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.