15.3. Half-Range Expansions

Try these first

1. 1.

   What is a Half-Range Sine Series?

   Hint

   Think about the symmetry of the graph.

2. 2.

   List the conditions for applying Half-Range Expansions.

   Hint

   Consider the behavior of the function at the endpoints.

3. 3.

   What type of function is suitable for a Half-Range Sine Series?

   • Even Functions
   • Odd Functions
   • Period Functions

   Hint

   Consider the symmetry of the function at the boundaries.

4. 4.

   True or False: Half-range expansions can only represent functions defined on full periods.

   • True
   • False

   Hint

   Remember, we discussed functions only on 0 < x < L.

5. 5.

   Given a function f(x) defined on the interval [0, 4] with f(0)=0, f(4)=0, express it using a Half-Range Sine Series.

   Hint

   Focus on the symmetric properties you use in establishing the sine series formula.

6. 6.

   Explain how you would apply Half-Range Cosine Series to model the temperature on a flat plate with even boundary conditions.

   Hint

   Remember, apply your knowledge of cosine functions and their symmetry at the boundaries.