352
Instructor’s Manual
This is the beginning of a broken line of segments spiraling away from the origin. At
the corner points, ƒ is alternatingly positive and negative and increases monotone in
absolute value.
10.
ƒ(
x
)
5
x
1
2
2
x
2
gives
z
(
t
)
5
x
2
t
[2
x
1
,
2
1]
5
[(1
2
2
t
)
x
1
,
x
2
1
t
],
hence
g
(
t
)
5
(1
2
2
t
)
2
x
1
2
2
x
2
2
t
,
g
9
(
t
)
52
4(1
2
2
t
)
x
1
2
2
1
5
0.
From this,
1
2
2
t
,
t
51
.
For this
t
,
z
(
t
)
5
[
2
,
x
2
11
]
.
From this, with
x
1
5
1,
x
2
5
1, we get successively
z
(1)
5
[
2
1
_
4
,1
1
1
_
2
1
1
_
8
]
T
z
(2)
5
[1, 1
1
2
•
1
_
2
1
1
_
8
1
2]
T
z
(3)
5
[
2
1
_
4
1
3
•
1
_
2
1
2
•
1
_
8
1
2]
T
etc.
The student should sketch this, to see that it is reasonable. The process continues
indefinitely, as had to be expected.
12. CAS Experiment.
(c) For ƒ(
x
)
5
x
1
2
1
x
2
4
the values converge relatively rapidly
to [0
0]
T
, and similarly for ƒ(
x
)
5
x
1
4
1
x
2
4
.
SECTION 22.2. Linear Programming, page 939
Purpose.
To discuss the basic ideas of linear programming in terms of very simple
examples involving two variables, so that the situation can be handled graphically and the
solution can be found geometrically. To prepare conceptually for the case of three or more
variables
x
1
,
•••
,
x
n
.
Main Content, Important Concepts
Linear programming problem
Its normal form. Slack variables
Feasible solution, basic feasible solution
Optimal solution
Comments on Content
Whereas the function to be maximized (or minimized) by Cauchy’s method was arbitrary
(differentiable), but we had no constraints, we now simply have a linear objective function,
but constraints, so that calculus no longer helps.
No systematic method of solution is discussed in this section; these follow in the next
sections.
1
}
8
x
1
2
1
}
2
1
}
4
x
1
1
}
8
x
1
2
1
}
2
1
}
4
x
1
2
im22.qxd
9/21/05
1:53 PM
Page 352