1. Rates of change in the Natural Sciences.
a) In your groups, experiment with quickly flashing a beam of light into your neighbors' eyes
and observe what happens to the areas of their pupils.Submission a:
Describe what happens.
Submission worksheet:
b) If R denotes the reaction of the body to some stimulus of strength x , the sensitivity S is defined to be the rate of change of the reaction with respect to x . A particular example is that when the brightness x of a light source is increased, the eye reacts by decreasing the area R of the pupil. The experimental formula
describes the dependence of R on x when R is measured in square millimeters and x is measured in appropriate units of brightness.
We do some work for you here by giving you the commands for entering the function R into Maple and getting a good graph:
>
plot(R(x),x=0..10,
labels=[`light stimuli`,`area of pupil: sq mm`],
title=` Pupil Reaction to light Stimuli`,tickmarks=[5,5],
titlefont=[TIMES,BOLD,12],view=[-2..10,0..40],thickness=3);
![[Maple Plot]](Derivatives Eye problem images/Derivatives Eye problem3.gif)
Submission b:
Why is
a decreasing function (give a physical/conceptual reason)? Does R(0) exist? If so, can you give a physical reason why this is the case, and what is its actual value? From the graph of
what can you say about the rate of change in the area of the pupil as the brightness increases? How is this different for low brightness versus high brightness?
Submission worksheet:
c) Sensitivity is defined to be the derivative of R with respect to x.
Submission c:
Use Maple to:
(1) find the derivative and
(2) get a good plot of Sensitivity. Use the model above to get a "good" plot - labeled axes, titled, etc.
(3) Think about what
you
understand about the word sensitivity. Why is the derivative of
R
a good mathematical description of sensitivity? Why is the graph of sensitivity contained in the fourth quadrant? Does
exist? What does this limit tell you about how the eye works?
Submission worksheet: