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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 y-intercept 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=3x-2 _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.... View Full Document