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# 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:

## Goal: Intuitive Appreciation (30 mins)

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)

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)

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)

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)

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.

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[^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].