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Unformatted text preview: then the exponential function has domain and range . Notice also that, since , the graph of is just the reflection of the graph of about the -axis. One reason for the importance of the exponential function lies in the following proper- ties. If x and y are rational numbers, then these laws are well known from elementary algebra. It can be proved that they remain true for arbitrary real numbers x and y . (See Appendix G.) LAWS OF EXPONENTS If a and b are positive numbers and x and y are any real numbers, then 1. 2 . 3. 4. EXAMPLE 1 Sketch the graph of the function and determine its domain and range. SOLUTION First we reflect the graph of [shown in Figures 2 and 5(a)] about the x-axis to get the graph of in Figure 5(b). Then we shift the graph of upward 3 units to obtain the graph of in Figure 5(c). The domain is and the range is . M EXAMPLE 2 Use a graphing device to compare the exponential function and the power function . Which function grows more quickly when x is large? SOLUTION Figure 6 shows both functions graphed in the viewing rectangle by . We see that the graphs intersect three times, but for the graph of x 4 0, 40 2, 6 t x x 2 f x 2 x V FIGURE 5 1 (a) y=2® x y _1 (b) y=_2® x y y=3 2 (c) y=3-2® x y , 3 y 3 2 x y 2 x y 2 x y 2 x y 3 2 x ab x a x b x a x y a xy a x y a x a y a x y a x a y 1 (0, 1) (a) y=a®, 0<a<1 (b) y=1® (c) y=a®, a>1 (0, 1) FIGURE 4 x y x y x y y y a x y 1 a x 1 a x 1 a x a x 0, y a x 54 |||| CHAPTER 1 FUNCTIONS AND MODELS www.stewartcalculus.com For review and practice using the Laws of Exponents, click on Review of Algebra. N For a review of reflecting and shifting graphs, see Section 1.3. SECTION 1.5 EXPONENTIAL FUNCTIONS |||| 55 stays above the graph of . Figure 7 gives a more global view and shows that for large values of x , the exponential function grows far more rapidly than the power function . M APPLICATIONS OF EXPONENTIAL FUNCTIONS The exponential function occurs very frequently in mathematical models of nature and society. Here we indicate briefly how it arises in the description of population growth. In Chapter 3 we will pursue these and other applications in greater detail. First we consider a population of bacteria in a homogeneous nutrient medium. Suppose that by sampling the population at certain intervals it is determined that the population doubles every hour. If the number of bacteria at time t is , where t is measured in hours, and the initial population is , then we have It seems from this pattern that, in general, This population function is a constant multiple of the exponential function , so it exhibits the rapid growth that we observed in Figures 2 and 7. Under ideal conditions (unlimited space and nutrition and freedom from disease) this exponential growth is typi- cal of what actually occurs in nature....
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- Spring '09
- Exponential Function