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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 6.1.57b
Chapter 6, Problem 6.1.57b

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) = 300+10x−0.01x²

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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 500 to 550 units, we need to find the change in total cost over this interval.
Recall that the additional cost when increasing production from \(x = a\) to \(x = b\) can be found by integrating the marginal cost function over the interval \([a, b]\). In this case, we want to compute \(\int_{500}^{550} C\prime(x) \, dx\).
Set up the definite integral using the given marginal cost function: \(\int_{500}^{550} \left(300 + 10x - 0.01x^{2}\right) \, dx\).
Integrate the function term-by-term: - The integral of \(300\) with respect to \(x\) is \$300x\(. - The integral of \)10x\( is \)5x^{2}\(. - The integral of \)-0.01x^{2}$ is \(-0.01 \times \frac{x^{3}}{3} = -\frac{0.01}{3} x^{3}\).
Evaluate the resulting antiderivative at the upper limit \(x=550\) and the lower limit \(x=500\), then subtract to find the total additional cost: \(\left[300x + 5x^{2} - \frac{0.01}{3} x^{3}\right]_{500}^{550} = \left(300 \times 550 + 5 \times 550^{2} - \frac{0.01}{3} \times 550^{3}\right) - \left(300 \times 500 + 5 \times 500^{2} - \frac{0.01}{3} \times 500^{3}\right)\).

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Key Concepts

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

Marginal Cost Function

The marginal cost function, C′(x), represents the rate of change of the total cost with respect to the number of units produced. It indicates the additional cost of producing one more unit at production level x. Understanding this helps in estimating cost changes for small increases in production.
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Definite Integral for Accumulated Change

To find the total additional cost when production increases over an interval, integrate the marginal cost function over that range. The definite integral of C′(x) from x = 500 to x = 550 gives the exact increase in total cost for producing those extra 50 units.
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Definition of the Definite Integral

Evaluating Polynomial Integrals

Since the marginal cost function is a polynomial, integrating it involves applying power rules to each term. Accurate integration and evaluation at the bounds are essential to compute the total additional cost correctly, ensuring precise results for the production increase.
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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]?

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.

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]? 

Textbook Question

Volumes without calculus Solve the following problems with and without calculus. A good picture helps.


b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?

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Textbook Question

Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).

b. If the length is doubled, is the required work doubled? Explain.

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Textbook Question

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

b. What is the displacement of the object over the interval [0,3]?