This lab continues the 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 maximum and minimum of a polynomial that approximates the oxygen consumption of a beautiful, but nasty insect. The problem is based on the research that was performed here at SDSU. The second problem examines two drug treatment regimes. For one drug regime, you use the derivative to find peak drug concentration.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 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: In this problem you will be graphing the given functions, then finding when they are equal to certain values.The exponentially decaying amount of drug should be a problem you can do by hand. The second part requires differentiating exponentials to find the maximum. You can have Maple help you with this or practice the techniques by hand. The most difficult part is trying to find when the second drug level is at 3 mg. This is most easily done using Maple's fsolve command, where you define the function B(t), then type fsolve(B(t) = 3, t = 0..20); You adjust the second part of the command to include the t value that you expect for when B(t) = 3 from examining the graph.