In 1913, Carlson [1] studied the growth of a culture of yeast, Saccharomyces cerevisiae. Over time this culture levels off, but its initial growth is exponential or Malthusian. A Malthusian growth model is given by

(We will study this model shortly.) Simply put, the population at the next time (n+1) is equal to the population at the current time (P_n) plus some growth term, which is simply proportional (r) to the current population. Thus, we have a growth function

Below is a table from Carlson's data showing the population and the rate of growth at that particular population

a. Plot these data. Use Excel's Trendline on the data points to find the best straight line passing through the origin. (Note you will need to use the option in trendline of setting the y-intercept = 0.) What is the slope of the line that best fits through the data?

b. In lecture (Function Review and Quadratics), we examined a linear model for mRNA synthesis. For the linear model (passing through the origin) given above, we can readily find the sum of squares function. Consider a data point ( P_i,g(P_i)). The error between this data point and our model is given by

Thus, e₁ = |10,700 - 18,300 r|. Similarly, you can find e₂, e₃, and e₄. The sum of squares function is given by

Find the expression for the quadratic function of the slope of the model, r (in simplest form). Sketch a graph of this quadratic for 0.3 < r < 0.7.

c. Find the r-coordinate of the vertex and compare this to the slope trendline gives you in Part a.
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[1] T. Carlson Über Geschwindigkeit und Grösse der Hefevermehrung in Würze. Biochem. Z. 57: 313-334, 1913.