A manufacturer’s daily production cost (in dollars) for producing units of a product…

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

A manufacturer’s daily production cost

C

(in dollars) for producing

y

units of a product is modeled by

C(y)=\begin{cases} 0.02y^2+4000, & 0\le y\le 500, \\ Ay+B, & y>500, \end{cases}

where

A

and

B

are constants chosen so that the cost and the marginal cost remain consistent when production exceeds

500

units. (a) Determine the conditions on

A

and

B

that guarantee

C

is continuous at

y=500

. (b) Write the condition that must hold for the marginal cost,

C'(y)

, to be continuous at

y=500

. (c) Solve simultaneously for

A

and

B

to satisfy both conditions from parts (a) and (b). (d) Using the values of

A

and

B

from part (c), calculate the approximate additional cost of increasing daily production from

510

520

units. Show each step clearly and state any approximation you employ. (e) Interpret the economic meaning of the marginal cost being continuous at

y=500

in the context of the manufacturer’s operations.

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