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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 3.6.29a
Chapter 3, Problem 3.6.29a

Consider the following cost functions.
a. Find the average cost and marginal cost functions.
C(x) = 1000+0.1x, 0≤x≤5000, a=2000

Verified step by step guidance
1
To find the average cost function, 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) = 1000 + 0.1x into the average cost formula: A(x) = (1000 + 0.1x) / x.
Simplify the expression for A(x) to get A(x) = 1000/x + 0.1.
To find the marginal cost function, take the derivative of the total cost function C(x) with respect to x. The marginal cost function MC(x) is given by MC(x) = dC(x)/dx.
Differentiate C(x) = 1000 + 0.1x with respect to x. Since the derivative of a constant is 0 and the derivative of 0.1x is 0.1, the marginal cost function MC(x) = 0.1.

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Key Concepts

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

Average Cost Function

The average cost function, denoted as AC(x), represents the total cost C(x) divided by the quantity produced x. It provides insight into the cost per unit of production, helping businesses determine pricing strategies. For the given cost function C(x) = 1000 + 0.1x, the average cost can be calculated as AC(x) = C(x)/x, which simplifies to (1000/x) + 0.1.
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Marginal Cost Function

The marginal cost function, denoted as MC(x), measures the additional cost incurred by producing one more unit of output. It is derived from the derivative of the total cost function C(x) with respect to x. For the cost function C(x) = 1000 + 0.1x, the marginal cost is found by calculating MC(x) = dC/dx, which results in a constant value of 0.1, indicating that each additional unit costs 0.1.
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Cost Function

A cost function describes the relationship between the quantity of output produced and the total cost incurred in production. It typically includes fixed costs, which do not change with output, and variable costs, which do. In the provided function C(x) = 1000 + 0.1x, the fixed cost is 1000, while the variable cost is represented by the term 0.1x, indicating that costs increase linearly with production.
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