#1 Predator Prey Due Feb. 12 |
#2 Due March 5. Work for a Moving Particle. | #3 Homogeneous
functions; motion Due: April 3 |
#4 Extremes of linear functions on
triangular regions. Due April 17 |
#5.Independent
Factors in Products. Due May 1 | 6.*** Curvature Due TBA |
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 to Example 1 of Stewart.
i.Suppose $f$ is differentiable and homogeneous of degree 2, show that $x D_x f(x,y) + y D_ y f(x,y)= 2 f(x,y)$. [Hint: If $g(t) = f(tx,ty)$ , find $g'(1)$.]
ii. Generalize your result to homogeneous differentiable functions of degree 3, 4, and $n$.
B. 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 $\int \int_R (3x^2+1) (4y^3 +2y +1) dA$ 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 $\int \int_R F(x,y) dA = \int_a^b g(x) dx \int_c^d h(y) dy$ where $R = [a,b] × [c,d]$..
C. Use part B to show that $\int \int_R exp(-x^2 - y^2) dA = [\int_{-m}^m exp(-x^2) dx]^2$ where $R = [-m,m] × [-m,m]$ .