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# Limit expression for the derivative of cos(x) at a minimum point

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Let g of x equal cosine of x. So let's actually graph that. So that would be the
graph right over there. I drew it ahead of time. So this is a segment of cosine
of x between x is equal to 0 and x equals pi. Obviously, it keeps on going
in both directions after that. And then they ask
us, what is the limit as h approaches 0 of g of pi
plus h minus g of pi over h? So let's just think
about it a little bit. So they are concerning
themselves with g of pi. So let's look at the
point pi comma g of pi. So that right there is the
point pi comma g of pi. And they're also concerning
themselves with g of pi plus h. So let's say that
this right over here is the x value pi plus h. And then this right over here
would be the point pi plus h, g of pi plus h-- would be
this point right over here. And they're essentially
trying to find the slope between
these two points. If we wanted to find the
slope between these points, it would be change in
our y value over change in x, or change in the vertical
over change in the horizontal. What is our change
in the vertical? What is the change
in the vertical? Well, the change
in the vertical, we'll take this y value
here-- so g of pi plus h. We'll subtract this y
value here-- minus g of pi. So that was our
change in the vertical over the change
in the horizontal. Well, what is that going to be? Well that's going to
be pi plus h minus pi. And this is exactly what we have
or, before we take the limit, what we have right over here. So this is going to be g
of pi plus h minus g of pi, all of that over--
these pi's cancel out-- all of that over h. Now, so this is the
slope, this slope the way I've-- it's a slope of the
secant line right over here. Now let's think
about what is going to happen as h gets
closer and closer to 0. So the way I've drawn
it right over here, that means that
this point is going to go further and
further to the left. This point is
going to go further and further and
further to the left. So pi plus h, as h approaches
0, is going to approach pi. If we assumed h was a
negative value, then we would be
approaching from here. But what is going to
happen as h becomes smaller and smaller and smaller? Well, the point pi
plus h, g of pi plus h is going to get closer
and closer to this point. And the slope of
those secant lines-- I know it's hard to see,
because it's small here-- they're going to
start getting closer and closer to the slope of
the tangent line right at x equals pi. So this right over here
really is another way of saying the slope of the
tangent line at x equals pi. Well, what does that look like? Well the slope of
the tangent line, we're at a minimum point at
x equals pi right over here. Cosine of pi we know
is a negative 1. That's a minimum value. It's one of the minimum values. It keeps going to
that for cosine of x. And so its tangent line is just
going to be a horizontal line. So we know that this right
over here is going to be 0. Now, there are other ways
you could have tackled it. We don't have the
tools right now to do it purely
algebraically-- to say cosine of pi plus h minus cosine of pi. There are ways to do it. But we're not going
to do that now. The other option that you
could do is use a calculator. So for example, you
could say, well, OK, let's just take very small
h's and just evaluate them. So we'll evaluate cosine of pi
plus some h minus cosine of pi over that small h. And we're going to get
smaller and smaller h's here. And so actually,
let's try to do that. That could be interesting. So let me clear this. So let's just take a
really, really small h's, h's getting really,
really close to 0. So let me make sure I'm in
Radian mode, first of all. I'm in Degree. So let me fix that. All right, perfect. And now let me
take cosine of pi. I'll take a reasonably small
h, 0.1, minus cosine of pi. I always forget where the pi is. So that's the numerator. And then let me divide
that by the same h, by 0.1. So this is just my previous
answer divided by 0.1. So I get 0.04. Now let me make h even smaller. And I'll actually do in
one expression, actually, this time. So cosine of-- actually,
make it a lot smaller-- of pi plus 0.0001-- so
1/10,000 more than pi, right over there--
minus cosine of pi. And now we are going to
divide by this h, 0.0001. And what do we get? 5 times 10 to the
negative 5th-- so you see clearly that we're getting
to some really, really, really, really small numbers, that this
expression is approaching 0.