Physics Gravitational force. 2. Strong or color force. 3. Electroweak force

Size: px
Start display at page:

Download "Physics Gravitational force. 2. Strong or color force. 3. Electroweak force"

Transcription

1 Phsics 360 Notes on Griffths - pluses and minuses No tetbook is perfect, and Griffithsisnoeception. Themajorplusisthat it is prett readable. For minuses, see below. Much of what G sas about the del operator applies onl in Cartesian coordinates. For eample, we ma onl think of del as a vector such that u is the dot product of and u if is epressed in Cartesian coordinates. Page 27, The unit vectors in Cartesian coordinates are constants and ma be pulled out of the integral, but unit vectors in other coordinate sstems are not constants! For eample, ˆr ma not be pulled out of an integral. Unit vectors are written as ˆ, ŷ, ẑ not î, ĵ, ˆk. This is consistent with what we do in ever other coordinate sstem, such as ˆr, ˆθ, ˆφ in spherical coordinates. (I regard this as a plus.) G is often slopp in writing vectors eperts can get awa with things that beginners cannot because the know where the are going. But it s a bad habit. Don t be slopp! Alwas write the vector sign, and be sure that there is a vector on both sides of an equals sign or neither. B convention, writing a vector without its vector sign means the magnitude of the vector: v v. If ou want to write the component, label it precisel, eg v or v r, for eample. Components have signs that indicate the direction of the vector; magnitudes are alwas positive. G is also slopp about eplaining where results come from. Results of comple integrals are sometimes just thrown down with no eplanation. In our homework and eams, ou should either (a) do the integral ourself, showing all the steps, or (b) look it up, and give the complete reference to where ou found the R result, and what substitutions have to be made. (For eample, if ou have r dr but the table has R d, ou will need to sa that ou are using result N.mm from Book X with = r.) (a) is definitel preferred, because ou will get better and better at doing integrals if ou practice them, and in the long run it will save ou time as well has help to develop our intuition. But go with (b) if the integral is reall hard, and the homework is due in 0 minutes! There are also a few Phsics errors. I will point these out as we go along. So let s get going! Phsics is a set of conceptual ideas that we epress in mathematical form in order to solve problems. To understand phsics well we have to use three different and equall important languages: English, pictures, and mathematics. Your solutions should contain adequate amounts of all three. The subject of this course is electricit and magnetism. In the 20th centur we learned that the fundamental forces of phsics are:. Gravitational force 2. Strong or color force 3. Electroweak force

2 () is studied using Newton s laws, or, for strong fields, Einstein s equations. (2) is studied using advanced mathematical concepts such as groups. (3) is the unification of electromagnetic theor and the weak nuclear force. Electromagnetic theor is the first unified field theor: the unification of electric and magnetic field theor, and is the subject of our stud this semester and net (in 460). In Phsics 230, we studied electric fields first, then magnetic fields. This is traditional, because the mathematics is a bit easier for electric fields, and we shall follow that plan in 360 too. But we have to remember that both are different aspects of one electromagnetic field. In Phsics 230, we started with the fields produced b a single source element (eg a point charge) and built up the field due due a distribution of sources using the principle of superposition. We can do this because the relation of the fields to their sources is linear. This is an eperimental fact. The electric field is a vector so when we add the fields due to different sources we are alwas doing vector addition. Fundamental principles and definitions. Charge is conserved. We believe that the total charge of the universe is eactl zero. Thus the onl wa to get a net charge Q in one place is to have a corresponding net charge Q somewhere else. Opposite charges attract, so these separated charges are alwas tring to get back together. This is the primar reason wh electrostatic forces are not ver obvious in everda life, or on the large scales of astronom.. Charge is quantized and appears in units of e/3, but we are doing classical theor this semester, so for the most part our charges will consist of so man fundamental units that we can consider charge to be a continuous quantit. We start with the definition of electric field. We place a test charge q at point P and measure the electric force F acting on q. Then the electric field at P is F E = lim q 0 q From Coulomb s law (another eperimental fact) we find that; 2

3 The electric field produced b a point charge Q is E (P )= kq r 2 ˆr where r is the vector with its tail on Q and its head at point P. A simple eample. Suppose we have a point charge Q =μc at the origin and a second charge 2Q at point P with coordinates ( mm, 2 mm, 3 mm). What is the electric field at an arbitrar point P with coordinates (,, z)? Given an problem, the solution consists of the following steps. MODEL First determine the important phsicical principles that govern the behavior of the sstem. Here the principle that we need is Coulomb s Law. We are also going to use the geometr of flat space, as epressed in the Pthagorean theorem. That seems so obvious that ou probabl wouldn t think of it as a separate principal. The more phsics ou learn, the more the first bits ou learned seem this "obvious", but we must be careful not to get complacent! We must be especiall careful not to think that things are obvious when the are actuall not even true! Asketchofthefield line diagram will help us understand the field. The net charge of the sstem is Q 2Q = Q Using 4 lines per Q, 4linesleave the +Q charge and g to the Q charge, while 4 come in from infinit to the Q charge. Wehavetohavesphericalsmmetratavergreatdistancefrom both charges, or ver close to either one. (You can download a program from m Phsics 230 web site that will allow ou to draw a prett accurate diagram in a plane for this sstem.). SETUP Now we are going to decide how to solve the problem, and get to the point where all we have left to do is some math. Let s start with a diagram. 3

4 The diagram shows the two charges and the fieldproducedbeach. Notice that E 2 points toward the negative charge. Now we can calculate the field using Coulomb s law: E (P )= kq r 2 ˆr = kq r r 3 The second form will be easiest to use here, since we will have to calculate ˆr as r/r. We must be ver clear as to what the smbol "r" means in this formula. It is the vector with its tail at the charge and its head at point P. For the first charge, Q, which is at the origin, r is the position vector of point P : r = ˆ + ŷ + zẑ and r = p z 2. For the second charge, 2Q, the vector r 2 hascomponents( q, 2,z 3) where all lengths are measured in mm., Then r 2 = ( ) 2 +( 2) 2 +(z 3) 2. Thus kq E = (ˆ + ŷ + zẑ) ( z 2 3/2 ) and 2kQ E 2 = h( ) 2 +( 2) 2 +(z 3) 2i [( ) ˆ +( 2) ŷ +(z 3) ẑ] 3/2 Finall, using the principle of superposition, we have E = E + E 2 4

5 E = SOLVE Now we are read to do the addition. kq ( z 2 ) (ˆ + ŷ + zẑ) 2kQ 3/2 h( ) 2 +( 2) 2 +(z 3) 2i (( ) ˆ +( 2) ŷ +(z 3) ẑ) 3/2 This is prett ugl. Let s do it one component at a time: E = kq ( z 2 ) 2( ) 3/2 h( ) 2 +( 2) 2 +(z 3) 2i 3/2 It s not going to be possible to make this look much nicer. We should alwas tr to simplif our answer as much as possible, but here I don t see how we can do much more. The other components will look similar. ANALYZE. This is a critical step. Does the answer have the right phsical dimensions? Let s check. The quantit in the curl brackets is length/(length) 3 =/(length) 2, so the answer is of the form k ³charge/length 2 which is correct. Does the answer have the right value in special cases? Well, if we let get ver big, let s see what we get. The relevant phsical value to compare with is the mm separation (along the ais) of the two charges. So let À mm. Then we rewrite the second fraction like this: 2( ) 2 h( ) 2 +( 2) 2 +(z 3) 2i = 3/2 h z 3 i 2 3/2 = 2 2 h z 3 2 i 3/2 Now let s also suppose that À, z. In this limit the quantit multipling 2/ 2 and we get ½ E = kq 2 2 ¾ 2 = kq 2 The and z components are much smaller: E E Our result is the electric fieldonthe ais due to a point charge Q at the origin. This is a particular eample of a general rule: RULE The electric field due to an charge distribution with net charge Q, measured at a ver great distance from it, equals the electric field due to a point charge Q located within the charge distribution. 5

6 Now let s set = z =0, keeping À mm, and see what the first order correction is. E = kq h i 2 3/2 ½ = kq 2 2 ¾ 2 f We re going to keep terms of order / 3 but drop terms of order / 4 and higher powers of /. So in the term f multipling 2/ 2 we keep terms of order /. f = h i 2 3/2 = 2 3/2 + O / 2 Now we use the binomial theorem to epand the denominator µ 2 3/2 µ =+ 3 µ µ O 2 Thus f = µ µ + 3 = But we have alread dropped terms of order / 2 in f, so here we must drop the 3/ 2 term to be consistent. (You can make some ver serious errors b using inconsistent levels of approimation!) Putting it all together, we have ½ E = kq 2 2 µ ¾ = kq ½ 2 4 ¾ 3 The first term is the "point source" term that we had before. The second term results from the fact that there are actuall two separate charges. It is a dipole field. (See LB $24.5 The field on the ais of the dipole is E = 2k p 3 We model our charge distribution as a point charge Q plus a dipole with charges Q at =mm and Q at the origin. The dipole moment p points from the negative charge to the positive charge, so in this case its component is negative. p = Q ( mm) This gives a dipole electric field on the -ais of E = 2kQ 3 6

7 But wait! We are off b a factor 2! That s because our point charge Q is not eactl at the origin, but displaced b a distance of mm to the right. So it contributes an etra dipole moment of p = Q ( mm). We ll learn more about how to compute dipole moments in Chapter 3 ( 3.4). Just to round off the discussion, let s find the fieldonthe ais ( = z = 0) with À 3 mm. E (0,,0) = = kq 2 kq ( 2 ) ŷ 2kQ 3/2 h( ) 2 +( 2) 2 +( 3) 2i ( ˆ +( 2) ŷ 3ẑ) 3/2 ŷ 2 ³ ³ ˆ + 2 ŷ 3 ẑ ³ 2 ³ 2 ³ /2 As before, let s drop terms in / 2 and higher inside the parentheses. ³ E (0,,0) = kq ³ ˆ + 2 ŷ 3 ẑ 2 ŷ 2 ³ 3/2 4 = kq ½ 2 ŷ 2 µ µ ˆ + 2 ŷ 3 µ ẑ + 3 ¾ = kq ½ µ 2 2 ˆ +ŷ ¾ ẑ = kq 2 ŷ + kq {2ˆ 2ŷ +6ẑ} 3 The leading term is the point charge field, as epected, and the correction is a dipole field, due to all three components of our dipole.. We ll come back to the details of this when we stud chapter 3. Eample 2. Find the electric fieldatapointonthe ais due to a filament of length 2 that stretches from = to = and has a uniform linear charge densit λ. Again we use superposition, but we start b modelling the line as a collection of differential elements, each of which we treat as a point charge.to which we can appl Coulomb s law. 7

8 Setup: A tpical differential element is at coordinate, with length d and charge λd = dq It prduces an electric field at P de = k dq r 2 ˆr that has both and components, as shown in the diagram. Now we can simplif our calculation b noting that our filament has mirror smmetr about the z plane. Thus we can find another element at coordinate, the "mirror image" of our first element, that produces an electric field of the same magnitude at P. This second field has an identical component but the eact opposite component, so when we add them together the net result is a field in the direction. Thus as we sum up the contributions from all our elements, we find that the total electric fieldmustbeinthe direction. It is important that we epress the distance r in terms of the variable. The electric field due to the pair of elements shown is de (P )=2 kλd ( ) cos θ Be careful here! is the -coordinate of our differential element (the one on the right) while is the coordinate of the point P. It would be better to give them labels that emphasize this difference, so we ll use 0 rather than. From the diagram cos θ = r Thus kλ d 0 de (P )=2 i 3/2 h( 0 ) Nowwesumtheelements. Thedifferential d 0 tells us what our integration variable will be. The limits of the integral are 0 to l, sincewearesumming 8

9 over pairs of elements, labelled b the coordinate of the element on the right in each pair. Solve: Z kλ d 0 E (0,,0) = 2 i 3/2 h( 0 ) Now k, λ and are all independent of 0, so we ma remove them from the integral. Z d 0 E (0,,0) = 2kλ i 3/2 0 h( 0 ) To do the integral, we use the tangent substitution. Let 0 = tan θ. (Note that this angle is actuall the same θ marked in the diagram.) Then ( 0 ) = 2 +tan 2 θ = 2 sec 2 θ, and so Z tan ( /) E = 2kλ = 2kλ = 2kλ To evaluate the sine, note that sin θ =cosθ tan θ = 0 Z tan ( /) 0 sec 2 θdθ 3 sec 3 θ cos θdθ sin θ tan ( /) 0 () tan θ sec2 θ = tan θ +tan 2 θ Thus E (0,,0) = 2kλ / (2) q+( /) 2 = 2kλ p (3) Analze: First note that our result has the right dimensions. Looking at the result in the form labelled (2), we see that the factor multipling 2kλ/ is dimensionless, because / is a dimensionless number. The linear charge densit λ is charge/length, so kλ/ is k charge/length 2, as required. Now let s look at the field a long wa from the line, so that À. Then in (2) we neglect the term ( /) 2 compared with the one in the denominator. The result is 2λ E = k 2p + 2 / ' k 2λ 2 2 = k Q 2 This is the electric field due to a point charge Q =2λ at the origin. another eample of RULE that we found above. Here is 9

10 Now let s see what happens if we let our line become infinitel long. The easiest wa to get the result is to go back to result () and let l. We note that lim w tan (w) =π/2, and sin π/2 =, so we have immediatel E = 2kλ (infinite line) From (3) we can see that this is also the result when we are ver close to a finite line ( ). We have found more than ou might imagine. Notice that our sstem has rotational smmetr about the ais in addition to the mirror smmetr we alread eploited. Thus we have found the electric fieldanwhereinthe z plane, and we ma write the result as E = 2kλ s /s ˆr q+( /s) 2 where s is the radial coordinate in a clindrical coordinate sstem with z-ais along the line of the charge. Now let s compare the three results (3), (??) and(??). We plot the dimensionless field variable e = E / (2kλ/ ) versus the dimensionless distance variable u = /. Then the three epressions are Eact (black line): e = u +u 2 Near (red line) : e = u Far (green line) : e = u e u The "near" result works ver well for u. 0.3 while the "far" result works well for u &.5. Eample 3. Now what if we have two infinite lines, with linear charge densities λ and λ?. What is the electric field produced b these lines at a 0

11 point in the plane midwa between them? MODEL We use the resut we have alread found for an infinite line, and add the two fields, as shown in the diagram. SETUP Putting the z ais along the line joining the two infinite filaments, and ais along the bisector, with point P having coordinates (0,) and the separation of the lines being 2L, we can see that the components cancel while the components add, to give SOLVE ANALYZE for À L, we get E = 2 2kλ = 4kλ cos θ ˆ L p L2 + ˆ 2 E 4kλ 2 ˆ This is the result for a line dipole. Let s summarize what we have learned so far about how electric fields var with distance from the source. source one two (dipole) point /distance 2 /distance 3 line /distance /distance 2 We ll finish here b writing the epression for E as an integral in its most general form. We label the point P at which we want to find E with the position vector r with respect to some origin O, and our differential element with position vector r 0. Then the vector pointing from the element dq = ρ ( r 0 ) dτ 0 to P is R = r r 0

12 and the electric field at P produced b the element is d E ( r) =k dq R 2 ˆR = k ρ ( r0 ) dτ 0 r r 0 3 ( r r0 ) and then we sum over all the elements b integrating over the primed coordinates: Z Z E ( r) = de ρ ( r 0 )( r r 0 ) = k r r 0 3 dτ 0 The volume V is all space, but in practice we can limit it to the volume where ρ is not zero. V 2

The Force Table Introduction: Theory:

The Force Table Introduction: Theory: 1 The Force Table Introduction: "The Force Table" is a simple tool for demonstrating Newton s First Law and the vector nature of forces. This tool is based on the principle of equilibrium. An object is

More information

11.4 Polar Coordinates

11.4 Polar Coordinates 11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.

More information

Triple Integrals. y x

Triple Integrals. y x Triple Integrals. (a) If is an solid (in space), what does the triple integral dv represent? Wh? (b) Suppose the shape of a solid object is described b the solid, and f(,, ) gives the densit of the object

More information

Analytic Trigonometry

Analytic Trigonometry CHAPTER 5 Analtic Trigonometr 5. Fundamental Identities 5. Proving Trigonometric Identities 5.3 Sum and Difference Identities 5.4 Multiple-Angle Identities 5.5 The Law of Sines 5.6 The Law of Cosines It

More information

Ch 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet.

Ch 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet. Ch Alg L Note Sheet Ke Do Activit 1 on our Ch Activit Sheet. Chapter : Quadratic Equations and Functions.1 Modeling Data With Quadratic Functions You had three forms for linear equations, ou will have

More information

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The

More information

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (

More information

Lab 5 Forces Part 1. Physics 211 Lab. You will be using Newton s 2 nd Law to help you examine the nature of these forces.

Lab 5 Forces Part 1. Physics 211 Lab. You will be using Newton s 2 nd Law to help you examine the nature of these forces. b Lab 5 Forces Part 1 Phsics 211 Lab Introduction This is the first week of a two part lab that deals with forces and related concepts. A force is a push or a pull on an object that can be caused b a variet

More information

= C. on q 1 to the left. Using Coulomb s law, on q 2 to the right, and the charge q 2 exerts a force F 2 on 1 ( )

= C. on q 1 to the left. Using Coulomb s law, on q 2 to the right, and the charge q 2 exerts a force F 2 on 1 ( ) Phsics Solutions to Chapter 5 5.. Model: Use the charge model. Solve: (a) In the process of charging b rubbing, electrons are removed from one material and transferred to the other because the are relativel

More information

Chapter 18 Quadratic Function 2

Chapter 18 Quadratic Function 2 Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are

More information

2.4 Orthogonal Coordinate Systems (pp.16-33)

2.4 Orthogonal Coordinate Systems (pp.16-33) 8/26/2004 sec 2_4 blank.doc 1/6 2.4 Orthogonal Coordinate Sstems (pp.16-33) 1) 2) Q: A: 1. 2. 3. Definition: ). 8/26/2004 sec 2_4 blank.doc 2/6 A. Coordinates * * * Point P(0,0,0) is alwas the origin.

More information

Review Topics for MATH 1400 Elements of Calculus Table of Contents

Review Topics for MATH 1400 Elements of Calculus Table of Contents Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

More information

Symmetry Arguments and the Role They Play in Using Gauss Law

Symmetry Arguments and the Role They Play in Using Gauss Law Smmetr Arguments and the Role The la in Using Gauss Law K. M. Westerberg (9/2005) Smmetr plas a ver important role in science in general, and phsics in particular. Arguments based on smmetr can often simplif

More information

VECTORS IN THREE DIMENSIONS

VECTORS IN THREE DIMENSIONS 1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude

More information

ES.182A Topic 36 Notes Jeremy Orloff

ES.182A Topic 36 Notes Jeremy Orloff ES.82A Topic 36 Notes Jerem Orloff 36 Vector fields and line integrals in the plane 36. Vector analsis We now will begin our stud of the part of 8.2 called vector analsis. This is the stud of vector fields

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted

More information

Trigonometric Functions

Trigonometric Functions TrigonometricReview.nb Trigonometric Functions The trigonometric (or trig) functions are ver important in our stud of calculus because the are periodic (meaning these functions repeat their values in a

More information

Let s try an example of Unit Analysis. Your friend gives you this formula: x=at. You have to figure out if it s right using Unit Analysis.

Let s try an example of Unit Analysis. Your friend gives you this formula: x=at. You have to figure out if it s right using Unit Analysis. Lecture 1 Introduction to Measurement - SI sstem Dimensional nalsis / Unit nalsis Unit Conversions Vectors and Mathematics International Sstem of Units (SI) Table 1.1, p.5 The Seven Base Units What is

More information

Pure Further Mathematics 2. Revision Notes

Pure Further Mathematics 2. Revision Notes Pure Further Mathematics Revision Notes October 016 FP OCT 016 SDB Further Pure 1 Inequalities... 3 Algebraic solutions... 3 Graphical solutions... 4 Series Method of Differences... 5 3 Comple Numbers...

More information

Welcome. to Electrostatics

Welcome. to Electrostatics Welcome to Electrostatics Outline 1. Coulomb s Law 2. The Electric Field - Examples 3. Gauss Law - Examples 4. Conductors in Electric Field Coulomb s Law Coulomb s law quantifies the magnitude of the electrostatic

More information

Roberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 3. Separable ODE s

Roberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 3. Separable ODE s Roberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 3 Separable ODE s What ou need to know alread: What an ODE is and how to solve an eponential ODE. What ou can learn

More information

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4

More information

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.

More information

15. Eigenvalues, Eigenvectors

15. Eigenvalues, Eigenvectors 5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does

More information

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator,

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator, GRADE PRE-CALCULUS UNIT C: QUADRATIC FUNCTIONS CLASS NOTES FRAME. After linear functions, = m + b, and their graph the Quadratic Functions are the net most important equation or function. The Quadratic

More information

Eigenvectors and Eigenvalues 1

Eigenvectors and Eigenvalues 1 Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and

More information

f(x) = 2x 2 + 2x - 4

f(x) = 2x 2 + 2x - 4 4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms

More information

Methods of Solving Ordinary Differential Equations (Online)

Methods of Solving Ordinary Differential Equations (Online) 7in 0in Felder c0_online.te V3 - Januar, 05 0:5 A.M. Page CHAPTER 0 Methods of Solving Ordinar Differential Equations (Online) 0.3 Phase Portraits Just as a slope field (Section.4) gives us a wa to visualize

More information

Ground Rules. PC1221 Fundamentals of Physics I. Coordinate Systems. Cartesian Coordinate System. Lectures 5 and 6 Vectors.

Ground Rules. PC1221 Fundamentals of Physics I. Coordinate Systems. Cartesian Coordinate System. Lectures 5 and 6 Vectors. PC1221 Fundamentals of Phsics I Lectures 5 and 6 Vectors Dr Ta Seng Chuan 1 Ground ules Switch off our handphone and pager Switch off our laptop computer and keep it No talking while lecture is going on

More information

MAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function

MAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,

More information

Table of Contents. Module 1

Table of Contents. Module 1 Table of Contents Module 1 11 Order of Operations 16 Signed Numbers 1 Factorization of Integers 17 Further Signed Numbers 13 Fractions 18 Power Laws 14 Fractions and Decimals 19 Introduction to Algebra

More information

1.2 Functions and Their Properties PreCalculus

1.2 Functions and Their Properties PreCalculus 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given

More information

We have examined power functions like f (x) = x 2. Interchanging x

We have examined power functions like f (x) = x 2. Interchanging x CHAPTER 5 Eponential and Logarithmic Functions We have eamined power functions like f =. Interchanging and ields a different function f =. This new function is radicall different from a power function

More information

Systems of Linear Equations: Solving by Graphing

Systems of Linear Equations: Solving by Graphing 8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From

More information

Lecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts. Lecture 5 Equations of Lines and Planes Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 Universit of Massachusetts Februar 6, 2018 (2) Upcoming midterm eam First midterm: Wednesda Feb. 21, 7:00-9:00

More information

Course 15 Numbers and Their Properties

Course 15 Numbers and Their Properties Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.

More information

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will

More information

Unit 12 Study Notes 1 Systems of Equations

Unit 12 Study Notes 1 Systems of Equations You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve

More information

Unit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents

Unit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents Unit NOTES Honors Common Core Math Da : Properties of Eponents Warm-Up: Before we begin toda s lesson, how much do ou remember about eponents? Use epanded form to write the rules for the eponents. OBJECTIVE

More information

Methods for Advanced Mathematics (C3) Coursework Numerical Methods

Methods for Advanced Mathematics (C3) Coursework Numerical Methods Woodhouse College 0 Page Introduction... 3 Terminolog... 3 Activit... 4 Wh use numerical methods?... Change of sign... Activit... 6 Interval Bisection... 7 Decimal Search... 8 Coursework Requirements on

More information

Mathematics. Mathematics 2. hsn.uk.net. Higher HSN22000

Mathematics. Mathematics 2. hsn.uk.net. Higher HSN22000 hsn.uk.net Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still

More information

1.6 ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM

1.6 ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM 1.6 ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM 23 How does this wave-particle dualit require us to alter our thinking about the electron? In our everda lives, we re accustomed to a deterministic world.

More information

14.1 Systems of Linear Equations in Two Variables

14.1 Systems of Linear Equations in Two Variables 86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination

More information

PES 1110 Fall 2013, Spendier Lecture 5/Page 1

PES 1110 Fall 2013, Spendier Lecture 5/Page 1 PES 1110 Fall 2013, Spendier Lecture 5/Page 1 Toda: - Announcements: Quiz moved to net Monda, Sept 9th due to website glitch! - Finish chapter 3: Vectors - Chapter 4: Motion in 2D and 3D (sections 4.1-4.4)

More information

10.2 The Unit Circle: Cosine and Sine

10.2 The Unit Circle: Cosine and Sine 0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment AP Calculus AB Summer Assignment Name: When you come back to school, you will be epected to have attempted every problem. These skills are all different tools that you will pull out of your toolbo this

More information

Section J Venn diagrams

Section J Venn diagrams Section J Venn diagrams A Venn diagram is a wa of using regions to represent sets. If the overlap it means that there are some items in both sets. Worked eample J. Draw a Venn diagram representing the

More information

Chapter 8 Notes SN AA U2C8

Chapter 8 Notes SN AA U2C8 Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of

More information

Mathematics. Mathematics 2. hsn.uk.net. Higher HSN22000

Mathematics. Mathematics 2. hsn.uk.net. Higher HSN22000 Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For

More information

2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:

2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a: SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert

More information

Lab 5 Forces Part 1. Physics 225 Lab. You will be using Newton s 2 nd Law to help you examine the nature of these forces.

Lab 5 Forces Part 1. Physics 225 Lab. You will be using Newton s 2 nd Law to help you examine the nature of these forces. b Lab 5 orces Part 1 Introduction his is the first week of a two part lab that deals with forces and related concepts. A force is a push or a pull on an object that can be caused b a variet of reasons.

More information

Vectors in Two Dimensions

Vectors in Two Dimensions Vectors in Two Dimensions Introduction In engineering, phsics, and mathematics, vectors are a mathematical or graphical representation of a phsical quantit that has a magnitude as well as a direction.

More information

x y plane is the plane in which the stresses act, yy xy xy Figure 3.5.1: non-zero stress components acting in the x y plane

x y plane is the plane in which the stresses act, yy xy xy Figure 3.5.1: non-zero stress components acting in the x y plane 3.5 Plane Stress This section is concerned with a special two-dimensional state of stress called plane stress. It is important for two reasons: () it arises in real components (particularl in thin components

More information

12.1 Systems of Linear equations: Substitution and Elimination

12.1 Systems of Linear equations: Substitution and Elimination . Sstems of Linear equations: Substitution and Elimination Sstems of two linear equations in two variables A sstem of equations is a collection of two or more equations. A solution of a sstem in two variables

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

More information

A. Real numbers greater than 2 B. Real numbers less than or equal to 2. C. Real numbers between 1 and 3 D. Real numbers greater than or equal to 2

A. Real numbers greater than 2 B. Real numbers less than or equal to 2. C. Real numbers between 1 and 3 D. Real numbers greater than or equal to 2 39 CHAPTER 9 DAY 0 DAY 0 Opportunities To Learn You are what ou are when nobod Is looking. - Ann Landers 6. Match the graph with its description. A. Real numbers greater than B. Real numbers less than

More information

4 The Cartesian Coordinate System- Pictures of Equations

4 The Cartesian Coordinate System- Pictures of Equations The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean

More information

Linear Equations and Arithmetic Sequences

Linear Equations and Arithmetic Sequences CONDENSED LESSON.1 Linear Equations and Arithmetic Sequences In this lesson, ou Write eplicit formulas for arithmetic sequences Write linear equations in intercept form You learned about recursive formulas

More information

Lesson 3: Free fall, Vectors, Motion in a plane (sections )

Lesson 3: Free fall, Vectors, Motion in a plane (sections ) Lesson 3: Free fall, Vectors, Motion in a plane (sections.6-3.5) Last time we looked at position s. time and acceleration s. time graphs. Since the instantaneous elocit is lim t 0 t the (instantaneous)

More information

Physics for Scientists and Engineers. Chapter 3 Vectors and Coordinate Systems

Physics for Scientists and Engineers. Chapter 3 Vectors and Coordinate Systems Phsics for Scientists and Engineers Chapter 3 Vectors and Coordinate Sstems Spring, 2008 Ho Jung Paik Coordinate Sstems Used to describe the position of a point in space Coordinate sstem consists of a

More information

Topic 3 Notes Jeremy Orloff

Topic 3 Notes Jeremy Orloff Topic 3 Notes Jerem Orloff 3 Line integrals and auch s theorem 3.1 Introduction The basic theme here is that comple line integrals will mirror much of what we ve seen for multivariable calculus line integrals.

More information

+ = + + = x = + = + = 36x

+ = + + = x = + = + = 36x Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

More information

DIFFERENTIAL EQUATIONS First Order Differential Equations. Paul Dawkins

DIFFERENTIAL EQUATIONS First Order Differential Equations. Paul Dawkins DIFFERENTIAL EQUATIONS First Order Paul Dawkins Table of Contents Preface... First Order... 3 Introduction... 3 Linear... 4 Separable... 7 Eact... 8 Bernoulli... 39 Substitutions... 46 Intervals of Validit...

More information

Topic 5.2: Introduction to Vector Fields

Topic 5.2: Introduction to Vector Fields Math 75 Notes Topic 5.: Introduction to Vector Fields Tetbook Section: 16.1 From the Toolbo (what you need from previous classes): Know what a vector is. Be able to sketch a vector using its component

More information

Vectors Primer. M.C. Simani. July 7, 2007

Vectors Primer. M.C. Simani. July 7, 2007 Vectors Primer M.. Simani Jul 7, 2007 This note gives a short introduction to the concept of vector and summarizes the basic properties of vectors. Reference textbook: Universit Phsics, Young and Freedman,

More information

MORE TRIGONOMETRY

MORE TRIGONOMETRY MORE TRIGONOMETRY 5.1.1 5.1.3 We net introduce two more trigonometric ratios: sine and cosine. Both of them are used with acute angles of right triangles, just as the tangent ratio is. Using the diagram

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

Separation of Variables in Cartesian Coordinates

Separation of Variables in Cartesian Coordinates Lecture 9 Separation of Variables in Cartesian Coordinates Phs 3750 Overview and Motivation: Toda we begin a more in-depth loo at the 3D wave euation. We introduce a techniue for finding solutions to partial

More information

4.7. Newton s Method. Procedure for Newton s Method HISTORICAL BIOGRAPHY

4.7. Newton s Method. Procedure for Newton s Method HISTORICAL BIOGRAPHY 4. Newton s Method 99 4. Newton s Method HISTORICAL BIOGRAPHY Niels Henrik Abel (18 189) One of the basic problems of mathematics is solving equations. Using the quadratic root formula, we know how to

More information

4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4

4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 297 local maimum or minimum. The second derivative is f 2 e 2 e 2 4 e 2 4 Since e and 4, we have f when and when 2 f. So the curve is concave downward

More information

MATH 021 UNIT 1 HOMEWORK ASSIGNMENTS

MATH 021 UNIT 1 HOMEWORK ASSIGNMENTS MATH 01 UNIT 1 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usuall, the part (A) questions ask for eplanations,

More information

AP Calculus BC Summer Assignment 2018

AP Calculus BC Summer Assignment 2018 AP Calculus BC Summer Assignment 018 Name: When you come back to school, I will epect you to have attempted every problem. These skills are all different tools that we will pull out of our toolbo at different

More information

2.1 Rates of Change and Limits AP Calculus

2.1 Rates of Change and Limits AP Calculus . Rates of Change and Limits AP Calculus. RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important

More information

8.1 Exponents and Roots

8.1 Exponents and Roots Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents

More information

Green s Theorem Jeremy Orloff

Green s Theorem Jeremy Orloff Green s Theorem Jerem Orloff Line integrals and Green s theorem. Vector Fields Vector notation. In 8.4 we will mostl use the notation (v) = (a, b) for vectors. The other common notation (v) = ai + bj runs

More information

and Discrete Charge Distributions

and Discrete Charge Distributions Module 03: Electric Fields and Discrete Charge Distributions 1 Module 03: Outline Review: Electric Fields Charge Dipoles 2 Last Time: Gravitational & Electric Fields 3 Gravitational & Electric Fields SOURCE:

More information

nm nm

nm nm The Quantum Mechanical Model of the Atom You have seen how Bohr s model of the atom eplains the emission spectrum of hdrogen. The emission spectra of other atoms, however, posed a problem. A mercur atom,

More information

Math-2. Lesson:1-2 Properties of Exponents

Math-2. Lesson:1-2 Properties of Exponents Math- Lesson:- Properties of Eponents Properties of Eponents What is a power? Power: An epression formed b repeated multiplication of the same factor. Coefficient Base Eponent The eponent applies to the

More information

Trigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric

Trigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric Trigonometric equations 6 sllabusref eferenceence Topic: Periodic functions and applications In this cha 6A 6B 6C 6D 6E chapter Simple trigonometric equations Equations using radians Further trigonometric

More information

Chapter 13. Overview. The Quadratic Formula. Overview. The Quadratic Formula. The Quadratic Formula. Lewinter & Widulski 1. The Quadratic Formula

Chapter 13. Overview. The Quadratic Formula. Overview. The Quadratic Formula. The Quadratic Formula. Lewinter & Widulski 1. The Quadratic Formula Chapter 13 Overview Some More Math Before You Go The Quadratic Formula The iscriminant Multiplication of Binomials F.O.I.L. Factoring Zero factor propert Graphing Parabolas The Ais of Smmetr, Verte and

More information

Math 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy

Math 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy Math 4 Spring 08 problem set. (a) Consider these two first order equations. (I) d d = + d (II) d = Below are four direction fields. Match the differential equations above to their direction fields. Provide

More information

2.5. Infinite Limits and Vertical Asymptotes. Infinite Limits

2.5. Infinite Limits and Vertical Asymptotes. Infinite Limits . Infinite Limits and Vertical Asmptotes. Infinite Limits and Vertical Asmptotes In this section we etend the concept of it to infinite its, which are not its as before, but rather an entirel new use of

More information

Vectors and the Geometry of Space

Vectors and the Geometry of Space Chapter 12 Vectors and the Geometr of Space Comments. What does multivariable mean in the name Multivariable Calculus? It means we stud functions that involve more than one variable in either the input

More information

Noncommuting Rotation and Angular Momentum Operators

Noncommuting Rotation and Angular Momentum Operators Noncommuting Rotation and Angular Momentum Operators Originall appeared at: http://behindtheguesses.blogspot.com/2009/08/noncommuting-rotation-and-angular.html Eli Lanse elanse@gmail.com August 31, 2009

More information

Higher. Integration 1

Higher. Integration 1 Higher Mathematics Contents Indefinite Integrals RC Preparing to Integrate RC Differential Equations A Definite Integrals RC 7 Geometric Interpretation of A 8 Areas between Curves A 7 Integrating along

More information

University of Alabama Department of Physics and Astronomy. PH 125 / LeClair Spring A Short Math Guide. Cartesian (x, y) Polar (r, θ)

University of Alabama Department of Physics and Astronomy. PH 125 / LeClair Spring A Short Math Guide. Cartesian (x, y) Polar (r, θ) University of Alabama Department of Physics and Astronomy PH 125 / LeClair Spring 2009 A Short Math Guide 1 Definition of coordinates Relationship between 2D cartesian (, y) and polar (r, θ) coordinates.

More information

LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

More information

INTRODUCTION TO DIOPHANTINE EQUATIONS

INTRODUCTION TO DIOPHANTINE EQUATIONS INTRODUCTION TO DIOPHANTINE EQUATIONS In the earl 20th centur, Thue made an important breakthrough in the stud of diophantine equations. His proof is one of the first eamples of the polnomial method. His

More information

2 4πε ( ) ( r θ. , symmetric about the x-axis, as shown in Figure What is the electric field E at the origin O?

2 4πε ( ) ( r θ. , symmetric about the x-axis, as shown in Figure What is the electric field E at the origin O? p E( r, θ) = cosθ 3 ( sinθ ˆi + cosθ ˆj ) + sinθ cosθ ˆi + ( cos θ 1) ˆj r ( ) ( p = cosθ sinθ ˆi + cosθ ˆj + sinθ cosθ ˆi sinθ ˆj 3 r where the trigonometric identit ( θ ) vectors ˆr and cos 1 = sin θ

More information

tan t = y x, x Z 0 sin u 2 = ; 1 - cos u cos u 2 = ; 1 + cos u tan u 2 = 1 - cos u cos a cos b = 1 2 sin a cos b = 1 2

tan t = y x, x Z 0 sin u 2 = ; 1 - cos u cos u 2 = ; 1 + cos u tan u 2 = 1 - cos u cos a cos b = 1 2 sin a cos b = 1 2 TRIGONOMETRIC FUNCTIONS Let t be a real number and let P =, be the point on the unit circle that corresponds to t. sin t = cos t = tan t =, Z 0 csc t =, Z 0 sec t =, Z 0 cot t =. Z 0 P (, ) t s t units

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define

More information

MATH Line integrals III Fall The fundamental theorem of line integrals. In general C

MATH Line integrals III Fall The fundamental theorem of line integrals. In general C MATH 255 Line integrals III Fall 216 In general 1. The fundamental theorem of line integrals v T ds depends on the curve between the starting point and the ending point. onsider two was to get from (1,

More information

8 Differential Calculus 1 Introduction

8 Differential Calculus 1 Introduction 8 Differential Calculus Introduction The ideas that are the basis for calculus have been with us for a ver long time. Between 5 BC and 5 BC, Greek mathematicians were working on problems that would find

More information

Differentiation and applications

Differentiation and applications FS O PA G E PR O U N C O R R EC TE D Differentiation and applications. Kick off with CAS. Limits, continuit and differentiabilit. Derivatives of power functions.4 C oordinate geometr applications of differentiation.5

More information

Practice Questions for Midterm 2 - Math 1060Q - Fall 2013

Practice Questions for Midterm 2 - Math 1060Q - Fall 2013 Eam Review Practice Questions for Midterm - Math 060Q - Fall 0 The following is a selection of problems to help prepare ou for the second midterm eam. Please note the following: anthing from Module/Chapter

More information

LESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

More information

PHYS-2010: General Physics I Course Lecture Notes Section IV

PHYS-2010: General Physics I Course Lecture Notes Section IV PHYS-010: General Phsics I Course Lecture Notes Section IV Dr. Donald G. Luttermoser East Tennessee State Universit Edition.3 Abstract These class notes are designed for use of the instructor and students

More information

5. Zeros. We deduce that the graph crosses the x-axis at the points x = 0, 1, 2 and 4, and nowhere else. And that s exactly what we see in the graph.

5. Zeros. We deduce that the graph crosses the x-axis at the points x = 0, 1, 2 and 4, and nowhere else. And that s exactly what we see in the graph. . Zeros Eample 1. At the right we have drawn the graph of the polnomial = ( 1) ( 2) ( 4). Argue that the form of the algebraic formula allows ou to see right awa where the graph is above the -ais, where

More information

Pre-AP Algebra 2 Lesson 1-1 Basics of Functions

Pre-AP Algebra 2 Lesson 1-1 Basics of Functions Lesson 1-1 Basics of Functions Objectives: The students will be able to represent functions verball, numericall, smbolicall, and graphicall. The students will be able to determine if a relation is a function

More information

ENR202 Mechanics of Materials Lecture 12B Slides and Notes

ENR202 Mechanics of Materials Lecture 12B Slides and Notes ENR0 Mechanics of Materials Lecture 1B Slides and Notes Slide 1 Copright Notice Do not remove this notice. COMMMONWEALTH OF AUSTRALIA Copright Regulations 1969 WARNING This material has been produced and

More information