Question

The cost in dollars of producing x units of a particular telephone is C(x) = x2 – 2500. Find the average rate of change of C with respect to x when the production level is changed from x = 100 to x = 103. Include units in your answer. Find the instantaneous rate of change of C with respect to x when x = 100. Include units in your answer.

Answer 1

The average rate of change over a given interval is the slope of the secant line through the endpoints of the interval.

$m = \dfrac{f(b)-f(a)}{b-a} = \dfrac{f(103)-f(100)}{103-100}\\\\ = \dfrac{(103)^2-2500-[(100)^2-2500]}{103-100} = \dfrac{609}{3}=203$

$203/unit.

The instantaneous rate of change at a certain point is the slope of the tangent line.

We differentiate C to get 2x. We substitute 100 for x to get that the instantaneous rate of change at 100 is $200/unit.