1.10. Summary

Try these first

1. 1.

   What does the First Shifting Theorem state?

   Hint

   Remember the relationship between multiplication by e and shifting.

2. 2.

   If $f(t) = t$, what is $\mathcal{L}\{e^{3t} t\}$?

   Hint

   Determine the basic Laplace Transform of t first.

3. 3.

   What is the First Shifting Theorem's main result?

   • \\( \\mathcal{L}\\{f(t)\\} = F(s) \\)
   • \\( \\mathcal{L}\\{e^{at} f(t)\\} = F(s + a) \\)
   • \\( \\mathcal{L}\\{e^{at} f(t)\\} = F(s - a) \\)

   Hint

   Focus on the direction of the shift.

4. 4.

   True or False: The condition $s > a$ must be satisfied for the theorem to apply.

   • True
   • False

   Hint

   Think about the implications of the exponential function.

5. 5.

   Derive the Laplace Transform for $e^{2t} \cos(5t)$ using the First Shifting Theorem.

   Hint

   Identify the base transform first, then apply the theorem.

6. 6.

   If $g(t) = e^{t} \sin(t)$, find $\mathcal{L}\{g(t)\}$.

   Hint

   Remember to isolate the transform of \\( \\sin(t) \\) before applying the exponent.