Question

Find the indicated Fourier series. Then differentiate your result repeatedly (both the function and the series) until you get a discontinuous function. Use a computer to plot \(f(x)\) and the derivative functions. For each graph, plot on the same axes one or more terms of the corresponding Fourier series. Note the number of terms needed for a good fit (see comment at the end of the section). $$f(x)=\left\\{\begin{array}{lr} 3 x^{2}+2 x^{3}, & -1< x< 0 \\ 3 x^{2}-2 x^{3}, & 0< x< 1 \end{array}\right.$$

Short answer

Find Fourier coefficients, differentiate until discontinuous, then plot.

Step-by-step solution

Determine the Fourier Coefficients

The first step is to determine the Fourier series coefficients: For the given function, distinguish between the even and odd coefficients: The general form of the Fourier series is \[ f(x) = a_0 + \frac{1}{2}\bigg[ \bigg(a_n \bigg)\text{cos}(n \frac{\text{Pi}}{L}x) + \bigg(b_n \bigg)\text{sin}(n \frac{\text{Pi}}{L}x)\bigg] \bigg] \bigg[ \frac{a}{b} \bigg] \]

Evaluate Both Fourier Coefficients

a(s) = ( 2/L ) ∫ F(x) dx b(s) = ( 2/L ) ∫ F(x) dx where x / t

Step 2. Differentiate the Result

Use the derived series and take derivatives several times until you get a discontinuous function. Differentiating the Fourier series term by term is valid because it is uniformly convergent.

Step 3. Calculate Specific Derivatives

Differentiate both the piecewise original function and the Fourier series representation. You must do this until the resulting function becomes discontinuous.

Plot the Functions

Generate plots for the original function and its derivatives, along with the first few terms of their Fourier series. Notice especially how many terms of the Fourier series are needed for a proper representation.

Key concepts

Fourier Coefficients

To begin with Fourier Series Analysis, we need to determine the Fourier coefficients. These coefficients are crucial in expressing a function using its Fourier series.
For a periodic function, the Fourier series is written as:
\[ f(x) = a_0 + \frac{1}{2}\bigg[ a_n \text{cos}(n \frac{\text{Pi}}{L}x) + b_n \text{sin}(n \frac{\text{Pi}}{L}x) \bigg]\]
The coefficients, \(a_n\) and \(b_n\), are obtained using integrals over one period of the function.
Specifically, they are calculated using:
\[a(s) = ( \frac{2}{L}) \bigg[ \bigint_{-L/2}^{L/2} F(x) \text{cos}(n \frac{\text{Pi}}{L}x) \bigg]\]and
\[b_n = ( \frac{2}{L}) \bigg[ \bigint_{-L/2}^{L/2} F(x) \text{sin}(n \frac{\text{Pi}}{L}x) \bigg]\].
With these coefficients, we build the Fourier series which approximates the original function.

Piecewise Functions

Piecewise functions define two or more functions applied to different intervals on the x-axis. For the given function:
\[ f(x)= \begin{cases} 3 x^{2}+2 x^{3}, & -1< x< 0 \ 3 x^{2}-2 x^{3}, & 0< x< 1 end{cases}\],it is clearly divided into two separate expressions over specific intervals.
Handling piecewise functions in Fourier analysis involves ensuring that the integration for Fourier coefficients is carried out over each interval properly.
This approach helps capture the behavior of the function over various sections, thereby aiding in developing an accurate Fourier series.

Differentiation

Differentiation of the Fourier series involves taking derivatives of the Fourier series term by term.
This process is significant in identifying discontinuities and analyzing how the function behaves when derivatives are applied.
Since our goal is to reach a discontinuous function, repeated differentiation is performed on both the piecewise function and its Fourier series.
Each differentiation should be approached carefully, ensuring uniform convergence. Uniform convergence guarantees that differentiating term by term is valid and the resulting series still converges to the intended function.

Convergence

Convergence is a critical concept in Fourier series analysis.
The Fourier series represents the original function as accurately as possible. A Fourier series converges to a function over an interval if the partial sums approach the function values as the number of terms increases.
Fourier series convergence is influenced by the nature of the original function, specifically regarding whether it is continuous or has discontinuities.
When differentiating, convergence ensures the series still provides a meaningful representation.
Uniform convergence, especially useful in differentiation, guarantees that the Fourier series form of a function remains faithful after applying derivatives.

Graphical Representation

Graphical representation of functions and their Fourier series is essential in visualizing how well the series approximates the original function.
Plotting the original piecewise function and comparing it with various harmonic terms (the first few terms of the Fourier series) shows how many terms are needed for a good fit.
As more terms are added, the Fourier series increasingly resembles the original function, highlighting convergence.
When derivatives are applied, plotting can reveal the points of discontinuity and provide deeper insights into how the original and derived functions behave under the Fourier series transformation.
Visualization enhances understanding and helps verify the theoretical analysis with practical observation.