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Chapter 10

 Approximations of Functions' Values

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 a quest to make computations of functions' values possible or more efficient, or to prove the possibility or impossibility of computing a value. In Chapters 8 and 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, in principle, exact values of accumulation functions. A significant area in advanced mathematics grew from attempts to generalize the idea of function and the ideas of derivative and antiderivative to functions that, with current methods, could not be computed.

In Chapter 9 we developed methods for determining closed form definitions of accumulation functions that are expressed in open form as integrals.  In Chapter 10 we will develop methods for approximating the values of accumulation functions that cannot be expressed 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.