2.7. Linearization and Newton’s Method http://www.ck12.org
2.7 Linearization and Newton’s Method
Learning Objectives
A student will be able to:
- Approximate a function by the method of linearization.
- Know Newton’s Method for approximating roots of a function.
Linearization: The Tangent Line Approximation
If f is a differentiable function atx 0 , then the tangent line,y=mx+b, to the curvey= f(x)atx 0 is a good
approximation to the curvey=f(x)for values ofxnearx 0 (Figure 8a). If you “zoom in” on the two graphs,
y=f(x)and the tangent line, at the point of tangency,(x 0 ,f(x 0 )), or if you look at a table of values near the point
of tangency, you will notice that the values are very close (Figure 8b).
Since the tangent line passes through point(x 0 ,f(x 0 ))and the slope is f′(x 0 ), we can write the equation of the
tangent line, in point-slope form, as
y−y 0 =m(x−x 0 )
y−f(x 0 ) =f′(x 0 )(x−x 0 )Solving fory,
y=f(x 0 )+f′(x 0 )(x−x 0 )