   Chapter 3.8, Problem 3E

Chapter
Section
Textbook Problem

# Suppose the tangent line to the curve y = f ( x ) at the point (2, 5) has the equation y = 9 − 2 x . If Newton’s method is used to locate a root of the equation f ( x ) = 0 and the initial approximation is x 1 = 2 , find the second approximation x 2 .

To determine

To find:

The second approximation x2

Explanation

1) Concept:

Use Newton’s method.

To find a root of y=f(x), start with an initial approximation x1 . After the first iteration of Newton’s method we will get x2, this x2 is actually the x-intercept of the tangent at that point (x1,f(x1)) and now draw a tangent at that point (x2,f(x2)) and the x-intercept at that tangent will be x3. Continue to do this till the value of xn tends to converge.

If initial approximation is given, use it for x1 to proceed.

2) Formula:

Newton’s formula for second approximation is x2=x1-f(x1)f'(x1)

3) Given:

y=f(x) At point (2,5) has the equation y=9-2x and initial approximation x1=2

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