Use Newton's method to find sqrt[4]10 correct to four decimal places. | Basic Mathematics 2081

Basic Mathematics20815 marksNumerical: Newton's method rootNewton's method for root finding

Use Newton's method to find $\sqrt[4]{10}$ correct to four decimal places. [5]

Target value: $\sqrt[4]{10}$ (fourth root of 10) - Required accuracy: 4 decimal places Let $x = \sqrt[4]{10}$ , so $x^4 = 10$ , giving: $f(x) = x^4 - 10 = 0$ $f'(x) = 4x^3$ $x{n+1} = xn - \frac{f(xn)}{f'(xn)} = xn - \frac{xn^4 - 10}{4xn^3} = \frac{3xn^4 + 10}{4xn^3}$ Since $1^4 = 1$ and

Target value: $\sqrt[4]{10}$ (fourth root of 10) - Required accuracy: 4 decimal places Let $x = \sqrt[4]{10}$, so $x^4 = 10$, giving: $$f(x) = x^4 - 10 = 0$$ $$f'(x) = 4x^3$$ $$x{n+1} = xn - \frac{f(xn)}{f'(xn)} = xn - \frac{xn^4 - 10}{4xn^3} = \frac{3xn^4 + 10}{4xn^3}$$ Since $1^4 = 1$ and