Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.1.52b

Chapter 6, Problem 6.1.52b

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.

Verified step by step guidance

Start by writing down the given growth rate function: \(N'(t) = r + A \sin\left(\frac{2\pi t}{P}\right)\), with \(P=10\), \(A=20\), and \(r=0\). So, \(N'(t) = 20 \sin\left(\frac{2\pi t}{10}\right)\).

Recall that the population function \(N(t)\) is the integral of the growth rate \(N'(t)\) with respect to time \(t\). Use the initial condition \(N(0) = 100\) to find the constant of integration after integrating.

Integrate \(N'(t)\): \(N(t) = \int 20 \sin\left(\frac{2\pi t}{10}\right) dt + C\). Use the substitution \(u = \frac{2\pi t}{10}\) to simplify the integral.

After integrating, express \(N(t)\) explicitly in terms of \(t\) and the constant \(C\). Then apply the initial condition \(N(0) = 100\) to solve for \(C\).

Analyze the expression for \(N(t)\) to determine if it ever reaches zero for \(t > 0\). Consider the amplitude and average value of the oscillations to conclude whether extinction (population reaching zero) occurs.

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Key Concepts

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

Differential Equations and Population Growth

A differential equation relates a function to its derivative, describing how a quantity changes over time. Here, N'(t) represents the growth rate of the population N(t). Solving or analyzing this equation helps determine the population's behavior, such as whether it reaches zero (extinction).

Oscillatory Functions and Periodicity

The growth rate includes a sinusoidal term A sin(2πt/P), which oscillates with period P. This means the growth rate fluctuates regularly over time, causing the population to increase and decrease in cycles. Understanding how these oscillations affect overall growth is key to predicting extinction.

Extinction Condition and Population Thresholds

Extinction occurs if the population N(t) reaches zero at any time after t=0. Analyzing whether the integral of the growth rate ever reduces N(t) to zero involves checking if negative growth phases outweigh positive growth, considering initial population and oscillations.

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