Suppose that the cost (in dollars) for a company to produce...

Suppose that the cost (in dollars) for a company to produce $ x $ pairs of a new line of jeans is $ C(x) = 2000 + 3x + 0.01x^2 + 0.0002x^3 $ (a) Find the marginal cost function. (b) Find $ C'(100) $ and explain its meaning. What does it predict? (c) Compare $ C'(100) $ with the cost of manufacturing the 101st pair of jeans.

Suppose that the cost (in dollars) for a company to produce $ x $ pairs of a new line of jeans is
$ C(x) = 2000 + 3x + 0.01x^2 + 0.0002x^3 $
(a) Find the marginal cost function.
(b) Find $ C'(100) $ and explain its meaning. What does it predict?
(c) Compare $ C'(100) $ with the cost of manufacturing the 101st pair of jeans.

Key Concepts

Comparison Between Marginal Cost and Actual Cost Increase

The marginal cost offers an approximate cost of producing one more unit based on an instantaneous rate of change, while the true cost of producing the next unit is determined by calculating the discrete difference in cost when production increases from one level to the next. This comparison highlights that marginal cost is an approximation, which is typically accurate for small changes in production but may differ from the exact additional cost when production increments are not infinitesimally small.

Interpretation of the Derivative at a Specific Production Level

Evaluating the derivative (or marginal cost) at a specific production level, such as 100 units, provides an estimate of the additional cost an extra unit will add when the production level is around that point. This value is interpreted as the cost to produce one more unit, assuming the production scale is near that evaluation point.

Cost Function

A cost function represents the total cost incurred by a business in producing a given number of units. It typically includes both fixed costs (costs that do not change with the level of production) and variable costs (costs that do vary with production). In mathematical terms, it is a function of the production quantity and is used to analyze cost behavior as production levels change.

Derivative as Instantaneous Rate of Change

The derivative of a function at a given point represents the instantaneous rate of change of that function with respect to its variable. In the context of cost functions, taking the derivative with respect to the quantity of goods produced gives a measure of how the cost changes for a very small change in the number of units produced.

Marginal Cost

Marginal cost is defined as the cost of producing one additional unit of a good. It is obtained by taking the derivative of the cost function. This concept is fundamental in economics and business for decision-making, as it informs a company about the cost impact of manufacturing one more unit, helping in optimizing production levels.

Transcript

Please give Ace some feedback