MATH 1170
MATHEMATICS FOR LIFE SCIENTISTS
Computer Assignment III
Due September 16, 2003
To make sure that your assignment looks nice and isn't too long, you can preview it
before printing. Here's how.
Go to File in Maple and select Print
In the Print box select Output to File by darkening in the little diamond. Click on
Print at the bottom of the box. This creates a Postscript version (.ps) of your Maple
le.
Go to your Local Window.
At the prompt type ghostview lename.ps where lename is what you have called
your Maple le (be sure to use the .ps ending rather than the .mws ending). Hit
return.
A new window will appear. Click where you want it.
Look at your pages and make sure that there are no gaping white spaces or other
embarrassing mistakes.
If your assignment is more than 4 pages, it's too long.
If there are problems with the way that it looks, go back into Maple and x and save
it, then repeat the above steps to review your corrections.
Those of you who don't believe in printing have the option of emailing me your Maple
worksheet. Ask me and I'll show you how.
PR OBLEMS
Exercise 1. We know that the discrete-time dynamical system
bt+1 = rbt
produces solutions that grow exponentially (if r > 1). We will compare these solutions
with a system that produces growth that is slower than exponential. The way to do
this, as we will study in much more detail soon, is to make the factor r that multiplies
the population a decreasing function of bt . A simple choice is
bt+1 =
In each case below, set r = 1:5.
rbt
1 + 0:001bt :
a. Dene updating functions for these two situations (give them dierent names)
and plot them on the same graph for values of b between 0 and 1000.
b. Use iterplot2 to plot the solutions of the two systems starting from the initial
condition b0 = 1. When do the two solutions begin to look dierent?
Exercise 2. Consider the following functions.
f1 (x) = cos(x ? )
2
cos(3x ? 2 )
f3 (x) =
3
cos(5x ? 2 )
f5 (x) =
5
cos(7x ? 2 )
f7 (x) =
:
7
a. Find the amplitude, period and phase of each.
b. Plot them together on one graph.
c. Plot the sum f1(x) + f3(x)
d. Plot the sum f1(x) + f3(x) + f5(x)
e. Plot the sum f1(x) + f3(x) + f5(x) + f7(x).
f. What does this sum look like?
g. Try to guess the pattern. An innite number of terms is a Fourier series, a sum
of cosine functions that add up to a square wave that jumps between values of
-1 and 1.
Exercise 3. Suppose that the amount of a drug in the body follows the updating function
dt+1 = h(dt ) = 0:25dt + 3:0g:
This means that the amount of drug decreases by 75% each day and is supplemented
by 3:0g each day.
a. Use Maple and the solve function to nd the inverse of h. The inverse is the
function that eats dt+1 and spits out dt. Convince Maple to tell you the formula
for the inverse. This is pretty tricky. Input this as a function with a command
like
> hinv := dnew -> some stuff;
b. Find the values of dt+1 when dt = 2:0, dt = 4:0, and dt = 6:0. Use the inverse
function on these values. Did they really come from the right place?
c. Find the values of dt when dt+1 = 2:0, dt+1 = 4:0, and dt+1 = 6:0. Do your
answers make sense?
d. Plot h and its inverse on one graph. Mark the points you found in parts b and c
on each line.