1
(Math 360) Free Response Final:
April 28, 2009
2
Rules
Preparation:
In preparing for this part of the nal exam you are allowed to use any books
as well as any source you nd on the internet. You are also allowed to talk
with anyone involved with this cla
Math 360: Homework 9
Due Friday, November 15
Midterm 2 Redux Question:
1. Prove that for any real constants a0 , a1 , a2 ,
x3 + a2 x2 + a1 x + a0
= .
x
x2 + 1
lim
In class on November 5th, we dened the functions sin x and cos x on the real line and establ
MATH 360 Homework 11
Due 23 April 2013
1. For a xed n N, consider the sequence of functions n : [0, 1] R, given by: if x = p is a rational
q
number in reduced form, with q n, then n (x) = q12 ; for other x (that is, the irrationals and any
rational with r
MATH 360 Homework 10
Due 12 April 2013
1. (Marsden-Homans problem 4.42) For x > 0, dene L(x) =
this denition of L and theorems about the denite integral.
x1
dt.
1t
Show the following using only
i. L(x) is a strictly increasing function.
ii. L(xy ) = L(x)
MATH 360 Homework 9
Due 29 March 2013
1. Show that f is dierentiable at x0 i lim
h0
f (x0 + h) f (x0 )
exists. Moreover if this limit exists, it
h
must be f (x0 ).
2. Let f : R R.
i. Suppose f is dierentiable at x0 (a, b). Show that
f (x0 ) = lim
h0
f (x0
MATH 360 Homework 1
Due 18 January 2012
Properties of the Real Numbers
1. Show that the following properties hold in any eld.
i. (Unique identities) If a + x = a for some a, then x = 0. If ax = a for some a, then x = 1.
ii. (Unique inverses) If a + x = 0,
MATH 360 Homework 2
Due 25 January 2013
1. (Versions of the Archimedean property) Prove that the following properties are equivalent in an ordered
eld, i.e. any ordered eld F which possesses one of the properties must possess the other two as well:
i. If
Math 360: Homework 10
Due Friday, November 22
1. Fix k 1, and suppose g (t) is real-valued and k -times dierentiable on some neighborhood of the
origin in R (we specically do not require that g (k) is continuous).
(a) Suppose that g (0) = g (0) = = g (k1)
Problem 1
Proof. a)
In general, we need only check the axioms when primes are involved, as
we know R is a eld.
(A3) Additive associativity - (a+b) + c = (a+b) + c) = (a+ (b+c)
= a + (b+c) = a + (b+c), and likewise for 2 or 3 primes
(A4)Additive identity:
1. Suppose f is twice dierentiable on [a, b], f (a) < 0, f (b) > 0, and
f (x) > 0, and 0 f (x) M for all x [a, b]. Let be the unique
point in (a, b) where f ( ) = 0.
f
a) Pick x1 (, b) and dene xn+1 = xn f (xn ) . Interpret this geometrixn
cally in terms
1. A - It is possible to place a linear order on C so that we do NOT get
an ordered eld, and impossible to place a linear order of C so that we DO
get an ordered eld.
2. D - |x + y | = |x| + |y | doesnt have to happen. For example, let
y = x = 0.
3. Yes,
MATH 360 Homework 8
Due 22 March 2013
1.
i. Give a precise meaning to the statement, Disconnections pull back. Is the statement true?
ii. Show that the image of a connected set under a continuous map is connected.
2. Dene S n = x Rn+1 x = 1 .
i. Show that
MATH 360 Homework 7
Due 15 March 2013
1. Suppose (V; kk) is a normed space. Show that kk : V 3 R is continuous with respect to the metric
induced by kk and the standard metric on R. (Hint. Use the reverse triangle inequality.)
2. Suppose (M1 ; d1 ) and (M
MATH 360 Homework 6
Due 1 March 2013
Denition. Given two metrics d and on the same set M , we say d is equivalent to if there are positive
real numbers C1 and C2 so that for any x, y M , we have
C1 (x, y ) d(x, y ) C2 (x, y )
1. Show that metric equivalen
MATH 360 Homework 6.5
Not due, but do!
Dene
R = (x1 , x2 , . . . , xk , . . .) xi R, all but nitely many xi = 0
Note that under pointwise operations, R is a vector space.
1. For x, y R , show that x, y
2
xi y i denes an inner product on R . Write d2 for t
MATH 360 Homework 5
Due 22 February 2013
Denition. A sequence (xn )n2N in a metric space is called eventually
constant if there is some N so that
! N guarantees x = x .
1. Let A & B be subsets of a metric space (M; d). Show that A is dense in B if and onl
MATH 360 Homework 4
Due 8 February 2013
Recall that our denition of a closed subset diers from Rudins:
Denition. A subset of a metric space A (M, d) is closed if its complement is open.
1.
i. Show that the union of a nite number of closed sets is closed.
MATH 360 Homework 3
Due 1 February 2013
1. Suppose (an )nNis a sequence of real numbers with an A and A > 0. Consider the sequence
bn = n + an n.
i. Let 0 < c < 1 . Show that there is some N N, depending on c and the sequence (an )nN , so that
2
n N guara
Solutions to First Exam, Math 360, Fall 2002
Question 1 Dene the Cantor set K to be the set of real numbers of
the form
an
,
3n
n=1
where each an cfw_0, 2. Is K countable or uncountable? Prove your answer.
Answer 1 K is uncountable. The proof proceeds muc
Solutions to Second Exam, Math 360, Fall 2002
Question 1
(a) If A is connected, prove cl(A) is connected.
(b) Show that if A is path-connected, cl(A) may not be path-connected.
Answer 1 Both parts of this problem are fairly easy if you remember
how this w
Review Sheet for Final Exam
Mathematics 360
December 16, 2002
The exam will be posted on the web site at 12:00 noon on Saturday, December 14 and
will be due in my oce (DRL 4N28) on Monday, December 16 at 1:00 pm.
You may use Marsden-Homan but no other boo
Fourier series can be obtained for any function defined on a finite range, as in the S-L section above. In practice they provide a way to do various calculations with, and to analyse the behaviour of, functions which are periodic, i.e. repeat the same val
Full range Fourier series
The idea is to write a function f (x) defined for a range of values of x of length 2 , say x , as a
series of trigonometric functions
f=1
2a0+
1
an cosnx+
1
bn sinnx. (7.1)
Here the 1
2a0 is really a cos0x = 1 term (the eigenfunc
Half range series; odd and even functions
We recall f (x) is odd f (x) = -f (-x) for all x. f (x) is even f (x) = f (-x) for all x. A general function can always be written as f (x) = 1 2 ( f (x)+ f (-x)+ 1 2 ( f (x)- f (-x) in which the first part on the
Completeness and convergence of Fourier series
We now give answers to two questions: can every function with period 2 be written this way, and does the
series always converge at all x? These ideas are referred to as completeness and convergence. To specif