Fourier Series Help: Piecewise Smooth | x=-1 to 1

[Sheridans]

Homework Statement

Hello,
Check each function to see whether it is piecewise smooth. If it is, state the value to which its Fourier series converges at each point x in the given interval and the end points

(a.) f(x)=|x|+x, -1<x<1
(it would be very helpful to see if i did this right, as the professor I have does not do examples and that is how I learn how to approach and solve problems)

Homework Equations

If f is piecewise smooth and is periodic with a period of 2a, then at each point x in the corresponding Fourier series to f converges and its sum is:

Fourier series= 0.5(f(x+)+f(x-)), where f(x+) is the limit from the right, and f(x-) is the limit from the left.

Criterion for piecewise smooth on interval a<x<b:
1) f is piecewise continuous (it is bounded and is continuous, except possibly for a finite number of jumps and removable discontinuities)
2)f'(x) exists except possibly at a finite number of points
3) f'(x) is piecewise continuous

The Attempt at a Solution

After sketching the function, it is continuous on the interval, f'(x) exists and it has a finite number of discontinuities (so it is piecewise continuous) Therefore, f(x) is piecewise smooth

f(x)= 2x, 0<x<1
0, -1<x<0 (spilt it up)

f(x)=.5(f(x+)+f(x-))=2x/2=x

endpoints: at x=-1 .5(f(-1+)+f(-1-))=-2/2=-1

at x=1, .5(f(1+)+f(1-))=2/2=1

Is this right? I think i went wrong somewhere. And do i have to actually find the Fourier series (which I know how to do, just thought it was not needed/was not even specified)?

Homework Statement

Homework Equations

The Attempt at a Solution

 

[ehild]

Your work looks correct, and no need to determine the Fourier series.

ehild

 

[Sheridans]

Thank you. I do have other problems but I am going to start fiending in the math learning center for this class. Professor expects us to do problems while showing no examples, and I am not the only one who has this problem in the class.

Once again thank you !

 

[ehild]

Show your other problems!

ehild