Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
21
![9911
----
2121₁22= [
-2
value) of h is b in the plane spanned by a, and a₂?
17. Let a =
18. Let y =
0
V₂
- [9] = []
-2
8
alue(s) of h is y in the plane generated by v₁ and v₂?
8
2 and v₂ =
-6
-[-2]
21. Let u =
7
Span (u, v) for all h and k.
, and b =
19. Give a geometric description of Span {V₁, V₂} for the vectors
12
- [3]
-9
and v=
-[:]
h
, and y
20. Give a geometric description of Span {V₁, V₂} for the vectors
in Exercise 16.
[²]
2
=
23. a. Another notation for the vector
h
-5
[3]
-3
Show that
- [3]. [*]
k
For what
22. Construct a 3 x 3 matrix A, with nonzero entries, and a vector
b in R³ such that b is not in the set spanned by the columns
of A.
25. Let A =
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
For what
[3]
b. The points in the plane corresponding to
lie on a line through the origin.
is in
is [-4 3].
[3]
c. An example of a linear combination of vectors v₁ and v₂
is the vector v₁.
and b =
and
d. The solution set of the linear system whose augmented
matrix is [a₁ a2 a3 b] is the same as the solution
set of the equation xa1 + x₂a2 + x3a3 = b.
e. The set Span (u, v) is always visualized as a plane
through the origin.
24 a. Any list of five real numbers is a vector in RS.
b. The vector u results when a vector u - v is added to the
vector V.
c. The weights C₁,.... Cp in a linear combination
CIVI + + Cpvp cannot all be zero.
d. When u and v are nonzero vectors, Span {(u, v) contains
the line through u and the origin.
1
vibn bi
e. Asking whether the linear system corresponding to
an augmented matrix [a₁ a2 a3 b] has a solution
amounts to asking whether b is in Span {a₁, a2, a3).
1 0-4
3 -2
0
-2 6 3
1
-[4
columns of A by a₁, a2, a3, and let W = Span {a₁, a2, a3).
Denote the
a. Is b in (a,, a2. a3)? How many vectors are in {a,. a2. a3}?
b. Is b in W? How many vectors are in W?
c.
Show that a, is in W. [Hint: Row operations are unnec-
essary.]
206
85
1
the set of all linear combinations of the columns of A.
26. Let A =
1.3 Vector Equations 33
-1
1
-2
V₂ =
let b =
----
10
3
3
a. Is b in W?
b. Show that the third column of A is in W.
and let W be
27.
A mining company has two mines. One day's operation at
mine #1 produces ore that contains 20 metric tons of cop-
per and 550 kilograms of silver, while one day's operation
at mine #2 produces ore that contains 30 metric tons of
20
and
copper and 500 kilograms of silver. Let v₁ =
550
Then v₁ and v₂ represent the "output per day"
30
500
of mine #1 and mine #2, respectively.
a. What physical interpretation can be given to the vector
5v₁ ?
b. Suppose the company operates mine #1 for x₁ days and
mine #2 for x₂ days. Write a vector equation whose solu-
tion gives the number of days each mine should operate in
order to produce 150 tons of copper and 2825 kilograms
of silver. Do not solve the equation.
c. [M] Solve the equation in (b).
28.
A steam plant burns two types of coal: anthracite (A) and
bituminous (B). For each ton of A burned, the plant produces
27.6 million Btu of heat, 3100 grams (g) of sulfur dioxide,
and 250 g of particulate matter (solid-particle pollutants). For
each ton of B burned, the plant produces 30.2 million Btu,
6400 g of sulfur dioxide, and 360 g of particulate matter.
a. How much heat does the steam plant produce when it
burns x₁ tons of A and x₂ tons of B?
b. Suppose the output of the steam plant is described by
a vector that lists the amounts of heat, sulfur dioxide,
and particulate matter. Express this output as a linear
combination of two vectors, assuming that the plant burns
x₁ tons of A and x₂ tons of B.
c. [M] Over a certain time period, the steam plant produced
162 million Btu of heat, 23,610 g of sulfur dioxide, and
1623 g of particulate matter. Determine how many tons
of each type of coal the steam plant must have burned.
Include a vector equation as part of your solution.
V
29. Let V₁...., Vk be points in R3 and suppose that for
j = 1,..., k an object with mass m, is located at point v, .
Physicists call such objects point masses. The total mass of
the system of point masses is
m = m₁ + ... + mk](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9aabfb62-88cc-47fc-a128-866ddac4a30f%2F197a83b9-813d-4939-9b20-093880c6b3a3%2Fu7tdn1k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:9911
----
2121₁22= [
-2
value) of h is b in the plane spanned by a, and a₂?
17. Let a =
18. Let y =
0
V₂
- [9] = []
-2
8
alue(s) of h is y in the plane generated by v₁ and v₂?
8
2 and v₂ =
-6
-[-2]
21. Let u =
7
Span (u, v) for all h and k.
, and b =
19. Give a geometric description of Span {V₁, V₂} for the vectors
12
- [3]
-9
and v=
-[:]
h
, and y
20. Give a geometric description of Span {V₁, V₂} for the vectors
in Exercise 16.
[²]
2
=
23. a. Another notation for the vector
h
-5
[3]
-3
Show that
- [3]. [*]
k
For what
22. Construct a 3 x 3 matrix A, with nonzero entries, and a vector
b in R³ such that b is not in the set spanned by the columns
of A.
25. Let A =
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
For what
[3]
b. The points in the plane corresponding to
lie on a line through the origin.
is in
is [-4 3].
[3]
c. An example of a linear combination of vectors v₁ and v₂
is the vector v₁.
and b =
and
d. The solution set of the linear system whose augmented
matrix is [a₁ a2 a3 b] is the same as the solution
set of the equation xa1 + x₂a2 + x3a3 = b.
e. The set Span (u, v) is always visualized as a plane
through the origin.
24 a. Any list of five real numbers is a vector in RS.
b. The vector u results when a vector u - v is added to the
vector V.
c. The weights C₁,.... Cp in a linear combination
CIVI + + Cpvp cannot all be zero.
d. When u and v are nonzero vectors, Span {(u, v) contains
the line through u and the origin.
1
vibn bi
e. Asking whether the linear system corresponding to
an augmented matrix [a₁ a2 a3 b] has a solution
amounts to asking whether b is in Span {a₁, a2, a3).
1 0-4
3 -2
0
-2 6 3
1
-[4
columns of A by a₁, a2, a3, and let W = Span {a₁, a2, a3).
Denote the
a. Is b in (a,, a2. a3)? How many vectors are in {a,. a2. a3}?
b. Is b in W? How many vectors are in W?
c.
Show that a, is in W. [Hint: Row operations are unnec-
essary.]
206
85
1
the set of all linear combinations of the columns of A.
26. Let A =
1.3 Vector Equations 33
-1
1
-2
V₂ =
let b =
----
10
3
3
a. Is b in W?
b. Show that the third column of A is in W.
and let W be
27.
A mining company has two mines. One day's operation at
mine #1 produces ore that contains 20 metric tons of cop-
per and 550 kilograms of silver, while one day's operation
at mine #2 produces ore that contains 30 metric tons of
20
and
copper and 500 kilograms of silver. Let v₁ =
550
Then v₁ and v₂ represent the "output per day"
30
500
of mine #1 and mine #2, respectively.
a. What physical interpretation can be given to the vector
5v₁ ?
b. Suppose the company operates mine #1 for x₁ days and
mine #2 for x₂ days. Write a vector equation whose solu-
tion gives the number of days each mine should operate in
order to produce 150 tons of copper and 2825 kilograms
of silver. Do not solve the equation.
c. [M] Solve the equation in (b).
28.
A steam plant burns two types of coal: anthracite (A) and
bituminous (B). For each ton of A burned, the plant produces
27.6 million Btu of heat, 3100 grams (g) of sulfur dioxide,
and 250 g of particulate matter (solid-particle pollutants). For
each ton of B burned, the plant produces 30.2 million Btu,
6400 g of sulfur dioxide, and 360 g of particulate matter.
a. How much heat does the steam plant produce when it
burns x₁ tons of A and x₂ tons of B?
b. Suppose the output of the steam plant is described by
a vector that lists the amounts of heat, sulfur dioxide,
and particulate matter. Express this output as a linear
combination of two vectors, assuming that the plant burns
x₁ tons of A and x₂ tons of B.
c. [M] Over a certain time period, the steam plant produced
162 million Btu of heat, 23,610 g of sulfur dioxide, and
1623 g of particulate matter. Determine how many tons
of each type of coal the steam plant must have burned.
Include a vector equation as part of your solution.
V
29. Let V₁...., Vk be points in R3 and suppose that for
j = 1,..., k an object with mass m, is located at point v, .
Physicists call such objects point masses. The total mass of
the system of point masses is
m = m₁ + ... + mk
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