integralCALC's Calculus I Survival Guide

by Krista King

of

integralCALC.

com©2010 integralCALC.All Rights Reserved.

Disclaimer

This e-book is presented solely for educational purposes. While best efforts have been used in

preparing this e-book, the author makes no representations or warranties of any kind and

assumes no liabilities of any kind with respect to the accuracy or completeness of the contents.

The author shall not be held liable or responsible to any person or entity with respect to any

loss or incidental or consequential damages caused, or alleged to have been caused, directly or

indirectly, by the information contained herein.

Every student and every course is different and the advice and strategies contained herein may

not be suitable for your situation. This e-book is intended for supplemental use only. You

should always seek help FIRST from your professor and other course material regarding any

questions you may have.

Introduction

Author’s Note

As a third grader, I learned my multiplication

tables faster than anyone in my class. I was

allowed to skip all of seventh-grade math and

go straight to eighth-grade (something I think

most people would pay a lot of money for,

considering how much math sucks for pretty

much everyone). As a junior, I finished all the

math courses my high school was offereing.

It wouldn’t be conceited to say that math was

a subject that came easier to me than it did to

others. Compared to my classmates, I was

always good at it. What can I say? I got my butt

kicked by every science class I ever took, but I

was always ahead of the curve when it came to

math. I guess it’s just the way my brain works.

And yet, despite the fact that it’s always been

easier for me, I’ve struggled with all kinds of

math concepts soooooo many times, and often

remember feeling totally and completely lost

in math classes.

You know that feeling when you’re reading

through an example in your textbook, hoping

with desperation that it will show you how to

do the problem you’re stuck on? You hang in

there for the first few steps, and you’re like,

“Okay awesome! I’m getting this!” And then by

about the fourth step, you start to lose track of

their logic and you can’t for the life of you

figure out how they got from Step 3 to Step 4?

It’s the worst feeling. This is the point where

most people give up completely and just resign

themselves to failing the final exam.

I’ve seen this same reaction in many of the

students I tutored in calculus while I was in

college. As hard as they tried to understand,

the professor and the textbook just didn’t

make sense, and they’d end up feeling

overwhelmed and defeated before they’d ever

really gotten started.

I wasn’t a math major in college, but I spent a

lot of time tutoring calculus students, and I’ve

come to the conclusion that for most people,

the way we teach math is fundamentally

wrong.

First, there’s a pretty good chance that you

won’t ever actually use what you learn in

calculus. Algebra? Definitely. Basic geometry?

Probably. But calculus? Not so much. Second,

even if it is worthwhile to learn this stuff,

trying to teach us how to work through

problems with proofs that are supposed to

illustrate how the original formulas are

derived, just seems ridiculous.

In my experience, most students get the most

benefit out of understanding the basic steps

involved in completing the problem, and

leaving it at that. Get in, get out, escape with

your life, and hopefully your G.P.A. still intact.

Sure, there’s a lot to be said for going more in

depth with the material, and I’d love to help

you do that if that’s your goal. For most people

though, a basic understanding is sufficient.

My greatest hope for this e-book and for

integralCALC.com is that they’ll help you in

whatever capacity you need them. If you’re

shooting for a C+, let’s get you a C+. I don’t

want to waste your time trying to give you

more than you need. That being said though,

most of the students I tutored who came in

shooting for a C+ came out with something

closer to a B+ or an A-. If you want an A,

attaining it is easier than you think.

No matter what your skill level, or the final

grade you’re shooting for, I hope that this e-

book will help you get closer to it, and better

yet, save you some stress along the way.

Remember, if there’s anything I can

ever do for you, please contact me

at integralCALC.com.

Words of Wisdom

There are two pieces of advice I’d like to give you before we get started.

1. Stay Positive More than anything, you have to stay positive.

Don’t defeat yourself before you even get

started. You’re smarter than you think, and

calculus is easier than you think it is. Don’t

panic.

Half of the people I’ve tutored over the years

needed a personal calculus cheerleader more

than they needed a tutor. They’d gingerly

proceed through a new problem… “Is this

right? Then if I… am I still doing it right?”

They’d doubt themselves at every step. And I

would just stand behind them and say “Yeah,

it’s right, you’re doing great, you’ve got it,

you’re right,” until they’d solved the problem

without my help at all.

So many students let themselves get worked

up and freaked out the moment something

starts to get difficult. It’s understandable, but

the more you can fight the fear that starts to

creep in, the better off you’ll be. So take a

deep breath. It’s going to be okay.

2. Use Your Calculator (Or Don’t) Your calculator can be your greatest ally, but it

can also be your worst enemy. As calculators

have gotten more powerful, students have

come to rely on them more and more to solve

their problems on both homework and exams.

Instead of relying on my calculator to solve

problems outright, I like to use it as a double-

check system. If you never learn how to do the

problem without your calculator, you won’t

know if what your calculator tells you is

correct. Nor will you be able to show any work

if you’re required to do so on an exam, which

could cost you big points.

Learning the calculus itself means you’ll be

able to show your work when you need to, and

you’ll actually understand what you’re doing.

Once you solve a problem, you should know

how to punch in the equation so that you can

look at the graph or solution to verify that the

answer you got is the same one your calculator

gives back to you.

What You Won’t Find

I’m not here to replace your textbook. Because

this is a quick-reference guide, you won’t find

chapter introductions full of calculus history

you don’t care about.

I’m also not here to replace your professor,

nor do I expect that you’re particularly excited

about learning calculus. If you are excited

about calculus, that’s awesome! So am I. But if

you’re not, this is the place to be, because, at

least in this e-book, you won’t find pointless

tangents where I geek out hard core and get

really excited about proofs, and you just get

bored and confused.

The purpose of this e-book is to serve as a

supplement to the rest of your course

material, not to completely replace your

professor or your textbook.

Even though I’ve tried to cover the most

common introductory calculus topics in

enough detail that you could get by with just

this e-book, neither of us can predict whether

your professor will ask you to solve a problem

with a different method on a test, or a specific

problem not covered here. The last thing I

want is for you to think that this e-book is a

replacement for going to class, miss that

information, and then do poorly on the test

because you didn’t get all the instructions.

What You Will Find

This e-book should give you the most crucial

pieces of information you’ll need for a real

understanding of how to solve most of the

problems you’ll encounter. I don’t want to be

your textbook, which is why this e-book is only

about thirty pages long. I want this to be your

quick reference, the thing you reach for when

you need a clear understanding in only a few

minutes.

For a specific list of topics covered in this e-

book, please refer to the Table of Contents.

(Clickable) Table of Contents

I. Foundations of Calculus

A. Functions

1. Vertical Line Test

2. Horizontal Line Test

3. Domain and Range

4. Independent/Dependent Variables

5. Linear Functions

a. Slope-Intercept Form

b. Point-Slope Form

6. Quadratic Functions

a. The Quadratic Formula

b. Completing the Square

7. Rational Functions

a. Long Division

B. Limits

1. What is a Limit?

2. When Does a Limit Exist?

a. General vs. One-Sided Limits

b. Where Limits Don’t Exist

3. Solving Limits Mathematically

a. Just Plug It In

b. Factor It

c. Conjugate Method

4. Trigonometric Limits

5. Infinite Limits

C. Continuity

1. Common Discontinuities

a. Jump Discontinuity

b. Point Discontinuity

c. Infinite/Essential Discontinuity

2. Removable Discontinuity

3. The Intermediate Value Theorem

II. The Derivative

A. The Difference Quotient

1. Secant and Tangent Lines

2. Creating the Derivative

3. Using the Difference Quotient

B. When Derivatives Don’t Exist

1. Discontinuities

2. Sharp Points

3. Vertical Tangent Lines

C. On to the Shortcuts!

1. The Derivative of a Constant

2. The Power Rule

3. The Product Rule

4. The Quotient Rule

5. The Reciprocal Rule

6. The Chain Rule

D. Common Operations

1. Equation of the Tangent Line

2. Implicit Differentiation

a. Equation of the Tangent Line

b. Related Rates

E. Common Applications

1. Speed/Velocity/Acceleration

2. L’Hopital’s Rule

3. Mean Value Theorem

4. Rolle’s Theorem

III. Graph Sketching

A. Critical Points

B. Increasing/Decreasing

C. Inflection Points

D. Concavity

E. - and -Intercepts

F. Local and Global Extrema

1. First Derivative Test

2. Second Derivative Test

G. Asymptotes

1. Vertical Asymptotes

2. Horizontal Asymptotes

3. Slant Asymptotes

H. Putting It All Together

IV. Optimization

V. Essential Formulas

Foundations of Calculus

Functions

Vertical Line Test Most of the equations you’ll encounter in

calculus are functions. Since not all equations

are functions, it’s important to understand

that only functions can pass the Vertical Line

Test. In other words, in order for a graph to be

a function, no completely vertical line can

cross its graph more than once.

This graph does not pass the Vertical Line Test because a vertical line would intersect it more

than once. Passing the Vertical Line Test also implies that

the graph has only one output value for for

any input value of . You know that an

equation is not a function if can be two

different values at a single value.

You know that the circle below is not a

function because any vertical line you draw

between and will cross the

graph twice, which causes the graph to fail the

Vertical Line Test.

You can also test this algebraically by plugging

in a point between and for , such as

.

At , can be both and . Since a

function can only have one unique output

value for for any input value of , the graph

fails the Vertical Line Test and is therefore not

a function. We’ve now proven with both the

graph and with algebra that this circle is not a

function.

Horizontal Line Test The Horizontal Line Test is used much less

frequently than the vertical line test, despite

the fact that they’re very similar. You’ll recall

that any function passing the Vertical Line Test

can only have one unique output of for any

single input of .

This graph passes the Horizontal Line Test

because a horizontal line cannot intersect it more than once.

Contrast that with the Horizontal Line Test,

which says that no value corresponds to two

different values. If a function passes the

Example

Determine algebraically whether or not

is a function.

Plug in for and simplify.

Horizontal Line Test, then no horizontal line

will cross the graph more than once, and the

graph is said to be “one-to-one.”

This graph does not pass the Horizontal Line

Test because any horizontal line between and would intersect it more

than once.

Domain and Range Think of the domain of a function as

everything you can plug in for without

causing your function to be undefined. Things

to look out for are values that would cause a

fraction’s denominator to equal and values

that would force a negative number under a

square root sign.

The range of a function is then any value that

could result for from plugging in every

number in the domain for .

Independent and Dependent Variables Your independent variable is , and your

dependent variable is . You always plug in a

value for first, and your function returns to

you a value for based on the value you gave

it for . Remember, if your equation is a

function, there is only one possible output of

for any input of .

Linear Functions You’ll need to know the formula for the

equation of a line like the back of your hand

(actually, better than the back of your hand,

because who really knows what the back of

their hand looks like anyway?). You have two

options about how to write the equation of a

line. Both of them require that you know at

least two of the following pieces of

information about the line:

1. A point

2. Another point

3. The slope,

4. The y-intercept,

If you know any two of these things, you can

plug them into either formula to find the

equation of the line.

Slope-Intercept Form The equation of a line can be written in slope-

intercept form as

,

where is the slope of the function and is

the -intercept, or the point at which the

graph crosses the -axis and where . The

slope, represented by , is calculated using

two points on the line, and ,

and the equation you use to calculate is

To find the slope, subtract the -coordinate in

the first point from the -coordinate in the

second point in the numerator, then subtract

the -coordinate in the first point from the -

coordinate in the second point in the

denominator.

Example

Describe the domain and range of the

function

In this function, cannot be equal to ,

because that value causes the

denominator of the fraction to equal .

Because setting equal to is the only

way to make the function undefined,

the domain of the function is all .