369 Chapter 6: Differential Equations
6.1 Slope Fields and Euler’s Method
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Which of the following is a solution of the differential equation
?
a.
b.
c.
d.
e.
____
2.
Which of the following is a solution of the differential equation
?
a.
b.
c.
d.
e.
____
3.
Which of the following is a solution of the differential equation
?
a.
b.
c.
d.
e.
____
4.
Find the particular solution of the differential equation
that satisfies
the initial condition
when x = 2, where
is the general solution.
a.
b.
c.
d.
e.
6.2 Slope Fields and Euler’s Method
370
____
5.
Use integration to find a general solution of the differential equation.
a.
b.
c.
d.
e.
____
6.
Use integration to find a general solution of the differential equation.
a.
b.
c.
d.
e.
____
7.
Use integration to find a general solution of the differential equation
.
a.
b.
c.
d.
e.
371
Chapter 6: Differential Equations
____
8.
Use integration to find a general solution of the differential equation
.
a.
b.
c.
d.
e.
____
9.
Use integration to find a general solution of the differential equation.
a.
b.
c.
d.
e.
6.2 Slope Fields and Euler’s Method
372
____
10.
Use integration to find a general solution of the differential equation .
a.
b.
c.
d.
e.
____
11.
Use integration to find a general solution of the differential equation.
a.
b.
c.
d.
e.
____
12.
Use the differential equation
and its slope field to find the slope at the
point
.
373 Chapter 6: Differential Equations
32
1
8
64
16
____ 13. Select from the choices below the slope field for the differential equation.
a. d.
b. e. none of the above
6.2 Slope Fields and Euler’s Method
374
c.
____ 14. Select from the choices below the slope field for the differential equation.
a. c.
b. d. none of the above
375
Chapter 6: Differential Equations
____
15.
Select from the choices below the slope field for the differential equation.
a. d.
b. e. none of the above
c.
6.2 Slope Fields and Euler’s Method
376
____ 16.
Sketch the slope field for the differential equation
and use the slope field
to sketch the solution that passes through the point
.
a.
b.
c.
y
d.
5
4
3
2
1
–5
–4
–3
–2
–1–1
1
2
3
4
5
x
–2
–3
–4
–5
y
e.
y
5
5
4
4
3
3
2
2
1
1
–5
–4
–3
–2
–1–1
1
2
3
4
5
x
–5 –4 –3 –2 –1–1
1
2
3
4
5
x
–2
–2
–3
–3
–4
–4
–5
y
–5
5
4
3
2
1
–5
–4
–3
–2
–1–1
1
2
3
4
5
x
–2
–3
–4
–5
377
Chapter 6: Differential Equations
____
17.
Sketch the slope field for the differential equation
and use the slope field
to sketch the solution satisfying the condition
.
a.
d.
b. e.
c.
6.2 Slope Fields and Euler’s Method
378
____ 18.
Use Euler’s Method to make a table of values for the approximate solution of the
following differential equation with specified initial value. Use 5 steps of size 0.15.
a.
b.
c.
d.
e.
____ 19.
At time
minutes, the temperature of an object is
. The temperature of the
object is changing at the rate given by the differential equation
. Use Euler’s
Method to approximate the particular solutions of this differential equation at
. Use a step size of
. Round your answer to one decimal place.
a.
123.5
b.
120.4
c.
121.9
d.
128.7
e.
116.4
379 Chapter 6: Differential Equations
6.1 Slope Fields and Euler’s Method
Answer Section
6.2 Differential Equations: Growth and Decay
380
6.2 Differential Equations: Growth and Decay
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Solve the differential equation.
a.
b.
c.
d.
e.
____ 2. Solve the differential equation.
a.
b.
c.
d.
e.
____ 3. Solve the differential equation.
a.
b.
c.
d.
e.
381
Chapter 6: Differential Equations
____
4.
Solve the differential equation
.
a.
b.
c.
d.
e.
____
5.
Solve the differential equation.
a.
b.
c.
d.
e.
____ 6. Write and solve the differential equation that models the following verbal statement: The
rate of change of with respect to is proportional to .
a.
b.
c.
d.
e.
6.2 Differential Equations: Growth and Decay
382
____
7.
Find the function
passing through the point
with the first derivative
.
a.
b.
c.
d.
e.
____
8.
Find the function
passing through the point
with the first derivative
.
a.
b.
c.
d.
e.
____ 9. Write and solve the differential equation that models the following verbal statement.
Evaluate the solution at the specified value of the independent variable, rounding your answer to four
decimal places:
The rate of change of is proportional to . When , and when , = 84. What is
the value of when ?
a.
b.
c.
d.
e.
383 Chapter 6: Differential Equations
____ 10. The rate of change of N is proportional to N. When and when
. What is the value of N when ? Round your answer to three decimal places.
2,129.520
2,099.520
2,049.520
491.383
262,440.000
____ 11. Find the exponential function that passes through the two given points. Round your
values of C and k to four decimal places.
a.
b.
c.
d.
e.
____ 12. The isotope has a half– life of 5,715 years. Given an initial amount of 11 grams of the
isotope, how many grams will remain after 500 years? After 5,000 years? Round your answers to
four decimal places.
7.2469 gm, 4.1988 gm
6.2117 gm, 3.5989 gm
10.3528 gm, 5.9982 gm
4.1411 gm, 2.3993 gm
12.4233 gm, 7.1979 gm
6.2 Differential Equations: Growth and Decay
384
____ 13.
The half-life of the radium isotope Ra-226 is approximately 1,599 years. If the initial
quantity of the isotope is 38 g, what is the amount left after 1,000 years? Round your answer to two
decimal places.
a.
24.63 g
b.
30.60 g
c.
25.13 g
d.
11.88 g
e.
12.32 g
____ 14. The isotope has a half-life of 5,715 years. After 2,000 years, a sample of the isotope
is reduced to 1.2 grams. What was the initial size of the sample (in grams)? How much will remain
after 20,000 years (i.e., after another 18000 years)? Round your answers to four decimal places.
1.0706 , 0.0947
2.4471 , 0.2164
1.5294 , 0.1352
2.1412 , 0.1893
1.9883 , 0.1758
____ 15. The isotope has a half-life of 24,100 years. After 10,000 years, a sample of the
isotope is reduced 1.6 grams. What was the initial size of the sample (in grams)? How large was
the sample after the first 1,000 years? Round your answers to four decimal places.
2.1332 , 2.0727
2.7731 , 2.6945
1.2799 , 1.2436
1.7065 , 1.6582
1.0666 , 1.0364
____ 16. The half life of the radium isotope Ra-226 is approximately 1,599 years. If the amount
left after 1,000 years is 1.8 g, what is the amount after 2000 years? Round your answer to three
decimal places.
1.167 g
0.939 g
1.800 g
0.490 g
2.334 g
____ 17. The half-life of the radium isotope Ra-226 is approximately 1,599 years. What
percent of a given amount remains after 800 years? Round your answer to two decimal places.
70.70 %
5.71 %
72.70 %
25.02 %
0.71 %
385 Chapter 6: Differential Equations
____ 18. The initial investment in a savings account in which interest is compounded continuously is
$813. If the time required to double the amount is years, what is the annual rate? Round your
answer to two decimal places.
7.49 %
7.70 %
13.34 %
6.29 %
8.89 %
____ 19. The initial investment in a savings account in which interest is compounded continuously is
$604. If the time required to double the amount is years, what is the amount after 15 years?
Round your answer to the nearest cent.
$1,917.58
$1,804.46
$1,907.37
$1,404.46
$8,278.18
____ 20. Find the principal that must be invested at the rate 8%, compounded monthly, so that
$1,000,000 will be available for retirement in 50 years. Round your answer to the nearest cent.
$250,000.00
$18,560.39
$717,324.37
$333,333.33
$21,321.23
____ 21. Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6%
compounded continuously. Round your answer to two decimal places.
1.16 years
11.55 years
1.39 years
11.90 years
11.58 years
____ 22. Suppose that the population (in millions) of Paraguay in 2007 was 6.7 and that the
expected continuous annual rate of change of the population is 0.024. Find the exponential growth
model for the population by letting correspond to 2000. Round your answer to four
decimal places.
a.
b.
c.
d.
e.
6.2 Differential Equations: Growth and Decay
386
____ 23.
Suppose that the population (in millions) of a Egypt in 2007 is 80.3 and that expected
continuous annual rate of change of the population is 0.017. The exponential growth model for the
population by letting
corresponds to 2000 is
. Use the model to predict the
population of the country in 2013. Round your answer to two decimal places.
83.08 million
87.42 million
90.45 million
88.92 million
81.68 million
____ 24. The number of bacteria in a culture is increasing according to the law of exponential
growth. After 5 hours there are 175 bacteria in the culture and after 10 hours there are 425 bacteria
in the culture. Answer the following questions, rounding numerical answers to four decimal places.
Find the initial population.
Write an exponential growth model for the bacteria population. Let t represent time in hours.
Use the model to determine the number of bacteria after 20 hours.
After how many hours will the bacteria count be 15,000?
a.
(i)
72.0588 ; (ii)
; (iii)
3,819.3668 ; (iv)
32.4162 hr
b.
(i)
74.2088 ; (ii)
; (iii)
5,194.0840 ; (iv)
34.6442 hr
c.
(i)
72.0588 ; (ii)
; (iii)
2,506.6327 ; (iv)
30.0817 hr
d.
(i)
77.8388 ; (ii)
; (iii)
7,945.5374 ; (iv)
36.7554 hr
e.
(i)
79.3988 ; (ii)
; (iii)
10,598.0009 ; (iv)
38.5348 hr
____
25.
A container of hot liquid is placed in a freezer that is kept at a constant temperature
of
. The initial temperature of the liquid is
. After 3 minutes, the liquid’s temperature is
. How much longer will it take for its temperature to decrease to
? Round your answer to
two decimal places.
1.89 minutes
2.84 minutes
3.16 minutes
1.26 minutes
3.47 minutes
387 Chapter 6: Differential Equations
6.2 Differential Equations: Growth and Decay
Answer Section
6.2 Differential Equations: Growth and Decay
388