Mathematics

Problem

For a small manufacturing company, the total cost

C

(in dollars) of producing

q

widgets is modeled for

0\le q\le600

by the differentiable function

C(q)=0.005q^{3}-0.4q^{2}+15q+2000,

where

q

is measured in widgets. (a) Calculate

C'(q)

and state the units of

C'(q)

. (b) Evaluate

C'(300)

and interpret the meaning of this value in the company’s context. (c) Using your answer to part (b), estimate the additional cost of increasing production from 300 to 301 widgets. Justify your estimate verbally.

Intuitive explanation

The key to understanding cost behavior is the marginal cost—the derivative $C'(q)$ tells you how much an additional widget costs to produce at any production level. By finding where $C'(q)$ is minimized, you'll locate the production quantity where efficiency is greatest (cost per extra widget is lowest). This is a classic optimization problem: take the derivative, then find critical points of that derivative to reveal the company's sweet spot for production.

Intuitive explanation

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