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I liked this simple calculus exercise

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Re: I liked this simple calculus exercise

#111

Post by gizmo686 »

kaba0 wrote: post #55
laurieg wrote: post #12

I did well in high school math. These days, when something involving algebra, trigonometry, geometry etc comes up I feel like I have a good understanding of it but my calculus seems weak to non-existent. I'm not sure if it's how I was taught, how I studied it or something else but calculus always seemed like a huge step change in difficulty. That said, I love how this article gives practical hints on how to replicate…

One of my professors used to say that “even a horse can do derivatives. Integration is the real deal”, another one said that you integrate by “look at it, deeply, deeply, deeply; and then solve it”. The point is, many part of high school math is actually really “algorithmic”. I was one of the few in my class who absolutely loved coordinate geometry over “normal” geometry, because I simply felt really comfortable with…

Most integration problems require numerical methods. The ones with with an analytic solution are just where someone at some point found a way of solving them.

Re: I liked this simple calculus exercise

#112

Post by ThrowawayTestr »

red_admiral wrote: post #46

Here's another exercise (resp. exam question) that tests understanding: given a sketch of a curve in a graph, roughly sketch the derivative (or integral). The number of otherwise good students who go "but I can't do the derivative without the formula?" suggests we need more questions like this.

Sketching equations and their derivatives was how we started pre-calc in high school

Re: I liked this simple calculus exercise

#113

Post by jerf »

kaba0 wrote: post #55
laurieg wrote: post #12

I did well in high school math. These days, when something involving algebra, trigonometry, geometry etc comes up I feel like I have a good understanding of it but my calculus seems weak to non-existent. I'm not sure if it's how I was taught, how I studied it or something else but calculus always seemed like a huge step change in difficulty. That said, I love how this article gives practical hints on how to replicate…

One of my professors used to say that “even a horse can do derivatives. Integration is the real deal”, another one said that you integrate by “look at it, deeply, deeply, deeply; and then solve it”. The point is, many part of high school math is actually really “algorithmic”. I was one of the few in my class who absolutely loved coordinate geometry over “normal” geometry, because I simply felt really comfortable with…

To be honest, we would probably better serve our students in general by presenting integration as a numerical approximation, doing enough symbolic stuff to demonstrate that some integrations can be solved that way, doing enough other stuff to demonstrate why there's a ton of integrations that have no closed-form solution with any conventional functions, and then moving on to more productive things rather than blow over a full semester grinding out integrations. Integration by parts is useful as a method for exploring the concept more deeply, and there's a few other such tricks to be used primarily for ensuring the concepts are understood better. But I'm not convinced there's a lot of value in all these integration tricks.

Derivatives are friendly enough that I feel like a certain amount of grinding is justifiable, and it's justifiable as practice for symbolic manipulations in general. But we're leaving a lot of useful stuff on the table while we're jamming down how to integrate with trig identities and other such things.

But it's all pie in the sky anyhow. The Curriculum Must Not Be Changed. The Curriculum Is Perfect. Nothing Can Be Dropped From The Curriculum. I don't know what miracle would have to be worked to get people to reconsider the curriculum from some sensible perspective of what students should be taught rather than the way that question happened to be answered about 100 years ago when the curriculum froze into place, but it probably involves the total destruction of the school system at this point. I can't even get people to process the idea that shoving incomprehensible combinations of 450-year-old words in what is effectively another language at children and telling them this is High True Art is a bad idea, what chance is there of prying away the utterly vital fact that cos(θ/2) = SqRt((1 + cos(θ))/2) out of The Curriculum?

Maybe if colleges continue dropping the SAT and the ACT we can start actually fixing these curricula.

Re: I liked this simple calculus exercise

#114

Post by mjd »

physicles wrote: post #49

When I saw the equation referred to as (*), I had a flashback to those problem sets with *hard and **harder problems. ** problems often required some real out-of-the-box thinking. I wasn't always able to solve those, but it was so satisfying when I did (usually after an hour or two of struggle).

I just love that little five-pointed star symbol.

I also like labeling formulas with playing card suits. I think they are easier for the reader to distinguish when they are looking back for the labeled formula.

A while back I had the idea of marking erroneous formulas with a red spade. I'm going to try doing that again because it's hilarious.

Re: I liked this simple calculus exercise

#115

Post by nerdponx »

angry_moose wrote: post #108
red_admiral wrote: post #46

Here's another exercise (resp. exam question) that tests understanding: given a sketch of a curve in a graph, roughly sketch the derivative (or integral). The number of otherwise good students who go "but I can't do the derivative without the formula?" suggests we need more questions like this.

I remember there being a distinct lack of "closing the loop" on concepts in university math. Day 1 of the class: The derivative calculates the slope of a function Day 2: The integral calculates the area under the curve of the function Days 3-89: Rote exercises deriving and integrating increasingly obscure functions Day 90: Final Exam Spending a few days at the end re-exploring the "big picture day-1" to tie together…

This, bigtime. I loved the 3blue1brown series an example of how to do this better.

Re: I liked this simple calculus exercise

#117

Post by andruby »

red_admiral wrote: post #46

Here's another exercise (resp. exam question) that tests understanding: given a sketch of a curve in a graph, roughly sketch the derivative (or integral). The number of otherwise good students who go "but I can't do the derivative without the formula?" suggests we need more questions like this.

My school teacher asked us this exact exercise several times. He always made sure to link abstract concepts to real applications, as well as showing us how the mathematical concepts were discovered.

(note: this was not in the US, but in the early 2000's in a small European country)

Re: I liked this simple calculus exercise

#119

Post by mabbo »

ziroshima wrote: post #118

This is surely a stupid question: In the article, the graph sure looks like a right triangle, with a base of 2 and a height of 1. Wouldn't the area under this curve (from 0-2) be ~1?

That's a graph of `x/2 - floor(x/2)`. The question is about `e^(x/2 - floor(x/2))`.

Re: I liked this simple calculus exercise

#120

Post by arbitrandomuser »

jrumbut wrote: post #21
nateb2022 wrote: post #6

Current Calc 2 student here. I would be braindead approaching this problem honestly, I don't think I'd even know how to begin; I'm hoping that's normal. Why would the exponent be equal to x/2 - floor(x/2) be equal to x/2 on the interval [0, 2)? And how does the graph of x/2 - floor(x/2) imply anything about the behavior of e^(x/2 - floor(x/2))? I'm hoping I just haven't learned enough yet?

It's definitely a tricky problem for a student. The reason that everyone likes it is that this is exactly the kind of problem you run into when you need to solve an integral in the real world. Once a year at least I run into a math situation like this. Obviously in some professions it will be much more (or less) often. Exponents, floor, ceiling, and absolute value are very frequently part of the problem. The approach…

> Often you'll end up using numerical methods to approximate the solution, but if you can come up with a closed form solution that's much nicer. Also even with the example in the post , thinking about the problem at first and exploiting it's periodicity before naively integrating numerically over the entire limit affords much faster computation. Although might not be specific to this example , utilising such symmetries can sometimes turn even a numerically intractable problem or a problem that requires days / 100s of gigabytes into something one can do on their laptop in hours.
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