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Calculus they won't teach you [video]

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Re: Calculus they won't teach you [video]

#91

Let me know, not if anyone here can do Calculus (that is ordinary; who can't? ), but rather whether you actually personally use Calculus, and please describe generally an example of needing Calculus to figure something out in the last 5 years. I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever…

I use calculus quite a lot, basically any time I need to go beyond an existing solution I could look up in a textbook. That is, all the interesting problems! For example in writing this article, especially details in footnotes 4 and 13: https://jbconsulting.substack.com/p/voiding-the-warranty-how...

When I want to write or tune a Kalman filter or some other control system, I need to work out the differential equations that govern a system, often from first principles if the system isn't a simple textbook example like an inverted pendulum. Sometimes I need to design that system myself or find out what change to its dynamics will fix a noise problem (eg. add a capacitor here, etc), and without an understanding of calculus I would be searching in the dark or stuck trying to look up known solutions in textbooks.

It also comes up a lot when I want to do probability or statistical calculations, because I end up needing to integrate various functions or products of functions to find means / medians / covariances / etc. Sometimes I can get away with leaning on Mathematica to do those integrals for me, but sometimes I can't, like when one of those functions is an arbitrary input like f(x). Even when I can use Mathematica to perform the integral, I need to know calculus for that to be any clearer than a magic spell. Without knowing calculus, how would I ever guess that I can solve my problem by typing "ln(x0-a)/arctan(x + a)" into wolfram alpha and then plotting the result?

Recently I had the need to do work backwards: find the function that satisfies certain properties like - being smooth and monotonic, zero at the origin, having a certain area within certain bounds, having a certain covariance, nowhere going to infinity, etc. This sounds very abstract but was the key to cracking a concrete signal processing problem.

And then there's all the optimization work, where I need to write numerical code that will look for the maxima or minima of some unknown function, to eg. design the best possible motor you can pack into a box of certain size and weight. 3rd-party libraries are certainly helpful but I usually find I need to supplement them with domain logic to keep it from getting stuck in a local optimum or rule out physically impossible search spaces.

A few months ago I needed to write a finite-element solver to handle soft, thin woven fabrics, where the material can support tension but not compression stresses and the material properties are different depending on the axis of the weave. I would have been at a total loss to get that done without calculus.

Even when I am pulling something out of a textbook, most engineering textbooks past the first-year-undergraduate level will presume some familiarity with calculus.

Not to mention properly thinking about current issues like inflation, CO2 emissions, etc, where the discussion bounces back and forth between integral and derivative quantities.

I'm not even a "math guy". I didn't do well in math in school, and I always struggle a lot to learn new math until I have an application in mind that I need it for. It's only because of how often I need it for my projects that I've ended up learning a lot of math, not the other way around.

Re: Calculus they won't teach you [video]

#92
post #77
post #30

Earlier quoted context omitted.

Another concrete example of the conflict is classical vs quantum physics which is nearly exclusively taught as the historical story of the quantum revolution or whatever you call it around 1900. It would be trivial to skip the historical story and just "yo here's a FET transistor, a solar cell, and a vacuum tube, and the math just works, so get used to it" but nobody is ever taught that way. Would be interesting for…

>On the other hand, in IT, this style of learning is considered normal. "This is how the kool kids do things today. The past does not matter and probably didn't exist anyway. The end." And yet nand2tetris is one of the most common suggestions for people who actually want to figure out how computers work. Because you actually are posed with problems that real world engineers used to struggle with, and are allowed to p…

But nand2tetris doesn't have a historical approach if i'm not mistaken.

Re: Calculus they won't teach you [video]

#93

Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).

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Re: Calculus they won't teach you [video]

#94
post #89

Earlier quoted context omitted.

There was another step: Leibniz' pre-limits formulation of calculus used hyperreal numbers: infinite and infinitesimals (small-delta) numbers. This version is conceptually simpler and for many people more intuitive than the limits-based explanation. It was the standard approach for many years. Infinitesimal calculus was discarded in late 19th century pedagogy in favor of limits as no rigorous proofs yet existed of it…

For the formal underpinnings of the infinitesimal approach, the phrase to look for is "Nonstandard Analysis": https://en.wikipedia.org/wiki/Nonstandard_analysis Fun stuff.

I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis?

I loved real analysis in college, for example I found cantors set kind of crazy. Does that "exist" in nonstandard analysis? Or are there other mind-bending implications?

Re: Calculus they won't teach you [video]

#95

Earlier quoted context omitted.

What's wrong with common core?

It could be worse but one thing that constantly bothers me is the stupid new vocabulary. One of the worst examples: "number sentence". It means what you, an adult, might guess it does (well, it means one of the things you might guess it means) but why use it at all ? They introduce it before the kids have a decent idea of what a normal sentence (what, a "word sentence", I suppose?) is. Many of the symbols in them are…

"Number sentence" does not appear in the common core standards. https://www.nctm.org/uploadedFiles/Standards_and_Positions/C...

Re: Calculus they won't teach you [video]

#96
post #26

Earlier quoted context omitted.

I don’t see why the history of discovery of something should be a particularly good way to rediscover it. As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth f…

“How do we know the world is round” is an interesting example to use. Eratosthenes got an estimate for the radius of the Earth that was pretty close using, I dunno, sticks and his eyeball or whatever ancient Greeks had. The history is useful there, I think. First off, it is motivating — that’s a pretty big question, really shows what math can accomplish without much hardware. Second, it provides some sort of useful c…

You can get a vaguely credible estimate by standing on a (small!) hill and measuring the distance to the horizon. Works better if you have a body of water handy, because bodies of water are nice and flat.

Re: Calculus they won't teach you [video]

#97

Let me know, not if anyone here can do Calculus (that is ordinary; who can't? ), but rather whether you actually personally use Calculus, and please describe generally an example of needing Calculus to figure something out in the last 5 years. I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever…

I can't. Promptly forgot it all which is why I will never bother to go get a graduate degree.

Re: Calculus they won't teach you [video]

#98
post #58

Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).

Integral calculus before differential calculus is how it is done in Apostol's "Calculus". Here are the first several chapter titles to give an idea of what is covered when. Introduction The Concepts of Integral Calculus Some Applications of Integration Continuous Functions Differential Calculus The Relation Between Integration and Differentiation The Logarithm, The Exponential, and The Inverse Trigonometric Functions…

My freshmen Calculus course used Apostol. I found it a great approach, and it was a lot of fun trying to prove ourselves the theorems in the book.

For many students, intuitively understanding the infinities and infinitesimals in integral calculus come easier than those in differentiation.

Re: Calculus they won't teach you [video]

#99
post #88
post #39

Earlier quoted context omitted.

Erastosthenes did it by comparing the angle of the shadow cast by a stick on the summer solstice at noon in two cities, Alexandria and Syene (now Aswan), which are about 800 km apart. What was special about Syene was that it is so close to the Tropic of Cancer line that the sun would hit the bottoms of wells on the solstice, meaning that he only needed one stick, in Alexandria, and the distance between the cities, to…

Darn, at some point I started imagining he had a protocol for synchronizing clocks over a distance. assuming the difference was in longitude. (E.g. Symmetric NTP over carrier pidgeon.)

Longitude was a much tougher problem to solve.

Re: Calculus they won't teach you [video]

#100
post #84
post #63

Earlier quoted context omitted.

Starting out with integration makes so much more sense, and it's way more graspable from a geometric standpoint (what's the area under this line?) as opposed to the weirdness (and relative abstraction) of the derivative.

I completely disagree. Also, there's basically one way to differentiate (not really, but kind of) wheres integration is so tricky there are many, many ways to do it.

Freshmen calculus I does not have to immediately jump into mechanically difficult integrals. You can leave those for Calculus II and concentrate on the fundamentals of integration and differentiation in Calc I.
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