#1 Predator Prey Due9-18 |
#2 Curvature Due | #3 In
Stewart: 15.3: 70,72 15. 4:38 Due |
#4 Homogeneous
functions; motion Due |
#5 Extremes of linear functions on
triangular regions. Due |
#6.Independent
Factors in Products. Due |
Equation 1: dR/dt = kR -
aRW dW/dt = -rW + bRW.
Read pages 589-591 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 Example 1 of Stewart.
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:
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:
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 with A,B, and C not all 0.
b. A planar
region bounded by a polygon with n sides.
Justify your statement and illustrate it with an example.
A. Find òò_{R } (3x^{2}+1) (4y^{3} +2y +1) dA where R = [1,3] × [0,2].
B. Suppose g and h are continuous real valued functions of one variable.
Let F(x,y) = g(x)h(y). Explain why òò_{R }F(x,y) dA = ò_{a}^{b }g(x) dx^{ }ò_{c}^{d }h(y) dy where R = [a,b] × [c,d].
C. Use part B to show that òò_{R }exp(-x^{2 }- y^{2}) dA = [ò_{-m}^{m }exp(-x^{2}) dx]^{2 } where R = [-m,m] × [-m,m] .