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Let's analyze each problem in this worksheet to apply Newton's Method. Newton's Method is an iterative method for finding approximations to the roots (or zeros) of a real-valued function. Given a function \( f(x) \) and an initial guess \( x_1 \), the method iterates according to the formula:

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

Now, let's go through each question.

### Problem 1
- **Given**: \( x_1 = 0 \)
- **Goal**: Use Newton's method to find \( x_2 \) and \( x_3 \), then explain or calculate the root estimate.

The first graph shows a function that starts positive, crosses the x-axis, and becomes negative. With an initial guess of \( x_1 = 0 \), the tangent line at \( x_1 \) would need to be calculated based on the slope (which is \( f'(x_1) \)), but it seems that the slope might be zero or undefined at \( x = 0 \) based on the curve's appearance. If the derivative \( f'(x_1) = 0 \), Newton's Method will fail at this point because division by zero in the formula is undefined. Therefore, in this case, Newton's Method might not be able to provide a meaningful estimate.

### Problem 2
- **Given**: \( x_1 = 1 \)
- **Goal**: Use Newton's method to find \( x_2 \) and \( x_3 \), then explain or calculate the root estimate.

The second graph shows a curve that approaches the x-axis asymptotically as \( x \) increases. Starting with \( x_1 = 1 \), we would again apply Newton's method using the formula above. Here, \( f(x) \) does not appear to cross the x-axis; it just approaches it. This means that Newton's Method may keep generating values without converging to a root because there isn't an actual root (the function does not cross zero). As a result, Newton's Method might not yield a finite estimate for the root.

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Would you like a more detailed step-by-step calculation of each iteration, or do you have specific questions on how Newton's Method would behave here? 

Here are five questions for further exploration:
1. What conditions make Newton's Method fail to converge?
2. How does the choice of the initial guess \( x_1 \) affect the convergence of Newton's Method?
3. Can Newton’s Method be adapted for functions with no real roots?
4. What are alternative root-finding methods if Newton’s Method fails?
5. How can you tell if Newton’s Method will diverge or oscillate instead of converging?

**Tip**: When using Newton's Method, always check the derivative at your starting point. If it's zero, the method cannot proceed from that point.

Given x₁ = 0 and x₁ = 1 for the respective functions in each problem, use Newton's method to find x₂ and x₃, and then label Newton’s method’s estimate for the root of each function or explain why Newton’s method will not be able to give an estimate.

Explore how to apply Newton's Method to find roots of functions in calculus. With given starting points x₁ = 0 and x₁ = 1, calculate subsequent approximations x₂ and x₃, and understand why Newton's Method may fail to provide an estimate for some functions.

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