BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.8, Problem 29E

(a)

To determine

To find: The marginal cost function if the cost function is C(x)=2000+3x+0.01x2+0.0002x3.

Expert Solution

Answer to Problem 29E

The marginal cost function is, C'(x)=3+0.02x+0.0006x2.

Explanation of Solution

The cost function is C(x)=2000+3x+0.01x2+0.0002x3.

Obtain the derivative of the cost function and obtain the marginal function.

C'(x)=ddx(2000+3x+0.01x2+0.0002x3)=3+0.02x+0.0006x2

Thus, the marginal cost function is, C'(x)=3+0.02x+0.0006x2.

(b)

To determine

To find: The value of C'(100) and interpret it. Explain what it predicts.

Expert Solution

Answer to Problem 29E

The value of C'(100)=$11/pair, which represents the rate of change of the production cost when the 100th pair of jeans is produced. It predicts the cost of the 101st pair of jeans.

Explanation of Solution

From part (a), the marginal cost function is, C'(x)=3+0.02x+0.0006x2.

Substitute x = 100 in C'(x) and find the value of C'(100).

C'(100)=3+0.02(100)+0.0006(100)2=3+2+6=11

Therefore, the value of C'(100)=$11/pair, which represents the rate of change of the production cost when the 100th pair of jeans is produced. It predicts the cost of the 101st pair of jeans.

(c)

To determine

To compare: The values of C'(100) and the cost of manufacturing the 101st pair of jeans.

Expert Solution

Answer to Problem 29E

The cost of manufacturing the 101st pair of jeans is approximately $11.07.

Explanation of Solution

From part (b), the value of C'(100)=$11/pair.

The cost of manufacturing the 101st pair of jeans is calculated by, C(101)C(100).

Substitute x = 100 in C(x) and find the value of C(100).

C(100)=2000+3(100)+0.01(100)2+0.0002(100)3=2000+300+100+200=2600

Substitute x = 101 in C(x) and find the value of C(101).

C(101)=2000+3(101)+0.01(101)2+0.0002(101)3=2000+303+102.01+206.0602=2611.0702

Substitute the respective values in C(101)C(100),

C(101)C(100)=2611.07022600=11.0702

The cost of manufacturing the 101st pair of jeans is approximately $11.07, which is very close to the marginal cost of C'(100).

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