Math 131A - Section 2 Spring 2010 Homework 4
All problems graded (out of 3 points for this short assignment). 10.1(b). bounded; (d) bounded; (f) bounded, non-increasing 10.2. Let (sn ) be a bounded no
Math 131A - Section 2 Spring 2010 Homework 6
The following problems were graded for this assignment: 14.7 (2 points), 14.12 (2 points), 15.6 (2 points), 17.9(b,c) (2 points), 17.10 (b,c) (2 points), 1
Math 131A Lecture 4: Homework 4, Due 10/24 in TA session
1-2. 10.6, 10.8
3. Let us consider the sequence x1 = 1 and xn+1 = 1 +
1
.
xn
(a) Show that xn [ 3 , 2] for n 2.
2
(b) Using (a), show that |xn+
Math 131A - Section 2 Spring 2010 Homework 5
For this assignment, the following problems were graded: 11.5 (3 points), 11.7 (3 points), 11.9(b) (2 points), 12.1 (2 points), 12.4 (2 points), 12.12 (3 p
Math 131A: Homework 2
(due Friday, January 15th)
1. Read sections 5.3, 6.1, 6.2, 6.3 up to theorem 6.3, of the Oxford Prelimary Notes ( this is a
link). 7.1 is a useful section too, though if you find
Math 131A: Homework 4
(due Friday, February 5th)
1. Let cfw_En nN be a collection of subsets of N. Consider the set
E = cfw_n N : n
/ En .
Show that En 6= E for any n N. Deduce that the set cfw_A : A
MATH 131A Analysis
Theorem Sheet
If you use a numbered result on a homework or exam question, you should cite it. You can
use the other results without citing them. There will be a copy of the numbere
Math 131A: Homework 6
(due Friday, February 19th)
1. Suppose (xn )
n=1 is a sequence of real numbers. Let f be the corresponding function, i.e. let
f : N R, n 7 xn .
Let g(x) =
1
x
on (0, ).
(a) What
Math 131A: Homework 5
(due Friday, February 12th)
1. Suppose a, b, L1 , L2 R, a < b, that f1 , f2 are real-valued functions who domains are subsets
of R, that (a, b) dom(f1 ) dom(f2 ), that limxa+ f1
Math 131A: Midterm 1 Preparation
(Midterm is on January 22nd)
1. Usually I try and make at least 50% of my midterms bookwork. That means if you know all
the definitions, all the theorem statements, an
Math 131A: Homework 8
(due Friday, March 11th)
Standard integration questions (only these need to be turned in)
1. Let f be integrable on [a, b], and suppose g is a function on [a, b] such that g(x) =
Math 131A: Homework 1
(due Friday, January 8th)
1. Read sections 1.1, 1.3, 2.2, 3.1, 5.1 up to but not including not, and, or, and 5.4 of the Oxford
Prelimary Notes ( this is a link).
2. (a) Guess a f
Math 131A: Homework 3
(due Friday, January 29th)
1. Use the algebra of limits and useful limits given in lecture notes (theorems 9.2-9.7 in the
textbook) to prove the following. Justify all steps.
2
+
Math 131A: Homework 7
(due Friday, March 4th)
1. Use the definition of the derivative to calculate the derivatives of the following functions at
the indicated points.
(a) f (x) = x3 at x = 2;
(b) g(x)
FINAL PRACTICE
KEVIN CARLSON
(1) Give examples of the following phenomena, or argue that they cannot occur.
(a) A continuous function f on a closed interval and a Cauchy sequence (xn )
such that (f (x
Math 131A: Midterm 2 Topics
(Midterm is on February 26th)
Things you should focus on
1. 8.1: Cauchy sequences.
2. 11: series. You do not need to be able to prove the tests, but should be able to state
1. (3.) Dene: rf(a:) is uniformly continuous on S Q R. (b) Suppose
f (cc) is uniformly continuous on S Q R and f(:z:) Z c > 0 for all
a: E S. Prove that the function 1/) dened by (1/f)(ac) : 1/f(:1:)
M 614%ij O?
1. (a) State the Intermediate Value Theorem. (0) Suppose that x)
is continuous on [09,0], that b 6 (me) and f (a) < f (c) < b). Prove
that there exist $1,332 6 [(1, c] with :31 7 932 such
1. Let (Sn) be a. sequence. (2) Dene: limTHw = s 6 R. (10)
Use the denition in part (a) to prove that if 3 Z 0 for all n and
limnoo = 5, then s 2 0.
(rm/cfw_#3623 gig/W49 A<o m4 M 6242
47% W &/J':NAWQ
1. (a) Dene: limw_,1,_ f($) = L E R. (b) Prove that if an) 2 0 for
all [B 6 (1,1) and limHL an) = L then L 2 0. 2. (a) Prove that if a, b 6 R, then
Hal - lbll S |ab|.
(b) Prove that is f(:c) is contin
1. State and prove the Archimedean Property. 2.
1i
minf 31,. = +00 then lime.r1 = +00.
(0) Dene: liminf 3 of a sequence (3). (b) Prove that if 3. (a) Dene: (Sn) converges to s 6 R. (b) Suppose that 3
HOMEWORK 3
MATH 131A
Chapter 2, Section 9: 9.4 9.15.
The following problems are due on Friday, October 28:
Chapter 2, Section 9: 9.6, 9.9, 9.10, 9.11, 9.12.
1