Question

Use Newton's method to approximate a root of the equation 5x^3 +8x+1=0 as follows.
Let x1 =−1 be the initial approximation.

The second approximation x2 is ?
and the third approximation x3 is ?

Answer 1

The formula for newton's method is:
\[x_{n+1} = x_{n} - f(x_{n})/f'(x_{n})\]

if
\[f(x) = 5x^3 + 8x + 1\]
\[f'(x) = 15x^2 + 8\]
So if \[x_{1} = -1\]
then \[x_{2} = x_{1} - f(x_{1})/f'(x_{1})\]
\[x_{2} = -1 - (5(-1)^3 + 8(-1) + 1)/(15(-1)^2 + 8)\]
\[x_{2} \approx -0.4782\]

then:
\[x_{3} = x_{2} - f(x_{2})/f'(x_{2})\]

When you work that out you'll get your answer for \[x_{3}\]

Answer 2

excellent, thanks