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.

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.