The function provided in the image is:

$f(x) = x^3 + 7$

You are tasked with finding the zero(s) of this function using Newton's Method and also with a graphing utility. Here's how to approach it step by step.

1. Newton's Method

Newton's method is an iterative approach for finding the roots of a function. The formula is:

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

Where:

• $f(x) = x^3 + 7$
• The derivative $f'(x) = 3x^2$

Steps to apply Newton's Method:

1. Start with an initial guess $x_0$. A reasonable guess might be $x_0 = -2$, since $x^3 + 7 = 0$ implies a negative root.

2. Compute: $x_{n+1} = x_n - \frac{x_n^3 + 7}{3x_n^2}$

3. Repeat the iteration until the difference between successive approximations is less than 0.001.

Iteration Example:

1. Start with $x_0 = -2$.
2. First iteration: $x_1 = x_0 - \frac{(-2)^3 + 7}{3(-2)^2} = -2 - \frac{-8 + 7}{12} = -2 - \frac{-1}{12} = -2 + 0.0833 = -1.9167$
3. Second iteration: $x_2 = -1.9167 - \frac{(-1.9167)^3 + 7}{3(-1.9167)^2}$ Continue these steps until two successive values differ by less than 0.001.

2. Graphing Utility

Using a graphing calculator or tool, plot the function $f(x) = x^3 + 7$ to find the zero visually. This should give an approximation close to what you obtain using Newton's method.

Would you like help with the further steps or clarification on the process?

Related questions:

1. How do we select an initial guess for Newton's method?
2. How does the rate of convergence of Newton's method compare to other root-finding methods?
3. Why does Newton's method require the function's derivative?
4. What are the limitations of Newton's method when applied to cubic functions?
5. Can Newton's method fail to find a root?

Tip: Always check the behavior of the function graphically before applying Newton's method, especially if multiple roots might exist.