A manufacturer’s pricing model for a specialty component is given by p(x)=p_0+ k x-500

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A manufacturer’s pricing model for a specialty component is given by

p(x)=p_0+\frac{k}{x-500}, \qquad x>500,

where

p(x)

is the price per unit (in dollars) when

x

units are produced,

p_0>0

is a base price, and

k

is a non-zero constant. (a) Identify the vertical asymptote of

p

and interpret its economic meaning in terms of production capacity. (b) Write the revenue function

R(x)=x\,p(x)

and evaluate

\displaystyle \lim_{x\to 500^{+}}R(x)

. Explain the implication of this limit for revenue as production approaches the capacity of

500

units. (c) Compute

R'(x)

for

x>500

and determine all critical points of

R

in that interval. (d) Discuss how the sign and magnitude of

k

influence the existence and location of any local maximum revenue occurring for

x>500

. Provide a justified conclusion based on your analysis in part (c).