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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 6.1.41b
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 guidance
1
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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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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Introduction 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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Initial Value Problems
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