AP Calculus AB · Contextual applications of differentiation
The cost, in dollars, of producing q liters of a chemical is given by a differentiable…
Problem
The cost, in dollars, of producing \(q\) liters of a chemical is given by a differentiable function \(C(q)\). A plant records \(C(50)=8200\) and \(C'(50)=95\). (a) Interpret \(C'(50)\) in the context of the problem. Use correct units in a complete sentence. (b) Use local linearization of \(C\) at \(q=50\) to estimate \(C(52)\) and to estimate the cost of producing the \(51\)st liter. (c) The average cost per liter at production level \(q>0\) is \(A(q)=\dfrac{C(q)}{q}\). Using only the given data, estimate \(A(50)\). Then, treating \(C'(50)\) as the marginal cost, determine whether estimated marginal cost is greater than, less than, or equal to estimated average cost at \(q=50\).
Hint
\(C'(50)\) is dollars per extra liter, not a total cost. Average cost is total cost divided by \(q\).
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