Let's practice our abstraction skills by simplifying concepts in Calculus.

Last time, we saw how abstraction simplifies ideas.

After removing enough detail, a photo of lions turns into the notion of quantity (where *n* happens to be 3 in this case).

Let's apply this to a function like f(x) = x^{2}. What's the simplified essence?

Here were my first thoughts as I worked through the idea:

**Abstraction 1: Multiple shapes**

First, look at a few representations of x^{2}. The common cliche is that it represents a square of side x, but we can be more creative. What about a rectangle with sides frac(1)(2)x and 2x? How about a portion of a circle? (If text(area) = π r^{2} then frac(1)(π) (32%) should leave us with r^{2}.)

**Abstraction 2: Examine the specific changes**

Next, look at the changes that happen with each of our shapes. The square gets equal lengths added to each side. The rectangle gets a "long, skinny" and "short, fat" added to each side. (The corners can be ignored for now.)

The changes to the circle are the simplest, with a small arc being added.

**Abstraction 3: Make the changes general**

Like the lion scenario above, we have a unique representation of each change ("three lion icons"). Let's make the changes generic (three lines) by finding a common format.

Here, we "melt down" each change until it resembles a straight line. Because the square, rectangle, and circle all represent x^{2}, the same line can describe the changes they undergo. Neat!

Now that's some nice abstractin', let's keep it going:

**Abstraction 4: Separate the line**

The orange "change line" is actually a transition between a starting and ending position. If we represent the start and end as blue dots, the height of the line is the amount of change between them.

Notice how we make an angled line as well: the input change (blue line) and output change (orange line) trace out the rate of change (green line).

**Abstraction 5: Show every state and angle**

Rather than picking specific starting and endpoint positions, graph *every* position (blue curve) and *every* rate of change (green line at each point).

The blue curve actually *generates* the green line: at any point, we can draw the tangent line and see the "change angle" to the next neighbor.

## The Number/Angle Abstraction

Here's where I get excited. On a graph, we're used to literal representations: we need a bigger line to represent a bigger change. But an angle (a certain ratio of height:width) represents every number in the same amount of space!

0:1 is 0 degrees

1:1 is 45 degrees

2:1 is 63.4 degrees (arctan(2) = 63.4)

100:1 is 89.4 degrees (arctan(100) = 89.4)

By using an angle, we've curled the number line into a format that fits into any space. Even a giant number like 10,000,000,000 can be written with the same effort as "1". Must bigger numbers take up more room?

We have a clean abstraction: **The curve shows every possible scenario, and the angle quantifies the rate of change**. In a way, the curve "writes down" its rate of change at every point.

Yowza. Maybe we discussed this in class, but I didn't think of it this way until trying to abstract each step.

This was a peek into an organic "aha-finding" technique: start with a specific idea, keep generalizing, and see what insights emerge.

Happy math.

## Other Posts In This Series

- A Gentle Introduction To Learning Calculus
- Understanding Calculus With A Bank Account Metaphor
- Prehistoric Calculus: Discovering Pi
- A Calculus Analogy: Integrals as Multiplication
- Calculus: Building Intuition for the Derivative
- How To Understand Derivatives: The Product, Power & Chain Rules
- How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms
- An Intuitive Introduction To Limits
- Why Do We Need Limits and Infinitesimals?
- Learning Calculus: Overcoming Our Artificial Need for Precision
- A Friendly Chat About Whether 0.999... = 1
- Analogy: The Calculus Camera
- Abstraction Practice: Calculus Graphs

## Leave a Reply

16 Comments on "Abstraction Practice: Calculus Graphs"

Great job, Kalid. Love the way you look at curvature & time. How would you look at “Contour”? Love aerodynamics! Please reply. Jamieson (Peter).

Hi Jamieson! Nice to see you. I don’t really know much about aerodynamics unfortunately, but physics is something I’d like to learn more.

Well done again , kalid sir

I need lapace transform intution but link u mentioned in better explained forum to ur old aha site is dead , plz help as u tube video and other material is not good enough for insight, I want to see ur take on this

Hi Aisha, I have some notes on Laplace here:

https://aha.betterexplained.com/t/laplace-transform/8

I am really excited to join a group of people who are intellectual curious and eager to share insights than the mere mechanics! I have a little bit background in financial math, option pricing. I hate being bothered by buzzwords without being able to understand so people can just keep bluffing and making me feel like a dumb . I would like to share a little bit of the insights I have regarding derivatives(buzzword1), option pricing (buzzword2), black-shole model (buzzword3). I personally feel it would be amazing to see the insights into each other’s field easily! If you guys like the idea, I can prepare a few slides and send to Kalid!

At the end of the day, there should be no secrets in most fields, and once you know what it is, it is really nothing difficult!

Shirly, the key is to divide and conquer. The question begs, whom? With all the different jargon (buzzwords), in so many fields, especially math, they are dividing and thus conquering us. It’s sad, but true that the more you learn, the more you realize how much you don’t know. I was so happy when I was stupid. Though Kalid is not making me angrierrr… with his vast amount of knowledge, insight, and educational prowess, he is making me more wise in the wealth of education I am receiving from him. I wish I had him as a teacher in all of my classes; starting in the second grade.

Yes, the more your learn, the easier to learn new things. But it is extremely difficult to a beginner with no help around. I felt happy sometimes frustrated by what’s been taught and would spent few days myself trying to figure out. The thought process is meaningful in that it strengthen your critical thinking capability. And if you are lucky enough to have a friend like Kalid to discuss, it would be fantastic.

But right now my trouble is, there are just so many things to learn, in different fields: physics, computing, engineering, finance, artificial intelligence … , all these are equally interesting. I am simple not able to catch up with everything. I really wish there are someone, given 5-10 minutes, led me into a particular fields, shed some lights, gave me some ideas. But typically what’s found in wikipedia are a few jargons linked to more jargons and circular referencing.

The problem with lots of information online is people are simply grouping together terms, words, jargons and pretend they are knowledge. But words by themselves are ABSOLUTELY NOT knowledge!!! It is so discouraging to a beginner. And the beginner i am talking about could be a well educated professional in one area trying to learn something in other fields.

Don’t you wish there could be some place you can go, spend 5-10 minutes, to really have some ideas about something?

Amen. Wise words, I am very, very interested in the group! I have been seeking many sites or something like this site betterexplained. There are not many unfortunately. And this site is not very active either but this has an enormous potential. I have always a “lost” feeling. I think you know what I am talking about. I have a lot of insights and I am willing to share them – math, my successful company, drawing and politics.

Excellent!I think many people share the same problems: unable to find useful information efficiently. Insights are scattered, buried deep by garbage info, sometimes i might just give up on learning. There are probably only a few people who are persistent enough be able to get what they want. But i think there must be a better way!! People needs to learn, people love to learn (some are lazy of course, but inherently, curious). But knowledge and wisdom are lost when people start quoting the same mechanics, with the ability to share info easily, garbage knowledge grow at a much faster speed, making it almost impossible to learn true knowledge. I only found this place a few weeks ago. Stay tuned Simon, i am chatting with Kalid recently to seek some collaboration. I will let you know the plan.

Thanks all! Yes, one of my goals for the site is to explore ways to make collaborative areas to find/surface the best analogies, diagrams, examples for the topics we’re trying to learn.

It is impossible for one individual, even a group of individual to come up with a suite of solutions to tackle the problem of learning.

My vision is: knowledge should be disseminated like utility: whether you want take a complete course, or you simply just want to google a term you overheard, be it “quantitative easing”, “fed rate hike”, “Fourier transform”, “Gilbert space”, etc. There is a way to explain the any key concepts l, the insights underpinning to give people some appreciation in 5-10 minutes. The very basic answer you need to provide is : what it is doing and why it is such a big deal, instead of purely the definitions and mechanics.

There is real demand for learning, and the world will be in a learning crisis soon (10 years later when tons of people lose their jobs due to machines, robots). And I cannot imagine a better way to teach people rather than this intuitive fashion. So the answer is to nurture a self-sustaining platform like youtube where everyone love to share true insights and knowledge, and overtime people will keep coming to this platform, simply because they need it and there is no better place to go.

Any people interested, please contact me at:

xzhu8216@hotmail.com

sorry, typo, should be “Hilbert space”

HI Kalid,

I really enjoyed your articles about complex number, complex multiplication, Fourier, Euler. It is my first time in life to realize complex number means rotation, and multiplication rules give you very simple rule to add angles.

Not sure if it is because I have not finished all your writing, maybe you have mentioned before. But I still have questions:

1) I see that complex number rotate, hooray, but there is a voice to myself: why would I want to rotate?

2) Then when I see the multiplication, then I realize it makes adding angles so easy. Hooray, but, wait a minute, why would I want to add angles at all?

3) Then I saw Fourier transform, I see where complex number is really used, yeah, but to be honest, isn’t it just a notation scheme? It is just a way to represent two numbers, two different things. Still, it is not convincing me why complex number is so magical;

Until this morning, it suddenly occurs to me, is the power of using complex showing up when handling convolution? Resolving into simple rules of adding angles makes it so much easier.

Please confirm if i am on the right track. And please point to me the right place you have already pointed out why complex number is useful.

Utility in math is an interesting issue :). Ultimately, math is a bunch of things we *can* do, ideas we can think of. Do we need to rotate? Add angles? Move in circles? Not really.

But sometimes we can construct a scenario where that idea is convenient. Complex numbers, which make rotations, angles, and circles easy, come to the rescue. Certain problems that looked impossible (Can negatives have square roots? Can an exponent follow a circular path?) become easy if we let ourselves rotate.

The Fourier Transform is probably the best example of complex numbers being useful. It’s very difficult to express the formula otherwise.

I never thought an x^2 has so many things hidden in it.I wish I could see things in a creative perspective

cool website, thanks for a clear presentation of maths.