Question

(a) Sketch at least three periods of the graph of the function represented by cosine series for f(x)in Problem 9.

(b) Sketch at least three periods of the graph of the exponential Fourier series of period2 for f(x)in Problem 9.

(c) To what value does the cosine series in (a) coverage at x=0? At x=1? At x=2? At x=-2?

(d) To what value does the exponential series in (b) converge at x=0? At x=1? At$\mathit{x}\mathbf{=}\frac{\mathbf{3}}{\mathbf{2}}$? At x=-2.

Short answer

(a) The graph of the function with at least three periods represented by cosine series for f(x) in Problem 9 is shown below:

(b) The graph of the function with at least three points of exponential Fourier with period 2 is shown below:

(c) The value of cosine series converges at $x=0,x=1,x=2,x=-2$is $f_e\left(0\right)=-1,f_e\left(1\right)=-\frac{1}{2},f_e\left(\frac{3}{2}\right)=-2,f_e\left(-2\right)=-1$.

(d) The value of the exponential series converges at$x=0,x=1,x=\frac{3}{2},x=-2$is$f_e\left(0\right)=-1,f_e\left(1\right)=-\frac{1}{2},f_e\left(\frac{3}{2}\right)=-2,f_e\left(-2\right)=-1$

Step-by-step solution

Define Function

A statement, rule, or law that specifies the relationship between the independent and dependent variables is known as a function in mathematics. There are functions everywhere in mathematics, and they are essential for developing connections between the disciplines.

Given parameter

Given the function of the problem 9:

$f\left(x\right)=\left\{\begin{array}{c}x\\ -2\end{array}\begin{array}{ccc}0& <x& <1\\ 1& <x& <2\end{array}\right\}$

Sketch the graph of the function with at least three periods that represent the cosine series

The graph of the function with at least three periods that represents the cosine series is shown in the below graph:

Sketch the graph of the exponential Fourier series of period with at least three periods

As the graph of the function is having even extension (observed by the above graph).

Thus, the exponential series will converge to it.

Then the graph of the of the exponential Fourier series of period with at least three periods.

Find the value of the cosine series in (a).

In the graph 1 , by Dirichlet condition

At x=0,

$f_c\left(0\right)=0$

At x=1 ,

$f_c\left(1\right)=-\frac{1}{2}$

At x=2,

$f_c\left(2\right)=-2$

At x=-2 ,

$f_c\left(-2\right)=-2$

So,$f_c\left(0\right)=0,f_c\left(1\right)=-\frac{1}{2},f_c\left(2\right)=-1,f_c\left(-2\right)=-2$

Find the value of the exponential series in (b)

In the graph 2 , by Dirichlet condition

At x=0 ,

$f_e\left(0\right)=-1$

At x=1 ,

$f_e\left(1\right)=-\frac{1}{2}$

At x=$\frac{3}{2}$ ,

$f_e\left(\frac{3}{2}\right)=-2$

At x=-2 ,

$f_e\left(-2\right)=-1$

So,$f_e\left(0\right)=-1,f_e\left(1\right)=-\frac{1}{2},f_e\left(\frac{3}{2}\right)=-2,f_e\left(-2\right)=-1$