Marginal cost & differential calculus | Applications of derivatives | AP Calculus AB | Khan Academy

Let's say I run some
type of a factory, and I've studied my operations. And I'm able to figure
out how my cost varies as a function of quantity over
a week, on a weekly period. And so to visualize
that, let me draw it. I could draw this cost function. So this is my cost axis. This right over here
could be my quantity axis. So that's quantity, or q,
let me just call that q. That's my q-axis.

And my function might
look something like this. It seems reasonable to me. Even if I produce nothing,
I still have fixed costs. I have to pay rent
on the factory. I have to probably pay people
even if we produce nothing. And so let's say that fixed
costs in the week is $1,000. And then as my quantity
increases, so do my costs. So if I produce 100
units right over here, then my cost goes up to $1,300. If I produce more than that,
you see my costs increase and they increase at
an ever faster rate. Now, I go into a lot more depth
on things like cost functions in the Economics
playlist, but what I want to think about in the
calculus context is what would the derivative of
this represent? What would the derivative of c
with respect to q, which could also written as c prime of
q, what does that represent? Well, if we think
about it visually, we know that we can think
about the derivative as the slope of
the tangent line.

So, for example,
that's the tangent line when q is equal to 100. So the slope of that tangent
line you could view as c prime, or it is c prime of 100. But what is that
slope telling us? Well, the slope is the
change in our cost divided by the change in our quantity. And it's the slope
of the tangent line. This is what we first
learned in calculus. As we get to smaller and
smaller and smaller changes in quantity, we
essentially take the limit as our change in
quantity approaches 0. That's how we get that
instantaneous change. So one way to think about it
is this is the instantaneous. This is the rate
right on the margin at which is our cost is changing
with respect to quantity. So if I were to produce just
another drop, another atom of whatever I'm
producing, at what rate is my cost going to increase? And the reason why I'm
saying right on the margin is we see that
it's not constant.

If our cost function
were aligned, we would have a constant slope. The tangent line
would essentially be the cost function. But we see it changes
right over here. The incremental
atom to produce here costs less than the incremental
atom right over here. The slope has gone up. And it might make sense. Maybe I'm using some
raw material out there in the world. And as I use more
and more of it, it becomes more and more scarce. And so the market price of
it goes up and up and up. But you might say,
well, why do I even care about the rate at which
my costs are increasing on the margin? Which is why this is
called marginal cost. Well, the reason why
you care about it is you might be trying to figure
out when do I stop producing? Let's say this is orange juice. If I know that next gallon is
going to cost me $5 to produce and I can sell it for $6,
then I'm going to do it. But if that next
gallon, if I'm up here, and I've already
produced a lot, and I'm taking all the oranges
off the market, and now I have to
transport oranges from the other side of the
planet or whatever it might be, and now if that incremental
gallon of oranges or gallon of orange juice costs
me $10 to produce, and I'm not going to be able
to sell it for more than $6, it doesn't make sense for
me to produce it anymore.

So in a calculus context, or
you can say in an economics context, if you can model your
cost as a function of quantity, the derivative of that
is the marginal cost. It's the rate at which
costs are increasing for that incremental unit. And there's other similar ideas. If we modeled our profit
as a function of quantity, if we took the derivative, that
would be our marginal profit. If we modeled revenue, that
would be our marginal revenue. How much does a
function increase as we increase our
input, as we increase our quantity on the margin?.

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