Question

For the following exercises, the revenue generated by selling x items is given by R(x) = 2x2 + 10x.

Find R′(10) and interpret.

Answer 1

The derivative of function f at is given by the formula:

f\'(a) = limh→0{f(a + h) – f(a)}/h

 

Consider the revenue generated by selling x items is as follows:

R(x) = 2x2 + 10x

 

Determine the derivative of R(x) = 2x2 + 10x at x = 10 as follows:

R\'(10) = lim{(R(10 + h) - R(10)}/h

           = lim[2(10 + h)² + 10(10 + h)− {2(10²)+10(10)}/h]

           = lim[2{102 + 2(10)(h) + h2} + 10(10 + h) – 2{2(102) + 10(10)}/h]

           = limh→0[2{100 + 20h + h2} + 10(10 + h) – (200 + 100)/h]

 

Further simplify:

R\'(10) = limh→0(200 + 40h + 2h2 + 100 + 10h – 300)/h

           = limh→0(2h2 + 50h)/h

           = 50

 

This implies the instantaneous rate of change of revenue when exactly 10 items are sold is $50.

This implies the instantaneous rate of change of revenue when exactly 10 items are sold is $50.