Textbook Question
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
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.1.41b
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.
Verified step by step guidance1
Identify the given rate of change of the population, which is the derivative of the population function: \(P\'(t) = 30(1 + \sqrt{t})\).
Recall that to find the population function \(P(t)\), you need to integrate the rate function \(P\'(t)\) with respect to \(t\): \(P(t) = \int P\'(t) \, dt + C\).
Set up the integral: \(P(t) = \int 30(1 + t^{1/2}) \, dt + C\).
Integrate each term separately: \(\int 30 \, dt\) and \(\int 30 t^{1/2} \, dt\).
Use the initial condition \(P(0) = 250\) to solve for the constant of integration \(C\) after performing the integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative as a Rate of Change
The derivative P′(t) represents the instantaneous rate of change of the population with respect to time. Understanding that P′(t) = 30(1 + √t) means the population grows faster as time increases, and this rate function is essential for finding the original population function P(t).
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Intro To Related Rates
Antiderivative and Indefinite Integration
To find the population function P(t) from its rate of change P′(t), we use antiderivatives or indefinite integrals. Integrating P′(t) with respect to t recovers P(t) up to a constant, which can be determined using initial conditions.
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05:04Introduction to Indefinite Integrals
Initial Conditions and Constant of Integration
When integrating, an unknown constant appears because differentiation loses constant terms. Using the initial population P(0) = 250 allows us to solve for this constant, ensuring the population function accurately reflects the starting value.
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05:03Initial Value Problems
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
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) = 3t²−6t on [0, 3]
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.