This lab begins by fitting data to the height of a ball thrown vertically, then uses that information to study the flight of the ball (height and velocity). This is a classic example relating height to its derivative, velocity. The second example starts with a basic model for the length of a species of fish as a function of age, then uses this information to study the derivative of the length, which gives growth rates for the fish as it ages. The last 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.

Problem 1: You start this problem by plotting the time and distance data. Excel's trendline is applied using the Trend/Regression Type, Polynomial with Order 2. It is very important that in the options section besides having Excel put the equation on the chart you check the option Set intercept = 0. This gives you the best quadratic fit that you will be using for the height, h(t), that is used in the rest of the problem. Since this is a quadratic equation, you should be able to answer the questions easily about maximum height and when it hits the ground.

The remainder of the problem has you computing average velocities and building your intuition about the derivative. There are a fair number of calculations, so you may want to take advantage of doing many of these calculations on the Excel spreadsheet, which with the pull down option, can rapidly give you many of the necessary answers.

Problem 2: This problem starts with the use of Solver to find the nonlinear least squares best fit to the data on the length of a species of fish as it ages. The remainder of this problem has you computing growth rates either directly from the data or from the best fitting function. Again you will be performing a fair number of repetitive computations, so it is advised that you use Excel's pull down feature to accelerate your calculations. This problem should help you build your intuition on how length is related to its derivative, the growth rate.

To evaluate a derivative of the von Bertalanffy model at a particular age we use the following sequence of Maple commands:

> L := t -> 172.5*(1 - exp(-0.5243*t));

> dL := diff(L(t),t);

> subs(t = 7.5,dL); evalf(%);

Problem 3: 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 5 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.

> subs(x=3,df); #This evaluates the derivative at x = 3.

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