Graphics Reference
In-Depth Information
binomial theorem. Sketch graphs of
y
nx
and
y
x
)
for
n
2 and
n
3 to illustrate Bernoulli's inequality. Put the graphs
for
n
1
(1
3 on separate diagrams. Use a computer
graphing packageor a graphics calculator if possibl.
2 and
n
Summary
-
properties of order
Trichotomy
For any two numbers,
a
and
b
, either
a
b
or
b
a
or
a
b
, and only oneof thseholds.
Transitivity
If
a
b
and
b
c
, then
a
c
.
Addition
If
a
b
and
c
d
, then
a
c
b
d
.
Multiplication
If
a
b
and 0
c
, then
ac
bc
.
If
a
b
and
c
0, then
bc
ac
.
Squares
0
a
, for all
a
.
Powers
If 0
a
b
, then
a
b
for all positive
integers
n
.
Suppose0
a
and 0
b
.If
a
b
for some
positive integer
n
, then
a
b
.
Reciprocals
If 0
a
b
, then 0
1/
b
1/
a
.
If
a
b
0, then 1/
b
1/
a
0.
Bernoulli
'
s inequality
If
1
x
, then (1
x
)
1
nx
, for all
positive integers
n
.
Arithmetic mean and geometric mean
30 If 2,
A
and 8 are three consecutive terms of an arithmetic
progression, what is the number
A
?
A
is called the
arithmetic mean
of 2 and 8.
If 2,
G
, 8 are three consecutive terms of a geometric progression of
positive terms, what is the number
G
?
G
is called the
geometric
mean
of 2 and 8.
Which is the greater,
A
or
G
?
31 Write down the arithmetic mean and the geometric mean of the
numbers 2 and 3. Check that (2
, and deduce that the
arithmetic mean of 2 and 3 is greater than their geometric mean.
Obtain the same result by using the inequality 0
(
3
2)
.
)
6
32 (
Euclid
)If
a
and
b
are positive, find the arithmetic mean and the
geometric mean of the numbers
a
and
b
. Which of themeans is
greater? Under what circumstances are the two means equal?
