Calculus, Single Variable: Early Transcendentals (3rd Edition)
3rd Edition
ISBN: 9780134766850
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Question
Chapter 9.4, Problem 40E
To determine
To solve: The given initial value problem.
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PART A (full explanation plz)
Certain populations that are present in a given habitat and are related in such a way that one species, known as the prey, has an ample food supply and the other species, known as the predator, feeds on the prey. This situation can be modeled with a system of differential equations known as the predator-prey or Lotka-Volterra equations. A solution of this system of equations is a pair of functions R (t ) and V (t) that describe the populations of the prey and predator as functions of time. Usually, it is impossible to find explicit formulas for R and V so graphical methods are used to analyze the equations.
(1) Suppose that the populations of aphids and ladybugs are modeled with a system of Lotka- Volterra equations given below
dA/dt=2A(1-0.0001A)-0.01AL
dL/dt= -.5L+.0001AL
where A(t) is the aphid population at time t and L(t)is the ladybug population at time t.
a) In the absence of ladybugs, what does the model predict about the aphids?
Chain Rule Use the Chain Rule to evaluate the following derivatives.
w'(t), where w = z ln (x2 + y2), x = 3et, y = 4et, and z = t
Chain Rule Use the Chain Rule to evaluate the following derivatives.
ws, wt, wss, wtt, and wst, where w = xyz, x = 2st, y = st2, andz = s2t
Chapter 9 Solutions
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Ch. 9.1 - What are the orders of the equations in Example 2?...Ch. 9.1 - Prob. 2QCCh. 9.1 - Prob. 3QCCh. 9.1 - Prob. 4QCCh. 9.1 - In Example 7, if the height function were given by...Ch. 9.1 - Prob. 1ECh. 9.1 - Prob. 2ECh. 9.1 - Prob. 3ECh. 9.1 - Prob. 4ECh. 9.1 - The solution to the initial value problem y(t) = 2...
Ch. 9.1 - Prob. 6ECh. 9.1 - Verifying general solutions Verify that the given...Ch. 9.1 - Prob. 8ECh. 9.1 - Prob. 9ECh. 9.1 - Prob. 10ECh. 9.1 - Prob. 11ECh. 9.1 - Prob. 12ECh. 9.1 - Verifying general solutions Verify that the given...Ch. 9.1 - Prob. 14ECh. 9.1 - Prob. 15ECh. 9.1 - Prob. 16ECh. 9.1 - Verifying solutions of initial value problems...Ch. 9.1 - Prob. 18ECh. 9.1 - Verifying solutions of initial value problems...Ch. 9.1 - Prob. 20ECh. 9.1 - Finding general solutions Find the general...Ch. 9.1 - Finding general solutions Find the general...Ch. 9.1 - Finding general solutions Find the general...Ch. 9.1 - Prob. 24ECh. 9.1 - Finding general solutions Find the general...Ch. 9.1 - Finding general solutions Find the general...Ch. 9.1 - Finding general solutions Find the general...Ch. 9.1 - Prob. 28ECh. 9.1 - Prob. 29ECh. 9.1 - General solutions Find the general solution of the...Ch. 9.1 - General solutions Find the general solution of the...Ch. 9.1 - Prob. 32ECh. 9.1 - Solving initial value problems Solve the following...Ch. 9.1 - Prob. 34ECh. 9.1 - Prob. 35ECh. 9.1 - Prob. 36ECh. 9.1 - Solving initial value problems Solve the following...Ch. 9.1 - Prob. 38ECh. 9.1 - Prob. 39ECh. 9.1 - Prob. 40ECh. 9.1 - Prob. 41ECh. 9.1 - Prob. 42ECh. 9.1 - Motion in a gravitational field An object is fired...Ch. 9.1 - Prob. 44ECh. 9.1 - Harvesting problems Consider the harvesting...Ch. 9.1 - Harvesting problems Consider the harvesting...Ch. 9.1 - Prob. 47ECh. 9.1 - Prob. 48ECh. 9.1 - Prob. 49ECh. 9.1 - Prob. 50ECh. 9.1 - Prob. 51ECh. 9.1 - Prob. 52ECh. 9.1 - Prob. 53ECh. 9.1 - Prob. 54ECh. 9.1 - Prob. 55ECh. 9.1 - Prob. 56ECh. 9.2 - Assuming solutions are unique (at most one...Ch. 9.2 - Prob. 2QCCh. 9.2 - Prob. 3QCCh. 9.2 - Prob. 4QCCh. 9.2 - Prob. 1ECh. 9.2 - Prob. 2ECh. 9.2 - Prob. 3ECh. 9.2 - Prob. 4ECh. 9.2 - Prob. 6ECh. 9.2 - Direction fields A differential equation and its...Ch. 9.2 - Prob. 8ECh. 9.2 - Prob. 9ECh. 9.2 - Prob. 10ECh. 9.2 - Prob. 11ECh. 9.2 - Prob. 12ECh. 9.2 - Prob. 13ECh. 9.2 - Prob. 14ECh. 9.2 - Prob. 15ECh. 9.2 - Prob. 16ECh. 9.2 - Increasing and decreasing solutions Consider the...Ch. 9.2 - Prob. 18ECh. 9.2 - Prob. 19ECh. 9.2 - Prob. 20ECh. 9.2 - Logistic equations Consider the following logistic...Ch. 9.2 - Logistic equations Consider the following logistic...Ch. 9.2 - Logistic equations Consider the following logistic...Ch. 9.2 - Logistic equations Consider the following logistic...Ch. 9.2 - Two steps of Eulers method For the following...Ch. 9.2 - Prob. 26ECh. 9.2 - Prob. 27ECh. 9.2 - Prob. 28ECh. 9.2 - Prob. 29ECh. 9.2 - Prob. 30ECh. 9.2 - Prob. 31ECh. 9.2 - Prob. 32ECh. 9.2 - Prob. 33ECh. 9.2 - Prob. 34ECh. 9.2 - Prob. 35ECh. 9.2 - Prob. 36ECh. 9.2 - Prob. 37ECh. 9.2 - Prob. 38ECh. 9.2 - Prob. 39ECh. 9.2 - Prob. 40ECh. 9.2 - Prob. 41ECh. 9.2 - Prob. 43ECh. 9.2 - Prob. 44ECh. 9.2 - Prob. 45ECh. 9.2 - Prob. 46ECh. 9.2 - Prob. 47ECh. 9.2 - Prob. 48ECh. 9.2 - Prob. 49ECh. 9.2 - Prob. 50ECh. 9.3 - Which of the following equations are separable?...Ch. 9.3 - Prob. 2QCCh. 9.3 - Prob. 3QCCh. 9.3 - Prob. 4QCCh. 9.3 - Prob. 1ECh. 9.3 - Prob. 2ECh. 9.3 - Prob. 3ECh. 9.3 - Prob. 4ECh. 9.3 - Solving separable equations Find the general...Ch. 9.3 - Prob. 6ECh. 9.3 - Prob. 7ECh. 9.3 - Prob. 8ECh. 9.3 - Solving separable equations Find the general...Ch. 9.3 - Prob. 10ECh. 9.3 - Prob. 11ECh. 9.3 - Prob. 12ECh. 9.3 - Prob. 13ECh. 9.3 - Prob. 14ECh. 9.3 - Solving separable equations Find the general...Ch. 9.3 - Prob. 16ECh. 9.3 - Prob. 17ECh. 9.3 - Prob. 18ECh. 9.3 - Prob. 19ECh. 9.3 - Prob. 20ECh. 9.3 - Solving initial value problems Determine whether...Ch. 9.3 - Solving initial value problems Determine whether...Ch. 9.3 - Prob. 23ECh. 9.3 - Prob. 24ECh. 9.3 - Solving initial value problems Determine whether...Ch. 9.3 - Prob. 26ECh. 9.3 - Solving initial value problems Determine whether...Ch. 9.3 - Prob. 28ECh. 9.3 - Prob. 29ECh. 9.3 - Prob. 30ECh. 9.3 - Prob. 31ECh. 9.3 - Prob. 32ECh. 9.3 - Prob. 33ECh. 9.3 - Prob. 34ECh. 9.3 - Solutions in implicit form Solve the following...Ch. 9.3 - Prob. 36ECh. 9.3 - Prob. 37ECh. 9.3 - Prob. 38ECh. 9.3 - Prob. 39ECh. 9.3 - Prob. 40ECh. 9.3 - Prob. 41ECh. 9.3 - Prob. 42ECh. 9.3 - Prob. 43ECh. 9.3 - Prob. 44ECh. 9.3 - Prob. 45ECh. 9.3 - Prob. 46ECh. 9.3 - Prob. 47ECh. 9.3 - Prob. 48ECh. 9.3 - Prob. 49ECh. 9.3 - Prob. 50ECh. 9.3 - Prob. 51ECh. 9.3 - Prob. 53ECh. 9.3 - Prob. 54ECh. 9.4 - Prob. 1QCCh. 9.4 - Prob. 2QCCh. 9.4 - Prob. 3QCCh. 9.4 - Verify that the solution of the initial value...Ch. 9.4 - Prob. 5QCCh. 9.4 - Prob. 1ECh. 9.4 - Prob. 2ECh. 9.4 - Prob. 3ECh. 9.4 - Prob. 4ECh. 9.4 - Prob. 5ECh. 9.4 - Prob. 6ECh. 9.4 - Prob. 7ECh. 9.4 - Prob. 8ECh. 9.4 - Prob. 9ECh. 9.4 - Prob. 10ECh. 9.4 - Prob. 11ECh. 9.4 - Prob. 12ECh. 9.4 - Prob. 13ECh. 9.4 - Prob. 14ECh. 9.4 - Prob. 15ECh. 9.4 - Prob. 16ECh. 9.4 - Prob. 17ECh. 9.4 - Prob. 18ECh. 9.4 - Prob. 19ECh. 9.4 - Prob. 20ECh. 9.4 - Prob. 21ECh. 9.4 - Stability of equilibrium points Find the...Ch. 9.4 - Prob. 23ECh. 9.4 - Prob. 24ECh. 9.4 - Loan problems The following initial value problems...Ch. 9.4 - Prob. 26ECh. 9.4 - Prob. 27ECh. 9.4 - Prob. 28ECh. 9.4 - Prob. 29ECh. 9.4 - Newtons Law of Cooling Solve the differential...Ch. 9.4 - Prob. 31ECh. 9.4 - Optimal harvesting rate Let y(t) be the population...Ch. 9.4 - Prob. 34ECh. 9.4 - Prob. 35ECh. 9.4 - Prob. 36ECh. 9.4 - Prob. 37ECh. 9.4 - Prob. 38ECh. 9.4 - Prob. 39ECh. 9.4 - Prob. 40ECh. 9.4 - Prob. 41ECh. 9.4 - Prob. 42ECh. 9.4 - Prob. 43ECh. 9.4 - Prob. 44ECh. 9.4 - Prob. 45ECh. 9.4 - Prob. 46ECh. 9.4 - Prob. 47ECh. 9.4 - Prob. 48ECh. 9.5 - Explain why the maximum growth rate for the...Ch. 9.5 - Suppose the tank is filled with a salt solution...Ch. 9.5 - Prob. 3QCCh. 9.5 - Explain how the growth rate function determines...Ch. 9.5 - What is a carrying capacity? Mathematically, how...Ch. 9.5 - Explain how the growth rate function can be...Ch. 9.5 - Prob. 4ECh. 9.5 - Is the differential equation that describes a...Ch. 9.5 - Prob. 6ECh. 9.5 - Prob. 7ECh. 9.5 - Describe the behavior of the two populations in a...Ch. 9.5 - Prob. 15ECh. 9.5 - Solving logistic equations Write a logistic...Ch. 9.5 - Prob. 17ECh. 9.5 - Prob. 18ECh. 9.5 - Prob. 19ECh. 9.5 - Prob. 20ECh. 9.5 - Solving the Gompertz equation Solve the Gompertz...Ch. 9.5 - Solving the Gompertz equation Solve the Gompertz...Ch. 9.5 - Prob. 23ECh. 9.5 - Prob. 24ECh. 9.5 - Prob. 25ECh. 9.5 - Prob. 26ECh. 9.5 - Prob. 31ECh. 9.5 - Prob. 32ECh. 9.5 - Prob. 33ECh. 9.5 - Prob. 34ECh. 9.5 - Prob. 35ECh. 9.5 - Prob. 36ECh. 9.5 - Prob. 37ECh. 9.5 - Prob. 38ECh. 9 - Explain why or why not Determine whether the...Ch. 9 - Prob. 2RECh. 9 - Prob. 3RECh. 9 - Prob. 4RECh. 9 - Prob. 5RECh. 9 - Prob. 6RECh. 9 - Prob. 7RECh. 9 - Prob. 8RECh. 9 - Prob. 9RECh. 9 - Prob. 10RECh. 9 - Prob. 11RECh. 9 - Prob. 12RECh. 9 - Prob. 13RECh. 9 - Prob. 14RECh. 9 - Prob. 15RECh. 9 - Prob. 16RECh. 9 - Prob. 17RECh. 9 - Prob. 18RECh. 9 - Prob. 19RECh. 9 - Direction fields The direction field for the...Ch. 9 - Prob. 21RECh. 9 - Prob. 22RECh. 9 - Prob. 23RECh. 9 - Prob. 24RECh. 9 - Prob. 25RECh. 9 - Logistic growth The population of a rabbit...Ch. 9 - Prob. 27RECh. 9 - Prob. 28RECh. 9 - Prob. 29RECh. 9 - Prob. 30RECh. 9 - Prob. 32RECh. 9 - Prob. 33RE
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