10.2.3.3. Differentiation

Try these first

1. 1.

   What is the Fourier Cosine Transform of a function's derivative?

   Hint

   Think about the relationship between differentiation and the transform.

2. 2.

   How does the Fourier Sine Transform relate to a function with a zero boundary?

   Hint

   Consider what happens along the boundaries of the scenario.

3. 3.

   What is the relationship between the Fourier Cosine Transform and the derivative of a function?

   • It's the same as the function.
   • FCT is zero.
   • It introduces a negative multiplier to the original transform.

   Hint

   Reflect on the definition and relationship of derivatives in transforms.

4. 4.

   When is the Fourier Sine Transform particularly useful?

   • True
   • False

   Hint

   Think of the bouncing wave examples we discussed.

5. 5.

   Propose an engineering problem involving heat transfer in a rod. Describe how you would apply both Fourier transforms.

   Hint

   Consider initial and boundary temperature distributions.

6. 6.

   Derive the displacement formula of a vibrating string using Fourier Sine Transform given its boundary conditions.

   Hint

   Identify fixed points and apply sine properties.