This project uses high school mathematics (the equation of a line) and the computer to give you an idea of the the scope of the problems we call "Projects." There is also a calculus moral to the story. The main approximation of calculus is to fit a linear function to small changes of a nonlinear function. Large-scale linear approximation makes sense in some problems but works in a different way. There is a pitfall in using a "local" approximation to make a long term prediction.

Global warming is a major environmental concern of our time. Carbon dioxide (CO₂) traps heat better than other gases in the atmosphere, so more CO₂ makes the atmosphere like a "greenhouse." Increased industrial production of CO₂ and decreased conversion of CO₂ to oxygen by plants are therefore thought to be causes of a general warming trend around the world. This is the greenhouse effect.

The computer program co2 program contains monthly average readings of CO₂ similar to those taken at Mauna Loa Observatory, Hawaii. Figure 1.1 shows a graph of the program's data from 1958 through 1967. (We couldn't obtain the raw data, so approximated it. Our graph looks similar to the published data.) The CO₂ level oscillates throughout the year, but there is a general increasing trend that appears to be fairly linear. It is natural to fit a linear function to this general increasing trend.

Figure 1.1: CO₂ Data

One way to make a linear approximation is simply to lay a transparent ruler on the graph and visually balance the wiggles above and below the edge of the ruler. That is a perfectly valid mathematical start, but the computer also has a "Fit" function and it produces the "approximation"

Figure 1.2: Computer Fit of CO₂ Data

The linear upward trend looks about like the yearly average of the data. The yearly averages are computed and fit to another approximating function

Figure 1.3: Yearly Averages of CO₂

Notice that the first fit to all the data is not the same as the fit to the annual averages of the data, but it is close. Both fits and sets of data are shown in Figure 1.4.

Figure 1.4: Data with Both Fits

1) Use the "eyeball" method of putting a ruler on the graph of the data where it balances the wiggles and then write the formula

for your approximation, a=?, b=?. You will notice that the "slope-intercept" formula h=ax+b is not the most convenient. If your ruler is at the points (1958,315) and (1968,320) then the change in h is 5 (ppm) for a change in x of 10 (years). How would you compute a? What obvious point does the line

pass through? We need the slope-intercept form for comparison with computer, but the point-slope formula for a line is most convenient in this part of the exercise. What is the point-slope form of the equation of a line?

2) How does your approximation compare with the two computer approximations f and g above?

3) Use all three approximating functions f, g, and h to predict the 1983 CO₂ average. How much increase in CO₂ over the 1958 level is your prediction?

When we did the preceding exercise, our calculation of the increase in CO₂ level over the 1958 level was 17 ppm, for a level of 333 ppm. The measured average turns out to be 343 for an increase of 27 ppm. That is an error of . The long-term data and prediction are shown in Figure 1.6.

The worrisome thing is that the average rate of increase from 1968 to 1983 is more than the linear trend of the 1958 to 1968 decade. What is your estimate of the present level of CO₂?

Figure 1.5: Linear Prediction of CO₂

You can read more in an article, Under the Sun - Is Our World Warming, in National Geographic, October 1990 (vol.178, nr.40). You can also use the computer to fit nonlinear functions to the data. For example, you might try a quadratic polynomial or exponential fit and use them to make predictions. Unfortunately, we do not know a scientific basis to choose a nonlinear function for our fit, so such predictions are only a curiosity.

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