This lab uses the derivative to study of maximum and minimum in three applications. Again you will find Maple useful to help solve these problems. The first problem parallels the work we did in lecture on the variation of body temperature of a human female over her menstrual cycle. The second 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. The last problem examines two drug treatment regimes.  You use the techniques developed in class for finding maxima and minima.

Problem 1: This problem examines the variation of the female body temperature over the period of a month. A cubic polynomial is fit to data, then the maximum and minimum are found. The point of inflection corresponds to the maximum increase in temperature, which also matches the point of ovulation or greatest fertility. This is a classic use of differential Calculus and follows the notes in the text. The problem uses Excel's Trendline with the polynomial curve fitting algorithm. The derivatives are most easily found with Maple, which allows easy computation of the minimum, maximum, and point of inflection.

Problem 2: This problem is similar though a little more difficult than the Lab on the O₂ consumption of kissing bugs or the one above on the monthly rhythms of body temperature for females. You start this problem by simply graphing the data and applying the log scale option to the vertical axis. Recall that to change the graph to having a Logarithmic scale, you can go to the Main menu and select Insert, then choose the Layout tab, under which you select Axes. From here you select the Primary Vertical Axis and choose More Primary Vertical Axis Options. This opens a box of Axis Options, where you can check the box for Logarithmic scale. It should be noted that this long progress can be simplified by going directly to the graph. You move the cursor until a box appears that says Vertical (Value) Axis, then you Right Click and the box of options to Format Axis appears, where you simply check the box Logarithmic scale. 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.

Problem 3: In this problem consider a decaying exponential and the difference of two decaying exponentials. This parallels the study of Prozac in the lecture notes. The exponentially decaying amount of drug is a problem you might want to practice by hand, as it is similar to problems that will arise in exams. The pharmacologial equation is used for drugs that arise from metabolism of one drug into another form with the difference of two decaying exponentials. The second drug concentration 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 some threshold level. This is most easily done using Maple's fsolve command. Note that you will need to find the first time when the drug becomes effective, then subtract this time from the time when the drug loses efficacy due to exponential decay. Graphing the function can help narrow the range where you use Maple's fsolve command. The computer methods are very similar to ones that you have already been using.