The second-degree Taylor polynomial of a function $f(x)$ centered at $a = 1$ is given by:

$T_2(x) = f(1) + f'(1)(x - 1) + \frac{f''(1)}{2!}(x - 1)^2$

Step-by-Step Solution:

1. Given:

   • $f(1) = 2$
   • $f'(1) = -1$
   • $f''(1) = 4$

2. Plugging in the values: $T_2(x) = 2 + (-1)(x - 1) + \frac{4}{2}(x - 1)^2$

3. Simplifying: $T_2(x) = 2 - (x - 1) + 2(x - 1)^2$

4. Final Expression: $T_2(x) = 2 - (x - 1) + 2(x - 1)^2$

Correct Answer:

$\boxed{T_2(x) = 2 - (x - 1) + 2(x - 1)^2}$

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Would you like me to provide more details or help with further clarification?
Here are five related questions to explore further:

1. What is the general formula for the $n$-degree Taylor polynomial?
2. How do you derive Taylor series for functions centered at different points?
3. Why is the second derivative divided by $2!$ in the polynomial?
4. What are the uses of Taylor polynomials in approximation?
5. How does the Taylor polynomial behave as the degree increases?

Tip: Taylor polynomials provide increasingly accurate approximations near the center point $a$.