This lab uses the derivative to study of maximum and minimum in two applications. Again you will find Maple useful to help solve these problems. The first problem is an application of the derivative to find the maxima and minima of polynomials. This application comes from research of Boyd Collier, formerly in the Biology Department at SDSU. It examines the oxygen consumption of a beautiful, but nasty insect. The second problem is very similar mathematically to the first as we study another cubic polynomial fitting data for human body temperature variation over a 24 hour period. The last problem is an application of the derivative to find the maximum and minimum of a polynomial that approximates the population of marine invertebrates in the Salton Sea. You use the techniques developed in class for finding maxima and minima.

Question 1: This problem begins as many other problems we have worked this semester, where you take the data from Professor Boyd Collier's "kissing bugs," and plot it in Excel. You use Excel's Trendline with a polynomial of degree 3 to find the best fit to the data. (Make sure that you have at least 3 significant figures in the coefficients formed by Trendline.) You will record this formula in your lab report.

To find the maximum and minimum of the cubic polynomial, you want to find the derivative of this function and set it equal to zero. The derivative of a cubic (you will soon see) is a quadratic, which you can solve by hand. This might be good practice for you to prepare for future exams. The easier way to work the differentiation part of this problem is to let Maple do the work for you. Below are a series of commands to show you how to find the minimum and maximum of a cubic polynomial in Maple.

The series of commands are to enter the function, differentiate the function, find when the derivative is zero, then use these values back in the original function. Here are those commands, which I will describe more in lab. (Notice you can use # in Maple to make comments. Anything following the # is ignored by Maple.)

> f := x -> 0.3*x^3-9.1*x^2+3.7*x+42.2; # This enters the function.

> df := diff(f(x),x); #This differentiates the function and assigns it to df.

> xm := fsolve(df=0,x); #This finds the x values at the extrema and assigns them to xm.

> f(xm[1]);f(xm[2]); #This finds the y values at the previous x values found above.

The rest of the problem is interpreting your results in terms of the biology of the problem. 

Question 2: This problem is handled exactly like the problem above. It is simply a different data set following the normal human body temperature variation over 24 hours (circadian).

Question 3: This problem is similar though a little more difficult than the Lab on the O₂ consumption of kissing bugs. You start this problem by simply graphing the data and applying the log scale option to the vertical axis. It probably looks best if you plot the data as points connected by straight lines, which is the fourth choice of the XY-scatter plot graphs.

In Part b, you take the log of the data and plot it. You simply apply Excel's Trendline find the best fit to the data with a 4^th order polynomial. (Do NOT forget to change the Excel Trendline equation to scientific notation and obtain 5-6 significant figures by choosing 4-5 decimal places!) For Part c, you will be able to differentiate this polynomial either with Maple or by hand. Since the derivative is a cubic equation, you will probably need Maple to find when the derivative of this polynomial is zero. This is another application with real data, where you find maxima and minima of the data using polynomial fits (though a polynomial is not likely to be the best choice in this case as populations fluctuate annually). For the most part, this problem is very much like the "kissing bug" problem with a bit more data and a higher order polynomial.