Consider the following cost functions.
b. Determine the average cost and the marginal cost when x=a.
C(x) = 500+0.02x, 0≤x≤2000, a=1000

Verified step by step guidance

1. To find the average cost, we need to divide the total cost function C(x) by the number of units x. The average cost function A(x) is given by A(x) = C(x) / x.
2. Substitute the given cost function C(x) = 500 + 0.02x into the average cost formula: A(x) = (500 + 0.02x) / x.
3. Simplify the expression for A(x): A(x) = 500/x + 0.02.
4. To find the average cost at x = a, substitute a = 1000 into the average cost function: A(1000) = 500/1000 + 0.02.
5. The marginal cost is the derivative of the cost function C(x) with respect to x. Differentiate C(x) = 500 + 0.02x to find C'(x), which is the marginal cost. Since C(x) is linear, C'(x) = 0.02. The marginal cost at x = a is simply C'(1000) = 0.02.

Key Concepts

Cost Functions

A cost function represents the total cost incurred by a firm in producing a certain quantity of goods, denoted as C(x). In this case, C(x) = 500 + 0.02x indicates that there is a fixed cost of 500 and a variable cost that increases linearly with the quantity produced (x). Understanding the structure of cost functions is essential for analyzing production costs.

Average Cost

The average cost is calculated by dividing the total cost by the quantity produced, expressed as AC(x) = C(x)/x. It provides insight into the cost per unit of production. For the given cost function, calculating the average cost at x = a (where a = 1000) will help determine how efficiently resources are being utilized at that production level.

Marginal Cost

Marginal cost refers to the additional cost incurred by producing one more unit of a good, mathematically represented as MC(x) = C'(x). It is derived from the derivative of the cost function. For the provided cost function, finding the marginal cost at x = a (x = 1000) will indicate how production costs change with incremental increases in output, which is crucial for decision-making in production.