Calculus In
5 Hours

Concepts Revealed so You Don't Have to Sit Through a Semester of Lectures

Dennis Jarecke

Calculus In 5 Hours: Concepts Revealed so You Don't Have to Sit Through a Semester of Lectures
by Dennis Jarecke

Published by Six Sycamores, LLC
14949 Lakewood Heights Boulevard, Lakewood, Ohio 44107

www.CalculusSolution.com

© 2017 Six Sycamores, LLC

All rights reserved. No portion of this book may be reproduced in any form without permission from the publisher, except as permitted by U.S. copyright law. For permissions contact:

Copyright@SixSycamores.com

ISBN: 978-0-9995254-0-1

Preface

Calculus is a vast and detailed subject that can take a lifetime to learn. So to teach Calculus in five hours I need to cut out a lot of material.

This book covers approximately 75% of the material taught in a college Calculus 1 course. Most of what's been cut out are function limits and trigonometric functions.

So why should you read this book?

Because you can learn the basics of Calculus quickly. Learning comes down to repetition. The more you're exposed to Calculus and the more familiar it becomes, the more likely you are to understand it.

By learning Calculus in five hours, you cram almost a semester's worth of material in your head quickly. When you cover it in your actual course, you'll be exposed to it a second time which allows you to learn it easier and faster at that point.

A second reason for reading this book is for a review. You may have learned Calculus years ago, but have forgotten it. Now you need or want to relearn it.

If this sounds like you, then this book will let you accomplish that. I don't assume you know Calculus. Instead, I take you from knowing nothing to having a broad overview of the main tools and concepts that are typically learned over the course of a college semester. And given your previous knowledge of Calculus, this will facilitate a quick relearning of the subject.

This book is organized into five parts and 32 chapters. The amount of time to go through each chapter is given in the table of contents and at the beginning of each part and chapter.

If this is your first time learning Calculus, go through the book sequentially to make sure you understand the concepts. If you're learning Calculus for the second time, feel free to skip around to get what you need.

And then when you're ready, head over to www.CalculusSolution.com for more lessons, Calculus specific calculators, and a library of fully worked out problems so you can get ahead with less effort.

For my wife and children,
Susan, Sara, Emily, Kiera, and Michael,
who put up with me sitting in the basement doing Calculus stuff.
I love you all very much.

To Dave Van Horn,
who understands the starts, stops, and frustrations of getting something done. Thanks for listening, answering questions, and being a big help to writers and entrepreneurs in the Cleveland area.

Special thanks to all those teachers and professors who had a hand in teaching me Calculus. I hope you know there are students who think what you do is really cool. I was one of them.
Thanks for making Calculus come alive.

Table of Contents

Title Page

Copyright

Preface

Acknowledgments

Contents

Your Biggest Obstacle to Learning Calculus

Part 1 - Functions and an Overview of Calculus (27 minutes)

Chapter 1 - Functions (10 minutes)

Chapter 2 - Straight Lines and Polynomials (7 minutes)

Chapter 3 - Combining Functions (7 minutes)

Chapter 4 - The Two Basic Concepts of Calculus (3 minutes)

Part 2 - Derivatives (127 minutes)

Chapter 5 - Understanding Derivatives (25 minutes)

Chapter 6 - Your First Four Derivatives (12 minutes)

Chapter 7 - The Tangent Line (3 minutes)

Chapter 8 - The Derivative of a Polynomial (16 minutes)

Chapter 9 - The Product Rule (12 minutes)

Chapter 10 - The Quotient Rule (13 minutes)

Chapter 11 - The Chain Rule (16 minutes)

Chapter 12 - Attaching Real World Meaning to Variables and Functions (7 minutes)

Chapter 13 - Attaching Real World Meaning to Derivatives (13 minutes)

Chapter 14 - Second Derivatives (10 minutes)

Part 3 - The Geometry of Functions (67 minutes)

Chapter 15 - The Mean-Value Theorem (7 minutes)

Chapter 16 - Increasing and Decreasing Functions (2 minutes)

Chapter 17 - Using Derivatives to Find Where a Function Increases and Decreases (15 minutes)

Chapter 18 - Critical Points (6 minutes)

Chapter 19 - Maxima and Minima (1 minute)

Chapter 20 - The Relationship Between Maxima, Minima, and Critical Points (5 minutes)

Chapter 21 - Using Derivatives to Find Maxima and Minima (12 minutes)

Chapter 22 - Concavity (5 minutes)

Chapter 23 - Using Second Derivatives to Determine Concavity (6 minutes)

Chapter 24 - Inflection Points (8 minutes)

Part 4 - Integrals (67 minutes)

Chapter 25 - Understanding Integrals (11 minutes)

Chapter 26 - Antiderivatives (20 minutes)

Chapter 27 - The Fundamental Theorem of Integral Calculus (17 minutes)

Chapter 28 - Rules of Integrals (7 minutes)

Chapter 29 - Negative Integrals (5 minutes)

Chapter 30 - Using Integrals to Find Average Values (7 minutes)

Part 5 - Exponential and Logarithm Functions (12 minutes)

Chapter 31 - The Exponential and Logarithm Functions (7 minutes)

Chapter 32 - The Calculus of Exponential and Logarithm Functions (5 minutes)

Glossary

List of Figures

Your Biggest Obstacle to Learning Calculus

The obvious goal of this book is to teach you Calculus and to teach it to you quickly. However, there's one problem I can't overcome in this book.

Most people have been brainwashed into thinking they're bad at math. This brainwashing comes from experiences in school, upbringing, peer pressure, and a popular culture that often portrays smart people as anti-social and weird.

You may feel you can't learn something as complex as Calculus, but the scientific research doesn't support this.

You were born with an ability to learn, discover, and become an expert at mathematics. Learning a language is harder than learning Calculus. The only difference is that you don't remember how difficult it was to learn to speak, read, and write English. Yet, you accomplished it, didn't you?

The reason you were able to do something so complex at a young age is because you were born with a massively powerful supercomputer in your head. This supercomputer is deep inside your brain, and it specializes in detecting and amplifying patterns, performing mathematical operations, learning new concepts, and coming up with ideas from thin air.

But surrounding your supercomputer is a filter that limits what goes in and what comes out. That filter is your self-image.

If your self-image is programmed to believe you're not good at math, then you'll never demonstrate a proficiency in mathematics. However, if your self-image is changed so you know you're good at mathematics, then you'll be successful at learning it.

This isn't a bunch of cheerleading psycho-babble. A person who believes they're bad at math can be hypnotized and convinced they're good at it. In a state of hypnosis they can accurately work out math problems they've been exposed to, but consciously thought they couldn't solve.

Specifically, math avoidant college students have been exposed to hypnotic techniques and compared to control groups of similar college students. By using well-documented techniques of Ericksonian hypnotherapy, math instructors are able to systematically modify students' beliefs about their ability to do mathematics. Once their beliefs have been changed, those students make more A's, B's, and C's than those students in the control group.

In other words, a person's self-image can be changed to make them believe they're good at math. This allows their supercomputer to accept mathematical instructions, do the calculations, and give the correct answer.

To be clear, hypnosis by itself won't magically give you mathematical abilities. It can only modify your conscious and artificially negative belief that you're bad at math. Mathematical training (at whatever level you desire or need to be good at) plus a positive belief in your ability to do math is what creates success.

In the absence of hypnosis, you can associate positive emotions with learning Calculus. Think of a time you accomplished something you were proud of. Now take that same feeling and imagine getting the highest grade you've ever gotten in a math class.

This will start the process of changing your self-image. And when you do make a higher grade, your self-image will have been reprogrammed to believe you're good at math.

So, learning Calculus is not a matter of "if." It's a matter of "when."

In this book I promise to teach you Calculus as fast as I know how. Your job is to believe the scientific research and experience of people just like you who were able to learn Calculus.

You have a powerful computer in your head, and it will let you learn Calculus if you put positive, successful emotions into your learning process.

Let's begin . . .

Part 1 - Functions and an Overview of Calculus (27 minutes)

In part one there are four chapters. In it I'll introduce functions, what we can do with them, and then introduce the two main types of Calculus: Differential Calculus and Integral Calculus.

Chapter 1: Functions (10 minutes)

Calculus deals with functions, so knowing what they are is important. You've already seen them in High School, but it never hurts to go over them again.

A function is a way of assigning one set of numbers to another set of numbers.

For example, let's say you're at home and start walking through town. As you walk you keep track of how far you are from your house, and you measure the height of the buildings as you go. (Let's say you measure the height of the buildings on your right-hand side.) At every distance from your home, there is a height of a building. This is a function, and it may look something like this:

If you want to describe your function like a mathematician would, label the horizontal axis the "x-axis," the vertical axis the "y-axis," and the function as "f(x)." A typical function looks something like:

In Calculus you describe functions in a way that is precise and allows you to do more advanced things with them. A rigorous definition of a function has four components:

1) A rule relating the x-axis with the y-axis.

Typically this is an algebraic, trigonometric or other relationship such as formula, formula, formula, or formula.

When writing down a function, you denote it as formula or sometimes by formula. For example, you would denote the function formula as formula or as formula.

2) For every value of formula there is only one value of formula.

In other words, you can't have two buildings on the same spot. This means formula is a function, but formula is not. Why?

Let's find the value of formula and formula at formula. For the function formula, formula at formula. However, for formula, formula can be either +2 or -2 at formula. At formula there are two possible values of formula for the function formula, but only one value for formula for the function formula. In other words, formula is ambiguous between two possible numbers, and formula is not. That's why formula is a function and formula is not.

Pictorially, this is a function:

And this is not a function:

If we wrote formula or formula where formula denotes the absolute value, then these would be functions because formula has only one value for every value of formula. This fact is so well known that you rarely see formula. You usually see formula which should always be taken to mean formula. Similarly, formula means formula.

3) The domain of the function is the portion of the x-axis for which the function is defined.

When I say the word "defined," I mean that formula is a known number. If I know that formula, then the function is defined at formula. If I don't know the value of formula at formula, then the function is undefined there.

Whenever the domain is not specified, you assume the domain is all real numbers for which formula is a real number. For example, for formula the domain is all real numbers.

However, for formula the domain is all positive numbers because the square root of a negative number is not a real number. It's an imaginary number.

When you need to explicitly define the domain of a function, you do it with a bracket that dictates what rule relating the formula and formula axes is used for which part of the domain. For example,

denotes a function whose domain is not made up of all real numbers. The domain is formula where formula is the union of the three intervals formula, formula, and formula

Oh yeah, an interval is a way of saying all numbers between two points on the x-axis. So formula is a fancy math way of saying all the numbers between 5 and 10. The difference between parentheses and brackets is that for parentheses the point is not included, but for brackets it is. So -3 is not included in the interval formula, but all points less than it are. The numbers 5 and 10 are included in the interval formula And you can have mixed intervals like formula which includes the number 20, but not the number 3.

4) The range of the function is the portion of the y-axis for which the function is defined.

The range is determined by knowing the domain and the rule relating the x-axis with the y-axis.

Here are five examples of the domain, range, and graph of a function:

Function formula

Domain (x-axis): All real numbers. In math notation you write: formula

Range (y-axis): All positive real numbers. In math notation you write: formula

Function formula

Domain (x-axis): All real numbers. In math notation you write: formula

Range (y-axis): All real numbers. In math notation you write: formula

Function formula

Domain (x-axis): All positive real numbers. In math notation you write: formula

Range (y-axis): All negative real numbers. In math notation you write: formula

Function formula

Domain (x-axis): All real numbers. In math notation you write: formula

Range (y-axis): All real numbers between -1 and 1. In math notation you write: formula

Function formula

Domain (x-axis): All real numbers between minus infinity and negative 3 (but not including negative three), AND all real numbers between positive 2 and positive infinity. In math notation you write: formula

Range (y-axis): All real numbers between negative infinity and negative 27, AND all real numbers between positive 4 and positive infinity. In math notation you write: formula

Now that you know what a function is and have seen some examples, let's talk about two important classes of functions: straight lines and polynomials.

Chapter 2: Straight Lines and Polynomials (7 minutes)

To understand Calculus you must first understand two types of important functions: straight lines and polynomials.

As you'll see in a minute, a straight line is a polynomial, but it's important to understand a straight line first because it's the basis for understanding Differential Calculus.

You've seen straight lines since the day you were born, but to understand them fully you must know the two fundamental characteristics of every straight line: slope and y-intercept.

The formula for a straight line is typically denoted by

where formula is the slope and formula is the y-intercept.

The slope (formula) is a measure of how "steep" the line is, and the y-intercept is where the line hits the y-axis. (By the way, you should memorize the last sentence.) Here's a picture of the slope and y-intercept for you:

Slope is a very important concept. It's defined as how much the straight line "rises" divided by the amount it "runs." So the larger the slope the "steeper" the line is.

When the slope is a positive number, it goes up ("rises") as the line "runs" from left to right. When the slope is negative, it goes down ("falls") as the line "runs" from left to right.

A broader class of functions is that of polynomials. A polynomial is a function of the form

where formula is a positive integer. The variables formula, formula, formula, formula are numbers.

Here are four examples of polynomials:

The straight line, formula.

And here are some functions that are not polynomials:

formula (The natural logarithm discussed in Chapter 31.)

formula (The exponential function discussed in Chapter 31.)

Trigonometric functions like formula and formula.

There are several terms associated with polynomials that are important to know:

The formula and formula are called coefficients.

formula is called the leading coefficient.

The degree of the polynomial is the highest integer for which formula is not zero.

For the four polynomial examples above, let's find the degree and leading coefficient of each:

Function | Leading Coefficient | Degree
formula | formula | 1
formula | 4 | 2
formula | 200 | 7
formula | 10 | 2

(If you have trouble viewing the table above, turn your device sideways.)

For the last function,

Only by expanding it can you determine the leading coefficient and degree.

Exercise: Go to www.CalculusSolution.com/Calculus-Calculator and graph the polynomials above.

In the next chapter I'll show you five different ways you can combine two functions. Pay close attention to function composition because that's the one you might not be familiar with.

Chapter 3: Combining Functions (7 minutes)

You can combine functions in different ways to create other functions. You can add, subtract, multiply, and divide two functions to create a new function. You did this in High School, but let's do some examples anyway.

Adding two functions: Let's add

and

To do this you add like terms. For example, group the formula terms to get formula. The answer is

Subtracting two functions: Let's subtract

from

Subtract like terms to get

Multiply two functions: Let's multiply

and

together. Okay, this one is time-consuming, but not hard. We get

Divide two functions: Let's be honest, doing this is hard and not useful in the context of learning Calculus in 5 hours. Let's skip it.

We can also combine two functions by function composition. Function composition is where we put one function inside another function. For example, we can put a function formula inside the function formula by doing formula.

Let's do an example.

Let's say that formula and formula. The first step to find formula is to replace formula by formula everywhere in formula like so:

Now, everywhere you see formula replace it with formula like this:

For the last step, do some algebra to clean things up. You get

Exercise: Find formula where formula and formula. (The answer is formula. Go to www.CalculusSolution.com/node/31550 for the solution.)

Finally, there are three more things you need to know:

The function composition formula is typically denoted by mathematicians as formula.

The function composition formula is typically denoted by mathematicians as formula.

formula is not always equal to formula.

To prove the last point, find formula for formula and formula in the exercise above. (You previously found formula, so now find formula.) The answer is formula. (See www.CalculusSolution.com/node/31826 for the solution.)

Therefore, formula for these two particular functions. Occasionally you'll find two functions where formula, but it's not guaranteed to be true in general.

Now that you've spent the last three chapters reviewing functions, it's time to reveal what Calculus is.

Here we go . . .

Chapter 4 - The Two Basic Concepts of Calculus (3 minutes)

Calculus comes in two flavors. The first is Differential Calculus, and the second is Integral Calculus.

Differential Calculus is where you find the slope of a function at a specific point. Integral Calculus is where you find the area underneath a function.

Let's talk about Differential Calculus first.

In High School you learned the slope of a straight line is the line's "rise" divided by its "run."

However, for any other function, the slope can be different depending on where you look. Differential Calculus is about finding all the slopes a function can have.

For example, the picture below shows that the function formula has a slope of -10 at formula, 0 at formula, and 4 at formula. So depending on where you look, the slope will be different.

The way to visualize the slope at a particular point is by drawing a straight line that has the same slope as formula at that point and touches formula there. This is called a tangent line, and it's always used to visually represent the slope of the function at a specific point.

The picture below shows the tangent lines at formula, formula, and formula. (In Chapter 7 I'll show you the exact formula for the tangent line, so don't worry too much about what it is right now.)

When dealing with functions that are not straight lines, it's common to stop calling the slope "slope" and start referring to it as a derivative. Derivatives are denoted by formula or formula. In other words,

The more accurate way to think about it is this. Once you have the derivative, formula, you can find the slope at any point on the function by putting into the derivative the value of formula where you want to find the slope. For example, if the derivative of formula is formula, then you can find the slope at formula by plugging 4 into the derivative to get the slope like so:

Now for Integral Calculus.

In High School geometry class you had to learn the area of common shapes like rectangles, circles, squares, and triangles.

Integral Calculus is about finding the area underneath a function. The area under a function is referred to as an integral, and it's denoted by

where formula and formula are the two points on the x-axis between which you find the area. See the picture below.

Congratulations! You've made it through Part 1. In Part 2 I'm going to go through derivatives and a real-world application.

Onward . . .

Part 2 - Derivatives (127 minutes)

In this part you'll learn about derivatives - what they are, how to get them, and what they're used for.

Chapter 5 - Understanding Derivatives (25 minutes)

To understand what a derivative is, you first need to know what a secant line is. A secant line is pictured below, and it's a straight line that touches the function at (at least) two points.

Because it's a straight line it has a slope:

Suppose you want to know the slope of formula at formula. You can have more and more secant lines that get closer and closer to formula, and then calculate the slope of each one. See the picture below.

Now, imagine one of those secant lines getting extremely close to formula and almost touching formula at one point. This imagined secant line is called a tangent line.

The slope of the tangent line is the slope of the function at that point. We usually stop calling it a slope and start referring to it as a derivative. In other words, a derivative is another word for slope.

Exercise: Calculate the value of the slope for each secant line in the picture above. (Answers: 10, 8, 6, 4, 2. At formula you had to divide by zero, didn't you? But you can't divide by zero so keep reading.)

The use of the Greek "formula" symbol is common in science, and it means "take the difference" or, equivalently, "the change in."

Let's say you have two variables, formula and formula, and took the difference formula. This difference is typically denoted by formula. In other words, formula. In plain English you refer to formula as "delta formula" or "the change in formula."

Because rise is defined as formula (the change in formula) and run is defined as formula (the change in formula), the slope of a secant line is:

Now things start to get complicated.

To find the slope at formula, slide the value of formula as close to formula as you can get without actually touching it. As you do this both the rise (formula) and the run (formula) get smaller and smaller. In other words,

If the rise (formula) approaches zero more quickly than the run, then the slope is zero. If the run (formula) approaches zero more quickly than the rise, then the slope becomes infinitely large. For most functions, both the rise and the run approach zero in such a way as to produce some non-zero and non-infinite slope. In other words, formula is a real number equal to the slope at formula.

How mathematicians do this is with function limits. Function limits are one of the most difficult concepts for beginning Calculus students to grasp. This is because function limits require a knowledge of logic which isn't taught in High School. Fortunately, you can still understand Calculus and make an A+ without being an expert at function limits.

Mathematicians use the formula symbol for denoting the process of calculating a function limit. For example, the act of sliding the value of formula towards formula in the secant slope is denoted by:

When the difference between formula and formula becomes extremely small, mathematicians stop denoting the difference by formula and instead denote it by formula. In other words, formula is a difference and formula is a very small difference.

Similarly, when the difference between formula and formula becomes extremely small, we stop denoting the difference by formula and instead denote it by formula.

This means that

You should recognize from the previous chapter that formula is the derivative. In other words,

Now, let me clarify and explain the notation you'll see in most Calculus textbooks.

First, we typically think of formula as being a distance formula away from formula. Mathematically, we write formula. This means that formula and formula. As formula gets closer and closer to formula, formula gets closer and closer to zero. Therefore,

Second, while I've talked about finding the value of the slope at formula, I didn't specify exactly where formula is on the x-axis. I didn't tell you the value of formula. When we go through the details of finding the function limit,

we usually don't have to specify where formula is at on the x-axis. This is not always true, but the functions you'll deal with in a beginning Calculus class are easy enough that you can find the derivative at formula and not actually specify where it is on the x-axis. Since the value of formula is not specified, everyone drops the subscript 1 and writes,

This formula for a derivative is how you'll see it in almost every introductory textbook on Calculus. It's not a new concept, but only a change in notation. Ideally, I should have initially said something like,

to show that the derivative was being calculated at formula. However, I didn't because I knew I was going to explain it later. And now I have.

It's important to emphasize that I haven't told you how to calculate a function limit. It's beyond the scope of trying to learn Calculus in five hours. Keep in mind the idea of formula sliding toward formula, and you'll be fine.

Now that you know what a derivative is, the next task is to learn some basic rules for finding it when given a specific function. The real math is in doing the function limits, but mathematicians have used function limits to find general rules for calculating derivatives so you don't have to do as much work. That's the focus of the next chapter.

Read on . . .

Chapter 6 - Your First Four Derivatives (12 minutes)

In this chapter I'm going to present the four most important rules of derivatives that you must know and memorize. These rules come from logical results and calculations of function limits. Mathematicians have used function limits to come up with these rules so you can find derivatives for a massive number of functions quickly and easily. Knowing and memorizing these formulas is easy. Let's get started.

Rule 1: The Derivative of a Constant

Let's assume that formula is a constant number like 2, 17, or 3.141592. The derivative of a constant is zero. In other words,

Remember that a derivative is a slope (rise over run). So how much does a constant "rise" when you "run"? It doesn't, and that's why the derivative is zero.

Rule 2: The Derivative of a Constant Times a Function

Let's assume formula is a constant number. If formula has the derivative formula, then the derivative of formula is

And here it is in prime notation:

Rule 3: The Derivative of the Sum of Two Functions

Let's assume formula is the sum of two functions formula and formula. That is, formula. If formula has the derivative formula and formula has the derivative formula, then the derivative of formula is

In other words, just add the derivatives of the individual functions. Sometimes it's helpful to see it in prime notation. Here it is:

You can now combine these in obvious and cool ways.

For example, what if you wanted to know the derivative of formula? Think of formula as formula where formula. This means you need to find the derivative of formula. From Rule 3 you know that

And from Rule 2 you know that

Plugging this back in you have

Finally, plug in formula for formula to get

With Rules 2 and 3 you discovered the derivative of the subtraction of two functions!

Okay, now for the one of the most important rules in Calculus - The Power Rule.

Rule 4: The Power Rule

where formula is an integer. (Remember an integer is a number like 1,3 or -100, but not 1.5 or 2.3.)

For example, the Power Rule means the derivative of formula is formula, the derivative of formula is formula, and the derivative of formula is formula. You have to know and memorize the Power Rule!

What if you want to find the derivative of formula? Think of this as formula where formula and formula. Rule 2 says that

This means that

but from the Power Rule

This means that

The exercise above is the basis for rule 5:

Rule 5: The Derivative of formula

where formula is an integer and formula is a constant. Remember, this comes from rules 2 and 4 so you only need to keep the first four rules in your head!

Next, we'll take a brief aside so you can learn more about the tangent line. We'll get its exact formula and see how it can be used to approximate the value of a function.

Keep reading . . .

Chapter 7 - The Tangent Line (3 minutes)

A Tangent Line is a straight line that touches formula at one point and has the same slope as the function at that point.

Assume we have a function formula and its derivative formula. Let's find the formula for the tangent line at any arbitrary point formula in terms of formula and formula.

Since the tangent line is a straight line, it has the formula formula. To accomplish our goal we need to find the value of formula and formula in terms of formula and formula. By definition, the tangent line's slope is equal to the function's slope at formula. Therefore, formula.

Now let's find formula in terms of formula and formula. Because we've defined the tangent line as touching formula at formula, the tangent line has the value formula at formula. Therefore, at formula we have formula. Now do some algebra to find formula:

Since formula we see that formula.

Plug formula and formula into formula to get the formula for the tangent line:

Now group terms with formula together. This gives you the formula for the tangent line seen in most textbooks:

This expresses the formula for the tangent line by how far formula is from formula. Once formula is known, it's multiplied by the slope of the line (formula) to find out how much higher or lower the tangent line is from formula. The distance of how much higher or lower the line is from formula is given by formula. That's because formula is how much the straight line rises per some amount of run. So we multiply a specific amount of run times the slope to get how much it rises in that amount of run.

Remember that the slope (formula) is always the same at every point on a straight line, and it's the amount a line will rise per the amount it runs. formula is an amount of run, and formula is the amount it rises when it runs for a formula amount. Add formula of rise to formula and you have the value of formula at formula. See the figure below.

The tangent line formula allows you to approximate the value formula knowing only its value at formula and it's derivative at formula. The approximation (denoted by formula instead of =) is:

The further away formula is from formula, the worse formula is at approximating formula. However, that's an advanced topic of Calculus best left for another day.

In the next chapter I'll reveal the formula for the derivative of a polynomial. You'll be surprised to find out you already know it.

Read on . . .

Remember, you can always resize or turn your device sideways if you have trouble seeing a formula or table.

Onward . . .

Chapter 8 - The Derivative of a Polynomial (16 minutes)

One of the easiest formulas to memorize is the derivative of a polynomial. Recall that the form of a polynomial is

Its derivative is:

The Derivative of a Polynomial

If formula is the polynomial

then its derivative is

For every term in the polynomial, multiply the coefficient by the exponent, and then subtracting one from the exponent. In other words, apply the Power Rule to every term in the polynomial.

You don't have to memorize this rule because it comes from the first four rules of derivatives you learned in Chapter 6.

Let's do a quick example by finding the derivative of

Using the rule above we know that

In the calculation we viewed formula as formula because formula. It's derivative is formula. In other words, the Power Rule is consistent with the rule that the derivative of a constant is zero.

Exercise: Determine the derivatives of the following polynomials:

formula at formula

formula at formula

formula at formula

formula at formula

(Answers: 144110, -4.6, -27.6, and 4655. Go to www.CalculusSolution.com/node/31729 for an interactive calculator to help you out.)

Remember, I showed you how to combine two functions by addition, subtraction, multiplication, division and function composition in Chapter 3. In Chapter 6 you learned the derivative of the addition and subtraction of two functions. In the next three chapters I'll give you the rules for finding the derivative of the multiplication, division, and function composition of two functions. These rules will make finding complex derivatives super easy, so keep reading.

Chapter 9 - The Product Rule (12 minutes)

Chapter 3 was about the five ways you can combine two functions. One of those ways was multiplication. The Product Rule is an easy way to find the derivative of the multiplication of two functions.

Let's say you have two functions, formula and formula, and you want to find the derivative of formula. Symbolically, you write

There are two ways to find it:

Multiply formula and formula together, and then find the derivative.

Find the derivative directly with the Product Rule. (You don't have to multiply formula and formula together first!)

The Product Rule is the second method, and it allows you to find the derivative without multiplying formula and formula together like you do in method 1.

The Product Rule

If you already know the derivatives of formula and formula (formula and formula), then the Product Rule states that

We can also write this in prime notation as

Exercise: Find the derivative of the multiplication of the following two functions:

formula and formula (Answer: www.CalculusSolution.com/node/31735)

formula and formula (Answer: www.CalculusSolution.com/node/31736)

formula and formula (Answer: www.CalculusSolution.com/node/31737)

What about the division of two functions? That's the topic of the next chapter . . .

Chapter 10 - The Quotient Rule (13 minutes)

Chapter 3 was about the five ways you can combine two functions. One of those ways was division. The Quotient Rule is an easy way to find the derivative of the division of two functions.

Let's say you have two functions, formula and formula, and you want to find the derivative of formula. Symbolically, you write

There are two ways you can find it:

Divide formula by formula, and then find the derivative. (Good luck with that!)

Find the derivative directly with the Quotient Rule. (You don't have to divide formula by formula first!)

The Quotient Rule

If you already know the derivatives of formula and formula (formula and formula), then the Quotient Rule states that

We can also write this in prime notation as

The Quotient Rule is a rule you have to memorize, so let's do something completely silly to help you memorize it. Let's call the function in the numerator (top function which is formula) "hi" for high, and call the function in the denominator (bottom function which is formula) "ho" for low. Instead of formula, let's print "D" and say "dee". For example, formula would be "D ho". We can now restate the Quotient Rule as

Say "ho dee hi minus hi dee ho over ho ho" out loud a few times, and you'll find you've memorized the Quotient Rule. Just remember that it's "hoDhi" in front of the minus sign, and not "hiDho".

Oh yeah, I think some math teachers use "lo" instead of "ho." That makes sense, of course, but this is how it was originally taught to me, and I can't get it out of my head. Choose lo or ho and stick with it. No matter which one you choose, you'll never forget the Quotient Rule.

Exercise: Find the derivative of the division of the following two functions:

formula and formula (Answer: www.CalculusSolution.com/node/31739)

formula and formula (Answer: www.CalculusSolution.com/node/31741)

formula and formula (Answer: www.CalculusSolution.com/node/31740)

I've shown you how to find the derivative of the addition, subtraction, multiplication, and division of two functions. But what about the derivative of a function composition? Wasn't that the fifth way to combine two functions in Chapter 3? Shouldn't we find that derivative since we've found the other four? Absolutely! That's the topic of the next chapter, and it will make some complex derivatives easy to calculate.

Chapter 11 - The Chain Rule (16 minutes)

Recall, in function composition you obtain formula by placing formula into every value of formula in formula to obtain formula. Then you put the formula for formula into every value of formula in formula.

Now, once you have the formula for formula, use the rules you already know to find its derivative. In prime notation we write the derivative of formula as formula.

For example, if formula and formula, then

Because formula is a polynomial we can determine its derivative. It is

Finding formula was difficult because you had to find the formula for formula first. But what if you could find formula without having to find formula first? Wouldn't that be easier? This is where the Chain Rule comes in handy.

Recall that there are two notations for denoting a derivative: formula and formula. It's helpful to state the Chain Rule using both notations:

The Chain Rule (in formula notation)

This notation is useful because it provides an easy way to keep the rule in your head. Let's call formula. Since formula the Chain Rule is equivalent to finding formula. Remember that anything divided by itself is formula. So formula. Therefore,

But formula because formula. Therefore,

What I just did is not a logical proof. It's a complex but useful way to keep the rule in your head.

In the formula notation the Chain Rule is:

The Chain Rule (in formula notation)

To get the hang of the Chain Rule, it helps to have a step-by-step procedure. Here it is along with the example of finding formula for formula and formula:

Step
Example

Determine formula and formula.

Place formula in for every value of formula in formula. Call this formula.

Place the function formula into every value of formula in formula. Continue to call this formula.

Multiply formula times formula. This is formula.

If needed, do some algebra to simplify or expand the answer.

It worked! Using the Chain Rule gave us the same answer.

In summary, there are two ways to find the derivative of formula:

Find formula and determine its derivative.

Apply the Chain Rule knowing the derivatives of formula and formula.

One of the most important places the Chain Rule is used is in the solution of problems of the form formula. Think of this as a function composition and find the derivative using the Chain Rule.

Let's say that formula. Then

and we know from the Chain Rule that

Try finding the derivative of formula by directly expanding it and then finding its derivative. Expanding it will take you forever, and then you'll still have to calculate the derivative! With the Chain Rule it only takes a second. Problems like this make you appreciate the Chain Rule.

Exercises:

Find the derivative of formula. (Answer www.CalculusSolution.com/node/31743.)

Find formula where formula and formula. (Answer www.CalculusSolution.com/node/31744.)

Find formula where formula and formula. Notice we did formula in the previous problem. (Answer www.CalculusSolution.com/node/31745.)

In two of the exercises you're asked to find the two function compositions formula and formula for formula and formula. Notice that formula. This comes from the general rule that the function compositions formula and formula are not guaranteed to be equal. So their derivatives aren't guaranteed to be equal either.

Okay, now you know what a derivative is and some of the most important rules for finding it. The next step is to connect functions and their derivatives to the real world.

Onward . . .

Chapter 12 - Attaching Real World Meaning to Variables and Functions (7 minutes)

If someone came up to you and said they had "four," what would your reaction be? For most people the response would be, "Four what?" The what is called a unit or dimension, and it lets you know what a number means.

A person could be referring to four inches, four feet, four dollars, or four apples. If someone said they had four oranges, then the unit would be oranges.

The most important system of units is the International System of Units which is typically abbreviated SI. This system is defined by seven fundamental units. Here are four of them:

Physical Quantity | Name | Abbreviation
Mass | Kilograms | formula
Length | Meters | formula
Time | Seconds | formula
Temperature | Kelvins | formula

More importantly, all physical quantities are made up of these fundamental units. For example:

Physical Quantity | Name | Abbreviation
Speed | Meters per Second | formula
Acceleration | Meters per Second-squared | formula
Force | Newtons | formula
Energy | Joules | formula

This means that the units of energy are kilograms times mass-squared divided by seconds-squared. That's a lot to say, so we call the units of energy Joules after James Prescott Joule who studied the nature of heat energy and its relationship to mechanical work.

Outside of communicating what we're talking about, units play a special role in how to describe the real world with mathematics. Ask yourself this simple question, "Is an apple the same as an orange?" No, it's not.

Similarly, for a mathematical equation to describe the real world, the units on the left side of an equal sign must be the same as the units on the right side of the equal sign.

The process of determining if the units on the left side of an equal sign are equal to the units on the right side of an equal sign is called dimensional analysis.

For example, the force of gravity between two objects is given by the equation

where formula is the distance between the two masses formula and formula. formula is the gravitational constant which has a value of formula. Finally, formula is the force in Newtons.

To see if this equation is physically valid, do a dimensional analysis. In a dimensional analysis you don't worry about putting in numbers. Place each quantity's unit in the equation like this:

Now cancel terms according to the rules of algebra. (Notice that the formula on the bottom will cancel with one of the formula's in the top. Also, the two formula's (formula) in the bottom will cancel with two on the top.) Doing this gives:

Both sides of the equal sign have the same unit of Newtons (formula) so it's a valid equation for describing the real world.

While dimensional analysis won't tell you if an equation is correct or not, it's always the first step to see if an equation can describe the real world. If the units on the left-hand side don't equal the units on the right-hand side, it can't be used to model the real world.

When you describe the real world, a variable (like formula) will have a unit, and a function (like formula) will also have unit. The units for formula and formula could be the same or they could be different from each other.

For example, scientists know the average height of a person as a function of their age. We can have the function formula represent the average height of a person when they are at an age of formula. So formula has a unit of feet, and the variable formula has a unit of years (the number of years alive). This function could look something like

The important thing to notice is that the units have been placed into the equation just like any number or variable would be. For example, if the number 3.5 was in an equation and you wanted to put in its unit, you would multiply the unit times 3.5. So if the unit was inches, 3.5 inches means 3.5 times inches where inches is treated like a variable. And once the unit is in the equation (by multiplying it times the number it is referring to), it is treated as a variable with the same algebraic rules applied to it as are applied to any other number or variable in the equation.

Let's say we want to see how tall someone will be at an age of 4 years. Let's go through the steps:

When all the calculations on the right are done, formula must be a number with units of feet. This emphasizes the point made previously that the units on the left side of the equal sign must equal the units on the right side of the equal sign.

Remember, when you add (or subtract) elements in a function they must all have the same unit. For example, if formula, then formula, formula, and formula must all have the same unit.

Now you know about units and how they're used to describe the real world. In the next chapter I'll tell you what the real world meaning of a derivative is.

Turn the page now . . .

Chapter 13 - Attaching Real World Meaning to Derivatives (13 minutes)

In the previous chapter you learned how variables and functions have a unit attached to them.

Now, pretend you drive from Chicago to New York city, and as you drive you record the distance you travel and the number of hours since you left.

This is a function, so let's denote it by formula. The variable formula is the distance measured in miles (miles is the unit of formula), and formula is the number of hours since you left (hours is the unit of formula).

By now you know that slope is the amount a straight line "rises" in some amount of "run." The slope of a secant line is:

In other words, formula is the amount formula changes per some amount of change in formula. We refer to this as a rate of change, and it is the real world meaning of a slope.

This is extremely important and cannot be emphasized enough. Slope is a rate of change - the amount formula changes during some amount of change in formula.

Let me explain this with the example of traveling from Chicago to New York. To get an intuitive feel for this you have to know what unit the slope has. The unit of the slope is the unit of formula divided by the unit of formula.

In the current example, the function is formula where formula is the distance measured in miles, and formula is the number of hours since you left Chicago. So the slope, formula, has a unit of miles per hour. This is speed! Speed is the change in your distance divided by the amount of time you traveled. Speed is the slope formula!

But you also know that a derivative is a slope - the slope of a tangent line at a specific point on a function. Therefore, a derivative is also a rate of change. It's how much formula is changing at the point formula.

So what's the difference between the slope of a derivative (tangent line) and the slope of a secant line? If both are rates of change, what do they mean? Is there a difference?

Yes, there is a difference. The slope of a secant line is an average rate of change between two points, and the derivative is an instantaneous rate of change at a single point.

Rate of Change of Functions

The average rate of change of a function formula between two points, formula and formula, is

where formula. The average rate of change is the slope of the secant line between formula and formula.

The function's instantaneous rate of change at a specific point, formula, is its derivative formula.

The instantaneous rate of change is the slope of the tangent line at formula.

Let me demonstrate the difference between an average rate of change and an instantaneous rate of change using our current example of driving from Chicago to New York.

What we want to know is what the real world difference between formula and formula is. Let's say you drove straight through without stopping, and it took you 12.5 hours to drive the 800 miles between the two cities. So,

This average rate of change is your average speed driving from Chicago to New York.

But your speed is NOT constant. You speed up to pass other cars, slow down for construction, and you make rest stops.

At any time you can look down at your speedometer to see how fast you're going. The speed you see when you look at your speedometer is formula.

The instantaneous rate of change, formula, is the instantaneous speed you see on your speedometer!

The rest of this chapter is meant to drive home the concept that slope is a rate of change and to emphasize there are two kinds of slopes. The slope of a secant line is an average rate of change, and the slope of a tangent line is an instantaneous rate of change.

First, take a look at the distance-time function pictured below. You'll see times when the car is speeding up, slowing down, and stopped.

Next, let's do a simple example. Suppose,

Using derivative rules one and three in Chapter 6 and the Chain Rule, you know that

You can now answer questions like:

What's the average speed between hours formula and formula?

What's the instantaneous speed at hour formula?

Let's answer the first question now. Here are the steps:

Answering the second question is easy. Just plug formula into formula like so:

Exercise: Assume you have a car whose distance-time function is formula. Answer the following:

What is its average speed between an hour and a half and a half hour after it starts its trip? (Answer: www.CalculusSolution.com/node/31750.)

What is its speed three quarters of an hour after it starts its trip? (Answer: www.CalculusSolution.com/node/31749.)

Now that you know formula is a car's speed, you may be wondering, "What's the car's acceleration?" Great question. Turn the page for the answer.

Chapter 14 - Second Derivatives (10 minutes)

Given any formula, you know how to find the derivative formula. Since formula is another function, can't you find its derivative? Yes!

When you find a derivative's derivative its called the second derivative, and it's denoted by either formula or formula.

To find a function's second derivative, apply any of the same rules for finding a derivative to the function formula. Here's a quick example:

The point is that I treated formula like any other function and applied the rules to it. There's nothing more to finding the second derivative than that.

Exercise: Find the second derivative of formula. (Answer: www.CalculusSolution.com/node/31752.)

What's important to understand about the second derivative is its meaning. Because it's the derivative of the first derivative, formula, the second derivative is a measure of how much the first derivative will rise as it runs from left to right on the x-axis. In other words, it's the rate of change of the first derivative.

Because the first derivative is the slope at a specific point on the function, the second derivative will measure how fast the slope is increasing or decreasing at that point. If the slope of a function is increasing, then formula. If the slope of a function is decreasing, then formula.

The second derivative tells you how the slope is changing. To understand this fully, consider the points formula and formula on the x-axis, and assume that formula. As we go from formula to formula, how will the slope of the function change? In other words, how does formula compare to formula? There are four cases to consider:

The current slope is positive (formula), and the second derivative is positive (formula).

The current slope is positive (formula), and the second derivative is negative (formula).

The current slope is negative (formula), and the second derivative is positive (formula).

The current slope is negative (formula), and the second derivative is negative (formula).

The pictures below deal with each of these four cases:

1) The current slope is positive, and the second derivative is positive.

The second derivative is how the first derivative changes. As we go from left to right on the graph, the slope will gain a positive amount. It becomes steeper.

2) The current slope is positive, and the second derivative is negative.

The second derivative is how the first derivative changes. As we go from left to right on the graph, the slope will lose a positive amount. It becomes less steep.

3) The current slope is negative, and the second derivative is positive.

The second derivative is how the first derivative changes. As we go from left to right on the graph, the slope will gain a positive amount. It becomes less negatively steep.

4) The current slope is negative, and the second derivative is negative.

The second derivative is how the first derivative changes. As we go from left to right on the graph, the slope will lose a positive amount. It becomes more negatively steep.

Because the second derivative is the rate of change of the first derivative, it explains two important concepts:

how fast the function is increasing or decreasing, and

how the function is curved.

(How a function is curved is called concavity. Don't worry. I explain it in Chapter 22.)

When I say that the second derivative explains how fast something is increasing or decreasing, it's best to think about the car example in the previous chapter.

Recall that a derivative is a rate of change. It tells you how much formula changes per change in formula at a specific point. It's an instantaneous rate of change as opposed to an average rate of change.

When you find the second derivative you're finding the derivative of a derivative. In other words, you're finding how fast the derivative changes per change in formula.

In the car example, you know that the derivative of distance, formula, is the speed you see on the speedometer. The second derivative is how fast your speed is changing. That's acceleration!

That's right, formula is the acceleration of your car. When formula is positive you're speeding up, and when it's negative you're slowing down.

To summarize:

formula is the distance a car has traveled at time formula,

formula is the speed you see on the speedometer at time formula, and

formula is the acceleration of the car as you push the gas peddle or apply the brake.

This is the most common example of using first and second derivatives to describe real-world events. Keep this in your head!

Congratulations! You know a lot about derivatives. You should feel proud of how much you've learned and accomplished.

This ends Part 2, and we're on to Part 3 where I show you how to use first and second derivatives to describe the way a function increases and decreases.

Oh yeah, this is a good time to take a break. You deserve it. But don't wait too long to come back. Learning is best done with speed and intensity.

Part 3 - The Geometry of Functions (67 minutes)

In this part I'll discuss what I call the Geometry of Functions. Just as we can describe a person's face as having two eyes, a nose, two ears, a chin, forehead, etc., we can also name and describe the visual parts of a function.

There's a lot of definitions and concepts in this part. I find it helpful to have an outline of how all the words are related. Hopefully, the outline below will help you keep everything straight.

Chapter 15 - The Mean-Value Theorem (10 minutes)

Increasing and Decreasing Functions

Chapter 16 - Increasing and Decreasing Functions (2 minutes)

Chapter 17 - Using Derivatives To Find Where A Function Increases and Decreases (18 minutes)

Chapter 18 - Critical Points (4 minutes)

Maxima and Minima

Chapter 19 - Maxima and Minima (1 minute)

Chapter 20 - The Relationship Between Maxima, Minima, and Critical Points (7 minutes)

Chapter 21 - Using Derivatives To Find Maxima and Minima (7 minutes)

Concavity

Chapter 22 - Concavity (8 minutes)

Chapter 23 - Using Second Derivatives To Determine Concavity (3 minutes)

Chapter 24 - Inflection Points (10 minutes)

Let's begin . . .

Chapter 15 - The Mean-Value Theorem (7 minutes)

The beauty of the Mean-Value Theorem is that you can look at a picture and realize it's true. So here it is:

Now let me state it in words:

The Mean-Value Theorem

If you make a secant line on formula between the points formula and formula, then there is a point between formula and formula (let's call it formula) where formula is the same slope as the secant line. In other words,

The steps for doing this are as follows:

Calculate the slope of the secant line with the formula formula

Find the derivative of formula

Set formula

Use High School algebra to find the value of formula. This is the point formula. (Note there could be more than one point where the slope equals the secant line slope.)

Exercise: Find the point e where the derivative equals the secant line slope for formula between formula and formula. (Answer:-34.50.)

You can play with this more at www.CalculusSolution.com/lc5h/mean-value-theorem where there's an interactive Mean-Value Calculator.

Next, I'm going to talk about increasing and decreasing functions. They're easy, so turn the page now.

Chapter 16 - Increasing and Decreasing Functions (2 minutes)

This is one of the easiest things to understand in Calculus because we only have to look at three pictures. An increasing function is one where, as we go from left to right on the graph, the function will be getting higher. The figure below illustrates this.

A decreasing function is one where, as we go from left to right on the graph, the function will be getting lower. Again, the figure below illustrates this.

Typically, when we talk about a function increasing or decreasing, we don't mean the function over the whole domain. The function can increase or decrease over different parts of the domain. The picture below shows a function increasing over an interval on the formula-axis, then decreasing over another, and then finally increasing over a third interval.

Like I said, this chapter was easy. Keep going to see how derivatives are used to find when a function is increasing or decreasing.

Chapter 17 - Using Derivatives to Find Where a Function Increases and Decreases (15 minutes)

In this chapter I'm going to show how derivatives are related to whether a function is increasing or not. But before we get to functions in general, let's focus on straight lines. Recall that a straight line's slope is

where formula and formula are any two points on the x-axis, and formula.

For an increasing straight line, formula is higher than formula. This means that both formula and formula are positive. Therefore, the straight line's slope will be positive.

Similarly, for a decreasing straight line, formula is lower than formula. In this case, formula will be negative while formula is positive. This means the decreasing straight line's slope will be negative.

Exercise: Determine whether the following straight lines are increasing or decreasing from their slope:

formula (Answer: The slope is +5, so the function is increasing.)

formula (Answer: The slope is negative (-8), so the function is decreasing.)

The problem with functions that are not straight lines (for example polynomials) is that you can't look at the slope of the secant line between formula and formula to see whether the function is increasing or decreasing between formula and formula.

For example, take a look at the picture below. It shows that the slope of the secant line between formula and formula is positive (increasing) while at the same time the function is always increasing.

However, take a look at a different function pictured below. It clearly shows that the secant line between formula and formula is increasing, but the function has places where it's both increasing and decreasing between formula and formula. You can't look at the slope of the secant line between formula and formula to see whether the function is increasing or decreasing between those two points.

So how can you know if a function is increasing or decreasing without looking at a picture? For an increasing function all of the derivatives between formula and formula will be positive. For a decreasing function all the derivatives between formula and formula will be negative. Memorize this.

Let's go back to our previous two pictures to demonstrate this. In the picture below the function is increasing. The important thing to note is that if we pick any arbitrary point between formula and formula, we'll find that formula is always positive.

However, if we find even one point where formula is negative, then we can't say the function is always increasing. There will be places between formula and formula where the function decreases. Take a look at the picture below. Between formula and formula the function is both increasing and decreasing. If we only looked at the slope of the secant line to determine if the function was increasing or decreasing, we would be deceived. The function is doing both. We can immediately show that it is also decreasing by find a point between formula and formula where the derivative is negative.

So let's review:

Using Derivatives to Find If formula Is Increasing or Decreasing Between Two Points, a and b

If formula for every point between formula and formula, then formula is increasing between formula and formula.

If formula for every point between formula and formula, then formula is decreasing between formula and formula.

Exercise: Show that formula is not a decreasing function between formula and formula by evaluating formula. (Answer at www.CalculusSolution.com/node/31756.)

Next up: critical points. You'll use them to find the tops of hills and the bottoms of valleys.

Onward . . .

Chapter 18 - Critical Points (6 minutes)

A critical point is where a function's derivative is zero. In other words, those points on the formula-axis where formula are critical points. At a critical point the function is neither increasing nor decreasing. Memorize this paragraph.

Let's do a quick example. Look at the function formula. In the graph below it looks like there are two points where the derivative is zero: formula and somewhere around formula.

Let's run through the process of finding where the derivative is zero.

First, find the derivative. It's formula.

Next, set the derivative equal to zero and find the values of formula that make it zero. Here's the algebra to do that:

Now, formula will be zero when formula and when formula. This means that formula has two critical points. One is at formula, and the other is at formula.

Exercise: Determine the critical points of formula. (Answer: It has only one critical point at x=-2. For the details go to www.CalculusSolution.com/node/31761.)

Critical points are easy. There's not much to them, but they're important because they help you identify the maxima and minima of a function. "What are those?" you ask. Keep reading.

Chapter 19 - Maxima and Minima (1 minute)

Think of a maximum as the top of a hill and a minimum as the bottom of a valley. See the picture below.

A function can have none, one, or more than one maximum or minimum. In the next chapter I'll show you how to use derivatives to find the maxima (plural of maximum) and minima (plural of minimum) of a function.

This was probably the easiest chapter in the book. In the next chapter I'll show you how to use critical points to find the maxima and minima of a function without looking at it.

Chapter 20 - The Relationship Between Maxima, Minima, and Critical Points (5 minutes)

With the picture below you should see that where there is a maximum, there is also a critical point. This comes from the fact that as we go from the part of the function that is increasing (positive slope) to the part that is decreasing (negative slope), there is some point in between where the slope is zero (a critical point).

Similarly, where there is a minimum, there is also a critical point. This comes from the fact that as we go from the part of the function that is decreasing (negative slope) to the part that is increasing (positive slope), there is some point in between where the slope is zero (a critical point).

This leads us to an important understanding of maxima, minima, and critical points. Where there is a maximum or minimum, there is also a critical point!

However, you need to be careful. Having a critical point does NOT mean you're guaranteed to have a maximum or minimum. The most common example of this is the function formula. It's derivative is formula, and it has a critical point at formula. However, as the graph of formula below shows, at formula there isn't a minimum or a maximum.

So what good does this relationship between maxima, minima, and critical points do for you? It allows you to find maxima and minima quickly.

The first step to finding a maximum or minimum is to find all the critical points. We know that if there are any maxima or minima on the function, they'll be at the critical points.

But once we've found all the critical points, we need to look at each one and do some additional testing to determine if it's a maximum, minimum, or neither as is the case for formula.

In the next chapter I'll show you how to determine if a critical point is a maximum or minimum or neither.

Turn the page now . . .

Chapter 21 - Using Derivatives to Find Maxima and Minima (12 minutes)

The first step in finding the maxima and minima of a function is to find all the critical points. We know if there are maxima or minima on the function, they will be at the critical points.

Next, look at each critical point to determine whether there is a maximum or minimum there.

In the previous chapter I showed that when you have a maximum the slope is positive right before the critical point and negative right after the critical point. So can't we look at the slope to the left and to the right of each critical point to find a maximum? Yes we can. Doing this is called the First Derivative Test.

Also, for a maximum this means that as we go from left to right on the graph the first derivative goes from being positive to being negative. So what do we call the change in the first derivative as we go from left to right? The second derivative! This means that the second derivative at the critical point is negative! We call this the Second Derivative Test.

To summarize, here are the two ways you can determine if a critical point is a maximum:

Determining If a Critical Point Is a Maximum

If formula is a critical point, then we can determine if formula is a maximum in one of two ways:

First Derivative Test: If formula immediately to the left of formula and formula immediately to the right of formula, then there is a maximum at formula.

Second Derivative Test: If formula, then there is a maximum at formula. This is because the first derivative is going from positive to negative so its change is negative.

Similarly, when you have a minimum the slope is negative right before the critical point and positive right after the critical point. So can't we look at the slope to the left and to the right of each critical point to find a minimum? Yes, and again this is called the First Derivative Test.

Also, for a minimum this means that as we go from left to right on the graph the first derivative goes from being negative to being positive. Again, the change in the first derivative as we go from left to right is the second derivative. This means that the second derivative at the critical point is positive! We call this the Second Derivative Test.

To summarize, here are the two ways you can determine if a critical point is a minimum:

Determining If a Critical Point Is a Minimum

If formula is a critical point, then we can determine if formula is a minimum in one of two ways:

First Derivative Test: If formula immediately to the left of formula and formula immediately to the right of formula, then there is a minimum at formula.

Second Derivative Test: If formula, then there is a minimum at formula. This is because the first derivative is going from negative to positive so its change is positive.

And what about the case where there is neither a maximum nor a minimum at the critical point like there was for formula?

Determining If a Critical Point Is Neither a Maximum nor a Minimum

If formula is a critical point, we can determine that it is a neither a maximum nor a minimum in one of two ways:

If formula immediately to the left of formula and immediately to the right of formula, then there is neither a maximum nor a minimum at formula.

If formula immediately to the left of formula and immediately to the right of formula, then there is neither a maximum nor a minimum at formula.

When there is a maximum or a minimum the first derivative changes sign. When there is a critical point, but not a maximum or minimum, the first derivative stays the same sign as you go from left to right across the critical point.

To summarize, here are the steps to determine the maxima and minima of a function:

How to Find Maxima and Minima

Graph the function to get a visual approximation of where the critical points, maxima, and minima are located. (You can do this at www.CalculusSolution.com/calculus-calculator.)

Determine the first and second derivatives. (Again, you can do this at www.CalculusSolution.com/calculus-calculator.)

Determine the critical points by finding the values of formula where formula.

For each critical point apply either the First Derivative Test or the Second Derivative Test to determine if the critical point is a maximum or a minimum. If the first derivative is the same sign on both sides of the critical point, then there is NOT a maximum or minimum at the critical point.

Exercise: Determine the maxima and minima of:

formula. (Answer: www.CalculusSolution.com/node/31764.)

formula. (Answer: www.CalculusSolution.com/node/31764.)

You know everything you need to know about maxima and minima. In the next chapter I'll shift to concavity.

Turn the page now . . .

Chapter 22 - Concavity (5 minutes)

The concept of concavity is best demonstrated with a picture:

In the picture above, the part of the graph that is concave up is like a cup. The part of the graph that is concave down is like a cup turned over. We could also say that a function "curves up" when it is concave up and "curves down" when it is concave down.

The concept of concavity is defined in terms of the first derivative:

Concavity

Mathematicians define concave up and concave down as:

formula is concave up when formula increases as we move from left to right on the graph.

formula is concave down when formula decreases as we move from left to right on the graph.

We can gain some intuition around the formal definition of concavity by looking at specific graphs and getting real numbers.

For example, let's look at the function formula whose derivative is formula by the Power Rule. By looking at the graph of formula below, you get the sense (visually) that formula is concave up.

Now let's look at the value of the derivative at -5, 0, and 2. (Note, there's nothing special about these values. I've arbitrarily chosen them.) As the function goes from left to right, formula is getting bigger (-10 to 0 to 4). Since the derivative is increasing formula is concave up.

The function formula is an excellent example of a function having the same concavity everywhere. But a function's concavity can change as you move along the x-axis.

A good example of this is formula. It's concave down when formula is negative and concave up when formula is positive. See the graph of formula below.

Concavity is an easy concept to grasp, but how do you know the concavity of an entire function? After all, you can't graph it from negative infinity to positive infinity. Keep reading, and I'll show you how to do it.

Chapter 23 - Using Second Derivatives to Determine Concavity (6 minutes)

Remember, concavity is defined as how the first derivative changes as you go from left to right on the graph.

One simple way to determine concavity is with the second derivative. Recall that the second derivative is the amount of change in the first derivative as you go from left to right. This is the exact definition of concavity in the previous chapter!

Using Second Derivatives to Determine Concavity

Concave Up: formula

Concave Down: formula

Let's do two quick examples.

First, in the previous chapter I talked about how formula is always concave up. We know that formula, which is always positive. So using the second derivative to determine the concavity worked for formula.

It's important to note that I've proven it's always concave up. Since you can't graph formula over the infinite domain of the x-axis, you can never prove it's always concave up by graphing it. The only way to prove it is with the second derivative like I just did.

Second, in the previous chapter we looked at formula and visually observed it's concave down when formula is negative and concave up when formula is positive. Let's use the second derivative to prove it.

We know that formula for formula. When formula is negative, formula. (A negative formula and a positive 6 always multiply to give a negative number. For example, at formula, formula.) By showing that formula is negative when formula is negative, we've shown that formula is concave down when formula is negative.

When formula is positive, formula. (A positive formula and a positive 6 always multiply to give a positive number. For example, at formula, formula.) By showing that formula is positive when formula is positive, we've shown that formula is concave up when formula is positive.

So, using the second derivative to determine concavity for formula worked!

Exercise: Determine the concavity of formula. Or more precisely, find the values of formula for which it is concave up and concave down. (Answer: www.CalculusSolution.com/node/31768.)

Now you know about the concavity of a function, but wouldn't it be nice to know where the concavity changes? That's the subject of the next chapter.

Chapter 24 - Inflection Points (8 minutes)

An inflection point is where a function's concavity changes. In other words, it's a point on the formula-axis where the function can go from being concave up to concave down, or from concave down to concave up. In the picture below there are four inflection points at formula.

The problem with inflection points is that you can't find them by looking at the graph. For example, you can't look at the graph and know that formula is an inflection point. However, there's an easy way to find the inflection points with the second derivative.

In the previous chapter I used the second derivative to determine the concavity. When formula the function is concave up, and when formula the function is concave down. In other words, if formula is an inflection point, then formula because the function has gone from positive to negative concavity or negative to positive concavity. This is worth repeating:

The Location of Inflection Points

If formula is an inflection point, then formula. Inflection points are located at places where the second derivative is equal to zero.

But be very careful with this. Just like a critical point is not guaranteed to be a maximum or minimum, not every point where formula is an inflection point.

To identify a maximum or minimum we first found all the critical points where formula. Then we had to look at each one to see whether it was a maximum, minimum, or neither.

Use the same type of thinking here. To identify all the inflection points, first find all the points where formula. Then look at each one to see if the concavity changes. If the concavity changes, then it's an inflection point. Otherwise, it's not.

For example, formula has a first derivative of formula and a second derivative of formula. The only point where formula is at formula. However, formula is not an inflection point. Here's why. The second derivative, formula, is always positive so the function is always concave up. It's concavity never changes.

So, a place where formula is not guaranteed to have an inflection point. (For formula, formula is a minimum, and not an inflection point even though formula there.)

Finally, our well known function formula is a great function to demonstrate concavity changes. In Chapter 23 I proved that formula is concave down when formula is negative, and it's concave up when formula is positive.

Let's use its second derivative to see if we can determine exactly where it changes concavity. An obvious guess is at formula, but let's prove it.

We know the first and second derivatives of formula are formula and formula. The second derivative is zero at formula. When formula is negative, formula is negative so the function is concave down to the left of formula. When formula is positive, formula is positive so the function is concave up to the right of formula. This means the concavity has changed as we go from left to right. Therefore, formula is an inflection point.

Exercise: Determine the inflection points of formula. (Answer: See www.CalculusSolution.com/node/31770.)

Congratulations, you've made it past all the derivatives stuff! Part 4 is the last big section, and it's all about integrals. Personally, I think integrals are easier. There are less rules, and they're intuitive to learn. Plus, there's a cool surprise in integrals that makes them easy to solve.

Part 4 - Integrals (67 minutes)

In this part you'll learn about integrals - what they are, how to calculate them, and a useful application. I've always liked integrals more than derivatives. They're easier and more intuitive in my opinion.

Let's begin . . .

Chapter 25 - Understanding Integrals (11 minutes)

As you learned in Chapter 4, an integral is a fancy name for the area under the function formula between points formula and formula. It's denoted by

So how do you determine the area under formula between points formula and formula?

The most common way is to approximate the area with a bunch of rectangles. See the picture below.

Remember, the area of a rectangle is its height times its width. For each rectangle in the picture, we can determine its area. Each rectangle has a width. But depending on where we evaluate the function, each rectangle can have several possible heights.

Each rectangle can have a height between the minimum and maximum height of the function over that part of the domain where the rectangle is. A logical analysis of integrals shows it doesn't matter which height you choose. Pick one and use it. Now add the areas of the rectangles to get an approximate value of the integral. Again, see the picture above.

Through a complicated process, mathematicians make the width of each rectangle smaller while, at the same time, adding more rectangles. Doing this gives increasingly better approximations of the integral. They continue doing this until they have an infinite number of rectangles with infinitesimally small widths. In other words,

As the number of rectangles becomes larger, the integral will become larger. However, the rectangles have a smaller area because their width is getting smaller. This will cause the integral to get smaller.

The difficulty in Integral Calculus is figuring out whether the integral is infinite, zero or somewhere in-between. For most functions that describe the real world, there's a sweet spot in this tug of war between infinity and zero that results in an actual area you can calculate.

Exactly how you go from this concept of adding areas of small rectangles to an exact area is beyond the scope of learning Calculus in five hours. However, this concept does explain where the integration symbol, formula, comes from. If you take the formula in the word "Summation" (for summation of the area of many rectangles) and stretch it out, you get formula. In other words, our symbol comes from

Finally, there are two types of integrals: definite and indefinite. A definite integral is the area between two specific points on the x-axis. You usually see the two points labeled formula and formula where formula.

We refer to formula as the lower limit of integration, formula as the upper limit of integration, and both of them collectively as the limits of integration.

An indefinite integral is an integral that doesn't specify where on the formula-axis the area is to be calculated. It results in a formula you can use later to find the area between two specific points, formula and formula.

Integral
Notation
Gives Us

Definite Integral

An actual number that is the area between formula and formula.

Indefinite Integral

A formula you can use later to determine the area between two specific points.

Exercise: Use the integral notation to write the following:

The indefinite integral of formula. (Answer: See www.CalculusSolution.com/node/31772.)

The area of formula between formula and formula. (Answer: See www.CalculusSolution.com/node/31773.)

OK, integrals are just areas, and I've mentioned that finding them is "a complicated mathematical process." You may be wondering, "Is there an easy way to calculate integrals that doesn't require a degree in mathematics?" There is, and it's really cool. But first you need to know what an antiderivative is.

Turn the page now to find out how easy antiderivatives are.

Chapter 26 - Antiderivatives (20 minutes)

Before I show you how to calculate an integral, I need to show you how to calculate an antiderivative. This won't make sense until the next chapter, but fortunately, antiderivatives are easy.

Here's the main thing to keep in your head: A function's antiderivative is another function (let's call it formula) such that formula.

To find a function's antiderivative, do the opposite of what you have to do to find its derivative.

Let's do a quick example.

Let's say that formula. What is its antiderivative? Well, do the opposite of what you would have done to find its derivative.

Recall the Power Rule for finding the derivative of formula. The steps are to multiply the exponent (formula) by formula, and then subtract 1 from formula in the exponent. To get the antiderivative, go backwards. First, add one to the exponent. Then take the new exponent, and divide the function by it. In other words,

You can check if this is correct by finding formula and seeing if it's equal to formula. So, find the derivative of formula:

It worked! formula, so we know our formula for the antiderivative is correct. Now we can answer the question. The antiderivative of formula is

Here's the second most important thing you need to know about antiderivatives. What if the antiderivative of formula is the function formula where formula is some constant number? Because of rules one and two in Chapter 6, we know that formula. So which is the correct antiderivative of formula? Is it formula or formula? The answer is that there isn't one antiderivative of formula. There are an infinite number of antiderivatives of formula, and they all differ from each other by a constant number.

For example, formula and formula are antiderivatives of formula, and they differ from each other by the constant number formula. formula

Because of this, we always add a constant to "doing the opposite of the derivative" step above and call that the antiderivative. In other words, for formula we have

where the formula reminds us that the antiderivative is not unique. We call formula the constant of integration for a reason that will become clear in the next chapter.

So here are the four main rules of antiderivatives:

Antiderivative Rule 1: The Antiderivative of Zero

The antiderivative of zero is any constant number, formula, because derivative rule one in Chapter 6 says formula. Remember, you can always check if an antiderivative is correct by finding its derivative. If formula, then you know formula is an antiderivative of formula.

Antiderivative Rule 2: The Antiderivative of formula

Let's assume that formula is any constant number. If formula has the antiderivative formula, then the antiderivative of formula is

because

Antiderivative Rule 3: The Antiderivative of formula

Let's assume that formula and formula are the antiderivatives of formula and formula, respectively. The antiderivative of formula is

because

Antiderivative Rule 4: The Antiderivative of formula

We've already discussed this, but let's restate it. The antiderivative of formula is

because

It's extremely important to note that this formula works for formula being any integer except formula. When formula, formula, and its antiderivative would be

But formula is not a number! So you can't use this formula to find the antiderivative of formula. In Chapter 32 I'll reveal what the antiderivative of formula is.

Okay, just one more rule:

Antiderivative Rule 5: The Antiderivative of a Polynomial

If formula is the polynomial

then its antiderivative is

because

Exercise: Determine the antiderivatives of

formula (Answer: See www.CalculusSolution.com/node/31783.)

formula (Answer": See http://www.CalculusSolution.com/node/31776.)

formula (Answer: See www.CalculusSolution.com/node/31777.)

formula (Answer: See www.CalculusSolution.com/node/31778.)

You may be wondering, "What's the use of this antiderivative stuff?" It turns out that antiderivatives make solving integrals super easy. Read on for one of the most important formulas in Calculus.

Chapter 27 - The Fundamental Theorem of Integral Calculus (17 minutes)

There are many ways to solve the integral

However, there's one technique that's so powerful and important it's called The Fundamental Theorem of Integral Calculus.

The Fundamental Theorem of Integral Calculus is the primary method we use to solve integrals. Its importance can't be overstated. Furthermore, it provides an elegant relationship between antiderivatives and integrals. Here it is:

The Fundamental Theorem of Integral Calculus

Given a function formula and any one of its antiderivatives, formula, then

for the definite integral, and

for the indefinite integral. formula is called the constant of integration.

Now you know why I presented antiderivatives in the previous chapter. The difficulty in calculating the integral has been reduced to finding the antiderivative. And you know how to do that!

Let's do a quick example. Find the area under formula between formula and formula. In other words, solve

In the previous chapter we found that the antiderivative of formula is formula, where formula is the constant of integration. Now all we need to do is evaluate formula. Here it is:

Notice how the constant of integration subtracted out. That's why it doesn't matter. It always subtracts from itself and never enters into the value of the definite integral.

When I talked about finding antiderivatives I said that antiderivatives are not unique. There are an infinite number of them, and they all differ from each other by a constant number.

It doesn't matter which antiderivative you use because they all differ by a constant number that will be subtracted out when you find the area. The constant of integration always subtracts from itself and never enters into the value of the definite integral.

Exercise: Determine the following integrals:

(Answer: See www.CalculusSolution.com/node/31782.)

(Answer: See www.CalculusSolution.com/node/31779.)

(Answer: See www.CalculusSolution.com/node/31780.)

(Answer www.CalculusSolution.com/node/31781.)

Next up is a bunch of rules for calculating integrals. They're simple and intuitive, but they still need to be stated since you'll use them a lot.

Onward . . .

Chapter 28 - Rules of Integrals (7 minutes)

When we calculate a definite integral like formula, we get a number that is the area underneath formula between 0 and 1. We can always add areas because they're just numbers. In other words, adding two areas like

can be done because the result of the two definite integrals is a number, and we know how to add two numbers.

But something cool happens when the limits of integration are the same. For example,

is the addition of two integrals with same limits of integration, 0 and 1.

This is the rule:

Rule 1: Adding Integrals

When adding two integrals and their limits of integration are the same, you can bring both functions under one integration symbol:

Looking at the previous example we have

What if we multiply a constant times the integral? Here's the rule:

Rule 2: Multiplying an Integral by a Constant

Let's assume that formula is a constant number:

From these two rules, and setting formula, you should feel comfortable that:

Rule 3: Subtracting Integrals

When subtracting two integrals and their limits of integration are the same, you can bring both functions under one integration symbol:

Rules 1 through 3 are specific cases of this more general rule:

Rule 4: Linear Combinations of Integrals

Let's assume that formula and formula are constant numbers, then

In other words, you only need to memorize this rule.

Finally, when two areas under formula touch, we can put both under one integral:

Rule 5: Adding Adjacent Areas

Let's assume that formula, then

This rule says that the area of section 1 plus the area of section 2 is the area of the entire region. See the picture below.

Exercise: If

and

then determine

(Answer: See www.CalculusSolution.com/node/31785.)

Integrals can be negative, so I'll talk about why that is in the next chapter.

Chapter 29 - Negative Integrals (5 minutes)

Even though we think of area as a positive number, it's possible for an integral to be negative. There are two things that can make an integral a negative number. First, when the function is negative the integral can be negative:

In general, you can't immediately tell if an integral will be positive or negative because a function can be both positive and negative between formula and formula. You have to calculate the integral to know for sure. See the picture below.

The second way an integral can become negative is when you change the limits of integration. From the Fundamental Theorem of Integral Calculus we know that

But by the Fundamental Theorem of Integral Calculus we also know that

Notice how I flipped formula and formula. Now let's do some simple algebra:

This leads us to an important rule:

Switching Limits of Integration

Switching the limits of integration negates the integral:

See the picture below.

So what is

If you think the width is zero so the integral is zero, you're correct. However, there's another way to solve this using the rule above about switching the limits of integration. If formula, then we have

Since the integral is just a number, let's call it formula. We have

So what is the value of formula that satisfies the equation formula? Zero! Because of this, you now know the following rule:

Integral When the Upper and Lower Limits of Integration Are the Same

For any function formula,

You're almost done with integrals! In the next and final chapter on integrals, I'll show you an important application that's simple because you already know how to do it.

Turn the page now . . .

Chapter 30 - Using Integrals to Find Average Values (7 minutes)

One of the most important applications of the integral is that of finding an average. In Elementary School you learned the average of a set of numbers is:

Let's do the same thing for a function. Break the formula-axis between formula and formula into formula equal width spaces. Now, approximate the average of formula like so:

In general, you can break the formula-axis into formula equal width spaces, and approximate the average of formula with the equation

Next, multiply and divide by formula, and do some algebra:

In the equation above formula, and it's the width of each rectangle. The height of each rectangle is the height of the function. For example, formula is the area of the rectangle between formula and formula. Remember, area is height times width. Since the second rectangle has a height of formula and a width of formula, the area is formula.

To get better and better approximations of the average, add more rectangles with smaller and smaller width. When you do this,

so that

This leads to a very important result:

The Average Value of a Function

The average value of formula between formula and formula is

Exercise: Determine the average value of formula between 0 and 2. (Answer: 4/3. See www.CalculusSolution.com/node/31788.)

OK, you're officially done with integrals, and you have only two more chapters to go. I've presented everything using only polynomials. However, there are two important functions you need to know about. I'll present them, their derivatives, and their integrals.

Oh yeah, and you'll know what the integral of formula is.

Part 5 - Exponential and Logarithm Functions (12 minutes)

In this last part I'll show you the exponential and natural logarithm functions. The natural logarithm is important because it allows you to integrate formula.

Chapter 31 - The Exponential and Logarithm Functions (7 minutes)

Two of the most important functions in mathematics are the natural logarithm function and the exponential function. These functions are used everywhere in science and math, so being familiar with them is important.

When I initially reviewed functions in Chapter 1, I talked about how a function has a domain, range, and rule that relates an formula value to a formula value, which is visually represented with a graph. I showed you several functions with their domain, range, and graph. Let's do the same thing here:

Function
Domain (x-axis)
Range (y-axis)

Exponential function:
formula
or
formula
All positive and negative numbers
All positive numbers

Natural logarithm:
formula
All positive numbers greater than zero
All positive and negative numbers

Here's the graph of the exponential function, formula:

And here's the graph of the logarithm function, formula:

The important thing to note is that the argument for the logarithm function can't be negative. So, for example, formula does not exist. Furthermore, neither does formula. As you get closer and closer to zero from the right, the logarithm function becomes more and more negatively large. It goes toward negative infinity.

The logarithm function formula is similar to the logarithm function formula you learned about in High School. In general, there's an infinite number of logarithm functions. What distinguishes one logarithm function from another is its base. The most generic notation is formula which denotes a logarithm function of base formula. When you see something like

it means

For example, formula.

If this is new to you, think about it this way. When you see formula you are really asking the question, "What is the value of formula such that formula?" It's the same type of thinking for logarithms. When you see formula you are asking, "What is the value of formula such that formula?"

Also, formula and formula are inverses of each other. This means

When you see formula this means a logarithm of base formula, and when you see formula this is a logarithm of base formula. We call formula the common logarithm, and we call formula the natural logarithm.

Because the logarithm functions and their bases are inverses of each other, we know that

and

The natural logarithm function has properties almost identical to those of the common logarithm:

Properties of the Natural Logarithm Function

The numbers formula and formula are positive numbers greater than zero, and formula is any number. The natural logarithm obeys the following:

The exponential function follows the well-known properties of exponents:

Properties of the Exponential Function

Finally, you may want to check out this blog post:

www.CalculusSolution.com/blog/exponentially-increasing-function

It gives you intuition about the logarithm and exponential functions that is often used in conversations to describe how fast something increases.

You're almost done! On to the final chapter where I reveal the derivatives and integrals of the exponential and logarithm functions. Plus, I'll finally reveal the integral of formula.

Chapter 32 - The Calculus of Exponential and Logarithm Functions (5 minutes)

There are several ways mathematicians develop the Calculus of formula and formula, but all of them are beyond the scope of learning Calculus in 5 hours. However, just knowing the derivatives of formula and formula can give you quite a bit.

The Derivative of formula

Well, that's easy to remember. The derivative of the exponential function is itself! More importantly, it is its own antiderivative, so from the Fundamental Theorem of Integral Calculus we know that:

The Integral of formula

Something even more important occurs for the derivative of the natural logarithm function. Here is its derivative:

The Derivative of the Natural Logarithm

The antiderivative of formula is formula! This is really important. In Chapter 26, Antiderivative Rule 4 would not allow us to find the antiderivative of formula because applying the formula made us divide by zero, which we can't do. But now we know its antiderivative and can state that

The Integral of formula When formula

Because the domain of the natural logarithm function is all positive numbers greater than zero, the above formula only works for formula and formula greater than zero. In general, you can have either formula and formula both greater than zero or both formula and formula less than zero to get

The Integral of formula

It's important to remember you can't use this formula when formula is negative and formula is positive. That's because formula becomes infinitely large as you approach formula, so the integral becomes infinitely large. Whenever we integrate over a point where the function gets infinitely large we call it an improper integral. Calculating improper integrals requires function limits and is beyond the scope of learning Calculus in 5 hours.

With the natural logarithm function and Antiderivative Rule 4, you know the integral of formula where formula is any integer and you're not integrating across formula for formula

Finally, we have the integral of formula:

The Integral of the Natural Logarithm

where formula and formula are greater than zero. (There's not too much that's interesting about this, but I thought you would find it strange if I didn't tell you about it when I told you about the integral of formula. So here it is in case you're curious.)

Congratulations, you're done! You've made it all the way through, and you should feel proud of how much you've accomplished.

Turn the page now for one last important message about learning Calculus . . .

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Glossary

Acceleration: How fast you are speeding up or slowing down. Mathematically, it's the second derivative of the distance-time function. See Chapter 14.

Antiderivative: Given a function formula, its antiderivative is another function formula such that formula. A function's antiderivative is used in the Fundamental Theorem of Integral Calculus to find integrals quickly and easily. See Chapters 26 and 27.

Area: The amount of 2-dimensional space taken up by an object. Circles, squares, and rectangles are examples of objects that have area. Three dimensional objects can also have area. For example, the surface of a ball is an area. An integral is the area underneath the function formula.

Average Rate of Change: The amount formula changes between two points on the x-axis. See Chapter 13.

Base: What distinguishes one logarithm from another is its base. See Chapter 31.

Brainwashed/brainwashing: The act of believing you're stupid and can't do Calculus despite scientific research that proves you can.

Calculus: Calculus is made up of Differential and Integral Calculus. Differential Calculus is about finding all the slopes a function can have. Integral Calculus is about finding the area underneath a function. See Chapter 4.

CalculusSolution.com: Online website for getting more Calculus information. Go to http://www.CalculusSolution.com.

Chain Rule: A quick way to find the derivative of the composition of two functions. See Chapter 11.

Coefficient: The numbers in front of terms like formula in a polynomial. See Chapter 2.

Composition: See Function Composition here in the glossary.

Concavity: Denotes the visual perspective of a function looking like a cup or a cup turned over. See Chapters 22 and 23.

Constant of Integration: A variable, usually denoted by formula, used to indicate that antiderivatives are not unique. They are subtracted out when the value of a definite integral is found. See Chapters 26 and 27.

Critical Point: : The places on a function where the slope is zero. Maxima and minima reside at critical points, but critical points are not guaranteed to have a maximum or minimum. See also the first and second derivative tests. See Chapters 18 and 20

Decreasing Function: As you move from left to right on the graph, the function will be getting lower. See Chapter 16.

Definite Integral: The area under a function between two points. It's typically denoted by formula. See Chapters 4 and 25.

Degree (of a polynomial): The highest integer formula for which formula isn't zero in the polynomial formula. See Chapter 2.

Delta (Greek formula): Used to denote the difference between two numbers. For example, formula. See Chapter 5.

Derivative: The slope of the function formula at formula. It's denoted by formula or formula. See Chapters 4 and 5.

Differential: A really small difference. When formula becomes really small we call it formula. Similarly, when formula becomes really small we call it formula. See Chapter 5.

Dimension: The dimension of a number or variable answers the question, "What is this?" See Chapter 12.

Dimensional Analysis: The process of checking to see if the dimension on the left-hand side of an equation is equal to the dimension on the right-hand side. See Chapter 12.

Domain: Where on the x-axis a function is defined. See Chapter 1.

Exponential Function: The function formula (also denoted by formula). See Chapter 30.

First Derivative: The slope of the function formula. It's denoted by formula or formula. Also see derivative here in the glossary. See Chapters 4 and 5.

First Derivative Test: A method for determining if a critical point is a maximum or a minimum. See Chapter 21.

Function: A mapping of one set of numbers to another. See Chapter 12.

Function Composition: Combining two functions by putting one inside the other. It's denoted by formula or formula. The derivative of a function composition can be quickly found with the Chain Rule. See Chapters 3 and 11.

Fundamental Theorem of Integral Calculus: An important and easy way to find the integral of a function using its antiderivative. See Chapter 27.

Graph: The visual representation of a function's rule that associates numbers on the x-axis with numbers on the y-axis. See Chapter 1.

"ho D hi minus hi D ho over ho ho": A simple way to remember the Quotient Rule. See Chapter 10.

Improper Integral: An integral where the function becomes infinite somewhere between the upper and lower limit of integration, or an integral whose upper or lower limit of integration is infinite. See Chapter 32.

Increasing Function: As you move from left to right on the graph, the function will be getting higher. See Chapter 16.

Indefinite Integral: A formula that can be used to find the definite integral between two points on the x-axis. It's typically denoted by formula. See Chapter 25.

Infinity: A word we use to describe something that is bigger than anything else. We denote it by formula.

Inflection Point: A point where a function's concavity changes. See Chapter 24.

Instantaneous Rate of Change: The amount formula changes at a specific point on the x-axis. The instantaneous rate of change of formula is its derivative formula. See Chapter 13.

Integer: A number like 1, 15, or -182. Numbers like 3.2 are not integers.

Integral: The area underneath a curve. There are two types of integrals. Definite and indefinite. The definite integral is a number that represents the area between two points. An indefinite integral is a formula you can use later to find the area between two points. See Chapter 25.

Integration: The act of solving an integral.

Interval: All the numbers between two other numbers. There are three types of intervals: open, closed, and mixed. Open intervals are denoted by formula and are all the numbers between formula and formula, but not formula and formula themselves. A closed interval is denoted by formula and is all the numbers between formula and formula including the numbers formula and formula themselves. Mixed intervals are a combination of open and closed intervals. Intervals are used to denote things like the domain and range of a function.

Inverses (function): A function's inverse is denoted by formula and is a function such that the function compositions equal formula. i.e. formula. In the official math notation we write formula. The natural logarithm and the exponential functions are inverses of each other. See Chapter 31.

Limit (function): A logical process of making one variable slide closer and closer to another variable or number in a function. See Chapter 5.

Limits of Integration: The two points on the x-axis between which you want to find the area underneath a function. See Chapter 25.

Logarithm Function: The function formula. See Chapter 30.

Lower Limit of Integration: formula in the integral formula. See Chapter 25.

Maxima: Plural of maximum.

Maximum: The part of a function that looks like the top of a hill. See Chapters 19, 20, and 21.

Mean-Value Theorem: The property of a function where the slope of a secant line drawn between any two points will be identical to the derivative somewhere between those two points. See Chapter 15.

Minima: Plural of minimum.

Minimum: The part of a function that looks like the bottom of a valley. See Chapters 19, 20, and 21.

Polynomial: A function of the form formula. Straight lines are polynomials. See Chapter 2.

Power Rule: A rule for finding the derivative of formula. If formula then formula. See Rule 4 in Chapter 6.

Product Rule: A quick and easy way to find the derivative of the product of two functions. See Chapter 9.

Psycho: Signing up for an 8:00 a.m. Calculus II class in college.

Quotient Rule: A quick and easy way to find the derivative of the division of two functions. See Chapter 10.

Range: Where on the y-axis a function is defined. See Chapter 1.

Rate of Change: How much something changes with respect to something else. See Chapter 13.

Rectangle: A geometric shape used to approximate an integral. Its area is its width times its height. See Chapter 25.

Rise: How much a straight line goes up per some amount it runs. It's the numerator of a straight line's slope. See Chapters 2 and 5.

Run: The denominator of a straight line's slope. See Chapters 2 and 5.

Secant: A straight line between two points on a function. The secant line's slope is the average rate of change of the function between those two points. See Chapters 5 and 13.

Second Derivative: The derivative of the first derivative. It has several uses:

Show how much a function's derivative is increasing or decreasing. See Chapter 14.

Concavity is defined by the second derivative. See Chapter 23.

Describing the physical act of acceleration. See Chapter 14.

Identifying maxima and minima of a function. See the Second Derivative Test below and Chapter 21.

Second Derivative Test: A method for determining if a critical point is a maximum or a minimum. See Chapter 21.

Slope: A measure of how much a line rises divided by how much it runs. See Chapter 2.

Speed: How much distance you travel per some amount of time. It's the first derivative of the distance-time function. See Chapter 13.

Steep: The amount of slope of a straight line. A slope of six is more steep than a slope of two. A slope of -6 is more steep than a slope of -2.

Straight Line: A function of the form formula where formula is the slope and formula is the y-intercept. See Chapter 2.

Tangent Line: A straight line whose slope is the value of a function's derivative at a specific point. See Chapters 5 and 7.

Union: The act of taking all the elements of two sets and putting them together. It's denoted by formula.

Unit: See dimension here in the glossary and Chapter 12.

Upper Limit of Integration: formula in the integral formula. See Chapter 25.

List of Figures

Chapter 1 - Functions

Building Height Function

Labeling a Function

Graph of formula Showing It Has Only One Value of formula for Every Value of formula

Graph of formula Showing It Is Not a Function

Function formula

Function formula

Function formula

Function formula

Function formula

Chapter 2 - Straight Lines and Polynomials

Straight line slope and y-intercept

Chapter 4 - The Two Basic Concepts of Calculus

Slopes of formula

Differential Calculus

Integral Calculus

Chapter 5 - Understanding Derivatives

The Secant Line

Tangent Line and Secant Lines

Chapter 7 - The Tangent Line

Tangent Line Slope and the Derivative

Explanation of the Tangent Line Formula

Chapter 13 - Attaching Real World Meaning to Derivatives

Graphical Intuition of Distance, Slope, and Speed

Chapter 14 - Second Derivatives

The Change in Positive Slope When the Second Derivative is Positive

The Change in Positive Slope When the Second Derivative is Negative

The Change in Negative Slope When the Second Derivative is Positive

The Change in Negative Slope When the Second Derivative is Negative

Chapter 15 - The Mean-Value Theorem

The Mean-Value Theorem

Chapter 16 - Increasing and Decreasing Functions

Definition of an Increasing Function

Definition of an Decreasing Function

Function That is Both Increasing and Decreasing

Chapter 17 - Using Derivatives to Find Where a Function Increases and Decreases

Increasing Straight Line Slope

Decreasing Straight Line Slope

Increasing Function With Increasing Secant Line

Increasing and Decreasing Function With Increasing Secant Line

Derivative of Increasing Function

Comparison of Slope and Secant Line of Increasing and Decreasing Function

Chapter 18 - Critical Points

Critical Points of formula

Chapter 19 - Maxima and Minima

Maxima and Minima

Chapter 20 - The Relationship Between Maxima, Minima, and Critical Points

Critical Point, Maximum, and Slope

Critical Point, Minimum, and Slope

formula Where the Critical Point is Neither a Maximum nor a Minimum

Chapter 21 - Using Derivatives to Find Maxima and Minima

Using the First and Second Derivative Tests to Find a Maximum

Using the First and Second Derivative Tests to Find a Minimum

Chapter 22 - Concavity

Definition of Concave Up and Concave Down

Concavity of formula

Concavity of formula

Chapter 24 - Inflection Points

Inflection Points

Chapter 25 - Understanding Integrals

Integral is the Area Under a Function

Approximating Integrals with Rectangles

Chapter 28 - Rules of Integrals

Adding Adjacent Areas:
formula

Chapter 29 - Negative Integrals

Negative Integral Because of Negative Function

Function With Positive and Negative Integral Areas

Negative Integral Because of Switched Limits of Integration

Chapter 30 - Using Integrals to Find Average Values

Approximate Average of a Function

Chapter 31 - The Exponential and Logarithm Functions

Graph of the Exponential Function

Graph of the Natural Logarithm Function