Newton's Method Solution 3

Processing math: 100%

SOLUTION 3: If we let

$x=100^{1/5}$ , then

$x^5=100$ and

$x^5-100=0$ , so let's define function

$f(x) = x^5-100$ , whose graph is given below.

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The derivative of

$f$ is

$f'(x) = 5x^4$ . Now use Newton's Method:

$x_{n+1} = x_{n} - { f(x_{n}) \over f'(x_{n}) } \ \ \ \ \longrightarrow$

$x_{n+1} = x_{n} - { x_{n}^5-100 \over 5x_{n}^4 } \ \ \ \ \longrightarrow$ (Let's simplify the right-hand side of this equation. First get a common denominator.)

$x_{n+1} = x_{n} \ { 5x_{n}^4 \over 5x_{n}^4 } - { x_{n}^5-100 \over 5x_{n}^4 } \ \ \ \ \longrightarrow$

$x_{n+1} = { 5x_{n}^5 - ( x_{n}^5-100 ) \over 5x_{n}^4 } \ \ \ \ \longrightarrow$

$x_{n+1} = { 4x_{n}^5 + 100 \over 5x_{n}^4 }$ I will choose to let

$x_{0}=2$ . Using Newton's Method formula for 6 iterations in a spreadsheet results in :

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Thus

$\ 100^{1/5} \$ to eight decimal places is

$\ 100^{1/5} \approx 2.511886432$ .

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