Supposewe magnifyour pictureand zoomin on the point The picturemightlook like this:
We note that the curvenow looksvery muchlike a straightline. If we wereto overlaythis view with a straight
line that intersectsthe curveat our picturewouldlook like this:
We can makethe followingobservations.First,this line wouldappearto providea goodestimateof the
valueof for -valuesvery closeto Second,the approximationsappearto be gettingcloser
and closerto the actualvale of the functionas we take pointson the line closerand closerto the point
This line is calledthe tangentline to at This is one of the basicsituationsthat we
will explorein calculus.
TangentLine to a Graph
Continuingour discussionof the tangentline to at we next wish to find the equationof the
tangentline. We knowthat it passesthough but we do not yet haveenoughinformationto generate
its equation.Whatotherinformationdo we need?(Answer:The slopeof the line.)
Yes, we needto find the slopeof the line. We wouldbe able to find the slopeif we knewa secondpointon
the line. So let’s choosea point on the line, very closeto We can approximatethe coordinates