Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.1.55a

Chapter 6, Problem 6.1.55a

55–58. Marginal cost Consider the following marginal cost functions.

a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.

C′(x)=200−0.05x

Verified step by step guidance

Understand that the marginal cost function \(C\prime(x)\) represents the rate of change of the total cost with respect to the number of units produced, \(x\). To find the additional cost incurred when production increases from 100 to 150 units, we need to integrate the marginal cost over this interval.

Set up the definite integral of the marginal cost function from \(x=100\) to \(x=150\): \(\displaystyle \int_{100}^{150} \left(200 - 0.05x\right) \, dx\)

Integrate the function \(200 - 0.05x\) with respect to \(x\). Recall that the integral of a constant \(a\) is \(ax\), and the integral of \(x\) is \(\frac{x^2}{2}\). So, \(\int \left(200 - 0.05x\right) dx = 200x - 0.05 \cdot \frac{x^2}{2} + C\)

Evaluate the definite integral by substituting the upper limit \(x=150\) and the lower limit \(x=100\) into the antiderivative, then subtract the two results: \(\left[200x - 0.025x^2\right]_{100}^{150} = \left(200 \times 150 - 0.025 \times 150^2\right) - \left(200 \times 100 - 0.025 \times 100^2\right)\)

Simplify the expression obtained in the previous step to find the additional cost incurred when production increases from 100 to 150 units.

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Key Concepts

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

Marginal Cost

Marginal cost represents the rate of change of the total cost with respect to the quantity produced. It is given by the derivative of the cost function, C'(x), and indicates the additional cost of producing one more unit at a certain production level.

Definite Integral for Accumulated Change

To find the total additional cost when production increases from one quantity to another, integrate the marginal cost function over that interval. The definite integral of C'(x) from x = a to x = b gives the total increase in cost between those production levels.

Interpreting and Applying the Marginal Cost Function

Understanding how to apply the marginal cost function involves evaluating or integrating it over a specific range. This requires recognizing that C'(x) = 200 - 0.05x means the cost per unit decreases as production increases, affecting the total additional cost calculation.

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