Math H104
Homework 1 Solutions
9/1/2016
Exercises:
1. Read Chapter I Notions from Set Theory in Rosenlicht.
2. Let f : X Y be a function.
(a) For a subset A X, show f 1 (f (A) A.
(b) Show that f is on
Identifying adjectives
Grade 1 Grammar Worksheet
Circle the adjective and underline the nouns it describes:
1. The bright sun warms our house.
2. Stinky cheese is my dads favorite.
Adjectives are
word
Using adverbs
Grade 3 Grammar Worksheet
Circle the adverbs.
urgently
today
cellphone
after
wrap
sparingly
softly
busily
truly
loyally
densely
green
often
bulldozer
turkey
equally
salty
carefully
prick
MATH H104 LECTURE XX, NOVEMBER 29, 2005
LECTURER: YUVAL PERES.
SCRIBE: KE LU
Read compelet proof of taylor approximation in text. Recap, points reached:
Rn (h) = f (x + h) Pn (h)
n
f (k) (x)
Pn (h) =
MATH H104 LECTURE 25, NOVEMBER 22, 2005
LECTURER: YUVAL PERES.
SCRIBE: JONATHAN GOLDMAN
Convergence of Series. Suppose > 0. Then
(1)
n=1
1 < for > 1
n = for 1
Blocking Test. Suppose cfw_an n1 is a seq
Math H104: Honors Introduction to Analysis
Fall 2005
Lecture 1: August 30
Lecturer: Yuval Peres
Scribe: Thomson Nguyen
Let Z be the set of all integers cfw_0, 1, 1, 2, 2, . . . and let N be the positi
MATH H104 LECTURE 1, AUGUST 30, 2005
LECTURER: YUVAL PERES.
SCRIBE: THOMSON NGUYEN
Let Z be the set of all integers cfw_0, 1, 1, 2, 2, . . . and let N be the positive integers
cfw_1, 2, 3, . . .. Deno
MATH H104 LECTURE 24, NOVEMBER 17, 2005
LECTURER: YUVAL PERES.
SCRIBE: DAVID WONG
Theorem 0.1 (Darbouxs Theorem). Given f dierentiable on [a, b], dened in a neighborhood of [a, b], f attains any value
MATH H104 LECTURE 20, NOVEMBER 10, 2005
LECTURER: YUVAL PERES. SCRIBE: BERNARD LIANG
Denition 20.1: A metric space X is separable if there is a countable set D X which is dense in X. (Examples: R is s
MATH H104 LECTURE, NOVEMBER 2, 2005
LECTURER: YUVAL PERES.
SCRIBE: ORIE SHELEF
Theorem 0.1. Let X,Y be metric spaces. If fn : X Y are continuous and fn converges
to f : X Y uniformly then f is continu
MATH H104 LECTURE 23, NOVEMBER 15, 2005
LECTURER: YUVAL PERES.
SCRIBE: BRIAN SHOTWELL
We move on to your 2nd (or 3rd) exposure to Calculus.
Denition 0.1. The derivative of a function f (x) is denoted
Math H104
Homework 2 Solutions
9/8/2016
Exercises:
1. Prove that if S R is non-empty and bounded below, then it has an infimum.
2. For S R a non-empty subset that is bounded above and x R, let xS be t
Homework 4 Solutions
Math H104
9/22/2016
Exercises:
1. Let A, B be subsets in a metric space (E, d).
(a) If A B, show A B and A B .
(b) Show A B = A B and (A B) = A B .
(c) In E = R with the usual met
Adjectives are
words that
describe nouns
Using adjectives
Grade 1 Grammar Worksheet
Circle the adjectives.
curly
photo
hollow
reading
jolly
bench
brown
walk
fast
angry
cross
card
fancy
flat
hose
Compl
Identifying adverbs
Grade 3 Grammar Worksheet
Adverbs usually
describe or
modify verbs.
Circle the adverbs and underline the verbs
that they describe:
1. We ate the popcorn quietly.
2. Bill handled th
Math H104
Midterm 1
Name:
Student ID Number:
Instructions:
1. You will have 80 minutes to complete the exam.
2. The exam is a total of 5 questions and each question is worth 10 points.
3. Unless state
Solutions
Math H104
Midterm 1
1. (a) (2 pts) The sequence (xn )nN converges to x in (E, d) if > 0, N N so that n N , d(xn , x) < .
p
(b) (8 pts) We claim the limit is 2. Let > 0. Choose N > 4/. Then f
The Completion of a Metric Space
Brent Nelson
Let (E, d) be a metric space, which we will reference throughout. The purpose of these notes is to guide
you through the construction of the completion of
HOMEWORK 10 SOLUTIONS
Due Wednesday, November 30th
The exercises are from Elementary Analysis, 2nd edition, by Kenneth Ross.
q
n+1
= 1, which implies
1. Exercise 23.2 (a) Solution: Observe that limn n
HOMEWORK 9 SOLUTIONS
Due Wednesday, November 16th
The exercises are from Elementary Analysis, 2nd edition, by Kenneth Ross.
Rx 2
1. Exercise 34.2 (a) Solution: Let F (x) = 0 et dt. Observe that as x 0
HOMEWORK 8 SOLUTIONS
Due Wednesday, November 2nd
The exercises are from Elementary Analysis, 2nd edition, by Kenneth Ross.
1. Exercise 32.2 (a) Solution: For an partition P of [0, b] we see that L(f,
HOMEWORK 6 SOLUTIONS
Due Wednesday, October 19th
The exercises are from Elementary Analysis, 2nd edition, by Kenneth Ross.
1. Exercise 20.4 Solution: Note that a part of the graph of f (x) = x sin x1
HOMEWORK 7 SOLUTIONS
Due Wednesday, October 26th
The exercises are from Elementary Analysis, 2nd edition, by Kenneth Ross.
1. Exercise 29.2 Solution: Let f (x) = cos x a continuous and differentiable
Math 104, Section 3
Fall 2015 Final
12/18/2015
Time Limit: 180 Minutes
Name (Print):
This exam contains ? pages (including this cover page and the grading chart page) and ? problems. Check to see if a
Using adjectives
Grade 3 Grammar Worksheet
Circle the adjectives.
eager
low
play
ocean
quiet
scrawny
horse
castle
handsome
odd
rich
long
phone
famous
driver
gentle
jump
clever
shallow
thankful
Complet
Identifying adverbs and adjectives
Grade 2 Grammar Worksheet
Circle the adverbs and underline the adjectives.
Skip the verbs and nouns!
Adverbs modify verbs
(doing words).
Adjectives describe
nouns (p
Using adverbs
Grade 3 Grammar Worksheet
Circle the adverbs.
weakly
shiny
excitedly
never
greatly
poorly
instantly
strictly
building
daily
ladder
suddenly clumsily
bashfully
dragon
later
outdoors
chopp
Using adjectives
Grade 3 Grammar Worksheet
Circle the adjectives.
small
boiling
mother
worried
witty
swift
calm
brother
pool
blow
narrow
high
funny
flaky
beautiful
cloud
dinner
proud
pilot
full
Comple
Parts of Speech: Verbs
Second Grade Vocabulary Worksheet
Verbs are words that tell you what something or somebody is doing or
being.
Circle the correct verb or verbs in each sentence.
Amy threw the ba