MT804 Analysis Homework IV
Solutions
November 16, 2008
7.1.3,
7.1.5,
7.1.6 ,
7.7.4,
7.7.5,
7.7.6,
7.7.11.
Exercise 7.1.3 Show that if the Riemann integral of f exists then it is unique.
Proof: Failure of uniqueness would mean there are two real numbers I
Solutions
Math 3320 (Fall 2015)
Page 1 of 6
Homework #9
November 11, 2015
Exercise 5.2.2
(a) Consider the functions f : R R and g : R R dened as follows:
f (x) =
0
1
if x = 0,
if x = 0;
g(x) =
and
1
0
if x = 0,
if x = 0.
Then, neither f nor g is continuou
Solutions
Math 3320 (Fall 2015)
Page 1 of 4
Homework #11
December 4, 2015
Exercise 7.4.5
Let f and g be integrable functions on [a, b].
(a) Let P be any partition of [a, b]. For any particular subinterval [xk1 , xk ], consider the
following quantities:
Mk
Solutions
Math 3320 (Fall 2015)
Page 1 of 4
Homework #4
September 30, 2015
Exercise 2.4.1
Let (xn ) be the sequence dened by x1 = 3 and xn+1 =
1
.
4 xn
(a) To show (xn ) is decreasing, notice that x1 = 3 > 1 = x2 . For the inductive step,
suppose xn > xn+
Solutions
Math 3320 (Fall 2015)
Page 1 of 4
Homework #8
November 4, 2015
Exercise 4.4.1
Let f : R R be dened by f (x) = x3 .
(a) Fix c R and notice that
|f (x) f (c)| = x3 c3 = |x c| x2 + xc + c2 .
We can obtain an upper bound on the last term by insistin
Solutions
Math 3320 (Fall 2015)
Page 1 of 2
Homework #2
September 16, 2015
Exercise 1.4.1
(a) Let a, b Q. So, there exist p1 , p2 , q1 , q2 Z with q1 , q2 = 0 such that a = p1 /q1 and
b = p2 /q2 .
i) Notice that ab =
p1 p2
q1 q2 .
Since p1 p2 , q1 q2 Z an
Solutions
Math 3320 (Fall 2015)
Page 1 of 4
Homework #3
September 23, 2015
Exercise 2.2.2
(a) First, notice that
2n + 1 2
3
3
=
.
5n + 4 5
25n + 20
25n
So, let > 0 be arbitrary. Choose N N such that N > 3/(25 ). Then, for any n N ,
2n + 1 2
3
3
<
= .
3
5n
Solutions
Math 3320 (Fall 2015)
Page 1 of 3
Homework #5
October 7, 2015
Exercise 2.6.2
(a) Consider the sequence (xn ) dened by xn = (1)n /n, which we know converges to zero.
This sequence is certainly not monotonic because it oscillates around zero. Howe
Solutions
Math 3320 (Fall 2015)
Page 1 of 5
Homework #1
September 9, 2015
Exercise 1.2.1
(a) For a contradiction, assume that there exist p Z and q N such that
p
q
2
= 3.
We may also assume that p and q have no common factors. Now, the above equation
impl
Solutions
Math 3320 (Fall 2015)
Page 1 of 3
Homework #6
October 21, 2015
Exercise 3.2.2
Dene the following sets:
A = (1)n +
2
:nN
n
and
B = cfw_x Q : 0 < x < 1 = Q (0, 1).
(a) Limit points of A are 1 and 1.
Limit points of B are [0, 1].
(b) A is neither
Math 804 Analysis I
Final Exam Solutions
December 20, 2008
Each problem is worth 30 points, for a total of 150 points. The rst three problems come from our textbook. You
may consult any book or website for help. However, you may not discuss or consult or
MT804 Analysis Homework I
Eudoxus
September 27, 2008
p. 119, Exercise 4.2.1 a,f,g. If necessary you can use parts b,c,d,e without
proof.
p. 123 prove Corollary 4.3.1
p. 125 Exercises 4.3.2, 4.3.3
Exercise 4.2.1
(a) Show that () = 0. More generally, show t
MT804 Analysis Homework II
Eudoxus
October 6, 2008
p. 135 4.5.1, 4.5.2 p. 136 4.5.3 (part a only) p. 140 4.6.1
Exercise 4.5.1 Use the Intermediate Value Theorem to prove that every polynomial of with real coefcients and odd degree has a zero in R.
Proof:
MT804 Analysis Homework III
Solutions
November 8, 2008
p.
p.
p.
p.
p.
p.
224 6.4.2 a,c
225 6.4.4
226 6.4.5
230 6.5.2
234 6.6.3b
234 6.6.4
Exercise 6.4.2a Use the Mean Value Theorem and the fact that (ex ) = ex to
prove that ex 1 + x for all x R
Proof: Let
Solutions
Math 3320 (Fall 2015)
Page 1 of 5
Homework #10
November 20, 2015
Exercise 7.2.2
Consider the function f (x) = 1/x on the interval [1, 4] and let P = cfw_1, 3/2, 2, 4.
(a) The following table might help with the requested computations:
k
[xk1 , x