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Use Newton’s Method to calculate the -intercept of the function , correct to five decimal places.

A reasonable estimate is .

Newton's Method for Approximating Roots

Newton’s Method uses an iterative process to approximate a root (-intercept) of a function.

Right side of upward parabola, vertex on negative y axis, 3 tick marks on x axis, all right of x intercept of parabola, labeled, from left to right, x sub 3, x sub 2, x sub 1, with vertical dashed segments, between the curve & x axis, 2 gray increasing lines, right line passes through x sub 2 & curve at x sub 1, left line passes through x sub 3 & curve at x sub 2.Begin with a point near the root. Write the equation of its tangent line at .

The -intercept of this tangent line is .
Solve for by setting .

so .

To calculate , solve for the root of the line tangent to at . This process continues until the desired accuracy is met.

Therefore, each new approximation, , can be found using . Newton’s Method was very important before calculators were invented and represents an excellent example of an iterative process. For most functions it works well, but it is always possible to find examples where it does not.