# Calculus, Better Explained

# A Guide To Developing Lasting Intuition

## Kalid Azad

# Contents

- Introduction
- Chapter 1 1 Minute Calculus: X-Ray and Time-Lapse Vision
- Chapter 2 Practicing X-Ray and Time-Lapse Vision
- Chapter 3 Expanding Our Intuition
- Chapter 4 Learning The Official Terms
- Chapter 5 Music From The Machine
- Chapter 6 Improving Arithmetic And Algebra
- Chapter 7 Seeing How Lines Work
- Chapter 8 Playing With Squares
- Chapter 9 Working With Infinity
- Chapter 10 The Theory Of Derivatives
- Chapter 11 The Fundamental Theorem Of Calculus (FTOC)
- Chapter 12 The Basic Arithmetic Of Calculus
- Chapter 13 Finding Patterns In The Rules
- Chapter 14 The Fancy Arithmetic Of Calculus
- Chapter 15 Discovering Archimedes’ Formulas
- Chapter 16 Afterword
- Chapter 17 Appendix: Learning Checklist
- Chapter 18 Appendix: Calculus Study Plan

# Introduction

Hi! It looks like you’re interested in learning Calculus. I like you already.

This book isn’t a collection of practice problems or formal theories. Hundreds of textbooks handle that quite well; this is the guide I wish they tucked into their front cover.

The goal is to help you:

**Grasp the essence of Calculus in hours, not months****Develop lasting, practical insights you can apply to your own life****Enjoy a subject often considered humorless****Solve an end-to-end problem on your own****Build an intuitive foundation for classroom study**

Most Calculus courses force you to build the car before driving it. Shouldn’t you master the physics of acceleration and the chemistry of gasoline before touching the wheel? (*Sure, if you want to kill someone’s interest in cars.*)

I’m here to yank you from class, put you in a go-kart, and let you ride
around Calculus Town. Yes, you’ll take control of the wheel (try to
avoid the pedestrians). Yes, you’ll make a few mistakes without *perfect* knowledge of the internals. But you’ll be having fun. True understanding comes from experiencing ideas
yourself, not having them lectured to you.

After some practice, you may ignite the curiosity to explore the details regular textbooks offer. (*How can I go faster? Handle corners better? Which fuel works best?*)

A few minutes into Chapter 1, and you’ll visualize what Calculus does. After an hour, you’ll analyze concepts using the Calculus lingo and mindset. A few days later, after internalizing the ideas, you’ll begin solving famous problems on your own.

So, how do we approach a notoriously difficult subject? Intuition-first.

## 1 Learning Strategy: Blurry To Sharp

What’s a better learning strategy: covering a subject in full detail from top-to-bottom, or progressively sharpening a quick overview?

After a single class, which strategy gives you a better understanding of
the material? Which helps you predict how later parts fit together? Which is more *fun*?

The linear, official, approach doesn’t work for me. Starting with a rough outline and gradually improving it keeps our interest and helps us see how the individual details are connected.

Most textbooks take the top-down approach, while this book is a blurry-to-sharp guide. The fine details are out there when you need them.

## 2 Learning Strategy: Appreciation To Performance

Next question: Do you need to be a musician to enjoy a song? No way.

There are several levels of music understanding:

**Intuitive Appreciation:**Just enjoying the music.**Natural Description:**Humming a tune you heard or made up.**Symbolic Description:**Reading and writing the sheet music.**Theory:**Explaining how harmonies work, why minor scales are somber, etc.**Performance:**Playing the official instruments.

Math knowledge is the same. Start by appreciating, even enjoying, the idea. Describe it with your own words, in English. Then, learn the official symbols to make communication easier (“2 + 3 = 5” is better than “Two plus three equals five”, right?).

When we’re ready, we “perform” math by doing the calculations
ourselves, and diving into the detailed theory. But only if you want! Decide for *yourself* what level of understanding you’d like to reach.

Here’s a guide to the book, depending on the level you’d like to reach:

- Intuitive Appreciation (Chapters 1-3)
- Technical Description (Chapters 4-5)
- Theory I (Chapters 6-8)
- Theory II (Chapters 9-14)
- Performance (Chapter 15)

The appendix includes a study plan if you wish to follow a formal course.

## 3 Learning Strategy: Study History

Calculus began when Archimedes realized shapes could be split into parts and rearranged. After a lifetime of effort, he discovered connections between spheres, circles and other shapes that were later etched onto his tomb.

Unfortunately, Calculus courses are taught out of order. The difficult, modern concepts are taught first (limits, developed 1800s) and the intuitive foundations are saved for the end (integrals, imagined 250 B.C.).

This book takes Archimedes’ approach: learn to *see* what Calculus can
do, then layer in the theory as needed. By then, your interest has been
piqued and you can read a modern textbook for the details (see
recommendations in the final chapter).

Please don’t feel obligated to reach the Performance level of Calculus. For most of us (myself included), Appreciation and Description give practical, day-to-day insights. The first few chapters of this course are all that’s needed to have a better intuition than I had despite dozens of engineering classes.

Sound good? Let’s dive in.

## 4 Email Updates

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## 5 Book Webpage

Nobody likes typing URLs by hand. For clickable links for to the URLs in this book, notes, and other resources, visit:

## 6 The Legal Stuff

All content is copyright (c) 2015 Kalid Azad, except the following images used under Wikipedia’s Creative Commons license:

- “Portrait of Isaac Newton” by Sir Godfrey Kneller http://commons.wikimedia.org/wiki/File:GodfreyKneller-IsaacNewton-1689.jpg
- “Tree Rings” by Arnoldius http://commons.wikimedia.org/wiki/File:Tree_rings.jpg
- “Acropoclipse” by Elias Politis http://commons.wikimedia.org/wiki/File:Acropoclipse.jpg
- “Geometric Series” by Indolences http://commons.wikimedia.org/wiki/File:Geometric_series_14_square2.svg

You may use any of this material for educational use with attribution; for commercial use, please contact me using http://betterexplained.com/contact.

Next → Lesson 1: 1 Minute Calculus (X-Ray and Time-Lapse Vision)