Math 121

Homework #2

Instructions. This is due on September 13 at the beginning of class. It is worth 20 points. Late papers lose 4 points if turned in Monday, plus 3 points per additional day or partial day late. You must work by yourself, but feel free to consult with me. Explain what you did and why you did it: good explanations are essential! Your paper should start with an introduction and finish with a conclusion. Start early, ask questions and do well.

Introduction. The pupils in your eyes can expand (dilate) and contract to control how much light is getting in. In bright light, your pupils contract and in low light they dilate (see page 73 for two pictures of my cat’s eyes). Each individual’s pupils has a smallest size (we’ll call it s) and a maximum size (we’ll call it m) that can be reached.In this assignment, we will explore one type of function that could model this phenomenon. We will let x represent the intensity of light, and f(x) will be the size of the pupils in mm.

Problems. Briefly explain why we want the following to be true:

· The larger x is, the smaller f(x) is.

· lim_x®0+f(x) = m

· lim_x®¥f(x) = s

For the function f(x) = (90x^-0.3 + 60) / (6x^-0.3 + 15), do the following.

· Find lim_x®0+f(x) (be sure to explain why you can’t just plug in 0) and interpret this answer in terms of the size of the pupils

· Find lim_x®¥f(x) and interpret this answer in terms of pupil size

· Graph the function using the window 0 £ x £ 10 and 0 £ y £ 15

· Comparing the graph to the limits you computed, identify two things that are misleading about the calculator graph. Briefly discuss how you might force the calculator to give more accurate graphs

Using the above work as a guide, find a function that could describe the pupil size of an animal whose pupils can shrink down to 2 mm and dilate to 10 mm. Explain how you chose this function.