10.1.4. Examples

Try these first

1. 1.

   Define the Fourier Cosine Transform.

   Hint

   Consider its utility in boundary value problems.

2. 2.

   Write the formula for the Fourier Cosine Transform of f(x) = e^(-ax).

   Hint

   Remember the integral setup you saw in class.

3. 3.

   What is the primary use of the Fourier Cosine Transform?

   • To transform functions defined on (-∞
   • ∞)
   • To analyze functions on semi-infinite domains
   • To express periodic functions only

   Hint

   Think about where these functions are applicable in engineering.

4. 4.

   Is the Fourier Cosine Transform linear?

   • True
   • False

   Hint

   Consider the linearity property discussed in class.

5. 5.

   Given the function f(x) = e^(−3x), derive the Fourier Cosine Transform and explain the meaning of your result in a context of heat transfer.

   Hint

   Apply the definition of the Fourier Cosine Transform.

6. 6.

   Consider the beam deflection problem. How would you set up the Fourier Cosine Transform of the beam equation? Explain your steps.

   Hint

   Start with the Euler-Bernoulli beam equation and think about boundary conditions.