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Problem 32 Medium Difficulty

The cost function for a certain commodity is
$ C(q) = 84 + 0.16q - 0.0006q^2 + 0.000003q^3 $
(a) Find and interpret $ C'(100). $
(b) Compare $ C'(100) $ with the cost of producing the 101st item.

Answer

a) done
b) done

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Video Transcript

here we have a cost function and were asked to find See Prime of 100. And what that is is what's called the marginal cost. And that would be the cost of producing the 101st item if you've already produced 100 items. So in order to find that we start by finding the derivative in general, see prime of Q and we would get 0.16 minus 0.12 q plus 0.0 Hopefully, we have the right number of zero's nine Q squared. Okay, so that then we're going to take 100 substituted into that equation. And this would be a great use of your calculator and you're going to get 0.13 0.13 So what does that mean? That is an estimate of the cost of the 101st item. Okay, so for part B were told to find the actual cost of the 101st item and the way to find the actual cost would be to find the cost for 101 items that would be the total for making 101 minus the cost for making 100 items and see what we get. And so we're going to use our calculator for this with the original equation equation. Put 101 in there and we get 97.13 and then we put 100 in there and we get 97. And when we subtract those noticed that we get 0.13 that was would be 13 cents. So our estimate of the cost of the 101st item was very, very correct. It was the same as far as we can tell.