# Calculus I

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About *Calculus I:*

Here is a listing of all the material that is available online.

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 to one degree or another.

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.

Limits Involving Infinity Here we will take a look at limits that involve infinity. This includes limits that are infinity and limits at infinity. We’ll also take a brief look at asymptotes.

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 a limit in this section. This section is not always covered in a standard calculus class.

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 function.

Derivatives of Inverse Trig Functions Here we will look at the derivatives of inverse trig functions.

Derivatives of Hyperbolic Trig Functions Here we will look at the derivatives of hyperbolic trig 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 on 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

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 that a function will be 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.

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 This will be the last application of derivatives that we’ll be covering. In this section we’ll see how to approximate solutions to an equation.

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.

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. The substitution rule problems were split into two sections for presentation on the web. This kept the page from getting too large.

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.

Work The final application we will look at is determining the amount of work required to move an object.

Extras

Proof of Product Rule The proof of the product rule for derivatives.

Proof of Quotient Rule The proof of the quotient rule for derivatives.

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.

Area and Volume Formulas Here are the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.

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 to one degree or another.

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.

Limits Involving Infinity Here we will take a look at limits that involve infinity. This includes limits that are infinity and limits at infinity. We’ll also take a brief look at asymptotes.

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 a limit in this section. This section is not always covered in a standard calculus class.

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 function.

Derivatives of Inverse Trig Functions Here we will look at the derivatives of inverse trig functions.

Derivatives of Hyperbolic Trig Functions Here we will look at the derivatives of hyperbolic trig 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 on 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

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 that a function will be 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.

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 This will be the last application of derivatives that we’ll be covering. In this section we’ll see how to approximate solutions to an equation.

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.

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. The substitution rule problems were split into two sections for presentation on the web. This kept the page from getting too large.

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.

Work The final application we will look at is determining the amount of work required to move an object.

Extras

Proof of Product Rule The proof of the product rule for derivatives.

Proof of Quotient Rule The proof of the quotient rule for derivatives.

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.

Area and Volume Formulas Here are the derivation of the formulas for finding area between two curves and finding the volume of a solid of revolution.