Question

Use Newton's method to approximate a root of the equation cos(x^2 - 7) = x^3. Let x1 = 1 be the initial approximation.

Answer 1

the equation needs to be rewritten

\[f(x) = \cos(x^2 - 7) - x^3 \]
the derivative is

\[f'(x) = -2x \sin(x^2 - 7) - 3x^2\]

now use the formula

\[x _{n + 1} = x _{n} - f(x _{n})/f'(x _{n})\]

and \[x _{1} = 1\]