Well hello everyone and welcome to our fourth lecture on Laplace transforms. Today we're going
to introduce a special type of function called a unit step function. Now mathematically modelling
real world scenarios often requires us to use non-standard functions in our equations,
equations like differential equations for example. One such function is the unit step function. Now
in coming lectures we'll see how to use Laplace transforms to solve differential equations
that involve the step function. But today we're going to introduce the step function and look at
plots of the step function when it's involved in linear combinations. We're looking at products
involving step functions and finally we'll have a look at determining the equations of functions
from the graphs. Now in our last lecture we saw how Laplace transforms can be applied to solve
an initial value problem where the differential equation has a standard
portion function on the right hand side. Now when we come to mathematically modelling real
world systems however the type of portion function on the right hand side of the differential equation
can make it difficult to solve an initial value problem using traditional time domain
methods such as the method of undetermined coefficients that we saw earlier on in this
course. Now Laplace transforms can simplify the process considerably so this becomes easier
quicker to obtain a solution of a given differential equation. And one function as I've said
that's often used to model real world quantities is the unit step function.
Now in engineering applications we often come across functions whose values change abruptly at
specified values of time. And one common example would be when a voltage is switched on or off
in an electrical circuit at a given value of time. The switching process can be described
mathematically by the unit step function. Now the unit step function is also known as the
heavy side function after the mathematician Oliver Heaviside who introduced it. So what does the
step function look like? The unit step function what does it look like? Well it's defined mathematically
like so u of t minus capital T it's equal to zero when t is less than capital T and it's equal to one
when t is greater than capital T. So if we're to plot this unit step function this is what it looks
like here. We're only defined it for positive values of time so it's zero all the way down here for
negative values of t. So here's our function here u of t minus capital T and up until the value
which we'll see later is called the critical value up until this critical value this function this
unit step function takes the value zero and then it suddenly jumps to take the value one
at this critical value t. So it's zero all the way up to the critical value and then it suddenly
jumps to take the value of one and it stays at one forever and ever and that's what this states.
For t less than capital T the unit step function takes the value zero as you can see here and for t
greater than capital T it takes the value one as you can see in this diagram here. So you can think
of it as a function that's been switched on when t is equal to capital T. I've included some
alternative notation for the unit step function at the foot of the slide here.
Okay so as I mentioned this value when the unit step function is switched on
this value of capital T is called the critical value of the step function and is where
the step function changes value, changes from zero to one. Now as you can see in this definition
here the value of the step function when t is equal to capital T is not defined. Some books will
give it the value one, others books will give it the value zero and some books will even give it
the value a half when t is equal to t. That's not really important for us. All we need to remember
is that up until the critical value this unit step function is equal to zero and once we hit that
critical value t equals capital T think of it as like a switch being flicked on and the unit step
function jumps up to have a value of one which it holds forever and ever. So let's have a look
at specific examples. So when our critical value is taken to be t is equal to three in other words
when capital T is equal to three this is what the function looks like in zero all the way up to three
then it suddenly gets switched on and it jumps up to take the value one and it'll hold that value
forever and ever and mathematically we would write it like this u of t minus three is equal to zero
for t strictly less than three and it's equal to one for t strictly greater than three and that's
what you can see in the diagram there. Now if we multiply our unit step function by a constant
the constant just represents the amplitude of our step function. Obviously when it's a unit step
function it's got amplitude one and we'll just refer to it as a step function when it's got some other
amplitude. So in this case with a critical value of capital T is equal to four the step function
has the value zero all the way up to four and then it's suddenly switched on and it jumps up
to take the value six because there's a six multiplying the unit step function. So that's
what's happening here so it's off all the way up until t is equal to four it suddenly jumps to the
value six at the value t equals four and it stays at six forever and ever. Now a special case might
be when the critical value t is equal to zero so in other words nothing happens it's a unit step
function as the value zero all the way up to time t equals zero and then it suddenly jumps to take
the value one and it stays at one forever and ever. Okay so in terms of step functions that I say
down here u of t minus zero is just equal to one it's just a constant that's equal to one.
So here's an example plot a step voltage of size 12 volts which starts at time t equals four
seconds so in this case we know that our units that will do this with a unit step function we
started with a step function who will take the value zero all the way up to the critical value
of t equals four and then suddenly gets switched on and it jumps up to take the value 12 and it
holds that value for all t after four so and that's what it looks like here. Okay so these are some
plots of step functions of different amplitudes and different critical values so let's have
a look at combining some step functions now so we've got two unit step functions here both of
them have the amplitude one and we're adding them together u of t minus one that's a unit step function
whose critical value is capital t is equal to one in other words it's zero all the way up to the
critical value of t equals one when it suddenly gets switched on and takes the value one because
the amplitude is one so unit step function takes value one forever and ever and to that we're going
to add another unit step function but this time the unit step function has a critical value of two
so once we hit time t equals two seconds this function will get switched on and it'll take the
value one forever and ever from from t equals two onwards so what happens when we add them together
well let's just have a look at this for if t is less than one both the functions are off we haven't
switched either of them on yet because look the critical values are capital t equals one and capital
t equals two so up at the t equals one nothing happens they're both switched off so this one will
have the value zero and this one will have the value zero so my overall function f of t would just be
zero plus zero which is zero so f of t is equal to zero the sum of these two unit step functions up
until the first critical value t equals one now once we hit t equals one this step function here
this one switches on and takes the value one whilst this one is still off this one will not come
into play until we hit the critical value t equals two so the overall function f of t will take the
value one here plus but this one will be zero so it will hold the value one all the way up until
the next critical value at t equals two and that's what you can see in the plot here so we said
earlier that both functions are zero up until t equals one so the overall function is zero so it
keeps going at zero all the way up until you hit the first critical value it then suddenly switches
and jumps up to take the value one now it holds that value all the way along until you hit the
next critical value at t equals two and it'll this one will switch on so now you got both functions
switched on both have a value one and you add them together one plus one gives you the value two
so this will now be the value of the function f of t forever and ever it'll be the hold the value
of two forever and ever on the next slide here i've sort of tried to explain this a bit in a bit
more detail so i'll just go quickly through this so as i said we've added together two unit step
functions u of t minus one in red and u of t minus two in blue this one switches on at the critical
value t equals one and this one will switch on at the critical value t equals two so let's have a look
then so here you can see along here between zero and one nothing happens because we haven't
reached either the critical value set both functions are off so in other words this u t minus one the
red one has the value zero and the blue one has the value zero so in here the sum of these functions
giving me f of t will be zero plus zero which is zero so down here i plot that so in this first
section between zero and one the overall function f of t is the sum of the two unit step functions
and that's zero then we reach the first critical value at t equals one and this function gets
switched on and it takes the value one this function hasn't been switched on yet so it's still off
so my overall function f of t will have the value one from the red function plus zero of the blue
function which is one so it'll take the value one and it'll hold it up until the next critical value
the next critical value will occur here at t equals two and this function this blue function
will get switched on and take the value one so that's what i've shown here the blue function
is zero all the way up to t equals two and then it suddenly gets switched on and it also has the
value one the red function we saw got switched on at t equals one and it jumped to take the value one
and it held the value one forever and ever so on this region from two onwards both functions have
the value one and when you add them together you get two and you can see that the overall function
which is sum of the two functions will have the value one plus one from t equals two onwards and
it'll hold that value forever and ever so that's adding two unit step functions together this is
the sort of function that comes out of it okay so here's one that's slightly more complicated now
i'm not going to go into as much detail on this one and hopefully you can see what's involved so
and first of all i'm going to examine my step functions now obviously this is a unit this unit
step function here is multiplied by two so this function will have amplitude two this one will
have amplitude minus three okay it'll have about three but we're subtracting that so we'll see what
effect that has that has shortly and this is just a unit step function so the first thing i want to
look at is i want to find the critical values the critical values are the all important values
where things happen where functions get switched on so the critical values in this case are five
10 and 15 so mark five 10 and 15 on my plot and let's have a look and see what happens well
up until t equals five there's nothing happening all the functions these three step functions
are all switched off none of them have come into play yet so up until up between zero and five
all these functions have the value zero so the overall function which is just a linear
combination of these three functions the two will have the value zero and i've shown that in blue here
when we hit t equals five this step function gets switched on
and it's going to have amplitude two these two though will not get switched on yet this one
gets switched on at t equals 10 and this one gets switched on on t is equal to 15 so so at this
instant t equals five this function gets switched on and takes the value two and that's what you
can see happening here so it jumps from the value zero the overall function jumps from having the
value zero to taking the value two now it's going to hold that value till we hit the next critical
value and we hit the next critical value at t equals 10 so once we hit that value this unit
step function gets switched on and it's multiplied by minus three so what's going to happen here is
we're going to be up here and we're going to drop down three units and it'll take the overall
function f of t will take the value minus one so that's what you can see happening here so the
overall function f of t had the value two all the way through the interval five up to 10 then
this function got switched on and it took the it it forced a drop of three units so you drop three
units from two and you go down to minus one so the overall function f of t will then hold the value
minus one on the next on this interval until we hit the next critical value and the next critical
value is hit at t equals 15 now when t is equal to 15 this function kicks in and it's got the value
one because that's the amplitude unit step function it's got the it's got the amplitude one and so we
add one to f of t to the f of t from from this interval here and that will bring us up to zero
and then the overall function f of t will hold the value zero forever and ever for all values of
time from t equals 15 onwards so that's what we've got that's that's the linear combination of three
step functions and this is the plot of it I mean just very quickly we've got three critical values
five 10 and 15 the first function gets switched on at t equals five so up until then there's nothing
happen when t equals five the step function has the value two so it will hold the value two up until
the next critical value which occurs at 10 and this corresponds to a drop because of the minus sign
by three units so we drop down by three units and then we hold that value of minus one over
this interval until you hit the next critical value which you hit when t is equal to 15 and this unit
step function of amplitude one comes into play and that will move us up one unit and then the
overall function f of t will take the value zero from t equals 15 onwards forever and ever so that's
a linear combination of three step functions so we've looked at sums and differences
of some step functions we'll now move on and look at a couple of different types of products that
involve step functions now one reason for using products that involve step functions is that it
enables us to switch functions other functions off and on at given values of time and we're going
to look at two different products in the first one we're going to look at any function some function
f of t been multiplied by a step function whose critical value is ut minus a now if you remember
the definition of a step function the definition of a step function tells us that the function is
equal to zero up until the critical value and it equals to one from the critical value onwards
now what does that mean well if you're multiplying some function something like perhaps t squared or
something like that by a step function whose critical value is at a now this function this step
function this unit step function here has the value zero up until you hit the critical value at t
equals a so if you multiply anything by this unit step function up until t is equal to a then
you're multiplied by zero so the whole function g of t will become zero and then from after values
of the critical value at a this unit step function takes the value one so our g of t will be equal
to f of t from a onwards so I'll just just recap on that up until the critical value this whole
function g of t will be zero and then from the critical value onwards it'll be equal to our function
f of t so in other words in our plot we'll have a plot of f of t zero up until t equals a and then it
starts at t equals a and goes on forever and ever after that the second type of product we look at is
where we've got a function existing between two values of t only between the two critical values
of two unit step functions and that gives us what we're going to call a mathematical window to the left
that's up to a this overall function g will be equal to zero then you'll have this window between a
and b where the function f of t will exist and then to the right from b onwards the function g of t
will be equal to zero so it only exists within a mathematical window between a and b so first of all
we'll look at type one so here's a function we've got a type one function here we've got some function
f of t multiplying a unit step function whose critical value is a and in our example our function
g of t the product of f of t and the step function is given by four t plus two times a step function
whose critical value is one that's u of t minus one and if you remember all that means is that the
step function has the value zero for t less than one and it's got the value one for t greater than one
so we're wanting to plot this function so what's actually going on here well remember what we said
in a type one function what's actually happening is multiplied by the step function switches
everything off up until the critical value of the step function so so up until t equals one
everything will be equal to zero so that's what i've done here so between t equals zero and the
critical value t equals one everything is off because the step function has the value zero
over that interval so zero times four t plus two is just zero and then suddenly a t equals one
it's like a switch inflict the unit step function kicks into action and has the value one so our
function g of t will take the value four t plus two from one onwards so what we need to do is we need
to draw the graph of the function four t plus two so it's a straight line it's a linear function
because it's t to the power one so to to draw a straight line all we need is any two points on the
line and so for example i could first of all take t equal to zero plug it in here and i would find
that f of zero is equal to two so my first point would be zero two i can then say t equal to one
and plug it in here and i get t f of t or f of one is equal to six so my second point would be one
six and i've plotted them on here and that would enable me to draw my line my function four t plus
and as it's a type one function multiplied by a unit step function basically sets that function
to zero up until the critical value the critical value is that t equals one so my function shown
here f of t is equal to four t plus two is going to be zero up the critical value then it's going
to jump as i've said before shown by a dotted line and then it's just going to take the value of the
function four t plus two for all values of t onwards so that's what it looks like i've just
projected the line back then here just to show where it cuts the where it would cut the vertical
axis so this line here this would be the graph of four t plus two but we're not interested in this
part of it because our switch hasn't been flicked on yet it gets flicked on at t equals one so that's
what that graph looks like okay so let's have a look at another so again we're going to do a type one
function we've got two t plus one times a step unit step function is critical value is two so
that's what our step function our unit step function looks like mathematically it's zero
for t less than two and it's one for t greater than two so first of all let's identify the value
of the critical of the unit the critical value of the step function the critical value is capital
t is equal to two it's that value there okay so that means that my function f of t is equal to two
t plus one will be off will be at zero up until the critical value of t equals two and the reason
for that is because the unit step is zero up until t equals two so you multiply anything by zero you
get zero so the function between my g of t function will be zero all the way up to t is equal to two
now as we said the unit step function gets switched on at t equals two and it'll take the value one
for for all t greater than two so you're going to have a function f of t is two t plus one
multiplying one so your overall function g of t will just be the graph of two t plus one and that's
for all t greater than two so what i need to do is i need to draw the graph of two t plus one again
it's a linear function straight line function so i just need two points on the line and i can select
any points that i like so for example i've chosen the values t equals two and t equals three now there's
a reason i chose t equals two the reason i chose t equals two is because that's the critical value so
i want to see where the function exactly jumps to at the left hand end if you like at the critical
value when the unit step function gets switched on so if i calculate the value of f of t when t is
equal to two it'll just be two times two plus one which is five and then i can also calculate the
value at t equals three and then f of three is equal to seven two times three plus one so that
means i've got my coordinate pair two points on the line two five and three seven just as i've
thrown just as i've included here in the diagram so that enables me then to draw my line two t plus
one and there it is and i've just projected back to show her cut the vertical axis but as we know
the overall function g of t is the product of f of t and the unit step function and the unit step
function has the ability to kill off the function all the way up to its critical value the critical
value is t equals two so as we said before our function g of t would be zero all the way up until
that critical value it'll then jump to take this value to take the value five when t equals two
and it'll just take the the form of the function two t plus one for all t
greater than two forever and ever so that's what it looks like the red graph here it goes along here
at zero suddenly jumps take the value five and then it just behaves as the function f of t
equals two t plus one okay so and here we are here's just another slightly more detailed plot of that
okay so let's have a look at another of these type one functions
in this case we've got sine t multiplied by a unit step function whose critical value is two pi
now pi is just a number so two pi is just a number so that we need we don't need to let that
concern us so I can draw the graph of sine t just to illustrate what's going on here so there's the
graph of sine t it's you know it's it's periodic and it oscillates between minus one and one
forever and ever so that's the graph of sine t shown in red here now um what do we know about
our step function we know that our step function has the critical value two pi so you multiply sine
t by a unit step function whose critical value is two pi the effect that that has is
orbital two pi this function our unit step function is equal to zero and therefore our whole function
g of t is equal to zero so that's what you can see here in blue the function g of t takes the value
zero all the way up to two pi and then suddenly the unit step function gets switched on and takes the
value one so overall function g of t will just be sine t times one or just sine t so it's basically
the the red graph comes into play then so it's just the graph of sine t from two pi onwards for
all values of t so the graph would keep going to the right so that's really all that's that's to it
that's all that's to it so basically um type one functions as we said a few slides ago I just go back
to find the slides basically it enables us to kill off a function up to some critical value and the
critical value is defined by the unit step function so in other words if we want to set a function f
of t equal to zero up until some specific some specific value we just set the critical value
in their step function equal to that specific value so our function f of t will start at t equals a
and it will continue like that forever and ever we now move on to look at our type two functions
and our type two functions I've referred to them as mathematical window so as I said in this case
we've got a function f of t multiplying the difference of two unit step functions and what
actually happens here is this forms as I've said a window which is zero to the left of a
it'll take the value one between a and b and then to the right of b it'll take the value zero
so the outside function f of t will only exist between a and b if you mark my f of t times one
between a and b so let's have a look at some examples of this so here's our function g of t
and that's a product of a constant function two and this window function here u of t minus four
minus ut minus ut minus eight so if we were going to sketch this what we would do is we would look
at our critical values four and eight so we mark these as four and eight now we know that what's
going to happen here is that everything is zero up until the first critical value and everything is
zero from the second critical value onwards and inside the window will basically have the plot
of whatever the multiplying function is and that in this case the constant function two so inside
the window this function g of t takes the value two first just going to be two times one and that's
what our function looks like as shown in this diagram down here so let's just have a little
bit more explanation of that so let's go let's go back here so here is our g of t as given in this
expression here so i've actually got it here so g of t is given here i've actually opened out the
brackets and i'm going to look at these as two separate functions two ut minus four critical
value four unit it's going to be a step function whose amplitude is two and will hold the value two
forever beyond t equals four similarly this one this one here is a unit step function of critical
value eight and it's been multiplied by two so we're going to get this function here zero all the
way up until t equals eight and then it suddenly gets switched on and takes the value two and holds
that value forever and ever so what we're doing here is we're doing red function minus blue function
so what happens here if we do that well let's have a look the red function between zero and four
is equal to zero so that's going to be zero all the way along there minus zero the blue function
is also zero over that interval so our resulting function our resulting function will also have
the value zero over that interval moving on let's see what happens now as we travel from four
up to eight which is the critical value of the second function well the red function
here my top function has the value uh two my blue function the other hand has the value zero so all
the way between four and eight we're going to have two minus zero which is equal to two so our g of t
our function will have the value two over that interval and then let's see what happens from
eight onwards so from eight onwards both functions have the value two and my overall function g of t
will just be two minus two all the way along here forever and ever and that's just zero
forever and ever so you can see what the function g of t looks like because sometimes called it's a
hat function so it's on a rectangular pulse so it's got the value zero all the way up to four it's
suddenly switches on and it holds the value two up until t equals eight when it then returns to zero
and it keeps the value zero forever and ever okay so let's now have a look at yet another example
of the type two so we've been asked to plot the function three of ut minus four minus ut minus two
minus ut minus four so basically we know this is going to be a mathematical window
and we know it's only defined between two and four so the left and right values of our window are
defined as two and four so our overall function g of t will take the value zero between zero and two
and between four and infinity forever and ever this you can see this could actually stretch all the
way down to minus infinity but we're only considering positive values of t so critical
values are two and four outside of these the function is equal to zero or g of t is equal to zero
inside the window here we've got the text of value one and we're multiplying that by three
so what we get is this constant line here between two and four so that's what our
hat function looks like a window function looks like and mathematically we would write that as g
of t is equal to zero for zero less than t is less than two it's equal to three for t less than
for two is less than t is less than four it's over that interval and it's equal to zero for t
strictly greater than four and that's over the interval between four and infinity so that's what
the function looks like okay so yet another example slightly more difficult this one because it's no
longer a constant function we're multiplying by but this shouldn't be too difficult it's just a linear
function so let's have a look at sketching this so first of all what we want to do is we want to
identify the critical values that will define our mathematical window the critical values are three
and five so I mark them down on my plot here I know that outside of these values that's less than
from zero to three and from five out to infinity the function g of t takes the value zero and that's
what I've done here now what I know is going to happen is that the multiplying function three t minus
one that's going to exist inside the window that's going to exist inside the window because this
put in here in the square brackets that's equal to one inside my window here so it's going to be three
t minus one times one which is just three t minus one so what I need to do is I need to be able to
sketch the graph of three t minus one inside this window and the way I'll choose to do it is I'll
get the coordinates of two points one at the left hand of the left hand end of the window
and the other one at the right hand end of the window so in other words when t is equal to three
this is to get the left hand value when t is equal to three I want f of three where f of t is just
three t minus one so t is equal to three I get three threes and nine minus one is eight so my
first coordinate pair are three eight so that is the left hand end of my window and that's the value
of my function g of t at the right hand end t is equal to five so I substitute that in here into f of t
so I get 15 minus one is 14 and that will give me the coordinate pair of five 14 at the right hand
end of my window and that will give me my straight line segment representing f of t is three t minus
one and therefore I've got the overall graph of g of t to the left of three it's got the value zero
inside the window it's equal to three t minus one and to the right of five it's equal to zero so there
is my windowed function my mathematical window this function three t minus one only exists inside
that window another example very similar we've got a mathematical window multiplied by a linear
function minus two t plus eight the critical values of the window are one and two so I mark them down
on my diagram I know that outside of that window my function g of t is going to take the value zero
so that's there between zero and one it's got the value zero and from two to infinity it's got the value
zero as well so what I need to do now is I need to plot the what's happening inside the window
well we know that this window here this function here ut minus one minus ut minus two has the value
one inside the window so it's basically going to be the graph of minus two t plus eight inside the
window the coefficient of t is negative so I know that the graph is going to be falling over that
window so all I need to do is get two points on this line here to sketch its graph and again
I choose the left and right values right hand side values that define my window so at t equals one
which comes from the critical value here my function f of t will be minus two times one plus
eight or in other words it's equal to six so i've got my coordinate pair one six at the left hand end
of the window at the right hand end of the window the critical value was t equals two so I plug that
in and I calculate the value of f of t it'll be minus two times two plus eight which is just four
and that gives me my second coordinate pair two four and I can draw the straight line segment
like so and that gives me my overall function g of t it's zero between um zero and one that's what
I've got here for zero for t less than one it takes a form of the graph minus two t plus eight in
other words it's just equal to f of t inside the window between one and two and then for t
bigger than two it takes the value zero and you can see all of that given mathematically over here
okay yet another function slightly different looking but we'll see that it actually boils
down to the same thing as we've been doing in the last two or three examples so sketch the graph
of the function g of t is equal to three minus t so that's going to be my f of t and i could one minus
ut minus three now this might not look like a unit step function but we did see this earlier on
we said that uh if we take the critical value of the step function as zero if we take our capital
t is equal to zero what we have we have u of t minus zero so our step function gets switched on
at t equals zero and it takes the value one from then onwards forever and ever and you can see that's
what's happening here we know it's zero all the way up to t equals zero it suddenly jumps
and gets switched on so that's a dotted line there all the way up to one and it just holds the value
of one forever and ever so i can replace the one here with u of t minus zero and that now looks like
my familiar mathematical window u of t minus a minus u of t minus b so i can now identify
the critical values that will define the left and right boundaries of my mathematical window
the first one is the left hand side so my critical value is zero so that'd be the
boundary of my uh window on the left hand side and on the right hand side the boundary of my window
is three so outside of that to the left of zero well my function g of t would be zero
and to the right of three my function g of t will also be zero inside the window my function g of t
would just take the value of this function which we i think called f of t earlier on
okay so what does that require us to do well again this is a straight line segment it's a linear
function t to the power one so i need to draw the graph of three minus t inside the window
and once again i'll do that by just selecting the critical values the the the the edges the
boundaries of the window that's t equals zero from here and t equals three from here
and i'll plug these in so when t is equal to zero my function f of t would be three minus zero is
just three so i get my first coordinate pair as zero three there we are here and at the other end
t is equal to three my function f of t will be three minus three which is zero so my uh
coordinate pair will be three zero and there it is so i've got two points on the line
and i could see my line inside the window i know it's going to have a negative gradient because
it's minus one t so it's going to be falling from left to right and there it is that's what it
that's what it looks like so that is the graph of my overall function g of t it's zero all the way
up until zero nothing's happening it suddenly switches on and over the window between zero and
three it takes the value defined by the function three minus t and then it's it then just returns
to zero outside the left hand boundary obtained from the critical value in here the left hand
boundary of my window so that's another example done i've got i've got another one here slightly
slightly trickier maybe what have i got here well i've actually got uh there's my mathematical
window critical values are two and four so i can easily mark my critical values and my overall
function g of t would be zero to the left of two which was my first critical value and be zero to
the right of four which was my second critical value obtained just by inspecting the mathematical
window now the function that i'm going to multiply by is actually product of two linear functions
i'm going to get a quadratic function and that's what i've got here that's what i'm showing you here
now uh we know a little bit about quadratic so we can get we can plot this we can plot this function
um so what happens here then so uh we've got a negative coefficient for the t squared term so
that means i just get my tablet that means that our graph the quadratic function is going to have
this shape some people say it's a sad face graph so that's going to be the shape of our graph now
what else can we find out well we can certainly identify where it cuts the x-axis because or the
t-axis sorry of course we've got two minus t into t minus four so if it cuts the t-axis
then this would be equal to zero so that'll happen when t is equal to two and when t is equal to four
and that's these two values marked down here okay now um
um what else can we determine well we can certainly because we know the quadratic graph we know it's
going to be symmetric about this midpoint here if we draw a vertical line through this peak here
through the this the apex through the the maximum of my quadratic function if i draw a vertical line
through it here i know the graph is going to be symmetric about this vertical line the left hand
boudre here is two the right hand boudre is four so that point in here must be equal to three
and if i know that t is equal to three i can substitute it into my equation so i get two minus
three times three minus four in other words minus one times minus one which is equal to one and that
will give me the amplitude or the other sorry or the the value of the function at its maximum at its
peak at its turning point if you like so i've been able to sketch the graph um and we know it only
exists inside that window so my overall function g of t then it's zero all the way up until the first
critical value of t equals two it's zero from the critical value of four onwards forever and ever
and inside this window it it's a quadratic function and i've been able to plot that by simply i identified
where it cut the t axis by just setting two minus t into t minus four equals zero so i found that it
cut the t axis at t equals two and t equals four i know that the quadratic it'll be it'll be um
it'll be symmetric about the line through the maximum and that's it there that occurs at t equals
three and i was able to calculate the value of the function at t equals three i found it's equal to
one so i was able to just sketch my graph in there so that's what it looks like okay so um we're
we're going to finish off now by
looking at how to determine the equations of functions from the graphs in the previous questions
we've been given the functions and we've been asked to sketch the graphs we're now going to go the
other way we've got the graphs and we want to determine the functions that gave us these graphs so
here's my graph i want the function of that that that represents that graph so um i've got
one two three critical values i can see there's critical values here because things are changing
at the critical values critical values represent where the function changes so i've got five 10
and 15 of these three critical values so between zero and five okay between zero and five this
function is equal to zero that's what it says between five and ten it's got the value two
that's what i'm saying here between 10 and 15 it's got the value minus one that's this says
and from 15 onwards it's got the value zero and that's what that condition says here so as i said
i've identified the critical values five 10 and 15 so that's going to give rise to three
unit step functions ut minus five ut minus 10 and ut minus 15 so obviously between zero and five
nothing happens so my overall function g of t will have the value zero on this interval
and that's what i say up here okay so we switch on at t equals five and the function g of t steps up
two units okay so that's that's what's happening here so to represent that i'll have
two ut minus five now it'll hold that value of two all the way up to the next critical value
and the next critical value occurs at t equals 10 and as you can see at t equals 10 there's a drop
of three units so we must have minus three times ut minus 10 because this is when the next this is
when the next change occurs at t equals 10 so we drop down from two to minus one that's a drop of
three units okay so we have minus three ut minus 10 and then finally when we hit the next critical
value at t equals 15 we can see that our function g of t jumps up one unit and takes the value zero
and it holds that value forever and ever and that's what i've said here g of t is equal to zero for t
greater than 15 so in other words we're going to add the one time we're going to add one times
the the unit's def function whose critical value is t equals 15 so you'll have one ut minus 15 and
that is my overall function i should have probably called it g of t down here not uh not f of t and
just correct that that's that's our function g of t i've called it g here and g here yes and g here
that should be of course g of t at the foot of the slide okay so let's uh do another one of these
just slightly uh different where we no longer got constant functions over each interval we've now got
still a linear this is a linear function with a negative gradient so let's see what we can do
with this thing so well we know it's a type two function we know this is a type two function because
we've got we've got two critical values one and four the function is zero outside of these values
and it only exists inside between one and four so we know that this this a type two function
in involving a mathematical window the mathematical window will be between one and four so uh our
overall function g of t will take the form of f of t which will be the function inside the window
times our mathematical window u of t minus a minus u of t minus b where a is the left hand
boundary of the window and b is the right hand boundary of the window now i can read these off
so i can certainly write down that my function g of t is some function f of t
given by this slanted line the sloping line here times my mathematical window who's got
that's got critical values one and four so it's ut minus one minus ut minus four so that just leaves
me to find what my function f of t actually is so as i say here uh what happens is by the
mathematical window we switch on f of t at equals one and we switch it off again at t equals four
so what i now need to do is if i can identify the coordinates of the points at t equals one
t equals four i will then have two points and that'll be enough to allow me to determine
the equation of the straight line segment inside the window so let's have a look from our diagram
we can read that at the left hand end of the window the line passes through the point one four
and at the right hand end of the window the point passes through the line passes through the point
four one okay so i can now go ahead and get the equation of this line once i know two points
on the line there they are given here okay so i'm going to use the standard form here of straight
line equation y equals mt plus c m is the gradient it's the change in the y direction divided by the
change in the t direction change in the vertical direction divided by the change in the horizontal
direction the change in the vertical direction well i would go from one up to four so i do one
minus four and the corresponding change in the t direction would be four minus one okay so you
know if you if you call this y the vertical one you can think of it as you can think of this one
as how it coordinates t1 y1 and t2 y2 and so the gradient will be y2 minus y1 so it's one minus four
divided by t2 minus t1 which is four minus one and if you work that out you find that the gradient
is minus one and that's really exactly what you would expect because for moving along three units
in the horizontal what's happened you've fallen by three units in the vertical direction so the
vertical change is minus three because it's falling divided by the horizontal change which is three
giving you a slope of minus one and then i can use either point in my equation so if i use the
point four one and the equation y equals mt plus c i know what m is we just found it it's minus one
okay i know what t is in this case it's equal to four and what i want to find is the value of c
if i work all of this out i can find that one is equal to minus four plus c or c is equal to five
so the overall equation of my function inside the window is y is equal to f of t remember this is f of t
f of t is equal to minus t plus five and so my overall function like i've written green g of t
zero for t less than one it's minus t plus five for one is less than t is less than four
and it's zero for t greater than four so from the plot of my function i was able to identify
that this required this was a mathematical window and i identified the critical values
appearing in the mathematical window one and four i set my function g of t to zero outside
these values as for all t less than one and all t greater than four and that left me with the task
of calculating the equation of the straight line segment inside the window and to do that i was able
just to read off the coordinate page and use these two points to calculate the equation
of the line inside the windows giving me minus t plus five
i mean you can check of course quite easily because you can check either end of the window my equation
f of t is minus t plus five when t is equal to one well f of t should be minus one plus five
should be equal to four yes it does when t is equal to four f of t should be minus four plus five
which is equal to one and yes it does so everything is fine everything checks out
okay so here's another example again we're gonna what looks like a mathematical window
i'm going to straight line segment inside that window we've got a linear function inside the
window so first thing we do is we identify the critical values which is a mathematical window
so we identify the critical values as t equals one and t equals two so we know that our overall
function is going to be zero outside the window to the left for t less than one and t greater
than two so um what's going on then so uh the function will switch on a t equals one and off a
t equals two so what we need to do is we need to find the equation of this this line here okay so
let's just find the coordinates of two points on the line so the first one is easy from the left
hand end of the window it's t is equal to one and you can read off that there uh
the vector value is zero so one zero is the first coordinate coordinate that we need the other one
at the other end of the window t is equal to two and f the f t or y whatever you want to call this
equal to three so my second pair is two three so what i want to do now is i want to calculate the
slope of this straight line segment it's a positive slope so i should get a positive value here when
i calculate m so it's vertical change divided by horizontal change so i'll just do three minus
zero just using the coordinate pair three minus zero divided by two minus one so just three over one
which is three so uh this one i've just i've just used this different formulation of a straight
line equation y minus b equals m t minus a where a b are the coordinates of either uh either paired
you can choose either pair to use so so let's have a look at this which one i'm going to use the
easier one i'm going to use one zero so i get uh so that's by a that's my b so i'll have y minus zero
is equal to three because that's gradient t minus one because my a value is one by choosing this
coordinate pair to use so i can simplify that to find that my equation is three t minus three
okay so that's the function that's going to multiply the mathematical window whose critical values
where t equals one and t equals two and there's my solution at the foot of the slide again you know
i could check if this was correct at the left hand end of the window t is equal to one so y would be
three minus three which is zero that's correct and at the other end uh t is equal to t is equal to
two so i would have three twos or six minus three give me the value three so that's absolutely
correct um so that's fine so that is the graph of my overall function which i've called f of t this time
um and i've called h of t been this function here okay right so uh maybe a couple more so this one
comes from Mobius to Tem the function f of t represent the graph below okay so just looking
at this the critical values are two and three so i know i'm going to have a mathematical window uh
ut minus three minus ut u of t minus two minus u of t minus three so the function inside here
switches on a t equals two and it switches off a t equals three so what we want now is we want the
equation of this straight line segment that lies inside the window okay so what do we do well we
need to read off some coordinate points and i do that by getting the left hand coordinate pair will
be two four and at the right hand end of the window we'll have three minus two so there we are so first
of all let's calculate the gradient so uh change in y divide by change in t so we'll get minus two
minus four divided by three minus two because minus six over one which is minus six and if i look at
my diagram i can see that that is correct because um my horizontal change i go from two to three so
i'm changing one in the horizontal direction there you are and in the vertical direction i've dropped
from four down to two so that's a change of six and it's falling so it's a negative change so we get
minus six over one vertical change or horizontal change give me minus six so you know you can either
do it like this or you could actually get the slope just by reading directly from the graph so here
again i'm going to use my equation y minus b equals mt minus a i know uh i know a i know b and i know
m so it's just a matter of plugging these in well i know a and b i've chosen to be the point two four
it's just easy to work with one with a negative in it so my a is two my b is four and my m is
minus six so i put all of these in i end up with this equation here and that will tell me that my
overall function is given by this equation here there's my mathematical window and it's multiplied
by minus six t plus 16 and that ensures that it only exists inside the window so again we can
check your values at either end of the window to see if this function is correct so on the left
hand side it's equal to two so i minus six times two is minus 12 plus 16 is four that's correct that's
the value there and the right hand end of the window where t is equal to three and i have minus six times
three is minus 18 plus 16 is minus two and that's exactly the value uh the function the diagram
takes at the value t three takes the value minus two so that's all correct we've been able to check
your answer so one more example then so again i want to determine the function f of t represent
this graph below again clearly it's a mathematical window uh for t less than two the function is
equal to zero and for t greater than four it's equal to zero it only exists between two and four
so that tells me that i should write down the equation for the mathematical window as ut minus
two minus ut minus four and again that leaves me the task of deriving the equation of that
straight line segment that exists inside the window it's uh going to have a positive gradient
because it's climbing you could actually just read the gradient off the diagram here so for two
steps two two steps in the t direction i would climb six in the vertical direction i would go from
minus two up to four over that window so the change in the vertical direction is six and the change in
the horizontal direction is two so the gradient should be three and it is and but you can verify
that by identifying the points two minus two at the left hand side and four four at the right hand
side and calculate the gradient in the standard way horizontal vertical change four minus minus two
divided by horizontal change four minus two and it does give you six or two which is three
which i got just by inspection really off the diagram so either way is perfectly okay and again
i use my equation for for for a straight line y minus b equals mt minus a i can choose either point
and i think it's sensible to choose the one without the negative value so i'm going to choose four four
as my point so a is four b is four i know m is three i can substitute all of that in
and i obtain my equation uh three t minus eight for the function that exists inside the window so
my overall function then my overall function i shown the diagram here will be my window ut minus
two minus ut minus four that ensures that everything outside less than two and greater than four is
equal to zero so if my overall function is equal to zero that's exactly what we see here and my it
ensures that the only thing that survives is inside the window where the where f t will take
the value of this function three t minus eight so let's just check that at the left hand end we've got
t is equal to two so you get six minus eight which is minus two yes that's correct two minus two
lies on the line at the right hand end t is equal to four so three times four minus eight 12 minus
eight is equal to four and as you can see the point four four does indeed lie on the line at the
right hand end of the window so that is correct so uh just to summarize what we've been doing so
in this lecture we've introduced the unit step function and we've looked at graphs of functions
that involve step functions we first of all considered functions involving sums and differences
of step functions and then we moved on to look at graphs of functions that involve different types
of products of step functions with other standard functions in our next lecture we're going to take
a look at calculating Laplace transforms of step functions and we're going to introduce the second
shifting theorem and this will enable us to transform time shifted functions now we need
these results later when we come to solving differential equations that involve step functions
to model applied voltages or forces so very quickly now I'll just say what you should be able to do
you should now be able to tackle mobius questions from number one to number 11 and from the notes
you should be able to attempt these tutorial questions so we can just stop at that we can
leave it at that and I'll stop recording