This comes at a very opportune time in my life. when, my ward enters their HS -Junior year and they are taking SVC. Question to the Author(in case they are monitoring this thread), is it appropriate for a High-schooler with just an intro to Python?
_Make: Calculus: Build models to learn, visualize, and explore_ by Joan Horvath, Rich Cameron
https://www.goodreads.com/book/show/61739368-make
It's a series with matching books for:
Geometry: https://www.goodreads.com/book/show/58059196-make
Trigonometry: https://www.goodreads.com/book/show/123127774-make
For example look at the diagram on[1]. It has unlabled axes of a shaded L-shaped box with a curve going through it and then it is followed by a bunch of equations where he derives the formula for integration by parts using a set of parametric equations with multiple substitutions etc. I know what integration by parts is and how it works really well. This is possibly the most confusing way you could possibly derive the formula and/or explain it, and this diagram really adds absolutely nothing unless you already know and understand the concept. If you have seen the exceptional diagrams and illustrations and clarity of explanation in a book like "Calculus" by James Stewart the contrast is so stark it really jumps out.
The normal way of explaining integration by parts starts the way the author does (with the product rule for derivitives) and shows a few examples of taking the derivitive of things using the product rule so you get an intuition for what the form of the resulting antiderivitive looks like. You then go through the derivation of the formula for integration by parts by using the product rule for derivitives, integrating both sides and splitting the resulting integral[2]. Its very clear and easy to follow. And if they want to actually help the student to do this they teach something like the tabular/"DI" method so the student isn't tearing their hair out getting the signs mixed up when integrating by parts multiple times.
[1] https://jverzani.github.io/CalculusWithJuliaNotes.jl/integra...
[2] My own notes on this derivation which I took when learning this are here. Note that this isn't me really trying to explain it to a beginner - it's just my personal notes but it's still a lot easier to follow than the example given https://publish.obsidian.md/uncarved/3+Resources/Public/Inte...
[1] because that indicates I don't understand it well enough myself and I need to work on it more and then fix the note.
Get the book Quick Calculus by Kleppner and Ramsey. Nothing else I used came close, for developing intuition about the concepts when they're fresh and new.
Once they have mastered that, any good book will do. James Stewart's is fantastic but so big and thick you may need to use it as a resource to intelligently select appropriate readings and problems from as they go, rather than just start at page 1 and assign every problem. But the main thing is to really get the basics of what a derivative and integral and limit are from the getgo, and nothing I tried compare to Quick Calculus for that.
Using this Julia book or some other similar book to select exercises based on your appropriate judgment from it to supplement Stewart would also be very cool if your student has an interest in programming. Let them complete then in python if they like, python should be fine. I recall implementing a numerical integrator and a symbolic differentiator in python at the same age, as a way of consolidating my knowledge of calculus, and found both to be very valuable and fun exercises. Symbolic differentiation seemed like magic especially, but it was just a matter of parsing and then continually adding rules for each rule I learned in math.
I don’t know what sort of course this one is. It starts with limits so maybe more the latter kind, but I don’t really see how Julia would be very useful there. My best guess about the suitability of the course is that it may assume mathematical maturity that you may not have, rather than anything about programming. Mathematical maturity tends to mean being able to cope with precise definitions and abstract concepts without deducing a load of wrong things, as well as being able to follow the kind of argument that is a mathematical proof.
But I have to correct you on the history. Essentially none of the material in a modern analysis course is the same as in Cauchy's day. You identify integration as coming after. But Cauchy didn't have a definition of a limit, axioms for the real numbers, a construction of the real numbers, set theory, the modern idea of open and closed sets, and so on and so forth.
Cauchy's approach was to define an infinitesmal as a sequence approaching 0, and then to use the old infinitesmal definition of a derivative. This almost works. One of the places where it falls apart is defining the chain rule when the inner derivative is 0, because the sequence may wind up with 0/0 infinitely often.
But it is from this definition that we get Cauchy sequences today.
Personally I have found that the best tools are 1) a firm grasp of elementary calculus (differential, series, and integral) and 2) exposure to simple numerical methods that apply to a broad range of problems.
Armed with this knowledge, in my opinion, Julia is a far superior language to python and its package ecosystem has a brighter future. Indeed, the language has been built with a focus on mathematical modeling and efficient numerical computation. It is a more natural starting point for those with an interest in mathematical modeling and engineering science, and will serve anyone who learns it better than python+numpy+sympy+scipy.
That might sell me alone.