Math 122 Calculus for Biology II
Fall Semester, 2012
Lab Help

05-Oct-11

San Diego State University


Laboratory Help Page

This page is designed to provide helpful information about the laboratory questions. You should start with your Cover Page and be sure that you have all the important contact information about your lab partner for coordination later in the week. On the cover page you begin by typing in the name of each team member and your computer numbers. For your WeBWork answers, it is important that you keep all calculations to 5 or 6 significant figures. Too few digits have been one of the leading causes of errors that we have found when approached by students.

Question 1: This problem shows another application related to our Malthusian growth and radioactive decay problems in differential equations. No special techniques are required for this problem.

Question 2: This problem shows an application related to the Malthusian growth and radioactive decay problems in differential equations section. The first part of the problem has you calculating some volumes and surface areas of different cells, then you use this information with the continuous Malthusian growth model to see how rapidly certain areas and volumes are reached. You are undoubtedly familiar with volumes or surface areas of spheres and cylinders (or can find the formulae on wikipedia); however, one of the cells examined in this lab is ellipsoid in shape. The volume of an ellipsoid has much in common with the volume of a sphere, and you will see that the volume formula is easy to work with. However, the surface area of general ellipsoid can not be found without invoking special functions. We consider a prolate ellipsoid, which is egg-shaped. It has the closed form solution given in your lab. It does use the inverse trigonometric function, arccos(x). Since cos(x) is not a one-to-one function, the inverse only exists over a limited range of angles. We will only consider what is called the principle arccos(x), which is sometimes denoted Arccos(x) with domain -1 < x < 1 and range < y < p. The function y = arccos(x) gives the angle whose cosine is x. An example is given by p/3 = arccos(1/2), since cos(p/3) = 1/2. The inverse cosine function is available on all scientific calculators. In Excel, it is written ACOS(x), while in Maple, it is arccos(x).

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! Don't forget that the doubling time and the initial condition ( 1 cell ) are different from your earlier calculations.

Question 3: This is an exercise to help you understand radioactive decay. The lecture notes give the solution to the basic radioactive decay problem, and this question uses that solution to date ancient objects. The techniques developed in class and the lecture notes provide all the basic material for this problem. About the only difference is the computing of bounds in age based on the error in readings of dpm and using the bounds on the age in Part e to find the error in the dpm readings. You can do each of these by simply putting in the extreme numbers. For example, if your object had a reading of 10.14 ± 0.3 dpm, then look at the ages from readings of 10.44 and 9.84. Similarly, in Part e. if the object has an age of 2,153 ± 200, you should find the expected readings in dpm from ages of 1953 and 2353 years old.