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To solve this problem using Newton's Method, we need to find the point where the two functions \( f(x) = x^2 \) and \( g(x) = \cos x \) intersect. This requires finding the root of the equation \( f(x) - g(x) = 0 \), or more specifically:

\[
h(x) = x^2 - \cos x = 0
\]

### Steps for Newton's Method

1. **Define the function and its derivative:**  
   - Let \( h(x) = x^2 - \cos x \).
   - The derivative \( h'(x) = 2x + \sin x \).

2. **Newton's Method Formula:**
   - Newton's Method formula is:
     \[
     x_{n+1} = x_n - \frac{h(x_n)}{h'(x_n)}
     \]
   - Here, we update \( x_n \) iteratively until the value converges to the desired accuracy.

3. **Initial Guess:**  
   Since we are asked to use technology for good initial approximations, visually inspecting or calculating values around \( x = 0 \) to \( x = 1 \) would be helpful. Let’s start with \( x_0 = 0.7 \) as an initial guess, based on the fact that \( x^2 \) and \( \cos x \) intersect around this value.

4. **Iterate to Convergence:**  
   Apply Newton's Method iteratively to get a value for \( x \) accurate to three decimal places.

Let me calculate the iterations for you:Using Newton's Method with an initial guess of \( x_0 = 0.7 \), the solution converges to approximately \( x = 0.824 \) to three decimal places. This means that the functions \( f(x) = x^2 \) and \( g(x) = \cos x \) are approximately equal when \( x \approx 0.824 \).

Would you like further details on the process or have any other questions?

Here are some related questions that might help deepen your understanding:

1. What is Newton's Method, and how does it work for finding roots?
2. How can you determine a good initial guess for Newton's Method?
3. Why is the convergence of Newton's Method sensitive to the choice of the initial guess?
4. What happens if we use Newton's Method on a function that has no roots?
5. How would we modify the process if we were asked for a different level of accuracy?

**Tip:** Newton's Method requires both the function and its derivative. Ensuring the derivative does not get too close to zero is key to avoiding large, unstable jumps in estimates.

In Exercises 13 – 16, use Newton's Method to approximate when the given functions are equal, accurate to 3 places after the decimal. Use technology to obtain good initial approximations.

13. f(x) = x^2, g(x) = cos x

Learn how to apply Newton's Method to approximate the intersection of the functions f(x) = x^2 and g(x) = cos(x) with three decimal accuracy. This example involves finding the root of the function h(x) = x^2 - cos(x) and iterating using derivatives for precision.

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