https://file.aiquickdraw.com/mathbot/file/1727269177216fo0g5ju2.jpg

Let's break down and solve each part of the problem step by step.

### (a) Write the function \( f(x) \) using your input variables \( A \) and \( B \).

The given function \( f(x) \) is piecewise, defined as:

\[
f(x) = 
\begin{cases} 
A & \text{for } -B \leq x < 0, \\
0 & \text{for } 0 \leq x < B.
\end{cases}
\]

This represents a function that takes the value \( A \) over the interval \( -B \leq x < 0 \), and 0 over the interval \( 0 \leq x < B \).

### (b) Sketch one period of the function for part (a).

To sketch one period:
- For \( -B \leq x < 0 \), the function value is \( A \).
- For \( 0 \leq x < B \), the function value is 0.
This creates a step-like waveform, which repeats with period \( 2B \) (assuming \( 2L = 2B \)).

### (c) Determine the length of the period \( 2L \) in units of \( x \).

Given that the function is defined over \( -B \leq x < B \), the length of the period is:

\[
2L = 2B
\]

So the period of the function is \( 2B \) in units of \( x \).

### (d) Describe the type of the function from part (b).

The function is a **discontinuous piecewise function**. Specifically, it is a **step function** that alternates between two constant values, \( A \) and 0, over each half-period. It has a discontinuity at \( x = 0 \) and repeats with a period of \( 2B \).

### (e) Determine the Fourier coefficients of this function \( f(x) \).

The Fourier coefficients are given as follows:

1. **\( a_0 \) (the average value of the function over one period):**

\[
a_0 = \frac{1}{2L} \int_{-L}^{L} f(x) \, dx = \frac{1}{2L} \left( \int_{-B}^{0} A \, dx + \int_0^{B} 0 \, dx \right)
\]

Evaluating the integrals:

\[
a_0 = \frac{1}{2B} \left( A \cdot B \right) = \frac{A}{2}
\]

2. **\( a_n \) (cosine coefficients):**

\[
a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n \pi x}{L}\right) \, dx = \frac{1}{B} \int_{-B}^{0} A \cos\left(\frac{n \pi x}{B}\right) \, dx
\]

Solving this integral, we get:

\[
a_n = \frac{2A}{n \pi} \sin\left(\frac{n \pi}{2}\right)
\]

This simplifies based on the value of \( \sin\left(\frac{n \pi}{2}\right) \), which alternates between 0, 1, and -1 for different values of \( n \).

3. **\( b_n \) (sine coefficients):**

\[
b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n \pi x}{L}\right) \, dx = \frac{1}{B} \int_{-B}^{0} A \sin\left(\frac{n \pi x}{B}\right) \, dx
\]

Evaluating this integral results in \( b_n = 0 \), since the sine function is odd and the function \( f(x) \) is symmetric about 0.

### (f) Write down the corresponding Fourier series for \( n = 1 \) to \( 6 \).

The Fourier series is:

\[
f(x) = \frac{A}{2} + \sum_{n=1}^{6} \frac{2A}{n \pi} \sin\left(\frac{n \pi}{2}\right) \cos\left(\frac{n \pi x}{B}\right)
\]

This uses the computed Fourier coefficients.

---

Would you like any further explanation on any of the parts? Here are five additional related questions to consider:

1. How would the Fourier series change if \( f(x) \) was symmetric about \( x = 0 \)?
2. What is the significance of the Fourier sine coefficients being zero for this function?
3. How can we generalize this Fourier series for any periodic piecewise function?
4. What would the graph of the Fourier series look like for small values of \( n \)?
5. How does the discontinuity at \( x = 0 \) affect the convergence of the Fourier series?

**Tip:** The Fourier series provides a powerful way to represent periodic functions as an infinite sum of sines and cosines, even for discontinuous functions like the one given.

A function with unknown period 2L is f(x) = { A, -B ≤ x < 0; 0, 0 ≤ x < B }, and the task is to compute its Fourier series coefficients.

Learn how to calculate the Fourier series of a piecewise function with unknown period 2L. This solution includes detailed steps for finding the Fourier coefficients a0, an, and bn, and writing the Fourier series for n = 1 to 6. Suitable for undergraduate students studying Fourier series and integration.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation