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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.
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Calculus 1 / AB
Rates of Change in the Natural and Social Sciences
Oregon State University
University of Michigan - Ann Arbor
University of Nottingham
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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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.
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