Calculus


#1

Calculus Study Guide

To “Learn Calculus”, pick a goal below. This is an honest, realistic and self-motivating learning plan.

  • Focus on bang for the buck: What’s the best way to spend 10 minutes? 2 hours?
  • Explore analogies, diagrams, and examples before diving into the technical details.
  • See learning as a progressive journey of discovery, not an all-or-nothing accomplishment.

Goal: Quick Insight (1 minute)

Calculus is the art of splitting patterns apart (X-rays, derivatives) and gluing patterns together (Time-lapses, integrals). Sometimes we can cleverly re-arrange the pattern to find a new insight.

A circle can be split into rings:

And the rings turned into a triangle:

Wow! We found the circle’s area in a simpler way. Welcome to Calculus.

Checkpoint:

  • Do you want to learn more more?

Goal: Intuitive Appreciation (30 mins)

Read:

Checkpoint: Describe, in your own words:

  • What Calculus does
  • X-Ray Vision
  • Time-lapse Vision
  • The tradeoffs when splitting a circle into rings, wedges, or boards
  • How to build a 3d shape from 2d parts

Goal: Technical Description (30 mins)

Read:

Checkpoint: Describe, in your own words:

  • Integral
  • Derivative
  • Integrand (a single step)
  • Bounds of integration

Skills:

  • Describe a Calculus action (splitting a circle into rings) using the official language
  • Enter the official language into Wolfram Alpha to solve the problem

Goal: Theory I (30 mins)

Read:

Checkpoint: Describe, in your own words:

  • How integrals/derivatives relate to multiplication/division

Skills:

  • Find the derivative/integral of a line
  • Find the derivative/integral of a constant
  • Find the derivative/integral of a square
  • Recognize the common notations for the derivative
  • Estimate the change in $f(x) = x^2$ using a step of size $dx$

Goal: Theory II (1 hour)

Read:

Checkpoint: Describe, in your own words:

  • How an infinite process can have a finite result
  • How a process with limited precision can point to a perfect result
  • The formal definition of the derivative
  • Estimate the change in $f(x) = x^2$ using a step of size $dx$, and let $dx$ go to zero. Verify the limit using Wolfram Alpha.
  • The Fundamental Theorem of Calculus (FTOC)

Derive and put into your own words:

  • The addition rule: $(f + g)’ = ?$
  • The product rule: $(f * g)’ = ?$
  • The inverse rule: $(\frac{1}{x})’ = ?$
  • The power rule: $ (x^n)’ = ? $
  • The quotient rule: $(\frac{f}{g})’ = ? $
  • Solve $ \frac{d}{dx} 3x^5 $ on your own and verify with Wolfram Alpha
  • Solve $ \int 2x^2 $ on your own and verify with Wolfram Alpha

Goal: Basic Problem Solving (1 hour)

Read:

Checkpoint: Describe how to turn the circumference of a circle into the area of a circle:

  • Explain your plan in plain English
  • Explain your plan using the official math notation
  • Apply the rules of Calculus to your equation and calculate the result
  • Verify the result using Wolfram Alpha
  • Repeat the steps above, turning the area of a circle into the volume of a sphere
  • Repeat the steps above, turning the volume of a sphere into the surface area of a sphere

Goal: Hey, I really need to pass this course! (12 weeks)

Gotcha. The best use of time is still spending a few hours on the above goals, to build a solid intuition. Then, begin your Calculus course, such as:

  • Elementary Calculus: An Infinitesimal Approach by Jerome Keisler (2002). This book is based on infinitesimals (an alternative to limits, which I like) and has plenty of practice problems. Available in print or free online.

  • Calculus Made Easy by Silvanus Thompson (1914). This book follows the traditional limit approach, and is written in a down-to-earth style. Available on Project Gutenberg and print.

  • MIT 1801: Single Variable Calculus. Includes video lectures, assignments, exams, and solutions. Available free online.

As you go through the traditional course, keep this in mind:

  • Review the intuitive definition. Rephrase technical definitions in terms that make sense to you.

  • It’s completely fine to use online tools for help. When stuck, get a hint, fix your mistakes, and try solving a new problem on your own.

  • Relate graphs back to shapes. Most courses emphasize graphs and slopes; convert the concepts to shapes to help visualize them.

  • Skip limits if you get stuck. Limits (and infinitesimals) were invented after the majority of Calculus. If you struggle, move on and return later.


#2

[^1]: Visit http://betterexplained.com/calculus/book for clickable URLs.

Analogy: Calculus flipbook.

Derivative is the pattern you are adding. Integral is [flipping] through all the results and seeing the overall timelapse. What are you building up to?

“Rate of change” => amount of “red” you are adding. Slope is a percentage relative to the size of the step you made. Are you making changes as big as your current step? Negative? etc.

Highlight the change you are seeing each time [value of derivative at this time].