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Linearizationand Newton’s Method
LearningObjectives
A studentwill be able to:
- Approximatea functionby the methodof linearization.
- KnowNewton’s Methodfor approximatingrootsof a function.
Linearization:The TangentLine Approximation
Iffis a differentiablefunctionatx 0 , then the tangentline,y=mx+b, to the curvey=f(x) atx 0 is a good
approximationto the curvey=f(x) for valuesofxnearx 0 (Figure8a). If you “zoomin” on the two graphs,
y=f(x) and the tangentline, at the pointof tangency, (x 0 ,f(x 0 )), or if you look at a tableof valuesnear the
pointof tangency, you will noticethat the valuesare very close(Figure8b).
Sincethe tangentline passesthroughpoint(x 0 ,f(x 0 )) and the slopeisf(x 0 ), we can writethe equation
of the tangentline, in point-slopeform,as
Solvingfory,