Test Bank | Ch9 – Mathematical Modeling Using Differential - Final Test Bank | Applied Calculus 7e by Hughes Hallett. DOCX document preview.

Back to Product

Test Bank | Ch9 – Mathematical Modeling Using Differential

Applied Calculus, 7e (Hughes-Hallett)

Chapter 9 Mathematical Modeling Using Differential Equations

9.1 Mathematical Modeling: Setting up a Differential Equation

1) Which of the following graphs best describes the speed of a car merging onto the freeway?

Four graphs plot a curve in the first quadrant of a coordinate plane. In graph 1, the curve is a bell shaped curve that increases concave up from a point on the positive vertical axis to a peak and then decreases concave up. In graph 2, the curve increases concave down from a point on the positive vertical axis. In graph 3, the curve decreases concave down from a point on the positive vertical axis to some extent, then decreases further concave up to a minimum, and then increases concave up to a higher vertical axis value. In graph 4, the curve decreases concave down from a point on the positive vertical axis.

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

2) Which of the following graphs best describes the balance of a banking account earning continuous compound interest?

I.A graph plots a curve in the first quadrant of a coordinate plane. The horizontal axis is labeled t. The curve increases concave up from a point on the positive vertical axis. II. A graph plots a curve in the first quadrant of a coordinate plane. The horizontal axis is labeled t. The curve increases concave up from a point on the positive vertical axis and then increase concave down.

III.A graph plots a curve in the first quadrant of a coordinate plane. The horizontal axis is labeled t. The curve is a bell shaped curve that increases concave up from a point on the positive vertical axis to a peak and then decreases concave up. IV. A graph plots a curve in the first quadrant of a coordinate plane. The horizontal axis is labeled t. The curve decreases concave down from a point on the positive vertical axis to a point on the positive t axis.

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

3) A population of rodents grows at a rate proportional to the size of the population. Which of the following is the the differential equation for the size of the population, P, as a function of time, t?

A) (dP/dt) = kP, with k positive B) (dP/dt) = kP, with k negative

C) (dP/dt) = (k/P), with k positive D) (dP/dt) = (k/P), with k negative

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

4) Carbon-14 decays at a rate proportional to the amount present. Which of the following is the differential equation for the amount, C, of carbon-14 present at time t?

A) (dC/dt) = kC, with k positive B) (dC/dt) = kC, with k negative

C) (dC/dt) = (k/C), with k positive D) (dC/dt) = (k/C), with k negative

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

5) The deer population, P, in an area is increasing at a rate of 30% per year due to breeding. At the same time, about 150 deer are shot by hunters each year. Which is the differential equation for the population of deer as a function of time t, in years?

A) (dP/dt) = 1.3P - 150 B) (dP/dt) = 150P - 0.3

C) (dP/dt) = 0.3P - 150 D) (dP/dt) = 150P - 1.3

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

6) Water is being pumped into a pool at a rate of 175 gallons per day, and is evaporating at a rate of 0.2% per day. Which is the differential equation for the amount, A, of gallons of water in the pool as a function of time, t, in days?

A) (dA/dt) = 175 - 1.002A B) (dA/dt) = 0.2 - 175A

C) (dA/dt) = 175 - 0.2A D) (dA/dt) = 175 - 0.002A

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

7) A person withdraws money from a trust fund at a rate of $18,000 per year, and the account is earning interest at a rate of 6.5% per year, compounded continuously. Write a differential equation for the balance, B, in the account as a function of time, t, in years and use it to calculate dB / dt if B = $100,000.

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

8) A bank account initially containing $1000 earns interest at a continuous rate of 4% per year. Deposits are made into the account at a constant rate of $500 per year. Which is the differential equation for the balance, B, in the account as a function of time, t, in years?

A) (dB/dt) = 0.04B + 500 B) (dB/dt) = 4B + 500

C) (dB/dt) = 0.04B + 1500 D) (dB/dt) = (1000)0.04B + 500

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

9) A drug is administered intravenously to a patient at a rate of 14 mg per day. About 45% of the drug in the patient's body is metabolized and leaves the body each day. Which is the differential equation for the amount of the drug, D, in the body as a function of time, t, in days?

A) (dD/dt) = 14 - 45D B) (dD/dt) = 14 - 0.45D

C) (dD/dt) = 45 - 14D D) (dD/dt) = 0.45 - 14D

Diff: 2 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

10) A quantity y satisfies the differential equation (dy/dt) = 0.03y. Thus, y is increasing when y is ________ (less/greater) than ________.

Part A: greater

Part B: 0

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Determine whether a quantity is increasing or decreasing given a differential equation.

11) A quantity Q satisfies the differential equation (dQ/dt) = 10 - 3q. Is Q increasing or decreasing at Q = 4?

Diff: 2 Var: 1

Section: 9.1

Learning Objectives: Determine whether a quantity is increasing or decreasing given a differential equation.

12) A quantity Q satisfies the differential equation (dQ/dt) = 8 - 2q. For what value of Q is the rate of change equal to 0?

Diff: 2 Var: 1

Section: 9.1

Learning Objectives: Determine whether a quantity is increasing or decreasing given a differential equation.

13) There is a theory that says the rate at which information spreads by word of mouth is proportional to the product of the number of people who have heard the information and the number who have not. Suppose the total population is N. Which of the following differential equations describe the rate, (dp/dt), at which the information spreads by word of mouth?

A) (dp/dt) = (kp/(p - N)) B) (dp/dt) = kp(N + p)

C) (dp/dt) = (kp/(N - p)) D) (dp/dt) = kp(N - p)

Diff: 2 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

14) A spherical raindrop evaporates at a rate proportional to its surface area. If V = volume of the raindrop and S = surface area, which of the following is a differential equation for (dV/dt)?

A) (dV/dt) = kS with k a negative constant B) (dV/dt) = (k/S) with k a negative constant

C) (dV/dt) = kS with k a positive constant D) (dV/dt) = (k/S) with k a positive constant

Diff: 1 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

15) A population of birds introduced onto an island without predators grows at a rate proportional to the size of the population. Write a differential equation for the size of the population, P, as a function of time. Is the constant of proportionality positive or negative?

A) (dP/dt) = kP, positive B) (dP/dt) = kP, negative

C) (dP/dt) = kt, negative D) (dP/dt) = kt, positive

Diff: 2 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

16) A quantity T satisfies the differential equation (dT/dt) = 7T + 19.

a) Is T increasing or decreasing when T = -5?

b) For what value of T is the rate of change of T equal to zero?

a) decreasing

b) -2.71

Diff: 2 Var: 1

Section: 9.1

Learning Objectives: Determine whether a quantity is increasing or decreasing given a differential equation.

17) A cup of green tea contains 32 mg of caffeine when you are using the tea leaves for the first time. A cup from the second brew contains 12 mg of caffeine, while a cup from the third brew contains only 4 mg of caffeine. Caffeine leaves the body at a continuous rate of about 17% per hour.

a) Write a differential equation for the amount, C, of caffeine in the body at time t hours after drinking the green tea.

b) Use the differential equation to find (dC/dt) at the start of the first hour (right after drinking the tea) for a cup from the first brew, and use your answer to estimate the change in caffeine in the body during the first hour.

c) Does the initial amount of caffeine in the body (whether from the first, second or third brew) change the differential equation?

a) (dC/dt) = -.017C

b) -.017(32) = -5.44 mg

c) no

Diff: 2 Var: 1

Section: 9.1

Learning Objectives: Determine whether a quantity is increasing or decreasing given a differential equation.; Write a differential equation to model a specific situation.

18) Water runs down a certain type of drainpipe at a rate proportional to the amount of water on the roof after a rainfall. Write a differential equation for the amount of water, W, on the roof at time t minutes after the rain stops.

A) (dW/dt) = -kW B) (dW/dt) = W - kt

C) (dW/dt) = k(W - A) D) (dW/dt) = -kt

Diff: 2 Var: 1

Section: 9.1

Learning Objectives: Write a differential equation to model a specific situation.

19) Consider the differential equation for the logistic model representing a population of tarantulas introduced into a new habitat: (dP/dt) = kP(200 - P). What is the carrying capacity?

A) 200 tarantulas B) k tarantulas

C) 200 - P tarantulas D) 1000 tarantulas

Diff: 3 Var: 1

Section: 9.1

Learning Objectives: Determine whether a quantity is increasing or decreasing given a differential equation.

9.2 Solutions of Differential Equations

1) Which one(s) of the following are solutions to the differential equation (dy/dx) = y?

A) y = 3x B) y = (e) with superscript (3x) C) y = x D) y = 3(e) with superscript (x) E) y = (x) with superscript (3)

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

2) Which one(s) of the following are solutions to the differential equation (dy/dx) = 3x?

A) y = 3x B) y = (e) with superscript (3x) C) y = x

D) y = 3(e) with superscript (x) E) y = (x) with superscript (3) F) none of the above

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

3) Which one(s) of the following are solutions to the differential equation (dy/dx) = 3y?

A) y = 3x B) y = (e) with superscript (3x) C) y = x D) y = 3(e) with superscript (x) E) y = (x) with superscript (3)

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

4) Which one(s) of the following are solutions to the differential equation (dy/dx) = (y/x)?

Select all that apply.

A) y = 3x B) y = (e) with superscript (3x) C) y = x D) y = 3(e) with superscript (x) E) y = (x) with superscript (3)

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

5) Which one(s) of the following are solutions to the differential equation (dy/dx) = 3(y/x)?

A) y = 3x B) y = (e) with superscript (3x) C) y = x D) y = 3(e) with superscript (x) E) y = (x) with superscript (3)

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

6) Is y = (x) with superscript (2) a solution to the differential equation x(dy/dx) - 3y?

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

7) Find the solution of the differential equation (dy/dx) = -4x + 3 satisfying y(1) = 5.

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

8) What is the solution of (dy/dx) = 4 - cos 2t when K(0) = -8?

A) K = 4t - (sin 2t/2) - 8 B) K = 4t + (sin 2t/2) - 8

C) K = 4t - (sin 2t/2) + 8 D) K = 4t + (sin 2t/2) + 8

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

9) What is the solution of (dy/dx) = - 9(e) with superscript (-t) when P(0) = 19?

A) P = 9(e) with superscript (-t) - 10 B) P = 9(e) with superscript (-t) + 10

C) P = -9(e) with superscript (-t) + 28 D) P = -9(e) with superscript (-t) - 28

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

10) Given that dy / dt = -0.3y and that y(0) = 17, estimate y(2) to 2 decimal places by first estimating y(1). Assume that the rate of growth given by dy / dt is approximately constant over each unit time interval.

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

11) Given that dy / dt = 3 + 0.2y and that y(0) = 18, estimate y(2) to 1 decimal place by first estimating y(1). Assume that the rate of growth given by dy / dt is approximately constant over each unit time interval.

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

12) Is y = (x) with superscript (k) a solution to the differential equation x(dy/dx) = 2ky?

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

13) If y = (Ce) with superscript (kt) is a solution to the differential equation (dy/dt) = 5y and y = 20 when t = 0, then k = ________ and C = ________.

Part A: 5

Part B: 20

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

14) Does y = 25 cos 5t satisfy ((dx) with superscript (2)y/d(t) with superscript (2)) + 25y = 0?

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

15) If y = (x) with superscript (2) + k is a solution to the differential equation 2y - x(dy/dx) = 14, then k = ________.

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

16) In Kenya, the population P for the recent past has obeyed the growth model (dP/dt) = ktP, with t the number of years since 2018. The solution to the differential equation is of the form P = A(e) with superscript (k(t) with superscript (2)/2). If the population in 2018 was 51.39 million and in 2020 was 53.77 million, then A = ________ and k = ________. Thus, in the year 2028, the population was approximately ________ million. Round all answers to 2 decimal places.

Part A: 51.39

Part B: 0.02

Part C: 139.69

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

17) Suppose P = C(e) with superscript (kt) satisfies the differential equation (dP/dt) = 0.09P. Then k = ________ and C = ________. If either cannot be determined from the information given, enter "cannot tell".

Part A: 0.09

Part B: cannot tell

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

18) Is y = (a/b) + (Ce) with superscript (-bt) the general solution to the differential equation (dy/dt) = a - by?

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

19) Fill in the missing values in the table, given that dy / dt = 8 + y. Assume the growth rate, given by dy / dt, is approximately constant over each time interval.

t

0

1

2

3

4

y

11

t

0

1

2

3

4

y

11

30

68

144

296

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

20) Consider the differential equation (dy/dt) = 1 - y. Is y = (Ce) with superscript (-t) - 1 the general solution to the differential equation? If it is not, answer "not the solution." If it is the general solution and if y(4) = 6, what is the constant C?

A) yes, C = 382 B) yes, C = 0.13

C) yes, C = 12.23 D) not the solution

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

21) For y = k(x) with superscript (2) - 9x to be a solution to the differential equation (dy/dx) = 4x - 9, what must be the constant k?

A) 2 B) 0

C) 4 D) k can be any number

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

22) Which of the following gives a solution to the differential equation (x) with superscript (2)(dy/dx) - xy =3?

table ( (first one: y = - (3/2x)_ _second one: y = 3(x) with superscript (2)) )

A) first one B) second one C) neither

Diff: 2 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

23) What is the general solution of (dy/dx) = sin x + (e) with superscript (x)?

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

24) y = (x) with superscript (4) is a solution to the differential equation y' = (4y/x).

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

25) Find the value of k for which y = (x) with superscript (2) + kx is a solution to the differential equation xy' - 2y = 9x.

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Verify that a function given by a formula is a solution of a differential equation.

26) Given that dy / dt and y(0) = 125, estimate y(3) by first estimating y(1) and y(2). Assume that the rate of growth given by dy / dt is approximately constant over each unit time interval.

Diff: 1 Var: 1

Section: 9.2

Learning Objectives: Use a differential equation to give approximate values of a solution.

9.3 Slope Fields

1) The following figure shows the slope field for the differential equation y' = -x/y. Estimate the equation of the solution curve that goes through the point (0, 1).

A graph plots a slope field on an x y coordinate plane. Both the axes range from negative 2 to 2, in increments of 2. The line segments representing the slope field are inclined up to the right in the second and fourth quadrants and are inclined down to the right in the first and third quadrants. In each quadrant, the steepness of the line segments decreases with decrease in distance from the y axis and increases with decrease in distance from the x axis.

Diff: 1 Var: 1

Section: 9.3

Learning Objectives: Understand the visualization of a differential equation with a slope field.

2) The following figure could be the slope field for the differential equation (dy/dx) = (x/y) for -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2.

A graph plots a slope field on an x y coordinate plane. Both the axes range from negative 2 to 2, in increments of 2. The line segments representing the slope field are inclined up to the right in the first and third quadrants and are inclined down to the right in the second and fourth quadrants. In each quadrant, the steepness of the line segments increases with increase in distance from the y axis and decreases with increase in distance from the x axis.

Diff: 1 Var: 1

Section: 9.3

Learning Objectives: Use a slope field to give a graphical solution of a differential equation.

3) Consider the slope field for dy / dx = y - x. What is the slope at the point (-2, -1)?

Diff: 1 Var: 1

Section: 9.3

Learning Objectives: Understand the visualization of a differential equation with a slope field.

4) Which of the following differential equations goes with the slope field in the figure?

A graph plots a slope field on an x y coordinate plane. Both the axes range from negative 4 to 4, in increments of 2. The line segments representing the slope field are inclined up to the right in the first and third quadrants and are inclined down to the right in the second and fourth quadrants. In each quadrant, the steepness of the line segments increases with increase in distance from the y axis and decreases with increase in distance from the x axis.

A) (dy/dx) = y B) (dy/dx) = (x/y) C) (dy/dx) = (x) with superscript (2) + (y) with superscript (2) D) (dy/dx) = x - y

Diff: 1 Var: 1

Section: 9.3

Learning Objectives: Understand the visualization of a differential equation with a slope field.

5) Which of the following slope fields goes with the differential equation y' = x?

Four graphs plot a slope field on an x y coordinate plane. Both the axes range from negative 5 to 5, in increments of 5. In graph 1, the line segments representing the slope field are inclined up to the right between y equals negative 5 and y equals 4 and are inclined down to the right between y equals 4 and y equals 5. The steepness of the line segments decreases on moving from y equals negative 5 to y equals 4 and increases above y equals 4. In graph 2, the line segments representing the slope field are inclined up to the right in all the four quadrants. The steepness of the line segments decreases with decrease in distance from the x axis in the first and second quadrants, and the steepness of the line segments increases with increase in distance from the x axis in the third and fourth quadrants. In graph 3, the line segments representing the slope field are inclined up to the right between x equals negative 5 and x equals negative 3.5 and between x equals 0 and x equals 3 and are inclined down to the right between x equals negative 3.5 and x equals 0 and between x equals 3 and x equals 5. The steepness of the line segments decreases on moving from x equals negative 5 to x equals 3.5, deceases on moving from x equals negative 3.5 to x equals 0, increases on moving from x equals 0 to x equals 3, and increases on moving from x equals 3 to x equals 5. In graph 4, the line segments representing the slope field are inclined up to the right in the first and fourth quadrants and are inclined down to the right in the second and third quadrants. In each quadrant, the steepness of the line segments decreases with decrease in distance from the y axis in the third and fourth quadrants and increases with increase in distance from the y axis in the first and fourth quadrants.

Diff: 1 Var: 1

Section: 9.3

Learning Objectives: Understand the visualization of a differential equation with a slope field.

6) Look at the slope field labeled I. Consider a solution curve for slope field (I). Which of the answer choices describes the long-run behavior of y at various starting points?

Four graphs plot a slope field on an x y coordinate plane. Both the axes range from negative 5 to 5, in increments of 5. In graph 1, the line segments representing the slope field are inclined up to the right between y equals negative 5 and y equals 4 and are inclined down to the right between y equals 4 and y equals 5. The steepness of the line segments decreases on moving from y equals negative 5 to y equals 4 and increases above y equals 4. In graph 2, the line segments representing the slope field are inclined up to the right in all the four quadrants. The steepness of the line segments decreases with decrease in distance from the x axis in the first and second quadrants, and the steepness of the line segments increases with increase in distance from the x axis in the third and fourth quadrants. In graph 3, the line segments representing the slope field are inclined up to the right between x equals negative 5 and x equals negative 3.5 and between x equals 0 and x equals 3 and are inclined down to the right between x equals negative 3.5 and x equals 0 and between x equals 3 and x equals 5. The steepness of the line segments decreases on moving from x equals negative 5 to x equals 3.5, deceases on moving from x equals negative 3.5 to x equals 0, increases on moving from x equals 0 to x equals 3, and increases on moving from x equals 3 to x equals 5. In graph 4, the line segments representing the slope field are inclined up to the right in the first and fourth quadrants and are inclined down to the right in the second and third quadrants. In each quadrant, the steepness of the line segments decreases with decrease in distance from the y axis in the third and fourth quadrants and increases with increase in distance from the y axis in the first and fourth quadrants.

A) No matter what the starting point, as x → ∞, y approaches the value of 4.

B) No matter what the starting point, as x → ∞, y oscillates within a certain finite range.

C) Depending on the starting point, as x → ∞, y approaches ±∞.

D) Depending on the starting point, as x → ∞, y approaches ∞ or 0.

Diff: 2 Var: 1

Section: 9.3

Learning Objectives: Use a slope field to give a graphical solution of a differential equation.

7) If the slope field for (dy/dx) has constant slopes where x is fixed, what do we know about (dy/dx)?

A) (dy/dx) depends only on y.

B) (dy/dx) depends only on x.

C) (dy/dx) must be a constant.

D) We can't determine anything about (dy/dx).

Diff: 2 Var: 1

Section: 9.3

Learning Objectives: Understand the visualization of a differential equation with a slope field.

8) Which of the following equations corresponds with the slope field shown below?

I. (dy/dx) = ((x - y)) with superscript (2)

II. (dy/dx) = ((x + y)) with superscript (2)

III. (dy/dx) = (x) with superscript (2) - (y) with superscript (2)

IV. None of them

A graph plots a slope field in a coordinate plane. A diagonal region is inclined up to the right from the third quadrant to the first quadrant. Along the diagonal region, the slope is horizontal. Two other diagonal regions above and below the previous diagonal region are inclined up to the right from the third quadrant to the first quadrant, along which the slope is positive. The slope is inclined upward to the right, near the diagonal region, and become vertical on moving away from these diagonal regions.

Diff: 2 Var: 1

Section: 9.3

Learning Objectives: Use a slope field to give a graphical solution of a differential equation.

9) Sketch a slope field for the differential equation (dy/dx) = (-x/y) using the points indicated on the axes.

21 points are plotted in an x y coordinate system. Both the axes range from negative 2 to 2, in increments of 1. The points, representing series 1, are plotted at (negative 2, 2), (negative 2, 1), (negative 2, 0), (negative 2, negative 1), (negative 2, negative 2), (negative 1, negative 2), (negative 1, 0), (negative 1, 2), (0, negative 1), (0, negative 2), (0, 1), (0, 2), (1, 2), (1, 1), (1, negative 1), (1, negative 2), (2, 2), (2, 1), (2, 0), (2, negative 1), and (1, negative 2).

Diff: 1 Var: 1

Section: 9.3

Learning Objectives: Use a slope field to give a graphical solution of a differential equation.

10) On the slope field for the differential equation (dy/dx) = xy, sketch the solution curve in the fourth quadrant that goes through the point (0, -1).

A graph plots a slope field in an x y coordinate system. The axes range from negative 5 to 5, in increments of 1. The line segments representing the slope field are horizontal at x equals 0. In the first and second quadrants, the slope field is positive and increasing, and negative and increasing, respectively. In the third and fourth quadrants, the slope field is positive and increasing, and negative and increasing, respectively. In the upper quadrants, the fields appear as u-shaped parallel. In the lower quadrants, the fields appear as inverted u-shaped parallel. All values are estimated.

A graph plots a slope field in an x y coordinate system. The axes range from negative 5 to 5, in increments of 1. The line segments representing the slope field are horizontal at x equals 0. In the first and second quadrants, the slope field is positive and increasing, and negative and increasing, respectively. In the third and fourth quadrants, the slope field is positive and increasing, and negative and increasing, respectively. In the upper quadrants, the fields appear as u-shaped parallel. In the lower quadrants, the fields appear as inverted u-shaped parallel. A red x-mark is plotted at about (0, negative 1.1). A concave down curve starts at the x-mark, and decreases through (1.5, negative 2.5). All values are estimated.

Diff: 1 Var: 1

Section: 9.3

Learning Objectives: Use a slope field to give a graphical solution of a differential equation.

11) The slope fields for y' = x - y and y' = y are shown in the following figure. Which slope field goes with the differential equation y' = y?

Two graphs, 1 and 2, plot a slope field on an x y coordinate plane. Both the axes range from negative 5 to 5, in increments of 5. In graph 1, a diagonal region is inclined up to the right from the third quadrant to the first quadrant through the origin. The line segments representing slope field in this region are horizontal. The line segments below the region are inclined up to the right, and the line segments above the region are inclined down to the right. Above the diagonal region, the steepness of the line segments decreases with decrease in distance from the y axis and decreases with increase in distance from the x axis. Below the diagonal region, the steepness of the line segments increases with increase in distance from the y axis, increases with increase in distance from the x axis in the third and fourth quadrants, and decreases with increase in distance from the x axis in the first quadrant. In graph 2, the line segments representing the slope field are inclined up to the right in the first and second quadrants and are inclined down to the right in the third and fourth quadrants. In each quadrant, the steepness of the line segments increases with increase in distance from the x axis.

Diff: 1 Var: 1

Section: 9.3

Learning Objectives: Understand the visualization of a differential equation with a slope field.

12) The following slope field has

A graph plots a slope field in an x y coordinate plane. Both the axes range from negative 5 to 5, in increments of 5. A horizontal region is extended from negative x axis to the positive x axis. Above the x axis, the slope is positive and increases as y axis value increases. Below the x axis, the slope is negative and decreases as y axis value decreases.

A) an unstable equilibrium solution.

B) a stable equilibrium solution.

C) no equilibrium solution.

Diff: 1 Var: 1

Section: 9.3

Learning Objectives: Use a slope field to give a graphical solution of a differential equation.

9.4 Exponential Growth and Decay

1) Find the particular solution to the differential equation (dP/dt) = 17P when P(0) = 15.

A) P(t) = 15(e) with superscript (17t) B) P(t) = 17(e) with superscript ((1/15)t)

C) P(t) = 15(e) with superscript ((1/17)t) D) P(t) = 17(e) with superscript (15t)

Diff: 1 Var: 1

Section: 9.4

Learning Objectives: Solve the differential equation dy/dt = ky.

2) Find a solution to the differential equation (dQ/dt) = - (Q/5) subject to the initial condition Q = 50 when t = 0.

Diff: 2 Var: 1

Section: 9.4

Learning Objectives: Solve the differential equation dy/dt = ky.

3) The solution to the differential equation (dy/dx) - (y/4) = 0 subject to the initial condition y(1) = 24 is y = (Ce) with superscript (kx), where k = ________ and C = ________. Round answers to 2 decimal places.

Part A: 0.25

Part B: 18.69

Diff: 1 Var: 1

Section: 9.4

Learning Objectives: Solve the differential equation dy/dt = ky.

4) The solution to the differential equation (dy/dx) = -8y subject to the initial condition that y = 80 when x = 0 is y = (Ce) with superscript (kx), where k = ________ and C = ________.

Part A: -8

Part B: 80

Diff: 1 Var: 1

Section: 9.4

Learning Objectives: Solve the differential equation dy/dt = ky.

5) What is the solution to the differential equation (dP/dt) = 0.15P, given that P = 25 when t = 0?

Diff: 1 Var: 1

Section: 9.4

Learning Objectives: Solve the differential equation dy/dt = ky.

6) A radioactive isotope decays at a continuous rate of approximately 20% per day. If A is the amount of the isotope and t is time in days, what is the differential equation for this situation and its general solution?

A) (dA/dt) = -0.2A; A = (Ce) with superscript (-0.2t) B) (dA/dt) = -0.8A; A = (Ce) with superscript (-0.8t)

C) (dA/dt) = 0.2A; A = (Ce) with superscript (0.2t) D) (dA/dt) = 0.8A; A = (Ce) with superscript (0.8t)

Diff: 1 Var: 1

Section: 9.4

Learning Objectives: Use the differential equation dy/dt = ky to model exponential growth and decay.

7) Money in a bank account grows continuously at an annual rate of 6%. Suppose $10,000 is put into an account at time t = 0. If B is the balance in the account after t years, what is the differential equation for B and its solution?

A) (dB/dt) = 1.06B; B = 10,000(e) with superscript (1.06t) B) (dB/dt) = 0.06B; B = 10,000(e) with superscript (0.06t)

C) (dB/dt) = -0.06B; B = 10,000(e) with superscript (-0.06t) D) (dB/dt) = 6B; B = 10,000(e) with superscript (6t)

Diff: 1 Var: 1

Section: 9.4

Learning Objectives: Use the differential equation dy/dt = ky to model exponential growth and decay.

8) An anti-inflammatory drug has a half-life in the human body of about 8 hours.

A. Use the half-life to find the value of k in the differential equation (dQ/dt) = -kQ, where Q is the quantity of the drug in the body t hours after the drug is administered. Round to 4 decimal places.

B. After how many hours will 35% of the original dose remain in the body? Round to 2 decimal places.

A. 0.0866

B. 12.12

Diff: 2 Var: 1

Section: 9.4

Learning Objectives: Use the differential equation dy/dt = ky to model exponential growth and decay.

9) The amount of medicine present in the blood of a patient decreases due to metabolism according to the exponential decay model. One hour after a dose was given, there were 3.7 nanograms/(cm) with superscript (3) present, and a hour later there were 2.5 ng/(cm) with superscript (3). After how many hours will there be less than 0.7 ng/(cm) with superscript (3) present, assuming no more medication is taken? Round to 1 decimal place.

Diff: 2 Var: 1

Section: 9.4

Learning Objectives: Use the differential equation dy/dt = ky to model exponential growth and decay.

10) For the first week, the spread of a rumor is proportional to the number of people who have heard the rumor. Find the particular solution to the differential equation for N, the number of people who have heard the rumor as a function of time in days, t, if 15 people have heard it at time t = 0, and 220 people have heard it at time t = 5.

A) N = 15(e) with superscript (0.8055t) B) N = 5(e) with superscript (0.8055t)

C) N = 15(e) with superscript (0.537t) D) N = 5(e) with superscript (0.537t)

Diff: 3 Var: 1

Section: 9.4

Learning Objectives: Use the differential equation dy/dt = ky to model exponential growth and decay.

11) A country experiences a continuous inflation rate of about 5.1% per year. If a t-shirt had a value of $11 in 2015, write a differential equation and use it to find what the t-shirt's value was in 2025. Round to the nearest cent.

Diff: 1 Var: 1

Section: 9.4

Learning Objectives: Use the differential equation dy/dt = ky to model exponential growth and decay.

12) If no more pollutants are dumped into a lake, the amount of pollution in the lake will decrease at a rate proportional to the amount of pollution present. If there are 400 units of pollution present initially and 184 units left after 8 years, use differential equations to find the number of units left after 9 years. Round to 1 decimal place.

Diff: 2 Var: 1

Section: 9.4

Learning Objectives: Use the differential equation dy/dt = ky to model exponential growth and decay.

13) On January 1, 1879, records show that 500 of a fish called Atlantic striped bass were introduced into the San Francisco Bay. In 1899, the first year fishing for bass was allowed, 100,000 of these bass were caught, representing 10% of the population at the start of 1899. Owing to reproduction, at any moment in time the bass population is growing at a rate proportional to the population at that moment. Write a differential equation satisfied by B(t), the number of Atlantic striped bass a time t, where t is in years since January 1, 1879 and 0 ≤ t < 20 and solve it for B(t).

A) B(t) = 500(e) with superscript (7.6t) B) B(t) = 500(e) with superscript (0.38t)

C) B(t) = 500(1 - (e) with superscript (0.38t)) D) B(t) = 500(20 - (e) with superscript (7.6t))

Diff: 2 Var: 1

Section: 9.4

Learning Objectives: Use the differential equation dy/dt = ky to model exponential growth and decay.

14) What is the general solution to the differential equation (dy/dt) = ky?

A) y = (Ce) with superscript (kt)

B) y = (Ce) with superscript (k/t)

C) y = (ke) with superscript (Ct)

D) y = (ke) with superscript (C/t)

E) none of the above

Diff: 1 Var: 1

Section: 9.4

Learning Objectives: Solve the differential equation dy/dt = ky.

15) You invest $4500 in your nephew's catering business. He guarantees you a minimum return at a continuous interest rate of 3%. Of course, if the business continues to thrive, you will earn at a higher rate.

a) Write a differential equation for the minimum amount, B, of your return on investment at time t.

b) Solve the differential equation.

c) Graph the solution.

a) (dB/dt) = 0.03B

b) B = 4500(e) with superscript (0.03t)

c)

A graph plots a curve on a coordinate plane. The horizontal axis is labeled t in years, and the vertical axis is labeled B in dollars. The curve increases concave up from the second quadrant to the first quadrant through a point, labeled Initial investment, on the positive vertical axis.

Diff: 3 Var: 1

Section: 9.4

Learning Objectives: Use the differential equation dy/dt = ky to model exponential growth and decay.

16) Find the solution to the differential equation (dQ/dt) = - (P/4), subject to the initial condition P(0) = 8.

A) P = 8(e) with superscript (- (1/32)t) B) P = 32(e) with superscript (- (1/4)t) C) P = 8(e) with superscript (- (1/4)t) D) P = 4(e) with superscript (- (1/8)t)

Diff: 2 Var: 1

Section: 9.4

Learning Objectives: Solve the differential equation dy/dt = ky.

9.5 Applications and Modeling

1) A certain bank account earns interest at the rate of 4% compounded continuously. Money is being withdrawn from the account in a continuous stream at a constant rate of $100,000 per year. Write a differential equation modeling how the balance B changes over time. Which of the following is the general solution, given an initial balance of (B) with subscript (0)?

A) B = ((B) with subscript (0) - 2,500,000)(e) with superscript (0.04t) + 2,500,000

B) B = ((B) with subscript (0) - 100,000)(e) with superscript (0.04t) + 100,000

C) B = (B) with subscript (0)(e) with superscript (0.04t) + 2,500,000

D) B = (B) with subscript (0)(e) with superscript (0.04t) + 100,000

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

2) A certain bank account earns interest at the rate of 4% compounded continuously. Money is being withdrawn from the account in a continuous stream at a constant rate of $100,000 per year. Use differential equations to determine what the minimum initial balance should be in order for the account never to be depleted.

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

3) What is the general solution of (dy/dx) = 100 - y?

Diff: 1 Var: 1

Section: 9.5

Learning Objectives: Solve the differential equation dy/dt = k(y-A)

4) The equilibrium solution for (dP/dt) = 0.5P - 50 is P = ________. This solution is ________ (stable/unstable).

Part B: unstable

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Find an equilibrium solution of a differential equation and determine whether it is stable or unstable.

5) The equilibrium solution for (dP/dt) = -0.8P + 144 is P = ________. This solution is ________ (stable/unstable).

Part B: stable

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Find an equilibrium solution of a differential equation and determine whether it is stable or unstable.

6) A person receives a drug intravenously at the rate of 3 mg per hour. The drug is eliminated from the body at a rate proportional to the amount present with a constant of proportionality of k = -0.4. What is the long term amount of the drug in the body, once the system has stabilized?

Diff: 1 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

7) A company earns a continuous annual rate of 11% of its net worth. At the same time, it has expenses of 7.1 million dollars per year. Write a differential equation for the company's worth, W, in millions of dollars as a function of time t, in years. What is the general solution to your differential equation?

A) W = 7.1 + (Ce) with superscript (0.11t) B) W = 64.5 + (Ce) with superscript (0.11t)

C) W = 71 + (Ce) with superscript (0.11t) D) W = 78.1 + (Ce) with superscript (0.11t)

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

8) A company earns a continuous annual rate of 11% of its net worth. At the same time, it has expenses of 7 million dollars per year. If the company's net worth at time t = 0 is 50 million, how many years will it take to go bankrupt? Round to the nearest year.

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

9) According to Newton, the rate at which the temperature of water in a swimming pool changes is directly proportional to the difference between the temperature L outside and the temperature T of the water in the pool. Suppose the temperature outside stays at a constant 24°C for two hours, and that during that time the temperature of the water increases from 20°C to 22°C. Which of the following shows the differential equation for this situation and its solution?

A) dT / dt = k(T - L); T(t) = 24 - 4(e) with superscript (-0.55t)

B) dT / dt = k(L - T); T(t) = 24 - 4(e) with superscript (-0.35t)

C) dT / dt = k(T - L); T(t) = 24 - 2(e) with superscript (-0.35t)

D) dT / dt = k(L - T); T(t) = 24 - 2(e) with superscript (-0.55t)

Diff: 3 Var: 1

Section: 9.5

Learning Objectives: Use Newton's Law of Heating and Cooling to write and solve a differential equation.

10) According to Newton, the rate at which the temperature of water in a swimming pool changes is directly proportional to the difference between the temperature L outside and the temperature T of the water in the pool. Suppose the temperature outside stays at a constant 26°C for two hours, and that during that time the temperature of the water increases from 20°C to 23°C. What is the equilibrium solution to the equation modeling this situation?

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Find an equilibrium solution of a differential equation and determine whether it is stable or unstable.; Use Newton's Law of Heating and Cooling to write and solve a differential equation.

11) A lake with constant volume V, in km3, contains a quantity of Q (km) with superscript (3) pollutant. Clean water enters the lake and causes a total outflow of r (km) with superscript (3) per year. The rate at which the pollutant decreases at any time t equals the product of the pollutant Q per volume V and the rate at which the water flows out of the lake. If V = 12 × (10) with superscript (3) (km) with superscript (3) and r = 70 (km) with superscript (3) per year, how many years will it take for the pollutant to decrease to half of its original quantity? Round to the nearest whole year.

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

12) What is the solution of the differential equation (dθ/dt) = 2θ - 4 if θ = 7 when t = 2?

A) θ(t) = 7(e) with superscript (2t) + 2 B) θ(t) = 7(e) with superscript (2t) + 4

C) θ(t) = (5/(e) with superscript (4))(e) with superscript (2t) + 2 D) θ(t) = (7/(e) with superscript (4))(e) with superscript (2t) + 4

Diff: 1 Var: 1

Section: 9.5

Learning Objectives: Solve the differential equation dy/dt = k(y-A)

13) The logistic model for a population's growth is(dP/dt) = 16P + 4(P) with superscript (2), where P is the size of the population in millions at any time t, measured in years. What is the size of the population when the rate of increase starts to decrease?

Diff: 1 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

14) Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between the temperature of the object and the temperature of the surrounding air. A detective discovers a corpse in an abandoned building, and finds its temperature to be 26°C. An hour later its temperature is 19°C. Assume that the air temperature is 9°C, that normal body temperature is 37°C, and that Newton's Law of Cooling applies to the corpse. How many hours has the corpse been dead at the moment it is discovered? Round to 2 decimal places.

Diff: 3 Var: 1

Section: 9.5

Learning Objectives: Use Newton's Law of Heating and Cooling to write and solve a differential equation.

15) On January 1, 1879, records show that 500 of a fish called Atlantic striped bass were introduced into the San Francisco Bay. In 1899, the first year fishing for bass was allowed, 100,000 of these bass were caught, representing 10% of the population at the start of 1899. Owing to reproduction, at any moment in time the bass population is growing at a rate proportional to the population at that moment. Assume that when fishing starts in 1899, the rate at which bass are caught is proportional to the square of the population with constant of proportionality (10) with superscript (-7). Write a differential equation satisfied by B(t), for t > 20.

A) (dB/dt) = 7.6B - (10) with superscript (-7)(B) with superscript (2) B) (dB/dt) = 7.6B - ((10) with superscript (-7)/(B) with superscript (2))

C) (dB/dt) = 0.38B - (10) with superscript (-7)(B) with superscript (2) D) (dB/dt) = 0.38B - ((10) with superscript (-7)/(B) with superscript (2))

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

16) On January 1, 1879, records show that 500 of a fish called Atlantic striped bass were introduced into the San Francisco Bay. In 1899, the first year fishing for bass was allowed, 100,000 of these bass were caught, representing 10% of the population at the start of 1899. Owing to reproduction, at any time the bass population is growing at a rate proportional to the population at that moment. Assume that when fishing starts in 1899, the rate at which bass are caught is proportional to the square of the population with constant of proportionality (10) with superscript (-8). What happens to the bass population in the long run?

A) It approaches 0. B) It grows without bound.

C) It approaches 7.6 ∙ (10) with superscript (7). D) It approaches 3.8 ∙ (10) with superscript (7).

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

17) A manufacturer of a chocolate beverage mixes liquid chocolate with milk in a large vat containing 350 liters of the mixture that is 20% chocolate initially. The differential equation for A, the amount of chocolate in the vat at time t, follows a "rate in - rate out" model. When chocolate flows in at a rate of 2 liters/min and the mixed beverage flows out at the same rate, you can model the situation with the differential equation (dA/dt) = 2 - (A/175) liters/min which can be rewritten as (dA/dt) = (-1/175)(A - 350).

a) What is the particular solution to this differential equation?

b) In the long run, how many liters of chocolate will be in the mixture contained in the vat?

a) A = 350 - 280(x) with superscript (-(1/175)t)

b) 350 liters

Diff: 3 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

18) In a study of the milk drinking habits of a certain population of children, it was found that children drank more and more slowly as they finished a 16 ounce container of milk. Suppose that the rate at which a child drinks is equal to the percentage left to drink, i.e. (dA/dt) = 100 - A. Find the particular solution given the initial condition A(0) = 0.

A) A = 100 - 100(e) with superscript (-t) B) A = 100 - (Ce) with superscript (-t)

C) A = 100(e) with superscript (-t) - 100 D) A = (Ce) with superscript (-t) - 100

Diff: 1 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

19) Pollutants are being dumped into a lake at a rate of 10 (m) with superscript (3) per day. About 16% of the lake's water leaves the lake each day and fresh, unpolluted water flows in to replace it. The differential equation for the amount of pollutant, Q, in the lake as a function of time, t, in days is (dQ/dt) = ________.

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

20) The general solution for the differential equation (dy/dx) = 0.5y = 2 is

A) y = 2 + (Ce) with superscript (0.5x) B) y = 4 + (Ce) with superscript (0.5x)

C) y = 0.5 + (Ce) with superscript (2x) D) y = 0.5 + (Ce) with superscript (4x)

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Solve the differential equation dy/dt = k(y-A)

21) Money in a bank account earns interest at a continuous rate of 6% per year, and payments are made continuously out of the account at the rate of $9000 per year. The account initially contains $100,000. Write a differential equation for the balance, B, in the account in t years and use it to find how many years it will take for the account to run out of money. Round to 1 decimal place.

Diff: 2 Var: 1

Section: 9.5

Learning Objectives: Use the differential equation dy/dt =k(y-A) to model a real world situation.

22) The body of a murder victim is found at 9:00 in the morning in a 70°F room. The temperature of the body when it is found is 86°F, and one hour later it is 80°F. If the victim had a normal temperature of 98.6°F when he died, how many hours had the victim been dead when the body was found? Round your answer to one decimal place.

Diff: 3 Var: 1

Section: 9.5

Learning Objectives: Use Newton's Law of Heating and Cooling to write and solve a differential equation.

9.6 Modeling the Interaction of Two Populations

1) Two species of insects coexist with each other. Both would do fine on their own. Species x does not do well in the presence of species y. Species y does not do well in the presence of species x. Which of the following systems of equations would best model this scenario?

A) table ( ((dx/dt) = y - xy__ (dy/dt) = x - xy) ) B) table ( ((dx/dt) = y + xy_ (dy/dt) = x + xy) )

C) table ( ((dx/dt) = x - xy__ (dy/dt) = y - xy) ) D) table ( ((dx/dt) = x + xy_ (dy/dt) = y + xy) )

Diff: 1 Var: 1

Section: 9.6

Learning Objectives: Interpret or write differential equations that model various types of species interactions.

2) Two movie theaters are across the street from each other. Each is doing well, but each would do better if the other were not there. Call the net worth of one theater x and the net worth of the other theater y. Which system of differential equations best models this scenario?

A) table ( ((dx/dt) = y + xy__ (dy/dt) = x + xy) ) B) table ( ((dx/dt) = y - xy___ (dy/dt) = x - xy) )

C) table ( ((dx/dt) = x + xy__ (dy/dt) = y + xy) ) D) table ( ((dx/dt) = x - xy___ (dy/dt) = y - xy) )

Diff: 2 Var: 1

Section: 9.6

Learning Objectives: Interpret or write differential equations that model various types of species interactions.

3) Trout are introduced into a stream. Trout is a predator species and therefore has an influence on the population size of other fish. The following figure shows how the trout and other fish populations vary over time. The progress of time is shown by the direction of the arrow. What happens to the size of the trout population at point P?

A graph plots a spiral curve in the first quadrant of a coordinate plane. The horizontal axis is labeled other, and the vertical axis is labeled trout. The spiral curve spreads out from a point labeled P and forms three loops on moving away from point P. An arrow, opposite to the curve direction, is marked along the curve.

A) It will be stable. B) It will be at a maximum.

C) It will be at a minimum. D) Nothing can be determined.

Diff: 1 Var: 1

Section: 9.6

Learning Objectives: Understand the predator-prey model.

4) Trout are introduced into a stream. Trout is a predator species and therefore has an influence on the population size of other fish. The first figure shows how the trout and other fish populations vary over time. The progress of time is shown by the direction of the arrow. Does the second figure accurately show how the two different populations vary over time?

Figure one: A graph plots a spiral curve in the first quadrant of a coordinate plane. The horizontal axis is labeled other, and the vertical axis is labeled trout. The spiral curve spreads out from a point labeled P and forms three loops on moving away from point P. An arrow, opposite to the curve direction, is marked along the curve. Figure two: "A graph plots two curves in the first quadrant of a coordinate plane. The horizontal axis is labeled t, and the vertical axis is labeled population. The first curve labeled trout starts at a point on the positive vertical axis and oscillates with decreasing amplitude. The second curve labeled other fish starts at a point above the starting point of the first curve on the positive vertical axis and oscillates with decreasing amplitude. The maxima of one curve is in vertical to the minima of another curve."

Diff: 2 Var: 1

Section: 9.6

Learning Objectives: Understand the predator-prey model.

5) Let f be the number of fruit tree blossoms (in ten thousands) and let b be the number of bees (in hundreds) in an orchard. Suppose f and b satisfy the differential equations

(df /dt) = - f + f b and (db/dt) = -b + f b,

which correspond to the slope field in the figure. Assume f = 2 and b = 1 when t = 0. What happens to the number of fruit tree blossoms and bees over time?

A graph plots a slope field in the first quadrant of an x y coordinate plane. The horizontal axis labeled b, ranges from 0 to 3, in increments of 1. The vertical axis labeled f, ranges from 0 to 3, in increments of 1. A diagonal region is inclined up to the right from the origin through the first quadrant. The line segments representing slope field in this region are inclined up to the right. The line segments below the diagonal region and below y equals 1 are inclined up to the right until x equals 1 with decreasing steepness and then inclined down to the right with increasing steepness. The line segments above the diagonal region and to the left of x equals 1 are inclined up to the right until y equals 1 with decreasing steepness and then inclined down to the right with decreasing steepness. The line segments between x equals 1 and y equals 1 and to the right of (1, 1) are inclined up to the right, with steepness increasing from x equals 1 toward the diagonal region and decreasing from y equals 1 toward the diagonal region.

A) The solution approaches the origin.

B) The solution approaches the line f = b.

C) f → ∞, but b does not.

D) b → ∞, but f does not.

Diff: 2 Var: 1

Section: 9.6

Learning Objectives: Understand the predator-prey model.

6) Let f be the number of fruit tree blossoms (in ten thousands) and let b be the number of bees (in hundreds) in an orchard. Suppose f and b satisfy the differential equations

(df /dt) = - f + f b and (db/dt) = -b + f b.

Assume f = 3 and b = 1 when t = 0. Use the differential equations to calculate df / dt and db / dt when t = 0, and use these to estimate:

A. the number of fruit tree blossoms when t = 1.

B. the number of bees when t = 1.

A. 30,000

B. 300

Diff: 2 Var: 1

Section: 9.6

Learning Objectives: Understand the predator-prey model.

7) Bees and flowers help each other, and each needs the other in order to survive. Which (if any) of the following systems of differential equations could model the interaction between bees and flowers, with either species being x or y?

A) (dx/dt) = -0.6x + 0.07xy, (dy/dt) = 0.2y - 0.04xy

B) (dx/dt) = 0.28x, (dy/dt) = -0.7y + 0.14xy

C) (dx/dt) = -0.5x + 0.31xy, (dy/dt) = -0.1y + 0.25xy

D) None of these

Diff: 2 Var: 1

Section: 9.6

Learning Objectives: Interpret or write differential equations that model various types of species interactions.

8) Owls need trees to survive, but trees don't care one way or the other about owls. Which (if any) of the following systems of differential equations could model the interaction between owls and trees, with trees as x and owls as y?

A) (dx/dt) = -0.3x + 0.05xy, (dy/dt) = 0.2y - 0.04xy

B) (dx/dt) = 0.33x, (dy/dt) = -0.4y + 0.14xy

C) (dx/dt) = -0.2x + 0.36xy, (dy/dt) = -0.1y + 0.3xy

D) None of these

Diff: 3 Var: 1

Section: 9.6

Learning Objectives: Interpret or write differential equations that model various types of species interactions.

9) Elk and buffalo are in competition with each other. Each would do fine without the other. Which (if any) of the following systems of differential equations could model the interaction between elk and buffalo, with either species being x or y?

A) (dx/dt) = -0.2x + 0.03xy, (dy/dt) = 0.5y - 0.07xy

B) (dx/dt) = 0.19x, (dy/dt) = -0.3y + 0.17xy

C) (dx/dt) = -0.1x + 0.22xy, (dy/dt) = -0.4y + 0.16xy

D) None of these

Diff: 2 Var: 1

Section: 9.6

Learning Objectives: Interpret or write differential equations that model various types of species interactions.

10) The fox eats the hare, so it needs it to survive. The hare would do fine without the fox. Which (if any) of the following systems of differential equations model the interaction between the fox and the hare, with the fox as x and the hare as y? Select all that apply.

A) (dx/dt) = -0.6x + 0.04xy, (dy/dt) = 0.6y - 0.03xy

B) (dx/dt) = 0.32x, (dy/dt) = -0.7y + 0.13xy

C) (dx/dt) = -0.5x + 0.35xy, (dy/dt) = -0.5y + 0.29xy

D) None of these

Diff: 2 Var: 1

Section: 9.6

Learning Objectives: Interpret or write differential equations that model various types of species interactions.

11) Two minor league baseball teams in the same city compete with each other for fan attendance. Both teams would do well in the absence of the other one, but each team hurts the other team's attendance at games. Create a system of differential equations to model this situation.

A) dx / dt = x – xy B) dx / dt = -x + xy

dy / dt = y - xy dy / dt = -y + xy

C) dx / dt = x – xy D) dx / dt = x

dy / dt = -y + xy dy / dt = y - xy

Diff: 2 Var: 1

Section: 9.6

Learning Objectives: Interpret or write differential equations that model various types of species interactions.

12) Solve the differential equation using separation of variables. (dy/dx) = xy + 3y + 5x + 15

A) y = ((x) with superscript (2)y/2) + (3(y) with superscript (2)/2) + (5(x) with superscript (2)/2) + 15x + C B) y = x(y) with superscript (2) - 5(x) with superscript (2) + C

C) x = 5, y = 0 D) y = (Ce) with superscript (((x) with superscript (2)/2)+3x) - 5

Diff: 2 Var: 1

Section: 9.6

Learning Objectives: Solve a differential equation by separation of variables.

9.7 Modeling the Spread of a Disease

1) Consider three strains of the flu modeled by the following sets of differential equations. Which has the infecteds being removed the slowest?

table ( (I. (dS/dt) = -0.01SI_ _(dI/dt) = 0.01SI - 0.3I_)(II. (dS/dt) = -0.02SI_ _(dI/dt) = 0.02SI - 0.5I_)(III. (dS/dt) = -0.03SI_ _(dI/dt) = 0.03SI - 0.4I_) )

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

2) At time t = 0, there are 200 students at a school, 3 of whom have the flu. Given the differential equation (dI/dt) = 0.0026SI - 0.5I, will the flu spread?

Diff: 1 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

3) At time t = 0, there are 400 students at a school, 3 of whom have the flu, and 200 of the students have been vaccinated against the flu. Given the differential equation (dI/dt) = 0.0026SI - 0.5I, will the flu spread?

Diff: 1 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

4) For a new strain of the flu, the differential equations are:

(dS/dt) = -0.0038SI and (dI/dt) = 0.0038SI - 0.8I.

What is (dI/dS)?

A) (210.5/S) - 1 B) (0.0048/S) - 1 C) (210.5/S) + 1 D) (0.0048/S) + 1

Diff: 1 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

5) For a new strain of the flu, the differential equations are:

(dS/dt) = -0.0038SI and (dI/dt) = 0.0038SI - 0.8I.

What is the threshold value for this strain of the flu? Round down to the nearest whole number.

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

6) Consider three strains of the flu modeled by the following sets of differential equations. Which is the most infectious?

table ( (I. (dS/dt) = -0.01SI_ _(dI/dt) = 0.01SI - 0.3I_)(II. (dS/dt) = -0.02SI_ _(dI/dt) = 0.02SI - 0.5I_)(III. (dS/dt) = -0.03SI_ _(dI/dt) = 0.03SI - 0.4I_) )

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

7) What is the threshold value for the strain of the flu modeled by the differential equations

table ( ((dS/dt) = -0.03SI _(dI/dt) = 0.03SI - 0.2I?) )

Round to the nearest whole number.

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

8) For the strain of the flu modeled by the differential equations

table ( ((dS/dt) = -0.01SI _(dI/dt) = 0.01SI - 0.3I,) )

does the disease spread if initially (S) with subscript (0) = 28?

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

9) In a boarding school of 1000 students, at least ________ students should be vaccinated against a flu strain satisfying the differential equations

table ( ((dS/dt) = -0.0025SI _(dI/dt) = 0.0025SI - 0.3I.) )

Diff: 1 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

10) For S and I satisfying the differential equations

table ( ((dS/dt) = -aSI _(dI/dt) = aSI - bI,) )

is I increasing or decreasing when S < (a/b)?

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

11) The following figure gives the slope field for dI / dS for an SIR epidemic model. Estimate the threshold value if the number of susceptibles is initially 300.

A graph plots a slope field in the first quadrant of an x y coordinate plane. The horizontal axis labeled S susceptibles, ranges from 0 to 400, in increments of 100. The vertical axis labeled I infecteds, ranges from 0 to 100, in increments of 100. The line segments representing slope fields are inclined up to the right between x equals 0 and x equals 40, with its steepness decreasing from left to right. The line segments are almost horizontal at x equals 150. The line segments are inclined down to the right after x equals 160, with its steepness increases from left to right.

A) About 70 B) About 150 C) About 200 D) About 40

Diff: 1 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

12) A fatal infectious disease is introduced into a growing population. Let S denote the number of susceptible people at time t and let I denote the number of infected people at time t. Suppose that, in the absence of the disease, the susceptible population grows at a rate proportional to itself, with constant of proportionality 0.4. People in the infected group die at a rate proportional to the infected population with constant of proportionality 0.27. The rate at which people get infected is proportional to the product of the number of susceptibles and the number of infecteds, with constant of proportionality 0.001. Which of the following systems of differential equations are satisfied by S and I?

A) (dS/dt) = 0.4SI - 0.27I, (dI/dt) = 0.4SI - 0.27I

B) (dS/dt) = 0.001SI - 0.4I, (dI/dt) = 0.27SI - 0.001I

C) (dS/dt) = 0.4SI - 0.001I, (dI/dt) = 0.001SI - 0.27I

D) (dS/dt) = 0.27SI - 0.001I, (dI/dt) = 0.001SI - 0.4I

Diff: 1 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

13) At time t = 0, there are 500 students in a school, 5 of whom have the flu. No one else has been exposed yet. Using the SIR model, (I) with subscript (0) = ________ and (S) with subscript (0) = ________.

Part A: 5

Part B: 495

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

14) At time t = 0, there are 700 students in a school, 5 of whom have the flu. No one else has been exposed yet. Using the SIR model and the differential equation (dI/dt) = 0.0014SI - 0.6I, will the flu spread?

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Understand the S-I-R model for the spread of a disease.

15) Mark all of the differential equations that are NOT separable.

A) (dy/dt) = 3t + 6y B) (dy/dx) = 6x + 4(y) with superscript (3)

C) (dy/dx) = 4(e) with superscript (xy) D) (dy/dx) = 6(e) with superscript (x+y)

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Solve a differential equation by separation of variables.

16) Find the general solution of the separable differential equation: (dy/dt) = 9yt.

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Solve a differential equation by separation of variables.

17) Solve the initial value problem using separation of variables, and then graph the solution.

(dy/dx) = (-x/y), y(0) = square root of (8)

A circle of radius 2.8 is graphed on an x y coordinate plane. Both the axes range from negative 4 to 4, in increments of 1. The circle is centered at the origin.

Diff: 2 Var: 1

Section: 9.7

Learning Objectives: Solve a differential equation by separation of variables.

© (2022) John Wiley & Sons, Inc. All rights reserved. Instructors who are authorized users of this course are permitted to download these materials and use them in connection with the course. Except as permitted herein or by law, no part of these materials should be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise.

Document Information

Document Type:
DOCX
Chapter Number:
9
Created Date:
Aug 21, 2025
Chapter Name:
Chapter 9 Mathematical Modeling Using Differential Equations
Author:
Hughes Hallett

Connected Book

Final Test Bank | Applied Calculus 7e

By Hughes Hallett

Test Bank General
View Product →

$24.99

100% satisfaction guarantee

Buy Full Test Bank

Quick Navigation

Preview Document Info Connected Book Recommendations

Benefits

Immediately available after payment
Answers are available after payment
ZIP file includes all related files
Files are in Word format (DOCX)
Check the description to see the contents of each ZIP file
We do not share your information with any third party