Maggie's Farm

Calculus Made Easy is the classic book along these lines. I read it as a freshman in high school and was able to help the seniors taking the AP calculus course. It was also used for a calculus class for business majors when I was a TA getting a Ph'D in math; the business students did better than the math students taking "real" calculus. The virtue of the book is that it teaches the idea of calculus and describes things in the way a mathematician or engineer typically thinks when they use it. It is light on rigor by choice, but mathematical rigor is a fancy suit for the corpse of the idea, the living part is having the idea and knowing where you want to go. The book is good for that.

Which Spivak? The books on differential geometry? Calculus on manifolds? The Latex stuff?

I'm not saying I don't like rigor, I love rigor, proofs are beautiful. But they don't come first, the idea comes first, and ideas rely on all sorts of things from analogies, to inspired guesses, to colliding blocks of color; there is a great deal of variation between people in that regard. But without ideas and direction students just wander in the wilderness without a compass.

I think a concentration on rigorous limits is a waste of time for beginners, better to present the concept with some pictures. A little topology under the covers helps the students see what all that epsilon-delta crap is about. And mind that Euler, Cauchy, and Gauss all used infinitesimals. Read some of Euler's demonstrations where expressions are raised to powers of infinite integers if you want to see how that was done.

It is also amusing to note that Riemann's thesis, the Riemann mapping theorem, famously contained a flawed proof that wasn't fixed up until some 50 years later. Yet no one doubted that the result was correct and the idea was quite wonderful.