# Notes for Geometry Conic Sections

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Nots for Gomtry Conic Sctions Th nots is takn from Gomtry, by David A. Brannan, Matthw F. Espln and Jrmy J. Gray, 2nd dition 1 Conic Sctions A conic sction is dfind as th curv of intrsction of a doubl con with a plan. Figur: Exampls: 1. non-dgnrat conic sctions: parabolas, llipss or hyprbolas; 2. dgnrat conic sctions: th singl point, singl lin and pair of lins. 1.1 Focus-Dirctrix Dfinition of th Non-Dgnrat Conics Th 3 non-dgnrat conics can b dfind as th st of points P in th plan satisfying: Th distanc of P from a fixd point F (calld th focus of th conic) is a constant multipl (calld its ccntricity, ) of th distanc of P from a fixd lin d (calld its dirctrix). Eccntricity: A non-dgnrat conic is 1. an llips if 0 < 1, 2. a parabola if = 1, 3. a hyprbola if > 1. 1

2 Parabola ( = 1): A parabola is dfind to b th st of points P in th plan whos distanc from a fixd point F is qual to thir distanc from a fixd lin d. W now driv a parabola in standard form: Lt F (a, 0) b th focus and d : x = a b th dirctrix. Lt P (x, y) b an arbitrary point on th parabola and lt M( a, y) b th foot of th prpndicular from P to th dirctrix. Figur: Sinc F P = P M, by th dfinition of th parabola, this follows that F P 2 = P M 2 (x a) 2 + y 2 = (x + a) 2 y 2 = 4ax. Th point (at 2, 2at), t R lis on th parabola. (2at) 2 = 4a at 2. Convrsly, w can writ th coordinats of ach point on th parabola in th form (at 2, 2at). For if w choos t = y y2, thn y = 2at and x = = (2at)2 = at 2. It follows 2a 4a 4a that thr is a on-to-on corrspondnc btwn th ral numbrs t and th points of th parabola. Parabola in standard form A parabola in standard form has quation y 2 = 4ax, whr a > 0. It has focus (a, 0) and dirctrix x = a; and it can b dscribd by th paramtric quations: x = at 2, y = 2at (t R). W call th x-axis th axis of th parabola in standard form, sinc th parabola is symmtric with rspct to this lin. W call th origin th vrtx of a parabola in standard form, sinc it is th point of th intrsction of th axis of th parabola with th parabola. A parabola has no cntr. 2

3 Exampl 1.1. Writ down th focus, vrtx, axis and dirctrix of th parabola E with quation y 2 = 2x. Solution. Focus: F = ( 1 2, 0), Axis: x-axis, Vrtx: (0, 0), Dirctrix: x = 1 2. Ellips (0 < 1): W dfin an llips with ccntricity zro to b a circl. W dfin an llips with ccntricity (whr 0 < < 1) to b th st of points P in th plan whos distanc from a fixd point F is tims thir distanc from a fixd lin. W now driv an llips in standard form: Lt F (a, 0), a > 0 b th focus and d : x = a. Lt P (x, y) b an arbitrary point on th parabola and lt M( a, y) b th foot of th prpndicular from P to th dirctrix. Figur: Sinc F P = P M, by th dfinition of th llips, this follows that F P 2 = 2 P M 2 (x a) 2 + y 2 = 2 (x a )2 x2 a 2 + y 2 a 2 (1 2 ) = 1. Lt b = a 1 2. Thn a + y2 2 b = 1. 2 Th quation is symmtrical in x and y. Th llips also has a scond focus F ( a, 0) and a scond dirctrix d : x = a. W call th sgmnt joining th points (±a, 0) th major axis of th llips and th sgmnt joining th points (0, ±b) th minor axis of th llips. b < a, th minor axis is shortr than th major axis. Th origin is th cntr of this llips. Not that (a cos t, b sin t) lis on th llips. (a cos t)2 (b sin t)2 + = cos 2 t + sin 2 t = 1. a 2 b 2 3

4 W can chck that x = a cos t, y = b sin t, t ( π, π] givs a paramtric rprsntation of th llips. Ellips in standard form An llips in standard form has quation a 2 + y2 b 2 = 1, whr a b > 0, b2 = a 2 (1 2 ), 0 < 1. It can b dscribd by th paramtric quations x = a cos t, y = b sin t, t ( π, π]. If > 0, it has foci (±a, 0) and dirctrix x = ± a. Exrcis 1.2. (x + a)2 + y 2 + (x a) 2 + y 2 = 2a. Hyprbola ( > 1): A hyprbola is th st of points P in th plan whos distanc from a fixd point F is tims thir distanc from a fixd lin d, whr > 1. W now driv a hyprbola in standard form: Lt F (a, 0), a > 0 b th focus and d : x = a. Lt P (x, y) b an arbitrary point on th parabola and lt M( a, y) b th foot of th prpndicular from P to th dirctrix. Figur: Sinc F P = P M, by th dfinition of th hyprbola, this follows that F P 2 = 2 P M 2 (x a) 2 + y 2 = 2 (x a )2 x2 a 2 y 2 a 2 ( 2 1) = 1. Lt b = a 2 1. Thn a 2 y2 b 2 = 1. 4

5 Th quation is symmtrical in x and y. Th llips also has a scond focus F ( a, 0) and a scond dirctrix d : x = a. W call th sgmnt joining th points (±a, 0) th major (transvrs) axis of th hyprbola and th sgmnt joining th points (0, ±b) th minor (conjugat) axis of th hyprbola. Th origin is th cntr of this hyprbola. W can chck that x = sc t, y = b tan t, t ( π, π) ( π, 3π ) givs a paramtric rprsntation of th hyprbola. Th hyprbola consist of 2 branchs. Whn x ±, th branchs gt closr and closr to th lins y = ± b x th asymptots of th a hyprbola. Hyprbola in standard form A hyprbola in standard form has quation a 2 y2 b 2 = 1, whr a b > 0, b2 = a 2 ( 2 1), > 1. it has foci (±a, 0) and dirctrix x = ± a. It can b dscribd by th paramtric quations x = a sc t, y = b tan t, t ( π 2, π 2 ) (π 2, 3π 2 ). Exampl 1.3. Dtrmin th foci F and F of th hyprbola E with quation 2y 2 = 1. Solution. ( ± ) 3/2, 0. Exrcis 1.4. (x + a)2 + y 2 (x a) 2 + y 2 = ±2a. 5

### Examples. Epipoles. Epipolar geometry and the fundamental matrix

Epipoar gomtry and th fundamnta matrix Epipoar ins Lt b a point in P 3. Lt x and x b its mapping in two imags through th camra cntrs C and C. Th point, th camra cntrs C and C and th (3D points corrspon

### New Basis Functions. Section 8. Complex Fourier Series

Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting

### Epipolar Geometry and the Fundamental Matrix

9 Epipolar Gomtry and th Fundamntal Matrix Th pipolar gomtry is th intrinsic projctiv gomtry btwn two viws. It is indpndnt of scn structur, and only dpnds on th camras intrnal paramtrs and rlativ pos.

### The Matrix Exponential

Th Matrix Exponntial (with xrciss) 92.222 - Linar Algbra II - Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial

### ME 612 Metal Forming and Theory of Plasticity. 6. Strain

Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.

### Question 3: How do you find the relative extrema of a function?

ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

### Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails

### Non-Homogeneous Systems, Euler s Method, and Exponential Matrix

Non-Homognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous first-ordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach

### AP Calculus AB 2008 Scoring Guidelines

AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.

### CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions

CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:

### A Note on Approximating. the Normal Distribution Function

Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and

### AP Calculus Multiple-Choice Question Collection 1969 1998. connect to college success www.collegeboard.com

AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos

### CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.

Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by

### Statistical Machine Translation

Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm

### The Normal Distribution: A derivation from basic principles

Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn

### Traffic Flow Analysis (2)

Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,

### The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.

Rsourc Allocation Abstract This is a small toy xampl which is wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl

### 5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This

### by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia

Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs

### Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective

Ovrviw Lctur 4 Projctions - 3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl - Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog

### SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* Rostov-on-Don. Russia

SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* Rostov-on-Don. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs

### Lecture 20: Emitter Follower and Differential Amplifiers

Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.

### Upper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing

Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif

### SPECIAL VOWEL SOUNDS

SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)

### Principles of Humidity Dalton s law

Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

### [ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)

MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (volts-sc/rad Motor torqu constant (lb-in/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris

### QUANTITATIVE METHODS CLASSES WEEK SEVEN

QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.

### Foreign Exchange Markets and Exchange Rates

Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls

### Section 7.4: Exponential Growth and Decay

1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart

### Introduction to Finite Element Modeling

Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation

### Unit 10: Quadratic Relations

Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand

### Algebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2

1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3-dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s1660-015-0741-1 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans Wi-Mao Qian

### Subatomic Physics: Particle Physics Study Guide

Subatomic Physics: Particl Physics Study Guid This is a uid of what to rvis for th xam. Th othr matrial w covrd in th cours may appar in ustions but it will always b providd if ruird. Rmmbr that, in an

### Lecture 3: Diffusion: Fick s first law

Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th

### Parametric Curves. EXAMPLE: Sketch and identify the curve defined by the parametric equations

Section 9. Parametric Curves 00 Kiryl Tsishchanka Parametric Curves Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t), y = g(t) (called

### Function Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015

Harold s s Cheat Sheet 8 December 05 Algebra Constant Linear Identity f(x) c f(x) x Range: [c, c] Undefined (asymptote) Restrictions: c is a real number Ay + B 0 g(x) x Restrictions: m 0 General Fms: Ax

### Architecture of the proposed standard

Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th

### Astromechanics Two-Body Problem (Cont)

5. Orbit Characteristics Astromechanics Two-Body Problem (Cont) We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the

### 5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power

Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Thursday, January 29, 2004 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and

### Basis risk. When speaking about forward or futures contracts, basis risk is the market

Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also

### Controversies about the value of the third cosmic velocity

.478/v46--9- A N N A L E S U N I V E R S I T A T I S A R I A E C U R I E - S K Ł O D O W S K A L U B L I N P O L O N I A VOL. LXVIII SECTIO AAA Controvrsis about th valu of th third cosic vlocity STANISŁAW

### Parametric Equations and the Parabola (Extension 1)

Parametric Equations and the Parabola (Extension 1) Parametric Equations Parametric equations are a set of equations in terms of a parameter that represent a relation. Each value of the parameter, when

### 55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

### Financial Mathematics

Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution

### CHAPTER 4c. ROOTS OF EQUATIONS

CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### 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

### Systems of Equations and Inequalities

CHAPTER CONNECTIONS Systms of Equations and Inqualitis CHAPTER OUTLINE.1 Linar Systms in Two Variabls with Applications 04. Linar Systms in Thr Variabls with Applications 16.3 Nonlinar Systms of Equations

### Quantum Graphs I. Some Basic Structures

Quantum Graphs I. Som Basic Structurs Ptr Kuchmnt Dpartmnt of Mathmatics Txas A& M Univrsity Collg Station, TX, USA 1 Introduction W us th nam quantum graph for a graph considrd as a on-dimnsional singular

### This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2.

INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

### Chapter 3: Capacitors, Inductors, and Complex Impedance

haptr 3: apacitors, Inductors, and omplx Impdanc In this chaptr w introduc th concpt of complx rsistanc, or impdanc, by studying two ractiv circuit lmnts, th capacitor and th inductor. W will study capacitors

### AC Circuits Three-Phase Circuits

AC Circuits Thr-Phs Circuits Contnts Wht is Thr-Phs Circuit? Blnc Thr-Phs oltgs Blnc Thr-Phs Connction Powr in Blncd Systm Unblncd Thr-Phs Systms Aliction Rsidntil Wiring Sinusoidl voltg sourcs A siml

### http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force

ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd

### GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

### Engineering Math II Spring 2015 Solutions for Class Activity #2

Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the x-axis. Then find

### Section 10.5 Rotation of Axes; General Form of a Conic

Section 10.5 Rotation of Axes; General Form of a Conic 8 Objective 1: Identifying a Non-rotated Conic. The graph of the equation Ax + Bxy + Cy + Dx + Ey + F = 0 where A, B, and C cannot all be zero is

### EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS

25 Vol. 3 () January-March, pp.37-5/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut

### June 2012. Enprise Rent. Enprise 1.1.6. Author: Document Version: Product: Product Version: SAP Version: 8.81.100 8.8

Jun 22 Enpris Rnt Author: Documnt Vrsion: Product: Product Vrsion: SAP Vrsion: Enpris Enpris Rnt 88 88 Enpris Rnt 22 Enpris Solutions All rights rsrvd No parts of this work may b rproducd in any form or

### a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

### Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,

### Axial flow rate (per unit circumferential length) [m 2 /s] R B, R J =R Bearing Radius ~ Journal Radius [m] S

NOTE 4 TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING Lctur 4 introducs th fundamnts of journal baring analysis. Th long and short lngth baring modls ar introducd. Th prssur fild in a short lngth baring

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

### Problem Solving Session 1: Electric Dipoles and Torque

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Dpatmnt of Physics 8.02 Poblm Solving Sssion 1: Elctic Dipols and Toqu Sction Tabl (if applicabl) Goup Mmbs Intoduction: In th fist poblm you will lan to apply Coulomb

### HOMEWORK FOR UNIT 5-1: FORCE AND MOTION

Nam Dat Partnrs HOMEWORK FOR UNIT 51: FORCE AND MOTION 1. You ar givn tn idntial springs. Dsrib how you would dvlop a sal of for (i., a mans of produing rpatabl fors of a varity of sizs) using ths springs.

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Imagine a car is traveling along the highway and you look down at the situation from high above: highway

Chapter 22 Parametric Equations Imagine a car is traveling along the highway you look down at the situation from high above highway curve (static) place car moving point (dynamic) Figure 22.1 The dynamic

### 3.1. Quadratic Equations and Models. Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models

3.1 Quadratic Equations and Models Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models 3.1-1 Polynomial Function A polynomial function of degree n, where n

### Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental

Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr zvilo@cs.bgu.ac.il Dpt. of Comm. Systms Enginring Bn Gurion Univrsity Br Shva Isral Boaz Patt-Shamir Dror Rawitz {boaz, rawitz}@ng.tau.ac.il

### Factorials! Stirling s formula

Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical

### Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Graphing Quadratic Functions

Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the

### 4.6 Transformation Between Geographic and UTM Coordinates

. ransfrmatin Btwn Ggraphic and UM rdinats.. nvrsin frm Ggraphic t UM rdinats Usd fr cnvrting and n an llipsid f knwn f and a, t UM crdinats. Ngativ valus ar usd fr wstrn lngituds. hs quatins ar accurat

### Graphs of Polar Equations

Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate

5.1 The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives The Unit Circle Terminal Points on the Unit Circle The Reference Number 2 The Unit Circle In this section we explore some

### Two vectors are equal if they have the same length and direction. They do not

Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

### On Resilience of Multicommodity Dynamical Flow Networks

On Rsilinc of Multicommodity Dynamical Flow Ntworks Gusta Nilsson, Giacomo Como, and Enrico Loisari bstract Dynamical flow ntworks with htrognous routing ar analyzd in trms of stability and rsilinc to

### Making and Using the Hertzsprung - Russell Diagram

Making and Using th Hrtzsprung - Russll Diagram Nam In astronomy th Hrtzsprung-Russll Diagram is on of th main ways that w organiz data dscribing how stars volv, ags of star clustrs, masss of stars tc.

### SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY

1 SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY ALEXA Vasil ABSTRACT Th prsnt papr has as targt to crat a programm in th Matlab ara, in ordr to solv, didactically

### Unit 9: Conic Sections Name Per. Test Part 1

Unit 9: Conic Sections Name Per 1/6 HOLIDAY 1/7 General Vocab Intro to Conics Circles 1/8-9 More Circles Ellipses 1/10 Hyperbolas (*)Pre AP Only 1/13 Parabolas HW: Part 4 HW: Part 1 1/14 Identifying conics

### Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY

Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl

### 42 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2. 1.6.1 Brief review of Conic Sections

2 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more sense. You

### All points on the coordinate plane are described with reference to the origin. What is the origin, and what are its coordinates?

Classwork Example 1: Extending the Axes Beyond Zero The point below represents zero on the number line. Draw a number line to the right starting at zero. Then, follow directions as provided by the teacher.

### Electronic Commerce. and. Competitive First-Degree Price Discrimination

Elctronic Commrc and Comptitiv First-Dgr Pric Discrimination David Ulph* and Nir Vulkan ** Fbruary 000 * ESRC Cntr for Economic arning and Social Evolution (ESE), Dpartmnt of Economics, Univrsity Collg

### APPROXIMATION OF QUANTUM GRAPH VERTEX COUPLINGS BY SCALED SCHRÖDINGER OPERATORS ON THIN BRANCHED MANIFOLDS

APPROXIMATION OF QUANTUM GRAPH VERTEX COUPLINGS BY SCALED SCHRÖDINGER OPERATORS ON THIN BRANCHED MANIFOLDS PAVEL EXNER AND OLAF POST Abstract. W discuss approximations of vrtx couplings of quantum graphs

### Parallel and Distributed Programming. Performance Metrics

Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:

### MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

### Simulation of the electric field generated by a brown ghost knife fish

C H A P T R 2 7 Simulation of th lctric fild gnratd by a brown ghost knif fish lctric fild CONCPTS 27.1 Th fild modl 27.2 lctric fild diagrams 27.3 Suprposition of lctric filds 27.4 lctric filds and forcs

### Hardware Modules of the RSA Algorithm

SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 11, No. 1, Fbruary 2014, 121-131 UDC: 004.3`142:621.394.14 DOI: 10.2298/SJEE140114011S Hardwar Moduls of th RSA Algorithm Vlibor Škobić 1, Branko Dokić 1,

### TI-83 Plus Conic Graphing

TI TI-83 Plus Conic Graphing Getting Started Start here How To Start and Quit Conic Graphing Use Conic Window and Conic Zoom Graph and Trace a Conic Section Examples Graphing a Circle Graphing an Ellipse