dp D .
p / p
p  0 p  D 
Now the elasticity is a function of p. Show that for the gasoline demand function the elasticity is
E(p)  
p
.
6p
7. The elasticity may be interpreted as the percent change in the demand that results for every
one percent change in the price. For example if E(p) = –2, a one-percent increase in price
produces a two-percent decrease in demand. If the price of gasoline is p = $4.50 and there is a
3.5% increase in the price, what is the elasticity and the corresponding percent change in the
number of gallons sold?
8. Graph the gasoline demand elasticity function for 0 ≤ p < 6.
9. When –∞ < E < –1, the demand is said to be elastic. When –1 < E < 0, the demand is said to be
inelastic. When E = –∞, the demand is perfectly elastic and when E = 0 the demand is perfectly
inelastic. Essential goods such as basic foods tend to have inelastic demands; discretionary
items, such as electronic equipment have elastic demands. Explain the meaning of these terms
in this context.
10. For what prices is the gasoline demand function elastic and inelastic?
11. The demand for processed pork in Canada is described by the function D(p) = 286 – 20p1. Graph
the demand function, compute the elasticity, and graph the elasticity. For what prices is the
demand function elastic and inelastic?
12. Show that the general linear demand function D(p) = a – bp, where a and b are positive real
numbers, has a decreasing elasticity for 0 ≤ p < a/b. Show that for the general linear demand
function, E = –1, when p = a/2b.
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MA3310 Calculus I
Study Guide
13. Not all demand functions are linear. Compute the elasticity for the exponential demand function
D(p) = ae–bp, where a and b are positive real numbers.
14. Show that the demand function D(p) = a/pb, where a and b are positive real numbers, has a
constant elasticity for all positive prices.
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