https://file.aiquickdraw.com/mathbot/file/1726710788172nnza1tcx.jpg

We are tasked with finding the Fourier series representation of the given periodic function \( f(t) \) based on the provided piecewise plot, up to the 5th harmonic. I will outline the detailed steps to calculate the Fourier series using the information in the figure and proceed to compute the Fourier coefficients. 

### Step 1: Analyze the given function
From the graph:
- The period \( T = 2\pi \), which means the fundamental angular frequency is \( \omega_0 = \frac{2\pi}{T} = 1 \).
- The function values based on the points marked in the graph are:
  - \( f(0) = 0 \)
  - \( f(\pi) = 1 \)
  - \( f(2\pi) = 0 \)
  - \( f(\pi/2) = 0.5 \)
  - \( f(3\pi/2) = -0.5 \)
  - The function appears to be piecewise linear.

The function \( f(t) \) is periodic with a period \( T = 2\pi \), so we will calculate the Fourier series in the form:
\[
f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(n \omega_0 t) + b_n \sin(n \omega_0 t) \right)
\]
where \( \omega_0 = 1 \) is the fundamental angular frequency.

### Step 2: Fourier coefficients formulas
We use the following formulas for the Fourier coefficients:
- \( a_0 \) is the average value over one period:
\[
a_0 = \frac{1}{T} \int_0^{T} f(t) \, dt
\]
- \( a_n \) (for \( n \geq 1 \)) is the cosine Fourier coefficient:
\[
a_n = \frac{2}{T} \int_0^{T} f(t) \cos(n \omega_0 t) \, dt
\]
- \( b_n \) (for \( n \geq 1 \)) is the sine Fourier coefficient:
\[
b_n = \frac{2}{T} \int_0^{T} f(t) \sin(n \omega_0 t) \, dt
\]

Given that the function is defined piecewise linearly, we will need to break the integral calculations into segments according to the different linear pieces of the graph.

### Step 3: Calculate \( a_0 \)
We begin by calculating \( a_0 \), which is the average value of the function over one period.

\[
a_0 = \frac{1}{2\pi} \int_0^{2\pi} f(t) \, dt
\]

Since the function is linear and symmetric over the interval \( [0, 2\pi] \), I will calculate the areas piecewise.

### Step 4: Calculate \( a_n \) and \( b_n \)
Next, we will compute \( a_n \) and \( b_n \) using the piecewise segments, and then express the Fourier series up to the 5th harmonic.

Let me perform the calculations for these coefficients.### Step 5: Results of Fourier coefficients

1. The DC term \( a_0 \) (the average value of the function):
   \[
   a_0 = \frac{1}{2}
   \]
   
2. The cosine coefficients \( a_n \) for the first five harmonics:
   \[
   a_1 = -\frac{4}{\pi^2}, \quad a_2 = 0, \quad a_3 = -\frac{4}{9\pi^2}, \quad a_4 = 0, \quad a_5 = -\frac{4}{25\pi^2}
   \]

3. The sine coefficients \( b_n \) for the first five harmonics are all zero:
   \[
   b_1 = b_2 = b_3 = b_4 = b_5 = 0
   \]

### Step 6: Fourier series up to the 5th harmonic
The Fourier series representation of the function \( f(t) \), up to the 5th harmonic, is:
\[
f(t) = \frac{1}{2} - \frac{4}{\pi^2} \cos(t) - \frac{4}{9\pi^2} \cos(3t) - \frac{4}{25\pi^2} \cos(5t)
\]

This series captures the function's behavior with its fundamental frequency and up to the 5th harmonic.

Would you like further details on any part of the solution or any clarification?

### Questions for further exploration:
1. How is the Fourier series affected if the period changes?
2. Can you explain the role of harmonics in approximating periodic functions?
3. What would happen if the function had discontinuities in the graph?
4. How would the Fourier series change if sine terms were non-zero?
5. What practical applications can this Fourier series be used for in engineering?

### Tip:
For smoother functions, higher harmonics contribute less, meaning we can often approximate the function well using only a few terms in the Fourier series.

Given the figure shown below calculate the Fourier series representation to at least 5th harmonics of its fundamental frequency take note please integrate over whole cycle show your detailed solution.

Explore a detailed step-by-step solution for calculating the Fourier series of a piecewise function up to the 5th harmonic, including how to compute various Fourier coefficients and their application.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation