Need help with some fourier series assignment problems

[Kenandmath]

This is an assignment and needed helped with the answers.Any help with this will be greatly appreciated.

5. Under what circumstances would I be better off using a Taylor series rather than a Maclaurin series to approximate a function?

6. Determine a Taylor Series approximation for f(x) where n = 4 and a = 3

Part II: Fourier series

Write the Fourier series for the following function: f(x) = {-2x -pi<x<0 }

{2x 0<x<pi }

1. Find a₀ using the formula a₀=1/2pi integral limit-pi to pi f(x)dx

definite integral limit -pi to 0 of -2x dx

definite integral limit 0 to pi of 2x dx

a₀= 1/2pi{definite integral limit of -pi to 0 of -2x dx + definite integral limit 0 to pi of 2x dx }=

2. Find a_k using the formula a_k=1/pi definite integral limit of -i pto pi f(x)cos(kx)dx.

Note: We can use the integration formula: indefinite integral ucos(u)du=cos(u)+usin(u). To do this we will set u = kx, which makes du = kdx. To complete the substitution we need to have a k attached to both the f(x) and dx portion of the integral. Thus, we are multiplying in two k’s. To balance this we must also divide by 2 k’s outside the integration. Thus, the constant outside the integration will look like 1/pik^2.

 

[HallsofIvy]

This is an assignment and needed helped with the answers.Any help with this will be greatly appreciated.

5. Under what circumstances would I be better off using a Taylor series rather than a Maclaurin series to approximate a function?

What is a "Taylor's series" or "MacLaurin series"? What is the difference?

6. Determine a Taylor Series approximation for f(x) where n = 4 and a = 3

Either this is a nonsense question or it is just asking you for basic definitions. (I would say "Taylor Polynomial" not "Taylor series approximation".)

Part II: Fourier series

Write the Fourier series for the following function: f(x) = {-2x -pi<x<0 }

{2x 0<x<pi }

1. Find a₀ using the formula a₀=1/2pi integral limit-pi to pi f(x)dx

So, what do you get when you do that?

definite integral limit -pi to 0 of -2x dx

definite integral limit 0 to pi of 2x dx

a₀= 1/2pi{definite integral limit of -pi to 0 of -2x dx + definite integral limit 0 to pi of 2x dx }=

What kind of "help" do you want? You are told exactly what to do. Are you not able to do the integration?

2. Find a_k using the formula a_k=1/pi definite integral limit of -i pto pi f(x)cos(kx)dx.

Note: We can use the integration formula: indefinite integral ucos(u)du=cos(u)+usin(u). To do this we will set u = kx, which makes du = kdx. To complete the substitution we need to have a k attached to both the f(x) and dx portion of the integral. Thus, we are multiplying in two k’s. To balance this we must also divide by 2 k’s outside the integration. Thus, the constant outside the integration will look like 1/pik^2.