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Section 4.13 : Newton's Method

For problems 1 & 2 use Newton’s Method to determine \({x_{\,2}}\) for the given function and given value of \({x_0}\).

  1. \(f\left( x \right) = {x^3} - 7{x^2} + 8x - 3\), \({x_{\,0}} = 5\) Solution
  2. \(f\left( x \right) = x\cos \left( x \right) - {x^2}\), \({x_{\,0}} = 1\) Solution

For problems 3 & 4 use Newton’s Method to find the root of the given equation, accurate to six decimal places, that lies in the given interval.

  1. \({x^4} - 5{x^3} + 9x + 3 = 0\) in \(\left[ {4,6} \right]\) Solution
  2. \(2{x^2} + 5 = {{\bf{e}}^x}\) in \(\left[ {3,4} \right]\) Solution

For problems 5 & 6 use Newton’s Method to find all the roots of the given equation accurate to six decimal places.

  1. \({x^3} - {x^2} - 15x + 1 = 0\) Solution
  2. \(2 - {x^2} = \sin \left( x \right)\) Solution