# Chapter Direct Method of Interpolation

Size: px
Start display at page:

Transcription

1 Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o find deriies nd inegrls of discree funcions. Wh is inerpolion? Mny imes, d is gien only discree poins such s (, y,, y, n yn, (, y n n. So, how hen does one find he lue of y ny oher lue of? Well, coninuous funcion f ( my be used o represen he n + d lues wih f ( pssing hrough he n + poins (Figure. Then one cn find he lue of y ny oher lue of. This is clled inerpolion. Of course, if flls ouside he rnge of for which he d is gien, i is no longer inerpolion bu insed is clled erpolion. So wh kind of funcion f ( should one choose? A polynomil is common choice for n inerpoling funcion becuse polynomils re esy o (A elue, (B differenie, nd (C inegre relie o oher choices such s rigonomeric nd eponenil series. Polynomil inerpolion inoles finding polynomil of order n h psses hrough he n + poins. One of he mehods of inerpolion is clled he direc mehod. Oher mehods include Newon s diided difference polynomil mehod nd he Lgrngin inerpolion mehod. We will discuss he direc mehod in his chper. (,..., ( 5..

2 5.. Chper 5. y (, y (, y (, y (, y Figure Inerpolion of discree d. f ( Direc Mehod The direc mehod of inerpolion is bsed on he following premise. Gien n + d poins, fi polynomil of order n s gien below n y n ( hrough he d, where,,..., n re n + rel consns. Since n + lues of y re gien n + lues of, one cn wrie n + equions. Then he n + consns,,,..., n cn be found by soling he n + simulneous liner equions. To find he lue of y gien lue of, simply subsiue he lue of in Equion. Bu, i is no necessry o use ll he d poins. How does one hen choose he order of he polynomil nd wh d poins o use? This concep nd he direc mehod of inerpolion re bes illusred using emples. Emple The upwrd elociy of rocke is gien s funcion of ime in Tble. Tble Velociy s funcion of ime. (s ( (m/s

3 Direc Mehod of Inerpolion 5.. Figure Grph of elociy s. ime d for he rocke emple. Deermine he lue of he elociy 6 seconds using he direc mehod of inerpolion nd firs order polynomil. Soluion For firs order polynomil inerpolion (lso clled liner inerpolion, he elociy gien by y ( + (, y f ( (, y Figure Liner inerpolion.

4 5..4 Chper 5. Since we wn o find he elociy 6, nd we re using firs order polynomil, we need o choose he wo d poins h re closes o 6 h lso brcke 6 o elue i. The wo poins re 5 nd. Then 5, ( 6. 78, ( gies ( 5 + ( ( + ( Wriing he equions in mri form, we he Soling he boe wo equions gies.9.94 Hence ( , 5 A 6, ( m/s Emple The upwrd elociy of rocke is gien s funcion of ime in Tble. Tble Velociy s funcion of ime. (s ( (m/s Deermine he lue of he elociy 6 seconds using he direc mehod of inerpolion nd second order polynomil. Soluion For second order polynomil inerpolion (lso clled qudric inerpolion, he elociy is gien by + + (

5 Direc Mehod of Inerpolion 5..5 y (, y (, y f ( (, y Figure 4 Qudric inerpolion. Since we wn o find he elociy 6, nd we re using second order polynomil, we need o choose he hree d poins h re closes o 6 h lso brcke 6 o elue i. The hree poins re, 5, nd. Then, ( , ( 6. 78, ( gies ( ( ( 4 ( 5 + ( 5 + ( ( + ( + ( Wriing he hree equions in mri form, we he Soling he boe hree equions gies Hence , A 6, ( ( ( m/s (

6 5..6 Chper 5. The bsolue relie pproime error second order polynomil is % obined beween he resuls from he firs nd Emple The upwrd elociy of rocke is gien s funcion of ime in Tble. Tble Velociy s funcion of ime. (s ( (m/s Deermine he lue of he elociy 6 seconds using he direc mehod of inerpolion nd hird order polynomil. b Find he bsolue relie pproime error for he hird order polynomil pproimion. c Using he hird order polynomil inerpoln for elociy from pr (, find he disnce coered by he rocke from s o 6s. d Using he hird order polynomil inerpoln for elociy from pr (, find he ccelerion of he rocke 6s. Soluion For hird order polynomil inerpolion (lso clled cubic inerpolion, we choose he elociy gien by (

7 Direc Mehod of Inerpolion 5..7 y (, y (, y (, y (, y Figure 5 Cubic inerpolion. f ( Since we wn o find he elociy 6, nd we re using hird order polynomil, we need o choose he four d poins closes o 6 h lso brcke 6 o elue i. The four poins re, 5, nd. 5. Then, ( , ( 6. 78, ( , ( gies ( ( ( ( 4 ( 5 + ( 5 + ( 5 + ( ( + ( + ( + ( (.5 + (.5 + (.5 + ( Wriing he four equions in mri form, we he Soling he boe four equions gies

8 5..8 Chper Hence ( ,.5 ( ( 6 +.4( ( m/s b The bsolue percenge relie pproime error for he lue obined for (6 beween second nd hird order polynomil is % c The disnce coered by he rocke beween s nd 6s cn be clculed from he inerpoling polynomil ( ,. 5 Noe h he polynomil is lid beween nd. 5 nd hence includes he limis of inegrion of nd 6. So 6 ( s( ( s 6 d 6 ( d m d The ccelerion 6 is gien by d ( 6 ( d 6 Gien h ( ,. 5 d ( ( d d ( d ,.5 ( ( ( m/s 6

9 Direc Mehod of Inerpolion 5..9 INTERPOLATION Topic Direc Mehod of Inerpolion Summry Tebook noes on he direc mehod of inerpolion. Mjor Generl Engineering Auhors Aur Kw, Peer Wrr, Michel Keels De June 7, Web Sie hp://numericlmehods.eng.usf.edu

### ENGR 1990 Engineering Mathematics The Integral of a Function as a Function

ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under

### Chapter 2. Motion along a straight line. 9/9/2015 Physics 218

Chper Moion long srigh line 9/9/05 Physics 8 Gols for Chper How o describe srigh line moion in erms of displcemen nd erge elociy. The mening of insnneous elociy nd speed. Aerge elociy/insnneous elociy

### (b) 10 yr. (b) 13 m. 1.6 m s, m s m s (c) 13.1 s. 32. (a) 20.0 s (b) No, the minimum distance to stop = 1.00 km. 1.

Answers o Een Numbered Problems Chper. () 7 m s, 6 m s (b) 8 5 yr 4.. m ih 6. () 5. m s (b).5 m s (c).5 m s (d) 3.33 m s (e) 8. ().3 min (b) 64 mi..3 h. ().3 s (b) 3 m 4..8 mi wes of he flgpole 6. (b)

### A 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m

PHYS : Soluions o Chper 3 Home Work. SSM REASONING The displcemen is ecor drwn from he iniil posiion o he finl posiion. The mgniude of he displcemen is he shores disnce beween he posiions. Noe h i is onl

### Average & instantaneous velocity and acceleration Motion with constant acceleration

Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission

### Physics 2A HW #3 Solutions

Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen

### e t dt e t dt = lim e t dt T (1 e T ) = 1

Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie

### Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl

### Physic 231 Lecture 4. Mi it ftd l t. Main points of today s lecture: Example: addition of velocities Trajectories of objects in 2 = =

Mi i fd l Phsic 3 Lecure 4 Min poins of od s lecure: Emple: ddiion of elociies Trjecories of objecs in dimensions: dimensions: g 9.8m/s downwrds ( ) g o g g Emple: A foobll pler runs he pern gien in he

### September 20 Homework Solutions

College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum

### 5.1-The Initial-Value Problems For Ordinary Differential Equations

5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil

### 3 Motion with constant acceleration: Linear and projectile motion

3 Moion wih consn ccelerion: Liner nd projecile moion cons, In he precedin Lecure we he considered moion wih consn ccelerion lon he is: Noe h,, cn be posiie nd neie h leds o rie of behiors. Clerl similr

### A Kalman filtering simulation

A Klmn filering simulion The performnce of Klmn filering hs been esed on he bsis of wo differen dynmicl models, ssuming eiher moion wih consn elociy or wih consn ccelerion. The former is epeced o beer

### PHYSICS 1210 Exam 1 University of Wyoming 14 February points

PHYSICS 1210 Em 1 Uniersiy of Wyoming 14 Februry 2013 150 poins This es is open-noe nd closed-book. Clculors re permied bu compuers re no. No collborion, consulion, or communicion wih oher people (oher

### Motion in a Straight Line

Moion in Srigh Line. Preei reched he mero sion nd found h he esclor ws no working. She wlked up he sionry esclor in ime. On oher dys, if she remins sionry on he moing esclor, hen he esclor kes her up in

### MAT 266 Calculus for Engineers II Notes on Chapter 6 Professor: John Quigg Semester: spring 2017

MAT 66 Clculus for Engineers II Noes on Chper 6 Professor: John Quigg Semeser: spring 7 Secion 6.: Inegrion by prs The Produc Rule is d d f()g() = f()g () + f ()g() Tking indefinie inegrls gives [f()g

### Name: Per: L o s A l t o s H i g h S c h o o l. Physics Unit 1 Workbook. 1D Kinematics. Mr. Randall Room 705

Nme: Per: L o s A l o s H i g h S c h o o l Physics Uni 1 Workbook 1D Kinemics Mr. Rndll Room 705 Adm.Rndll@ml.ne www.laphysics.com Uni 1 - Objecies Te: Physics 6 h Ediion Cunel & Johnson The objecies

### INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).

INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely

### f t f a f x dx By Lin McMullin f x dx= f b f a. 2

Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes

### Phys 110. Answers to even numbered problems on Midterm Map

Phys Answers o een numbered problems on Miderm Mp. REASONING The word per indices rio, so.35 mm per dy mens.35 mm/d, which is o be epressed s re in f/cenury. These unis differ from he gien unis in boh

### Properties of Logarithms. Solving Exponential and Logarithmic Equations. Properties of Logarithms. Properties of Logarithms. ( x)

Properies of Logrihms Solving Eponenil nd Logrihmic Equions Properies of Logrihms Produc Rule ( ) log mn = log m + log n ( ) log = log + log Properies of Logrihms Quoien Rule log m = logm logn n log7 =

### Contraction Mapping Principle Approach to Differential Equations

epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of

### Mathematics 805 Final Examination Answers

. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se

### CHAPTER 2 KINEMATICS IN ONE DIMENSION ANSWERS TO FOCUS ON CONCEPTS QUESTIONS

Physics h Ediion Cunell Johnson Young Sdler Soluions Mnul Soluions Mnul, Answer keys, Insrucor's Resource Mnul for ll chpers re included. Compleed downlod links: hps://esbnkre.com/downlod/physics-h-ediion-soluions-mnulcunell-johnson-young-sdler/

### 1. Consider a PSA initially at rest in the beginning of the left-hand end of a long ISS corridor. Assume xo = 0 on the left end of the ISS corridor.

In Eercise 1, use sndrd recngulr Cresin coordine sysem. Le ime be represened long he horizonl is. Assume ll ccelerions nd decelerions re consn. 1. Consider PSA iniilly res in he beginning of he lef-hnd

### FM Applications of Integration 1.Centroid of Area

FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is

### Chapter 2 PROBLEM SOLUTIONS

Chper PROBLEM SOLUTIONS. We ssume h you re pproximely m ll nd h he nere impulse rels uniform speed. The elpsed ime is hen Δ x m Δ = m s s. s.3 Disnces reled beween pirs of ciies re ( ) Δx = Δ = 8. km h.5

### 2k 1. . And when n is odd number, ) The conclusion is when n is even number, an. ( 1) ( 2 1) ( k 0,1,2 L )

Scholrs Journl of Engineering d Technology SJET) Sch. J. Eng. Tech., ; A):8-6 Scholrs Acdemic d Scienific Publisher An Inernionl Publisher for Acdemic d Scienific Resources) www.sspublisher.com ISSN -X

### IX.2 THE FOURIER TRANSFORM

Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 7 IX. THE FOURIER TRANSFORM IX.. The Fourier Trnsform Definiion 7 IX.. Properies 73 IX..3 Emples 74 IX..4 Soluion of ODE 76 IX..5

### 4.8 Improper Integrals

4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls

### Lagrangian Interpolation

Lagrangian Inerpolaion Maor: All Engineering Maors Auhors: Auar Kaw, Jai Paul hp://numericalmehods.eng.usf.edu Transforming Numerical Mehods Educaion for STEM Undergraduaes hp://numericalmehods.eng.usf.edu

### A Simple Method to Solve Quartic Equations. Key words: Polynomials, Quartics, Equations of the Fourth Degree INTRODUCTION

Ausrlin Journl of Bsic nd Applied Sciences, 6(6): -6, 0 ISSN 99-878 A Simple Mehod o Solve Quric Equions Amir Fhi, Poo Mobdersn, Rhim Fhi Deprmen of Elecricl Engineering, Urmi brnch, Islmic Ad Universi,

### On the Pseudo-Spectral Method of Solving Linear Ordinary Differential Equations

Journl of Mhemics nd Sisics 5 ():136-14, 9 ISS 1549-3644 9 Science Publicions On he Pseudo-Specrl Mehod of Solving Liner Ordinry Differenil Equions B.S. Ogundre Deprmen of Pure nd Applied Mhemics, Universiy

### 0 for t < 0 1 for t > 0

8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside

### MTH 146 Class 11 Notes

8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he

### 2/5/2012 9:01 AM. Chapter 11. Kinematics of Particles. Dr. Mohammad Abuhaiba, P.E.

/5/1 9:1 AM Chper 11 Kinemic of Pricle 1 /5/1 9:1 AM Inroducion Mechnic Mechnic i Th cience which decribe nd predic he condiion of re or moion of bodie under he cion of force I i diided ino hree pr 1.

### IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704

Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 6, 8 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion

### The solution is often represented as a vector: 2xI + 4X2 + 2X3 + 4X4 + 2X5 = 4 2xI + 4X2 + 3X3 + 3X4 + 3X5 = 4. 3xI + 6X2 + 6X3 + 3X4 + 6X5 = 6.

[~ o o :- o o ill] i 1. Mrices, Vecors, nd Guss-Jordn Eliminion 1 x y = = - z= The soluion is ofen represened s vecor: n his exmple, he process of eliminion works very smoohly. We cn elimine ll enries

### ME 141. Engineering Mechanics

ME 141 Engineeing Mechnics Lecue 13: Kinemics of igid bodies hmd Shhedi Shkil Lecue, ep. of Mechnicl Engg, UET E-mil: sshkil@me.bue.c.bd, shkil6791@gmil.com Websie: eche.bue.c.bd/sshkil Couesy: Veco Mechnics

### 2D Motion WS. A horizontally launched projectile s initial vertical velocity is zero. Solve the following problems with this information.

Nme D Moion WS The equions of moion h rele o projeciles were discussed in he Projecile Moion Anlsis Acii. ou found h projecile moes wih consn eloci in he horizonl direcion nd consn ccelerion in he ericl

### Introduction to LoggerPro

Inroducion o LoggerPro Sr/Sop collecion Define zero Se d collecion prmeers Auoscle D Browser Open file Sensor seup window To sr d collecion, click he green Collec buon on he ool br. There is dely of second

### Lecture 3: 1-D Kinematics. This Week s Announcements: Class Webpage: visit regularly

Lecure 3: 1-D Kinemics This Week s Announcemens: Clss Webpge: hp://kesrel.nm.edu/~dmeier/phys121/phys121.hml isi regulrly Our TA is Lorrine Bowmn Week 2 Reding: Chper 2 - Gincoli Week 2 Assignmens: Due:

### Brock University Physics 1P21/1P91 Fall 2013 Dr. D Agostino. Solutions for Tutorial 3: Chapter 2, Motion in One Dimension

Brock Uniersiy Physics 1P21/1P91 Fall 2013 Dr. D Agosino Soluions for Tuorial 3: Chaper 2, Moion in One Dimension The goals of his uorial are: undersand posiion-ime graphs, elociy-ime graphs, and heir

### IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704

Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion

### MATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)

MATH 14 AND 15 FINAL EXAM REVIEW PACKET (Revised spring 8) The following quesions cn be used s review for Mh 14/ 15 These quesions re no cul smples of quesions h will pper on he finl em, bu hey will provide

### Physics for Scientists and Engineers I

Physics for Scieniss nd Engineers I PHY 48, Secion 4 Dr. Beriz Roldán Cueny Uniersiy of Cenrl Florid, Physics Deprmen, Orlndo, FL Chper - Inroducion I. Generl II. Inernionl Sysem of Unis III. Conersion

### Solutions to Problems from Chapter 2

Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5

### Minimum Squared Error

Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > for ll smples y i solve sysem of liner inequliies MSE procedure y i = i for ll smples

### Physics Worksheet Lesson 4: Linear Motion Section: Name:

Physics Workshee Lesson 4: Liner Moion Secion: Nme: 1. Relie Moion:. All moion is. b. is n rbirry coorine sysem wih reference o which he posiion or moion of somehing is escribe or physicl lws re formule.

### RESPONSE UNDER A GENERAL PERIODIC FORCE. When the external force F(t) is periodic with periodτ = 2π

RESPONSE UNDER A GENERAL PERIODIC FORCE When he exernl force F() is periodic wih periodτ / ω,i cn be expnded in Fourier series F( ) o α ω α b ω () where τ F( ) ω d, τ,,,... () nd b τ F( ) ω d, τ,,... (3)

### Minimum Squared Error

Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > 0 for ll smples y i solve sysem of liner inequliies MSE procedure y i i for ll smples

### INSTANTANEOUS VELOCITY

INSTANTANEOUS VELOCITY I claim ha ha if acceleraion is consan, hen he elociy is a linear funcion of ime and he posiion a quadraic funcion of ime. We wan o inesigae hose claims, and a he same ime, work

### What distance must an airliner travel down a runway before reaching

2 LEARNING GALS By sudying his chper, you will lern: How o describe srigh-line moion in erms of erge elociy, insnneous elociy, erge ccelerion, nd insnneous ccelerion. How o inerpre grphs of posiion ersus

### REAL ANALYSIS I HOMEWORK 3. Chapter 1

REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs

### THE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI

THE 2-BODY PROBLEM ROBERT J. VANDERBEI ABSTRACT. In his shor noe, we show ha a pair of ellipses wih a common focus is a soluion o he 2-body problem. INTRODUCTION. Solving he 2-body problem from scrach

### WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x

WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile

### PARABOLA. moves such that PM. = e (constant > 0) (eccentricity) then locus of P is called a conic. or conic section.

wwwskshieducioncom PARABOLA Le S be given fixed poin (focus) nd le l be given fixed line (Direcrix) Le SP nd PM be he disnce of vrible poin P o he focus nd direcrix respecively nd P SP moves such h PM

### ( ) ( ) ( ) ( ) ( ) ( y )

8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll

### Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:

Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial

### CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES

CHAPTER PARAMETRIC EQUATIONS AND POLAR COORDINATES. PARAMETRIZATIONS OF PLANE CURVES., 9, _ _ Ê.,, Ê or, Ÿ. 5, 7, _ _.,, Ÿ Ÿ Ê Ê 5 Ê ( 5) Ê ˆ Ê 6 Ê ( 5) 7 Ê Ê, Ÿ Ÿ \$ 5. cos, sin, Ÿ Ÿ 6. cos ( ), sin (

### LAPLACE TRANSFORMS. 1. Basic transforms

LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming

### EE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?

EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of

### Chapter Floating Point Representation

Chaper 01.05 Floaing Poin Represenaion Afer reading his chaper, you should be able o: 1. conver a base- number o a binary floaing poin represenaion,. conver a binary floaing poin number o is equivalen

### Question Details Int Vocab 1 [ ] Question Details Int Vocab 2 [ ]

/3/5 Assignmen Previewer 3 Bsic: Definie Inegrls (67795) Due: Wed Apr 5 5 9: AM MDT Quesion 3 5 6 7 8 9 3 5 6 7 8 9 3 5 6 Insrucions Red ody's Noes nd Lerning Gols. Quesion Deils In Vocb [37897] The chnge

### LAB # 2 - Equilibrium (static)

AB # - Equilibrium (saic) Inroducion Isaac Newon's conribuion o physics was o recognize ha despie he seeming compleiy of he Unierse, he moion of is pars is guided by surprisingly simple aws. Newon's inspiraion

### 3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon

3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of

### AQA Maths M2. Topic Questions from Papers. Differential Equations. Answers

AQA Mahs M Topic Quesions from Papers Differenial Equaions Answers PhysicsAndMahsTuor.com Q Soluion Marks Toal Commens M 600 0 = A Applying Newonís second law wih 0 and. Correc equaion = 0 dm Separaing

### t s (half of the total time in the air) d?

.. In Cl or Homework Eercie. An Olmpic long jumper i cpble of jumping 8.0 m. Auming hi horizonl peed i 9.0 m/ he lee he ground, how long w he in he ir nd how high did he go? horizonl? 8.0m 9.0 m / 8.0

### Kinematics and kinematic functions

Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables and vice versa Direc Posiion

### Phys 221 Fall Chapter 2. Motion in One Dimension. 2014, 2005 A. Dzyubenko Brooks/Cole

Phys 221 Fall 2014 Chaper 2 Moion in One Dimension 2014, 2005 A. Dzyubenko 2004 Brooks/Cole 1 Kinemaics Kinemaics, a par of classical mechanics: Describes moion in erms of space and ime Ignores he agen

### Honours Introductory Maths Course 2011 Integration, Differential and Difference Equations

Honours Inroducory Mhs Course 0 Inegrion, Differenil nd Difference Equions Reding: Ching Chper 4 Noe: These noes do no fully cover he meril in Ching, u re men o supplemen your reding in Ching. Thus fr

### Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle

Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,

Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:

### Motion on a Curve and Curvature

Moion on Cue nd Cuue his uni is bsed on Secions 9. & 9.3, Chpe 9. All ssigned edings nd execises e fom he exbook Objecies: Mke cein h you cn define, nd use in conex, he ems, conceps nd fomuls lised below:

### Ch.4 Motion in 2D. Ch.4 Motion in 2D

Moion in plne, such s in he sceen, is clled 2-dimensionl (2D) moion. 1. Posiion, displcemen nd eloci ecos If he picle s posiion is ( 1, 1 ) 1, nd ( 2, 2 ) 2, he posiions ecos e 1 = 1 1 2 = 2 2 Aege eloci

### Physics 100: Lecture 1

Physics : Lecure Agen for Toy Aice Scope of his course Mesuremen n Unis Funmenl unis Sysems of unis Conering beween sysems of unis Dimensionl Anlysis -D Kinemics (reiew) Aerge & insnneous elociy n ccelerion

### Chapter 10. Simple Harmonic Motion and Elasticity. Goals for Chapter 10

Chper 0 Siple Hronic Moion nd Elsiciy Gols or Chper 0 o ollow periodic oion o sudy o siple hronic oion. o sole equions o siple hronic oion. o use he pendulu s prooypicl syse undergoing siple hronic oion.

### ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 2

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER Seion Eerise -: Coninuiy of he uiliy funion Le λ ( ) be he monooni uiliy funion defined in he proof of eisene of uiliy funion If his funion is oninuous y hen

### 3, so θ = arccos

Mahemaics 210 Professor Alan H Sein Monday, Ocober 1, 2007 SOLUTIONS This problem se is worh 50 poins 1 Find he angle beween he vecors (2, 7, 3) and (5, 2, 4) Soluion: Le θ be he angle (2, 7, 3) (5, 2,

### An integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples.

Improper Inegrls To his poin we hve only considered inegrls f(x) wih he is of inegrion nd b finie nd he inegrnd f(x) bounded (nd in fc coninuous excep possibly for finiely mny jump disconinuiies) An inegrl

### 15. Vector Valued Functions

1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,

### 1.0 Electrical Systems

. Elecricl Sysems The ypes of dynmicl sysems we will e sudying cn e modeled in erms of lgeric equions, differenil equions, or inegrl equions. We will egin y looking fmilir mhemicl models of idel resisors,

### ME 391 Mechanical Engineering Analysis

Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of

### 14. The fundamental theorem of the calculus

4. The funmenl heorem of he clculus V 20 00 80 60 40 20 0 0 0.2 0.4 0.6 0.8 v 400 200 0 0 0.2 0.5 0.8 200 400 Figure : () Venriculr volume for subjecs wih cpciies C = 24 ml, C = 20 ml, C = 2 ml n (b) he

### Physics 101 Lecture 4 Motion in 2D and 3D

Phsics 11 Lecure 4 Moion in D nd 3D Dr. Ali ÖVGÜN EMU Phsics Deprmen www.ogun.com Vecor nd is componens The componens re he legs of he righ ringle whose hpoenuse is A A A A A n ( θ ) A Acos( θ) A A A nd

### Chapter Direct Method of Interpolation More Examples Civil Engineering

Chpter 5. Direct Method of Interpoltion More Exmples Civil Engineering Exmple o mximie ctch of bss in lke, it is suggested to throw the line to the depth of the thermocline. he chrcteristic feture of this

### How to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.

How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle

### Development of a New Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations

IOS Journl o Memics IOSJM ISSN: 78-78 Volume Issue July-Aug PP -9 www.iosrjournls.org Developmen o New Sceme or e Soluion o Iniil Vlue Problems in Ordinry Dierenil Equions Ogunrinde. B. dugb S. E. Deprmen

### Module 2 F c i k c s la l w a s o s f dif di fusi s o i n

Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms

### Let us start with a two dimensional case. We consider a vector ( x,

Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our

### MEI Mechanics 1 General motion. Section 1: Using calculus

Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a 6 4 4 When =, v 4 a 6 4 6. (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy

### Sections 2.2 & 2.3 Limit of a Function and Limit Laws

Mah 80 www.imeodare.com Secions. &. Limi of a Funcion and Limi Laws In secion. we saw how is arise when we wan o find he angen o a curve or he velociy of an objec. Now we urn our aenion o is in general

### AP Calculus BC Chapter 10 Part 1 AP Exam Problems

AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a

### The order of reaction is defined as the number of atoms or molecules whose concentration change during the chemical reaction.

www.hechemisryguru.com Re Lw Expression Order of Recion The order of recion is defined s he number of oms or molecules whose concenrion chnge during he chemicl recion. Or The ol number of molecules or

### Chapter Lagrangian Interpolation

Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and

### Science Advertisement Intergovernmental Panel on Climate Change: The Physical Science Basis 2/3/2007 Physics 253

Science Adeisemen Inegoenmenl Pnel on Clime Chnge: The Phsicl Science Bsis hp://www.ipcc.ch/spmfeb7.pdf /3/7 Phsics 53 hp://www.fonews.com/pojecs/pdf/spmfeb7.pdf /3/7 Phsics 53 3 Sus: Uni, Chpe 3 Vecos

### Math 2214 Solution Test 1A Spring 2016

Mah 14 Soluion Tes 1A Spring 016 sec Problem 1: Wha is he larges -inerval for which ( 4) = has a guaraneed + unique soluion for iniial value (-1) = 3 according o he Exisence Uniqueness Theorem? Soluion

### Y 0.4Y 0.45Y Y to a proper ARMA specification.

HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where