If you don't use math, you can skip this one. When I took calculus I always wondered about the choice of topics. For example, they would teach us obscure tricks for integrating weird-looking functions that obviously had some kind of significance, but they never explained what it was.
As it turned out, I do use calculus all the time, so I feel as qualified as anyone to give some perspective on the "why?" of the table of contents. So here goes. This is Finney and Thomas, 7th edition (1988).
Before I get rolling, I've heard some people say calculus is old-fashioned and should be de-emphasized in favor of "finite math", which is a catch-all of linear algebra, probability, finite differences - anything but calculus. Yes, calculus is increasingly irrelevant and only of interest in case you want to learn the laws of physics, and who cares about those any more? If you just want to throw around Greek letters to put a gloss of credibility on your ravings, you can skip the calculus and make a fine living as an economist.
1. The Rate of Change of a Function. Provides the entire motivation for differential calculus. A whole chapter just to beat the notion into you that a curve has a different slope at every point. They have to do this because you've just spent two years or more studying linear functions that have the same slope everywhere. Chapter finishes up with a little bit of theory about continuity. But aren't all functions of any interest continuous? What possible application could there be for a function with jumps or holes in it? Well, it turns out, a lot. The boundary between two materials, a shock wave, a piecewise interpolation - discontinuous and you had better understand what it means.It would help if they explained this.
2. Derivatives. Differentiation is so easy that it takes maybe a week to learn, but what does it mean? There are functions like polynomials that, if you keep taking the derivative, eventually give you zero. Then there are other functions like sines and cosines that you can keep differentiating and they just sort of cycle. You use the two in different ways.
3. Applications of Derivatives. This chapter is all context, so no more explanation is needed.
4. Integration. The integral of a function is both the area under the curve, and the inverse of differentiation. If that doesn't blow your mind, you're not paying attention.
5. Applications of Definite Integrals. All context.
6. Transcendental Functions. It's all about the weird little function exp(x) and its inverse, ln(x). A whole chapter just to talk about one function? Yes; it's only the most important function in math. A function that is its own derivative? Doesn't seem that big a deal until you realize that the laws of physics are differential equations, and solving differential equations using functions that stay the same when you differentiate is so easy it's fun. It's so easy that even when we can't do it, we do it anyway. That is a math joke.
7. Methods of Integration. This is the "long division" of calculus, a chapter I think could be scrapped, except for integration by parts, which is never explained well. All it is is the integral of the product rule of differentiation. The rest of the chapter is integrating those weird functions like even powers of sine and cosine, and rational functions. Those would be a lot easier to take if they'd just explain that the powers of sine come about when you do Fourier series, and the rational functions come about when you do Laplace transforms. They could also explain that nobody does that stuff by hand any more, but then the kids wouldn't study.
8. Conic sections. Pointless unless you're planning on becoming a 17th-century astronomer.
9. Hyperbolic functions. Why? They're less useful than, say, Bessel functions, which they would never think of covering in a freshman course.
10. Polar coordinates. There's no point in covering this until the student is ready for vectors in polar coordinates. Trying to discuss the topic without vectors is a waste of time and should be skipped.
11. Infinite series. Fascinating, but only mathematicians ever worry about whether a series converges. When we use series solutions in the sciences, we say they converge if the first few terms get smaller and smaller, which is a totally different concept from what they're talking about in this chapter. Plus, we use series that in fact don't converge (asymptotic series), so why worry about convergence? I am not sure I want to say it should be left out. It really is a fascinating topic, with some wildly counterintuitive results and crazy stuff like
π = 4-4/3+4/5-4/7+...
12. Power series. Just do Taylor series, which are undeniably vital, and skip the business about indeterminate forms and such.
The rest of the book is vector calculus and differential equations. It's funny that the much-feared course Differential Equations is just one chapter at the back of the freshman calculus book. I was so scared I got a C+ in it, but it's just linear, ordinary differential equations and once you start using the ideas it's trivial. What is really funny is that they actually introduce series solutions of differential equations. In a freshman calculus book. That people like pre-med majors are going to use. What do they think, a cardiologist is going to use a series approximation to your EKG trace? They should leave it until the real differential equations course.