Laboratory Help Page

This page is designed to provide helpful information about the laboratory questions. Begin this lab and every lab by introducing yourself to your partner. Detemine the times when you can meet together during the week before the lab is due on Friday, Oct. 5. If your schedules are totally incompatible, then notify me immediately.

You will probably want to download your specific lab page (and may want to convert it to a Word document). On the cover page you begin by typing in the name of each team member and your group number.

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 last 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 asks you to repeat some of the calculations required to produce the graphs that appear in the derivative of the trigonometric functions section on tides. Much of the most difficult work has already been done for you and is provided by a downloadable Excel worksheet. On this sheet there are two pages. One page has the data that was downloaded from the NOAA website on tide tables. This is then converted to data in hours for the entire month of October 2000 and the height of the tides in feet for San Diego. (You do NOT do anything to this particular sheet!) The second sheet contains the converted data transferred to columns A and C for use in your calculations for this lab. The mathematical model is in column D (which you will not change, but simply fill down), which sums together the four main forces that are in columns E through H. The entry in E2 is provided to help guide you in how you construct the entries in F2, G2, and H2. Create the entries in F2, G2, and H2, using the appropriate parameters from the entries in the table on the spreadsheet in cells L1 to O7. Note that these parameters already have an initial “guess” values for A_i and f_i in cells L1 to O7. The computed height of the model now appears in D2, which sums cells E2 through H2. The square of the error is computed in cell I2.

The next step is to fill down all the entries in columns D through I to row 110. The sum of the square of the errors appears in J2. Now use Solver ( found under Tools), Set Target Cell as J2, equal to Min by changing cells N3:O7. By selecting all the cells, Excel will find the combination of all these constants which gives the smallest difference from the actual values. This is the least squares fit that we have used before.

To create the graphs for either a week or a day, you need to first set in column A (below the current information or on a separate sheet) the appropriate range of hours for the desired graph, stepping 1 hour at a time. Do not forget to account for the fact that Oct. 1 goes from 0-24 hours. Column B will be found by taking the entries in column A and dividing by 24. In column C, you will want to put the model (which can be done by copying the model from above with appropriate reference changes or simply putting it in column D with the accompanying entries from columns E through H). Use chart wizard to create your graph and simply add the data from the appropriate range of the data already given to you.

To find the minima and maxima, you are going to use Maple. You enter the function in Maple from the numbers on your Excel spreadsheet. (You may find it easier to type in h1 := t -> A1*cos(2*Pi*t/P1 + phi1); using the correct values for A1, P1, and phi1, then do the same for h2, h3, and h4, and finally setting h := t -> A0 + h1(t) + h2(t) + h3(t) + h4(t);) Use the information on the spreadsheet to narrow the range of search for the key values (in hours) of the highest and lowest tides (or any other high or low tide), then proceed as you have always done to differentiate your function for the height of the tides and set the derivative equal to zero (diff and fsolve commands). Finally, you will need to convert your decimal times into times with hours and minutes to compare to the tide tables information. The remainder of the questions should follow readily from the answers provided above.

Question 3: This problem works with real data on Paramecium. This again is an Excel problem. You want to start by putting the populations from days 0-11 in column A. Next you put the populations from days 1-12 in column B. You graph these two columns using scatter plot in Excel (which you should have done before). Leave these as data points clearly distinct on the graph. Next you use the Trendline polynomial fit as instructed in the Lab question. For Part b, you want to choose 3 other columns. In the first column, enter the data for the day number. In the second column, enter the population data. In the third column, you first enter the starting population, then just as you did in the question for Malthusian growth in the first lab, you enter the formula (as Part a. for the formula) and fill down. You can easily graph these 3 columns for your result. Use points on the graph for data and lines for the theoretical model.