This preview shows pages 1–4. Sign up to view the full content.
MAT 1330: Calculus for the Life Sciences I
01.10.2008
Assignment 1, SOLUTIONS
Pawel Lorek
University of Ottawa
Problem 1:
[4 points] Suppose that every morning a patient receives the same dose
of drug. From the dose, the drug concentration in his blood increases by 2. Over the
course of 24 hours between doses, 75% of the drug in the blood is removed.
(a) Write the linear DTDS for the drug concentration,
x
t
+1
=
f
(
x
t
)
,
and ±nd
x
4
when
x
0
= 88
.
(b) Draw the updating function and start the cobwebbing process at
x
= 0
.
2
.
(c) Find the equilibrium explicitly.
(d) Is the equilibrium stable? Use the stability criterion from class and compare with
your cobwebbing.
Solution
Because formulation of the Problem was ambiguous two solutions are accepted: with
DTDS
x
t
+1
= 0
.
25
x
t
+ 0
.
5 and with DTDS
x
t
+1
= 0
.
25
x
t
+ 2
Solution with
x
t
+1
= 0
.
25
x
t
+ 0
.
5
(a) Just after receiving the medication the concentration is
x
n
+ 2, but after 24 hours,
before next dose is applied, 75% of drug is removed, i.e. we have
x
n
+1
= (
x
n
+ 2)
−
0
.
75(
x
n
+ 2) = 0
.
25
·
x
n
+ 0
.
5
We have
x
0
= 88
x
1
= 0
.
25
·
x
0
+ 0
.
5 = 0
.
25
·
88 + 0
.
5 = 22
.
5
x
2
= 0
.
25
·
x
1
+ 0
.
5 = 0
.
25
·
22
.
5 + 0
.
5 = 6
.
125
x
3
= 0
.
25
·
x
2
+ 0
.
5 = 0
.
25
·
6
.
125 + 0
.
5 = 2
.
03125
x
4
= 0
.
25
·
x
3
+ 0
.
5 = 0
.
25
·
2
.
03125 + 0
.
5 = 1
.
0078125
(b) See Figure 1.
(c) Equilibrium
x
*
must ful±lls
x
*
=
f
(
x
*
) i.e.
x
*
=
0
.
25
·
x
*
+ 0
.
5
substract 0
.
25
·
x
*
from both sides
0
.
75
·
x
*
=
0
.
5
divide both sides by 0.75
x
*
=
0
.
5
0
.
75
=
1
2
·
4
3
=
2
3
(d) The slope of updating function is equal to
1
4
. And

1
4

<
1, so the Slope Criterion
implies that
x
*
=
2
3
is a stable equilibrium. Cobwebbing, we should obtain the same
equilibrium starting at
x
0
n
= 0
.
2.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
n
x
n+1
x
n+1
= 0.25 x
n
+ 0.5
y = x
y=f(x) = 0.25 x+0.5
Figure 1: Problem 1: Cobwebbing with
x
0
= 0
.
2
Solution with
x
t
+1
= 0
.
25
x
t
+ 2
The analysis is similar, here are just results:
(a)
x
0
= 88
, x
1
= 24
, x
2
= 8
, x
3
= 4
, x
4
= 3,
(b) similar,
(c)
x
*
=
8
3
,
(d) the same.
2
Problem 2:
[4 points] A group of patients is given a certain dose of a drug once. Two
measurements of concentration of the drug in the blood are taken 24 hours apart to de
termine the rate at which the drug is removed from the blood stream. The measurements
are given below.
patient
initial measurement
Fnal measurement
1
3
1
2
4.5
1.5
3
0.6
0.2
4
1.8
0.6
(a) Write a DTDS of the form
x
t
+1
=
ax
t
for drug removal and Fnd the value of
a.
(b) ±or patient 1, how long will it take until the drug concentration is below 0.1?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/25/2011 for the course MAT 1330 taught by Professor Dumitriscu during the Spring '08 term at University of Ottawa.
 Spring '08
 DUMITRISCU
 Calculus

Click to edit the document details