I’ve gotten back into my quest to learn calculus, but I’ve run into a little hiccup in the form of what appears to be a fundamental error in the book I’m using (Silvanus P. Thompson’s Calculus Made Easy, Second Edition (the PDF version at that link).

On page 18, we see this paragraph:

What does (dx)^2 mean? Remember that dx meant a bit – a little bit – of x. Then (dx)^2 will mean a little bit of a little bit of x; that is, as explained above (p. 4), it is a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms.

Emphasis mine. This is the part that really made me go “Huh?”, but I figured that the author knew more than I do, and decided to leave it.

Then later, on pages 19 & 20, we are given an example of differentiating y=x^3, and just to make sure I was understanding things correctly I tried to work it out. I decided I would have x=3 and dx=1. Here is the example from the text:

This ends with dy/dx = 3x^2, but that part won’t copy for some reason.

But, there’s a problem. If, as I did, you set x=3 and dx=1, then you end up with y=27 and y+dy=64, making dy=37. But, if you take (3x^2)(dx) when x=3 and dx=1, you end up with dy=27.

So, I tried working it out without discarding (dx)^2 and (dx)^3, and got:

y + dy = x³ + 3x²(dx) + 3x(dx)² + (dx)³

y = x³, so they cancel and give:

dy = 3x²(dx) + 3x(dx)² + (dx)³

dy = 3(3²)(1) + 3(3)(1²) + (1)³

dy = 27 + 9 + 1 = 37

The right answer.

A 10 out of 37 error is not negligible as the book asserts. So, can anyone who knows calculus tell me who is wrong here? Me, or the book? And why?

Help!

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One of the things I regret not having done yet is learning calculus. I know I’m perfectly capable of it – math has always been pretty easy for me, if I actually put my mind to it and focus – but I was psychologically unprepared for college. When I did take calculus, not only was I heavily distracted from school by some personal issues, but I also got shafted with a professor who was unintelligible ( a “visiting professor” from India – she spoke English well, but her accent was so thick she was barely intelligible in normal conversation, much less when rattling off (at warp speed) mathematical terms and formulae to a bunch of freshthings). Needless to say, I didn’t learn squat. I ended up changing majors right after that and not needing to take calculus anymore.

So, I recently decided to teach myself. I’m starting off with Calculus Made Easy, available free at Project Gutenberg. It seems to be a good text, and comes highly recommended. I’m also considering supplementing it with the Khan Academy, which I’ve also heard good things about.

I’ve already run into one problem, though. Since it’s been almost 15 years since I’ve had to do any kind of higher math (very basic algebra, at most), I’m finding myself, even in the first couple of chapters, having to go back and look up or review a lot of what I learned in high school – even some basic geometry.

As you might imagine, for a proudly self-professed geek like me, this is both irritating and somewhat embarrassing.

So, does anyone have any recommendations for a good review source for someone in my position?

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[Also, I added a new category: Self-education. I’ll try to post occasional updates, in case anyone is interested.]