Math

CalculusSingle Variable CalculusIn a seasonal-growth model , a periodic function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for example, be caused by seasonal changes in the availability of food. (a) Find the solution of the seasonal-growth model d P d t = k P cos ( r t − ϕ ) P ( 0 ) = P 0 (b) where k , r , and ϕ are positive constants. (c) By graphing the solution for several values of k , r , and ϕ , explain how the values of k , r , and ϕ affect the solution. What can you say about lim t → ∞ P ( t )?BuyFind*launch*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266636

Chapter 9.4, Problem 23E

Textbook Problem

In a **seasonal-growth model**, a periodic function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for example, be caused by seasonal changes in the availability of food.

- (a) Find the solution of the seasonal-growth model

- (b) where
*k*,*r*, and*ϕ*are positive constants. - (c) By graphing the solution for several values of
*k*,*r*, and*ϕ*, explain how the values of*k*,*r*, and*ϕ*affect the solution. What can you say about lim_{t}_{→∞}*P*(*t*)?

Single Variable Calculus

Ch. 9.1 - Show that y=23ex+e2x is a solution of the...Ch. 9.1 - Verify that y = t cos t t is a solution of the...Ch. 9.1 - (a) For what values of r does the function y = erx...Ch. 9.1 - (a) For what values of k does the function y = cos...Ch. 9.1 - Which of the following functions are solutions of...Ch. 9.1 - (a) Show that every member of the family of...Ch. 9.1 - (a) What can you say about a solution of the...Ch. 9.1 - (a) What can you say about the graph of a solution...Ch. 9.1 - A population is modeled by the differential...Ch. 9.1 - The Fitzhugh-Nagumo model for the electrical...

Ch. 9.1 - Explain why the functions with the given graphs...Ch. 9.1 - The function with the given graph is a solution of...Ch. 9.1 - Match the differential equations with the solution...Ch. 9.1 - Suppose you have just poured a cup of freshly...Ch. 9.1 - Psychologists interested in learning theory study...Ch. 9.1 - Von Bertalanffys equation states that the rate of...Ch. 9.1 - Differential equations have been used extensively...Ch. 9.2 - A direction field for the differential equation y...Ch. 9.2 - A direction field for the differential equation...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Use the direction field labeled I (above) to...Ch. 9.2 - Use the direction field labeled III (above) to...Ch. 9.2 - Sketch a direction field for the differential...Ch. 9.2 - Sketch a direction field for the differential...Ch. 9.2 - Sketch the direction field of the differential...Ch. 9.2 - Sketch the direction field of the differential...Ch. 9.2 - Sketch the direction field of the differential...Ch. 9.2 - Sketch the direction field of the differential...Ch. 9.2 - (a) Use Eulers method with each of the following...Ch. 9.2 - A direction field for a differential equation is...Ch. 9.2 - Use Eulers method with step size 0.5 to compute...Ch. 9.2 - Use Eulers method with step size 0.2 to estimate...Ch. 9.2 - Use Eulers method with step size 0.1 to estimate...Ch. 9.2 - (a) Use Eulers method with step size 0.2 to...Ch. 9.2 - The figure shows a circuit containing an...Ch. 9.3 - Solve the differential equation. 1. dydx=3x2y2Ch. 9.3 - Solve the differential equation. 2. dydx=xyCh. 9.3 - Solve the differential equation. 3. xyy = x2 + 1Ch. 9.3 - Solve the differential equation. 4. y + xey = 0Ch. 9.3 - Solve the differential equation. 5. (ey 1)y = 2 +...Ch. 9.3 - Solve the differential equation. 6....Ch. 9.3 - Solve the differential equation. 7. ddt=tsecet2Ch. 9.3 - Solve the differential equation. 8....Ch. 9.3 - Solve the differential equation. 9. dpdt=t2pp+t21Ch. 9.3 - Solve the differential equation. 10. dzdt+et+z=0Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find an equation of the curve that passes through...Ch. 9.3 - Find the function f such that f(x) = xf(x) x and...Ch. 9.3 - Solve the differential equation y = x + y by...Ch. 9.3 - Solve the differential equation xy = y + xey/x by...Ch. 9.3 - (a) Solve the differential equationy=2x1y2. (b)...Ch. 9.3 - Solve the equation ey y + cos x = 0 and graph...Ch. 9.3 - Find the orthogonal trajectories of the family of...Ch. 9.3 - Find the orthogonal trajectories of the family of...Ch. 9.3 - Find the orthogonal trajectories of the family of...Ch. 9.3 - Find the orthogonal trajectories of the family of...Ch. 9.3 - An integral equation is an equation that contains...Ch. 9.3 - An integral equation is an equation that contains...Ch. 9.3 - An integral equation is an equation that contains...Ch. 9.3 - Find a function f such that f(3) = 2 and...Ch. 9.3 - Solve the initial-value problem in Exercise 9.2.27...Ch. 9.3 - In Exercise 9.2.28 we discussed a differential...Ch. 9.3 - In Exercise 9.1.15 we formulated a model for...Ch. 9.3 - In an elementary chemical reaction, single...Ch. 9.3 - In contrast to the situation of Exercise 40,...Ch. 9.3 - A sphere with radius 1 m has temperature 15C. It...Ch. 9.3 - A glucose solution is administered intravenously...Ch. 9.3 - A certain small country has 10 billion in paper...Ch. 9.3 - A tank contains 1000 L of brine with 15 kg of...Ch. 9.3 - The air in a room with volume 180 m3 contains...Ch. 9.3 - A vat with 500 gallons of beer contains 4% alcohol...Ch. 9.3 - A tank contains 1000 L of pure water. Brine that...Ch. 9.3 - When a raindrop falls, it increases in size and so...Ch. 9.3 - An object of mass m is moving horizontally through...Ch. 9.3 - Allometric growth in biology refers to...Ch. 9.3 - A model for tumor growth is given by the Gompertz...Ch. 9.3 - According to Newtons Law of Universal Gravitation,...Ch. 9.4 - A population grows according to the given logistic...Ch. 9.4 - A population grows according to the given logistic...Ch. 9.4 - The Pacific halibut fishery has been modeled by...Ch. 9.4 - Suppose a population P(t) satisfies...Ch. 9.4 - Suppose a population grows according to a logistic...Ch. 9.4 - The table gives the number of yeast cells in a new...Ch. 9.4 - The population of the world was about 6.1 billion...Ch. 9.4 - (a) Assume that the carrying capacity for the US...Ch. 9.4 - One model for the spread of a rumor is that the...Ch. 9.4 - Biologists stocked a lake with 400 fish and...Ch. 9.4 - (a) Show that if P satisfies the logistic equation...Ch. 9.4 - For a fixed value of M (say M = 10), the family of...Ch. 9.4 - The table gives the midyear population of Norway,...Ch. 9.4 - Consider a population P = P(t) with constant...Ch. 9.4 - Let c be a positive number. A differential...Ch. 9.4 - There is considerable evidence to support the...Ch. 9.4 - Another model for a growth function for a limited...Ch. 9.4 - In a seasonal-growth model, a periodic function of...Ch. 9.4 - Suppose we alter the differential equation in...Ch. 9.4 - Graphs of logistic functions (Figures 2 and 3)...Ch. 9.5 - Determine whether the differential equation is...Ch. 9.5 - Determine whether the differential equation is...Ch. 9.5 - Determine whether the differential equation is...Ch. 9.5 - Determine whether the differential equation is...Ch. 9.5 - Solve the differential equation. 5. y + y = 1Ch. 9.5 - Solve the differential equation. 6. y y = exCh. 9.5 - Solve the differential equation. 7. y = x yCh. 9.5 - Solve the differential equation. 8. 4x3y + x4y =...Ch. 9.5 - Solve the differential equation. 9. xy+y=xCh. 9.5 - Solve the differential equation. 10. 2xy+y=2xCh. 9.5 - Solve the differential equation. 11. xy 2y = x2,...Ch. 9.5 - Solve the differential equation. 12. y + 2xy = 1Ch. 9.5 - Solve the differential equation. 13....Ch. 9.5 - Solve the differential equation. 14....Ch. 9.5 - Solve the initial-value problem. 15. x2y + 2xy =...Ch. 9.5 - Solve the initial-value problem. 16....Ch. 9.5 - Solve the initial-value problem. 17. tdudt=t2+3u,...Ch. 9.5 - Solve the initial-value problem. 18. xy + y = x ln...Ch. 9.5 - Solve the initial-value problem. 19. xy = y + x2...Ch. 9.5 - Solve the initial-value problem. 20....Ch. 9.5 - Solve the differential equation and use a...Ch. 9.5 - Solve the differential equation and use a...Ch. 9.5 - A Bernoulli differential equation (named after...Ch. 9.5 - Use the method of Exercise 23 to solve the...Ch. 9.5 - Use the method of Exercise 23 to solve the...Ch. 9.5 - Solve the second-order equation xy + 2y = 12x2 by...Ch. 9.5 - In the circuit shown in Figure 4, a battery...Ch. 9.5 - In the circuit shown in Figure 4, a generator...Ch. 9.5 - The figure shows a circuit containing an...Ch. 9.5 - In the circuit of Exercise 29, R = 2 , C = 0.01 F,...Ch. 9.5 - Let P(t) be the performance level of someone...Ch. 9.5 - Two new workers were hired for an assembly line....Ch. 9.5 - In Section 9.3 we looked at mixing problems in...Ch. 9.5 - A tank with a capacity of 400 L is full of a...Ch. 9.5 - An object with mass m is dropped from rest and we...Ch. 9.5 - If we ignore air resistance, we can conclude that...Ch. 9.5 - (a) Show that the substitution z = 1/P transforms...Ch. 9.5 - To account for seasonal variation in the logistic...Ch. 9.6 - For each predator-prey system, determine which of...Ch. 9.6 - Each system of differential equations is a model...Ch. 9.6 - The system of differential equations...Ch. 9.6 - Lynx eat snowshoe hares and snowshoe hares eat...Ch. 9.6 - A phase trajectory is shown for populations of...Ch. 9.6 - A phase trajectory is shown for populations of...Ch. 9.6 - Graphs of populations of two species are shown....Ch. 9.6 - Graphs of populations of two species are shown....Ch. 9.6 - Populations of aphids and ladybugs are modeled by...Ch. 9.6 - In Example 1 we used Lotka-Volterra equations to...Ch. 9 - (a) What is a differential equation? (b) What is...Ch. 9 - What can you say about the solutions of the...Ch. 9 - What is a direction field for the differential...Ch. 9 - Explain how Eulers method works.Ch. 9 - What is a separable differential equation? How do...Ch. 9 - What is a first-order linear differential...Ch. 9 - (a) Write a differential equation that expresses...Ch. 9 - (a) Write the logistic differential equation. (b)...Ch. 9 - (a) Write Lotka-Volterra equations to model...Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - (a) A direction field for the differential...Ch. 9 - (a) Sketch a direction field for the differential...Ch. 9 - (a) A direction field for the differential...Ch. 9 - (a) Use Eulers method with step size 0.2 to...Ch. 9 - Solve the differential equation. 5. y = xesin x y...Ch. 9 - Solve the differential equation. 6. dxdt=1t+xtxCh. 9 - Solve the differential equation. 7. 2yey2y=2x+3xCh. 9 - Solve the differential equation. 8. x2y y =...Ch. 9 - Solve the initial-value problem. 9. drdt+2tr=r,...Ch. 9 - Solve the initial-value problem. 10. (1 + cos x)y...Ch. 9 - Solve the initial-value problem. 11. xy y = x ln...Ch. 9 - Solve the initial-value problem y = 3x2ey, y(0) =...Ch. 9 - Find the orthogonal trajectories of the family of...Ch. 9 - Find the orthogonal trajectories of the family of...Ch. 9 - (a) Write the solution of the initial-value...Ch. 9 - (a) The population of the world was 6.1 billion in...Ch. 9 - The von Bertalanffy growth model is used to...Ch. 9 - A tank contains 100 L of pure water. Brine that...Ch. 9 - One model for the spread of an epidemic is that...Ch. 9 - The Brentano-Stevens Law in psychology models the...Ch. 9 - The transport of a substance across a capillary...Ch. 9 - Populations of birds and insects are modeled by...Ch. 9 - Suppose the model of Exercise 22 is replaced by...Ch. 9 - Barbara weighs 60 kg and is on a diet of 1600...Ch. 9 - Find all functions f such that f is continuous and...Ch. 9 - A student forgot the Product Rule for...Ch. 9 - Let f be a function with the property that f(0) =...Ch. 9 - Find all functions f that satisfy the equation...Ch. 9 - Find the curve y = f(x) such that f(x) 0, f(0) =...Ch. 9 - A subtangent is a portion of the x-axis that lies...Ch. 9 - A peach pie is taken out of the oven at 5:00 pm....Ch. 9 - Snow began to fall during the morning of February...Ch. 9 - A dog sees a rabbit running in a straight line...Ch. 9 - (a) Suppose that the dog in Problem 9 runs twice...Ch. 9 - A planning engineer for a new alum plant must...Ch. 9 - Find the curve that passes through the point (3,...Ch. 9 - Recall that the normal line to a curve at a point...Ch. 9 - Find all curves with the property that if the...Ch. 9 - Find all curves with the property that if a line...

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