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Your document is an assignment for the MAT3700 course, focusing on Fourier Series. Here's a breakdown of the tasks:

### Questions from the Assignment:
1. **(15 Marks)**: Obtain the Fourier series expansion for a periodic function \( f(t) \) with period \( 2\pi \), defined as:
   \[
   f(t + 2\pi) = f(t), \quad -\pi < t < \pi.
   \]

2. **(25 Marks)**: Consider a function \( f(t) \) defined over \( 0 < t < 4 \):
   \[
   f(t) = \begin{cases}
   t, & 0 < t < 2, \\
   4 - t, & 2 < t < 4.
   \end{cases}
   \]
   - 2.1: Find the half-range **cosine** Fourier expansion.
   - 2.2: Find both **cosine and sine** half-range Fourier expansions.

These cover topics from **Fourier Series for periodic functions**, **even/odd functions**, and **half-range expansions** based on your study material.

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Would you like help in solving specific parts of the questions, or would you like a detailed explanation of the Fourier Series and how to approach this assignment?

### Follow-up Questions:
1. What steps are involved in deriving a Fourier series for a periodic function?
2. How do you find the Fourier coefficients for a function defined over a specific range?
3. What is the significance of even and odd functions in Fourier expansions?
4. How do half-range cosine expansions differ from full Fourier series expansions?
5. How do you identify whether to use sine or cosine in half-range expansions?

### Tip:
In half-range expansions, use only sine functions for odd extensions and cosine functions for even extensions.

Obtain the Fourier series expansion for a periodic function f(t) with period 2π, and find the half-range cosine and sine Fourier expansions for a specific piecewise function.

Explore how to find the Fourier series expansion for a periodic function and the half-range cosine and sine Fourier expansions for a piecewise-defined function. This problem includes detailed steps for understanding Fourier Series concepts, suitable for university-level students.

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