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

17Calculus Article - You Can Read and Learn From Math Books

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It is just not possible to read a math textbook like you would a read a novel and be able to understand and learn the material.

Have you ever read through a section, paragraph or even just a sentence of a math textbook and at the end you have no idea what you just read? If so, you are not alone. And most teachers don't know these techniques or they learned them by trial and error on their own and expect you to do the same.
[There could be other reasons you have never been taught to do this. See the basic learning tools page for more info.]

Using supplementary books can boost your learning significantly. For suggestions on how to select and use supplementary books, read the discussion on the college books page .

On this page, we give you specific guidelines on how you can read and learn from math books as well as recommendations of books that we have found helpful for precalculus, calculus, differential equations and math proofs. Some of the books we recommend are free and the rest of them are reasonably priced on Amazon.
Remember - - If you don't need a specific edition of a textbook, buying a previous version will probably save you some money while giving you good, up-to-date content.

Book Recommendations

Of course, books are the most important resource for learning calculus. Are you surprised that we said that? You may be thinking that your instructor is the most important resource. We believe that is not the case. Most of your learning will occur outside the classroom (while you work practice problems), away from any teacher or tutor or fellow students. (In fact, your fellow students are more important resources than your instructor. See the More Help page for more on that.)
We have an entire page dedicated to helping you find the right books and ways to save money on books that you have to buy, the books page.
Here are some suggestions to get you started.

We believe that every serious calculus student needs these 3 books, Deep Work: Rules for Focused Success in a Distracted World, How to Ace Calculus: The Streetwise Guide and How to Ace the Rest of Calculus.

Deep Work: Rules for Focused Success in a Distracted World

Deep Work: Rules for Focused Success in a Distracted World

This book could catapult your learning, if you apply the techniques and insights carefully and radically.
Deep work is necessary as a student to succeed but few students do it. This leaves a huge chasm of possibility for you to stand out and achieve the seemingly extraordinary feat of acing calculus. This book not only explains deep work but also how to implement it in your life.
Why You Need To Work Deeply (from chapter 1)
- Deep Work Helps You Quickly Learn Hard Things
- Deep Work Helps You Produce at an Elite Level
Order this book now [ Deep Work: Rules for Focused Success in a Distracted World ] and read it during your next semester break. It will be time well spent.

How To Ace Calculus

Two of the best books I've found are How To Ace Calculus for Calculus 1 and it's sequel How To Ace The Rest Of Calculus for Calculus 2 and 3.

How to Ace Calculus: The Streetwise Guide
How to Ace the Rest of Calculus

Larson Calculus

Additionally, if your textbook is not clear, the Larson Calculus family of textbooks are the best.

Larson ETF 3th Edition

Larson Calculus ETF - - The main textbook we recommend is Larson Calculus ETF*. It is well written and logically organized. If you don't need the latest edition for a class, you may be able to get an earlier edition for as little as $10. If you are struggling through another textbook, you may want to consider buying a used (or early edition, I recommend the 3rd and 4th) copy of this book. The book has great examples and solutions to odd problems can be found at CalcChat.com.

*ETF = Early Transcendental Functions refers to how early in the textbook the author discusses using calculus techniques on trigonometric, exponential and hyperbolic functions. See the 17calculus bookstore page for a complete discussion on the Larson Calculus and Precalculus family of books.

Differential Equations

For differential equations students, the best series of books are the Dover books.

Ordinary Differential Equations (Dover Books on Mathematics)
Partial Differential Equations for Scientists and Engineers (Dover Books on Mathematics)
Introduction to Partial Differential Equations with Applications (Dover Books on Mathematics)

Some Websites to Read

The 10 Best Sites To Rent Or Buy College TextBooks Cheaply

3 Websites To Purchase Electronic College Textbooks


Do not purchase any books containing the words 'dummy', 'dummies' or 'idiot'. You are none of these and putting that message in your brain will just make calculus harder for you. Additionally, do not buy books that say that calculus is easy or simple. It's not. But it is possible to ace calculus. And this website gives you the tools to do just that.

Additional Recommendations

We provide a complete list of books that we recommend on the bookstore page.

Learn From Math Books

What are going to tell you on this page will help you skip the trial and error part of learning how to utilize math books, which most people have to go through, and jump right into being able to learn and understand math directly from textbooks. Many schools are implementing a teaching method called flipped teaching and these math classes require you to know how to read math textbooks. Part of the beauty of these techniques is that you begin to learn on your own and become a more independent student. So, if you have a teacher sometime that is not very good, you will still be able to learn.

So, what do you need to do? We have listed below the main techniques that we think will help you.

Before starting to read, scan the section to see what the important points are. It would be nice if your instructor provided an overview of exactly what you are supposed to do and learn for each section but that usually won't happen. So you need to do it and you will need to learn everything, unless your instructor tells you otherwise. Assume everything is going to be on the exam and assume you will need to know everything for your next class or sometime in the future.
Before we go any further, let's watch this short video that gives specific advice on how to get started learning from a textbook and remember what you read.

Matt DiMaio - How To ABSORB TEXTBOOKS Like A Sponge [8min-16sec]

video by Matt DiMaio

Okay, now that you have the big picture of what to look for in the current section, here are some recommendations on the specifics of studying the chapter.

1. Get out a pencil (not a pen), an eraser and a notebook. You will use these to write notes as you go along. These notes don't have to be organized or clean.

2. Slow down and read each word and sentence carefully. This a hard one since we are so used to reading quickly so that we can get through with whatever we are doing, so that we can go to the next task as quickly as possible. You need time for your mind to become aware of what the book is saying and process new terms and ideas.

3. When going through examples, carefully process each step until you understand what they are doing. If there are things that are going on that you don't understand or it seems like the book is skipping steps (which all books do), then write out the step in your notebook, filling in the missing steps. Process it by writing it out until you understand the step. Then go to the next step. If you get stuck, write a note to yourself in your notebook and see if subsequent steps help you understand. If not, then get some help from your study group, fellow student, tutor or instructor.

4. Do not skip the proofs. Although most instructors skip the proofs and do not even require you to read them, read them anyway. The proofs give you an idea of when and how to use theorems and push your learning so that you can understand even more of the material. As you go through a proof, write out each step and fill in the gaps, similar to examples. However, with proofs, it helps to rewrite the proof in your own words.

5. Do not skip graphs and pictures. They are excellent ways to help remember concepts. If you understand a graph and how it relates to an equation or concept, you now have two ways to remember the material.

6. Write down important terms in your notebook along with the definition, in your own words. So what are the important terms? When you are first learning, you don't know. So write down the obvious like bold, italicized or highlighted terms and concepts. Include theorems or anything with proofs. Then write down what you THINK are important terms. You will find out later if they are or not. Don't be afraid to write down too much.

7. Read other textbooks and supplemental math books. What?! Read more than just what I am required to read?! Yes, because the point of reading is to understand and there is not any one textbook that will help every student. We have some suggestions in the How To Save On and Use College Books page but they are just to get you started. This is important and will help you a lot.

8. Work a few practice problems even if you don't have to. As a student, your time is limited. You have a lot to do and not much time. But if you take the time to do a little bit of work over and above what you are asked to do, you save even more time later since you will not have to relearn the material. You will already know it and have some experience with it.

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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calculus motivation - music and learning

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Book Recommendations

Learn From Math Books

Calculus, Better Explained: A Guide To Developing Lasting Intuition

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