Calculus I (Math 2413)
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.
- 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.
- 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.
- 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.
- 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.
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 problems 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
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 Intermediate 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.
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 who 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 at 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, possibly 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.
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 derivatives to the business field.
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 at 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.
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 this section 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.