In Chapter 8 we studied a technique for representing periodic functions as combinations of trigonometric functions. In this chapter we take up the equally important problem of representing functions — periodic or not — by polynomials. These polynomial representations will enable us to address — if not completely answer — such questions as

• How is it possible for your computer or calculator to calculate numeric values of logarithmic, exponential, trigonometric, and inverse trigonometric functions?

If you have come to take those built-in functions for granted, you may prefer to focus instead on these questions:

• How accurate are those computations?
• Can you believe all the digits a computer or calculator produces?

We left unanswered in Chapter 9 the question of how to calculate values of the error function in a fraction of the time that would be required for numerical integration. [In the graphic below, we show both a computer-drawn graph of the error function on the interval from `-3` to `3`, as well as a graphing calculator (TI-89) with the same graph. A computer algebra system with a built-in erf function can draw this graph as quickly as it graphs a sine function. The calculator, which has to find the function values by integration, takes four minutes to draw the graph.] What we learn about polynomial representations of simpler functions will show us how to find approximating polynomials for the error function as well, and that will turn out to be the tool we need for rapid calculation.

Throughout this chapter you will see the fundamental concepts of derivative and integral being put to use in a variety of ways, and we will also make frequent use of their connecting link, the Fundamental Theorem of Calculus.