The total cost (in dollars) of producing hundred widgets at a factory is modeled by…

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

Problem

The total cost (in dollars) of producing

x

hundred widgets at a factory is modeled by the polynomial function

C(x)=0.04x^{4}-0.3x^{3}+0.9x^{2}+50.

Assume

x

is between 0 and 5 (i.e., up to 500 widgets). (a) Derive an expression for the marginal cost function

C'(x)

and state its units. (b) Compute

C'(3)

and interpret its meaning in the production context. (c) Write the equation of the line normal to the graph of

y=C(x)

at the point where

x=3

. Express your answer in point-slope form. (d) Explain how the sign of

C'(3)

relates to the behavior of total cost with respect to production level near

x=3

. Support your explanation with appropriate reasoning.