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Chapter 10

 Approximating Functions' Values

in Chapter 9 you used the Fundamental Theorem of Calculus to devise techniques for producing closed form definitions of accumulation functions that are defined in open form as integrals. The two representations (closed and open form) are equivalent. Closed form definitions provide a formula for calculating the function's values; the open form definition reminds you that the function you are evaluating is an accumulation function.

You found closed form definitions of accumulation functions with the aim of computing their values efficiently. Unfortunately, the vast majority of accumulation functions that you will meet in applied settings cannot be represented in closed form. A seemingly simple case is $r_f(x)=\sin(3\cos(5x))$. Figure 10.0.1 shows Wolfram Alpha Pro's report when asked for an antiderivative of $\sin(3\cos(5x))$.

Figure 10.0.1. The result of Wolfram Alpha Pro's attempt to determine an antiderivative of sin(3cos(5x)). It could not determine one.

Wolfran Alpha's indability to find a closed form antiderivative of $r_f(x)=\sin(3\cos(5x))$ is not a shortcoming of Wolfram Alpha. Instead it reflects the fact that $r_f(x)=\sin(3\cos(5x))$ does not have a closed form antiderivative.

But $r_f(x)=\sin(3\cos(5x))$ does have antiderivative! An antiderivative of $r_f(x)=\sin(3\cos(5x))$ is $$F(x)=\int_a^x r_f(t)dt$$

The problem is that we do not know how to compute values of $F$ efficiently.

Section 10.1 addresses the problem of how to lessen the computational effort to approximate values of accumulation functions. In addressing this problem we will lay groundwork for Section 10.2. In this section we develop general methods for approximating values of functions that we do not know how to compute.

A Little History

Much of the mathematics that exists today arose from a centuries-long quest to develop methods for computing values of common functions like sine, cosine, tangent, exponential, and logarithmic functions. Today we take it for granted that we can evaluate them on a calculator.

Before the advent of electronic digital technologies, people had to compute these values by hand. Indeed, prior to 1900 CE, a computer was "a person who calculates." The efforts of these computers were recorded in tables of values, such as for trigonometric functions and logarithmic functions, that other computers (people) could use in their computations. Astronomers, physicists, chemists, and engineers needed these tables to make precise computations.

Sometimes tables of values were too cumbersome for quick computations. In the 1960's, engineers in NASA's project to put humans on the moon often used slide rules to quickly estimate products and quotients of large or small numbers or to estimate values of trigonometric, logarithmic, and exponential functions.

Mathematical discoveries often had their roots in the quest to make computations of functions' values more efficient. In Chapter 9 we used insights gained through the Fundamental Theorem of Calculus to find closed form representations of accumulation functions that enabled us to efficiently calculate values of open form integrals that we defined in Chapter 8.

In Chapter 9 we developed methods for determining closed form definitions of accumulation functions that are defined in open form as integrals.  As we saw, and will see again, there are functions that do not have closed form antiderivatives, such as $f(x)=\cos(x)e^{\cos(x)}$. Indeed, in a real analysis course you will learn that functions that have closed form antiderivatives are exceedingly rare (have measure 0) in the class of real-valued functions.

In Chapter 10 we will develop methods for approximating the values of any accumulation function, including those that cannot be defined in closed form. We also will extend these ideas and methods to create ways to compute values of functions that cannot be defined algebraically (such as trigonometric, exponential, and logarithmic functions). These latter methods are behind your calculator's ability to produce accurate computations quickly.