Stewart – Calculus ET 8e Chapter 9 Form F
10. A sum of is invested at interest. If is the amount of the investment at time t for
the case of continuous compounding, write a differential equation and an initial condition satisfied
by .
11. Consider a population with constant relative birth and death rates and , respectively,
and a constant emigration rate m, where , and . Then the rate of change of
the population at time t is modeled by the differential equation
where
.
Find the solution of this equation with the rate of change of the population at time that
satisfies the initial condition .
12. Solve the initial-value problem.
____ 13. Solve the differential equation. Select the correct answer.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 9 Form F
14. Solve the initial-value problem.
____ 15. Determine whether the differential equation is linear. Select the correct answer.
a. the equation is linear
b. the equation is not linear
16. In the circuit shown in Figure, a generator supplies a voltage of volts, the
inductance is 2 H, the resistance is 40 , and . Find the current 0.2 s after the switch is
closed. Round your answer to two decimal places.
17. Solve the differential equation.
18. Solve the differential equation.
19. Let be the performance level of someone learning a skill as a function of the training time t.
The graph of P is called a learning curve. We propose the differential equation
as a reasonable model for learning, where r is a positive constant. Solve it as a linear differential
equation.
Stewart – Calculus ET 8e Chapter 9 Form F
____ 20. A phase trajectory is shown for populations of rabbits and foxes . Describe how each
population changes as time goes by.
Select the correct statement.
a. At the number of rabbits rebounds to 500.
b. At the number of foxes reaches a maximum of about 2400.
c. At the population of foxes reaches a minimum of about 30.
Stewart – Calculus ET 8e Chapter 9 Form F
Answer Key
Stewart – Calculus ET 8e Chapter 8 Form G
____ 1. Which equation does the function satisfy? Select the correct answer.
a.
b.
c.
d.
e.
2. Solve the initial-value problem.
____ 3. Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the
temperature difference between the object and its surroundings. Suppose that a roast turkey is
taken from an oven when its temperature has reached and is placed on a table in a room
where the temperature is . If is the temperature of the turkey after t minutes, then
Newton’s Law of Cooling implies that
.
This could be solved as a separable differential equation. Another method is to make the change
of variable . If the temperature of the turkey is after half an hour, what is the
temperature after 35 min? Select the correct answer.
a.
b.
c.
d.
e.
4. Solve the differential equation.
5. A curve passes through the point and has the property that the slope of the curve at every
point P is times the y-coordinate P. What is the equation of the curve?
6. Find the orthogonal trajectories of the family of curves.
Stewart – Calculus ET 8e Chapter 8 Form G
7. Solve the differential equation.
____ 8. Suppose that a population develops according to the logistic equation
,
where t is measured in weeks. What is the carrying capacity? Select the correct answer.
a.
b.
c.
d.
e.
9. One model for the spread of an epidemic is that the rate of spread is jointly proportional to the
number of infected people and the number of uninfected people. In an isolated town of
inhabitants, people have a disease at the beginning of the week and have it at the end
of the week. How long does it take for of the population to be infected?
10. Let c be a positive number. A differential equation of the form
where k is a positive constant is called a doomsday equation because the exponent in the
expression is larger than the exponent 1 for natural growth. An especially prolific breed of
rabbits has the growth term . If such rabbits breed initially and the warren has rabbits
after months, then when is doomsday?
Stewart – Calculus ET 8e Chapter 8 Form G
____ 11. The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s range from 35
to 40 million per year and death rates range from 15 to 20 million per year. Let’s assume that the
carrying capacity for world population is 100 billion. Use the logistic model to predict the world
population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the
initial population is small compared to the carrying capacity, you can take k to be an estimate of
the initial relative growth rate.) Select the correct answer.
a. 78.3 billion
b. 27.0 billion
c. 17.1 billion
d. 59.2 billion
e. 32.9 billion
12. A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this
bacterium in a nutrient-broth medium divides into two cells every . The initial
population of a culture is cells. Find the number of cells after hours.
13. Solve the initial-value problem.
14. Solve the differential equation.
____ 15. Solve the initial-value problem. Select the correct answer.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 8 Form G
____ 16. Determine whether the differential equation is linear. Select the correct answer.
a. the equation is linear
b. the equation is not linear
17. In the circuit shown in Figure, a generator supplies a voltage of volts, the
inductance is 2 H, the resistance is 40 , and . Find the current 0.2 s after the switch is
closed. Round your answer to two decimal places.
18. Solve the differential equation.
Stewart – Calculus ET 8e Chapter 8 Form G
____ 19. Solve the differential equation. Select the correct answer.
a.
b.
c.
d.
e.
20. We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we
modify those equations as follows:
,
Find the equilibrium solution.
Stewart – Calculus ET 8e Chapter 8 Form G
Answer Key
Stewart – Calculus ET 8e Chapter 9 Form H
____ 1. Which equation does the function satisfy? Select the correct answer.
a.
b.
c.
d.
e.
2. A population is modeled by the differential equation.
For what values of P is the population increasing?
3. Use Euler’s method with step size 0.25 to estimate , where is the solution of the
initial-value problem. Round your answer to four decimal places.
4. Solve the differential equation.
____ 5. Solve the differential equation. Select the correct answer.
a.
b.
c.
d.
e.
6. Find the orthogonal trajectories of the family of curves.
Stewart – Calculus ET 8e Chapter 9 Form H
7. Suppose that a population develops according to the logistic equation
,
where t is measured in weeks. What is the carrying capacity?
8. A sum of is invested at interest. If is the amount of the investment at time t for
the case of continuous compounding, write a differential equation and an initial condition satisfied
by .
____ 9. Let c be a positive number. A differential equation of the form
where k is a positive constant is called a doomsday equation because the exponent in the
expression is larger than the exponent 1 for natural growth. An especially prolific breed of
rabbits has the growth term . If such rabbits breed initially and the warren has rabbits
after months, then when is doomsday? Select the correct answer.
a.
b.
c.
d.
e.
10. A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this
bacterium in a nutrient-broth medium divides into two cells every . The initial
population of a culture is cells. Find the number of cells after hours.
Stewart – Calculus ET 8e Chapter 9 Form H
11. Consider a population with constant relative birth and death rates and , respectively,
and a constant emigration rate m, where , and . Then the rate of change of
the population at time t is modeled by the differential equation
where
.
Find the solution of this equation with the rate of change of the population at time that
satisfies the initial condition .
____ 12. Solve the initial-value problem. Select the correct answer.
a.
b.
c.
d.
e.
13. Solve the differential equation.
14. Solve the initial-value problem.
____ 15. Determine whether the differential equation is linear. Select the correct answer.
a. the equation is linear
b. the equation is not linear
16. Solve the differential equation.
Stewart – Calculus ET 8e Chapter 9 Form H
17. Solve the differential equation.
18. Let be the performance level of someone learning a skill as a function of the training time t.
The graph of P is called a learning curve. We propose the differential equation
as a reasonable model for learning, where r is a positive constant. Solve it as a linear differential
equation.
19. We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we
modify those equations as follows:
,
Find the equilibrium solution.
Stewart – Calculus ET 8e Chapter 9 Form H
____ 20. A phase trajectory is shown for populations of rabbits and foxes . Describe how each
population changes as time goes by. Select the correct answer.
Select the correct statement.
a. At the number of rabbits rebounds to 500.
b. At the number of foxes reaches a maximum of about 2400.
c. At the population of foxes reaches a minimum of about 30.
Stewart – Calculus ET 8e Chapter 9 Form H
Answer Key