Calculate two iterations of Newton's Method to approximate a zero of the function using the given initial guess. (Round your answers to four decimal places.)

make a table for f(x) = cos x,    x1 = 1.9

rows:
n
1
2
columns: (xn), f(xn), f'(xn), f(xn)/f'(xn), xn-f(xn)/f'(xn)

To apply Newton's Method, the formula for each iteration is:

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

We are given the function \( f(x) = \cos(x) \), and we need its derivative \( f'(x) \). 

Since \( f(x) = \cos(x) \), we know that:

\[
f'(x) = -\sin(x)
\]

We are starting with the initial guess \( x_1 = 1.9 \).

Let's calculate two iterations of Newton's Method, rounding values to four decimal places. I'll create a table for this.

### First, we summarize the required columns:

- \( x_n \) (current estimate)
- \( f(x_n) \) (function value at \( x_n \))
- \( f'(x_n) \) (derivative value at \( x_n \))
- \( \frac{f(x_n)}{f'(x_n)} \)
- \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \) (next estimate)

### Initial Guess:
- \( x_1 = 1.9 \)

Let’s calculate the values:

#### Iteration 1 (n = 1):
- \( f(x_1) = \cos(1.9) \)
- \( f'(x_1) = -\sin(1.9) \)
- Calculate \( \frac{f(x_1)}{f'(x_1)} \)
- Update \( x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \)

#### Iteration 2 (n = 2):
- \( f(x_2) = \cos(x_2) \)
- \( f'(x_2) = -\sin(x_2) \)
- Calculate \( \frac{f(x_2)}{f'(x_2)} \)
- Update \( x_3 = x_2 - \frac{f(x_2)}{f'(x_2)} \)

Let’s compute these values now.Here is the table with two iterations of Newton's Method for approximating a zero of \( f(x) = \cos(x) \) starting with \( x_1 = 1.9 \):

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
n & x_n & f(x_n) & f'(x_n) & \frac{f(x_n)}{f'(x_n)} & x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \\
\hline
1 & 1.9000 & -0.3233 & -0.9463 & 0.3416 & 1.5584 \\
2 & 1.5584 & 0.0124 & -0.9999 & -0.0124 & 1.5708 \\
\hline
\end{array}
\]

- After the first iteration, \( x_2 \approx 1.5584 \).
- After the second iteration, \( x_3 \approx 1.5708 \).

This process is converging to a root of \( \cos(x) = 0 \), which is close to \( x = \frac{\pi}{2} \approx 1.5708 \).

---

Would you like more details, or do you have any questions?

### Follow-up Questions:
1. How does Newton's Method work with other types of functions?
2. What happens if we start with a different initial guess?
3. Can Newton's Method fail to converge?
4. Why is the derivative necessary in Newton's Method?
5. What is the geometric interpretation of Newton's Method?

### Tip:
For faster convergence, the initial guess should be close to the actual root.

Calculate two iterations of Newton's Method to approximate a zero of the function using the given initial guess. (Round your answers to four decimal places.)

Make a table for f(x) = cos x, x1 = 1.9.

Explore how to use Newton's Method to approximate the root of the function cos(x) starting with an initial guess of 1.9. This detailed solution includes step-by-step calculations over two iterations, focusing on key calculus and trigonometric concepts. Suitable for Grades 11-12 and college-level students.

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