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Chapter 8: Problem 10

Make a sketch of the population function (as a function of time) that results from the following growth rate functions. Assume the population at time \(t=0\) begins at some positive value.

Short Answer

Expert verified
Answer: The general form of the population function is \(P(t) = \int g(t) dt + P_0\), where \(g(t)\) is the growth rate function, \(P_0\) is the initial population, and \(t\) is time.

Step by step solution

01

Express the Growth rate functions.

In this exercise, we are not given specific growth rate functions to work with. So, let's denote the growth rate function as \(g(t)\), which represents the rate of change in population with respect to time, i.e., \(\frac{dP(t)}{dt=}=g(t)\).
02

Find the Population function.

To find the population function \(P(t)\), integrate the growth rate function with respect to time. In other words, \(P(t) = \int g(t) dt\). After finding the integral, we get the population function including a constant of integration, \(C\).
03

Determine the initial condition.

As per the exercise's instruction, the population at time \(t = 0\) begins at some positive value. To obtain a specific value, we're going to assume the population at \(t=0\) is \(P_0\), which represents a general initial population. Therefore, the initial condition is given as \(P(0) = P_0\).
04

Solve for the constant of integration.

Using the initial condition obtained in Step 3, substitute it into the population function equation. So, \(P(0) = \int g(0) dt + C\). This will give us the value of the constant of integration, \(C\), in terms of \(P_0\).
05

Finalize the Population function.

Now, by substituting the solved value of \(C\) into the population function, we get the desired population function \(P(t)\), considering \(P_0\) and the growth rate function \(g(t)\).
06

Sketch the Population function.

Finally, sketch the population function in the form \(P(t) = \int g(t) dt + P_0\) over time. Take some example values for \(P_0\) and different forms of growth rate functions \(g(t)\) (like exponential, linear, logistic growth) to get insights into how the population function will behave over time.

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Most popular questions from this chapter

Solve the differential equation for Newton's Law of Cooling to find the temperature in the following cases. Then answer any additional questions. A pot of boiling soup \(\left(100^{\circ} \mathrm{C}\right)\) is put in a cellar with a temperature of \(10^{\circ} \mathrm{C}\). After 30 minutes, the soup has cooled to \(80^{\circ} \mathrm{C}\). When will the temperature of the soup reach \(30^{\circ} \mathrm{C} ?\)

Solve the differential equation for Newton's Law of Cooling to find the temperature in the following cases. Then answer any additional questions. An iron rod is removed from a blacksmith's forge at a temperature of \(900^{\circ} \mathrm{C}\). Assume that \(k=0.02\) and the rod cools in a room with a temperature of \(30^{\circ} \mathrm{C}\). When does the temperature of the rod reach \(100^{\circ} \mathrm{C} ?\)

Solve the differential equation for Newton's Law of Cooling to find the temperature in the following cases. Then answer any additional questions. A cup of coffee has a temperature of \(90^{\circ} \mathrm{C}\) when it is poured and allowed to cool in a room with a temperature of \(25^{\circ} \mathrm{C}\). One minute after the coffee is poured, its temperature is \(85^{\circ} \mathrm{C}\). How long must you wait until the coffee is cool enough to drink, say \(30^{\circ} \mathrm{C} ?\)

A differential equation of the form \(y^{\prime}(t)=f(y)\) is said to be autonomous (the function \(f\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(f\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=\sin y$$

Suppose an object with an initial temperature of \(T_{0}>0\) is put in surroundings with an ambient temperature of \(A\) where \(A<\frac{T_{0}}{2}\). Let \(t_{1 / 2}\) be the time required for the object to cool to \(\frac{T_{0}}{2}\) a. Show that \(t_{1 / 2}=-\frac{1}{k} \ln \left[\frac{T_{0}-2 A}{2\left(T_{0}-A\right)}\right]\) b. Does \(t_{1 / 2}\) increase or decrease as \(k\) increases? Explain. c. Why is the condition \(A<\frac{T_{0}}{2}\) needed?
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