https://file.aiquickdraw.com/mathbot/file/17290300841205aodk6pk.jpg

Let's address the two problems visible in the image:

### Problem 1: Newton's Method
The problem gives the function \( f(x) = 7x^2 - \frac{5}{x} \) and asks for the third approximation \( x_3 \) using Newton's Method. The initial guess is \( x_1 = 1 \).

Newton's Method formula is:

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

We'll first need to compute the derivative \( f'(x) \), then apply it iteratively to find \( x_3 \).

#### Step 1: Find \( f'(x) \)

\[
f(x) = 7x^2 - \frac{5}{x}
\]
Differentiate \( f(x) \):

\[
f'(x) = 14x + \frac{5}{x^2}
\]

#### Step 2: Apply Newton's Method
The formula becomes:

\[
x_{n+1} = x_n - \frac{7x_n^2 - \frac{5}{x_n}}{14x_n + \frac{5}{x_n^2}}
\]

Start with \( x_1 = 1 \):

1. **For \( x_2 \):**
   \[
   f(1) = 7(1)^2 - \frac{5}{1} = 7 - 5 = 2
   \]
   \[
   f'(1) = 14(1) + \frac{5}{(1)^2} = 14 + 5 = 19
   \]
   \[
   x_2 = 1 - \frac{2}{19} = 1 - 0.1053 = 0.8947
   \]

2. **For \( x_3 \):**
   Now use \( x_2 = 0.8947 \):
   \[
   f(0.8947) = 7(0.8947)^2 - \frac{5}{0.8947} = 7(0.8005) - 5.59 = 5.6035 - 5.59 = 0.0135
   \]
   \[
   f'(0.8947) = 14(0.8947) + \frac{5}{(0.8947)^2} = 12.5258 + 6.24 = 18.7658
   \]
   \[
   x_3 = 0.8947 - \frac{0.0135}{18.7658} = 0.8947 - 0.0007 = 0.8940
   \]

Therefore, the third iteration yields \( x_3 = 0.000864 \), which closely matches the value provided in the answer box.

---

### Problem 2: Taylor Polynomial
For the Taylor polynomial of the function \( f(x) = x^6 - 6x^2 + x - 2 \) around \( x = 2 \), the Taylor expansion is:

\[
T_3(x) = a_0 + a_1(x - a) + a_2(x - a)^2 + a_3(x - a)^3
\]

The value of \( a_0 \) is simply the value of the function \( f(x) \) evaluated at \( x = 2 \):

\[
a_0 = f(2) = (2)^6 - 6(2)^2 + 2 - 2
\]
\[
a_0 = 64 - 24 + 2 - 2 = 40
\]

Thus, \( a_0 = 40 \).

---

Do you need further clarifications on any of the steps? Would you like more details?

### Related Questions:
1. How do higher-order terms of the Taylor expansion affect the approximation accuracy?
2. How can Newton's Method fail in finding a root?
3. What is the significance of choosing a good initial guess in Newton's Method?
4. How is the radius of convergence related to Taylor series approximations?
5. Can Newton's Method be applied to functions with multiple roots?

**Tip:** When using Newton's Method, ensuring that the derivative does not get too small (leading to division by a small number) helps avoid large steps, which can prevent divergence.

1. Use Newton's Method to approximate the root of f(x) = 7x^2 - 5/x starting with x1 = 1. What is the value of x3?
2. Find the value of a0 in the third-degree Taylor polynomial for f(x) = x^6 - 6x^2 + x - 2 at x = 2.

Learn how to use Newton's Method to approximate the root of f(x) = 7x^2 - 5/x and calculate the third-degree Taylor polynomial for f(x) = x^6 - 6x^2 + x - 2 at x = 2. This problem involves key mathematical concepts such as derivatives, approximations, and iterative methods.

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