2007-10-26 MVT and Indefinite Integrals

# 2007-10-26 MVT and Indefinite Integrals - -Tangent parallel...

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Unformatted text preview: ---Tangent parallel to.chord-~/B,.,.,,,,TheMeanValueTheoremSupposey=f(x)is "tontinuousona closedinterval[a, b]and differentiableon theinterval'sinterior(a, b).Then there is at least one pointc in(a, b)at which01/\ay=f(x)ISlopef(b)-f(a)Ib-aII1bI(b)-I(a)=f'(c).b-a(1)ProofWe pictur~ the graph ofIas a curve in the plane and drawa linethroughthe pointsA(a, I(a»andB(b, I(b»The line is the graphof thecA(a.f(a»"---'"IIIIIIIIaB(b.f(b»x. Geometrically,theMean ValueTheoremsays thatsomewherebetweenAandBthecurve has at least one tangentparallelto chordAB.+oXfunctiong(x)=I(a)+I(b)-I(a)/,_ "(x-a)(2)(point-slopeequation). The verticaldifferencebetween the graphsofIandgatxis-."xyB<fIIIIIIIIA'hex)=f(x)-./'i.g(.\)-g(x)abh(x)=f(x)-.g(x)-f(b)-f(a)=f(x)-f(a)-.(x-a).-a(3)Thefunctionhsatisfies the hypothesesof Rolle'stheorem on[a. b].It iscontinuouson[a.b]and differentiableon(a, b)becausebothfandgare. Also,h(a)=h(b)=0 because the graphs offandgboth pass through A andB.There-fore,h'=0 at some point c in(a. b).(a....
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2007-10-26 MVT and Indefinite Integrals - -Tangent parallel...

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