This is your second to the last lab before the Lab Final. As usual, you will have one week to work on this lab, being due on April 27.

Question 1: This problem is meant to help motivate the idea of a derivative, which we will be studying over most of the rest of this course. We will see that velocity is simply the derivative of position.

The first part of this problem uses Excel's trendline with a quadratic (polynomial order 2) fit to the data. Ask yourself if you want to check the box to set the y-intercept to zero by examining the form of the quadratic equation. (Is the point (0, 0) a solution of the equation?) Finding the maximum height of the ball and when the ball hits the ground shows that you understand how to interpret the model given by this quadratic function.

The remaining parts of the problem use the quadratic function found in Part a. You may want to set up a spreadsheet in Excel to perform these calculations rather than doing all of them by hand. You could enter t₁ in column A, t₂ in column B, the function depending on column A in column C, then copy this to column D to get the function h(t) depending on column B. Finally, column E could compute v_ave by the formula that is given to you. You may want to do a couple of them by hand to make sure you understand the process (as you may have to do this on an exam). Graphing your results in Part c. should provide some understanding of the derivative of the quadratic function h(t).

Question 2: The first part of this problem gives you more experience with allometric modeling using heights and weights of girls. Part c is related to the previous problem giving you more intuition on rate of gain of weight, which is again a derivative. The rate of gain of weight is similar to the computations we did for the lecture notes.

Question 3: This problem is a review of the Malthusian growth models, on which (for the most part) you have been doing quite well. The first part of the problem has you calculating some volumes and surface areas of different cells, then you use this information with the discrete Malthusian growth model to see how rapidly certain areas and volumes are reached. Finally, you will learn how powerful exponential growth is by trying to analyze Michael Crichton's quote from his book the Andromeda Strain. The hard part of this problem is getting your units correct, so you must be extremely careful when you perform these calculations. This problem can be readily done on a calculator, so it is ideal to do at home! I might recommend talking this problem over with your lab partner, then each try the calculation independently to see if you can get the same answer. (You may want to put some intermediate calculations to get partial credit.) Don't forget that the doubling time and the initial condition (1 cell) are different from your earlier calculations.