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Department
of Mathematics & Computer Science
Mathematics 31S. Calculus II with Applications to t=
he
Life Sciences.  =
;
Fall 2011
Course Outline
Time and
Place: =
MWF
11:00 am - 10:50 am
Andrew 253
Instructor: =
&=
nbsp; Dr.
Adolfo J. Rumbos
Office:<=
span
style=3D'mso-spacerun:yes'> =
=
Andrew
259.
Phone / e-mail: =
ext. 18713 /
Office Hours: =
&=
nbsp; MWF
9:00 am - 10:15 am or by appointment
Courses Website: =
http://pages.pomona.edu/~ajr04747/
Text:<=
span
style=3D'mso-tab-count:3'> &=
nbsp; &nbs=
p; Essential Calculus with Applications by Richard A. Silverman
Prerequisites:<=
span
style=3D'mso-spacerun:yes'> Math
30 (grade of C- or better), or passing score in placement exam.
Course Description. In this c=
ourse
we study integral and differential calculus in the context of problems aris=
ing
in the life sciences. We will=
be
dealing mainly with problems that come up in population biology concerning =
the
description of the evolution in time of the size of the population of a giv=
en
species, as well as the interaction of several species living in a common
environment. Analysis of this=
type
of problems leads naturally to diff=
erential
equations. These are
expressions involving an unknown function (which one seeks to find) and its
derivatives. We will spend the
first part of the course learning how to analyze the differential equations
that come up in the study of the problems mentioned above. Some of the equations can be solved
using integral calculus, but others cannot be solved easily, and so the best
one can do is to use approximations, in particular, linear approximations, =
to
analyze them. We will see that sometimes those approximate solutions to the
equations actually tell us a lot about the system we are studying.
The specific topics to be covered are listed in the attached Tentative Schedule of Lectures and
Examinations.
Assigned
Grading Policy. Grades will be =
based
on the homework, two 50-minute examinations, plus a comprehensive final
examination. The grades will =
be
computed as follows:
=
homework
&=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; 20%
=
Two
50-minute exams &nbs=
p; &=
nbsp; &nbs=
p; 50%
=
final
examination &n=
bsp;  =
; &n=
bsp;  =
; 30%
Time: =
Friday,
December 16 &=
nbsp;
9:00 am
Place: Andrew
253
Math
31S  =
; &n=
bsp;  =
; &n=
bsp;  =
; &n=
bsp;  =
; Fall
2011
Tentative Schedule of Lecture=
s and
Examinations
W Aug.<=
span
style=3D'mso-spacerun:yes'> 31 A
conservation principle: One-compartment dilution
F &=
nbsp; Sep. 2 &=
nbsp; Recovering
a function from its rate of change
M &n=
bsp; Sep. 5 &=
nbsp; What
is a differential equation?
W Sep. 7 &=
nbsp; Review
of integration: The Fundamental Theorem of Calculus
F &=
nbsp; Sep. 9 &=
nbsp; The
natural logarithm function
M &n=
bsp; Sep. 12 The
natural logarithm function (continued)
W Sep. 14 The
exponential function
F &=
nbsp; Sep. 16 The
exponential function (continued)
M &n=
bsp; Sep. 19 Solving
first order differential equations
W Sep. 21 Separat=
ion
of variables
F &=
nbsp; Sep. 23 Linear
first order differential equations
M &n=
bsp; Sep. 26 Linear
first order differential equations with constant coefficients
W Sep. 28 Applications of =
first
order differential equations
F &=
nbsp; Sep. 30 Qualita=
tive
analysis of a first order equation.
M &n=
bsp; Oct. 3 &=
nbsp; Qualitative
analysis (continued)
W Oct. 5 &=
nbsp; Models
of population growth
F &=
nbsp; Oct. 7 &=
nbsp; Models
of population growth (continued)
M &n= bsp; Oct. 10 Review<= o:p>
W Oct. 12 Exam
=
1
F &=
nbsp; Oct. 14 The
logistic model of population growth
M &n=
bsp; Oct. 17 Fall
recess: No Classes
W Oct. 19 The
logistic model (continued)
F &=
nbsp; Oct. 21 Solving the logi=
stic
model: Partial fractions
M &n=
bsp; Oct. 24 Partial
fractions (continued)
W Oct. 26 Lineari=
zation
F &=
nbsp; Oct. 28 Integra=
tion
by parts
M &n=
bsp; Oct. 31 Integra=
tion
by parts (continued)
W Nov. 2 Principle =
of
linearized stability
F &=
nbsp; Nov. 4 Systems of
differential equations
M &n=
bsp; Nov. 7 Solving
systems of differential equations
W Nov. 9 Phase-plane
analysis: nullclines, equilibrium points and stability
F &=
nbsp; Nov. 11=
Phase-plane
analysis (continued)
M &n=
bsp; Nov. 14=
Population
models of two interacting species
W Nov. 16=
Predator-Prey
models: The Lotka-Volterra equations
F &=
nbsp; Nov. 18=
M &n=
bsp; Nov. 21=
Predator-prey
models continued
W Nov. 23=
Competition
and cooperation
F &=
nbsp; Nov. 25=
Thanksgivi=
ng
recess
M &n=
bsp; Nov. 28=
The
principle of competitive exclusion.
W Nov. 30=
Review
F &=
nbsp; Dec. 2 E=
xam
2
M &n=
bsp; Dec. 5 Revi=
ew
W Dec. 7 Revi=
ew
F &=
nbsp; Dec. 16=
Final
Examination