Well, I've been requested to do some videos on concativity and
maxima and minimum points and inflection points, and all the
like that you normally see in your first semester of
a calculus course.
So, I thought I would start doing them.
But before we kind of go into the actual problems that you
might see in your book, I just want to give you a
broad intuition.
Because in calculus class, people-- there are all these
rules, when, you know, the second derivative is positive,
you're concave this way.
When the second derivative is negative, you're
concave this way.
And people memorize the rules, and they take the tests, and if
they memorize it correctly they do well on the tests.
And then they forget it.
Maybe they memorize it again by the time of the final exam.
But by the time they are 30 years old, they will
have forgotten it.
And that, frankly, is useless.
Unless your whole goal is just to pass exams.
So I want to give you some intuition.
And hopefully, you will never forget this intuition.
So when you're first learning calculus there's much more
to functions than what I'm going to do in this video.
But there's kind of 3 types of interesting points on most of
the curves we'll talk about.
So let's say, let me draw a curve.
Let's say the curve looks something like this.
You know, if this is the x-axis-- that's too thick--
that's the y-axis.
The curve could look something like this.
I don't know.
Let me see.
It could look like this.
And so on.
And if you have a curve like this-- and you'll see all sorts
of curves when you do your calculus, because you have
multiple degree polynomials, et cetera-- what are the
interesting points?
Well, the interesting points tend to be when
the slope is equal to 0.
So, in this, where does the slope equal 0?
What are the interesting points here?
Let me make sure I'm using the right tool.
Right there the slope is 0, right?
Because if you were to take the tangent right there, it would
look something like that.
Right there the slope is 0, as well, because if you took the
tangent it would look something like that.
And then right there, as well, the slope is 0.
And the line, it really isn't a tangent.
It crosses the graph.
But as we could see, the graph kind of flattens out there.
So these are the three points that you tend to deal with in
your first semester calculus class when you're graphing
and looking for extrema.
And this is called a maximum.
And in this case, it's a local maxima.
If, you know, if this was the highest point for the entire
function, it would be a global maximum.
This is a local minima.
And if it was, you know, if it was the lowest point of the
entire graph-- which is the way I drew it-- maybe this graph
keeps going up like that, and keeps going up like this.
So this would be the global minimum point.
And then this is called an inflection point, because it's
neither a maximum, minimum, but you kind of see that the
graph kind of inflects.
So let's study them a little bit more and see if we can get
a little intuition on what happens to the graph, the first
derivative, and the second derivative, at each
of these points.
Let's start with a maximum point, because maybe that's--
well, let's start with a minimum point, because that's
what we see the most.
You know, when we used to study y equals x squared, or regular
parabolas, that was the most common thing we saw.
And then of course, if we have a negative of a parabola,
it becomes like that.
But anyway, let's study a minimum point where the
graph is concave upward.
Let's say we have a point like this.
So that's the graph of the function.
I don't know what that is, that's f of x.
So what is happening to the slope as we go from--
roughly-- from here to there?
Well, let's try a few points and see if we can, by drawing
the tangent lines, we get intuition for the slope.
So let's see, if I pick this point, just randomly, the slope
looks something like that.
I'm trying to do a tangent.
If I take this point-- I'm just moving to the right a little
bit, I'm just increasing my x value-- the slope will
look something like that.
So it's negative here, but if you look here, the
slope is less negative.
The slope is less negative there.
And if we go to the minimum point there,
the slope flattens out.
And so, of course, the derivative there would be 0.
And then as we pass that point, what happens?
Well, here, the slope is increasing a little bit more.
Now the slope is actually positive.
So the slope was negative, less negative, 0, a little positive.
And then when you go here, it goes more positive.
So let's see if we can graph that.
Just to get an intuition.
Obviously I haven't written any numbers down, so we don't know
what the exact value of the slope is.
So what does the slope look like?
So here it's really negative.
So the slope is going to be something really negative here.
I'm just picking a random point.
And then, at this point right here, it was less negative.
So the slope is going to be less negative.
And then here, roughly here, the slope is at 0.
Right?
The slope was 0.
And then the slope gets a little bit more positive.
And then over there, it gets a little bit more positive.
So the slope, if we were to actually graph the derivative
of the function at that point, it would be an upward sloping.
And so I'm assuming this is a second degree polynomial.
If this was third degree, then this would actually be a curve.
So the slope will actually look something like this.
And does that make sense?
Well, sure.
The graph of this could be y is equal to, I don't
know, ax squared plus b.
And then the derivative, or, you know, f of x is equal
to ax squared plus b.
And so the derivative of that, of course, would
be-- what is that?
2ax plus 0.
So this would just be a line.
And it actually goes through 0, 0.
But it didn't have to.
But hopefully that gives you a little intuition.
But now let's move to the second derivative.
So the slope of the slope.
Let me draw it down here.
The second derivative.
My wife is coming home, so I have to finish these videos
before she gets here, otherwise I have to continue it tomorrow.
She called 5 minutes ago, so I have to hurry.
Anyway, so this is f prime, prime of x.
Or the second derivative.
So what's happening to the second derivative here?
Well, the second derivative is just the slope
of the derivative.
So what's the slope?
Well, it's just a constant, in this case, it's just a
constant upward moving slope.
So the second derivative-- actually, if we were to just
evaluate that-- it would be 2a.
So it's going to look something like this.
And I can actually-- it's going to look-- it's just going to be
a constant number, and it's going to be positive.
So, in general, when you study your extrema points, and say,
whether it's a minimum or a maximum, et cetera, et cetera.
The way that you think about it is, well, if it's a minimum
point-- like this-- if it's a minimum point, what is
happening to the slope around it?
Well, the slope has to be constantly increasing, right?
Even though, for example.
Even though, as we go from here to here, the value
of f of x is decreasing.
The rate at which it's decreasing is getting less.
So the slope is actually increasing.
It's getting less and less negative.
And then it increases even more.
So you know you're at a minimum point when the
slope is increasing.
And similarly, you know that it's concave upwards when the
second derivative is positive.
So hopefully that helps you a little bit, getting the
intuition of what happens when you're at a minimum point and
you're concave upwards.
Well, what happens when you have the opposite?
Well, pretty much everything becomes the opposite.
Let's say that this is your curve.
I know that this is the x-axis.
If that's your curve, well, here you have
a very high slope.
Then your slope becomes a little bit less, then it
becomes 0, then it becomes a little bit more negative, then
it becomes more negative.
Then it goes more negative.
The rate of the slope becomes more and more negative the
whole time that we're kind of going through this
maximum point, or this concave downward.
So what will the slope look like?
Well, in this example, if I were to draw the derivative,
the slope is going to look something like this.
Because back here, the slope is-- actually, a really high
number is going to be up here.
And then as we get close-- as we move further and further,
increasing values of x-- the slope decreases.
It becomes 0 right around here.
And then it just keeps decreasing.
And then similarly, if we were to draw the second derivative,
what will that look like?
Well, the second derivative is just going to be the
slope of this function.
So it's just going to be a negative number, if we assume
that this is constant.
So the second-- if this is the x-axis, the second
derivative is going to be something like that.
It's going to be some negative number.
I'm assuming this is 0, 0.
Anyway, I'm about to run out of-- well, I'm not about to
run out of time, but I think I don't want to rush it.
So, that's the maximum, minimum points.
And in the next video, I'll give you a little intuition
on what happens with inflection points.
And then we will do a bunch of examples that you will normally
see in a calculus textbook.
But I just wanted to do this for you because there's nothing
magical about a lot of the rules that you'll learn.
You know, when they say, oh, you take the second derivative.
And if the second derivative is positive, it's concave upwards.
And if the second derivative is negative, it's
concave downwards.
And you might, you know, fill your brain with all
of these memorizations.
But it's hopefully-- if you think about it this
way-- fairly intuitive.
And hopefully, it should be fairly intuitive that whenever
the first derivative is 0, your slope is 0.
And you're at a point that is interesting, at
least in this context.
But anyway, I will do the next video on inflection points, and
then I'll do a bunch of actual problems.
See you soon.