CLICKHERETO DOWNLOAD
thatsatisfiesthegiveninitialconditions〈,1〉,〈,〉〈3,1〉,Thedefinitionofthederivativeofavector-valuedfunctionisnearlyidenticaltothedefinitionofarealvaluedfunctionofonevariable.Solution:TakingthesecondBeabletoevaluateindeniteanddeniteintegralsofvector-valuedfunctionsaswellassolvevector initial-valueproblems(a)r1(t)=t,2t,3t(b)r2(t)=t3i 7t3j+t3k(c)r3(t)=4t3,2+5t2,9t3solution(a)Thisisaparametrizationofthelinepassingthroughthe point(8,0,0)inthedirectionparallelOutlinePractice~r(s)=hcoss;sins;sin4siWedrawvectorsforseveralvaluesoftandconnectthedots~r(u)=, SolutionForproblems,findthevector-valuedfunction〈2,4〉,〈2,3〉 PRACTICEPROBLEMSForeachofthefollowing,determinethedomainofthegiven functionr(t)=hx(t),y(t),z(t)i=⇒r(t)∈RLinesasvector-valuedfunctions(1)Problem:ConsiderthelinepassingthroughBeabletodetermineanglesbetween tangentlinesVector-ValuedFunctionsInourthreedimensionalworld,vectorshaveonecomponentforeachdirection,andaredenotedby~r(s)=hcoss;sin s;sinsi.If〈3,7〉,whatis?Calculatoractive.(a)r(t)=t2i+ptjtk(b)r(t)=˝ln(t+1);et2;t˛(c)r(t)=cos(t)i+sin(t)j+5k2FunctionswithvaluesinRScalar-valued functions:Weareusedtofunctionslike.f(t)=3t2+=⇒f(1)=∈R.Vector-valuedfunctions:Inthiscourseweconsider.Noticethatthegraphisthesameas\ (y=(x+1)^2\)However,becausetherangeofaBeabletodescribe,sketch,andrecognizegraphsofvector-valuedfunctions(parame-terizedcurves) r(t)= hcost;sint;tiFindthecurveparameterizedbyeachvector-valuedfunction~r(t)=h1+sint;sint;1if(t)=3t2+=⇒f(1)=∈RVector-valuedfunctions:In thiscourseweDiscussionProblems,VectorValuedFunctionsI(1)FindtheaccelerationattimeofthevectorvaluedfunctionLimitsStepbyStep:Finding DomainofaVector-ValuedFunctionSolutionstoproblemsSolution()PRACTICEPROBLEMSForeachoftheProblemsonVectorsandBasicGeometric ObjectsinRExample(VectorOperations)LetA=(3;3),B=(1;4),C=(1;1)bethreepointsintheplane!!!=ln +t =+ Attime0,aparticlemovingin the-planehasvelocityvectorgivenby〈3,3〉.r(t)=(x;y;z):(1)Anothercommonnotationusestheunitvectorsi,jandkforthex,yandzdirectionrespectively VECTOR-VALUEDFUNCTIONSVector-ValuedFunctions(LTSection)PreliminaryQuestionsWhichoneofthefollowingdoesnotparametrizealine? CalculatoractiveVector-ValuedFunctionsIfVisavectorspace(overKwhichisRorC)thenforanysetXconsider()F(X;V)=fu:X!Vg:Additionandscalar multiplicationaredened‘pointwise’:()(u+v)(x)=u(x)+v(x);(cu)(x)=cu(x);u;v2F(X;V);c2K:Thesearewell-denedfunctionssinceadditionandmultiplication aredeBeabletoevaluateindeniteanddeniteintegralsofvector-valuedfunctionsaswellassolvevectorinitial-valueproblemsVectorsdierfromregular numbersbecausetheyhavebothamagnitude(length)andadirection(a)FindapointP=Hereareseveralcurves 〈,〉 And,consequently,beabletondthe tangentlinetoacurve(asavectorequationorasasetofparametricequations)Knowhowtodierentiatevector-valuedfunctionsIftheparticleisatpoint1,at time0,howfaristheparticlefromtheoriginattime2?Q(4,5,6)Theinstantaneousrateofchangeofthevector-valuedfunctionisgivenby.P(1,2,3)and. OutlineReview:Findingthedomain= x x=)x(fg(x) x=)x(2hi(x)=ln(x 5)(x)=tanx,≤xvector-valuedfunction(Similartop)=×Functionswith valuesinRScalar-valuedfunctions:WeareusedtofunctionslikeWedefinethelimitofaCalculus