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I liked this simple calculus exercise

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Re: I liked this simple calculus exercise

#131

Post by deanCommie »

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…

It's interesting because I was going to argue with you until I applied the same idea to a subject I couldn't care less about if I tried in school, but now think is deeply fascinating: History.

See, I loved math. So all through Calculus I could easily remember the big picture. Every time I practiced an Integral I imagined curves and calculating the areas under them and the visual problem and the relevance of what the curve represented in real life and the massive amount of applications it could be used for in the real world lit up my neurons like fireworks in the sky.

(Then I went into computer science, and wound up never needing calclus again, based on the type of work I happen to be doing, but alas)

But where I really would have wished a constant reinforcement of the "big picture" is History. Cuz it always seemed so pointless and useless. Why are we studying these old dead people, and everything they did. Who cares? They're old, and they're dead, and nothing they did matters to us anymore.

Until you grow up, and go from your 20s to your 40s, and suddenly realize oh shit we're living THROUGH history. We're creating history NOW. We're making choices, and we're making mistakes just like those old dead people in history. Old dead people that weren't really any less developed or evolved primates than us. Just equally victims of their circumstance like us, and also agents of change like us.

Suddenly history seems much more significant.

Re: I liked this simple calculus exercise

#132

Post by [deleted] »

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.

[deleted]

Re: I liked this simple calculus exercise

#133

Post by never_inline »

aashutoshrathi wrote: post #23

We used to get lot of such tricky stuff during the preparation of IIT-JEE here in India, and I'm telling you if you don't understand Area under curve is integral, you can't touch most of the questions. But I get your point, if you are interested in such questions, you should checkout IIT JEE mathematics question, you'll love them

I am from a rural region where IIT-JEE is not that popular, and I liked working these physics and maths problems initially.

Unfortunately the competition has become so intense you practically need coaching (which is expensive) and dedicate lot of time, at that point it becomes grunt work. There are many "tricks" and "shortcuts" taught in these coachings which doesn't exists in normal NCERT syllabus. Needless to say I didn't do very well.

Re: I liked this simple calculus exercise

#134

Post by bluepod4 »

hgsgm wrote: post #78
bluepod4 wrote: post #60

Earlier quoted context omitted.

I was aware of the floor function (and the corresponding ceiling function) since I’m a software engineer. But I wasn’t aware that you could graph it. It never came up in high school or college math. And I never thought about it. Of course, it makes sense now that I’ve seen it.

Can you name a single-variable function that software engineers use that can't be graphed? How would such a function be useful?

No. I think the term “function” is overloaded here.

My point was that I viewed it solely as a _programming_ function and not a _mathematical_ function (even though it exists in math libraries), hence my last sentence “Of course, it makes sense now that I’ve seen it.”

Out of all the functions in math libraries that I’ve used, floor/ceiling are the only ones where I had this idea for some reason. It was obvious to me that Math.sin(x) and Math.abs(x) can be graphed. I’ve seen those graphs over and over again. But whenever I used the floor or ceiling functions, I just thought in terms of rounding up or down with a predetermined rule to finish whatever piece of code I was working on.

But as others have pointed out, I have actually worked with the floor function as a mathematical function in several math classes. I have seen the graphs. They just weren’t called floor or ceiling functions.

I just never made the connection that they were the same and I don’t recall any computer science professor or TA “bridging the gap” to what was learned in the math classes.

Re: I liked this simple calculus exercise

#135

Post by bluepod4 »

wholinator2 wrote: post #76
bluepod4 wrote: post #60

Earlier quoted context omitted.

I was aware of the floor function (and the corresponding ceiling function) since I’m a software engineer. But I wasn’t aware that you could graph it. It never came up in high school or college math. And I never thought about it. Of course, it makes sense now that I’ve seen it.

It's also sometimes referred to as a step-function though i believe that name encompasses many more functions than just the floor

Yeah, in college I’ve worked with step/unit functions in both my “differential equations” and “signal and systems” classes for sure.

But it never occurred to me and it was never presented by professors or TAs that they were essentially the same as floor/ceiling functions.

Or maybe I just forgot since it’s been more than a decade?

Re: I liked this simple calculus exercise

#137

Post by sizzle »

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…

I wish I could give this comment hold, straight up put me off math because of this lack of real world application in lower level calc and adhd brain craving meaning and substance.

Re: I liked this simple calculus exercise

#138

Post by postsantum »

deanCommie wrote: post #131
angry_moose wrote: post #108

Earlier quoted context omitted.

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…

It's interesting because I was going to argue with you until I applied the same idea to a subject I couldn't care less about if I tried in school, but now think is deeply fascinating: History. See, I loved math. So all through Calculus I could easily remember the big picture. Every time I practiced an Integral I imagined curves and calculating the areas under them and the visual problem and the relevance of what the…

Funny enough, I ended up with the opposing view - history doesn't matter, narratives do. During the last year I learned about propaganda, psy ops and myths much more than I knew before. And at the risk of sounding as an edgy teenager, I can vouch for the phrase "History is written by victors" which stopped being a platitude for me and became a part of my worldview

Re: I liked this simple calculus exercise

#139

Post by tgv »

itengelhardt wrote: post #50
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?

Man, thank you for writing floor(x/2) because I've been looking at the original post and didn't know what those silly brackets are supposed to mean.

Surprising how many people have never seen floor/ceiling notation.

Re: I liked this simple calculus exercise

#140

Post by mrguyorama »

pavlov wrote: post #82
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 learned basics of calculus as a teenager making animations with a pirated copy of Adobe After Effects. It’s a motion graphics package where you can animate any property of an element over time using both absolute values and velocity curves (i.e. the derivative). It shows both curves next to each other, so you can tweak either and see how the other changes. Seeing graphics animate according to a derivative that you…

Geogebra and friends are more powerful and open-ended, education focused systems. I played around with it a bit in freshman year Geometry.
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