To apply the secant method to the function $F(x) = x^2 - 2x + 6$, we need to use the following iterative formula:

$x_{n+1} = x_n - \frac{F(x_n)(x_n - x_{n-1})}{F(x_n) - F(x_{n-1})}$

We are given two initial guesses:

• $x_0 = 4$
• $x_1 = 4.5$

Step 1: Compute $F(x_0)$ and $F(x_1)$

The function is $F(x) = x^2 - 2x + 6$.

• $F(4) = 4^2 - 2(4) + 6 = 16 - 8 + 6 = 14$
• $F(4.5) = 4.5^2 - 2(4.5) + 6 = 20.25 - 9 + 6 = 17.25$

Step 2: Apply the secant method formula

$x_2 = 4.5 - \frac{17.25(4.5 - 4)}{17.25 - 14}$ Simplifying: $x_2 = 4.5 - \frac{17.25(0.5)}{3.25}$ $x_2 = 4.5 - \frac{8.625}{3.25} = 4.5 - 2.654$ $x_2 \approx 1.846$

Thus, the first iterative update is approximately 1.84.

Final Answer: 1.84

Would you like further details or have any questions?

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Here are five related questions:

1. How does the secant method compare to the Newton-Raphson method?
2. What is the condition for convergence of the secant method?
3. Can the secant method fail, and if so, under what circumstances?
4. How many iterations are typically required for the secant method to converge?
5. What are the advantages of the secant method over the bisection method?

Tip: The secant method does not require the calculation of the derivative of the function, which makes it more efficient than methods like Newton-Raphson for some problems.