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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 6.1.57a
Chapter 6, Problem 6.1.57a

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) = 300+10x−0.01x²

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1
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 find the total change in cost over this interval.
Set up the integral of the marginal cost function \(C\prime(x) = 300 + 10x - 0.01x^{2}\) from \(x = 100\) to \(x = 150\). This integral will give the total additional cost incurred when production increases from 100 to 150 units.
Write the integral expression for the additional cost as: \(\int_{100}^{150} (300 + 10x - 0.01x^{2}) \, dx\).
Integrate the function term-by-term: - The integral of \(300\) with respect to \(x\) is \$300x\(. - The integral of \)10x\( with respect to \)x\( is \)5x^{2}\(. - The integral of \)-0.01x^{2}\( with respect to \)x$ is \(-0.01 \times \frac{x^{3}}{3} = -\frac{0.01}{3} x^{3}\).
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 to find the total 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 Function

The marginal cost function, denoted as C′(x), represents the rate of change of the total cost with respect to the number of units produced. It gives the approximate additional cost of producing one more unit at production level x. Understanding this function is essential to estimate incremental costs over production intervals.
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Properties of Functions

Definite Integral for Total 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 exact total increase in cost between producing a and b units.
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Definition of the Definite Integral

Application of Integration in Economics

Integration is used in economics to accumulate marginal values into total quantities. Here, integrating the marginal cost function helps convert a rate of change into a total cost change, linking calculus concepts directly to practical cost analysis in production.
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Real World Application
Related Practice
Textbook Question

Work in a gravitational field For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm / (x+R)², where G = 6.7×10^−11 N m²/kg² is the gravitational constant, M = 6×10^24 kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378×10⁶ m is the radius of Earth, and x≥0 is the distance above the surface of Earth (in meters).


a. How much work is required to launch a rocket with a mass of 500 kg in a vertical flight path to a height of 2500 km (from Earth’s surface)?

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Textbook Question

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

a. ∫a^b √1+16x⁴ dx

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 = x³/3, for −1≤x≤1

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The distance traveled by an object moving along a line is the same as the displacement of the object.

Textbook Question

Given the velocity function of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.

Textbook Question

Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels. 


a. Find Q(t), the total amount of oil extracted by the nation after t years.