- #1
ConnorM
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- 1
Homework Statement
Here is an imgur link to my assignment: http://imgur.com/N0l2Buk
I also uploaded it as a picture and attached it to this post.
Homework Equations
[itex]u_c (t) =
\begin{cases}
1 & \text{if } t \geq c \\
0 & \text{if } t < c
\end{cases}[/itex]
The Attempt at a Solution
Question 1.1 -
[itex]L[tu(t)] = \int_0^∞ tu(t)e^{-st} \,dt[/itex]
Using the definition of the step function, [itex] t \geq 0, u(t) = 1[/itex]
*Is it right to assume that [itex] c = 0 [/itex]?*
[itex]L[tu(t)] = \int_0^∞ t(1)e^{-st} \,dt[/itex]
[itex]L[tu(t)] = \int_0^∞ te^{-st} \,dt[/itex]
[itex]L[tu(t)] = 1/s^2 [/itex]
I'm not sure if this is correct. Should it be solved using the rule, [itex] L[tf(t)] = -F'(s)[/itex]
Question 1.2 -
Let [itex] r_1 (t), r_2 (t) [/itex] be the two ramp functions
Let [itex] u_1 (t), u_2 (t) [/itex] be the two unit-step functions
[itex]r_1 (t) =
\begin{cases}
t & \text{if } 0 \leq t < 1
\end{cases}[/itex]
[itex]r_2 (t) =
\begin{cases}
t+1 & \text{if } 1 \leq t < 2
\end{cases}[/itex]
[itex]u_2 (t) =
\begin{cases}
3 & \text{if } 2 \leq t < 4
\end{cases}[/itex]
I'm not quite sure what to do for the unit-step functions. Could someone help me figure out what they should be?
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