# I have a few minutes for Calculus, what can I learn?

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

- Explore analogies/diagrams before the technical details
- Treat learning as a journey, not an all-or-nothing destination
- Allow for limited motivation: what can we learn in minutes, not weeks?

## 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:

- Lesson 1 - Use X-Ray and Time-Lapse Vision
- Lesson 2 - Practice Your X-Ray and Time-Lapse Vision
- Lesson 3 - Expanding Our Intuition

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:

- 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

## +1 hour: Theory II

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

## +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:

**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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