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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.6.31b
Chapter 3, Problem 3.6.31b

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.
Substitute the given cost function C(x) = -0.01x² + 40x + 100 into the average cost formula: A(x) = (-0.01x² + 40x + 100)/x.
Simplify the expression for A(x) by dividing each term in the numerator by x: A(x) = -0.01x + 40 + 100/x.
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.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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