Laboratory Help Page

This page is designed to provide helpful information about the laboratory questions. Once again introduce yourself to your partner (who may be the same this time). Detemine the times when you can meet together during the week before the lab is due on Friday, Sept. 22. If your schedules are totally incompatible, then notify me immediately.

Question 1: This question allows you to experiment with trigonometric functions to build intuition before we work formally with them in class. This problem examines properties like the period and oscillatory behavior of the trigonometric functions, sine and cosine. It should also reinforce a few of the Maple commands that you used in the first Lab. (Don't forget the summary of the Maple commands on the special Maple help sheet.) Maple should help with the differentiation and finding values of extrema and intercepts.

Question 2: This problem is one of the classic problems in using Calculus to find an optimum. You may want to take a sheet of paper and cut squares out of the 4 corners to give you some intuition into this problem. Recall that the volume of a rectangular box is the product of the length times the height times the width. The surface area will clearly be the area of the sheet of cardboard minus the area of the pieces cut out of the cardboard. By trying to cut out square corners from a sheet of paper, you should be able to figure the limits of the domain. (Obviously, one is x = 0 by not doing any cutting from the paper.) In Part c. you may want to connect the answer in Part a. (value of x) to the function in Part b. to get the value of S at the maximum V.

Question 3: This problem is very similar to the material in the lecture notes, where crows drop whelks in an optimal foraging manner. This question examines the foraging behavior of Glaucous-winged gulls eating butter clams. The first part of the problem uses a powerful feature of Excel called the solver under Tools. Below are more detailed instructions for using this solver. After obtaining the parameters for fitting the data, then the remainder of Part b. should be relatively straightforward application of the optimization techniques in the lecture notes, i.e., use the lectures notes to guide your computations. The last part of the problem notes that other biological factors must be entering into the gull foraging behavior, so this part requires you to think more biologically for the discussion rather than precise mathematical formulae. (It would be an interesting project for someone to run experiments to determine the precise amount of kleptoparasitism amongst these gulls, and then find a mathematical formula (penalty function) for the loss of food to other gulls in the area.)

Here are more detailed instructions on how to find the least squares best fit to the data.

Put the height data in column A, and the number of drops in column B. In cell F1, put the value of 1, which is your starting guess for the value a. In cell F2, put the value 0, which is your starting guess for the value b.
In column C, use the values of a and b to calculate the number of drops, =1+$F$1/(A1-$F$2).  This can be copied down the column. The $ are used to make an absolute reference to a cell, instead of the default updating reference.
In column D, find the square of the difference between the theory and experiment =(B1-C1)^2 for each data entry.  Then you find the sum of the squares by going to the cell at the bottom of these values in the D column and clicking the summation sign  S on your toolbar.

If you recall from our early lecture in Math 121, the most common scientific method of fitting data is to adjust your parameters in the problem to minimize the sum of the squares. This is known as finding the minimum of the Least Squares Estimate or the Least Squares Best Fit.

To find this Least Squares Best Fit, you'll want to use Excel's solver under Tools. (If you do not find the solver under Tools, then go to the Add-in line under Tools and check the box labeled "Solver Add-in.") Click on Tools, and choose solver.
The target cell is the cell that you have just filled with the summation (D11 or D12, if you labeled your columns). Chose Equal to: min to minumize this value. By changing cells needs to point to the values of a and b, in cells F1 and F2.
Press solve. Solver should find the best values for a and b, and put the calculated values of the function N(H) into column C. You can now graph columns B and C against column A to give you the graph desired for this problem.

To calculate the energy, use column G and put in suitable values for H (0.5 every 0.1). Go to cell G1 and put in the value 0.5, then in G2 you insert =G1+0.1. In column H use your values of a and b to find the calculated value of N(H)  =1+$F$1/(G1-$F$2).  Use column I to calculate the energy function =1*G1*H1. Use the fill down feature to complete your values of the functions. By graphing the values in column I against those in column G, you can produce the energy graph. You may want to use Maple to find the actual value of the minimum as I would like your answer to 4 significant figures.