Chapter 9.6, Problem 37E

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Chapter
Section

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

# Revenue The revenue from the sale of a product is R =   1500 x +   3000 ( 2 x +   3 ) − 1   −   1000  dollars where x is the number of units sold. Find the marginal revenue when 100 units are sold. Interpret your result.

To determine

To calculate: The marginal revenue when 100 units are sold if the revenue from the sale of a product is R=1500x+3000(2x+3)11000 where x is the number of units sold. There are 100 units sold.

Explanation

Given Information:

The revenue from the sale of a product is R=1500x+3000(2x+3)âˆ’1âˆ’1000 where x is the number of units sold. There are 100 units sold.

Formula used:

According to the power rule, if f(x)=xn, then,

fâ€²(x)=nxnâˆ’1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

gâ€²(x)=cfâ€²(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

fâ€²(x)=uâ€²(x)+vâ€²(x)

According to the product rule, if f(x)=u(x)â‹…v(x), then

fâ€²(x)=uâ€²(x)â‹…v(x)+vâ€²(x)â‹…u(x)

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nunâˆ’1dudx

Calculation:

Consider the provided function,

R=1500x+3000(2x+3)âˆ’1âˆ’1000

Consider (2x+3) to be u,

R=1500x+3000uâˆ’1âˆ’1000

Differentiate both sides with respect to x,

dRdx=ddx(1500x+3000uâˆ’1âˆ’1000)=ddx(1500x)+ddx(3000uâˆ’1)âˆ’ddx(1000)=1500ddx(x)+3000ddx(uâˆ’1)âˆ’ddx(1000)

Simplify using the power rule and rule for constants,

dRd</

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