Functions defined by an algorithm 

Forreference,seehere

• ExamplesinAnalysis
  • TheEulermethodforan approximatesolutionofthedifferential equationy'=y
• • Approximateanti-derivative
• Trajectories: Roadrunner andCoyote
  • FunctionswithvaluesinIR
  • FunctionswithvaluesinIR²

Examplesin Analysis

TheEuler methodforan approximatesolutionofthedifferential equationy'=y

Inthisexample,thebasicsteps tocreateanduseafunctiondefined byanalgorithmaredemonstrated.
Selecttheentryinthealgebra tabmenu.

Theprogram

Awindowopenswithanembryonicprogram.
 

Structure

Wemodifyline3andchoose Repeatinthelistontheleft.

 

 

Declarations

Wechangethedeclarationofthevariable y.

 

Withthecorrespondingmenuentry, aninputfieldopensforthe initialvalue.

Wereplace0by1theOK.

Wedeclareanewvariablep.

  Weselecttheletterinthelistontheleftandweinitializelikefor y

LoopBody

WechoosetheentryAffect VariableintheLoop

Abarsimilartotheinitializationappears.Thevariablehastobe selectedinthealreadydeclaredvariables.


Thelineisinsertedaftertherepeat line
Wealsointroduceline6(below).

StopCondition

Wemodifyline7.
Wekeepthevariablexand wechangetheformula0into 2.
Thusthefunctionwillbedefinedover[0;2]

Execution

Beforedefiningthefunction,we havetocheckthattheprogramrunscorrectly.
Wechoosetostoptheprogrambefore 500iterations.
Ifourprogramiscorrectit muststopafter200iteration;this limitof500willnotbereached.
Wechoosealsoafunctionwithvaluesin IR(checkx’!y (x)).
Ifourprogramiscorrectxvaluesareincreasing.

 

 

WeactivatethemenuentryTest twice(oneforchoosingthevariabletodisplay,theothertorun).
Wecheckthattheprogramrunscorrectly. Thevaluesofx,yandpinthe iterationaredisplayedin columns.
Wefillthecommentfieldto reminduswhatthefunction,thenOK tocreatethefunction.

Usingthefunction

Thefunctionappearsinthe listof  functionswiththecommentthatwehavechosen.
Abuttonwiththeletteralso appearsinthegraphic tab.Byactivatingthisbutton,the functionisgraphed.
Therealsoexistsanentryin thecurve menuofthegeometrytab.

 

Wecanchoosetodefinethe exponentialfunctionontheinterval[0, 2]andnotethat thecurvesaresuperimposed.
Amore accuratecomparisoncanbemadeusing thetable.

Wecanalsocreatethefunction f-g,toestimate thedifference.

Amessagewarnsthatthe function'suseislimitedtographics.

Indeed,wenotethatthe functionhhasadecentplot butCasyopéecannotcompute symbolicvalueash(2).
Anapproximatevalueofh(2) couldbeobtainedusing thetableorthepassingEvaluate Formula.

Approximate anti-derivative

Inthisexample,wewillshowhowCasyopée objectscanbeusedinan algorithm.

 

CasyopéeObjects

Createaparameterpfor thestepofthemethod,sothat wecanchangeit.

Createafunctionw,which canalsobechanged later.

Editingafunctiondefinedbyan algorithm

Byrightclicking  onthelineinthelistoffunctions correspondingtothefunctionf,we willmodifyitsprogram.

Deletepdeclarationatline 2.Thuspisnowthe parameter.

Modifyalsolines4and 5andthecommentsandtest thealgorithm.
Leprogram terminates,nous

Theprogramterminates,wecanapply thechangetothefunction(OK).

Usingthefunction

Byanimatingtheparameterp, weseehowthefunction obtainedfitsthefunctiondefinedby
Belowforthevaluesofp1, 1/2and1/20.



Itisalsopossibletochangethe functionw.Herewejusttake w(x)=xetU(x)=1+x²/2.

Trajectories:  RoadrunnerandCoyote

Functionswith valuesinIR

TheCoyoteispostedat20 mdistancefromtheexitofatunnel.
Roadrunnerexitsthetunnel onastraighttrailat aspeedof2m.s-1.
Coyote'srocketallows himtohaveaspeedof3m.s-1.
"Givenadirectionchosenbythe Coyote,canhecatchRoadrunner? ".

Functionsdefiningthetrajectories

Wedefineaparameterfor theslopeofthetrajectoryof Coyoteandanother forthetraveltime.

Roadrunner'strajectoryis thecurveofthefunctionb definedbythefollowingalgorithm.

Coyotte'strajectoryisthecurve ofthefunctioncdefinedby thefollowingalgorithm.

Usingthefunction

Oncethetwofunctionsandthe curvesarecreatedintheGeometry tabforagivenvalue ofa, weanimatethe parameterT

Withaslopeof9/10coyote passesBeepBeep!Missed again.

Functionswith valuesinIR²

ImaginenowthatCoyoteruns constantlytowardsBipBip, withaspeedof5m.s-1 and Roadrunner1m.s-1.
WemodifyCoyotte'salgorithm andfixT=20.

Wefindthatthealgorithm stoppedattime5:Coyote outpacedRoadrunnerandturned around.
Wechoosetheothercheckbox thatcorrespondstoafunction withvaluesinIR².

Nowthetimedidgoup to20andthevaluesofx,abscissaof Coyotearenolongermonotonic.

ThedisplayintheGeometrytabas aparametric curveshowsCoyote'schaoticbehavior.

 

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