Course Homepage Preface On Learning
Build Your Intuition
1. 1-Minute Summary 2. X-Ray Vision 3. 3d Intuition
Learn The Lingo
4. Integrals, Derivatives 5. Computer Notation
Basic Understanding
6. Improved Algebra 7. Linear Changes 8. Squared Changes
Deeper Understanding
9. Infinity 10. Derivatives 11. Fundamental Theorem
Figure Out The Rules
12. Add, Multiply, Invert 13. Patterns In The Rules 14. Take Powers, Divide
Put It To Use
15. Archimedes' Formulas Summary
Welcome to an intuition-first calculus course.
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7 min read

3. Expanding Our Intuition

Hope you thought about the question from last time: how do we take our X-Ray strategies into the 3rd dimension?

Here's my take:

Circles and spheres

The 3d steps can be seen as the 2d versions, swept out in various ways. For example, spin the individual rings (like a coin) to create a shell. Imagine slices pushed through a changing mold to make wedges. Lastly, imagine we spin the boards to make plates, like carving a wooden sphere with a lathe (video).

The tradeoffs are similar to the 2d versions:

An orange is an interesting hybrid: from the outside, it appears to be made from shells, growing over time. And inside, it forms a symmetric internal structure -- a better way to distribute seeds, right? We could analyze it both ways.

Exploring The 3d Perspective

In the first lesson we had the vague notion the circle/sphere formulas were related:

calculus circles

Well, now we have an idea how:

Circles and spheres

Wow! These descriptions are pretty detailed. We know, intuitively, how to morph shapes into alternate versions by thinking "time-lapse this" or "X-Ray that". We can move backwards, from a sphere back to circumference, or try different strategies: maybe we want to split the circle into boards, not rings.

The Need For Math Notation

You might have noticed it's getting harder to explain your ideas. We're reaching for physical analogies (rings, boards, wedges) to explain our plans: "Ok, take that circular area, and try to make some discs out of it. Yeah, like that. Now line them up into the shape of a sphere…".

I love diagrams and analogies, but should they be required to explain an idea? Probably not.

Take a look how numbers developed. At first, we used very literal symbols for counting: I, II, III, and so on. Eventually, we realized a symbol like V could take the place of IIIII, and even better, that every digit could have its own symbol (we do keep our metaphorical history with the number 1).

This math abstraction helped in a few ways:

Multiplication started as a way to count groups and measure rectangular area. But when you write "$15/hour × 2.5 hours = $37.50" you probably aren't thinking of "groups" of hours or getting the area of a "wage-hour" rectangle. You're just applying arithmetic to the concepts.

In the upcoming lessons we'll learn the official language to help us communicate our ideas and work out the rules ourselves. And once we've internalized the rules of calculus, we can explore patterns, whether they came from geometric shapes, business plans, or scientific theories.


1) What's your grandma-friendly version of the above?

2) Any aha moments or areas to clarify?

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Class Discussion

Guided class discussions are available in the complete edition.