\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

Math Is NOT Easy

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Yes, that's right. Math is not easy. It was never intended to be easy. You may see books entitled Math Made Easy or 10 Easy Steps To Understanding Calculus. Those books will not help you.
You may also see books entitled Math For Dummies or something similar. Those books won't help you either.

But math, and more specifically calculus, is understandable. And You CAN Ace Calculus. But it will not be easy or fast and you will have to work for it. But it WILL be worth it. Why study calculus? Find your answer here.

Don't think you can do it? Then watch these videos. This guy is awesome. What would you do if you were born with no arms and legs?

Nick Vujicic --- I Love Living Life. I Am Happy

Nick Vujicic --- No Arms, No Legs, No Worries! [ Part One ]

Nick Vujicic --- No Arms, No Legs, No Worries! [ Part Two ]

Nick Vujicic --- No Arms, No Legs, No Worries! [ Part Three ]

Do You Want To Understand Calculus?

There is one thing that will absolutely determine whether you do well in calculus or not. That one thing is your attitude. Wait! Don't stop reading. This is not a pep-talk. This is about how your mind works. Every action and reaction you make in your life starts in your mind. What you think about your ability to master calculus (or anything, for that matter) defines how far you will go. You CAN choose what to think. Here is a video that will help you understand this a bit better.

David Foster Wallace - This Is Water (commencement speech excerpt)

One thing he mentions in the video is that you, as a college student, do not really have a concept for the daily grind and boredom of day-to-day life yet. I think that is true to some degree but you get a taste of it when you have to take a course that is hard, that you don't like and that you think will never end. I'm sorry to say that, for you, this may be calculus. But you CAN do something about it. You are not a victim.

If you use the word can't then that will limit you more than almost any other factor in your life. Here is an article on how to remove this word from your vocabulary called Don't Say It: How to Get 'Can't' Out of Your Vocabulary.

Nothing about you can keep you from understanding calculus, IF YOU WORK HARD ENOUGH, have the algebra and trig background, and, most importantly, don't give up. If you give up, you have a zero percent chance of learning calculus.

Still don't believe me? Keep trying, even if you don't feel like it. Do you think Nick ever wanted to give up? You bet! But he didn't and look at him today.

Story Of Pushing Against The Rock

A man was sleeping at night in his cabin when suddenly his room filled with light and G-d appeared. G-d told the man He had work for him to do, and showed him a large rock in front of his cabin. G-d explained that the man was to push against the rock with all his might.

This the man did, day after day. For many years he toiled from sun up to sun down, his shoulders set squarely against the cold, massive surface of the unmoving rock, pushing it with all his might. Each night the man returned to his cabin sore, and worn out, feeling that his whole day had been spent in vain.

"G-d" he said, "I have labored long and hard in Your service, putting all my strength to do that which you have asked. Yet, after all this time, I have not even been able to budge that rock. What is wrong? Why am I failing?"

G-d responded compassionately, "My friend, When I asked you to serve Me and you accepted, I told you that your task was to push against the rock with all your strength, which you have done. Never once did I mention to you that I expected you to move it. Your task was to push. And now you come to Me with your strength spent, thinking that you have failed. But, is that really so?"

"Look at yourself. Your arms are strong and muscled, your back sinewy and brown, your hands are callused from constant pressure, and your legs have become massive and hard. Through opposition you have grown much and your abilities now surpass that which you used to have. Yet you haven't moved the rock. But your calling was to be obedient and to push and to exercise your faith and trust in My wisdom. This you have done. I, My friend, will now move the rock."

Something To Think About: Sometimes persistence itself produces things in you that you may not expect.

Try Again!

Motivation Questions

1. Why should I study calculus when I will never use it?

This is a great question and one that, as an instructor, I love to answer. When you ask this question it shows that you are thinking about your future and paying attention to the path you are on (or maybe you are just fed up and tired of math?). For whatever reason, I am going to give you my take on why you are not wasting your time learning calculus.

First of all, you are probably right. You will most likely never use the material you learn in calculus unless you are a math major. Even as an engineer, you may never directly use a lot of what you are learning in calculus, although you will probably use a lot more than other people.

To understand my logic, let me give you a little bit of background on how your mind works. As you go through life, you read things, learn things, hear things, see things and have experiences happen to you, each little input to your brain shapes your thinking, both what you think and how you think. Without going into too much detail, your brain is building patterns or structures. To do that, you have chemical changes in your brain.

Studying and learning calculus, changes the chemistry of your brain and shapes those structures which allows you to put other information into those structures that you will use. For example, if you want to drink water, you need a glass, cup, bottle or even your hands to get the water into your mouth. If you decide to bypass those options, you still need the pipes to get water to you. If you go directly to a river, you need the riverbank to move the flow of water to a location close to you. You need some kind of structure to get water to your mouth.

The same idea works with calculus. You need the structure of calculus to be able to understand other math, engineering, physics, chemistry, business and many other topics.

Now, let's go back to the topic of structures. If you are struggling with calculus, listen to what you are telling yourself about it. Are you telling yourself that you can't do it? Are you reminding yourself constantly how much you hate it? Do you think about how you will never use calculus so you are wasting your time? If so, you are building a set of concrete structures that makes learning the material more difficult, if not impossible. All of us have to do things in our lives that we don't want to do, that we don't like to do and that we think are a waste of time. But successful people are the ones that stop saying those things to themselves, change what they are thinking, dig deep and do what they have to do regardless of how they feel.

Do you want some techniques on how to study math? Check out this page.

Finally, if you don't believe anything I've said above, there is another reason to study calculus. Everyone needs to learn persistence and how to deal with frustration. This could be the time for you.

You CAN Ace Calculus

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)


\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)


\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)


\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)


\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)


\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)


\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)


\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)


\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)


\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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