This would make for a good mental math interview problem. Thanks.
I liked this simple calculus exercise
61–70 of 163 posts
Re: I liked this simple calculus exercise
#62Earlier quoted context omitted.
I would classify a trick as something that happens to work but isn't rigorous. Like treating dy/dx as a fraction sometimes works, but only under certain conditions.
What's your favourite example of where it doesn't work? Physics is full of quasi-infinistesimal quantities and I always like good counter examples (ideally without invoking something like the blamange function or similar....)
Off the top aren't there differential equations that are inseparable? ie you cannot just pretend x and y are x(t) and y(t).
Re: I liked this simple calculus exercise
#63Slightly off-topic: didn't know what ⌊x/2⌋ is Google: x squared (???) GPT: The expression ⌊x/2⌋ represents the greatest integer that is less than or equal to x/2. It is called the floor function of x/2. For example, if x=5, then ⌊x/2⌋ = ⌊5/2⌋ = 2. If x is an even integer, then ⌊x/2⌋ = x/2. If x is an odd integer, then ⌊x/2⌋ = (x-1)/2.
It's the floor operator. Also known as round down.
Re: I liked this simple calculus exercise
#64Here'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.
The problem is that most teachers cannot come up with questions that go outside the small number of cases which the student trains on. I mean, coming up with fundamentally new questions is very hard work once the low hanging fruit is gone.
Re: I liked this simple calculus exercise
#65That technique won't scale if you're trying to integrate some crazy complex function.
Re: I liked this simple calculus exercise
#66That technique won't scale if you're trying to integrate some crazy complex function.
A "calculus tyro", as a general rule, is not trying to integrate some crazy complex function as part of a homework assignment.
Re: I liked this simple calculus exercise
#67Slightly off-topic: didn't know what ⌊x/2⌋ is Google: x squared (???) GPT: The expression ⌊x/2⌋ represents the greatest integer that is less than or equal to x/2. It is called the floor function of x/2. For example, if x=5, then ⌊x/2⌋ = ⌊5/2⌋ = 2. If x is an even integer, then ⌊x/2⌋ = x/2. If x is an odd integer, then ⌊x/2⌋ = (x-1)/2.
It's the floor operator. Also known as round down.
And introduced by Iverson for APL in 1962: https://www.johndcook.com/blog/2021/04/15/floor-ceiling-brac... and https://en.wikipedia.org/wiki/Floor_and_ceiling_functions .
Re: I liked this simple calculus exercise
#68This reminds me of an exercise I'll never forget from my Math Methods course: finding the derivative of arcsin(x). It seems almost impossible because, just looking at it, there seems to be nothing you can do to simplify it. Then, out of sheer nothing-else-to-do-ism, you take the sin() of it and realize sin(arcsin(x)) = x. Take the derivative of both sides, apply chain rule and draw a right triangle and you have the a…
One technique for finding the derivative of the sin function is to find the derivative of arcsin first. The arcsin function can be expressed as the area of a certain figure, and therefore admits an expression as an integral. From this, the derivative of arcsin is immediate. Finally, apply your technique to arcsin(sin(x)) = x and obtain the derivative of sin.
A similar technique finds the derivative of exp(x) from ln(x), by defining the latter as the integral of 1/x.
Re: I liked this simple calculus exercise
#69Earlier quoted context omitted.
I would classify a trick as something that happens to work but isn't rigorous. Like treating dy/dx as a fraction sometimes works, but only under certain conditions.
What's your favourite example of where it doesn't work? Physics is full of quasi-infinistesimal quantities and I always like good counter examples (ideally without invoking something like the blamange function or similar....)
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Re: I liked this simple calculus exercise
#70Earlier quoted context omitted.
I would classify a trick as something that happens to work but isn't rigorous. Like treating dy/dx as a fraction sometimes works, but only under certain conditions.
What's your favourite example of where it doesn't work? Physics is full of quasi-infinistesimal quantities and I always like good counter examples (ideally without invoking something like the blamange function or similar....)
It basically works without issue in 1d, simply because dx and dy can be considered modular forms and dy is exactly dx times the derivative dy/dx. You can even put an integral sign in front of them and calculate the corresponding integral.
Where this doesn't work is if you have more than 1 dimension. Then you need to deal with the added complexity of integrating modular forms and the fact that in 2D you don't have df = (df/dx) dx but df = (df/dx) dx + (df/dy) dx. The chain rule also changes into a matrix product, rather than a simple dz/dx = dy/dx dz/dy.