10.9.2. Sine Transform of First Derivative

Try these first

1. 1.

   What is the Sine Transform of the first derivative?

   Hint

   Look for the formula relating f' and f.

2. 2.

   Why is the Sine Transform useful in civil engineering?

   Hint

   Think about the types of problems engineers face.

3. 3.

   What does the Sine Transform of the first derivative equate to?

   • F {f′(x)} = sF {f(x)} - f(0)
   • F {f′(x)} = -sF {f(x)}
   • F {f′(x)} = F {f(x)}

   Hint

   Remember the boundary term in the transformation.

4. 4.

   True or False: The Sine Transform is useful when the function value at one boundary is zero.

   • True
   • False

   Hint

   Think about the types of boundary conditions suitable for the Sine Transform.

5. 5.

   Consider a material with temperature distributed along its length according to the function f(x) = T_0 e^{-kx}. Use the Sine Transform to find the derivative of temperature concerning the distance x.

   Hint

   Differentiate f(x) first before applying the transform.

6. 6.

   Analyzing a cantilever beam, the deflection can be modeled by f(x). If the load at the end is constant, use the Sine Transform to express the first derivative of the deflection at the free end.

   Hint

   Consider what happens at the fixed end and free end of the beam.