Mathematical Methods Quiz

The functionf(x) =e^x is

• even

• odd

• both even and odd

• neither even nor odd

The Fourier series of an odd function, in the interval (−π,π), contains only

• sine terms

• cos terms

• odd terms

• even terms

The Fourier series of a real periodic function, in the interval (−π,π), has only

P: cosine terms if the function is even

Q: sine terms if the function is even

R: cosine terms if the function is odd

S: sine terms if the function is even

Which of the above statements are true?

• P and S

• P and R

• Q and S

• Q and R

The Fourier coefficienta_0 of a functionf(x) in the interval (0,2π) is given by

• 1/π ∫ from -π to π f(x)dx

• 1/π ∫ from 0 to 2π f(x)dx

• 1/π ∫ from 0 to 2π f(x) sin(nx) dx

• 1/π ∫ from 0 to 2π f(x) cos(nx) dx

The Fourier coefficienta_n if a functionf(x) in the interval (0,2π) is

• 1/π ∫ from 0 to 2π f(x) sin(nx) dx

• 1/π ∫ from 0 to 2π f(x) cos(nx) dx

• 1/π ∫ from -π to π f(x) sin(nx) dx

• 1/π ∫ from -π to π f(x) cos(nx) dx

The value of the Fourier coefficienta_0 if the functionf(x) =x in (0,2π) is

• 2π

• -2π

• π/2

• 2π/3

The value of the Fourier coefficienta_0 if the functionf(x) =x in (−π,π) is

• 2π

• -2π

• π/2

• 0

For the functionf(x) =x^2 in the interval (0,2π), the value of the Fourier coefficientb_n is

• -4π/n

• 4π/n

• π/n

• 2π/n

For the functionf(x) = { 0; −π < x < 0; x; 0 < x < π the value of the Fourier coefficienta_0 is

• π/2

• π

• -π/2

• 3π/2

An example of an even function is

• f(x) =x^2

• f(x) =x+x^3

• f(x) = sinx+ cosx

• f(x) =e^x