Math 117 Fall 2014 Lecture 8
(Sept. 15, 2014)
Reading: 314 Notes: Sets and Functions 1; Bowman 1.A.
Sets.
Definition 1. A set is a collection of objects. Each object is called a member (or element)
of
Math 117 Fall 2014 Homework 4 Solutions
Due Thursday Oct. 9 3pm in Assignment Box
Question 1. (10 pts) Prove the following statements by denition.
n!
a) (2 pts) limn!1 nn = 0.
h
1
b) (2 pts) limn!1 p
Math 117 Fall 2014 Lecture 23
(Oct. 15, 2014)
Reading:
Let fan g be a sequence. Recall:
If fan g converges, then fan g is bounded. Note that limn!1 an = 1 are called fan g
diverges to +1/1.
If fan g i
Math 117 Fall 2014 Lecture 19
(Oct. 6, 2014)
Reading:
In the following a; b; L; M 2 R. The cases of one or more of them are +1 or 1 are left
as exercises. Please make sure you work on these cases some
Math 117 Fall 2014 Lecture 21
(Oct. 9, 2014)
Reading: Bowman 2.E.
Note that convergence means converging to a number. It does not include the cases an !
+1 and an ! 1. Please make sure you think about
Math 117 Fall 2014 Midterm 2 Review Problems
Midterm 2 coverage:
Lectures 12 - 25 and the exercises therein.
Required sections in Dr. Bowman's book and my 314 notes.
Homeworks 3 - 5.
The exercises bel
Math 117 Fall 2014 Lecture 18
(Oct. 3, 2014)
Reading:
Some leftovers.
a; L 2 R. Denition for limx!af (x) = L is not true.
9" > 0 8 > 0 9x0 < jx aj < ;
jf (x) Lj > ":
(1)
Remark 1. Note the dierence be
Math 117 Fall 2014 Lecture 24
(Oct. 16, 2014)
Reading:
Let fan g be a bounded sequence. Recall:
Can dene the set of accumulation points:
A(fan g) := fa 2 Rj 9ank ! ag:
Have seen:
p
A
n 2
= [0; 1]:
Math 117 Fall 2014 Lecture 20
(Oct. 8, 2014)
Reading:
In the following a; b; L; M 2 R. The cases of one or more of them are +1 or 1 are left as
exercises. Please make sure you work on these cases some
Math 117 Fall 2014 Lecture 25
(Oct. 17, 2014)
Reading:
Recall: For a bounded sequence fan g,
limsup an := lim sup ak ;
n!1 k>n
n!1
liminf an := lim
n!1
h
i
inf ak :
n!1 k>n
(1)
Theorem 1. Let fan g; f
Math 117 Fall 2014 Homework 2
Due Thursday Sept. 18 3pm in Assignment Box
Question 1. (5 pts)
a) (2 pts) Find two irrational numbers a; b such that both a + b and a b are rational.
b) (3 pts) Can you
Math 117 Fall 2014 Homework 1 Solutions
Due Thursday Sept. 11 3pm in Assignment Box
Question 1. (5 pts) Prove that 11 is prime but 57 is not.
Proof.
11 is prime.
First for any number n > 11, nj 11. No
Math 117 Fall 2014 Lecture 6
(Sept. 11, 2014)
What is ?
The ratio between the circumference and diameter of a circle; or the ratio between
the area and the square of the radius of a circle. But why ar
Math 117 Fall 2014 Lecture 9
(Sept. 17, 2014)
Reading: 314 Notes: Sets and Functions 2.1 (Open and Closed Sets: Optional); Bowman 1.G.
Operations on two sets (cont.)
Example 1. Prove that (A B) \ (B A
Math 117 Fall 2014 Lecture 4
Reading:
(Sept. 8, 2014)
p
Required reading: Dr. Bowman's book 1.B.
Optional reading: 1.C.
p
2 is irrational, that is 2 is not rational.
p
Notation. The symbolic way of wr
Math 117 Fall 2014 Lecture 5
(Sept. 10, 2014)
Prehistory.
Before the invention of logarithm, people calculated multiplications through the following
trigonometric identities:
sin(A B) = sin A cos B co
Math 117 Fall 2014 Lecture 7
(Sept. 12, 2014)
Reading: Bowman 1.H, 1.I, 1.J.
A bit more about numbers.
N: natural numbers; Z: integers; Q: rational numbers; R: real numbers.
Q is dense in R; Qc is den
Math 117 Fall 2014 Midterm Exam 1 Solutions
Sept. 26, 2014 10am - 10:50am. Total 20+2 Pts
NAME:
ID#:
There are ve questions.
Please write clearly and show enough work.
Question 1. (5 pts) Prove that
Math 117 Fall 2014 Lecture 2
Number systems:
(Sept. 4, 2014)
N = f1; 2; 3; :g: Natural numbers;
Z = f: 2; 1; 0; 1; 2; :g: Integers;
Q: Rational numbers;
R: Real numbers;
C: Complex numbers;
And many m
Math 117 Fall 2014 Midterm 1 Review
Midterm 1 coverage:
Lectures 1 - 11 and the exercises therein.
Homeworks 1 & 2.
Required sections in Dr. Bowman's book and my 314 notes.
The exercises below are to
Math 117 Fall 2014 Lecture 3
(Sept. 5, 2014)
How detailed should your answers to Homework 1 be:
For example, to prove that the product of two odd numbers is still odd, you should write
something like:
We see that the old denition still applies.
A (c, d) where a (c, d).
1
Example 3. Study limx1 x3 .
1
Solution. Clearly the limit should be 1. The problem now is that f (x) = x3 is not
dened at x = 0.