Applied rates of change: marginal costs

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.