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CALCULUS   I

Paul Dawkins

Calculus I

Preface ........................................................................................................................................... iii
Outline ........................................................................................................................................... iv
Review............................................................................................................................................. 2
Introduction .............................................................................................................................................. 2
Review : Functions ................................................................................................................................... 4
Review : Inverse Functions .................................................................................................................... 14
Review : Trig Functions ......................................................................................................................... 21
Review : Solving Trig Equations ............................................................................................................ 28
Review : Solving Trig Equations with Calculators, Part I .................................................................... 37
Review : Solving Trig Equations with Calculators, Part II ................................................................... 48
Review : Exponential Functions ............................................................................................................ 53
Review : Logarithm Functions ............................................................................................................... 56
Review : Exponential and Logarithm Equations .................................................................................. 62
Review : Common Graphs ...................................................................................................................... 68
Limits ............................................................................................................................................ 80
Introduction ............................................................................................................................................ 80
Rates of Change and Tangent Lines ...................................................................................................... 82
The Limit ................................................................................................................................................. 91
One‐Sided Limits ...................................................................................................................................101
Limit Properties .....................................................................................................................................107
Computing Limits ..................................................................................................................................113
Infinite Limits ........................................................................................................................................121
Limits At Infinity, Part I .........................................................................................................................130
Limits At Infinity, Part II .......................................................................................................................139
Continuity ...............................................................................................................................................148
The Definition of the Limit ....................................................................................................................155
Derivatives .................................................................................................................................. 170
Introduction ...........................................................................................................................................170
The Definition of the Derivative ...........................................................................................................172
Interpretations of the Derivative .........................................................................................................178
Differentiation Formulas ......................................................................................................................183
Product and Quotient Rule ...................................................................................................................191
Derivatives of Trig Functions ...............................................................................................................197
Derivatives of Exponential and Logarithm Functions ........................................................................208
Derivatives of Inverse Trig Functions ..................................................................................................213
Derivatives of Hyperbolic Functions ....................................................................................................219
Chain Rule ..............................................................................................................................................221
Implicit Differentiation .........................................................................................................................231
Related Rates .........................................................................................................................................240
Higher Order Derivatives ......................................................................................................................254
Logarithmic Differentiation ..................................................................................................................259
Applications of Derivatives ....................................................................................................... 262
Introduction ...........................................................................................................................................262
Rates of Change......................................................................................................................................264
Critical Points .........................................................................................................................................267
Minimum and Maximum Values ...........................................................................................................273
Finding Absolute Extrema ....................................................................................................................281
The Shape of a Graph, Part I ..................................................................................................................287
The Shape of a Graph, Part II ................................................................................................................296
The Mean Value Theorem .....................................................................................................................305
Optimization ..........................................................................................................................................312
More Optimization Problems ...............................................................................................................326
i
http://tutorial.math.lamar.edu/terms.aspx

Calculus I
Indeterminate Forms and L’Hospital’s Rule ........................................................................................341
Linear Approximations .........................................................................................................................347
Differentials ...........................................................................................................................................350
Newton’s Method ...................................................................................................................................353
Integrals ...................................................................................................................................... 364
Introduction ...........................................................................................................................................364
Indefinite Integrals ................................................................................................................................365
Computing Indefinite Integrals ............................................................................................................371
Substitution Rule for Indefinite Integrals ............................................................................................381
More Substitution Rule .........................................................................................................................394
Area Problem .........................................................................................................................................407
The Definition of the Definite Integral .................................................................................................417
Computing Definite Integrals ...............................................................................................................427
Substitution Rule for Definite Integrals ...............................................................................................439
Applications of Integrals ........................................................................................................... 450
Introduction ...........................................................................................................................................450
Average Function Value ........................................................................................................................451
Area Between Curves ............................................................................................................................454
Volumes of Solids of Revolution / Method of Rings ............................................................................465
Volumes of Solids of Revolution / Method of Cylinders .....................................................................475
More Volume Problems .........................................................................................................................483
Work .......................................................................................................................................................494
Extras .......................................................................................................................................... 498
Introduction ...........................................................................................................................................498
Proof of Various Limit Properties ........................................................................................................499
Proof of Various Derivative Facts/Formulas/Properties ...................................................................510
Proof of Trig Limits ...............................................................................................................................523
Proofs of Derivative Applications Facts/Formulas .............................................................................528
Proof of Various Integral Facts/Formulas/Properties .......................................................................539
Area and Volume Formulas ..................................................................................................................551
Types of Infinity .....................................................................................................................................555
Summation Notation .............................................................................................................................559
Constants of Integration .......................................................................................................................561

ii
http://tutorial.math.lamar.edu/terms.aspx

Calculus I

Preface

Here are my online notes for my Calculus I course that I teach here at Lamar University. Despite
the fact that these are my “class notes” they should be accessible to anyone wanting to learn
Calculus I or needing a refresher in some of the early topics in calculus.

I’ve tried to make these notes as self contained as possible and so all the information needed to
read through them is either from an Algebra or Trig class or contained in other sections of the
notes.

Here are a couple of warnings to my students who may be here to get a copy of what happened on
a day that you missed.

1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn
calculus I have included some material that I do not usually have time to cover in class
and because this changes from semester to semester it is not noted here. You will need to
find one of your fellow class mates to see if there is something in these notes that wasn’t
covered in class.

2. Because I want these notes to provide some more examples for you to read through, I
don’t always work the same problems in class as those given in the notes. Likewise, even
if I do work some of the problems in here I may work fewer problems in class than are
presented here.

3. Sometimes questions in class will lead down paths that are not covered here. I try to
anticipate as many of the questions as possible when writing these up, but the reality is
that I can’t anticipate all the questions. Sometimes a very good question gets asked in
class that leads to insights that I’ve not included here. You should always talk to
someone who was in class on the day you missed and compare these notes to their notes
and see what the differences are.

4. This is somewhat related to the previous three items, but is important enough to merit its
own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!
Using these notes as a substitute for class is liable to get you in trouble. As already noted
not everything in these notes is covered in class and often material or insights not in these
notes is covered in class.
iii
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Calculus I

Outline

Here is a listing and brief description of the material in this set of notes.

Review
Review : Functions – Here is a quick review of functions, function notation and
a couple of fairly important ideas about functions.
Review : Inverse Functions – A quick review of inverse functions and the
notation for inverse functions.
Review : Trig Functions – A review of trig functions, evaluation of trig
functions and the unit circle. This section usually gets a quick review in my
class.
Review : Solving Trig Equations – A reminder on how to solve trig equations.
This section is always covered in my class.
Review : Solving Trig Equations with Calculators, Part I – The previous
section worked problem whose answers were always the “standard” angles. In
this section we work some problems whose answers are not “standard” and so a
calculator is needed. This section is always covered in my class as most trig
equations in the remainder will need a calculator.
Review : Solving Trig Equations with Calculators, Part II – Even more trig
equations requiring a calculator to solve.
Review : Exponential Functions – A review of exponential functions. This
section usually gets a quick review in my class.
Review : Logarithm Functions – A review of logarithm functions and
logarithm properties. This section usually gets a quick review in my class.
Review : Exponential and Logarithm Equations – How to solve exponential
and logarithm equations. This section is always covered in my class.
Review : Common Graphs – This section isn’t much. It’s mostly a collection
of graphs of many of the common functions that are liable to be seen in a
Calculus class.

Limits
Tangent Lines and Rates of Change – In this section we will take a look at two
problems that we will see time and again in this course. These problems will be
used to introduce the topic of limits.
The Limit – Here we will take a conceptual look at limits and try to get a grasp
on just what they are and what they can tell us.
One-Sided Limits – A brief introduction to one-sided limits.
Limit Properties – Properties of limits that we’ll need to use in computing
limits. We will also compute some basic limits in this section
iv
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Calculus I
Computing Limits – Many of the limits we’ll be asked to compute will not be
“simple” limits. In other words, we won’t be able to just apply the properties and
be done. In this section we will look at several types of limits that require some
work before we can use the limit properties to compute them.

Infinite Limits – Here we will take a look at limits that have a value of infinity
or negative infinity. We’ll also take a brief look at vertical asymptotes.
Limits At Infinity, Part I – In this section we’ll look at limits at infinity. In
other words, limits in which the variable gets very large in either the positive or
negative sense. We’ll also take a brief look at horizontal asymptotes in this
section. We’ll be concentrating on polynomials and rational expression
involving polynomials in this section.
Limits At Infinity, Part II – We’ll continue to look at limits at infinity in this
section, but this time we’ll be looking at exponential, logarithms and inverse
tangents.
Continuity – In this section we will introduce the concept of continuity and how
it relates to limits. We will also see the Mean Value Theorem in this section.
The Definition of the Limit – We will give the exact definition of several of the
limits covered in this section. We’ll also give the exact definition of continuity.

Derivatives
The Definition of the Derivative – In this section we will be looking at the
definition of the derivative.
Interpretation of the Derivative – Here we will take a quick look at some
interpretations of the derivative.
Differentiation Formulas – Here we will start introducing some of the
differentiation formulas used in a calculus course.
Product and Quotient Rule – In this section we will took at differentiating
products and quotients of functions.
Derivatives of Trig Functions – We’ll give the derivatives of the trig functions
in this section.
Derivatives of Exponential and Logarithm Functions – In this section we will
get the derivatives of the exponential and logarithm functions.
Derivatives of Inverse Trig Functions – Here we will look at the derivatives of
inverse trig functions.
Derivatives of Hyperbolic Functions – Here we will look at the derivatives of
hyperbolic functions.
Chain Rule – The Chain Rule is one of the more important differentiation rules
and will allow us to differentiate a wider variety of functions. In this section we
will take a look at it.
Implicit Differentiation – In this section we will be looking at implicit
differentiation. Without this we won’t be able to work some of the applications
of derivatives.
v
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Calculus I
Related Rates – In this section we will look at the lone application to derivatives
in this chapter. This topic is here rather than the next chapter because it will help
to cement in our minds one of the more important concepts about derivatives and
because it requires implicit differentiation.

Higher Order Derivatives – Here we will introduce the idea of higher order
derivatives.
Logarithmic Differentiation – The topic of logarithmic differentiation is not
always presented in a standard calculus course. It is presented here for those how
are interested in seeing how it is done and the types of functions on which it can
be used.

Applications of Derivatives
Rates of Change – The point of this section is to remind us of the
application/interpretation of derivatives that we were dealing with in the previous
chapter. Namely, rates of change.
Critical Points – In this section we will define critical points. Critical points
will show up in many of the sections in this chapter so it will be important to
understand them.
Minimum and Maximum Values – In this section we will take a look at some
of the basic definitions and facts involving minimum and maximum values of
functions.
Finding Absolute Extrema – Here is the first application of derivatives that
we’ll look at in this chapter. We will be determining the largest and smallest
value of a function on an interval.
The Shape of a Graph, Part I – We will start looking at the information that the
first derivatives can tell us about the graph of a function. We will be looking at
increasing/decreasing functions as well as the First Derivative Test.
The Shape of a Graph, Part II – In this section we will look at the information
about the graph of a function that the second derivatives can tell us. We will
look at inflection points, concavity, and the Second Derivative Test.
The Mean Value Theorem – Here we will take a look that the Mean Value
Theorem.
Optimization Problems – This is the second major application of derivatives in
this chapter. In this section we will look at optimizing a function, possible
subject to some constraint.
More Optimization Problems – Here are even more optimization problems.
L’Hospital’s Rule and Indeterminate Forms – This isn’t the first time that
we’ve looked at indeterminate forms. In this section we will take a look at
L’Hospital’s Rule. This rule will allow us to compute some limits that we
couldn’t do until this section.
Linear Approximations – Here we will use derivatives to compute a linear
approximation to a function. As we will see however, we’ve actually already
done this.
vi
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Calculus I
Differentials – We will look at differentials in this section as well as an
application for them.
Newton’s Method – With this application of derivatives we’ll see how to
approximate solutions to an equation.
Business Applications – Here we will take a quick look at some applications of

Integrals
Indefinite Integrals – In this section we will start with the definition of
indefinite integral. This section will be devoted mostly to the definition and
properties of indefinite integrals and we won’t be working many examples in this
section.
Computing Indefinite Integrals – In this section we will compute some
indefinite integrals and take a look at a quick application of indefinite integrals.
Substitution Rule for Indefinite Integrals – Here we will look at the
Substitution Rule as it applies to indefinite integrals. Many of the integrals that
we’ll be doing later on in the course and in later courses will require use of the
substitution rule.
More Substitution Rule – Even more substitution rule problems.
Area Problem – In this section we start off with the motivation for definite
integrals and give one of the interpretations of definite integrals.
Definition of the Definite Integral – We will formally define the definite
integral in this section and give many of its properties. We will also take a look
at the first part of the Fundamental Theorem of Calculus.
Computing Definite Integrals – We will take a look at the second part of the
Fundamental Theorem of Calculus in this section and start to compute definite
integrals.
Substitution Rule for Definite Integrals – In this section we will revisit the
substitution rule as it applies to definite integrals.

Applications of Integrals
Average Function Value – We can use integrals to determine the average value
of a function.
Area Between Two Curves – In this section we’ll take a look at determining the
area between two curves.
Volumes of Solids of Revolution / Method of Rings – This is the first of two
sections devoted to find the volume of a solid of revolution. In this section we
look that the method of rings/disks.
Volumes of Solids of Revolution / Method of Cylinders – This is the second
section devoted to finding the volume of a solid of revolution. Here we will look
at the method of cylinders.
More Volume Problems – In this section we’ll take a look at find the volume of
some solids that are either not solids of revolutions or are not easy to do as a
solid of revolution.
vii
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Calculus I
Work – The final application we will look at is determining the amount of work
required to move an object.

Extras
Proof of Various Limit Properties – In we prove several of the limit properties
and facts that were given in various sections of the Limits chapter.
Proof of Various Derivative Facts/Formulas/Properties – In this section we
give the proof for several of the rules/formulas/properties of derivatives that we
saw in Derivatives Chapter. Included are multiple proofs of the Power Rule,
Product Rule, Quotient Rule and Chain Rule.
Proof of Trig Limits – Here we give proofs for the two limits that are needed to
find the derivative of the sine and cosine functions.
Proofs of Derivative Applications Facts/Formulas – We’ll give proofs of many
of the facts that we saw in the Applications of Derivatives chapter.
Proof of Various Integral Facts/Formulas/Properties – Here we will give the
proofs of some of the facts and formulas from the Integral Chapter as well as a
couple from the Applications of Integrals chapter.
Area and Volume Formulas – Here is the derivation of the formulas for finding
area between two curves and finding the volume of a solid of revolution.
Types of Infinity – This is a discussion on the types of infinity and how these
affect certain limits.
Summation Notation – Here is a quick review of summation notation.
Constant of Integration – This is a discussion on a couple of subtleties
involving constants of integration that many students don’t think about.
viii
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Calculus I
1
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