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This image shows a Taylor polynomial problem for the function \( f(x) = \sqrt{x+1} \), where the Taylor polynomial of degree 3 is centered at \( x = 3 \). The formula for the Taylor polynomial is given as:

\[
T_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3
\]

### Steps to Correct the Mistakes in the Problem:

1. **Finding \( f(3) \):**
   \[
   f(x) = \sqrt{x+1}, \quad f(3) = \sqrt{3+1} = \sqrt{4} = 2
   \]
   You have an error symbol here, meaning the value of \( f(3) \) must be corrected to \( 2 \).

2. **Finding \( f'(3) \):**
   \[
   f'(x) = \frac{1}{2}(x+1)^{-\frac{1}{2}}, \quad f'(3) = \frac{1}{2}(3+1)^{-\frac{1}{2}} = \frac{1}{2 \times 2} = \frac{1}{4}
   \]
   So, \( f'(3) = \frac{1}{4} \).

3. **Finding \( f''(3) \):**
   \[
   f''(x) = -\frac{1}{4}(x+1)^{-\frac{3}{2}}, \quad f''(3) = -\frac{1}{4}(3+1)^{-\frac{3}{2}} = -\frac{1}{4 \times 8} = -\frac{1}{32}
   \]
   So, \( f''(3) = -\frac{1}{32} \).

4. **Finding \( f'''(3) \):**
   \[
   f'''(x) = \frac{3}{8}(x+1)^{-\frac{5}{2}}, \quad f'''(3) = \frac{3}{8}(3+1)^{-\frac{5}{2}} = \frac{3}{8 \times 32} = \frac{3}{256}
   \]
   So, \( f'''(3) = \frac{3}{256} \).

### Therefore, the corrected Taylor polynomial is:

\[
T_3(x) = 2 + \frac{1}{4}(x - 3) + \frac{-1}{64}(x - 3)^2 + \frac{1}{512}(x - 3)^3
\]

Would you like further details or clarifications on any of these calculations? 

### Related questions:
1. What is the significance of the Taylor polynomial in approximating functions?
2. How does the accuracy of the Taylor polynomial improve with higher-degree terms?
3. How are the derivatives used to construct the Taylor polynomial?
4. Can this method be used for functions with singularities at certain points?
5. What happens to the Taylor polynomial as \(x\) approaches the center \(a\)?

### Tip:
Always verify the function's derivatives carefully, as errors in higher-order derivatives can drastically affect the resulting polynomial.

Find the third-degree Taylor polynomial of the function f(x) = sqrt(x+1) about x = 3. The Taylor polynomial formula is provided as: T_3(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + (f'''(a)/3!)(x-a)^3.

Learn how to find the third-degree Taylor polynomial for the function f(x) = sqrt(x+1) around x = 3. The problem involves calculating f(3), f'(3), f''(3), and f'''(3) to construct the polynomial. Detailed steps and derivatives are provided.

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