Question

Express f(t) in terms of the unit step function u_c(t):

Answer 1

\[f(t)=
\left\{\begin{matrix}
t^2, & 0 \le t <2\\
1 & t \ge 2
\end{matrix}\right.\]

My solution was \(t^2u_0(t)-3u_2(t)\), but the solution that I found was \(t^2+u_2(t)(1-t^2)\). Aren't they technically the same thing? How did they get the \(1-t^2\) part?

Answer 2

@ganeshie8 @Michele_Laino @IrishBoy123

Answer 3

My line of thought was that at t=2, the first quadratic function ends at 4. Then it jumps from 4 to 1, hence the \(-3u_2(t)\)

Answer 4

they are not same, your function looks wrong

Answer 5

Darn.. I understand the first part of the solution that I found, but where is the \(1-t^2\) coming from?

Answer 6

lets look at your function :
\(t^2u_0(t) -3u_0(t) \)

the first term, \(t^2u_0(t) = t^2\) for \(t\gt 0\)

you need to kill this for \(t\ge 2\)

Answer 7

maybe lets graph the given piecewise function quick

Answer 8

Oh so in order to cancel it out, we must subtract by that function

Answer 9

So that we're only left with 1

Answer 10

Exactly!

Answer 11

lets graph and see