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### Problem Description:
The problem asks for the first three terms in the Taylor series for a function \( f(x) \) centered at \( x = 0 \). The function \( f(x) \) is given by:

\[
f(x) = \left( 1 - x \right)^{-\frac{1}{3}}
\]

We are to find the Taylor series approximation up to the second degree, i.e., \( f(x) \approx 1 + ax + bx^2 \).

### Solution:
To find the Taylor series expansion, we follow these steps:

1. **Determine \( f(0) \):**
   \[
   f(0) = \left( 1 - 0 \right)^{-\frac{1}{3}} = 1
   \]

2. **First Derivative \( f'(x) \):**
   \[
   f'(x) = -\frac{1}{3} \cdot \left( 1 - x \right)^{-\frac{4}{3}} \cdot (-1) = \frac{1}{3} \cdot \left( 1 - x \right)^{-\frac{4}{3}}
   \]
   Therefore:
   \[
   f'(0) = \frac{1}{3} \cdot 1 = \frac{1}{3}
   \]

3. **Second Derivative \( f''(x) \):**
   \[
   f''(x) = \frac{1}{3} \cdot -\frac{4}{3} \cdot \left( 1 - x \right)^{-\frac{7}{3}} \cdot (-1) = \frac{4}{9} \cdot \left( 1 - x \right)^{-\frac{7}{3}}
   \]
   Therefore:
   \[
   f''(0) = \frac{4}{9} \cdot 1 = \frac{4}{9}
   \]

4. **Taylor Series Expansion:**
   The Taylor series is given by:
   \[
   f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots
   \]
   Substituting the values we have:
   \[
   f(x) \approx 1 + \frac{1}{3}x + \frac{4}{9} \cdot \frac{1}{2}x^2 = 1 + \frac{1}{3}x + \frac{2}{9}x^2
   \]

### Final Answer:
The Taylor series approximation up to the second degree is:
\[
f(x) \approx 1 + \frac{1}{3}x + \frac{2}{9}x^2
\]

Would you like more details or have any questions about this process? Here are some related questions for further exploration:

1. What is the general formula for the nth derivative of \( f(x) = (1 - x)^{-\frac{1}{3}} \)?
2. How does the radius of convergence affect the Taylor series for \( f(x) = (1 - x)^{-\frac{1}{3}} \)?
3. Can you compute the Taylor series for \( f(x) = (1 - x)^{-\frac{1}{2}} \) using the same method?
4. What is the significance of the Taylor series in approximating functions?
5. How can we use Taylor series to approximate functions at points other than \( x = 0 \)?

**Tip:** When expanding a function using Taylor series, make sure to calculate each derivative carefully, as errors in the derivatives will propagate through the entire expansion.

Find the first three terms in the Taylor series for f(x) centered at x = 0 where f(x) = (1 - x)^(-1/3).

Explore the first three terms in the Taylor series for f(x) = (1 - x)^(-1/3) centered at x = 0. This detailed solution covers the calculation of derivatives and the application of the Taylor series formula. Ideal for undergraduate calculus students.

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