What is the laplace transform of:
f(t) = 0, 0<t<2
t, 2<t
thanks!
To find the Laplace transform of the given function f(t), we can follow these steps:
Step 1: Divide the function into two separate intervals:
- For 0 < t < 2, f(t) = 0
- For t > 2, f(t) = t
Step 2: Take the Laplace transform of each part separately.
1) Laplace transform for 0 < t < 2:
The Laplace transform of a constant function (in this case, f(t) = 0) is straightforward. The Laplace transform of a constant 'C' is C/s (where 's' is the complex variable).
So, for 0 < t < 2, the Laplace transform of f(t) = 0 is:
L[f(t)] = L[0] = 0/s = 0
2) Laplace transform for t > 2:
To find the Laplace transform of f(t) = t, we can use the table of Laplace transforms or apply the basic formula for the Laplace transform of t^n, where 'n' is a positive integer:
L[t^n] = n! / s^(n+1)
In this case, n = 1, so we have:
L[f(t)] = L[t] = 1 / s^2
Step 3: Combine the Laplace transforms of both intervals:
Since f(t) is defined to be 0 for 0 < t < 2 and t for t > 2, we can write the overall Laplace transform as follows:
L[f(t)] = 0 (for 0 < t < 2) + (1 / s^2) (for t > 2) = 0 + (1 / s^2) = 1 / s^2
Therefore, the Laplace transform of f(t) is 1/s^2.