00:01
Hi, we are given the piecewise function f of t defined as 4 if t is greater than or equal to 0, but less than 2, and it is t squared if t is greater than or equal to 2.
00:19
We have two tasks.
00:22
The first task is to write f of t in terms of the unit step function.
00:39
And consequently compute for the laplace transform of f of t using this transformation.
00:54
Okay, so in general, given a step function, j of t equals g0 of t if t is greater than or equal to zero, but less than or equal to, let's say, t sub 1, and it is g sub 1 of t.
01:13
If t is greater than or equal to t sub one okay this should be less than strictly less than then then g of t can be written as g nut of t plus u of t minus t sub one times g sub one of t minus g not of t where u of t minus t sub 1 is the unit step function defined as 0 if t is less than t sub 1, and it is 1 if t is greater than or equal to t sub 1.
01:58
So using this formula, we have f of t equals our f sub not, in this case is 4.
02:11
Have 4 plus our t sub 1 is equal to 2 so we have u t minus 2 times our f sub 1 is t squared and minus f sub 0 which is 4 so this is now the unit step function transformation of our function f of p now we compute for the laplace transform of f of f so della plus transform of f of t is equal to della plus transform of 4 plus u of t minus 2 times t squared minus 4...