To apply Newton's Method for solving the equation $f(x) = x^3 + 2x + 5$, starting at $x_0 = -1.45$, follow these steps:

Newton's Method Formula:

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

• $f(x) = x^3 + 2x + 5$
• $f'(x) = 3x^2 + 2$

Step 1: Calculate $f(x)$ and $f'(x)$

$f(x) = x^3 + 2x + 5$ $f'(x) = 3x^2 + 2$

Step 2: Perform Iterations

Start with $x_0 = -1.45$.

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Iteration 1 ($x_1$):

$f(-1.45) = (-1.45)^3 + 2(-1.45) + 5 = -3.051125$ $f'(-1.45) = 3(-1.45)^2 + 2 = 8.3025$ $x_1 = -1.45 - \frac{-3.051125}{8.3025} = -1.082725$

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Iteration 2 ($x_2$):

$f(-1.082725) = (-1.082725)^3 + 2(-1.082725) + 5 = -0.679437$ $f'(-1.082725) = 3(-1.082725)^2 + 2 = 5.514036$ $x_2 = -1.082725 - \frac{-0.679437}{5.514036} = -0.959274$

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Iteration 3 ($x_3$):

$f(-0.959274) = (-0.959274)^3 + 2(-0.959274) + 5 = -0.049126$ $f'(-0.959274) = 3(-0.959274)^2 + 2 = 4.759453$ $x_3 = -0.959274 - \frac{-0.049126}{4.759453} = -0.948923$

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Results:

• $x_1 = -1.082725$
• $x_2 = -0.959274$
• $x_3 = -0.948923$

Would you like me to expand on the method, verify calculations, or explain further?

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Relative Questions:

1. How does Newton's method converge, and what are its limitations?
2. Why is the derivative important in Newton's method?
3. Can Newton's method fail to converge? If so, under what conditions?
4. How can initial guesses affect the convergence of Newton's method?
5. What are the differences between Newton's method and other root-finding techniques?

Tip:

Always check $f'(x) \neq 0$ at each step to avoid division by zero.