MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A manufacturing company’s total cost, in dollars, for producing q widgets is modeled by a twice-differentiable function C(q) on the closed interval [200,800], where q is measured in units. The company’s records show that C(200)=12400 and C(800)=34000.
(a) Compute the average cost per widget produced over the interval 200≤q≤800.
(b) Show that there exists a quantity q=k with 200<k<800 at which the marginal cost C′(k) equals the average cost per widget you found in part (a).
(c) Interpret the economic meaning of the equality C′(k)= (average cost) in the context of production decisions.
(d) Suppose that for quantities below k the marginal cost is less than the average cost per widget. Explain how this information could influence the company’s decision about whether to increase or decrease production from the current level of k widgets.