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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

Consider the following curves on the given intervals.  


b. Use a calculator or software to approximate the surface area.


y=tan x , for 0≤x≤π/4; about the x-axis 

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Textbook Question

A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7. 

b. How much work is done in stretching the spring from its equilibrium position (x=0) to x=1.5?

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Textbook Question

40–43. Population growth


When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.


b. Find the population P(t) at any time t≥0.

Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

b. The distance traveled between t=0 and t=5

Textbook Question

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.


b. Find the function that gives the total blood pumped between t=0 and a future time t>0.

Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


b. Suppose P=10, A=20, and r=0. If the initial population is N(0)=100, does the population ever become extinct? Explain.