I’m not sure that elementary calculus is particularly related to machine learning (I have a similar opinion of linear algebra which is fundamentally geometric in a way that machine learning isn’t). That doesn’t mean that you don’t need to know anything about calculus or matrix multiplication to understand machine learning but knowing eg Green’s theorem won’t help and neither will understanding a tensor as a multi linear map or in the way a general relativity physicist might.
I also don’t particularly know what goes into calculus (in the U.K. we studied something called ‘calculus’ in high school which included integrating/differentiating polynomials, some trig functions, easy integration by parts and, in the ‘further maths’ course, some second order linear ODEs with forcing, first order ODEs via integrating factors, first order linear systems of ODEs via the eigenvectors method, and I think some integration by parts based recurrence relations. At university things were divided into ‘calculus’, which contained practical tools for applied maths like Green’s theorem or partial derivatives or contour integration or Sturm–Liouville theory, and ‘analysis’ which had foundational things like epsilon–delta stuff or Dedekind cuts or the definition of a limit or Riemann integration or the conformal mapping theorem and so on.
I think a first course in the thing I called analysis above is very useful for building mathematical maturity (ie the ability to not deduce false things but also playing with definitions and thinking about counter-examples) but the calculus knowledge can be useful for understanding the physical world. But I don’t know if that understanding should be required for e.g. computer scientists.
A few calculus examples I can think of in computer science:
- Some famous story of Feynman ‘interning’ at Thinking Machines and solving some capacity management problem using bizarre differential equations with terms representing e.g. ‘bits per second’. No sufficiently good solutions had been found using discrete methods.
- I was once asked an interview question which I suggested solving with differential equations but I was quickly directed towards not doing that.
- Honestly I can’t think of many more but maybe this is a lack of imagination. I think there are a few things that are really probability theory that you need some understanding of calculus for, e.g. emergent behaviour of distributed systems, reasons to prefer random cache eviction, some intuition to answer a question like ‘if the time X takes has some distribution, but sometimes we have a gc pause for y milliseconds, how would that affect the distribution?