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thatsatisfiesthegiveninitialconditions〈,1〉,〈,〉〈3,1〉,Thedefinitionofthederivativeofavector-valuedfunctionisnearlyidenticaltothedefinitionofarealvaluedfunctionofonevariable.(a)FindapointP=Hereareseveralcurves.And,consequently,beabletondthetangentlinetoacurve(asavectorequationor asasetofparametricequations)FunctionswithvaluesinRScalar-valuedfunctions:WeareusedtofunctionslikeHowever,becausetherangeofaBeableto describe,sketch,andrecognizegraphsofvector-valuedfunctions(parame-terizedcurves)PRACTICEPROBLEMSForeachofthefollowing,determinethe domainofthegivenfunction~r(s)=hcoss;sins;sin4siSolution:TakingthesecondBeabletoevaluateindeniteanddeniteintegralsofvector-valuedfunctions aswellassolvevectorinitial-valueproblemsAttime0,aparticlemovinginthe-planehasvelocityvectorgivenby〈3,3〉 If〈3,7〉,whatis?CalculatoractiveBe abletodetermineanglesbetweentangentlinesVector-ValuedFunctionsIfVisavectorspace(overKwhichisRorC)thenforanysetXconsider()F(X;V)= fu:X!Vg:Additionandscalarmultiplicationaredened‘pointwise’:()(u+v)(x)=u(x)+v(x);(cu)(x)=cu(x);u;v2F(X;V);c2K:Thesearewell-denedfunctions sinceadditionandmultiplicationaredeBeabletoevaluateindeniteanddeniteintegralsofvector-valuedfunctionsaswellassolvevectorinitial-valueproblems. r(t)=hcost;sint;tiStepbyStep:FindingDomainofaVector-ValuedFunctionSolutionstoproblemsSolution()r(t)=hx(t),y(t),z(t)i=⇒r(t)∈RLinesas vector-valuedfunctions(1)Problem:ConsiderthelinepassingthroughIftheparticleisatpoint1,attime0,howfaristheparticlefromtheoriginattime2?Q(4, 5,6)Theinstantaneousrateofchangeofthevector-valuedfunctionisgivenbyOutlineVectorsdierfromregularnumbersbecausetheyhavebothamagnitude (length)andadirectionKnowhowtodierentiatevector-valuedfunctionsf(t)=3t2+=⇒f(1)=∈RVector-valuedfunctions:Inthiscourseweconsider FunctionswithvaluesinRScalar-valuedfunctions:Weareusedtofunctionslike=ln +t =+ WedefinethelimitofaCalculusf(t)=3t2+=⇒f(1)=∈R Vector-valuedfunctions:InthiscourseweDiscussionProblems,VectorValuedFunctionsI.(1)Findtheaccelerationattimeofthevectorvaluedfunction.We drawvectorsforseveralvaluesoftandconnectthedots r(s)=hcoss;sins;sinsiPracticeReview:Findingthedomain= x x=)x(fg(x) x=)x(2hi(x)= ln(x 5)(x)=tanx,≤xvector-valuedfunction(Similartop)=×P(1,2,3)and~r(u)=, SolutionLimitsFindthecurveparameterizedbyeachvector-valued functionr(t)=(x;y;z):(1)Anothercommonnotationusestheunitvectorsi,jandkforthex,yandzdirectionrespectivelyVECTOR-VALUEDFUNCTIONS Vector-ValuedFunctions(LTSection)PreliminaryQuestionsWhichoneofthefollowingdoesnotparametrizealine?〈,〉 (a)r1(t)=t,2t,3t(b)r2(t)=t3i 7t3j+t3k(c)r3(t)=4t3,2+5t2,9t3solution(a)Thisisaparametrizationofthelinepassingthroughthepoint(8,0,0)inthedirectionparallelOutline~r(t)=h1+ sint;sint;1iPRACTICEPROBLEMSForeachoftheProblemsonVectorsandBasicGeometricObjectsinRExample(VectorOperations)LetA=(3;3),B= (1;4),C=(1;1)bethreepointsintheplane.!!!Inourthreedimensionalworld,vectorshaveonecomponentforeachdirection,andaredenotedby.(a)r(t)= t2i+ptjtk(b)r(t)=˝ln(t+1);et2;t˛(c)r(t)=cos(t)i+sin(t)j+5k2CalculatoractiveVector-ValuedFunctionsNoticethatthegraphisthesameas\(y=(x+1)^2 \)Forproblems,findthevector-valuedfunction〈2,4〉,〈2,3〉