Elementary Analysis Math 140BWinter 2007
Homework answersAssignment 10; February 19, 2007
Exercise 28.4, page 211
1
Let f (x) = x2 sin x for x = 0 and f (0) = 0.
(a) Show that f is dierentiable at each a = 0 and calculate f (a). Solution: If x = 0,
then
1
Name:
May 30th , 2013
Quiz #6 - Last Quiz! - Solutions
1. Let f (x) = 3x 1. Today we are computing
2
f (x) dx
0
using Darboux integration.
(a) First set up a regularly-spaced partition Pn = cfw_x0 , ., xn (i.e. as if
you were doing a Riemann integral).
S
Name:
May 23rd , 2013
Quiz #5: Solutions
1. The Taylor series for sin(x) is
(1)n x2n+1
(2n + 1)!
n=0
and it converges to sin(x) for all x R. Briey justify why the series
converges.
Solution: Let r R>0 be any positive real number. Then on the interval
(r,
Name:
May 16th , 2013
Quiz #4: Solutions
1. Suppose that f (x) and g (x) are two functions such that f (0) = g (0) and
for x [0, ), f (x) g (x). Prove that for all x [0, ), f (x) g (x).
Proof: Let h(x) = g (x) f (x). WTS h(x) 0 for all x [0, ). Observe
th
Name:
April 18th , 2013
Quiz #1
1. Let f (x) = sin
1
x
for x = 0. Prove that lim f (x) does not exist, where
x0S
S = R\cfw_0.
Recall that the denition says that the limit exists lim f (x) and equals
xaS
some value L if and only if for any sequence xn a, f
Name:
April 9th , 2013
Practice Quiz #1: Solutions
The solutions I write will generally be longer than what you might write.
This is because I want my solutions to be as complete as possible. If yours is
shorter, it may still be okay! Also, there may be m
SOLUTIONS FOR HOMEWORK 6
A. [This is a bonus problem very little partial credit is given] Compute
1
lim
n n n
n
k.
k=1
Hint. Does this sum remind you of a Riemann or Darboux sum?
1
The function f (t) = t is integrable on [0, 1], with 0 f = [2t3/2 /3]1 =
SOLUTIONS FOR HOMEWORK 3
20.6. We have f (x) = x3 /|x|. The natural domain of f is dom (f ) = cfw_x
R : x = 0 = (, 0) (0, ). We can write
f (x) =
x2
x>0
.
x2 x < 0
Therefore,
lim f (x) = lim x2 = +,
x+
x+
lim f (x) = lim x2 = +,
x
lim f (x) =
x+0
lim f (
SOLUTIONS FOR HOMEWORK 1
17.1. (a) dom (f + g) = dom (f g) = (, 4]. x dom (f g) i
x2 4 (f (g(x) = f (x2 ) must be dened), hence dom (f g) = [2, 2].
dom (gf ) = (, 4], since g(f (x) is dened whenever f (x) is dened.
(b) f g(0) = f (g(0) = f (0) = 2. g f (0
SOLUTIONS FOR HOMEWORK 4
29.3. (a) By Mean Value Theorem, there exists c (0, 2) s.t. f (c) = (f (1) f (0)/(2
0) = 1/2.
(b) By Mean Value Theorem, there exists d (1, 2) s.t. f (d) = (f (2) f (1)/(2 1) = 0.
1/7 (0, 1/2), hence, by Intermediate Value Theore
SOLUTIONS FOR HOMEWORK 2
19.1. (a) f is continuous on [0, ], hence uniformly continuous (Theorem
19.2).
(b) f is continuous on [0, 1], hence uniformly continuous (Theorem 19.2).
(c) f is uniformly continuous on (0, 1). Indeed, consider the function f ,
de
SOLUTIONS FOR HOMEWORK 7
Before proceeding, we need to recall a few results from Chapter II:
lim n1/n = 1, lim(n!)1/n = +.
1/n
If (tn ) is a sequence of positive numbers, and lim tn+1 /tn = c, then lim tn
Theorem 12.2, Corollary 12.3).
= c (see
If
ak conv
SOLUTIONS FOR HOMEWORK 5
A. [This is a bonus problem] If x is a non-zero rational number, write x = p/q, where p Z
and q N have no common factors. For such x, let f (x) = 1/(q + 1)2q ). For x irrational,
1
let f (x) = 0. Also, let f (0) = 1. Prove that f
PRACTICE PROBLEMS FOR FINAL SOLUTIONS
The comprehensive nal will be given on Wednesday, June 9, 8:00
10:00. In preparing for the test, you can practice solving the problems from
the list below. In addition, take a look at the homework assignments (at
leas
PRACTICE PROBLEMS FOR FINAL
The comprehensive nal will be given on Wednesday, June 9, 8:00
10:00. In preparing for the test, you can practice solving the problems from
the list below. In addition, take a look at the homework assignments (at
least one prob
MATH 140B: SOLUTIONS FOR THE FINAL
1 (10 points): Prove that x = cos x for some x (0, /2).
The function f (x) = x cos x is continuous on [0, /2]. Furthermore, f (0) =
1 < 0, and f (/2) = /2 1. By Intermediate Value Theorem, there exists
c (0, /2) s.t. f (
PRACTICE PROBLEMS FOR MIDTERM SOLUTIONS
The test will be given on Wednesday, April 28. It will cover Sections
17-20 and 28-29 (everything covered in this quarter, up to and including
Mean Value Theorem).
In preparing for the test, you can practice solving
MATH 140B: PRACTICE PROBLEMS FOR MIDTERM
The test will be given on Wednesday, April 28. It will cover Sections
17-20 and 28-29 (everything covered in this quarter, up to and including
Mean Value Theorem).
In preparing for the test, you can practice solvin
Name:
May 30th , 2013
Quiz #6 - Last Quiz!
1. Let f (x) = 3x 1. Today we are computing
2
f (x) dx
0
using Darboux integration.
(a) First set up a regularly-spaced partition Pn = cfw_x0 , ., xn (i.e. as if
you were doing a Riemann integral).
(b) Compute U
Name:
April 18th , 2013
Quiz #1
1. Let f (x) = sin
1
x
for x = 0. Prove that lim f (x) does not exist, where
x0S
S = R\cfw_0.
2. Let g (x) = x sin
1
x
, for x = 0. Prove that lim f (x) exists (for S =
x0S
R\cfw_0), and show what it equals.
1
Name:
April 25th , 2013
Quiz #2
1. Find the radius of convergence and the exact interval of convergence for
the power series.
xn
3n2
n=1
2. Dene a sequence of functions on (0, ) by fn (x) =
1
xn .
(a) Find f (x) = lim fn (x) for x (1, ). (Dont be confused
Elementary Analysis Math 140BWinter 2007
Homework answersAssignment 22; March 18, 2007
Exercise 32.2
Let f (x) = x for rational x and f (x) = 0 for irrational x.
(a) Calculate the upper and lower Darboux integrals for f on the interval [0, b].
Solution: F
Elementary Analysis Math 140BWinter 2007
Solutions to First Midterm; February 7, 2007
Problem 1 (25 points) Let fn (x) = x2 /(nx + 1) if 0 < x < , n = 1, 2, . . .
(a) Does fn converge uniformly on (0, 1)? Justify your answer.
Solution: YES;
fn (x) =
x2
x2
Elementary Analysis Math 140BWinter 2007
Homework answersAssignment 8; February 7, 2007
Exercise 25.12, page 191
Suppose that gk is a series of continuous functions gk on [a, b] that converges
k=1
uniformly to g on [a, b]. Prove that
b
b
g (x) dx =
a
k=1
Elementary Analysis Math 140BWinter 2007
Homework answersAssignment 7; January 29, 2007
Exercise 26.2, page 199
(a) Observe that
n=1
nxn =
x
(1x)2
for |x| < 1.
Solution: By Example 1 on page 195, for |x| < 1,
nxn1 =
n=1
1
.
(1 x)2
Therefore, for |x| < 1,
Elementary Analysis Math 140BWinter 2007
Homework answersAssignment 4; January 22, 2007
Exercise 24.14, page 183
Let fn (x) =
nx
1+n2 x2
for x R.
(a) Show that fn 0 pointwise on R.
Solution: For any n, fn (0) = 0 so that if f denotes the pointwise limit f
Elementary Analysis Math 140BWinter 2007
Homework answersAssignment 5; January 29, 2007
Exercise 25.4, page 190
Let (fn ) be a sequence of functions on a set S R, and supporst that fn f uniformly on S .
Prove that (fn ) is uniformly Cauchy on S .
Solution
Elementary Analysis Math 140BWinter 2007
Homework answersAssignment 1; January 17, 2007
Exercise 23.4, page 176
n
For n = 0, 1, . . ., let an = 4+2(5 1)
1/n
n
1/n
(a) Find lim sup an , lim inf an , lim sup
an+1
an
and lim inf
an+1
an
.
Solution:
1/n
1/n
6
Elementary Analysis Math 140BWinter 2007
Homework answersAssignment 3; January 22, 2007
Exercise 24.13, page 183
Prove that if (fn ) is a sequence of uniformly continuous functions on an interval (a, b),
and if fn f uniformly on (a, b), then f is also uni
Elementary Analysis Math 140BWinter 2007
Homework answersAssignment 9; February 7, 2007
Exercise 27.2, page 204
Show that if f is continuous on R, then there exists a sequence (pn ) of polynomials
such that pn f uniformly on each bounded subset of R.
Solu