# I have [X] 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.

## Leave a Reply

31 Comments on "Calculus Learning Guide"

this is why you are loved

Thanks ghosty :)

You are one of my favorite people, Kalid! :D

Really appreciate it Kenny, thanks!

Like that Beatles song: It’s getting better all the time…

Thank you.

Do you have any thoughts on differential equations?

Hi Steve, I have some notes here: http://aha.betterexplained.com/t/differential-equations/1065 . My key intuition: imagine you’re a football coach with a playbook. Based on your plan, you want to know where every player is 1 minute into the game. If there are interactions (A follows B, B avoids C, D chases C…) the prediction isn’t analytically solvable, and you have to actually simulate the game to see where people end up.

I’ll have to spend time here. Thank you!

Wow.

I have been planning to learn calculus(self study) for a long time.I just start to read Apostol’s Calculus vol-1 which I bought it a year ago but never touched.Wanted to get an intuitive concept of what derivatives and integrals are.I feel like this is a great place to learn .

Thank you so much Kalid.

Thanks Ajith, hope you enjoy it!

Kalid, 69 year old [former Federal Ave. grandson (if yu remember )] who thanks you a lot for your help in assisting me/us in understanding math stuff.

also, I am gently suggesting my son, who seems confused and frustrated about the worth of what higher math is trying to accmplish, that he look and think about the thinking you share. thanks again. kobukmike

Thanks Mike, glad you’re enjoying it :). Happy you’re sharing it with your son, there’s a lot of frustrations with poor teaching experiences (I had them). Hopefully this approach can help.

I’m trying to expand on the x-ray and time lapse ideas into 3D, so I’m attempting to do a thought experiment for the surface area of a sphere being the addition of “rings” and its volume being the addition of discs. So my feeling is that for a unit sphere that would be: A= integral(2pi(sqrt(1-x^2))dx and V=integral(pi(1-x^2)dx). both over x=-1 to x=1. Where am I going astray?

Astray on the area, the volume seems correct.

Wolfram input: 2*pi*integral ((r^2-x^2))dx x=0 to 1

any combination.. no sorry.. any permutation of 26 letters is not enough to describe your works!! an ideal teacher…

Appreciate it Sonali!

I learned calculus during my teenage years, and then spent 5 years doing calculus in order to obtain my degree. How I wished this site was around back then, it would have saved a huge pain in the arse back then. (Not to mention all the argument with my friends why the heck am I messing around with letters all the time I’m supposed to be doing math)

Impressive job Kalid, you’re the man. Keep it up, I’m saving this site for my son’s reference.

Thanks so much, glad you’re enjoying it :).

Kalid, May I use the material, with aknowledgement, for teaching?

Yes, of course!

Hey Kalid. Maybe you have been told by a lot of people that you are amazing. You know what? IT’S TRUE!!

I am an Indian 15 year old student, and i thought of googling ‘calculus – explained simply’ after my teacher said that he’d test us on it the next day.

I am so glad and satisfied with your explanations, that i even told my mother — a pol-science ‘person’ who apparently steers five miles away whenever she sees a Maths book — that even she could easily grasp these concepts!

And guess what, that thing u said about Sweetness of sugar – how we see the chemistry behind it and ignore the message of nature—-that quote is my current whatsApp status!

But then ofcourse. Thank you Kalid.

:D

Thanks so much, I’m glad it had such a positive impact for you!

absolutely kalid. and see i even checked to see if u’ve replied ! :-D

No, Thompson’s “Calculus Made Easy” does _not_ follow the traditional limit approach. It is a classic text that adopts the infinitesimal approach.

Why hadn’t I find this last year? :'(

Thank you for all your efforts Kalid, you are great!!!

Oliver from Hungary.

Thanks!

Hi, I love your approach to tutoring. The visualizations are phenomenal. I wish I had known about this when I was taking Calc 1. Are you planning to release a series on Calc 2 or 3 in the future?

Thanks Jen! No definite dates but I want to cover all of Calculus eventually :).

And THIS is how our Education system should be…. Awesome explanation man.. Helped me understand stuff I didn’t in my math class :)