10.9. Fourier Transforms of Derivatives

Try these first

1. 1.

   What is the formula for the Fourier Cosine Transform of a first derivative?

   Hint

   Think about how you relate derivatives to the transform.

2. 2.

   State the additional term involved in the Fourier Sine Transform of a derivative.

   Hint

   Consider the significance at the boundary.

3. 3.

   What is the Fourier Cosine Transform of the first derivative?

   • sF {f(x)}
   • -sF {f(x)}
   • F {f(x)} + f(0)

   Hint

   Think about the relationship with the rate of change.

4. 4.

   The Fourier Sine Transform of the first derivative includes which additional term?

   • True
   • False

   Hint

   Reflect on the boundary implications.

5. 5.

   Given a function f(x) = e^(-ax), derive the Fourier Cosine Transform of its derivative.

   Hint

   Utilize standard Fourier Transform pairs in your derivations.

6. 6.

   Consider a system described by u(x,t) where u(0,t) = 0. Apply the Fourier sine transform to determine the solution form.

   Hint

   Reflect on the implications of zero-value conditions on sinusoidal oscillations.