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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