# HW 1-solutions.pdf - kong (jk43969) – HW 1 – osborn –...

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kong (jk43969) – HW 1 – osborn – (54610)1Thisprint-outshouldhave16questions.Multiple-choice questions may continue onthe next column or page – find all choicesbefore answering.00110.0pointsEvaluate the integralI=integraldisplaye2x4-4xx2dx.1.I=e3-203+ 4 ln 22.I=e33-203+ 4 ln 2correct3.I=e33+203-ln 24.I=e3+203-ln 25.I=e3+203+ ln 26.I=e33-203-4 ln 2Explanation:After division,x4-4xx2=x2-4x.ThusI=integraldisplaye2parenleftBigx2-4xparenrightBigdx=bracketleftBig13x3-4 lnxbracketrightBige2.Consequently,I=e33-203+ 4 ln 2.00210.0pointsIffis a linear function whose graph hasslopemandy-interceptb, evaluate the inte-gralI=integraldisplay60f(x)dx .1.I= 3m+b2.I= 6m+b3.I= 3m4.I=32m+b5.I=32m6.I= 6mcorrectExplanation:Since the graph offhas slopemandy-interceptb,f(x) =mx+b ,sof(x) =m .But then by the Fundamental Theorem ofCalculus,integraldisplay60f(x)dx=integraldisplay60m dx=bracketleftBigmxbracketrightBig60.Consequently,I= 6m.Alternatively, we could use the Fundamen-tal Theorem of Calculus directly:integraldisplay60f(x)dx=f(6)-f(0) = 6m .00310.0pointsIffis a continuous function such thatintegraldisplayx1f(t)dt=4xx2+ 6+47,find the value off(0).1.f(0) =832.f(0) =563.f(0) = 0
kong (jk43969) – HW 1 – osborn – (54610)24.f(0) =125.f(0) =23correct6.f(0) =19Explanation:By the Fundamental Theorem of Calculus,ddxparenleftBigintegraldisplayx1f(t)dtparenrightBig=f(x).So by the Quotient Rule,f(x) =ddxparenleftBig4xx2+ 6+47parenrightBig=24-4x2(x2+ 6)2.Consequently,f(0) =23.00410.0pointsFind the derivative ofFwhenF(x) =integraldisplay2xx(t2-2t)dt .1.F(x) = 5x2+ 4x2.F(x) = 5x2+ 6x3.F(x) = 5x2-4x4.F(x) = 7x2-4x5.F(x) = 7x2-6xcorrect6.F(x) = 7x2+ 6xExplanation:One version of the FTC tells us thatddxparenleftbiggintegraldisplayxaf(t)dtparenrightbigg=f(x)for each fixeda. More generally,ddxparenleftBiggintegraldisplayg(x)af(t)dtparenrightBigg=g(x)f(g(x))for each fixedaand differentiable functiong.To apply this toFwriteF(x) =integraldisplay2xx(t2-2t)dt=integraldisplay0x(t2-2t)dt+integraldisplay2x0(t2-2t)dt=-integraldisplayx0(t2-2t)dt+integraldisplay2x0(t2-2t)dt .

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