integralcalc's calculus i survival guide
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Preview the first seven pages of integralCALC's Calculus I Survival Guide E-book from integralcalc.comTRANSCRIPT
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 .