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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 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

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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.
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Example 3: Maximizing Profit

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.
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Definition of the Definite Integral

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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Related Practice
Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


a. Determine when the motion is in the positive direction and when it is in the negative direction. 


v(t) = 50e^−2t on [0, 4]

Textbook Question

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = 1/x, for 1 ≤ x ≤ 10

Textbook Question

A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.

a. How much work is required to compress the spring and lift the mass 0.5 m?

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Textbook Question

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.


a. Determine the position function, for t≥0, using the antiderivative method


v(t) = 6−2t on [0, 5]; s(0)=0

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Textbook Question

Region R is revolved about the line x=4 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point y in [1, 3]?

Textbook Question

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC.

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