资源描述:
《经典教材从差分方程到微分方程-微积分基础教程_》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
1、DifferenceEquationsSection1.1toCalculus:AreasAndTangentsDifferentialEquationsThestudyofcalculusbeginswithquestionsaboutchange.Whathappenstothevelocityofaswingingpendulumasitspositionchanges?Whathappenstothepositionofaplanetastimechanges?Whathappenstoapopulationofowlsasitsrateofrepr
2、oductionchanges?Mathematically,oneisinterestedinlearningtowhatextentchangesinonequantityaffectthevalueofanotherrelatedquantity.Throughthestudyofthewayinwhichquantitieschangeweareabletounderstandmoredeeplytherelationshipsbetweenthequantitiesthemselves.Forexample,changingtheangleofele
3、vationofaprojectileaffectsthedistanceitwilltravel;byconsideringtheeffectofachangeinangleondistance,weareabletodetermine,forexample,theanglewhichwillmaximizethedistance.Relatedtoquestionsofchangeareproblemsofapproximation.Ifwedesiretoapproxi-mateaquantitywhichcannotbecomputeddirectly(
4、forexample,theareaofsomeplanarregion),wemaydevelopatechniqueforapproximatingitsvalue.Theaccuracyofourtech-niquewilldependonhowmanycomputationswearewillingtomake;calculusmaythenbeusedtoanswerquestionsabouttherelationshipbetweentheaccuracyoftheapproxi-mationandthenumberofcalculations
5、used.Ifwedoublethenumberofcomputations,howmuchdowegaininaccuracy?Asweincreasethenumberofcomputations,dotheapproximationsapproachsomelimitingvalue?Andifso,canweuseourapproximatingmethodtoarriveatanexactanswer?Notethatonceagainweareaskingquestionsabouttheeffectsofchange.Twofundamental
6、conceptsforstudyingchangearesequencesandlimitsofsequences.Forourpurposes,asequenceisnothingmorethanalistofnumbers.Forexample,1111,,,,...248mightrepresentthebeginningofasequence,wheretheellipsisindicatesthatthelististocontinueonindefinitelyinsomepattern.Forexample,the5thterminthisseq
7、uencemightbe11=,1624the8thterm11=,12827and,ingeneral,thenthterm12n−1wheren=1,2,3,....Noticethatthesequenceiscompletelyspecifiedonlywhenwehavegiventhegeneralformofaterminthesequence.Alsonotethatthislistofnumbersis1CopyrightcbyDanSloughter20002Calculus:AreasAndTangentsSection1.1approa
8、ching0,whichwewouldcallthe