
Estimating Solutions to Parametric Differential Equations: A
PredatorPrey Model.
Due Friday, Feb. 23 .

Read Section 10.7 of Stewart especially Example 1, parts a and b related
to
Equation 1: dR/dt = kR  aRW
dW/dt = rW + bRW.
Read pages 624626 on Euler's method together with materials from Flashman
on Euler's method .

Suppose R(0)=100 and W(0)=10 in Equation 1.
Estimate R(4) and W(4) using Euler's method
with n = 4 with the following choices for the constants a,b,k, and
r.
In each case discuss the quality of your estimate and the relation of these
to part a and b ot the problem.

a = b = 1, k = 2, r = .5 .
a = .2, b =.5, k = r = 1.

Curvature for graphs of functions in the
plane.
Due Wednesday, March 7.

Paramatrize the graph of y = f(x) with x = t and
y = f(t).
Prove that the curvature of the graph of f at (x, f(x)
is given by the formula
k (x) = f '' (x) / [1 + (f'(x)^{2}]
^{3/2
}.

Find where the graph of y=ln(x) has its largest curvature.

Homogeneous functions and the motion
controlled by a heat seeking process.
Due Wednesday April 4.
1. a. A real valued function of two variables, f::R^{2}>R
is called homogeneous of degree m if for any real numbers
x,y,and t,
f(tx,ty) = t ^{m }f(x,y).
Suppose f is differentiable and homogeneous of degree m, show that
x D _{x} f(x,y) + y D _{y }f(x,y) = m
f(x,y).
[Hint: If g(t) = f(tx,ty) , find g'(1).]
b. Generalize your result to homogeneous differentiable
functions of 3 and 4 variables.
2. The temperature distribution on a metal plate at time t is given
by the function H _{t} of x and y:
H_{ t}(x,y) = [(xcos( pt
))^{ 2} + (ysin( pt ))^{ 2}
].
A manually controlled particle moves along the plate in straight lines
periodically adjusting the direction vector of the line it follows so that
at the time the direction is adjusted the new direction vector is in the
direction of most rapid temperature increase at that instant and the particle
has speed equal to that most rapid rate.
a) Describe the motion of the particle if it is located
at (0,0) when t=0 and adjustments are made when
t = 0,1,2,3,4,... etc.
b) Describe the motion of the particle if it is located
at (0,0) when t=0 and adjustments are made when
t = 0,1/2,1,3/2,2,... etc.

Extremes of linear functions on triangular regions.
Due
41301 Now 416!
Suppose T is a triangular region in the plane.
Suppose L(x,y) = Ax + By where A and B are constants that
are not both 0. Use the LaGrange theory to explain why the maximum and
minimum values of L for the region T will occur at one vertices of the
triangle.
State and justify the analogous result for planar regions
bounded by quadrilaterals and pentagons.
Apply this work to find the maximum and minimum
values of L(x,y) = 3x + 5y when (x,y) satisfy all the following (linear)
inequalities:
x  2y < = 12; 12 > = x + 2 y;
4 < = x 2y ; and 1 < = x. [ revised 411 old version
had 12 < = x + 2 y]
3 Bonus Points: Generalize this problem to one of the following
situations:
a. A tetrahedron in
space with L(x, y, z) = Ax + By + Cz.
b. A planar region
bounded by a polygon with n sides.
Justify your statement and illustrate it with an example.

Independent Factors in Products.
Due 5201.