Chapter 9.9, Problem 19E

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

If the cost function for a commodity is C ( x )   =   x 3 − 4 x 2 + 30 x +  2 0 dollars find the marginal cost at x =  4 units and tell what this predicts about the cost of producing 1 additional unit and 3 additional units.

To determine

To calculate: The marginal cost, at x=4 units, for the cost function, C(x)=x34x2+30x+20. Also, find the cost for producing additional 1 unit and 3 units.

Explanation

Given Information:

The cost function is C(x)=x3âˆ’4x2+30x+20.

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)

The derivative of a constant value, k, is

ddx(k)=0

Calculation:

Consider the marginal cost function to be MCÂ¯. Consider the provided cost function,

C(x)=x3âˆ’4x2+30x+20

The marginal cost of a cost function is found by differentiating the cost function.

MCÂ¯=Câ€²(x)

Thus, differentiate both sides of the cost function with respect to x,

Câ€²(x)=ddx(x3âˆ’4x2+30x+20)MCÂ¯=ddx(x3)È¡

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