This is a realistic learning plan for Calculus based on the ADEPT method.

I have a few 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:

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?

+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) = x2 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.

using a step of size , and let 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

on your own and verify with Wolfram Alpha Solve int 2x2 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



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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: