Question bank(unit-3)

                               Question bank(unit-3)

                               Question bank(unit-3)

Question-1: Find the fourier series of the function f(x) = x      where 0 < x < 2 π

Sol. We know that, from fourier series,

                                   f(x) =

First we will find ,

Now,

And ,

Put these value in fourier series, we get

Question-2: Find the fourier series for f(x) = x / 2 over the interval 0 < x < 2π

And has period 2π

Sol. First we will find

                                              = 

                                                   =

                                                   =

                                                  = π

                                               =   π

Similarly,

Which gives,       =   0

Now,

We get,    

We know that, the fourier series

Put these values in fourier series, we get

Question-3: Find the Fourier series of f(x) = x in the interval

Solution:

Here ;

It’s Fourier series is given by

 … (1)

Where

&

Hence the required Fourier series is

1. Question-4: Find the Fourier series for

in the interval

Hence deduce that

Solution:

Here ;

Hence it’s Fourier series is,

… (1)

Where

&

Hence equation (1) becomes

Put we get

i.e.

Question-5: Find a Fourier series expansion in the interval for

;

;

Solution:

Here

;

;

Hence it’s Fourier series expansion is,

… (1)

Where

And

Hence equation (1) becomes

Question-6: Find a Fourier series of

;

;

Solution:

Here

;

;

Here f(x) is odd function Hence we get half range sine series i.e.

… (1)

Where

Hence equation (1) becomes,

Question-7: Find the fourier expression of f(x) = x³ for –π < x <  π.

Sol.

Here, we can see that f(x) Is an odd function

So that,

                          and    

We will use here ,

We get the value of f(x),

Question-8: Find the Fourier series expansion of the periodic function of period 2π.

                                             f(x) = x² ,    -π≤x≤π

Sol. The given function is even, so that,

We will find

The fourier series will be ,

Question-9: Find a Fourier series for

;

Solution:

Here

;

Since f(x) is even function hence

It’s Fourier series is

… (1)

Where

Hence equation (1) becomes,

Question-10: Find half range cosine series of in the interval and hence deduce that

a)    

b)   

Solution:

Here

;

Hence it’s half range cosine series is,

… (1)

Where

Hence equation (1) becomes,

… (2)

Put x = 0, we get

Hence the result

Put we get,

i.e.