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Let's solve this step by step. We need to find the third-order Taylor polynomial approximation of the function \( f(x) = \frac{1}{x} \) about \( x = 1 \).

### Step 1: Compute the derivatives of \( f(x) \) and evaluate them at \( x = 1 \).

1. **Function value** \( f(x) = \frac{1}{x} \):

   \[
   f(1) = \frac{1}{1} = 1
   \]

2. **First derivative** \( f'(x) = -\frac{1}{x^2} \):

   \[
   f'(1) = -\frac{1}{1^2} = -1
   \]

3. **Second derivative** \( f''(x) = \frac{2}{x^3} \):

   \[
   f''(1) = \frac{2}{1^3} = 2
   \]

4. **Third derivative** \( f'''(x) = -\frac{6}{x^4} \):

   \[
   f'''(1) = -\frac{6}{1^4} = -6
   \]

### Step 2: Fill in the results.

- \( f(1) = 1 \)
- \( f'(1) = -1 \)
- \( f''(1) = 2 \)
- \( f'''(1) = -6 \)

Would you like to proceed with constructing the third-order Taylor polynomial using these values?

Find the third order Taylor polynomial approximation to 1/x about x = 1.

Learn how to find the third-order Taylor polynomial approximation of 1/x about x = 1. This problem covers important calculus concepts, such as derivatives and polynomial approximation, using the Taylor series expansion. The function and its derivatives are computed step-by-step, providing a clear solution.

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