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Calculus they won't teach you [video]

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

#81

Post by kuhewa »

whimsicalism wrote: post #69
Maursault wrote: post #65

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…

> that is ordinary; who can't? In my experience, almost everyone out of college who I've met who claims to know calculus.

At least if you've actually learned the concept of derivatives and integrals, it wouldn't be that hard to refresh enough for a given use case. I'd agree that most of us that have taken a year of university calculus don't know calculus.

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

#83

Post by 77pt77 »

tzs wrote: post #58
jwsteigerwalt wrote: post #42

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…

And it's not a good book.

Curiously his book on mathematical analysis is excellent, except that many of the exercises were downright wrong.

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

#84

Post by 77pt77 »

dvt wrote: post #63
tzs wrote: post #58

Earlier quoted context omitted.

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…

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.

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

#85

Post by dvt »

77pt77 wrote: post #84
dvt wrote: 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.

I totally understand your position, and I completely agree that integration is way more complex, often doesn't work, and very quickly gets you in the Gamma function danger zone, but I still think it should be taught first (maybe just looking at elementary functions) because it's way more intuitive in my opinion.

There's a reason it was developed first, historically speaking.

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

#86

Post by 8note »

dvt wrote: post #63
tzs wrote: post #58

Earlier quoted context omitted.

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…

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.

Derivatives have nice physical problems to solve like "find the maximum" and "find the relationship between these two variables" which you can find by measuring a linear slope a lot of times. Learning derivatives starts with linear regression, which you do when learning about lines on a Cartesian plane.

By the time you know about drawing lines on planes to find the integral area underneath, you've already figured out the derivative so you know what direction to move your pen in next while drawing the line

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

#87

Post by dvt »

8note wrote: post #86
dvt wrote: 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.

Derivatives have nice physical problems to solve like "find the maximum" and "find the relationship between these two variables" which you can find by measuring a linear slope a lot of times. Learning derivatives starts with linear regression, which you do when learning about lines on a Cartesian plane. By the time you know about drawing lines on planes to find the integral area underneath, you've already figured out…

> problems to solve like "find the maximum" and "find the relationship between these two variables"

I think these are actually pretty abstract, so it's hard exactly to grasp what a derivative is until maybe Calc II or III -- imo, the main problem is that it just seems like mathematical black magic: you need to define a limit, you need to use that definition to subtract two slopes, the complexity of which magically kind of vanishes, and you end up with a "method" for spitting out derivatives. Whereas finding area under a curve seems like a common sense thing to do: you take smaller and smaller slices of a thing and add it all up.

I do agree that if you really learn anti-derivatives (which, by all accounts, is harder), you basically get derivatives for free.

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

#88

Post by a9h74j »

not2b wrote: post #39
bee_rider wrote: post #33

Earlier quoted context omitted.

“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…

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.)

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

#89

Post by makr17 »

pjlegato wrote: post #62
blagie wrote: post #47

Earlier quoted context omitted.

That's the historical progression, not the teaching progression. That's the point. Integrals were conceived over 2000 years ago. Derivatives came Newton-ish. Formalism with limits came last.

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.

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

#90

Post by a9h74j »

sifar wrote: post #20
msla wrote: post #14

Earlier quoted context omitted.

Things were usually discovered in a confusing way that kind of made sense at the time, but had false starts and conceptual limitations driven by people still being in an older mindset when they were trying to discover it. A good teacher sands off those edges, leading to an oversimplified history, but not starting from the firm conceptual footing we only got later on still makes things overly difficult to grasp. Good…

>> Things were usually discovered in a confusing way that kind of made sense at the time, but had false starts and conceptual limitations driven by people still being in an older mindset when they were trying to discover it To know this gives us some perspective on our own biases/limitations and not take things as gospel.

> some perspective on our own biases/limitations

This is sometimes prescribed in such form as: "Fraction X of your reading should be of authors writing at least 100 years ago."

Similar: In defense of the reading of old books by C.S. Lewis ...

https://www.youtube.com/watch?v=jFHQDIE7CSw

"All contemporary writers [of any period] share, to some extent, the contemporary outlook."

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