The Chase Problem (Part 2) David C. Arney


 Edward Porter
 1 years ago
 Views:
Transcription
1 The Chae Problem Par David C. Arne Inroducion In he previou ecion, eniled The Chae Problem Par, we dicued a dicree model for a chaing cenario where one hing chae anoher. Some of he applicaion of hi kind of chaing were given in he eample of he previou ecion: miile inerceping oher miile, aniank round eeking a ank, and orpedo racking an enem hip. In hi ecion, we eend and refine hi fir model, build a coninuou model for hi problem, and build more effecivene and ophiicaion ino our chae algorihm. The Problem Our problem i o deermine he movemen pah for he chaer, given ha we know he locaion and omeime more informaion of he arge. We ar wih he aumpion ha he chaer know he arge poiion eacl. The chaer poiion i repreened in wodimenional Careian coordinae b,. We alo aume ha he chaer move a a conan peed given b and he arge poiion in wo dimenion i given b he parameric relaionhip,. We ar wih he echnique ha he chaer move direcl oward he arge. Laer we ll allow for he chaer o lead he arge. A he locaion of he arge change, he chaer coninuall adju i pah o coninue o move direcl oward he arge. The Model We can model hi procedure wih a em of differenial equaion, one for each pace dimenion we model. Since we ll perform our modeling in wo pace dimenion and, we ll build em of wo differenial equaion. Our modeling proce for a coninuou ime changing even i o e up relaionhip ha epre he derivaive of he changing variable, in erm of funcion of variable. Le draw a diagram of wha happen during a ime inerval. We will ake he limi a in order o ge coninuou feedback and coninuou movemen for he chaer. Figure how he locaion of he chaer and he arge a ome ime. The movemen made b he chaer over ha inerval i indicaed. The final poiion of he chaer i given b,. 9
2 n, n o n, o n o o n, o n o Figure : Movemen b he chaer during he ime inerval from ep n o ep n. We ee wo imilar riangle in Figure. Like he dicree cae developed in Par, he relaion beween hee imilar righ riangle enable u o wrie our model. Recall ha we can e up proporional equaion relaing he ide of he riangle wih he hpoenue of he riangle. Fir le deermine formula for each of he ide of our wo riangle. The larger riangle in Figure ha i horizonal ide of lengh . The verical ide ha lengh . Therefore, b he Phagorean Theorem he hpoenue ha lengh. The maller riangle ha ide  and . We need o deermine he lengh of he hpoenue in erm of value oher han and. We alo know ha he chaer move a peed. Therefore, he lengh of he hpoenue can be approimaed b he diance raveled during he ime inerval or. We wrie ou he equaion relaing he ide of he riangle wih he hpoenue of he riangle. Fir he horizonal ide and hpoenue of boh riangle produce he relaionhip: The verical ide and hpoenue produce: n n n n n n n n We now have difference quoien on he lef ide of Equaion and. We ake he limi a of and o produce he differenial equaion 3
3 lim lim d d d d 3 Thi i our chae/movemen model equaion 3 and, which will provide a mean of deermining he movemen of he chaer, when we know he movemen of he arge. Thi em of differenial equaion i nonlinear and mu be olved numericall. Therefore, we will need a numerical olver on a compuer or calculaor o deermine he pah of he chae. Man ofware package ha ue Euler mehod or he RungeKua mehod are available. I i alo poible o implemen hee algorihm b convering he differenial equaion o difference equaion and implemening ieraion on a pread hee. Remember our aumpion: he chaer move a a conan peed and he chaer alwa ee he arge. When we olve our differenial equaion wih a numerical mehod, we acuall model he chaer moving oward he arge for a e ime inerval,, ued in he numerical cheme. We uuall e he ime inerval o be ver mall o aure accurac of he oluion and o approimae accurael he coninuou movemen of he chaer. We need o deermine when o op our calculaion. In he previou ecion, we dicued everal of he facor involved in hi deciion. There i no need o coninue afer he chaer ha caugh he arge. We need a opping crieria ha reflec caching he arge. We will aume ha caching he arge mean ju being cloe enough or wihin he olerance of he opping crieria denoed b ε. We have choice for deermining hi olerance value. I could be a fied value or a funcion of he peed and ime inerval. We implemen opping crieria in our model b deermining he diance, denoed d, beween he chaer and arge afer each ieraion. The value of d i deermined b he diance formula beween wo poin, d We op when d < ε or afer a pecified amoun of ime ha epired > M. Le look a an eample o ee how hi work. Eample : Aniank Round 5 A oldier locaed a poin,3 launche a racking aniank round wih peed 3.5 a a ank a ime following a ellipical coure given b he following parameric equaion: 8 3co and in 6 We will deermine he round pah baed on i eeker uing our racking model of moving direcl 3
4 oward he ank. If he kill radiu of he round i.5 uni, our opping crieria i ε.5. Subiuing he known value and funcion ino our model, Equaion 3 and, produce d d co 7 8 3co in d d 3.5 in 8 8 3co in Saring wih our iniial condiion, and 3, we ue a RungeKua olver wih τ.6 unil we achieve our opping crieria of d < ε.5. Thi produce he oluion for he pah of he round. The graph of he pah for boh he round and he ank, unil heir impac a a ime lighl grea han 5 econd, are given in Figure. Noice how he round curve around o follow he ank and evenuall cache i. z n, b n 6 z, a n, n Figure. Graph of he pah of he round olid curve and he ank doed curve, from launch o impac. Doe our oluion make ene? Doe he chaer move in an efficien pah oward he arge? Doe he chaer op when he opping crieria i achieved? In general he anwer o hee queion are e. I appear ha we have a good model, bu i ma no be he be. I could help if he round wa able o lead he ank o i could cach i faer. We ll r implemening a lead algorihm for hi chae problem laer in hi ecion. 3
5 The Modeling Proce Le review our modeling proce for hi problem. Our behavior of inere, he movemen of he chaer, i coninuou in naure. We modeled hi movemen wih a coninuou differenial equaion. Our oluion mehod for hi model, he RungeKua numerical mehod, i dicree and give an approimae oluion o he coninuou model. We hen convered he dicree equence of locaion of he pah o a coninuou pah b connecing he poin in he graph of Figure. We how hi inerpla beween dicree and coninuou repreenaion in our modeling proce in Figure 3. Behavior Model Soluion Mehod Verificaion Mehod movemen differenial RungeKua graph in Figure of chaer equaion numerical cheme dicree coninuou coninuou coninuou Figure 3. Inerpla beween dicree and coninuou in he modeling proce of he differenial equaion chae model. The lead algorihm How do we ge he chaer o lead he arge? We need o ake ino accoun boh he peed and he veloci of he arge, hen ue ha informaion o predic where he arge will be when he chaer cache he arge. We ue he Talor polnomial o do hi. The Talor polnomial i an approimaion o a funcion. For he funcion f and uing he ar poin a a, we wrie he Talor polnomial of degree n a f f a f a a n f a f a f a a a 3... a! 3! n! n 9 We can ue hi approimaion for he wo funcion and repreening he wo dimenion of he arge pah. Fir, le ue he degree polnomial approimaion, which ake ino accoun he locaion and he veloci bu no he acceleraion. We mu be epediiou in our elecion of a and in Equaion 9, and e n. To ge our approimaion in he proper form, we ue τ and a, and herefore, a τ. Then, we can wrie τ τ and τ τ. 33
6 The value of τ i he value of he ime advance o he locaion where he arge i prediced o be, in order o have a proper lead. The phanom locaion o aim for i impl he poin τ, τ. Therefore, we modif he model in Equaion 3 and, uing hi phanom lead poin in place of, and he formula in Equaion o obain d τ d τ τ d τ d τ τ The geomer of hi lead algorihm i hown in Figure. lead poin τ, τ, Targe curren locaion Chaer movemen direcion o, o Chaer curren locaion Figure : Pah of chaer when heading for he lead poin. We need an algorihm o deermine he value for τ. How much hould we lead? We could do hi everal wa. One wa i o hink of τ a he ime needed o cach he arge. We will approimae hi cach ime b uing he ime for he chaer o reach he arge curren locaion. Therefore, he formula for τ i impl he curren diance beween he chaer and he arge given in Equaion 5 divided b he peed. We wrie hi a τ 3 3
7 35 Then our new lead model i formed b ubiuing Equaion 3 ino Equaion and : d d d d 5 Le r hi model in our previou cenario of an aniank round. Eample : Aniank Round reviied A oldier locaed a,3 launche a racking aniank round wih peed 3.5 a a ank a ime, following a ellipical coure given b: 8 3 co and in. 6 We deermine he round pah baed on i eeker uing our new racking model of leading he ank. We ue ε.5 and ubiue he known value and funcion ino Equaion and 5. Saring wih he iniial condiion, and 3, we ue a RungeKua olver wih τ.6 unil we achieve our opping crieria of d < ε.5. Thi produce he oluion for he pah of he round. The graph of he pah for boh he round and he ank unil heir impac are given in Figure 5. B comparing hi graph wih ha of Figure, we ee ha he lead algorihm ave ime and chae diance b urning ooner and more harpl o cach he ank.
8 .5.5 z n, b n.5 6 z, a n, n Figure. Graph of he pah of he round olid curve and he ank doed curve, from launch o impac. I appear ha we have developed a beer model b leading he arge. I i he be we can do? Are here beer lead model? Of coure, hee are leading queion. There probabl are beer wa o compue τ. And we could keep more han ju one erm of he Talor Polnomial Equaion 9. If we keep wo erm, n, we would ge new approimaion for our lead locaion of he arge. We would now be aking ino accoun he acceleraion of he ank, a well a i veloci. The new model i wrien a follow: τ τ and τ τ. 7 Subiuion of hee formula ino Equaion 3 and, along wih uing Equaion 3 for τ, creae a model wih ver large, me differenial equaion. We won r o how hem here, bu we ll how an eample which compare he variou model we have dicued. Eample 3: To Lead or no o Lead Thi ime he oldier firing he aniank round i locaed a he origin,,. He launche a racking round wih peed 5 a a ank a ime. The ank follow an ocillaing coure given b: 3 3 and in3 8 We deermine he round pah baed on i eeker uing hree differen racking model: moving direcl oward he ank, leading he ank b uing he veloci one derivaive erm in he Talor polnomial a in Equaion and 36
9 5, and 3 leading he ank b uing veloci and acceleraion including erm in he Talor Polnomial hown in Equaion 7. Our opping crieria i ε.5. The pah of he round racking direcl for he ank i given in Figure 5. The round cache he ank a 3.9 econd. The pah of he round when leading he arge uing he ank veloci i hown in Figure 6. Thi pah i more direc and cache he ank in.8 econd. Finall, he new lead algorihm uing boh veloci and acceleraion produce he graph in Figure 7. Thi model produce a cach a.7 econd. Thi la mehod i no much faer han he veloci onl model. Someime i doen help or ma even hinder o lead he arge, bu, in general, he more informaion ou ue he quicker ou can cach our arge. z n, b n z, a n, n 5 Figure 5: Pah of arge doed curve and chaer olid curve uing he model of moving direcl oward he arge. z n, b n z, a n, n 5 Figure 6: Pah of arge doed curve and chaer olid curve uing he model of leading he ank b uing he veloci. 37
10 z n, b n z, a n, n 5 Figure 7: Pah of arge doed curve and chaer olid curve uing he model of leading he ank b uing boh he veloci and acceleraion. In hi ecion, we have udied and olved a challenging problem wih man applicaion. Our model and i oluion have performed well in he eample we have olved. We now know wha happen when we lead he arge, inead of moving direcl oward i. There are ill man queion we have no addreed. Wha abou he maneuverabili of he chaer? Can i alwa urn fa enough o make he necear move of he algorihm? How hould he arge move o evade he chaer? Thee are difficul queion ha meri furher ud and more ophiicaed mahemaical model. Good luck o hoe who ud hi imporan problem wih numerou miliar applicaion. Eercie. An enem ank, currenl a locaion,, i moving in a zigzag paern awa from our locaion wih parameric equaion: 3 and in. You launch a ank racking round moving a a peed of from our locaion a,. The guidance em of he radarconrolled round alwa move direcl oward he ank arge. a Wrie a em of differenial equaion ha model he movemen of he round oward he arge. b Ue a numerical cheme o olve he equaion and plo he oluion for <<. c Wha i he diance beween he ank and he round a? I he round cloer a or?. Uing he ame general cenario a eercie for an enem ank aring a, and moving wih equaion: 3 and in, ou launch a racking round moving a peed of from our locaion a,. Thi new improved round ha a guidance em ha lead he ank b conidering i veloci. a Wrie a em of differenial equaion b ubiuing ino Equaion 5 and 6 ha model he movemen of he round oward he arge. b Ue a numerical cheme o olve he equaion and plo he oluion for <<. 38
11 c Wha i he diance beween he ank and he round a? I he round cloer a or? 3. A hip locaed a,5 deec a orpedo a 5,6 and begin he evaive maneuver of moving direcl awa from he orpedo a a conan peed of 8. a Wha are he parameric equaion, uing ime a he parameer, for he pah of he hip wih repreening he ar ime of hi pah? b If he orpedo follow he hip wih a peed of, wha are he differenial equaion ha govern he moion of he orpedo?. A hip locaed a,5 deec a orpedo a 5,6 and begin an evaive maneuver defined b he equaion 3 and 5 3in. a If he orpedo follow he hip wih a peed of, wha are he differenial equaion ha govern he moion of he orpedo uing he 3 chae algorihm decribed in hi ecion direc inercep, lead uing veloci, lead uing veloci and acceleraion? b Solve he 3 model in par a and deermine which algorihm guide he orpedo cloe o he hip afer 3 econd. 5. Dicu he dichoom of dicree and coninuou mahemaic. Include in our dicuion eample of behavior and funcion ha are naurall dicree and behavior and funcion ha are naurall coninuou. Reference Dunbar, Seven R., Minimodule: Chae Problem, The UMAP Journal, vol. 5, no., 99, pp Fellman, Bruce, Gue Who Coming o Dinner: Mechanim which Help Inec Ecape Ba, Naional Wildlife, vol.3, Feb 993, pp. 5. Johnon, Elgin and Mahew, Jerold, Projec Baed Calculu a Iowa Sae, Compuer Algebra Sem in Educaion Newleer, no., Sepember 99, pp. 7. Simmon, George, Differenial Equaion wih wih Applicaion and Hiorical Noe, nd ediion, New York: McGrawHill, 99. Yae, Rober C., Differenial Equaion, New York: McGrawHill, 95 pp. 9. Yavin, Y. and Pacher, M. P. edior, PuruiEvaion Differenial Game III, Compuer & Mahemaic wih Applicaion, vol. 6, no. 6, Sep 93, pp. 5 enire volume dedicaed o hi ubjec. 39
Newton's second law in action
Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In
More information6.003 Homework #4 Solutions
6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d
More information2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity
.6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This
More informationCSE202 Greedy algorithms
CSE0 Greedy algorihm . Shore Pah in a Graph hore pah from Princeon CS deparmen o Einein' houe . Shore Pah in a Graph hore pah from Princeon CS deparmen o Einein' houe Tree wih a mo edge G i a ree on n
More information1. The graph shows the variation with time t of the velocity v of an object.
1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationAcceleration Lab Teacher s Guide
Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion
More informationChapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr
Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i
More information( ) in the following way. ( ) < 2
Sraigh Line Moion  Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and manmade. Wrie down several of hem. Horizonal cars waer
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationChapter 2 Kinematics in One Dimension
Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how
More informationTopic: Applications of Network Flow Date: 9/14/2007
CS787: Advanced Algorihm Scribe: Daniel Wong and Priyananda Shenoy Lecurer: Shuchi Chawla Topic: Applicaion of Nework Flow Dae: 9/4/2007 5. Inroducion and Recap In he la lecure, we analyzed he problem
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationChapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edgedisjoint paths in a directed graphs
Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edgedijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationA Curriculum Module for AP Calculus BC Curriculum Module
Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy.
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationDigital Data Acquisition
ME231 Measuremens Laboraory Spring 1999 Digial Daa Acquisiion Edmundo Corona c The laer par of he 20h cenury winessed he birh of he compuer revoluion. The developmen of digial compuer echnology has had
More informationSection 7.1 Angles and Their Measure
Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationName: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling
Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: Solving Exponenial Equaions (The Mehod of Common Bases) Solving Exponenial Equaions (Using Logarihms)
More information4.2 Trigonometric Functions; The Unit Circle
4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.
More informationHANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
More informationand Decay Functions f (t) = C(1± r) t / K, for t 0, where
MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae
More informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
More informationAP Calculus AB 2007 Scoring Guidelines
AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationCircle Geometry (Part 3)
Eam aer 3 ircle Geomery (ar 3) emen andard:.4.(c) yclic uadrilaeral La week we covered u otheorem 3, he idea of a convere and we alied our heory o ome roblem called IE. Okay, o now ono he ne chunk of heory
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED
More informationChapter 8 Natural and Step Responses of RLC Circuits
haper 8 Naural an Sep Repone of R ircui 8. The Naural Repone of a Parallel R ircui 8.3 The Sep Repone of a Parallel R ircui 8.4 The Naural an Sep Repone of a Serie R ircui Key poin Wha o he repone curve
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More informationLAB 6: SIMPLE HARMONIC MOTION
1 Name Dae Day/Time of Lab Parner(s) Lab TA Objecives LAB 6: SIMPLE HARMONIC MOTION To undersand oscillaion in relaion o equilibrium of conservaive forces To manipulae he independen variables of oscillaion:
More informationRotational Inertia of a Point Mass
Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationChapter 3. Motion in Two or Three Dimensions
Chaper 3 Moion in Two or Three Dimensions 1 Ouline 1. Posiion, eloci, acceleraion. Moion in a plane (Se of equaions) 3. Projecile Moion (Range, Heigh, Veloci, Trajecor) 4. Circular Moion (Polar coordinaes,
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO
Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,
More informationSAMPLE LESSON PLAN with Commentary from ReadingQuest.org
Lesson Plan: Energy Resources ubject: Earth cience Grade: 9 Purpose: students will learn about the energy resources, explore the differences between renewable and nonrenewable resources, evaluate the environmental
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More informationA Mathematical Description of MOSFET Behavior
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
More informationChapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.
Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,
More informationRenewal processes and Poisson process
CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial
More informationMotion Along a Straight Line
Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his
More informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
More informationMOTION ALONG A STRAIGHT LINE
Chaper 2: MOTION ALONG A STRAIGHT LINE 1 A paricle moes along he ais from i o f Of he following alues of he iniial and final coordinaes, which resuls in he displacemen wih he larges magniude? A i =4m,
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationMarkov Models and Hidden Markov Models (HMMs)
Markov Models and Hidden Markov Models (HMMs (Following slides are modified from Prof. Claire Cardie s slides and Prof. Raymond Mooney s slides. Some of he graphs are aken from he exbook. Markov Model
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More information2. Waves in Elastic Media, Mechanical Waves
2. Waves in Elasic Media, Mechanical Waves Wave moion appears in almos ever branch of phsics. We confine our aenion o waves in deformable or elasic media. These waves, for eample ordinar sound waves in
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More informationMortality Variance of the Present Value (PV) of Future Annuity Payments
Morali Variance of he Presen Value (PV) of Fuure Annui Pamens Frank Y. Kang, Ph.D. Research Anals a Frank Russell Compan Absrac The variance of he presen value of fuure annui pamens plas an imporan role
More informationSection A: Forces and Motion
I is very useful o be able o make predicions abou he way moving objecs behave. In his chaper you will learn abou some equaions of moion ha can be used o calculae he speed and acceleraion of objecs, and
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationModule 3 Design for Strength. Version 2 ME, IIT Kharagpur
Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67  FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1  TRANSIENTS
EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67  FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1  RANIEN Uni conen 2 Undersand he ransien behaviour of resisorcapacior (R) and resisorinducor (RL) D circuis
More informationFortified financial forecasting models: nonlinear searching approaches
0 Inernaional Conference on Economic and inance Reearch IPEDR vol.4 (0 (0 IACSIT Pre, Singapore orified financial forecaing model: nonlinear earching approache Mohammad R. Hamidizadeh, Ph.D. Profeor,
More informationTEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED
Radioacive Daing Science Objecives Sudens will discover ha radioacive isoopes decay exponenially. Sudens will discover ha each radioacive isoope has a specific halflife. Sudens will develop mahemaical
More information11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.
11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure
More informationBasic Assumption: population dynamics of a group controlled by two functions of time
opulaion Models Basic Assumpion: populaion dynamics of a group conrolled by wo funcions of ime Birh Rae β(, ) = average number of birhs per group member, per uni ime Deah Rae δ(, ) = average number of
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More information1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z 1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z
o ffix uden abel ere uden ame chool ame isric ame/ ender emale ale onh ay ear ae of irh an eb ar pr ay un ul ug ep c ov ec as ame irs ame lace he uden abel ere ae uden denifier chool se nly rined in he
More informationWeek #9  The Integral Section 5.1
Week #9  The Inegral Secion 5.1 From Calculus, Single Variable by HughesHalle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,
More informationChapter 15: Superposition and Interference of Waves
Chaper 5: Superposiion and Inerference of Waves Real waves are rarely purely sinusoidal (harmonic, bu hey can be represened by superposiions of harmonic waves In his chaper we explore wha happens when
More informationTHE PRESSURE DERIVATIVE
Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationFullwave rectification, bulk capacitor calculations Chris Basso January 2009
ullwave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal
More informationApplication of kinematic equation:
HELP: See me (office hours). There will be a HW help session on Monda nigh from 78 in Nicholson 109. Tuoring a #10 of Nicholson Hall. Applicaion of kinemaic equaion: a = cons. v= v0 + a = + v + 0 0 a
More information4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F
efore you begin: Use black pencil. Wrie and bubble your SU ID Number a boom lef. Fill bubbles fully and erase cleanly if you wish o change! 20 Quesions, each quesion is 10 poins. Each quesion has a mos
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationRC Circuit and Time Constant
ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisorcapacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he
More informationInterference, Diffraction and Polarization
L.1  Simple nerference Chaper L nerference, Diffracion and Polarizaion A sinusoidal wave raveling in one dimension has he form: Blinn College  Physics 2426  Terry Honan A coshk x w L where in he case
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationLecture III: Finish Discounted Value Formulation
Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationVector Autoregressions (VARs): Operational Perspectives
Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101115. Macroeconomericians
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More informationEconomics 140A Hypothesis Testing in Regression Models
Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1
More informationAP1 Kinematics (A) (C) (B) (D) Answer: C
1. A ball is hrown verically upward from he ground. Which pair of graphs bes describes he moion of he ball as a funcion of ime while i is in he air? Neglec air resisance. y a v a (A) (C) y a v a (B) (D)
More information