Anaximenes
1. Anaximenes postulates air as the arch. This may seem to be a throwback to Thales, a
step backward after Anaximander; but I will argue that is is not.
2. Anaximenes is offering a new world view: a. Anaximander thought of the basic stuffs and
CM221A
ANALYSIS I
Solutions to Sheet 3
1. Since the sums and products of continuous functions are continuous functions, it is sucient to prove the statement for xn , where n is a non-negative integer. Let us x c R and show that xn is continuous at c. We h
CM221A
ANALYSIS I
Solutions to Sheet 4
1. We have
sin x tan x = . If x < /2 and x /2 then sin x 1 1 + tan x sin x + cos x
and cos x 0. Applying the theorem about the limits of sums and quotients, we obtain lim tan x 1 = = 1. 1 + tan x 1+0
x/20
2. The comp
CM221A
ANALYSIS I
Solutions to Exercise Sheet 5
Calculate the derivatives of each of the following two functions. Find all of the local and global maximum and minimum values, if there are any, of the functions on the intervals given. Sketch the graphs of
CM221A
ANALYSIS I
Solutions to Exercise Sheet 7
1. Find the maximum and minimum values of the function f (x) = 2x5 x3 on the interval [0, 1], and sketch the graph of the function on this interval. Solving f (x) = 0, we nd the only solution in the interval
DEPARTMENT OF MATHEMATICS SYLLABUS Course # & Name:
MAT 146: Algebraic Combinatorics Herbert Wilf, "Generatingfunctionology", ISBN: 1-56881-279-5, 2005, $39 or available for free at http:/www.math.upenn.edu/%7Ewilf/Down ldGF.html UPC Approval Date: 2/28/0
Math 61 Answers
March 7, 2013
1 2011 Midterm
1. a) Let Rn stand for the number of messages you can send with $n. We split it
into cases depending on the rst letter, getting:
Rn = Rn1 + Rn2 + Rn3 .
b)You dont need to nd an explicit formula. You just need t
Math 61
Final
March 24, 2006
Name:
,
Please put your last name rst and print clearly
Signature:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
You must put all your answers in the spaces provided on the page of the
problem . Please do not use the spaces o
Selected Solutions
March 17, 2013
1 2012 Final
1. One way to do this is to rephrase it as: mRn if m2 = n2 mod 5. We can know
make a table of squares mod 5.
x x2
2
1
0
1
2
3
4
mod 5
4
1
0
1
4
4
1
So there are 3 equivalence classes, and they are cfw_2, 2, 3
Math 61
Midterm I
May 2, 2012
Name:
,
Please put your last name rst and print clearly
Signature:
TA section you are attending:
(Tues or Thurs, name of TA, section letter)
1.
2.
3.
4.
5.
Total
You must put all your answers in the spaces provided on the pag
Math 61
Midterm II
March 4, 2011
Name:
,
Please put your last name rst and print clearly
Signature:
TA section you are attending:
(Tues or Thurs, name of TA, section letter)
1.
2.
3.
4.
5.
Total
You must put all your answers in the spaces provided on the
Math 61
Final
March 18, 2011
Name:
,
Please put your last name rst and print clearly
Signature:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
You must put all your answers in the spaces provided on the page of the
problem . Please do not use the spaces o
Math 61
Midterm II
June 1, 2012
Name:
,
Please put your last name rst and print clearly
Signature:
TA section you are attending:
(Tues or Thurs, name of TA, section letter)
1.
2.
3.
4.
5.
Total
You must put all your answers in the spaces provided on the p
Review sheet for Final Math 61( Feb 8)
You may bring one 4 by 6 card with notes, both sides.
The exam will be in two rooms: Rolfe 1200 and Haines 118. Students with
last names beginning with A-P should go to Rolfe 1200 and students with
last names beginni
Review sheet for Midterm II Math 61 (March 8)
You may bring one 4 by 6 card with notes, both sides .
Topics to be covered:
I. Permutations and combinations, 6.1, 6.2, 6.3.
II. Probability 6.5, 6.6
III. Recurrence relations: How to set up and how to solve
Math 61
Final
June 12, 2012
Name:
,
Please put your last name rst and print clearly
Signature:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
You must put all your answers in the spaces provided on the page of the
problem . Please do not use the spaces on thi
CM221A
ANALYSIS I
Solutions for Sheet 2
1. No. The formula lim (n an ) = n does not make sense.
n
2. We have 2n (sin n) 2n . Since 2n 0, the Sandwich Theorem implies that the sequence converges to 0. 3. lim n because 4. lim n + 1 n = nlim n+1 c2 n 1 c2 n
CM221A
ANALYSIS I
Solutions to Sheet 1
1. By the Archimedean Property (see course notes), we have 1 < n for some n N. Multiplying both parts of this inequality by n1 , we obtain n1 < . 2. Obviously, each an is positive and, given this, 0 < an+1 < an . The
CM221A
ANALYSIS I
Exercise Sheet 6
1. Find the maximum and minimum values of the function f (x) = 2x5 x3 on the interval [0, 1], and sketch the graph of the function on this interval. 2. Prove that the derivative of the function f (x) = (sin x)2 + x2 only
INTEGRATION LECTURE NOTES by E B Davies 1. Introduction
b
If f is a real-valued function on a bounded interval [a, b] the integral a f (x) dx is intended to be a measure of the area under the graph of the function. If the function takes both positive and
CM221A 2010
ANALYSIS I
NOTES ON WEEK 1
NOTATION (not examinable) We shall use the following standard notation N is the set of positive integer numbers, N = cfw_1, 2, . . .. Z is the set of integer numbers, Z = cfw_. . . , 2, 1, 0, 1, 2, . . .. Q is the se
CM221A
ANALYSIS I
NOTES ON WEEK 2
You must remember and be able to use all the denitions and theorems stated in this weeks notes. Their proofs can be found in the CM115 lecture notes. The proofs are not examinable.
CONVERGENT SEQUENCES Let cfw_an be a se
CM221A
ANALYSIS I
NOTES ON WEEK 3
UPPER AND LOWER LIMITS One says that + is an accumulation point of a sequence cfw_an if there is a subsequence of cfw_an which converges to +. Similarly, is an accumulation point of cfw_an if there is a subsequence whi
CM221A
ANALYSIS I
NOTES ON WEEK 4
TWO CONVERGENCE TESTS Ratio Test Theorem. Assume that an = 0 for all n, and that limn |an+1 /an | exists and is equal to c. If c < 1 then the series an is absolutely convergent. n=1 If c > 1 then the series diverges. (If
CM221A
ANALYSIS I
NOTES ON WEEK 5
FUNCTIONS OF A REAL VARIABLE Let R be a nonempty subset of the real line. A (real-valued) function f on is a mapping R, that is, an association x f (x) of each element x of to some real number f (x) which is called the va
CM221A
ANALYSIS I
NOTES ON WEEK 6
There will be no lectures on the week starting 8 November. Use the opportunity to revise the material covered in lectures. Many of you need to go through solutions to the class test and exercise sheets to see what you got
CM221A DIFFERENTIATION
ANALYSIS I
NOTES ON WEEK 8
Let f be a function dened on an interval I . We say that f (x) = k if > 0 x, > 0 : 0 < |x y | < x, Equivalently > 0 x, > 0 : 0 < | | < x, and f (x + ) f (x) k < . f (x) f (y ) k < . xy
f (x) f (y ) = k. y
CM221A
ANALYSIS I
NOTES ON WEEK 9
FINDING MAXIMAL AND MINIMAL VALUES Denition. Let f be a function dened on an interval (a, b). We say that f has a local maximum at a point c (a, b) if there exists > 0 such that f (c) f (x) for all x (c , c + ). Similarly
CM221A
ANALYSIS I
NOTES ON WEEK 10
n times dierentiable functions. We say that f is n times dierentiable on (a, b) if each derivative of order up to n exists at every point of the interval (the derivative of order two is the derivative of the rst derivati
CM221A
ANALYSIS I
Exercise Sheet 1
1. Prove that, for every > 0, there exists n N such that n1 < . 2. Let a1 = 1 and an+1 = an for all n 2. Prove that the sequence is an + 2 decreasing and nd its limit.
3. Making references to any theorems about convergen