James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 1.2 Mathematical Models: A Catalog of Essential Functions 25
T
20
10
T=_10h+20
(b) The graph is sketched in Figure 3. The slope is m − 210°Cykm, and this represents
the rate of change of temperature with respect to height.
(c) At a height of h − 2.5 km, the temperature is
T − 210s2.5d 1 20 − 2 5°C
■
0
FIGURE 3
1 3
h
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.
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 2012. 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 represents
time (in years) and C represents the CO 2 level (in parts per million, ppm).
Year
CO 2 level
(in ppm)
Table 1
Year
CO 2 level
(in ppm)
1980 338.7 1998 366.5
1982 341.2 2000 369.4
1984 344.4 2002 373.2
1986 347.2 2004 377.5
1988 351.5 2006 381.9
1990 354.2 2008 385.6
1992 356.3 2010 389.9
1994 358.6 2012 393.8
1996 362.4
C
400
390
380
370
360
350
340
(ppm)
1980 1985 1990 1995 2000 2005 2010 t
FIGURE 4 Scatter plot for the average CO 2 level
Notice that the data points appear to lie close to a straight line, so it’s 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? One possibility is the line that passes
through the first and last data points. The slope of this line is
We write its equation as
393.8 2 338.7
2012 2 1980 − 55.1 − 1.721875 < 1.722
32
or
C 2 338.7 − 1.722st 2 1980d
1 C − 1.722t 2 3070.86
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