We're going to find the Laplace Transform
which is big F of s of f of t
where f of t is the given step function.
To do this, we'll write f of t in terms of the
Heaviside step function, the unit step function shown here.
Let's first look at the graph of f of t.
F of t equals two if t is less than four,
which is this piece of the graph.
F of t equals five when t is greater than or equal
to four and less than seven, which is this piece.
And finally f of t equals seven if t is greater than
or equal to seven, which should be this piece of the graph.
So again, we want to write f of t in terms
of the Heaviside step function, a unit step function
which is u sub c of t which can also be expressed
as u of the quantity t minus c
which equals zero if t is less than c,
and is equal to one if t is greater than or equal to c.
And then we use the Laplace transform formula.
The Laplace transform of u sub c of t equals
e raised to the power of negative cs divided by s.
So f of t first takes on the function value of two,
so we'll have two times
I'm going to write a difference here
that's equal to one when t is less than four
and equal to zero when t is greater than or equal to four.
We can write this difference as one minus u sub four of t.
Remember, u sub four of t is going to be equal to one
when t is greater than or equal to four
and equal to zero when t is less than four.
So when t is less than four, this difference
will be one minus zero, giving us a function value
of two times one or two.
And when t is greater than or equal to four
u sub four of t is one, so we'd have two times zero
which brings us to the next function value of five.
So now we'll have plus five.
Now I'm going to write a difference here
that's equal to one over this interval
otherwise equal to zero, which would be
u sub four of t
minus u sub seven of t.
Again, over the interval from four to seven
including four and not including seven,
this difference would be one because
u sub four of t would be one,
u sub seven of t would be zero.
Notice how this difference would also still be zero.
And then when t is greater than seven
this difference would be zero,
again taking us to the next function value of seven.
So we have plus seven.
We want the function value to be seven
when t is greater than or equal to seven.
We can just say seven times u sub seven of t.
Again, when t is greater than or equal to seven,
u sub seven of t is one.
This difference would be zero, this difference would be zero
giving us the function value of seven.
And now let's simplify this by distributing the two,
distributing the five, and combining like terms.
So we'd have f of t equals two minus two
times u sub four of t plus five times u sub four of t
minus five times u sub seven of t
plus seven times u sub seven of t.
Now let's go ahead and combine like terms here,
as well as here.
So we have f of t equals two,
and then plus three times u sub four of t
plus two times u sub seven of t.
We will take a look at this problem again
and come up with this form of f of t from the beginning,
but I did want to show it this way
because regardless of how complicated
this piece-wise defined function gets,
this first set-up will always work.
And now we'll take the Laplace transform
on both sides of the equation.
The Lap lace transform of f of t
equals the Laplace transform of two
plus three times u sub four of t
plus two times u sub seven of t.
Let's go ahead and continue on the next slide.
The Laplace transform of f of t is equal to big f of s.
Using the properties of Laplace transform, let's
write the right side as
the Laplace transform of two
plus three times the Laplace transform of u sub four of t
plus two times the Laplace transform of u sub seven of t.
So we have big f of s equals
the Laplace transform of two, remember,
would be two times one over s.
So we just have two divided by s
plus three times the Laplace transform of u sub c of t,
using this formula here where c is equal to four
would be e raised to the power of negative four s
divided by s.
So we have three times e to the negative 4 s divided by s,
and then plus two times the Laplace transform
of u sub seven of t would be times e
raised to the power of negative seven s divided by s.
Notice how we do have a common denominator,
so we'd have big f of s equals.
The denominator is s.
The numerator would be two
plus three times e to the negative four s
plus two times e to the negative seven s.
So this would be the Laplace transform
of the given function f of t.
I hope you found this helpful.