Find the Taylor polynomial of order 3 generated by f at a.

f(x) = x2, a = 3
Group of answer choices

P3(x) = 1 + 6(x - 3) + 9(x - 3)2 + 12(x - 3)3

P3(x) = 9 + 6(x - 3) + (x - 3)2

P3(x) = 1 + 18(x - 3) + 81(x - 3)2 + 324(x - 3)3

P3(x) = 9 + 6(x - 3) + 9(x - 3)2 + 12(x - 3)3

To find the Taylor polynomial of order 3 generated by \( f(x) = x^2 \) at \( a = 3 \), we need to use the formula for the Taylor series expansion:

\[ P_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 \]

where \( n = 3 \) in this case.

1. **Compute \( f(a) \):**
   \[ f(3) = 3^2 = 9 \]

2. **Compute the first derivative \( f'(x) \):**
   \[ f'(x) = 2x \]
   \[ f'(3) = 2 \cdot 3 = 6 \]

3. **Compute the second derivative \( f''(x) \):**
   \[ f''(x) = 2 \]
   \[ f''(3) = 2 \]

4. **Compute the third derivative \( f'''(x) \):**
   \[ f'''(x) = 0 \]
   \[ f'''(3) = 0 \]

Now, substitute these values into the Taylor series formula:

\[ P_3(x) = f(3) + f'(3)(x - 3) + \frac{f''(3)}{2!}(x - 3)^2 + \frac{f'''(3)}{3!}(x - 3)^3 \]

\[ P_3(x) = 9 + 6(x - 3) + \frac{2}{2}(x - 3)^2 + \frac{0}{6}(x - 3)^3 \]

\[ P_3(x) = 9 + 6(x - 3) + (x - 3)^2 \]

Thus, the correct Taylor polynomial of order 3 for \( f(x) = x^2 \) at \( a = 3 \) is:

\[ P_3(x) = 9 + 6(x - 3) + (x - 3)^2 \]

Therefore, the correct choice is:
\[ \boxed{P_3(x) = 9 + 6(x - 3) + (x - 3)^2} \]

Would you like more details or have any questions? 

Here are some additional questions to deepen your understanding:

1. How do you derive the formula for the Taylor series expansion?
2. Why is the third derivative \( f'''(x) \) zero in this case?
3. What is the significance of each term in the Taylor polynomial?
4. How does the accuracy of a Taylor polynomial change with the order \( n \)?
5. Can you explain the process of finding higher-order derivatives for more complex functions?
6. What are some practical applications of Taylor series in real-world scenarios?
7. How do you find the radius of convergence for a Taylor series?
8. What happens to the Taylor series as \( x \) approaches \( a \)?

**Tip:** Always verify each term in the Taylor polynomial by computing the necessary derivatives carefully, ensuring all calculations align with the given function and point of expansion.

Learn how to find the Taylor polynomial of order 3 for the function f(x) = x^2 at a = 3. This problem involves using the Taylor series expansion formula to compute the polynomial, including detailed steps for derivatives and substitutions.

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