of usingthe function ; hence Recallthat for pointsvery closeto the points
on the line are closeapproximatepointsof the function.Usingthis approximation,we can computethe slope
of the tangentas follows:
(Note:We choosepointsvery closeto but not the pointitself,so
).In particular, for we have and Hencethe equationof the
tangentline, in pointslopeform is We can keepgettingcloserto the actualvalue
of the slopeby taking closerto or closerand closerto as in the followingtable:
P(x,y) m
(1.2, 1.44) 2.2
(1.15,1.3225) 2.15
(1.1, 1.21) 2.1
(1.05,1.1025) 2.05
(1.005,1.010025) 2.005
(1.0001,1.00020001)2.0001As we get closerto we get closerto the actualslopeof the tangentline, the value We call the
slopeof the tangentline at the point the derivativeof the function at the point
Let’s makea coupleof observationsaboutthis process.First,we can interpretthe processgraphicallyas
findingsecantlines from to otherpointson the graph.Fromthe diagramwe see a sequenceof these
secantlinesand can observehow they beginto approximatethe tangentline to the graphat The
diagramshowsa pair of secantlines,joining with points and
Second,in examiningthe sequenceof slopesof thesesecants,we are systematicallyobservingapproximate
slopesof the functionas point gets closerto Finally, producingthe tableof slopevaluesabove
was an inductiveprocessin whichwe generatedsomedata and then lookedto deducefrom our data the
valueto whichthe generatedresultstended.In this example,the slopevaluesappearto approachthe value