Use​ Newton's method to estimate the two zeros of the following function. Start with x 0equalsnegative 2 for the left zero and x 0equals3 for the right zero. Then for each​ case, find x 2.
x minus x squared plus 8
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Part 1
If x 0equalsnegative 2​, then x 2equals
  
enter your response here.
​(Type an integer or a simplified​ fraction.)

Newton's method is an iterative method to approximate the roots of a function. It is defined as:

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

### Given Function:
\[
f(x) = x - x^2 + 8
\]

#### Step 1: Find \( f'(x) \):
\[
f'(x) = 1 - 2x
\]

#### For \( x_0 = -2 \):
1. Calculate \( f(x_0) \) and \( f'(x_0) \):
   \[
   f(-2) = -2 - (-2)^2 + 8 = -2 - 4 + 8 = 2
   \]
   \[
   f'(-2) = 1 - 2(-2) = 1 + 4 = 5
   \]

2. Compute \( x_1 \):
   \[
   x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = -2 - \frac{2}{5} = -2 - 0.4 = -2.4
   \]

3. Calculate \( f(x_1) \) and \( f'(x_1) \):
   \[
   f(-2.4) = -2.4 - (-2.4)^2 + 8 = -2.4 - 5.76 + 8 = -0.16
   \]
   \[
   f'(-2.4) = 1 - 2(-2.4) = 1 + 4.8 = 5.8
   \]

4. Compute \( x_2 \):
   \[
   x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} = -2.4 - \frac{-0.16}{5.8} = -2.4 + 0.027586 = -2.372414
   \]

Thus, for \( x_0 = -2 \), \( x_2 \approx -2.372 \).

---

#### For \( x_0 = 3 \):
1. Calculate \( f(x_0) \) and \( f'(x_0) \):
   \[
   f(3) = 3 - 3^2 + 8 = 3 - 9 + 8 = 2
   \]
   \[
   f'(3) = 1 - 2(3) = 1 - 6 = -5
   \]

2. Compute \( x_1 \):
   \[
   x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 3 - \frac{2}{-5} = 3 + 0.4 = 3.4
   \]

3. Calculate \( f(x_1) \) and \( f'(x_1) \):
   \[
   f(3.4) = 3.4 - (3.4)^2 + 8 = 3.4 - 11.56 + 8 = -0.16
   \]
   \[
   f'(3.4) = 1 - 2(3.4) = 1 - 6.8 = -5.8
   \]

4. Compute \( x_2 \):
   \[
   x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} = 3.4 - \frac{-0.16}{-5.8} = 3.4 - 0.027586 = 3.372414
   \]

Thus, for \( x_0 = 3 \), \( x_2 \approx 3.372 \).

---

### Final Answers:
1. For \( x_0 = -2 \), \( x_2 \approx -2.372 \).
2. For \( x_0 = 3 \), \( x_2 \approx 3.372 \).

Would you like a detailed explanation of the iterations?  
Here are 5 related questions:  
1. What is the significance of Newton's method in finding roots?  
2. Can Newton's method fail or converge slowly? When?  
3. What happens if we use a different starting value for this problem?  
4. How can we improve the accuracy of \( x_2 \) further?  
5. How does Newton's method compare to other root-finding methods like the bisection method?  

**Tip:** Always check \( f'(x) \neq 0 \) when using Newton's method to avoid division by zero!

Use Newton's method to estimate the two zeros of the following function. Start with x0 = -2 for the left zero and x0 = 3 for the right zero. Then for each case, find x2.

Learn how to use Newton's method to estimate the zeros of the function x - x^2 + 8. This problem includes detailed steps to find x2 starting from initial values x0 = -2 and x0 = 3. Explore iterative algorithms and calculus concepts suitable for advanced high school or college-level students.

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