Welcome back.
Well, I've been requested to do several problems by
our friend [? Akosh. ?]
So I thought I would keep doing them.
I'm skipping around a little bit.
Because I think if I did all of them, it would just
generate too many videos.
But I encourage all of you all to let me know if you feel that
there's something that I might have missed.
But anyway, I'm just going through some of the
problems he gave.
So this one-- and I actually almost find it funny, because
they use such formal language for something that's actually
a pretty intuitive concept.
Well they say, use the monotonicity theorem to find
where the given function is increasing, and when
it is decreasing.
Monotonicity theorem.
It make sound very serious.
Well all the monotonicity theorem-- at least if I'm
remembering things properly-- all it says is when the
derivative is positive, when f prime of x is greater than 0,
your function is increasing.
And when the derivative is negative, your function
is decreasing.
And why am I kind of disparaging of this?
Well what's the derivative?
The derivative is the slope.
And you learned in Algebra 1, that if the slope is
positive, the graph of the line is increasing.
And if the slope is negative, the graph of
the line is decreasing.
And the only thing that's different now is that this
function isn't necessarily a line, it's a function.
So it could be a curve.
But having a positive slope means that the value of the
function is increasing.
For every change it x, there's a positive change in y.
And similarly, a negative slope says that for every change in
x, no matter how small, there's a negative change in y.
So there's nothing fancy here.
But let's see, they gave some examples.
So they want us to use this theorem to find where the
given function is increasing, and when it is decreasing.
So the first function they give is f of x is
equal to 3x plus 3.
What's the derivative of this? f prime of x is equal to 3.
For any value of x this is a positive.
Right?
F prime of x is positive for all values of x.
So this thing, using the monotonicity theorem
is positive.
Sorry, it's increasing for all values of x.
And you could have done that in the 9th grade.
How?
Well you would have said, this is the slope y-intercept.
The slope is 3.
So this is an increasing function.
The function will look something like this.
The y-intercept is 3.
And the slope is 3.
So this function looks something like that.
And so it's increasing over from minus infinite to positive
infinite over that interval, if you wanted to be fancy.
But this is really just a lot of fancy terminology to talk
about, I think, something fairly straightforward.
Anyway, the next one looks a little bit more interesting.
H of z is equal to z to the fourth over 4, minus-- and I
hope I'm reading this right-- 4 z to the third over 6.
So let's see if we can figure out when this is increasing,
and when this is decreasing, when the function is
increasing or decreasing.
So what's the derivative of this?
H prime of z is equal to 4.
This will cancel out.
So it's z to the third, 4z to the third over 4.
So it's z to the third minus 3 times 4 is 12z squared over 6.
So it's 2 2z squared.
So we just have to figure out when is this function greater
than 0, and when is this function less than 0.
And to figure this out, we really just have to break out
our algebra toolkit to figure out well one, when does
this function equal 0?
So let's set z-cubed minus 2z squared is equal to 0.
We could factor out a z-squared.
And then we have z minus 2 is equal to 0.
So we know that either z-squared is equal to 0, or
z minus 2 is equal to 0.
So the points at which the derivative is 0, so H prime
of 0 is equal to 0, and H prime of 2.
And that's just our Algebra 2.
So H prime of 2 is equal to 0.
And so we just have to figure out what happens in the
interval maybe when we're below 0, between 0 and
2, and then above 2.
So what is the derivative?
So when we're below 0, let's just take a point.
Let's say H prime of negative 1 is equal to what?
That is equal to negative 1 to the third is negative 1, minus
2 times negative 1 squared.
Well negative 1 squared is just positive 1.
So it's minus 2.
So it equals minus 3.
OK, let's take a point in between these two.
Well 1 is in between them.
So let's just take H prime of 1 is equal to 1 minus 2.
Lets make sure I'm doing that right.
Because it's 1 squared.
So it equals negative 1, H prime of 1.
And then what happens when we go of z values greater than 2.
So let's try 3.
H prime of 3 is equal to what?
H prime of 3 is equal to 27-- what's minus 2 times 9.
It's equal to 9, right, 18, right.
So it's positive.
So, what do we know?
We know that in all fairness, this is kind of an
interesting problem.
Because the derivative is negative.
So if we were to draw the derivative of this, what are
the interesting points?
It's 0.
So that's the x-axis.
That's 0.
And then we had 2.
And I'm going to draw the derivative.
I'm not drawing the actual function.
Because the monotonicity theorem, we care about
whether the derivative is positive or negative.
So what this tells us is that the derivative, it
was a negative when we're to left of 0, right?
Because we just took a test point.
Because we know it's 0 here.
But then what happened?
It's not like it got positive then.
It went back negative again when we tried to
derivative it once.
It probably does something like this.
Now what would it be?
Then it goes up at 2 where it's greater than 0.
Right.
I was confusing myself for a second.
Right.
So the derivative, it starts.
But it never goes above right there.
So the derivative probably has a bit of a maximum
point there or something.
This is the derivative, remember.
That's why I was confusing myself.
And then it probably flips around, and then
goes above there.
But from the monotonicity theorem, what do we care about?
We care about the intervals where the derivative
is positive, and the derivative is negative.
So if the derivative is positive for all of
these values, for z is greater than 2.
So we could say, using the monotonicity theorem, this
function is increasing when z is greater than 2.
So let me write that.
And then we can say that the function is decreasing.
We could say flat or decreasing, right?
Because the slope of the function is 0 right here.
And remember this is the graph of the derivative.
So we could say flat or decreasing when
z is less than 2.
You normally do consider flat, monotonic in one direction.
So you could say monotonic increasing would still
include something that kind of flattens out.
Well anyway, I'm pushing ten minutes.
And I don't think I have time for the next problem, which I
might do in the next video.
But hopefully you found that a little bit helpful,
and not too confusing.
I will see you in the next video.