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The problem involves expanding the function \( f(x) = 1 - x \) using Fourier series over the interval \( 0 < x < 2 \). Let's break down each part step by step.

### (a) Full-Range Fourier Series Expansion
The full-range Fourier series represents a function as a sum of sine and cosine terms over the interval \( [-L, L] \), where the function is extended periodically. Here, \( L = 1 \), since the interval length is 2.

The general form of the full-range Fourier series is:
\[
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left( \frac{n\pi x}{L} \right) + b_n \sin\left( \frac{n\pi x}{L} \right) \right)
\]
Where:
- \( a_0 = \frac{1}{L} \int_0^{2L} f(x) \, dx \)
- \( a_n = \frac{1}{L} \int_0^{2L} f(x) \cos\left( \frac{n\pi x}{L} \right) \, dx \)
- \( b_n = \frac{1}{L} \int_0^{2L} f(x) \sin\left( \frac{n\pi x}{L} \right) \, dx \)

Here \( f(x) = 1 - x \), and \( L = 1 \), so the interval is \( [0, 2] \).

We now compute the coefficients.

1. **\( a_0 \) (DC term):**
   \[
   a_0 = \frac{1}{1} \int_0^2 (1 - x) \, dx = \int_0^2 (1 - x) \, dx
   \]
   \[
   a_0 = \left[ x - \frac{x^2}{2} \right]_0^2 = (2 - 2) - (0) = 0
   \]

2. **\( a_n \) (Cosine coefficients):**
   \[
   a_n = \int_0^2 (1 - x) \cos(n\pi x) \, dx
   \]
   For each \( n \), we would need to solve this integral, which will depend on the value of \( n \).

3. **\( b_n \) (Sine coefficients):**
   \[
   b_n = \int_0^2 (1 - x) \sin(n\pi x) \, dx
   \]
   This also depends on the value of \( n \).

### (b) Half-Range Cosine Series Expansion
In this case, we use the half-range cosine series since the function is even. The form is:
\[
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left( \frac{n\pi x}{L} \right)
\]
The coefficients are given by:
- \( a_0 = \frac{2}{L} \int_0^L f(x) \, dx \)
- \( a_n = \frac{2}{L} \int_0^L f(x) \cos\left( \frac{n\pi x}{L} \right) \, dx \)

For \( L = 2 \) and \( f(x) = 1 - x \), we calculate the coefficients similarly to part (a).

### (c) Half-Range Sine Series Expansion
For the half-range sine series (for odd extensions), the form is:
\[
f(x) = \sum_{n=1}^{\infty} b_n \sin\left( \frac{n\pi x}{L} \right)
\]
The coefficients are given by:
- \( b_n = \frac{2}{L} \int_0^L f(x) \sin\left( \frac{n\pi x}{L} \right) \, dx \)

Again, substituting \( L = 2 \) and \( f(x) = 1 - x \), we compute the sine series.

### (d) Graphs
For the graphical part, you are required to:
1. Plot the original function \( f(x) = 1 - x \).
2. Plot the periodic extensions of the function for the full-range Fourier series, half-range cosine series, and half-range sine series for \( -8 \leq x \leq 8 \).

This will illustrate how the Fourier series approximates the original function.

---

Would you like more details on how to compute each of the integrals for the Fourier coefficients? I can guide you through specific steps for any of the expansions if you'd like. Here are a few follow-up questions that might help:

1. Do you need a detailed breakdown of the integration for \( a_n \) and \( b_n \)?
2. Would you like to see the calculations for specific values of \( n \) in the full-range Fourier series?
3. Are you looking for the graphical representation of the Fourier series approximations?
4. Should I explain more about the symmetry considerations for the half-range expansions?
5. Do you want to focus on any specific part of the question first?

**Tip**: Fourier series can be tricky to compute manually for complex functions, but understanding the role of each term helps in interpreting the periodic behavior!

The function f(x) = 1 - x is to be represented by a Fourier series expansion over the finite interval 0 < x < 2. Obtain a suitable full-range Fourier series expansion, half-range cosine series expansion, and half-range Fourier sine series expansion. Draw graphs of f(x) and of the periodic functions represented by each of the three series for -8 ≤ x ≤ 8.

Learn how to represent the function f(x) = 1 - x using a full-range Fourier series expansion, half-range cosine series, and half-range sine series. This problem involves calculating the Fourier coefficients and graphing the periodic extensions over the interval -8 ≤ x ≤ 8.

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