We begin this chapter by studying three different models for the velocity of raindrops falling under the influence of gravity. The first model assumes nothing else -- just a constant acceleration -- and we quickly see that this is an unrealistic model. The other two models take air resistance into consideration, and we find that one of these appropriately models very small raindrops, the other very large ones.

This last model is an initial value problem for which we have no tools to find a symbolic solution, that is, an explicit formula for velocity as a function of time. (Those tools will appear later in the course.) We can, however, generate approximate numerical solutions by picking up where we left off in Chapter 4, using the differential to approximate the difference in function values from one time to the next.

Recall that the differential is a rise along the tangent line to a curve, and rise \(=\) slope \(\times\) run. Using the rise on the tangent line to approximate the rise on a curve is the heart of an iterative numerical method for solving initial value problems: Euler's Method.

We have already used iterative methods to solve different types of problems. For example, in Chapter 2 we saw that we could record the growing balance in an interest-bearing bank account by an iterative formula that calculated each year's balance from the previous year's balance. (Replacing that iterative formula by an explicit formula led us to exponential functions.) In Chapter 4, we saw that algebraic equations in one unknown can be solved by an iteration called Newton's Method.

The idea of iteration -- repeating the same type of step over and over -- is a powerful concept that we will use many more times in this course. Most iterations would not be practical to carry out with pencil and paper, but the concept is ideally suited to computers. Indeed, this concept is at the heart of most applications of electronic computing technology.

We turn next to modeling repetitive behavior, for which none of the functions studied thus far -- exponential, logarithmic, and algebraic functions -- provide adequate models. Polynomial functions may have peaks and valleys, but not in a regular, repeating pattern.

Where do we see repetitive phenomena? Our hearts (if healthy) return to the same state every second or so. The Sun rises and sets in a regular cycle of about 24 hours. The electric current provided by our local utility (if properly regulated) alternates in direction 60 times every second. The planets, asteroids, and some comets travel around the Sun in repeating orbits. The hands of our analog clocks and watches traverse a complete circle every minute, hour, or half-day. And all the operations in our computers are timed by internal clocks that cycle millions of times every second.

To model such repetitive processes, we need to identify and study basic repetitive functions, the ones that will play the role of building blocks -- as do power functions in the study of falling bodies and exponential functions in the study of biological growth or radioactive decay.

We will see that these basic functions satisfy second-order differential equations -- and that we can expand the idea of Euler's Method to find approximate solutions of second-order initial value problems. The insight we gain from graphs of approximate solutions will lead us to formulas for the basic repetitive functions (sine and cosine) and their derivatives, thus bringing us full-circle in this chapter. (If you see that as a pun, we definitely intended it.)

Having introduced sine and cosine, which should be familiar from trigonometry, we take up some additional problems that involve related trigonometric functions, their inverses, and the derivatives of these functions. At the end of the chapter we summarize what we have learned about derivatives and differential calculus of our several families of functions that have turned out to be useful for modeling important phenomena.