Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902
|
|
|
- Sheryl Joseph
- 8 years ago
- Views:
Transcription
1 Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902
2 Different Graphs, Similar Properties E F J K A B I C D N O G H V R W L M U S P Q X T Y What do the five graphs above have in common? For each graph, write its degree list and add the numbers together. What do you notice? If we add any other edge to each of these graphs, what would happen to the numbers we came up with earlier? Why?
3 Sum of Degrees of Vertices Theorem Theorem (Sum of Degrees of Vertices) Suppose a graph has v vertices with degrees d 1, d 2, d 3,..., d v. Add together all degrees to get a new number d 1 + d 2 + d d v = D V. Then D V = 2e. In words, for any graph the sum of the degrees of the vertices equals twice the number of edges. Stated in a slightly different way, D V = 2e says that D V is ALWAYS an even number. (Using the Sum of Degrees of Vertices Theorem) It is impossible to make a graph with v = 6 where the vertices have degrees 1, 2, 2, 3, 3, 4. This is because the sum of the degrees D V is D V = = 15 D V is always an even number but 15 is odd!
4 Using the Sum of the Degrees of Vertices Formula 1 Consider the following scenarios: A graph has 4 vertices with degrees 0, 0, 0, and 0. What does this graph look like? A graph has 1 vertex with degree 2. What does this graph look like? A graph has 4 vertices with degrees 2, 3, 3, and 4. How many edges are there? A graph has 4 vertices with degrees 2, 2, 2, and 4. Can you say what this graph looks like?
5 Using the Sum of the Degrees of Vertices Formula 2 Consider the following scenarios: Is it possible to have a graph with vertices of degrees: 1 and 1? Is it possible to have a graph with vertices of degrees: 1 and 2? Is it possible to have a graph with vertices of degrees: 1, 1, 2, 3? Is it possible to have a graph with vertices of degrees: 1, 1, 2, 3, 3?
6 Planar Graphs Definition (Planar Graph) A graph is Planar if it is isomorphic to another graph whose edges do not cross. Based on the definition, to determine if a graph is planar we will often have to consider isomorphic versions of the graph. (Using Isomorphisms to Show a Graph is Planar) The graph on the left is definitely planar (edges do not cross). Even though the graph to the right does not appear to be planar, it is because it is isomorphic to the graph on the left!
7 Faces of a Planar Graph Definition (Faces of a Planar Graphs) In any planar graph, drawn with no intersections, the edges divide the planes into different regions. The regions enclosed by the planar graph are called interior faces of the graph. The region surrounding the planar graph is called the exterior (or infinite) face of the graph. When we say faces of the graph we mean BOTH the interior AND the exterior faces. We usually denote the number of faces of a planar graph by f. Be Careful! Before you count the faces of a planar graph, it is important to first draw an isomorphic version of the graph so that no edges cross!
8 Counting Faces of a Planar Graph It is really tempting to try to count the faces in the graph below: Remember, the idea of faces only applies to planar graphs drawn with no edge crossings. Instead we need to redraw the graph to get an isomorphic version: In the bottom version it is easy to see there are 3 interior faces and 1 exterior face for a total of 4 faces.
9 Degree of a Face Definition (Degree of a Face) For a planar graph drawn without edges crossing, the number of edges bordering a particular face is called the degree of the face. In the graph below each of the two interior faces have degree 3. The infinite (exterior) face has degree 4. Adding up degrees gives a result that is curiously similar to one before. Theorem (Sum of the Degrees of Faces) In any planar graph, the sum of the degrees of all faces is equal to twice the number of edges. In symbols, D F = 2e.
10 Counting Faces and Degrees 1 C A B D E F Is this graph planar? How many faces does this graph have? What is the degree of each face? List v, e, and f for this graph.
11 Counting Faces and Degrees 2 C A B D E F Is this graph planar? How many faces does this graph have? What is the degree of each face? List v, e, and f for this graph.
12 Counting Faces and Degrees 3 A B D C E F Is this graph planar? How many faces does this graph have? What is the degree of each face? List v, e, and f for this graph.
13 Making Graphs Look Planar Here are a couple more graphs that don t look planar at first. Can you count the faces? Left Graph (Cube) Right Graph (K 3,2 ) Vertices v 8 5 Edges e 12 6 Faces f 6 3 These numbers are giving more evidence of some big ideas with planar graphs.
14 Euler s Formula Theorem (Euler s Formula) Take any connected planar graph drawn with no intersecting edges. Let v be the number of vertices in the graph. Let e be the number of edges in the graph. Let f be the number of faces in the graph. Then v e + f = 2. Check the planar graphs on the previous slide: Cube v e + f = = 2 K 3,2 v e + f = = 2 Let s see what Euler s Formula can do for us!
15 Using Euler s Formula 1 C B A D E F What is v? What is e? If the graph is planar, what must f be? Redraw the graph with vertices/edges moved around so the graph appears planar.
16 Using Euler s Formula 2 A connected planar graph has 8 vertices and 12 edges. How many faces are there? A connected planar graph has 6 vertices and 4 faces. How many edges are there? A connected planar graph has 8 vertices with degrees: 1, 1, 2, 2, 3, 3, 4, 4. How many edges are there? A connected planar graph has 8 vertices with degrees: 1, 1, 2, 2, 3, 3, 4, 4. How many faces are there?
17 Using Euler s Formula 3 A connected planar graph has 6 faces and 12 edges. How many vertices are there? A connected planar graph has 4 faces with degrees: 3, 3, 4, 4. How many edges are there? A connected planar graph has 4 faces with degrees: 3, 3, 4, 4. How many vertices are there? Draw a connected planar graph with 4 faces with degrees: 3, 3, 4, 4. Is there only one such isomorphic graph?
18 Using Euler s Formula 4 B A C E D F What is v? What is e? Though it doesn t look it, this is graph is planar. What is f?
19 Useful Facts for Planar Graphs We can combine the following theorems to answer questions about planar graphs. Theorem (Sum of the Degrees of Vertices) In any graph, the sum of the degrees of all vertices is equal to twice the number of edges. along with what we just looked at Theorem (Sum of the Degrees of Faces) In any planar graph, the sum of the degrees of all faces is equal to twice the number of edges. along with Euler s Formula Theorem (Euler s Formula) For a connected planar graph with vertices v, edges e, and faces f, the following must hold: v e + f = 2.
20 Another Euler s Formula Find v, e, and f for these two graphs. Find c for these two graphs. Combine c, e, f, and v to make a new Euler s Formula!
21 A Chance to Discover Euler s General Formula Note that Euler s Formula only applies to connected graphs, i.e., graphs that have one component where we write c = 1. See if you can extend Euler s Formula for c = 2 in the graph below! Theorem (Euler s General Formula) For any planar graph with vertices v, edges e, faces f and components c, the following must hold: v e + f c = 1.
22 Another Euler s Formula 2 A connected planar graph has vertices whose degrees are 3, 3, 4, 4, 5, 6, 7. How many vertices are there? A connected planar graph has vertices whose degrees are 3, 3, 4, 4, 5, 6, 7. How many edges are there? A connected planar graph has vertices whose degrees are 3, 3, 4, 4, 5, 6, 7. How many faces are there? A disconnected planar graph with c = 2 has vertices whose degrees are 3, 3, 4, 4, 5, 6, 7. How many faces are there? A disconnected planar graph with c = 4 has vertices whose degrees are 3, 4, 5, 6, 7, 8, 9. How many faces are there?
Sum of Degrees of Vertices Theorem
Sum of Degrees of Vertices Theorem Theorem (Sum of Degrees of Vertices Theorem) Suppose a graph has n vertices with degrees d 1, d 2, d 3,...,d n. Add together all degrees to get a new number d 1 + d 2
CS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin
CS311H Prof: Department of Computer Science The University of Texas at Austin Good Morning, Colleagues Good Morning, Colleagues Are there any questions? Logistics Class survey Logistics Class survey Homework
Euler Paths and Euler Circuits
Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and
Class One: Degree Sequences
Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of
Introduction to Graph Theory
Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate
Planar Graphs. Complement to Chapter 2, The Villas of the Bellevue
Planar Graphs Complement to Chapter 2, The Villas of the Bellevue In the chapter The Villas of the Bellevue, Manori gives Courtel the following definition. Definition A graph is planar if it can be drawn
On planar regular graphs degree three without Hamiltonian cycles 1
On planar regular graphs degree three without Hamiltonian cycles 1 E. Grinbergs Computing Centre of Latvian State University Abstract. Necessary condition to have Hamiltonian cycle in planar graph is given.
Simple Graphs Degrees, Isomorphism, Paths
Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week Multi-Graph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1
136 CHAPTER 4. INDUCTION, GRAPHS AND TREES
136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics
2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and
Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.
Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.
Math 2443, Section 16.3
Math 44, Section 6. Review These notes will supplement not replace) the lectures based on Section 6. Section 6. i) ouble integrals over general regions: We defined double integrals over rectangles in the
Midterm Practice Problems
6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator
OPTIONS TRADING AS A BUSINESS UPDATE: Using ODDS Online to Find A Straddle s Exit Point
This is an update to the Exit Strategy in Don Fishback s Options Trading As A Business course. We re going to use the same example as in the course. That is, the AMZN trade: Buy the AMZN July 22.50 straddle
The Clar Structure of Fullerenes
The Clar Structure of Fullerenes Liz Hartung Massachusetts College of Liberal Arts June 12, 2013 Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 1 / 25
WHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
Chapter 6: Graph Theory
Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.
Social Media Mining. Graph Essentials
Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures
Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
Removing Even Crossings
EuroComb 2005 DMTCS proc. AE, 2005, 105 110 Removing Even Crossings Michael J. Pelsmajer 1, Marcus Schaefer 2 and Daniel Štefankovič 2 1 Department of Applied Mathematics, Illinois Institute of Technology,
Examination paper for MA0301 Elementær diskret matematikk
Department of Mathematical Sciences Examination paper for MA0301 Elementær diskret matematikk Academic contact during examination: Iris Marjan Smit a, Sverre Olaf Smalø b Phone: a 9285 0781, b 7359 1750
10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
Chapter 4.1 Parallel Lines and Planes
Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about
V. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005
V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer
Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
Geometry Review Flash Cards
point is like a star in the night sky. However, unlike stars, geometric points have no size. Think of them as being so small that they take up zero amount of space. point may be represented by a dot on
Answer: (a) Since we cannot repeat men on the committee, and the order we select them in does not matter, ( )
1. (Chapter 1 supplementary, problem 7): There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the
Incenter Circumcenter
TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is
SURFACE AREA AND VOLUME
SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has
Grade Level: High School
Lesson I: Triangles- Exterior Angle Theorem KEY WORDS: Triangles, exterior-angle theorem, and remote interior angles. Grade Level: High School SUMMARY: With this investigation students will discover the
A permutation can also be represented by describing its cycles. What do you suppose is meant by this?
Shuffling, Cycles, and Matrices Warm up problem. Eight people stand in a line. From left to right their positions are numbered,,,... 8. The eight people then change places according to THE RULE which directs
Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits
Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique
INCIDENCE-BETWEENNESS GEOMETRY
INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
Graph Theory Problems and Solutions
raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is
Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
Arrangements And Duality
Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,
x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
Pythagorean Theorem: Proof and Applications
Pythagorean Theorem: Proof and Applications Kamel Al-Khaled & Ameen Alawneh Department of Mathematics and Statistics, Jordan University of Science and Technology IRBID 22110, JORDAN E-mail: kamel@just.edu.jo,
PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh
Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh Graph Theory: Graph theory can be defined as the study of graphs; Graphs are mathematical structures used to model pair-wise relations between
Circle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
Geometry 1. Unit 3: Perpendicular and Parallel Lines
Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples
Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
The minimum number of distinct areas of triangles determined by a set of n points in the plane
The minimum number of distinct areas of triangles determined by a set of n points in the plane Rom Pinchasi Israel Institute of Technology, Technion 1 August 6, 007 Abstract We prove a conjecture of Erdős,
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes
Jim Lambers MAT 169 Fall Semester 009-10 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that
Chapter 6 Notes: Circles
Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment
Just the Factors, Ma am
1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive
Chapters 6 and 7 Notes: Circles, Locus and Concurrence
Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of
MATH 13150: Freshman Seminar Unit 10
MATH 13150: Freshman Seminar Unit 10 1. Relatively prime numbers and Euler s function In this chapter, we are going to discuss when two numbers are relatively prime, and learn how to count the numbers
Sample Problems. Practice Problems
Lecture Notes Circles - Part page Sample Problems. Find an equation for the circle centered at (; ) with radius r = units.. Graph the equation + + = ( ).. Consider the circle ( ) + ( + ) =. Find all points
If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
Labeling outerplanar graphs with maximum degree three
Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics
Topological Treatment of Platonic, Archimedean, and Related Polyhedra
Forum Geometricorum Volume 15 (015) 43 51. FORUM GEOM ISSN 1534-1178 Topological Treatment of Platonic, Archimedean, and Related Polyhedra Tom M. Apostol and Mamikon A. Mnatsakanian Abstract. Platonic
Discrete Mathematics Problems
Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 E-mail: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................
Number Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
CMPSCI611: Approximating MAX-CUT Lecture 20
CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to
6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
Lecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
Rubiks Cube or Magic Cube - How to solve the Rubiks Cube - follow these instructions
The solution to the Rubiks Cube or Magic Cube - How to solve the Rubiks Cube - follow these instructions The first step to complete the Rubiks Cube. Complete just one face of the cube. You will at some
Lecture 3: Derivatives and extremes of functions
Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16
POLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs
Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph
A summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs:
summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs: efinitions: efinition of mid-point and segment bisector M If a line intersects another line segment
Greatest Common Factor and Least Common Multiple
Greatest Common Factor and Least Common Multiple Intro In order to understand the concepts of Greatest Common Factor (GCF) and Least Common Multiple (LCM), we need to define two key terms: Multiple: Multiples
ALGEBRA. Find the nth term, justifying its form by referring to the context in which it was generated
ALGEBRA Pupils should be taught to: Find the nth term, justifying its form by referring to the context in which it was generated As outcomes, Year 7 pupils should, for example: Generate sequences from
MECHANICAL PRINCIPLES HNC/D MOMENTS OF AREA. Define and calculate 1st. moments of areas. Define and calculate 2nd moments of areas.
MECHANICAL PRINCIPLES HNC/D MOMENTS OF AREA The concepts of first and second moments of area fundamental to several areas of engineering including solid mechanics and fluid mechanics. Students who are
7.2 Quadratic Equations
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
MA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
Tenacity and rupture degree of permutation graphs of complete bipartite graphs
Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China
Mathematical Induction. Lecture 10-11
Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
Math 1526 Consumer and Producer Surplus
Math 156 Consumer and Producer Surplus Scenario: In the grocery store, I find that two-liter sodas are on sale for 89. This is good news for me, because I was prepared to pay $1.9 for them. The store manager
Section 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.
Patterns in Pascal s Triangle
Pascal s Triangle Pascal s Triangle is an infinite triangular array of numbers beginning with a at the top. Pascal s Triangle can be constructed starting with just the on the top by following one easy
Connectivity and cuts
Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every
n 2 + 4n + 3. The answer in decimal form (for the Blitz): 0, 75. Solution. (n + 1)(n + 3) = n + 3 2 lim m 2 1
. Calculate the sum of the series Answer: 3 4. n 2 + 4n + 3. The answer in decimal form (for the Blitz):, 75. Solution. n 2 + 4n + 3 = (n + )(n + 3) = (n + 3) (n + ) = 2 (n + )(n + 3) ( 2 n + ) = m ( n
1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
Some Lecture Notes and In-Class Examples for Pre-Calculus:
Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
No Solution Equations Let s look at the following equation: 2 +3=2 +7
5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are
The GED math test gives you a page of math formulas that
Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding
Geometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
E XPLORING QUADRILATERALS
E XPLORING QUADRILATERALS E 1 Geometry State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Statement of Purpose: The activities in this
This puzzle is based on the following anecdote concerning a Hungarian sociologist and his observations of circles of friends among children.
0.1 Friend Trends This puzzle is based on the following anecdote concerning a Hungarian sociologist and his observations of circles of friends among children. In the 1950s, a Hungarian sociologist S. Szalai
Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013
Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is
Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York.
1: 1. Compute a random 4-dimensional polytope P as the convex hull of 10 random points using rand sphere(4,10). Run VISUAL to see a Schlegel diagram. How many 3-dimensional polytopes do you see? How many
Chapter 7. Permutation Groups
Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral
Algebra 1 Course Title
Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b.
PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.
Everyday Mathematics. Grade 4 Grade-Level Goals CCSS EDITION. Content Strand: Number and Numeration. Program Goal Content Thread Grade-Level Goal
Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation
Soil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay
Soil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay Module - 2 Vibration Theory Lecture - 8 Forced Vibrations, Dynamic Magnification Factor Let
CSC 373: Algorithm Design and Analysis Lecture 16
CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements
EdExcel Decision Mathematics 1
EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation
Plots, Curve-Fitting, and Data Modeling in Microsoft Excel
Plots, Curve-Fitting, and Data Modeling in Microsoft Excel This handout offers some tips on making nice plots of data collected in your lab experiments, as well as instruction on how to use the built-in
Everyday Mathematics. Grade 4 Grade-Level Goals. 3rd Edition. Content Strand: Number and Numeration. Program Goal Content Thread Grade-Level Goals
Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation
Angles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
Solving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
Logo Symmetry Learning Task. Unit 5
Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to