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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 6.1.40b
Chapter 6, Problem 6.1.40b

40–43. Population growth


Starting with an initial value of P(0)=55, the population of a prairie dog community grows at a rate of P′(t)=20−t/5 (prairie dogs/month), for 0≤t≤200, where t is measured in months.


b. Find the population P(t), for 0≤t≤200.

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1
Identify the given rate of change of the population, which is the derivative of the population function: \(P\'(t) = 20 - \frac{t}{5}\).
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: \(P(t) = \int \left(20 - \frac{t}{5}\right) dt + C\).
Integrate each term separately: the integral of 20 with respect to \(t\) is \$20t\(, and the integral of \(-\frac{t}{5}\) with respect to \)t$ is \(-\frac{1}{5} \cdot \frac{t^2}{2} = -\frac{t^2}{10}\).
Combine the results and include the constant of integration \(C\): \(P(t) = 20t - \frac{t^2}{10} + C\). Use the initial condition \(P(0) = 55\) to solve for \(C\) by substituting \(t=0\) and \(P(0)=55\).

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Key Concepts

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

Initial Value Problems

An initial value problem involves finding a function that satisfies a given differential equation and meets a specified initial condition. Here, the population function P(t) must satisfy P'(t) = 20 - t/5 and the initial value P(0) = 55, which anchors the solution to a specific starting point.
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Initial Value Problems

Antiderivatives and Integration

To find P(t) from its rate of change P'(t), we integrate P'(t) with respect to t. Integration reverses differentiation, allowing us to recover the original function up to a constant, which is then determined using the initial condition.
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Antiderivatives

Applying Initial Conditions to Determine Constants

After integrating the rate function, the solution includes an arbitrary constant. Using the initial condition P(0) = 55, we substitute t = 0 into the integrated function to solve for this constant, ensuring the population model accurately reflects the starting population.
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Initial Value Problems Example 1
Related Practice
Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.

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Textbook Question

Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.


b. Find the displacement of the object on the interval 0≤t≤6.

Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point x in [0, 4]?

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) = 50e^−2t on [0, 4]

Textbook Question

Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.

b. How much work is required to compress the spring 0.2 m from its equilibrium position?

Textbook Question

Consider the following curves on the given intervals.  


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


y=cos x, for 0≤x≤π/2; about the x-axis