Suppose the cost, in dollars, of producing x items is...

Suppose the cost, in dollars, of producing $x$ items is $C(x)=100 x+2000 / x$ Determine (a) the cost of producing 50 items, (b) the cost of producing the 50th item, the marginal cost when $x=50,$ (c) the average cost function, (d) the average cost of producing 50 items, (e) estimate the level of production which minimizes the cost and estimate the minimum cost.

Key Concepts

Differentiation

Differentiation is the mathematical process used to compute the rate of change of functions, such as the cost function. It is crucial for determining marginal costs and for conducting optimization analyses to find cost minimizing or maximizing levels.

Average Cost Function

The average cost function is obtained by dividing the total cost by the number of items produced, yielding the cost per unit. This concept helps in assessing economies of scale by showing how costs change as production levels vary.

Optimization

Optimization in this context involves using calculus techniques to determine the production level that minimizes total cost. This typically involves finding the derivative of the cost function, setting it equal to zero, and using tests like the second derivative test to confirm a minimum.

Marginal Cost

Marginal cost is the derivative of the cost function with respect to the number of items produced. It approximates the additional cost incurred from producing one more unit, and is used in decision-making to determine the cost-effectiveness of increasing production.

Cost Function

A cost function represents the total cost incurred in producing x number of items. It typically includes fixed and variable components, and serves as the basis for analyzing production decisions, budgeting, and optimization in operations research and economics.

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