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Unformatted text preview: (5/16/07) Section 0.3 Power and exponential functions Overview: As we will see in later chapters, many mathematical models use power functions y = x n and exponential functions y = b x . The definitions and basic properties of these functions, which are studied in precalculus courses, are reviewed in this section. We will discuss applications of these functions in later chapters, beginning with Chapter 4, where we study their derivatives. We also describe here how formulas for functions can be modified to translate, reflect, expand, and contract their graphs, The section closes with notes on the history of analytic geometry. Topics: Power functions y = x n and their graphs Vertical and horizontal translation Reflection, magnification, and contraction Exponential functions y = b x and their graphs Laws of exponents Historical notes Power functions A power function is a function of the form y = x n , where x is the variable and n is a constant. If n is a positive integer, then x n equals the product of n x s, as in the formula x 3 = x x x . If n is a positive fraction, p/q , then x n = x p/q is the q th root of the p th power of x , which also equals the p th power of the q th root of x . In the case of n = 5 3 , for example, we have x n = x 5 / 3 = 3 x 5 = bracketleftbig 3 x bracketrightbig 5 . If n is a negative integer or fraction, so that n =- m with m a positive integer or fraction, then x n equals 1 /x m , as in the formulas x- 3 = 1 x 3 = 1 x x x x- 5 / 3 = 1 x 5 / 3 = 1 3 x 5 = 1 bracketleftbig 3 x bracketrightbig 5 . To have formulas and identities involving x n apply with zero exponents, x is defined to be 1 for all x . If n is not an integer or a fraction, it is irrational and has an infinite decimal expansion, which is used to define x n for positive x . The irrational number 2, for example, has the decimal expansion 2 = 1 . 4142135623 ... , and if we want to define 10 2 , we let n 1 = 1 . 4 be the number obtained by taking only one digit after the decimal point in the expansion of 2, let n 2 = 1 . 41 be the number obtained by taking two digits after the decimal point, and so forth. This gives us an infinite string of rational numbers n 1 ,n 2 ,n 3 ,... that approaches 2. We say that 2 is the limit of the numbers n 1 ,n 2 ,n 3 ,... . The numbers 10 n 1 , 10 n 2 , 10 n 3 ,... are defined because the exponents n 1 ,n 2 ,n 3 ,... are rational. And, just as the numbers n 1 ,n 2 ,n 3 ,... approach their limit 2, the numbers 10 n 1 , 10 n 2 , 10 n 3 ,... approach their limit, which is defined to be 10 2 . The first seven of the numbers 10 n j are calculated in the next question....
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