To solve this using Newton's Method, the formula for the next approximation $x_{n+1}$ is:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Given:

• $f(x) = x^4 - 2$,
• Initial guess $x_1 = 2$.

Step 1: Compute $f'(x)$

The derivative of $f(x)$ is: $f'(x) = 4x^3$

Step 2: Evaluate $f(x_1)$ and $f'(x_1)$

For $x_1 = 2$: $f(2) = 2^4 - 2 = 16 - 2 = 14$ $f'(2) = 4(2^3) = 4(8) = 32$

Step 3: Compute $x_2$

Using the Newton's method formula: $x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}$ Substitute the values: $x_2 = 2 - \frac{14}{32} = 2 - 0.4375 = 1.5625$

Final Answer:

$x_2 = 1.5625$

Let me know if you'd like further clarifications or additional examples! 😊

Related Questions:

1. How does Newton's method work for non-polynomial functions?
2. What happens if the initial guess is far from the actual root?
3. Can Newton's method fail to converge? If so, why?
4. How do you choose a good initial guess for Newton's method?
5. What are some alternative methods for finding roots of functions?

Tip:

When using Newton's Method, always check $f'(x)$ at your approximations. If it gets close to zero, the method can diverge or fail!