Douglas Murray McGregor (1906 1 October 1964) was a management professor at the MIT Sloan
School of Management and president of Antioch College from 1948 to 1954.[1] He also taught at the
Indian Insti
1.
2.
3.
4.
5.
6.
See the text. The time complexity for a bubble sort is O(n^2).
Omitted.
n 1 times
See the text. The time complexity for a merge sort is O(nlogn).
Omitted.
Consider a list with two el
1.
The constant factor is ignored in big O notation, because it has no impact on the
growth rate of the time complexity function. A nondominating term is ignored in Big O notation,
because as the inpu
Intro to Calculus Name:
Date:
Test Review: Optimization, Related Rates
How can you tell the difference between an optimization problem and a related rates problem? What
are you asked to nd in each?
Li
Volumes - Shell Method
1. Use the shell method to nd the volume of the solid generated by revolving the planar
region about the yaxis:
2
(a)y=x ,y=0,x=2
<b)y=x2,y=4x~x2
1 _ 2
(CURft! x /2 ,y=0,x=0,x=
Section 1.3: Exponential Function
Do you like story problems?
Graph the function y =2' and y =T'. What do you get?
-11-1-1-1
:/=1-1+1
:1=1-1=,1_t1-. .-,
._ [
LI-1= f
~!-1-i -!~-t-t-+-
Graph the funct
Quiz through 4.3 Key
Directions: Please answer the questions below (without calculators) very
carefully.
Name:
1) What are the
global minimum
and maximum (if they exist) of the function
2
f (x) = e(
APPENDIX
I
Common Errors In Proofs
The following list describes some common errors made by students while creating
proofs. The list is intended to help you avoid making the same errors.
1. An example
CS 1151 INTRODUCTION) TO DISCRETE STBUQTUBES 1 Spring M
(E) Q E] R6911cfw_51579727fQReZEE$7MyaZ>7ZayHa
Answelj
(1') REFLEXIVE: m (111') ANTISYMMETRICi m
Sincd(y,y)Re. (y,z)eRaJAHz,g/QSRA butyz.
Q3 an INTRODUCTION TO DISCRETE STRUCTURES 1 Spring M
Solutions to HQIIIBMKLEk 5
[bid Datej Wednesdayl 04i27[0q
You are supposed to solve within gour stud group 1ng rst thinking about the problems fo
Math 150s
Proof and Mathematical Reasoning
Jenny Wilson
A Primer on Mathematical Proof
A proof is an argument to convince your audience that a mathematical statement is true. It can be a calculation,
Q3 1):] INTRODUCTION TO DISCRETE STRUCTURES 1 Spring M
Q] (g) EQrtthnethaI1311011aneguivalencgrelaliengivdanexamplefQR 2111de
sethaeiiheRUQrRAiemtransim
Answeii
Claim 3. i = R LJ R2 215i not an eq
11/13/2017
Sets
CSCE 235
Handout 11: Common Mistakes in Proof by Induction
February 28, 2002
There are many types of problems that we have talked about in this class: the summation of a series of numb
03%
INTRODUCTION To DISCRETE mm 1
(g) 9%) sd:cfw_(a,bj | aeiLobeiiLoazzby
L
O
Answen
Let us illustrate thd set Ed ill the (Zartesian planel
000000000000
000000000000
000000000000
000000000000
0000
QSm
INTRODUCTION TO DISCRETE STRUCTURES 1 Spring M
(g) Show1 every la, b) 6 SJ alsd belongs td Rd. Since Si is not dened recursively1 we
cannot use stnuetural induction) however we can rephrase the
Q3 me INTRODUCTION TO DISCRETE SanieTunEd 1 Spring M
0 Every postitive integer n can be written as a sum Qf distinetl powers Q 2. Jig 13 =
3 d 01
8 4 1 2 2 2 2
Hinti W hen considering k + M in the i
game
INTRODUCTION TO DISCRETE STRUCTURES 1 Spring M
Answeri
Eirst meneed tdconsider the twd elements ofS denedjn the basic part eithemlenitien
of Si
BASIQ STEP: Both 0 and t hale length one and are
Common Mistakes in Mathematical Induction
Zhang Yichi
October 4, 2012
1
No Basis Step
2
Wrong Inductive Step
Examples 1
Prove that for all integers n 1, 22n 1 is divisible by 3.
Proof. Basis Step: Wan
CS 1):] INTRODUCTION TO DISCRETE STRUCTURES 1 Spring M
Q] 9%) 512mm] | qubCMa humandbisidmsibiebw
Answeii
Let us illustrate thd set ill the (Zartesian plane.
A A
16 00000000 13
11/13/2017
Some Induction Examples : nrich.maths.org
Copyright University of Cambridge. All rights reserved.
Some Induction Examples
Stage: 5
Article by Alison Kiddle
Published December 2013.
You are
Math 150s
Proof and Mathematical Reasoning
Jenny Wilson
Converse
The conditional statement
A = B
is not equivalent to the statement
B = A,
which is called its converse.
Example of a True Implication w
Math 150s
Proof and Mathematical Reasoning
Jenny Wilson
Other ways to write this:
If A is true, then B is true.
A is true only if B is true.
B is true whenever A is true.
B is true if A is true.
Math 150s
Proof and Mathematical Reasoning
Jenny Wilson
A Primer on Mathematical Proof
A proof is an argument to convince your audience that a mathematical statement is true. It can be a calculation,
ERRORS IN MATHEMATICAL WRITING
5
Good: We proved that if a2 is even, then a is even. Suppose a8 is
even. Then, by successively applying the result to a4 , a2 , and a, we
see that a is even.
Watch you
Math 347
Worksheet: Some Common Errors in Proof-Writing
A.J. Hildebrand
Worksheet: Some Common Errors in Proof-Writing
Each of the following statements has a logical, or notational, or language, error
ERRORS IN MATHEMATICAL WRITING
3
Good: Choose z C.
3. Equations and expressions
If an equation or expression is important, either for later reference
or for emphasis, display it on its own line. If y