Consider the following cost functions.
b. Determine the average cost and the marginal cost when x=a.
C(x) = − 0.01x²+40x+100, 0≤x≤1500, 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) = -0.01x² + 40x + 100 into the average cost formula: A(x) = (-0.01x² + 40x + 100)/x.
3. Simplify the expression for A(x) by dividing each term in the numerator by x: A(x) = -0.01x + 40 + 100/x.
4. To find the average cost at x = a = 1000, substitute x = 1000 into the simplified average cost function: A(1000) = -0.01(1000) + 40 + 100/1000.
5. The marginal cost is the derivative of the cost function C(x) with respect to x. Compute the derivative C'(x) = d/dx(-0.01x² + 40x + 100) and then evaluate C'(1000) to find the marginal cost at x = 1000.

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, the function C(x) = -0.01x² + 40x + 100 describes how costs change with varying levels of production (x). Understanding the structure of this function is essential for analyzing average and marginal costs.

Average Cost

The average cost is calculated by dividing the total cost by the quantity produced, expressed as AC(x) = C(x)/x. This metric provides insight into the cost per unit of production, helping firms assess efficiency. For the given cost function, calculating the average cost at x = 1000 will reveal how much it costs, on average, to produce each unit 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), where C'(x) is the derivative of the cost function. This concept is crucial for decision-making in production, as it helps firms determine the cost-effectiveness of increasing output. Evaluating the marginal cost at x = 1000 will indicate the cost impact of producing one additional unit at that level.