f(t)=4e^(-3(t-4)) a) Find L[(df(t))/dt] by differentiating f(t) b) using the theorem for differentiation

ruigE

ruigE

Answered question

2020-11-08

Given that f(t)=4e−3(t−4)
a) Find L[df(t)dt] by differentiating f(t) and then using the Laplace transform tables in lecture notes.
b) Find L[df(t)dt] using the theorem for differentiation
c) Repeat a) and b) for the case that f(t)=4e−3(t−4)u(t−4)

Answer & Explanation

Brittany Patton

Brittany Patton

Skilled2020-11-09Added 100 answers

Step 1
According to the given information, the given function is:
f(t)=4e−3(t−4)
Step 2
For part (a) it is required to find Ldf(t)dt by differentiating f(t) and then using Laplace transform.
So,
df(t)dt=4e−3(t−4)(−3)=−12e−3(t−4)
L(dfdt)=L(−12e−3(t−4))=L(−12e−3te12)
=−12e12L(e−3t)[L(e−at)=1s+a]
L(dfdt)=−12e12(1s+3)
Step 3
Now, it is required to find L(df(t)dt) using theorem for differentiation.
L(dfdt)=sL(f(t))−f(0)
=s(L(4e−3(t−4)))−4e−3(0−4)
=s(L(4e−3te12))−4e12
=4se12L(e−3t)−4e12
=4se12(1s+3)−4e12
=4e12(ss+3−1)
=4e12(s−(s+3)s+3)
=4e12(−3s+3)
L(dfdt)=−12e12s+3
Step 4
Now, for part (c) it is required to repeat part (a) and (b) for the function:
f(t)=4e−3(t−4

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