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SSG1.4-5%20Power_Exp

SSG1.4-5%20Power_Exp - 86 1.4 POWER FUNCTIONS AND SCALING...

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86 1.4. POWER FUNCTIONS AND SCALING LAWS 1.4 Power Functions and Scaling Laws Why can an ant lift one hundred times its weight while a typical man can only lift about 0.6 of his weight? Why is getting wet life-threatening for a fly but not a human? Why can a mouse fall from the top of a sky scraper and still scurry home, while a person will almost certainly be killed? Why are elephants legs so much thicker relative to their length than are gazelle legs ? A class of functions called power functions provide a means to answering these questions. Power Functions and Their Properties We begin with a definition. Power Functions A function f ( x ) is a power function if it is of the form y = f ( x ) = ax b where a and b are real numbers. The variable x is called the base , the parameter b is called the exponent and the parameter a the constant of proportionality . Note that 5 7 x 1 and x 3 are power functions, while 3 x is not because, in this latter case, the exponent rather than the base is the variable. Example 1. Graphing power functions Graph each of the following sets of functions and discuss how they differ from one other and what properties they have in common. a. y = x 2 , y = x 4 , and y = x 6 . b. y = x 3 , y = x 5 , and y = x 7 . c. y = x 1 / 2 , y = x , and y = x 3 / 2 . d. y = 1 x and y = 1 x 2 . Solution. a. Graphing y = x 2 , y = x 4 , and y = x 6 gives © 2010 Schreiber, Smith & Getz
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1.4. POWER FUNCTIONS AND SCALING LAWS 87 All of these graphs tend to “bend” upward and are “U-shaped.” All three of these graphs intersect at the points (0 , 0), ( 1 , 1) and (1 , 1). On the interval [ 1 , 1] the function with the smallest exponent grows most rapidly as you move away from x = 0, and on the intervals ( −∞ , 1) and (1 , ) the function with the largest exponent increases most rapidly. b. Graphing y = x 3 , y = x 5 , and y = x 7 gives All of these graphs are “seat shaped”, bending downward for negative x and bending upward for positive x . All three of these graphs intersect at the points (0 , 0), ( 1 , 1) and (1 , 1). On the interval [ 1 , 1] the function with the smallest exponent grows most rapidly, and on the intervals ( −∞ , 1) and (1 , ) the function with the largest exponent grows most rapidly. c. Graphing y = x 1 / 2 , y = x , and y = x 3 / 2 gives © 2010 Schreiber, Smith & Getz
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88 1.4. POWER FUNCTIONS AND SCALING LAWS We graphed over the domain [0 , ) of y = x 1 / 2 and y = x 3 / 2 (these two functions are only real for x 0). All of these graphs increase as x increases, and pass through the points (0 , 0) and (1 , 1). The graph of x 1 / 2 becomes steeper and steeper at 0, while the graph of x 3 / 2 becomes flatter and flatter. Moreover, the graph of x 1 / 2 bends downward, while the graph of x 3 / 2 bends upward. d. Graphing y = 1 x and y = 1 x 2 gives Both of these functions “blow up” (i.e. have a vertical asymptotes) at x = 0 and pass through the point (1 , 1). Parts of the graphs lying above the x -axis, bend upwards, while parts lying below bend downwards.
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