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The cost in dollars of producing
x gallons of wood stain is given by C of x is equal to
3,200 plus 0.1x minus 0.001x squared plus 0.0004x
to the third power. What is a formula for the
marginal cost function C prime of x? So we really just have to
take the derivative of C with respect to
x, to think about how does C change as x changes. As our quantity increases,
how does our cost change? So what's the derivative
of this, right over here? Well, the derivative of 3,200
with respect to x is just 0. The derivative of 0.1x is 0.1. The derivative of negative
0.001x squared is going to be negative 0.002x. And then finally, the
derivative of this is going to be-- let's see,
this is 4 ten thousandths. So 3 times 4 ten thousandths
is 12 ten thousandths. So 0 point 1, 2, 3, 4. Yep, that's 0.0012x squared. So that right over there is
equal to the marginal cost function. So if you see this
in an exercise, you just want to type
in the expression here. But this, of course, is
equal to C prime of x. To the nearest penny,
what is the marginal cost of producing the
101st gallon of stain? So they say C prime of 100
is equal to blank dollars per gallon. So what do they
want us to do here? What they really
want us to do is approximate what the incremental
cost of the next gallon is going to be using the
marginal cost function. So for example, let's say
that this right over here, so this is our x-axis,
this is 100 gallons. And let's say this is the
next gallon, it's 101 gallons. And our cost function might
look something like this. When you calculate
C prime of 100, is going to give you
the slope at this point, is going to give you the
slope of the tangent line at this point. That's C prime of 100. So let me write this. Slope is equal to
C prime of 100. And so what they really
want us to figure out, if we want to know the exact
actual cost of producing the 101st gallon, what we would
do is we we'd say well, look, this value right over
here is C of 100. This value up here, this
value over here is C of 101. Let me do that in
a different color. C of 101. If you just want to calculate
the exact cost of producing that next extra unit, you would
take C of 101 minus C of 100. And so that's what they want to
do next to do the exact cost. But we can approximate it
using the marginal cost, the derivative of our cost
function right at this point. We could figure out what this
slope is, C prime of 100, and then multiply it
times that one extra unit. If you take a slope and you
multiply it times-- remember, slope is change in y or
change in the vertical axis over the change in
the horizontal axis. If you then multiply it times a
change in the horizontal axis, which in this case
is one unit, it's going to give you the resulting
change in the vertical axis. So in this first one, if
you take C prime of 100 and then multiply
that times 1 unit, so you're still just going
to get C prime of 100, you're going to get this
distance right over here, which we can view
is an approximation for this large one. I exaggerated the difference
between this curve and this tangent line. But let's just
actually calculate it. And I'll get my
calculator out to do it. So C prime of 100 is going
to be-- C prime of 100 is-- C prime, so it's
0.1 minus 0.002 times 100 plus 0.0012
times 100 squared, which is going to be 10,000,
I'll just type that in. So it's $11.90. So this is going to be $11.90. So that's this right over here. We just took this slope times 1. We're doing one extra gallon,
and we we're getting this approximation, which is
this distance being $11.90. Now let's actually calculate
this thing right over here. Let's see, we could
actually simplify it a little bit before I even
type it into my calculator. This expression is
going to be 3,200 plus 0.1 times 101
minus 0.001 times 101-- I'm going to switch colors for
contrast-- times 101 squared plus 0.0004 times 101
to the third power. And then we're going
to subtract 3,200. We're going to
subtract 0.1 times 100. We're going to then add
0.001 times 100 squared. And then we're going to
subtract-- let me change colors here-- subtract 0.0004 times
100 to the third power. So let's get the calculator
out and calculate that. So 3,200s will cancel out, so
I don't have to even type that in. And let's see, this
is going to be-- we could write this as 0.1
times 101 minus 100. Well, that's just going to be
1, I don't even have to do that. So this is just going to be 0.1. So then we're going to have
minus 0.001 times 101 squared minus 100 squared. And then we have plus 0.0004
times 101 to the third power minus 100 to the
third power, gives us-- they said to the
nearest penny-- so $12.02. So this is going to be $12.02. So that was a pretty
good approximation using the derivative
and multiplying by the incremental
number of units. That was a pretty
good approximation for the actual
difference, which was the actual cost of producing
that next unit was $12.02. And you might say, once
again, why is it discrepancy? Remember, the marginal
cost of this derivative is just for the
next drop, right at that-- it's the
instantaneous rate of change. Well, each incremental drop was
getting more and more and more expensive, because the curve
does not have a constant slope, does not have a
constant rate of change. So we use this
as, you could view this is really a
linear approximation. And this right over
here is the exact value.