MATHS NOTES

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. 

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.

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

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.

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.

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 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.

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.