Chapter 10 Equations and Facts
Equation of a Sphere
r^2=(x-a)^2+(y-b)^2+(z-c)^2
Vectors have both _ and _
magnitude and direction
A+B (both vectors) =
(A1+B1)i + (A2 +B2)j + (A3+B3)k
A Unit Vector has a magnitude of _
one
The three standard unit vectors a
Your analysis, based on the concepts covered in this course, will address each of the following:
1. Business Strategy Analysis: Develop an understanding of the business and
competitive strategies of the company. Which of the three generic competitive
stra
Mathematics 580: Real Analysis
Midterm Review
I. Denitions and statements of theorems.
1. Given a sequence, cfw_ xn in R, give two equivalent, yet dierent, definitions of lim supn xn and lim inf n xn .
2. Given a countable family of sets cfw_En , give tw
MATHEMATICS 580
FINAL EXAM
DUE THURSDAY, DECEMBER 12
Please deliver your paper by 5:00 PM Thursday December 12 to my oce in 149
Gordon Palmer hall. If I am not there you may slide it under my door. In working
the exam, you may consult me, your lecture not
MATHEMATICS 580: REAL ANALYSIS I
MIDTERM, OCTOBER 9, 2013
I. Theory
1. Let (X, d) be a metric space.
(a) What is the Borel -algebra, BX ?
The Borel -algebra is the smallest -algebra that contains all the open sets in X .
(b) Give three dierent families of
Mathematics 580: Real Analysis
Midterm Review
I. Denitions and statements of theorems.
1. Given a sequence, cfw_ xn in R, give two equivalent, yet dierent, definitions of lim supn xn and lim inf n xn .
lim sup xn = lim sup xk = supcfw_ x : cfw_ xnk with
MATHEMATICS 580
HOMEWORK 1 SOLUTIONS
DUE SEPTEMBER 4
1. We will show that the sequence cfw_xn is Cauchy, hence convergent since X is
complete. Since the series converges, given > 0 we can nd N such that m
n N implies m n d(xk , xk+1 ) < . By the triangl
MATHEMATICS 580
HOMEWORK 2
SOLUTIONS
1. Dene the function F (x) = d(x, f (x). Then given x, y K we have
F (x) = d(x, f (x) d(x, y ) + d(y, f (y ) + d(f (y ), f (x)
2d(x, y ) + d(y, f (y ) = 2d(x, y ) + F (y ).
So F (x) F (y ) 2d(x, y ). By interchanging
MATHEMATICS 580
FINAL EXAM
DUE THURSDAY, DECEMBER 12
Please deliver your paper by 5:00 PM Thursday December 12 to my oce in 149
Gordon Palmer hall. If I am not there you may slide it under my door. In working
the exam, you may consult me, your lecture not
MATHEMATICS 580
HOMEWORK 5
SOLUTIONS
1. The point of this exercise is to show that the measurability assumption of the
function f (x, y ) on the product space is important. Consider the measure spaces
(X, M, ) = (Y, N , ) = ([0, 1], L, m).
Notice both of
MATHEMATICS 580
HOMEWORK 4
SOLUTIONS
1. Let f be the Cantor function and dene (x) = x + f (x).
(a) Prove that is a continuous bijection from [0, 1] to [0, 2], and therefore has
an inverse function 1 : [0, 2] [0, 1] that is continuous. (Hint: Show that
is
MATHEMATICS 580
HOMEWORK 3
SOLUTIONS
1. Suppose (X, M) is a measurable space, g : X R is measurable, and f : R R
is Borel measurable cfw_x R : f (x) > a BR for all a R. Prove that h = f g
is measurable. (Hint: use exercise 4 (b) on homework assignment 2.)
Math 238 Project
Due by April 26, 2013
Note: You will get credit for the project provided it appears that you made a good faith eort to
answer the questions below. You should fully work one of the two problems, but try to do both!
You can document your wo
Review for Final
Note: The only topic which might be on the exam, but is not on this review, is direction elds.
In particular, I decided not to put any questions about systems on the exam. The review over
represents section 7.6 compared to the nal, but th
Homework #9 Solution
3.71 A warehouse contains ten printing machines, four of which are defective. A company randomly selects
ve of the machines for purchase. What is the probability that all ve of the machines are not defective?
ans: There are four machi