Graphics Reference
In-Depth Information
For positive
a
,
a
1/
a
2.
Completing the square
Provided
a
0,
ax
bx
c
(
ax
b
/2)
/
a
c
b
/4
a
.
Provided
a
is positive,
ax
bx
c
0 for all
values of
x
b
4
ac
.
(1
1/
n
)
(1
1/
n
)
increases with
n
and
2
(1
1/
n
)
3.
nth roots
When
a
1, (
n
1)(
a
1)
n
(
a
1).
n
decreases with
n
, when
n
3.
Absolute value
a
when 0
a
, and
We define
a
a
when
a
0.
52 Sktch thegraph of theabsolutevaluefunction
y
x
.
The results that follow on absolute value may be scanned rather than
proved. The results are essential for the rest of the topic, but the proofs
are not notably instructive. We will repeatedly appeal to questions 61
and 63.
53 Provethat max(
a
,
a
)
a
.
54 Provethat
a
a
and
a
a
, for all
a
. Thus
a
a
a
,
for all
a
.
55 Provethat 0
a
for all
a
. Deal with
a
positiveand
a
negative
separately.
a
a
56 Provethat
for all
a
.
57 Provethat
a
a
for all
a
.
Beware of presuming that
2 must be
negative, but
a
need not be. Another way of writing this result is
a
a
a
has to be negative.
.
58 Provethat
ab
a
ยท
b
for all
a
and
b
.
59 Provethat
a
/
b
a
/
b
, provided
b
0.
