   Chapter 9.8, Problem 36E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Revenue Suppose that the revenue (in dollars) from the sale of a product is given by R   = 70 x +   0.5 x 2 −   0.001 x 3 where x is the number of units sold. How fast is the marginal revenue M R ¯ changing when x = 100?

To determine

To calculate: The instantaneous rate of change of the marginal revenue for x=100. The revenue (in dollars) from the sale of x number of units is provided by function R=70x+0.5x20.001x3.

Explanation

Given Information:

The revenue (in dollars) from the sale of x number of units is provided by function R=70x+0.5x20.001x3 and the number of units sold is x=100.

Formula used:

Power of x rule for function f(x)=xn is f(x)=nxn1, where n is a real number.

Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

Constant function rule for a constant c is such that, if f(x)=c then f(x)=0.

The instantaneous rate of change of the marginal revenue is the second derivative of the revenue function.

Calculation:

Consider the function,

R=70x+0.5x20.001x3

Differentiate both sides of the function with respect to x,

dRdx=ddx(70x+0.5x20.001x3)R=ddx(70x)+ddx(0.5x2)ddx(0.001x3)=70ddx(x)+0.5ddx(x2)0.001ddx(x3)

Use the power of x rule and constant rule,

R=70(x11)+0

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