Question

In Problem 26 and 27, 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 $\mathit{f}\left(x\right)$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).

$\mathbf{26}\mathbf{.}\mathit{f}\left(x\right)\mathbf{=}\{\begin{array}{ll}3x^2+2x^3,& -1<x<0\\ 3x^2-2x^3,& 0<x<1\end{array}$

Short answer

Answer

The Fourier series of the function, up to the third derivative (when the function becomes discontinuous) are shown below.

Step-by-step solution

Given information

The given function is.

Definition of Fourier series

A Fourier series is a sum that representsa periodic function as a sum of sine and cosine waves. It can be mathematically presented as.

The corresponding Fourier coefficients are as shown below.

Find the derivatives of the given function

The functionis differentiable at a, the limit L exists, then this limit is called the derivative ofat a and is denoted as.

The function and its derivatives are.

Plot the graph of the function and its derivatives

The graph of the function and its derivatives are shown below.

Evaluate the value of coefficients

The function is even, so.The other coefficients are:

Solve further for the value ofas shown below.

Thus, the Fourier series of the function, up to the third derivative (when the function becomes discontinuous) are shown below.