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Calculus Learning Guide

This is a realistic learning plan for Calculus based on theADEPT method:

  • Explore high-level themes (analogies, diagrams) before the low-level details
  • See learning as a journey, not an all-or-nothing destination.
  • Acknowledge our limited motivation. What Aha! moments can we have in minutes, not weeks?

I have [X] minutes for Calculus, what can I learn?

1 minute: The Big Aha!

Level 1: Appreciation

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:

Calculus Learning Guide

And the rings turned into a triangle:

Calculus Learning Guide

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

Checkpoint:

  • Do you want to learn more more?

+20 minutes: Intuitive Appreciation

Level 2: Natural Description

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

+20 minutes: Technical Description

Level 3: Symbolic Description

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

+30 minutes: Theory I

Level 4: Basic Theory

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

+1 hour: Theory II

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

+1 hour: Basic Problem Solving

Level 5: Basic Performance

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
Calculus Learning Guide

Read the lessons above in Calculus, Better Explained, now an Amazon bestseller.

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+12 weeks: I need to pass a course!

Level 5: Advanced Performance

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.

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