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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 0.1 X minus 0.001 x squared plus zero point zero zero zero for X 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 zero the derivative of 0.1 X is 0.1 the derivative of negative 0.001 x squared is going to be negative zero point zero zero two X and then finally the derivative of this is going to be let's see this is for 10,000 so three times four ten thousands is twelve ten thousands so zero point one two three four I up this 12 ten thousandth x squared so that right over there is equal to the marginal cost function so if you see this had an exercise you just want to type in the expression here but this of course is equal to this of course is equal to C prime of X to the nearest to the nearest penny what is the marginal cost of producing the hundred first one hundred first gallon of stain so they say C prime of a hundred is equal to blank dollars per gallon so what are they wanting is what do they want us to do here well 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 let's say that this right over here so this is the number of this is our x-axis this is a hundred gallons and let's say this is the next gallon as 101 gallons and our cost function might look something like this so our cost function might look something like this when you calculate C prime of 100 so C prime of 100 is going to give you is going to give you the slope at this point it's going to give you this the slope 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 101st gallon what we would do is we'd say well look this value right over here is C of 100 this value up here this value R over here is C of C of 101 we've done in a different color C of 1 C of 101 if you 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 us to do next to do the exact cost but we can approximate it using the marginal the marginal cost the derivative of our cost function right at this point we can figure out what this slope is C prime of 100 and then multiply it times that one extra unit 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 one unit so you're still just going to get C prime of 100 you're going to get this distance right over here you're going to get this distance right over here which we can view as 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 point zero zero two times 100 times 100 plus point zero zero 1 2 times 100 squared which is going to be 10,000 volts type that in so it's $11.90 so this is going to be 11 90 1190 so that's this right over here we just took this slope times 1 we're doing one extra gallon and we're getting this approximation which is this distance being 11 dollars 98 the thing right over here and see 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 32 hundred thirty-two hundred plus 0.1 times 101 minus 0.001 times 101 on the switch colors for contrast times 101 squared plus zero point zero zero zero four times 101 to the third power and then we're going to subtract 3,200 we're going to subtract zero point one times 100 we're going to then add zero point two zero zero one times 100 squared and then we're going to subtract let me to change colors here subtract zero point zero zero zero four times 100 to the third 100 to the third power so let's get the calculator out and calculate that so 3,200 s will cancel out so I don't have to type 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 one I don't have to do that so it's going to be these two are going to cut this is just going to be 0.1 so then we're going to have minus minus 0.001 times 101 squared 101 squared minus 100 squared minus 100 squared and then we have plus plus 0z point zero zero zero four times 101 to the third power minus 100 to the third power minus 100 to the third power gives us 12 they sit to the nearest penny so twelve dollars and two cents so this is going to be twelve dollars and two cents so that was a pretty good approximation using the derivative and multiplying by the number of that 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 twelve dollars and two cents and you might say once again why is a discrepancy remember the marginal cost is derivative is just for the next very the next drop right at that instance the instantaneous rate of change well each drop was getting each incremental drop is 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 can view this as really a linear approximation and this of this right over here is the exact value