Hi, welcome to Calculus 1.
Where we will be looking at a field of mathematics known as Calculus.
Basically, calculus is split into two parts,
we have differential calculus and integral calculus.
And all that involves is the study of curved lines or curved surfaces.
I like to open with this quote by Leonhard Euler,
that says that nothing takes place in the world
who's meaning is not that of some maximum or minimum.
Now, whether you look at this philosophically or mathematically,
it holds through to our lives and to the world of mathematics.
Everything that we do has a beginning or an end, and a maximum or minimum.
And that basically constructs calculus.
It was put together or modern-day calculus was put together
by Newton and Leibniz who brought it together around in the 17th century.
And those are the formulas and the methods
that we will be looking at through this course.
So why do we study Calculus?
The simple answer to that, is that it's everywhere.
Whether you look at demographics. How does population grow?
Or how does a bacteria grow or tumor grow?
In order to model that you will have to use some form of Calculus
because it is the rate study of the rate of change.
In medicine, looking at things like mitosis.
How do things grow over time? I recently had a child,
so I often wonder and look at the rate of change of the growth of a baby,
and that is calculus of a curved surface if we were to model that on a graph.
In engineering, there's no escaping it, it's all around us,
you can see curved lines and curved surfaces in bridges
or in construction or in architecture.
You can witness it yourself when you're sat on roller coasters.
You can feel a positive gradient as you go up,
you can feel it flattened to zero gradients as you reach the top.
And then, to negative gradient when you go down.
So essentially, you can feel calculus, you can feel gradients.
Thanks to these mathematicians that have come up with all these fun rides for us.
I've recently had an interest in studying radioactivity in medicine,
and I'm intrigued by what it's doing in the world of medical development.
For example, the pet scans, that use radioactive substance within the body
and then measures signals such as gamma rays
to give you a scan of whatever it is that's causing trouble or the blockages.
Iodine therapy is used for treating things like hypothyroidism
and again it's essential for medical students
and people to understand how radioactive half-life works
in order to understand how those processes within the body work.
So, let's start. We are going to start this course
just thinking like the ancients had perhaps.
They look at a curve and they think about it.
They think about how to calculate the gradient of a curve.
We are well versed and hopefully know beforehand
how to calculate the gradient of a straight line,
but we'll quickly go over that to develop into further ideas.
With its straight line, you can calculate the gradient,
the midpoint and the lengths on graphs,
but we are going to use those concepts
and apply them onto curved lines in this lecture.
The method that we will use is called differentiation from the first principle
or some call it the actual definition of a derivative
and the application of it is obviously going to be looking at gradients
of straight lines and curves.