i nt egr al CAL C' sCal c ul usI Sur v i v al Gui de
gof n i K a st i r com K . C y L b CA al r g e nt i
CALC. al egr nt 0i Š201 ed. v er sRes ght lRi Al
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 ebook 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.
really excited about proofs, and you just get bored and confused.
Instead of relying on my calculator to solve problems outright, I like to use it as a doublecheck 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.
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.
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
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 ebook, please refer to the Table of Contents.
(Clickable) Table of Contents I. Foundations of Calculus A. Functions
C. Continuity 1. Common Discontinuities
1. Vertical Line Test
a. Jump Discontinuity
2. Horizontal Line Test
b. Point Discontinuity
3. Domain and Range
c. Infinite/Essential Discontinuity
2. Implicit Differentiation a. Equation of the Tangent Line b. Related Rates E. Common Applications 1. Speed/Velocity/Acceleration
4. Independent/Dependent Variables
2. Removable Discontinuity
2. L’Hopital’s Rule
5. Linear Functions
3. The Intermediate Value Theorem
3. Mean Value Theorem
a. Slope-Intercept Form b. Point-Slope Form 6. Quadratic Functions
II. The Derivative A. The Difference Quotient
4. Rolle’s Theorem III. Graph Sketching
1. Secant and Tangent Lines
A. Critical Points
a. The Quadratic Formula
2. Creating the Derivative
B. Increasing/Decreasing
b. Completing the Square
3. Using the Difference Quotient
C. Inflection Points
7. Rational Functions a. Long Division B. Limits 1. What is a Limit? 2. When Does a Limit Exist?
B. When Derivatives Don’t Exist
D. Concavity
1. Discontinuities
E. - and -Intercepts
2. Sharp Points
F. Local and Global Extrema
3. Vertical Tangent Lines C. On to the Shortcuts!
1. First Derivative Test 2. Second Derivative Test
a. General vs. One-Sided Limits
1. The Derivative of a Constant
b. Where Limits Don’t Exist
2. The Power Rule
1. Vertical Asymptotes
3. The Product Rule
2. Horizontal Asymptotes
a. Just Plug It In
4. The Quotient Rule
3. Slant Asymptotes
b. Factor It
5. The Reciprocal Rule
c. Conjugate Method
6. The Chain Rule
3. Solving Limits Mathematically
4. Trigonometric Limits 5. Infinite Limits
D. Common Operations 1. Equation of the Tangent Line
G. 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.
between and will cross the graph twice, which causes the graph to fail the Vertical Line Test.
Horizontal Line Test
You can also test this algebraically by plugging in a point between and for , such as .
This graph does not pass the Vertical Line Test because a vertical line would intersect it more than once.
Example Determine algebraically whether or not is a function.
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.
Plug in for
You know that the circle below is not a function because any vertical line you draw
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.
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 .
and simplify. 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
Horizontal Line Test, then no horizontal line will cross the graph more than once, and the graph is said to be “one-to-one.”
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 .
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. 2. 3. 4.
A point Another point The slope, 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 slopeintercept 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.