Approximate the positive zero(s) of the function $f ( x ) = x ^ { 3 } - \cos x \text { to three decimal }$ places. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. $0.001$
A)$- 0.967$
B) $0.865$
C) $0.842$
D) $- 0.865$
E) $0.916$

Question

Approximate the positive zero(s) of the function $f ( x ) = x ^ { 3 } - \cos x \text { to three decimal }$ places. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. $0.001$ 
A)$- 0.967$
B) $0.865$
C) $0.842$
D) $- 0.865$
E) $0.916$

Quizplus · Accepted Answer

Newton's Method formula is $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, where $f'(x)$ is the derivative of $f(x)$. For $f(x) = x^3 - \cos(x)$, $f'(x) = 3x^2 + \sin(x)$. Starting with a reasonable guess, such as $x_0 = 1$, and applying the formula iteratively will lead to a value that approximates the positive zero of the function. After a few iterations, the value will converge to approximately $0.865$, satisfying the condition that two successive approximations differ by less than $0.001$.

Approximate the Positive Zero(s) of the Function f(x)=x3−cos⁡x to three decimal f ( x ) = x ^ { 3 } - \cos x \text { to three decimal }f(x)=x3−cosx to three decimal

Approximate the Positive Zero(s) of the Function $f ( x ) = x ^ { 3 } - \cos x \text { to three decimal }$