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mathematics
calculus

Questions and Answers of Calculus

Graph a rate-of-change function that has a slope of 0 at the equilibrium but the equilibrium is unstable. What is the sign of the second derivative at the equilibrium? What is the sign of the third
Try to draw a phase-line diagram with two stable equilibria in a row. Use the Intermediate Value Theorem to sketch a proof of why this is impossible. The fact that the rate-of-change function is
Why is it impossible for a solution of an autonomous differential equation to oscillate? The fact that the rate-of-change function is continuous means that many behaviors are impossible for an
Consider the equationfor both positive and negative values of x. Find the equilibria as functions of a for values of a between -1 and 1. Draw a bifurcation diagram and describe in words what happens
Consider the equationfor both positive and negative values of x. Find the equilibria as functions of a for values of a between -1 and 1. Draw a bifurcation diagram and describe in words what happens
Consider the equationfor both positive and negative values of x. Find the equilibria as functions of a for values of a between -1 and 1. Draw a bifurcation diagram and describe in words what happens
From the following graphs of the rate of change as a function of the state variable, identify stable and unstable equilibria by checking whether the rate of change is an increasing or decreasing
Consider the equationfor both positive and negative values of x. Find the equilibria as functions of a for values of a between -1 and 1. Draw a bifurcation diagram and describe in words what happens
Use the stability theorem to check the phase-line diagrams for the following models of bacterial population growth.1. The model in Section 5.1, Exercise 27 and Section 5.2 Exercise 21.2. The model in
Use the stability theorem to check the phase-line diagrams for the following models of bacterial population growth.1. The model in Section 5.1, Exercise 37 and Section 5.2, Exercise 27.2. The model
A reaction-diffusion equation describes how chemical concentration changes due to two factors simultaneously, reaction and movement. A simple model has the formThe first term describes diffusion, and
A reaction-diffusion equation describes how chemical concentration changes due to two factors simultaneously, reaction and movement. A simple model has the formThe first term describes diffusion, and
A reaction-diffusion equation describes how chemical concentration changes due to two factors simultaneously, reaction and movement. A simple model has the formThe first term describes diffusion, and
Use the stability theorem to evaluate the stability of the equilibria of the following autonomous differential equations.Compare your results with the phase line in Section 5.2, Exercise 15.
A reaction-diffusion equation describes how chemical concentration changes due to two factors simultaneously, reaction and movement. A simple model has the formThe first term describes diffusion, and
Apply the stability theorem for autonomous differential equations to the following equations. Show that your results match what you found in your phase-line diagrams, and give a biological
Apply the stability theorem for autonomous differential equations to the following equations. Show that your results match what you found in your phase-line diagrams, and give a biological
Apply the stability theorem for autonomous differential equations to the following equations. Show that your results match what you found in your phase-line diagrams, and give a biological
Apply the stability theorem for autonomous differential equations to the following equations. Show that your results match what you found in your phase-line diagrams, and give a biological
Apply the stability theorem for autonomous differential equations to the following equations. Show that your results match what you found in your phase-line diagrams, and give a biological
The equation from the previous problem but with water entering at a rate of 4.0 cm/s. Apply the stability theorem for autonomous differential equations to the following equations. Show that your
Apply the stability theorem for autonomous differential equations to the following equations. Show that your results match what you found in your phase-line diagrams, and give a biological
The equation from the previous problem but with substrate added at rate R = 0.5. Apply the stability theorem for autonomous differential equations to the following equations. Show that your results
Apply the stability theorem for autonomous differential equations to the following equations. Show that your results match what you found in your phase-line diagrams, and give a biological
Use the stability theorem to evaluate the stability of the equilibria of the following autonomous differential equations.Compare your results with the phase line in Section 5.2, Exercise 16.
Apply the stability theorem for autonomous differential equations to the following equations. Show that your results match what you found in your phase-line diagrams, and give a biological
Consider the logistic differential equation (Section 5.1, Exercise 27) with harvesting proportional to population size, or db/dt = b(1 - b - h) where h represents the fraction harvested. Graph the
Suppose μ = 1 in the basic disease model dI/dt = αI(1 - I) - μI. Graph the two equilibria as functions of α for values of a between 0 and 2, using a solid line when an equilibrium is stable and a
Consider a version of the equation in Section 5.2, Exercise 29 that includes the parameter r,Graph the equilibria as functions of r for values of r between 0 and 3, using a solid line when an
Consider a variant of the basic disease model given byGraph the equilibria as functions of Î± for values of Î± between 0 and 5, using a solid line when an equilibrium is stable and a dashed line
Analyze the stability of the positive equilibrium in the model from Exercise 44 when α = 4, the point where the bifurcation occurs. Right at a bifurcation point, the stability theorem fails because
Analyze the stability of the disease model when α = μ = 1, the point where the bifurcation occurs in Exercise 42. Right at a bifurcation point, the stability theorem fails because the slope of the
Use the stability theorem to evaluate the stability of the equilibria of the following autonomous differential equations.Compare your results with the phase line in Section 5.2, Exercise 17.
Use the stability theorem to evaluate the stability of the equilibria of the following autonomous differential equations.Compare your results with the phase line in Section 5.2, Exercise 18.
Find the stability of the equilibria of the following autonomous differential equations that include parameters.Suppose that a > 0.
Use separation of variables to solve the following autonomous differential equations. Check your answers by differentiating.
Use separation of variables to solve the following pure-time differential equations. Check your answers by differentiating.
Find the solution of the differential equation db/dt = bp in the following cases. At what time does it approach infinity? Sketch a graph. p = 2 (as in the text) and b(0) = 100.
Find the solution of the differential equation db/dt = bp in the following cases. At what time does it approach infinity? Sketch a graph. p = 2 and b(0) = 0.1.
Find the solution of the differential equation db/dt = bp in the following cases. At what time does it approach infinity? Sketch a graph. p = 1.1 and b(0) = 100.
Find the solution of the differential equation db/dt = bp in the following cases. At what time does it approach infinity? Sketch a graph. p = 1.1 and b(0) = 0.1.
The autonomous differential equationwith x(0) = 1. This describes a population with per capita production that decreases like 1/1 + x.The following autonomous differential equations can be solved
The autonomous differential equationwith x(0) = l. This describes a population with per capita production that decreases like 1/1 + x2. Describe in words how the solution differs from that in
Show that the solution comes out the same if N > K. Check the above results about the solution of the logistic equation from Example 5.4.7.
Differentiate to check that the solution given in Example 5.4.7 solves the differential equation. Check the above results about the solution of the logistic equation from Example 5.4.7.
Using the method used to find the solution derived for Newton's law of cooling, find the solution of the chemical diffusion equation dC/dt = β(Г - C) with the following parameter values and initial
Use separation of variables to solve the following autonomous differential equations. Check your answers by differentiating.
Using the method used to find the solution derived for Newton's law of cooling, find the solution of the chemical diffusion equation dC/dt = β(Г - C) with the following parameter values and initial
Suppose the initial condition is y(0) = 4. Find the solution with separation of variables and graph the result. What really happens at time t = 2? And what happens after this time? How does this
Suppose the initial condition is y(0) = 16. Find the solution with separation of variables and graph the result. When does the solution reach 0? What would the depth be at this time if draining
Suppose a population is growing at constant rate λ, but that individuals are harvested at a rate of h, following the differential equation db/dt = λb - h. For each of the following values of λ and
Suppose a population is growing at constant rate λ, but that individuals are harvested at a rate of h, following the differential equation db/dt = λb - h. For each of the following values of λ and
Use separation of variables to solve for C in the following models describing chemical diffusion, and find the solution starting from the initial condition C = Г.1. The model in Section 5.1,
Separation of variables can help to solve some non autonomous differential equations. For example, suppose that the per capita production rate is λ(t), a function of time, so that db/dt =λ(t)b.
Use separation of variables to solve the following autonomous differential equations. Check your answers by differentiating.
Separation of variables can help to solve some non autonomous differential equations. For example, suppose that the per capita production rate is λ(t), a function of time, so that db/dt =λ(t)b.
Separation of variables can help to solve some non autonomous differential equations. For example, suppose that the per capita production rate is λ(t), a function of time, so that db/dt =λ(t)b.
Separation of variables can help to solve some non autonomous differential equations. For example, suppose that the per capita production rate is λ(t), a function of time, so that db/dt =λ(t)b.
The model from Section 5.1, Exercise 29 with initial condition b(0) = 100.
The model from Section 5.1, Exercise 30 with initial condition b(0) = 10.000.
Find the solution of Equation 5.1.4 as a special case of the logistic equation and compare with the solution in Equation 5.1.6. Our model of selection (Equation 5.1.4) looks much like the logistic
Suppose that μ = 2 - 2p and λ = 1. Find the equilibria and their stability. What method would you use to try to find the solution after separating variables? Our model of selection (Equation 5.1.4)
Create the new variable y = et H and find a differential equation for y. There are many important differential equations for which separation of variables fails, but which can be solved with other
Identify the type of differential equation, and solve it with the initial condition H(0) = 0. There are many important differential equations for which separation of variables fails, but which can be
Graph your solution and the ambient temperature when β is small, say β = 0.1. Describe the result. There are many important differential equations for which separation of variables fails, but which
Use separation of variables to solve the following autonomous differential equations. Check your answers by differentiating.
Graph your solution and the ambient temperature when β is large, say β = 1.0. Why are the two curves so much farther apart? There are many important differential equations for which separation of
Use separation of variables to solve the following autonomous differential equations. Check your answers by differentiating.
Use separation of variables to solve the following autonomous differential equations. Check your answers by differentiating.
Use separation of variables to solve the following pure-time differential equations. Check your answers by differentiating.
Use separation of variables to solve the following pure-time differential equations. Check your answers by differentiating.
Use separation of variables to solve the following pure-time differential equations. Check your answers by differentiating.
Consider the following special cases of the predator-prey equationsWrite the differential equations and say what they mean.ˆˆ = Î· = 0.
Start from a = 250 and b = 500. Take four steps, with a step length of Î”t = 0.05. How do your results compare with those in Exercise 8?Apply Euler's method to the competition equationsstarting
Suppose Î± = 0.3 and Î±2 = 0.1. Start from H = 60 and A = 20. Take two steps, with a step length of Î”t = 0.1.Apply Euler's method to Newton's law of coolingwith the given parameter values and
Suppose Î± = 0.3 and Î±2 = 0.1. Start from H = 0 and A = 20. Take two steps, with a step length of Î”t = 0.1.Apply Euler's method to Newton's law of coolingwith the given parameter values and
Supposes Î± = 3.0 and Î±2 = 1.0. Start from H = 60 and A = 20. Take two steps, with a step length of Î”t = 0.25. Do the results look reasonable?Apply Euler's method to Newton's law of
Suppose Î± = 3.0 and Î±2 = l.0. Start from H = 0 and A = 20. Take two steps, with a step length of Î”t = 0.5. Do the results look reasonable?Apply Euler's method to Newton's law of coolingwith
Write the velocity v in terms of the derivative of the position x, and the acceleration in terms of the derivative of the velocity v. Use the spring equation to write the derivative of the velocity
Friction also creates acceleration proportional to the negative of the velocity (see Section 2.10, Exercise 39). A simple case obeys the equationWrite this as a pair of coupled differential equations
We know that one solution of the basic spring equation in Exercise 15 is x(t) = cos(t). Find v(t) and check that the solution matches the system of equations. What are the initial position and
One solution of the spring equation with friction in Exercise 16 is x(t) = e–t cos(t). Find v(t) and check that the solution matches the system of equations. What are the initial position and
Use Euler's method with Î”t = 0.2 for five steps to estimate x(1) and v(1) for the model in Exercise 15 with initial position x(0) = 1 and initial velocity v(0) = 0. Compare with the results in
Consider the following special cases of the predator-prey equationsWrite the differential equations and say what they mean.Î´ = 0.
Use Euler's method with Î”t = 0.2 for five steps to estimate x(1) and v(1) for the model in Exercise 16 with initial position x(0) = 1 and initial velocity v(0) = -1. Compare with the results in
Non autonomous differential equations can be written as a system of coupled autonomous differential equations by writing a separate differential equation for the variable t.1. Write the differential
Per capita growth of prey = 1.0 - 0.05p Per capita growth of predators = -1.0 + 0.02b. Consider the above type of predator-prey interactions. Graph the per capita rates of change and write the
Per capita growth of prey = 2.0 - 0.01p Per capita growth of predators = 1.0 + 0.01b. How does this differ from the basic predator-prey system (Equation 5.5.1)? Consider the above type of
Per capita growth of prey = 2.0 - 0.0001p2 per capita growth of predators = -1.0 + 0.01b Consider the above type of predator-prey interactions. Graph the per capita rates of change and write the
per capita growth of prey = 2.0 - 0.01p per capita growth of predators = -1.0 + 0.0001b2 Consider the above type of predator-prey interactions. Graph the per capita rates of change and write the
Write systems of differential equations describing the above situation. Feel free to make up parameter values as needed.1. Two predators that must eat each other to survive.2. Two predators that must
Consider the following special cases of the predator-prey equationsWrite the differential equations and say what they mean.Î· = 0.
Suppose that the size of the first cell is 2.0 μL and the second is 5.0 μL.Follow these steps to derive the equations for chemical exchange between two adjacent cells of different size. Suppose the
Suppose that the size of the first cell is 5.0 μL and the second is 12.0 μL.Follow these steps to derive the equations for chemical exchange between two adjacent cells of different size. Suppose
The size of the room is 10.0 times that of the object, but the specific heat of the room is 0.2 times that of the object (meaning that a small amount of heat can produce a large change in the
The size of the room is 5.0 times that of the object, but the specific heat of the room is 2.0 times that of the object (meaning that a large amount of heat produces only a small change in the
There are many important extensions of the two-dimensional disease model (Equation 5.5.2) that include more categories of people, and that model processes of birth, death, and immunity.1. Suppose
Consider the following special cases of the predator-prey equationsWrite the differential equations and say what they mean.ˆˆ = 0.
Write the following models of frequency dependence as systems of autonomous differential equations for a and b.1. The situation in Section 5.1, Exercise 37.2. The situation in Section 5.1, Exercise
Suppose that types a and b do not interact (equivalent to setting b = 0 in the differential equation for a).If each state variable in a system of autonomous differential equations does not respond to

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