Calculus Study Guide
To “Learn Calculus”, pick a goal below. This is an honest, realistic and selfmotivating 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 allornothing accomplishment.
Goal: Quick Insight (1 minute)
Calculus is the art of splitting patterns apart (Xrays, derivatives) and gluing patterns together (Timelapses, integrals). Sometimes we can cleverly rearrange 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:
 Lesson 1  Use XRay and TimeLapse Vision
 Lesson 2  Practice Your XRay and TimeLapse Vision
 Lesson 3  Expanding Our Intuition
Checkpoint: Describe, in your own words:
 What Calculus does
 XRay Vision
 Timelapse 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:
 Lesson 6  Improving Arithmetic And Algebra
 Lesson 7  Seeing How Lines Work
 Lesson 8  Playing With Squares
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:
 Lesson 9  Working With Infinity
 Lesson 10  The Theory Of Derivatives
 Lesson 11  The Fundamental Theorem Of Calculus (FTOC)
 Lesson 12  The Basic Arithmetic Of Calculus
 Lesson 13  Finding Patterns In The Rules
 Lesson 14  The Fancy Arithmetic Of Calculus
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 downtoearth 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.