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World Population (Historic) Before the technology was available to fit many kinds of models to data, researchers and others were restricted to using linear models. Because exponential data are common in many fields of study, it has always been important to be able to fit an exponential model to data. The table shows past and predicted world population. World Population $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (billions) } \end{array} \\ \hline 1850 & 1.1 \\ \hline 1930 & 2.0 \\ \hline 1975 & 4.0 \\ \hline 2013 & 8.0 \\ \hline \end{array} $$ a. Construct a scatter plot of the data. Comment on the curvature. b. Change the data so that they represent the year and the natural log of the population. Construct a scatter plot of the new data. c. Use the technique discussed in this section to find the best-fitting linear function for the changed data in part \(b\). d. If \(a\) and \(b\) are the parameters of the linear function \(y=a x+b\) found in part \(c,\) graph the function \(y=e^{b}\left(e^{A}\right)^{x}\) on the scatter plot of the original data. e. Use technology and an exponential regression routine to find the best exponential model for the population data. Compare it with the model in part \(d\) and reconcile any differences.

a. Write the Lagrange system of partial derivative equations. b. Locate the optimal point of the constrained system. c. Identify the optimal point as either a maximum point or a minimum point. $$ \left\\{\begin{array}{l} \text { optimize } f(x, y)=x^{2} y \\ \text { subject to } g(x, y)=x+y=16 \end{array}\right. $$

Parasite Development The average time for a \(C .\) grandis egg to develop into an adult can be modeled as $$ g(w, x)=25.6691-0.838 x+2.4297 w+0.0084 x^{2} $$ \(-0.0726 w^{2}-0.0181 x w\) days where the relative humidity is held constant at \(x \%,\) the eggs are exposed to \(w\) hours of light each day, and the temperature is held constant at \(30^{\circ} \mathrm{C}\) (Source: J. A. Morales-Ramos, S. M. Greenberg, and E. G. King, "Selection of Optimal Physical Conditions for Mass Propagation of Catolaccus grandis," Environmental Entomology, vol. \(25,\) no. 1 \((1996),\) pp. \(165-173)\) a. Calculate the point where the partial derivatives of \(g\) are both equal to zero. b. What type of critical point is the point in part \(a\) ?