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hi welcome to calculus one 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 it involves is
the study of curved lines or curved
surfaces I like to open with this quote
by Leonard Euler that says that nothing
takes place in the world whose meaning
is not that of some maximum or minimum
now whether you look at this
philosophically or mathematically it
holds true to our lives and to the world
of mathematics everything that we do has
a beginning or an end and a maximum or a
minimum and that basically construct
calculus it was put together while
modern-day calculus was put together by
Newton and Lightning's who brought it
together around 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 a 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 or 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 rollercoasters
you can feel a positive gradient as you
go up you can feel it flatten to zero
gradient as you reach the top and then
to negative gradient when you go down
so essentially you can feel calculus you
can feel gradients and thanks to these
mathematicians that have come up with
all these
fun rides for us I 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 a radioactive substance within
the body and then measure signals such
as gamma rays to give you a scan of
whatever it is that's causing trouble or
the blockages are you deem 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 it's straight line you can
calculate the gradient the midpoint and
the length of on graphs but we're going
to use those concepts and apply them
onto curved lines in this lecture the
method that we will use is called
differentiation from 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
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