So let’s look at our final example.
It’s talking about pressure that’s being
exerted on the walls of an artery
for a specific patient.
So we’re talking about blood pressure here.
And t is the number of seconds since
the beginning of the cardiac cycle.
Now, if you look at what
the question is asking,
it say’s find t when the
pressure is highest.
So it’s asking us for the time when the
pressure for this function is at its maximum.
Now, it’s really amazing that
you were able to do this now
because you don’t need
any more information.
You can just use this function
and your own knowledge
as to when you get maximums
and when you get minimums
and actually work out and
predict what the time should be
when the pressure is highest.
Let’s have a look the function and see
what we can do to answer this question.
I have a function of t,
which equals to 90 plus 15 sine
2.5 pi t.
So this is just a function,
it could be anything.
So any general function, this isn’t a
rule, but that’s what the pressure is.
It could be anything, but we’re just going
to learn what to do to deal with it.
It’s says find t when the
pressure is highest,
so when this value or
this pressure is maximum.
There must be a lot of theory going
through your minds right now.
How can you calculate the maximum
or the minimum of a function?
Think about it.
We’ve done this before when we
looked at stationary points.
We spoke about maximum and minimum points.
We spoke about how to find
maximum and minimum points
and we also spoke about how to prove
that they are maximums or minimums.
So if we remind ourselves
about stationary points,
remember that if you have any type
of a function, so any type of graph,
these here are your
And we said that these points
occur when the gradient is 0.
So if this is Y-X graph, we can say
that it occurs when dy/dx equals to 0.
That’s how you can find
a stationary point.
And then you can move on to
showing that this is the highest
or when the pressure
is the highest.
So let’s try this out
on this question.
We need to differentiate
this, so we need to do dP/dt
and we need to equal it to 0
to get the stationary points.
Also, a reminder of how
to differentiate trig.
So we know that sine and
cos and cos minus sine.
This direction to differentiate,
this direction to integrate.
We’ve spoken about this quite a few times.
So let’s start to differentiate now,
so I’ll say that dP/dt
equals to any constant
differentiates to 0.
So that goes away.
We have the 15, which can stay,
because it’s just a number
Sine, look at sine, differentiates to cos.
So this goes to cos of 2.5 pi t.
But there is an important thing
that you shouldn’t forget.
And that is to observe that this is a
function of t inside of a function.
So you have to remember how to
differentiate using the chain rule.
So we’ve differentiated the entire function
sine of something goes
to cos of that thing.
And then you have to multiply it
with the differential of the inside.
So the differential of 2.5 pi t.
Remember pi is just a number,
so it’s just the same as 10t.
So if you were differentiating
10t, you just get an answer of 10.
If you’re differentiating 2.5 pi t,
you just get an
answer of 2.5 pi.
So we’re now saying that
our gradient is 15,
let’s put that here.
2.5 pi cos
of 2.5 pi t.
And remember that at stationary
points, that needs to equal to 0.
You can rewrite your
2.5M as a fraction,
so you can write this
as 15, 5 over 2 pi,
cos of 2.5 pi t
equals to 0.
So we can work this out.
So if you times that through,
we’ve got 15 multiplied by 5,
which gives you 75 over 2 pi.
And then we’ve got cos
of 2.5 pi t equals to 0.
We can now start to deal
with this fraction here.
We’ve got 2 at the bottom,
so because it’s dividing,
you can multiply it
on the other side.
And when you multiply 0 times
2, that will give you 0.
So we can say that just
disappeared, I’ll do that here.
So we’ve got 75 pi cos of
2.5 pi t equals to 0.
You can then take this 75 pi
and since it’s multiplying here, it’s
going to divide on the other side,
which will also give you 0,
leaving you with cos
2.5 pi t equals to 0.
So remember that basically that entire
term here just goes to the other side,
and because it’s multiplying and
dividing with 0, it just goes to zero.
Continuing on from here, so we now
need to figure out what t is,
so we can take cos to the other side,
so we can rewrite this as 2.5 pi
t equals to cos inverse of 0.
Cos inverse of 0, if you’re using your
calculator, will give you an answer of 1.
Or you can remind yourselves what that
equals to just by drawing a cos curve.
So a cos graph looks like this.
So you know that when cos equals to 0,
so when we’re here, the answer is 1.
So you will get 2.5 pi t equals to 1.
Just make some more
space for myself here.
So when I rearrange this equation,
I’ll get t equals to 1 over 2.5 pi,
2.5 is just 5 over 2, so you
can write this as 5 over 2 pi.
And if you bring that 2 up, you get 2 over
5 pi is your time.
And you can work that out, let’s just
put that as t is 2 over 5 pi in seconds
and you can work that out as a decimal
and round up or you could keep it in
its whole number form as it is now.
So we’ve just concluded that
because that’s a stationary point
and that gives the time of
when the pressure is highest
is when t equals to 2
over 5 pi seconds.
And so we've now done some medical
examples using lots and lots of calculus.
Now, remember no matter what field
you’re planning of going into,
whether it’s a medical
field or any other field,
maths will make you
a better student.
It will make you a better problem solver
because you’ll always be
able to look at a problem
and look at it systematically and
go through it method by method.
Practicing mathematics is like
exercising for your brain.
The more you do it, the
better you’ll get at it.
So my advice if you’re
sitting in any type of
math exam is to practice
as much as you can.
Don’t leave it until the
last day or the last week,
just consistently practice.
So that you’re mind
will become sharper
and you’re becoming better
with all the methods.
Enjoy the challenges, that's
what you’re doing maths for.
You’re doing it to look at problems
and to try and solve them.
Enjoy the kind of questions
that are a lot harder to solve,
that take a lot more thinking, that take
a lot more methods and algebra to solve.
Because even if you don’t use
it in your tests or your exams,
it’s making you a
it’s making you a better person
at solving different problems.
So once you’re geared at certain methods,
push yourselves with
slightly harder questions
and challenge yourselves
further to look at questions
that need a more advanced
level of algebra.
Employers and degrees are
increasingly wanting maths
and are impressed with
students that do maths
only because it’s considered
to be a difficult subject
because it makes you
think in a logical way
that lots of other subjects
perhaps don’t manage to
develop that skill as
well as mathematics does.
It makes you think in a methodical
way so that you look at a problem,
you think of everything that you know,
think of all the things that you need
and you put them together
to find your solutions.
Make sure you sleep well for mathematics
because it’s important that
your brain is well rested.
It’s not like reading a book.
It is your brain continuously
working and remembering.
And remember when things get easy
and you feel that you’re just doing
the same thing over and over again
and you’re really good at it,
that’s the time to move forward
to a further challenge
and to a question that’s
of more difficulty.
Your analytical skills are developing
because you’re looking at results
and you’re looking at data and
you’re looking at equations
and you’re understanding
what they’re saying.
When you look at the rate of
change of population, so dP by dt,
you know what that means now.
You understand that as
population changing with time,
you’re actually analyzing more information
that you were able to before.
You’re able to visualize what
these functions look like,
so you can look at a quadratic
and you know what it looks like
in XY space or in
which again helps you analyze
data and helps you understand
whether your numbers or your function is
growing or whether it’s becoming smaller.
You're understanding gradients whether
they’re positive or negative.
What kind of rate of change can you see
in the growth or decay of an object?
Is it getting bigger?
Is it getting smaller?
And it’s all analytical skills
that you’re developing.
It’s all problem solving
skills that you’re developing.
So my advice at the end is
try and enjoy mathematics.
When you look at a challenge,
don’t panic, think about it.
Let your brain do the work.
Think of all the processes.
Think of everything that you know.
Think of things that you might need to
learn further and put it all together.
Stay calm and do it one step at a time
because remember doing maths is
learning how to solve problems.
And if it isn’t a problem, it isn’t really
too much of math or too much of challenge.