ANSWERS TO HOMEWORK SET 2
Proboem 4.1
(a) We use Mathematica to draw the vector eld in the case r1 = r2 = 1,
a1 = a2 = 1, b1 = b2 = 1/2. Any choice of r1 , r2 , a1 , a2 , b1 , b2 which satisfy the
given inequalities produces a similar graph.
Figure 1. Vec
Math 4428. MATHEMATICAL MODELING. Spring 2017
Homework #3 (due on Wednesday, April 5, at the beginning of class).
50 points are divided between problems 14. In your solutions, whether or
not you are using a computer software, you need to justify each step
Appendix A. Simplex Method
Consider the problem of maximization of the objective function
y = f (x) = cT x = (c, x) = c1 x1 + + cn xn
on a feasible region S defined by the constraints
Ax b, i.e. gi (x) = ai1 x1 + + ain xn bi (i = 1, . . . , m), and xi 0 f
Math 4428, Sec. 2. MATHEMATICAL MODELING.
Spring 2017
Short Solutions to Homework #1.
#1. (12 points.) Reformulate the pig market problem (Example 1.1 on p.4 in the textbook), assuming that
a pig gains pounds per day, where 0 10.
(a). Find the best time x
Appendix C. Properties of Symmetric Matrices
We will treat vectors x Rn as matrices n 1. Then (x, y) = xT y, where
transposition. Therefore,
T
means the
(Ax, y) = (Ax)T y = xT AT y = (x, AT y).
(1)
In particular, if A is symmetric matrix, i.e. AT = A, the
Math 4428, Sec. 2. MATHEMATICAL MODELING.
Spring 2017
Short Solutions to Homework #2.
#1. (10 points.) Using Newtons iteration method, find the roots of the equation x ln x = 1 correct to six
decimal places. Start with the initial point x0 = 1, and write
ANSWERS TO SELECTED HOMEWORK PROBLEMS
Problem 1.2. The sensitivity of the best time to sell to the cost per day of
keeping the pig is -0.5625. The sensitivity of the maximal prot to the cost per day
of keeping the pig is -0.027.
In the case of using the n
ANSWERS TO HOMEWORK SET 3
Proboem 7.7
(a) 1 0.99952 0.0507.
(b) 1(1n/1000)52 , for n = 1, 2, ., 9. The following table shows the probability
p of coming out a winner for the year if we buy n = 1, 2, ., 9 tickets per week.
1
2
3
n
p .051 .099 .145
4
.188
5
Math 4428
Mathematical Modeling
Midterm Exam
Name:
Student I.D. #:
Instructions:
There are 2 problems for a total of 80 points.
You have 50 minutes to complete the exam.
Show all your work. Unsupported answers will receive little credit.
Question Point
ANSWERS TO HOMEWORK SET 1
Problem 1.5
(a) Step 1: Ask the question.
Variables:
x =population (whales)
E =level of eort (boat-days)
g =growth rate (whales per year)
h =harvest rate (whale per year).
Assumptions:
g =0.08x(1 x/400, 000)
h =0.00001Ex
g =h, x
Introduction to Dynamic Models
March 7, 2014
Dynamic models describe the processes that evolve over time, e.g.,
electrical circuits, chemical reactions, population growth, investments and
annuities, space ight, etc.
Introduction to Dynamic Models
March 7,
Computational Methods for Optimization
February 21, 2014
1
One Variable Optimization
2
Multivariable Optimization
3
Linear Programming
4
Discrete Optimization
Computational Methods for Optimization
February 21, 2014
2 / 101
Section 1
One Variable Optimiza
Appendix B. Ordinary Dierential Equations
1. Examples
Example 1.1. Consider the rotation of a two-dimensional plane 0x1 x2 about the origin 0 with
satisfies |v| = |x|, v x. Therefore, choosing the
unit angular velocity. Then v = dx
dt = x
)
)
(
(
x2
x1
,