Discussion
Laplace transform of a function f(t) = t4 as a function of 's' is __________.
Answer: B
The Laplace transform is \(24/s^{5}\).
The standard result for a power of t is
\[ \mathcal{L}\{t^{n}\} = \frac{n!}{s^{n+1}} \]
With \(n = 4\), \(n! = 24\) and the power of s is \(n + 1 = 5\), so \(\mathcal{L}\{t^{4}\} = 24/s^{5}\).
120 is \(5!\), the factorial for \(t^{5}\), so \(120/s^{5}\) and \(120/s^{4}\) use the wrong factorial. \(24/s^{4}\) has the right numerator but forgets that the power of s is one more than the power of t.
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