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Unformatted text preview: A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear function and a table of sam ple values. Notice that whenever x increases by 0.1, the value of increases by 0.3. So increases three times as fast as x . Thus the slope of the graph , namely 3, can be interpreted as the rate of change of y with respect to x . EXAMPLE 1 (a) As dry air moves upward, it expands and cools. If the ground temperature is and the temperature at a height of 1 km is , express the temperature T (in C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? SOLUTION (a) Because we are assuming that T is a linear function of h , we can write We are given that when , so In other words, the yintercept is . We are also given that when , so The slope of the line is therefore and the required linear function is (b) The graph is sketched in Figure 3. The slope is , and this represents the rate of change of temperature with respect to height. (c) At a height of , the temperature is M If there is no physical law or principle to help us formulate a model, we construct an empirical model , which is based entirely on collected data. We seek a curve that fits the data in the sense that it captures the basic trend of the data points. T 10 2.5 20 5 C h 2.5 km m 10 C km T 10 h 20 m 10 20 10 10 m 1 20 h 1 T 10 b 20 20 m b b h T 20 T mh b 10 C 20 C V x y y=3x2 _2 FIGURE 2 y 3 x 2 f x f x f x 3 x 2 SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS     25 x 1.0 1.0 1.1 1.3 1.2 1.6 1.3 1.9 1.4 2.2 1.5 2.5 f x 3 x 2 FIGURE 3 T=_10h+20 T h 10 20 1 3 EXAMPLE 2 Table 1 lists the average carbon dioxide level in the atmosphere, measured in parts per million at Mauna Loa Observatory from 1980 to 2002. Use the data in Table 1 to find a model for the carbon dioxide level. SOLUTION We use the data in Table 1 to make the scatter plot in Figure 4, where t repre sents time (in years) and C represents the level (in parts per million, ppm). Notice that the data points appear to lie close to a straight line, so its natural to choose a linear model in this case. But there are many possible lines that approximate these data points, so which one should we use? From the graph, it appears that one possi bility is the line that passes through the first and last data points. The slope of this line is and its equation is or Equation 1 gives one possible linear model for the carbon dioxide level; it is graphed in Figure 5. Although our model fits the data reasonably well, it gives values higher than most of the actual levels. A better linear model is obtained by a procedure from statistics CO 2 Linear model through first and last data points FIGURE 5 340 350 360 1980 1985 1990 C t 1995 2000 370 C 1.5545 t 2739.21 1 C 338.7 1.5545 t 1980 372.9372....
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 Spring '09
 ganong

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