   Chapter 9.4, Problem 53E Mathematical Applications for the ...

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

Solutions

Chapter
Section Mathematical Applications for the ...

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

Cost and average cost Suppose that the total cost function, in dollars, for the production of x units of a product is given by C ( x ) = 4000 + 55 x + 0.1 x 2 Then the average cost of producing x items is C ( x ) ¯ = total cost x = 4000 x + 55 + 0.1 x (a) Find the instantaneous rate of change of average cost with respect to the number of units produced at any level of production.(b) Find the level of production at which this rate of change equals zero.(c) At the value found in part (b), find the instantaneous rate of change of cost and find the average cost. What do you notice?

(a)

To determine

To calculate: The instantaneous rate of change of average cost with respect to number of units produced. The total cost function for x units produced is given by C(x)=4000+55x+0.1x2, and the average cost of producing x items is C(x)¯=4000x+55+0.1x.

Explanation

Given Information:

The total cost function for x units produced is given by C(x)=4000+55x+0.1x2. The average cost of producing x items is C(x)¯=4000x+55+0.1x.

Formula Used:

According to sum rule of derivatives,

If

f(x)=u(x)+v(x)

Then,

f(x)=u(x)+v(x)

According to power rule,

If f(x)=xn, then f(x)=nxn1.

According to constant function rule,

If f(x)=c, then f(x)=0.

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).

Calculation:

Consider the provided function for average cost of producing x items is C(x)¯=4000x+55+0.1x

Differentiate the function C(x)¯=4000x+55+0

(b)

To determine

To calculate: The level of production at which rate of change is zero, where the total cost function for x units produced is given by C(x)=4000+55x+0.1x2, and the average cost of producing x items is C(x)¯=4000x+55+0.1x.

(c)

To determine

To calculate: The instantaneous rate of change of cost and average cost in part (b), where the total cost function for x units produced is given by C(x)=4000+55x+0.1x2, and the average cost of producing x items is C(x)¯=4000x+55+0.1x.

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