In this chapter you will find a collection of familiar topics. You need not spend a great deal of time here. Rather, review the material as necessary, until you are comfortable with all of the topics discussed. We have primarily included material that we consider essential for the study of calculus that you are about to begin. We must emphasize that understanding is always built upon a solid foundation. While we do not intend this chapter to be a comprehensive review of precalculus mathematics, we have tried to hit the highlights and provide you with some standard notation and language that we will use throughout the text.
As it grows, a chambered nautilus creates a spiral shell. Behind this beautiful geometry is a surprising amount of mathematics. The nautilus grows in such a way that the overall proportions of its shell remain constant. That is, if you draw a rectangle to circumscribe the shell, the ratio of height to width of the rectangle remains nearly constant.
There are several ways to represent this property mathematically. In polar coordinates (which we present in Chapter 9), we study logarithmic spirals that have the property that the angle of growth is constant, producing the constant proportions of a nautilus shell. Using basic geometry, you can divide the circumscribing rectangle into a sequence of squares as in the figure. The relative sizes of the squares form the famous Fibonacci sequence 1, 1, 2, 3, 5, 8, . . . , where each number in the sequence is the sum of the preceding two numbers.
The Fibonacci sequence has an amazing list of interesting properties. (Search on the Internet to see what we mean!) Numbers in the sequence have a surprising habit of showing up in nature, such as the number of petals on a lily (3), buttercup (5), marigold (13), black-eyed Susan (21) and pyrethrum (34). Although we have a very simple description of how to generate the Fibonacci sequence, think about how you might describe it as an algebraic function. A plot of the first several numbers in the sequence (shown in Figure 0.1) should give you the impression of a graph curving up, perhaps a parabola or an exponential function.
In this chapter, we discuss methods for deciding exactly which function provides the best description of these numbers.
Two aspects of this problem are important themes throughout the calculus. One of these is the attempt to find patterns to help us better describe the world. The other theme is the interplay between graphs and functions. By connecting the powerful equation-solving techniques of algebra with the visual images provided by graphs, you will significantly improve your ability to make use of your mathematical skills in solving real-world problems.