Graphing
Related
Topics
Knowledge
Summary:
RULES
&
DEFINITIONS
TO
KEEP
IN
MIND:
Minimum
and
Maximum
Values:
1) f(x)
has
an
absolute
maximum
and
a
global
maximum
at
x=c
if
f(x)
<
f(c)
for
every
x
in
that
particular
domain
2) f(x)
has
a
relative
maximum
and
a
local
maximum
at
x=c
if
f(x)
<
f(c)
for
every
x
in
some
open
interval
around
x=c
3) f(x)
has
an
absolute
minimum
and
a
global
minimum
at
x=c
if
f(x)
>
f(c)
for
every
x
in
that
particular
domain
4) f(x)
has
a
relative
minimum
and
a
local
minimm
at
x=c
if
f(x)
>
f(c)
for
every
x
in
some
open
interval
around
x=c
Critical
Points:
We
say
that
x=c
is
a
critical
point
of
the
function
f(x)
if
f(c)
exists
and
if
either
of
the
following
are
true:
f’(c)=0
or
f’(c)
does
not
exist.
Note:
we
require
f(c)
to
exist
in
order
for
x=c
to
actually
be
a
critical
point.
Find
the
critical
numbers
by
obtaining
first
derivative
and
setting
function
to
0:
1.
Where
y’=0
or
does
not
exist
2.
Where
y’
changes
from
+
to
-‐
there’s
a
local
max
3.
Where
y’
changes
from
-‐
to
+
there’s
a
local
min
Absolute
Extrema:
Finding
absolute
extrema
of
f(x)
on
[a,b]
1. Verify
that
the
function
is
continuous
on
the
interval
[a,b].
2. Find
all
the
critical
points
of
f(x)
that
are
in
the
interval
[a,b].
3. Evaluate
the
function
at
the
critical
points
found
in
step
1
and
at
the
end
points
4. Identify
the
absolute
extrema
Fermat’s
Theorem:
If
f(x)
has
a
relative
extrema
at
x=c
and
f’(c)
exists
then
x=c
is
a
critical
point
of
f(x).
It
will
be
a
critical
point
such
that
f’(c)=0
Mean
Value
Theorem:
Suppose
that
f(x)
is
a
function
that
satisfies
both
of
the
following:
1. f(x)
is
continuous
on
the
closed
interval
[a,b]
2. f(x)
is
differentiable
on
the
open
interval
(a,b)
▯ ▯ ▯ ▯▯(▯)
Then
there’s
a
number
c
such
that
a
<
c
0
for
all
x
in
(a,
b),
then
f
is
increasing
on
[a,
b]
2. If
f
'(x)
<
0
for
all
x
in
(a,
b),
then
f
is
decreasing
on
[a,
b]
3. If
f
'(x)
=
0
for
all
x
in
(a,
b),
then
f
is
constant
on
[a,
b]
Concavity:
Let
f
be
a
function
whose
second
derivative
exists
on
an
open
interval
I.
1. If
f
''(x)
>
0
for
all
x
in
I,
then
the
graph
of
f
is
concave
upward
2. If
f
''(x)
<
0
for
all
x
in
I,
then
the
graph
of
f
is
concave
downward
3. If
f
''(x)
=
0
for
all
x
in
I,
then
the
graph
of
f
is
a
line,
neither
concave
upward
or
downward
Inflection
Points:
Find
points
of
inflection
by
obtaining
f''(x),
setting
it
to
0,
and
solving
for
x.
Asymptotes:
For
horizontal
asymptotes
check
the
limit
as
x
approaches
infinity .
For
vertical
asymptotes
when
the
function
is
rational
set
the
denominator
equal
to
zero
and
solve .
First
Derivative
Test:
This
is
a
method
for
determining
whether
an
inflection
point
is
a
minimum,
a
maximum,
or
neither.
Suppose
that
x=c
is
a
critical
point
of
f(x)
then,
1. If
f’(x)
>
0
to
the
left
of
x=
c
and
f’(x)
<
0
to
the
right
of
x=
c,
then
x=
c
is
a
relative
maximum.
2. If
f’(x)
<
0
to
the
left
of
x=
c
and
f’(x)
>
0
to
the
right
of
x=
c,
then
x=
c
is
a
relative
minimum.
3. If
f’(x)
is
the
same
sign
on
both
sides
of
x=
c
then
x=
c
is
neither
a
relative
maximum
nor
a
relative
minimum.
Second
Derivative
Test:
The
second
derivative
test
relates
the
concepts
of
critical
points,
extreme
values,
and
concavity
to
give
a
very
useful
tool
for
determining
whether
a
critical
point
on
the
graph
of
a
function
is
a
relative
minimum
or
a
relative
maximum.
Suppose
that
x=
c
is
a
critical
point
of
f’(c)
such
that
f’(c)=0
and
that
f’’(x)
is
continuous
in
a
region
around
x=
c.
Then,
1. If
f’’(c)
<
0
then
x=
c
is
a
relative
maximum
2. If
f’’(c)
>
0
then
x=
c
is
a
relative
minimum
3. If
f’’(c)
=
0
then
x=
c
can
be
a
relative
maximum,
relative
minimum
or
neither
Examples:
Year:
2002
Question:
16
If
the
two
curves
y=
4lnx
and
y=
cx
(where
c
is
a
positive
constant)
have
exactly
one
point
in
common,
what
must
be
the
value
of
c?
y=
cx 2
P
(a,
b)
1y=
4lnx
If
the
two
curves
have
exactly
one
point
in
common,
say
P
(a,
b),
then
the
two
curves
must
be
tangential
at
P.
y=
cx
y’=
2cx
y’
at
P=
2ca
▯ ▯
Therefore:
2c
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