x 091 0991 09991 110001 11001 1101 111 fbfa ba 19 199 1999 20001 2001 201 21 weseethatf012bTheequationofthetangentlineisyf1f01x1ory12x1Inp ID: 298017
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4Example2.1.3(a)Findf0(1)forf(x)=x2:(b)Findtheequationofthetangentlinetothegraphoff(x)atthepoint(1;f(1)):Solution.Completingthefollowingchart x [0.9,1] [0.99,1] [0.999,1] [1,1.0001] [1,1.001] [1,1.01] [1,1.1] f(b)�f(a) b�a 1.9 1.99 1.999 2.0001 2.001 2.01 2.1 weseethatf0(1)=2:(b)Theequationofthetangentlineisy�f(1)=f0(1)(x�1)ory�1=2(x�1):Inpoint-interceptform,wehavey=2x�1 Example2.1.4(NumericalEstimationoftheDerivative)Findapproximatevaluesforf0(x)ateachofthex�valuesgiveninthefol-lowingtable x 0 5 10 15 20 f(x) 100 70 55 46 40 Solution.Thederivativecanbeestimatedbyusingtheaveragerateofchangeorthedi erencequotientf0(a)f(a+h)�f(a) h:Ifaisaleft-endpointthenf0(a)isestimatedbyf0(a)f(b)�f(a) b�awhereb�a:Ifaisaright-endpointthenf0(a)isestimatedbyf0(a)f(a)�f(b) a�b