In Exercises 3 – 8, the roots of f(x) are known or are easily found. Use 5 iterations of Newton’s Method with the given iniƟal approximation to approximate the root. Compare it to the known value of the root.
f’(x) = 2x + 1
\[ x_1 = 0 - \frac{f(0)}{f'(0)} = 0 - \frac{0^2 + 0 -2}{2(0) + 1} = 2 \]
\[ x_2 = 2 - \frac{f(3)}{f'(3)} = 2 - \frac{3^2 + 3 -2}{2(3) + 1} = 0.571 \]
\[ x_3 = 0.571 - \frac{f(0.571)}{f'(0.571)} = 0.571 - \frac{0.571^2 + 0.571-2}{2(0.571) + 1} = 1.088 \]
\[ x_4 = 1.088 - \frac{f(1.088)}{f'(-1.088)} = 1 - \frac{1.088^2 + 1.088-2}{2(1.088 ) + 1} = 1.002 \]
\[ x_5 = 1.002 - \frac{f(1.002)}{f'(1.002)} = 1 - \frac{1.002^2 + 1.088-2}{2(1.002) + 1} = 1.000001332 \]