458 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions EXAMPLE 11 Using Data to Model Exponential Growth Data from recent years indicate that future amounts of carbon dioxide in the atmosphere may grow according to the table. Amounts are given in parts per million. (a) Make a scatter diagram of the data. Do the carbon dioxide levels appear to grow exponentially? (b) One model for the data is the function y = 0.001942e0.00609x, where x is the year and 1990 … x … 2275. Use a graph of this model to estimate when future levels of carbon dioxide will double and triple over the preindustrial level of 280 ppm. Year Carbon Dioxide (ppm) 1990 353 2000 375 2075 590 2175 1090 2275 2000 Data from International Panel on Climate Change (IPCC). SOLUTION (a) We show a calculator graph for the data in Figure 22(a). The data appear to resemble the graph of an increasing exponential function. (b) A graph of y = 0.001942e0.00609x in Figure 22(b) shows that it is very close to the data points. We graph y2 = 2 # 280 = 560 in Figure 23(a) and y2 = 3 # 280 = 840 in Figure 23(b) on the same coordinate axes as the given function, and we use the calculator to find the intersection points. y = 0.001942e0.00609x 300 1975 2100 2300 (b) Figure 22 300 1975 2100 2300 (a) Figure 23 y1 = 0.001942e 0.00609x y2 = 840 1975 −500 2100 2300 (b) y1 = 0.001942e 0.00609x y2 = 560 1975 −500 2100 2300 (a) The graph of the function intersects the horizontal lines at x-values of approximately 2064.4 and 2130.9. According to this model, carbon dioxide levels will have doubled during 2064 and tripled by 2131. S Now Try Exercise 107. Graphing calculators are capable of fitting exponential curves to scatter diagrams like the one found in Example 11. The TI-84 Plus displays another (different) equation in Figure 24(a) for the atmospheric carbon dioxide example, approximated as follows. y = 0.00192311.0061092x This calculator form differs from the model in Example 11. Figure 24(b) shows the data points and the graph of this exponential regression equation. 7 y1 = 0.001923(1.006109) x 300 1975 2100 2300 (b) Figure 24 (a)
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