Thinking About Equations || Estimation and Approximation

✍ Scribed by Bernstein, Matt A.; Friedman, William A.

• Publisher: John Wiley & Sons, Inc.
• Year: 2009
• Tongue: English
• Weight: 500 KB
• Edition: 1
• Category: Article
• ISBN: 0470186208
• DOI: 10.1002/9780470495032.ch5

✦ Synopsis

ESTIMATION AND APPROXIMATION

Most problems in science and engineering can be modeled with equations, but only a few yield equations for which we can fi nd an exact, analytic solution. In this chapter, we discuss estimation and approximation techniques that can be applied to many of the remaining problems. Even if the exact solution is attainable, an approximate solution might suffi ce and is usually much easier to fi nd. The array of estimation and approximation approaches is very broad, and here we limit ourselves to a few selected methods to give a fl avor for this topic.

One use of the word " estimation " is to describe the process with which we make quick, mental calculations. Section 5.1 contains an example illustrating how using powers of two provides simple and accurate numerical estimates for geometric sequences (also known as geometric progressions). Another use of the word " estimation " is to describe the technique of making order -ofmagnitude guesses for numerical values based on simple principles and commonsense reasoning. These techniques are illustrated with example problems in Sections 5.2 and 5.3 . The study of such problems has been advocated to help remedy " innumeracy " (i.e., numerical illiteracy) or a lack of " number sense " manifested by diffi culty in determining whether the order of magnitude of a numerical quantity is plausible.

The remaining three sections of this chapter deal with approximation methods. Several analytic methods that can be used to approximate the value of defi nite integrals are discussed in Section 5.4 . Basic aspects of perturbation