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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 6.1.55b
Chapter 6, Problem 6.1.55b

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


b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.


C′(x)=200−0.05x

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1
Identify the marginal cost function given: \(C\' (x) = 200 - 0.05x\), which represents the rate of change of the cost with respect to the number of units produced.
Understand that the additional cost incurred when production increases from 500 to 550 units can be found by integrating the marginal cost function over the interval from \(x = 500\) to \(x = 550\).
Set up the definite integral to find the additional cost: \(\int_{500}^{550} (200 - 0.05x) \, dx\).
Integrate the function: find the antiderivative of \(200 - 0.05x\), which is \(200x - 0.025x^2\).
Evaluate the antiderivative at the upper and lower limits and subtract: calculate \(\left[200x - 0.025x^2\right]_{500}^{550}\) to find the additional cost.

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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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Definite Integral for Accumulated Change

To find the total additional cost over an interval, integrate the marginal cost function over that range. The definite integral of C'(x) from x = a to x = b gives the total increase in cost when production increases from a to b units.
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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 interval to find actual cost changes. This requires recognizing that marginal cost varies with production level and using calculus tools to compute total cost increments.
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Related Practice
Textbook Question

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.


b. What is the height of a cylindrical shell at a point x in [0, 4]?

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Textbook Question

For the given regions R₁ and R₂, complete the following steps.


b. Find the area of region R₂ using geometry and the answer to part (a).


R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.

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Textbook Question

Displacement and distance from velocity Consider the velocity function shown below of an object moving along a line. Assume time is measured in seconds and distance is measured in meters. The areas of four regions bounded by the velocity curve and the t-axis are also given.

b. What is the displacement of the object over the interval [2, 6]? 

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Textbook Question

Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N.

b. How much work is done in compressing the spring 0.5 m from its equilibrium position?

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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.


b. Find the displacement over the given interval. 


v(t) = 50e^−2t on [0, 4]

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Textbook Question

Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.

b. How much work is required to compress the spring 0.2 m from its equilibrium position?

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