We have now looked at a huge chunk of calculus. We have looked at lots and lots of different
ways of differentiating and finding the gradients and also some of their applications.
Remember that we have learned how to differentiate from first principles which was the definition
of a derivative and we’ve used that definition through this course. We then looked at different
types of equations and how to differentiate them such as a function of a function for where we
used the chain rule. We looked at two functions multiplying together so we used the product rule then.
Two functions dividing each other and we used the quotient rule. We then moved on to differentiating
Trig and e(x)'s and ln(x)'s with their general results. We then looked at new types of equations.
So two new types of equations that we introduced are parametric equations and implicit equations.
We discussed how we’d go about differentiating those equations. We’re finally at a point
where we can start to look at some more applications of differentiation. One of the major ones
being stationary points and we’ll discuss why we do it and how we do it in a minute. Stationary points,
we’re going to discuss stationary points in this lecture. We’ll discuss what they are and also
the mathematics behind them and why we use them. The techniques we’re going to use here
are all different types of differentiation methods, so any type of equation whether they’re
multiplying, whether they’re a function of a function. We now use all the skills that we have learned
and then we put them together into application now. The actual outcome of this will be
that we'll be able to find maximum and minimum points on curves, which really is quite interesting
because if you think about it, once you know the gradients, the maximum, minimum points of curves,
you can actually sketch an entire curve accurately without needing computers or calculators.
So let’s move straight into calculating or discussing what stationary points are. Now, this is quite
nice because we actually experience stationary points in our everyday lives. They’re part of
applied mathematics. So they’re used extensively in things like mechanics and in fluid dynamics,
and also in things like molecular modelling if you’re thinking about the medical field.
Stationary points are essentially points which are flat. A common example or a common experience
that you might have is when you’re going up roads. So when they’re steep, you go up a certain gradient
and then it flattens. You suddenly reach a point which is flat that is stationary that has zero gradient.
If you enjoy rollercoasters, you can experience, you can actually sense gradients. So as you go up,
you feel positive gradient, you might suddenly have a moment where everything looks straight.
That is a stationary point. It’s that when you reach zero gradient. Then you have negative gradient
as you go down. Engineers and medicine professionals work together at designing these kinds
of recreational activities for us which are safe for our bodies but also give us that thrill and also
use all the mathematics to reach a safe, optimum stationary point. Let’s have a look at what
they actually are mathematically. Imagine, I have a curve. Let’s make it an interesting curve.
Let’s just say it goes up and down and does something like that. You’ll see that each part of this curve
is different. If you think about the roller coaster example once again, you’ve got a positive gradient
here as you go up. You might come down so you have a negative gradient there, a positive gradient
here once again, and a positive gradient here. The interesting points to ask now are these little
bits here. So what’s happening here, there, as the gradients change? So you can see at this point here
that the gradient is going from positive to negative. That’s an interesting point for us.
Here, you can see that the gradient goes from negative to positive. Then here, which makes it an interesting
point, you have positive gradient, flat, and then positive again. Now these parts here are actually
known as stationary points. We call them stationary points because the gradient flattens here.
If we’re thinking about this in mathematical terms, at stationary points, your gradient, dy/dx = 0.
That’s what makes these points so special because they are the only points which have a flat gradient.
If you’re ever looking for stationary points in a curve, all you have to do is equal your gradient
to zero in order to calculate where exactly these points are. It really is quite interesting
when we start to do it as calculations. Let’s summarize what we’ve just said here. Remember that
at stationary points, the gradient is equal to zero. The gradient to us as mathematicians
means that dy/dx = 0. You know all types of methods of calculating the gradient. You know different
functions and how to calculate the gradient. All you have to do if you’re looking for stationary
points is to equal the gradient to zero here. The next thing you might notice is that these are
different types of stationary points. If you observe closely, you’ll see that you have a stationary
point that looks like this, and a stationary point that looks like this, and a stationary point
that looks like this. So, how do we distinguish between those three types of stationary points?
Yes, we can find the coordinates of the stationary points when we equal our gradient to zero.
But how do we actually find those three types of points? Let me just see if I can show you
the three types of points. You have a point that looks like that, a point that looks like this, and a point
that looks like this. Remember that this point could also look like so. Think about this point here.
If you think about this in the graph here, this is the highest this curve is reaching.
Essentially, we call this a maximum point or a maximum stationary point. Here, we’re reaching
a low point in this curve. So this point here, we call a minimum point. These points here are special
in nature because they go flat for a little while and then they follow the same gradient.
So if you observe here, we have a positive gradient here, and then it goes flat,
and then it goes positive again. Alternatively, we could say that it could go negative, and then flat,
and then negative as well. So both things could apply here. Now, these points are called points of inflection.
This needs a little bit more investigation. So, we’ll do some calculations and show you how to find points
of inflections. But remember that the entire thing we’re talking about here is gradients.
How do you find gradients of curves? It’s dy/dx. How do you dy/dx? We’ve been doing that through
this course so you know lots and lots of different methods of dy/dx-ing or finding the gradient.
So, just to summarize this little page here, we are saying that stationary points occur when
the gradient is zero. The gradient to us as mathematicians is dy/dx of a curve. We’re saying that
there are three types of stationary points: a maximum point, a minimum point, or a point
of inflection. We are going to move one step forward and actually discuss how to calculate
maximum, minimum, and points of inflection.