MATH-112-200: MATH 112: Calculus 1 (Online)

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Introduction to this Course Manual
Thank you for purchasing the lecture packet for MATH 112: Calculus 1.
As part of your materials, you should have received these items:
•  this introductory letter
•  copies of the course lectures (the pages following this letter)
Begin your online course by reading the syllabus; it contains the
information you need to successfully complete the course. As you begin,
you will notice that each lesson includes a brief introduction, learning
outcomes for the lesson, reading assignments, and “lecture” material.
The lecture material is included in this lecture packet.
 The lessons also have online homework and Speedback assignments
associated with them; these are available only through the online course.
 This packet is designed to give you the best experience for reading
the detailed lecture materials, taking notes, and carrying the course with
you wherever you go. For the full course experience, you need to use
both online and paper resources.
 Best wishes for your success in this course!MATH 112: Calculus 1
4
Table of Contents
Chapter 1: Getting Started and Reviewing Prerequisites.............8
Lesson 1.1: Getting Off to a Good Start........................................10
Tour the Text...................................................................................11
How to Read a Mathematics Text...................................................11
Lesson 1.2: Linear Functions.........................................................15
Pre-test Instructions....................................................................... 18
Chapter 2: Limits and Derivatives .................................................19
Lesson 2.1: Tangent and Velocity..................................................22
Discussion Material....................................................................... 23
Lesson 2.2: The Limit of a Function..............................................26
Discussion Material....................................................................... 27
Lesson 2.3: Calculating Limits Using the Limit Laws..................29
Discussion Material....................................................................... 31
Lesson 2.4: The Precise Definition of the Limit...........................32
Discussion Material....................................................................... 34
Lesson 2.5: Continuity....................................................................37
Discussion Material....................................................................... 39
Practice .......................................................................................... 40
Lesson 2.6: Limits at Infinity..........................................................41
Discussion Material....................................................................... 42
Practice .......................................................................................... 45
Lesson 2.7: Derivatives and Rates of Change..............................46
Discussion Material....................................................................... 47
Practice .......................................................................................... 485
Table of Contents
Lesson 2.8: The Derivative as a Function.....................................49
Discussion Material....................................................................... 50
Midcourse Exam 1 Preparation.....................................................52
Chapter 3: Differentiation Rules ...................................................53
Lesson 3.1: Derivatives of Polynomials and Exponential Functions.................................................................................................54
Discussion Material....................................................................... 56
Practice .......................................................................................... 57
Lesson 3.2: The Product and Quotient Rules ...............................58
Discussion Material....................................................................... 59
Lesson 3.3: Derivatives of Trigonometric Functions....................61
Discussion Material....................................................................... 62
Lesson 3.4: The Chain Rule............................................................65
Discussion Material....................................................................... 66
Practice .......................................................................................... 67
Lesson 3.5: Implicit Differentiation ..............................................68
Discussion Material....................................................................... 69
Lesson 3.6: Derivatives of Logarithmic Functions.......................72
Discussion Material....................................................................... 73
Lesson 3.7: Rates of Change in the Natural and Social Sciences..
76
Discussion Material....................................................................... 78
Practice .......................................................................................... 78
Lesson 3.8: Exponential Growth and Decay.................................79
Discussion Material....................................................................... 80
Practice .......................................................................................... 80MATH 112: Calculus 1
6
Lesson 3.9: Related Rates.............................................................81
Discussion Material....................................................................... 82
Lesson 3.10: Linear Approximations and Differentials...............84
Discussion Material....................................................................... 85
Practice .......................................................................................... 86
Lesson 3.11: Hyperbolic Functions...............................................87
Discussion Material....................................................................... 88
Practice .......................................................................................... 88
Midcourse Exam 2 Preparation.....................................................89
Chapter 4: Application of Differentiation .....................................90
Lesson 4.1: Maximum and Minimum Values................................91
Discussion Material....................................................................... 93
Practice .......................................................................................... 94
Lesson 4.2: The Mean Value Theorem..........................................95
Discussion Material....................................................................... 97
Lesson 4.3: How Derivatives Affect the Shape of a Graph..........98
Discussion Material....................................................................... 99
Practice ........................................................................................ 101
Lesson 4.4: Indeterminate Forms and L’Hospital’s Rule..........102
Discussion Material..................................................................... 103
Lesson 4.5: Summary of Curve Sketching................................. 107
Discussion Material..................................................................... 108
Lesson 4.7: Optimization Problems............................................110
Discussion Material......................................................................111
Lesson 4.9: Antiderivatives.........................................................114
Discussion Material......................................................................1157
Table of Contents
Midcourse Exam 3 Preparation.................................................. 117
Chapter 5: Integrals.....................................................................118
Lesson 5.1: Areas and Distances...............................................119
Discussion Material..................................................................... 120
Lesson 5.2: The Definite Integral ...............................................124
Discussion Material..................................................................... 125
Lesson 5.3: The Fundamental Theorem of Calculus.................129
Discussion Material..................................................................... 130
Lesson 5.4: Indefinite Integrals and the Net Change Theorem135
Discussion Material..................................................................... 136
Lesson 5.5: The Substitution Rule..............................................138
Discussion Material..................................................................... 139
Homework Transfer Request Instructions................................... 143
Final Exam Preparation...............................................................1448
Chapter 1
Getting Started and Reviewing
Prerequisites
A
lgebra. CalCulus teaChers will tell you that it is usually not
the calculus that students struggle with—its the algebra! That is
why your textbook and this course begin with a review of algebra. The
good news is, when you have mastered algebra, adding the calculus skills
will be much easier.
Famous mathematician story: Norbert Wienerwas perhaps the
greatest American mathematician in the first half of the twentieth
century, revered among his colleagues for his brilliance. He was
also famous for his absentmindedness.
After a few years at MIT, Norbert Wiener moved to a larger house.
His wife, knowing his nature, figured that he would forget his new
address and be unable to find his way home after work. So she wrote
the address of the new home on a piece of paper which she made
him put in his shirt pocket. At lunchtime that day, the professor
had an inspiring idea. He pulled the paper out of his pocket and
used it to scribble down some calculations. Finding a flaw, he threw
the paper away in disgust. At the end of the day he realized he
had thrown away his address. He now had no idea where he lived.9
Unit 1: Getting Started and Reviewing Prerequisites
Putting his mind to work, he came up with a plan. He would go to
his old house and await rescue. His wife would surely realize that
he was lost and go to his old house to pick him up. Unfortunately,
when he arrived at his old house there was no sign of his wife,
only a small girl standing in front of the house. “Excuse me little
girl,“ he said, “but do you happen to know where the people who
used to live here moved to?“ “Its okay, daddy,“ said the little girl.
“Mommy sent me to get you.”
P.S. Norbert Wiener’s daughter was recently tracked down by a
mathematics newsletter. She denies he forgot who she was, but
admits he forgot where the house was.
—Colin Adams, Joel Hass, Abigail Thompson, How to Ace Calculus: The
Streetwise Guide, [New York: W.H. Freeman and Company, 1998], p. 8.
The point of the story? We all have strengths and shortcomings. If
you ever find that you have “gotten lost“ in this course, don’t get stuck,
get help. It can come from any source you choose, but remember that
the TAs assigned to this course are prepared and available to answer
your questions. Also, there are several excellent Web sites that
you may use. Rather than recommending any particular one,
we suggest that you type “calculus help“ in the search engine
of your choice; it will return many good sources. You will also
find helpful classroom-style lectures on various video sites.