math117 lecturenotes7

math117 lecturenotes7 - 3. g 2.4 -3pplioat‘xows +0 mama!...

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[Euler, 1755, Institutiones calculi dlflemtialis, Pars Sol‘-‘ : The fiinction is defined for all x (since the denominator does not vanish on R). Proceed as follows. 0 Step 1. Information fi'om y’: ,__1-(1+X2)—x-2x __ l—x2 (1+x2)2 (1+x2)2 y = O :> x0 = i1 (stationary points), (l—x2)>0 for —1<x<+1 :> y increasing there, (1-x2)<0 for —oo<x<—l :9 y decreasing there. and 1<x<oo 0 Step 2. Information from y” : _ —6x +2x3 — 2 3 , (using Quotient and Chain rules). (1 + x ) and so =0 for x=0 and x=i-\/3—, —6x+2x3=2x(x2—3) >0 for —\/§<x<0 and J5<x<oo, <0 elsewhere. 0 Step 3. Sketch the graph using this information: Maxima, minima, inflection points of Euler’s example ...
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math117 lecturenotes7 - 3. g 2.4 -3pplioat‘xows +0 mama!...

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