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?