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AP Calc: FUN‑4 (EU), FUN‑4.B (LO), FUN‑4.B.1 (EK), FUN‑4.C (LO), FUN‑4.C.1 (EK)

You've opened up a
shoe factory and you're trying to figure out how many
thousands of pairs of shoes to produce in order to
optimize your profit. And so let's let x equal the
thousands of pairs produced. Now let's think
about how much money you're going to make per pair. Actually, let me say
how much revenue, which is how much you actually
get to sell those shoes for. So let's write a
function right here. Revenue as a function of x. Well, you have a
wholesaler who's willing to pay you $10
per pair for as many pairs as you're willing to give him. So your revenue as a function
of x is going to be 10 times x. And since x is in thousands
of pairs produced, if x is 1, that means 1,000 pairs produced
times 10, which means $10,000. But this will just give you 10. So this right over here is
in thousands of dollars. So if x is 1, that means
1,000 pairs produced. 10 times 1 says
r is equal to 10, but that really means $10,000. Now, it would be a nice
business if all you had was revenue and no costs. But you do have costs. You have materials, you
have to build your factory, have to pay your
employees, you have to pay the electricity bill. And so you hire a
bunch of consultants to come up with what your
cost is as a function of x. And they come up
with a function. They say it is the number of the
thousands of pairs you produce cubed minus 6 times the
thousands of pairs you produce squared plus 15 times
the thousands of pairs that you produce. And once again,
this is also going to be in thousands of dollars. Now, given these functions
of x for revenue and cost, what is profit as a
function of x going to be? Well, your profit
as a function of x is just going to be equal to
your revenue as a function of x minus your cost as
a function of x. If you produce a
certain amount and let's say you bring in, I don't
know, $10,000 of revenue and it costs you $5,000
to produce those shoes, you'll have $5,000 in profit. Those numbers aren't the
ones that would actually you would get from
this right here. I'm just giving you an example. So this is what you
want to optimize. You want to optimize
p as a function of x. So what is it? I've just said it here
in abstract terms, but we know what r of
x is and what's c of x. This is 10x minus
all of this business. So minus x to the third
plus 6x squared minus 15x. I just subtracted x
squared, you subtract 6x squared it becomes positive,
you subtract a 15x it becomes negative 15x, and then we can
simplify this as-- let's see, we have negative x
to the third plus 6x squared minus 15x plus
10x, so that is minus 5x. Now if we want to optimize this
profit function analytically, the easiest way is
to think about what are the critical points
of this profit function and are any of those
critical points minimum points or
maximum points? And if one of them is a
maximum point, then we can say, well, let's produce that many. That is going to be--
we will have optimized or we will figure out the
quantity we need to produce in order to optimize our profit. So to figure out
critical points, we essentially have to find
the derivative of our function and figure out when does
that derivative equal 0 or when is that
derivative undefined? That's the definition
of critical points. So p prime of x is going
to be equal to negative 3x squared plus 12x minus 5. And so this thing is going
to be defined for all x. So the only critical
points we're going to have is when the first derivative
right over here is equal to 0. So negative 3x squared plus 12x
minus 5 needs to be equal to 0 in order for x to
be a critical point. So now we just have
to solve for x. And so we just are essentially
solving a quadratic equation. Just so that I don't
have as many negatives, let's multiply both
sides by negative 1. I just like to have a
clean first coefficient. So if we multiply both
sides by negative 1, we get 3x squared minus
12x plus 5 is equal to 0. And now we can use the quadratic
formula to solve for x. So x is going to be equal
to negative b, which is 12, plus or minus the square root. I always need to make my
radical signs wide enough. The square root of
b squared, which is 144, minus 4 times a, which
is 3, times c, which is 5. All of that over 2a. So 2 times 3 is 6. So x is equal to
12 plus or minus the square root of, let's see,
4 times 3 is 12 times 5 is 60. 144 minus 60 is 84. All of that over 6. So x could be equal to 12 plus
the square root of 84 over 6 or x could be equal to 12 minus
the square root of 84 over 6. So let's figure out
what these two are. And I'll use a calculator. I'll use the calculator
for this one. So I get, let's see, 12 plus the
square root of 84 divided by 6 gives me 3.5--
I'll just say 3.53. So approximately
3.-- Actually, let me go one more digit, because
I'm talking about thousands. So let me say 3.528. So this would literally
be 3,528 shoes, because this is in
thousands, or pairs of shoes. And then let's do the
situation where we subtract. And actually we can look
at our previous entry and just change this
to a subtraction. Change that to not a
negative sign, a subtraction. There you go. And we get 0.4725. Let me remember that. 0.4725. Approximately equal to 0.4725. I have a horrible
memory, so let me review that I wrote
the same thing. 4725. Yep. All right. Now these are all
we know about these, or these are both
critical points. These are points at which
our derivative is equal to 0. But we don't know whether
they're minimum points, they're points at which
the function takes on a minimum value, a
maximum value, or neither. To do that, I'll use the
second derivative test to figure out if our
function is concave upwards or concave downwards or
neither at one of these points. So let's look at the
second derivative. So p prime prime
of x is going to be equal to negative 6x plus 12. And so if we look at-- let me
make sure I have enough space. So if we look at p
prime prime of 3.528. So let's see if I can
think this through. So this is between 3 and 4. So if we take the lower
value, 3 times negative 6 is negative 18 plus 12 is
going to be less than 0. And if this was 4 it'd
be even more negative, so this thing is going
to be less than 0. Don't even have to use my
calculator to evaluate it. Now what about this
thing right over here? 0.47. Well, 0.47, that's roughly 0.5. So negative 6 times
0.5 is negative 3. This is going to be nowhere
close to being negative. This is definitely
going to be positive. So p prime prime of
0.4725 is greater than 0. So the fact that the second
derivative is less than 0, that means that my
derivative is decreasing. My first derivative
is decreasing when x is equal to
this value, which means that our
graph, our function, is concave downwards here. And concave downwards means
it looks something like this. And so you can see what it
looks something like that, the slope is
constantly decreasing. So if you have an interval
where the slope is decreasing and you know the point where
the slope is exactly 0, which is where x is equal to
3.528, it must be a maximum. So we actually do take on a
maximum value when x is 3.528. On the other side we
see that over here we're concave upwards. The graph will look something
like this over here. And if the slope is 0 where
the graph looks like that, we see that that
is a local minimum. And so we definitely
don't want to do this. We would produce
472 and 1/2 units if we were looking to minimize
our profit, maximize our loss. So we definitely
don't want to do this. But let's actually
think about what our profit is going
to be if we produce 3.528 thousands of
shoes, or 3,528 shoes. Well, to do that we just
have to input it back into our original profit
function right over here. So let's do that. So I get my calculator out. So my original profit
function is right over there. So I want to be able
to see that and that. So I get negative 3.528 to the
third power plus 6 times 3.528 squared minus 5 times
3.528 gives me-- and we get a drum roll now--
gives me a profit of 13.128. So let me write this down. The profit when I produce 3,528
shoes is approximately equal to or it is equal to, if I produce
exactly that many shoes, it's equal to 13.128. Or actually it's approximately,
because I'm still rounding 13.128. So if I produce 3,528
shoes in a given period, I'm going to have a
profit of $13,128. Remember, this right over
here is in thousands, this right over here
is 13.128 thousands of dollars in profit,
which is $13,128. Anyway, we are now going to
be rich shoe manufacturers.

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