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Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Department of Mathematics
Pomona College
Course Outline for Mathematics 32S
Calculus III with Applications to the Life Sciences
Spring 2015
Time
Place:
Instructor:
Office:
Phone/email:
Courses Website:
Office Hours:
Text:
Prerequisites:
MWF 11:00 am  11
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Math 32S. Rumbos
Spring 2015
1
Solutions to Assignment #6
1. Give a formula dening the vector eld F (x, y) = f (x, y)i + g(x, y)j, where
f and g are real valued functions dened on the plane, whose picture is shown
below.
y
x
Solution: F (x, y) = xi + y j,
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Math 32S. Rumbos
Spring 2015
1
Review Problems for Exam 1
1. Sketch the curve C parametrized by
x = sin2 (t);
y = cos2 (t),
for
2
.
2
t
2. A curve C is parametrized by the dierentiable path given by
(t) = (3t2 , 2 + 5t), for t R.
r
Sketch the curve C in
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Math 32S. Rumbos
Spring 2015
1
Solutions to Assignment #4
1. A particle moves in the xyplane along a path determined by the parametric
equations
x = t;
for t R.
y = t3 t,
Compute the velocity and acceleration of the particle.
Solution: Set r(t) = (t, t3 t
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Math 32S. Rumbos
Spring 2015
1
Solutions to Assignment #2
1. A curve C in the xyplane is parametrized by the equations
x(t) = t + 2
and
y(t) = t + 1,
for t R.
Sketch the graph of C.
Solution: The equations
x = 2 + t;
y = 1t
(1)
are the parametric equation
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Math 32S. Rumbos
Spring 2015
1
Solutions to Assignment #3
1. Give a parametrization of the portion of the graph of y =
(1, 1) the point (16, 4).
x from the point
Sketch the curve.
Solution: A parametrization is given by
(x(t), y(t) = (t, t),
for 1
t
16.
A
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Math 32S. Rumbos
Spring 2015
1
Solutions to Assignment #5
1. Let J denote an open interval in R, and : J R2 be a dierentiable path
r
. For xed a J, dene
with continuous derivative r
t
s(t) =
( ) d
r
for all t J.
a
Show that s is dierentiable and compute
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Math 32S. Rumbos
Spring 2015
1
Solutions to Assignment #1
Background and Denitions.
In Section 2.1 of the class lecture notes, we derived the KermackMcKendrick SIR
model for the spread of an infections disease in a population,
dS = SI;
dt
dI
(1)
= SI I;
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
(D
SecRom Sh kes/ Wearaw
R m4 ' m o as
? Give an @19wa SWFace 5) We in 93 NW He
Fouows :
V r H 1 1} I : evhmg
V968 +l«e,$+ekes,.m Jew F=<am w >.» S we 6 SEW
fjl+:;=ol:
o as ._y CCW éav'eo' Mew viewed 'Rom above.
.°_. m K
f if) d: = f (Sana
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Math 32S. Rumbos
Spring 2015
1
Solutions to Assignment #7
1. Sketch the ow of the vector eld
F (x, y) = xi y j.
Solution: The ow of the given vector eld are curves that are obtained as
solutions to the dierential equations
dx
dt = x;
(1)
dy
= y,
dt
Sol
Calculus III with Applications to the Life Sciences
MATH 32

Spring 2015
Math 32S. Rumbos
Spring 2015
1
Topics for Exam 1
1. Paths and Curves
1.1 Parametrized curves
1.2 Dierentiable paths
1.3 Tangent lines to curves
1.4 Linear approximations to paths
1.5 Arc length
2. Vector Fields
2.1 Twodimensional vector elds
2.2 Flow of a
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