5.1 - MATH
1081
 Wednesday,
February
23
 


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Unformatted text preview: MATH
1081
 Wednesday,
February
23
 
 Chapter
5
–
Section
1
 
 INCREASING
AND
DECREASING
 FUNCTIONS
 
 Homework
#6
(due
2/28):
 Section
4.4
#32,
38,
58
 
 
 
 
 
 
 
 
 Section
4.5
#42,
58
 
 
 
 
 
 
 
 
 Section
5.1
#26,
48,
52
 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 
 
 1 . T h e 
G e n e ra l iz e d 
P o w e r
 R u l e 
 d 

 
 f ( x ))n = n ⋅ ( f ( x ))n−1 ⋅ f ' ( x )
 ( dx [ 2 .
 
 T h e 
 G e n e ra l iz e d 
 E x p o n e n t ia l 
 F u n c t io n 
R u l e 
 ( b a s e 
 e ) 
 d f (x) 

 
 
 e = e f ( x ) ⋅ f ' ( x )
 dx 
 
 3 .
 
 T h e 
 G e n e ra l iz e d 
 L o g a rit h m ic 
 F u n c t io n 
R u l e 
 ( b a s e 
 e ) 
 d 1 

 
 
 ⋅ f ' ( x) 
 [ln( f ( x ))] = dx f ( x) [ Now
that
we
have
seen
all
the
differentiation
rules
relevant
to
 the
functions
we
will
see,
we
will
return
to
analyzing
functions
 using
the
derivative.
 
 
 As
we
have
discussed
many
times,
the
derivative
function
is
a
 “slope”
function,
where
the
input
is
the
 x 
in
the
domain
of
a
 function
 f 
and
the
output
is
the
slope
of
that
curve
 f 
at
 x .
 
 
 In
particular,
by
finding
the
derivative
of
certain
functions,
we
 have
been
able
to
discuss
the
velocity
of
a
falling
object,
the
 marginal
cost,
revenue,
and
profit,
and
other
general
rates
of
 change.
 One
additional
application
(actually
first
introduced
in
the
text
 in
Section
4.2),
in
case
you
missed
it,
is
average
cost.

To
 motivate
this
definition,
suppose
that
it
costs
a
total
of
$5000
 to
produce
100
units
of
a
certain
product.

What
is
the
average
 cost
of
production
per
unit?
 
 
 In
general
then,
if
 C ( x ) 
is
the
total
cost
to
produce
 x 
units,
then
 the
average
cost
function
is
given
by

 
 C ( x) 
 
 
 
 
 
 
 C ( x) = .
 x 
 We
can
likewise
define
an
average
revenue
function
and
an
 average
profit
function.

Then
we
can
also
find
marginal
 average
cost,
revenue,
and
profit.
 In
Section
5.1,
we
want
to
use
the
derivative
to
analyze
where
a
 function
is
increasing
and
where
it
is
decreasing.

Without
 formally
defining
the
terms
“increasing”
and
“decreasing”,
 where
would
you
say
the
following
function
is
increasing?

 Decreasing?
 
 
 Note
that
when
we
talk
about
“where”
a
function
has
a
certain
 behavior,
we
answer
in
terms
of
the
values
of
 x 
at
which
the
 value
of
the
function
is
doing
what
we
want.

 
 In
particular,
we
say
that
a
function
 f ( x ) 
is
increasing
on
an
 interval
( a, b ) 
if
for
every
 x1 
and
 x2 
in
the
interval
 



 f ( x1 ) < f ( x2 ) 
whenever
 x1 < x2 .
 The
graph
of
the
function
is
moving
upward
from
left
to
right.
 
 
 Likewise,
we
say
a
function
is
decreasing
on
( a, b ) 
if
 



 f ( x1 ) > f ( x2 ) 
whenever
 x1 < x2 .
 The
graph
is
moving
downward
from
left
to
right.
 
 Theorem:

 Suppose
that
a
function
 f 
has
a
derivative
at
each
point
in
an
 open
interval;
then
 
 a. If
 f ' ( x ) > 0 
for
each
 x 
in
the
interval,
then
 f 
is
 increasing
on
that
interval;
 
 b. If
 f ' ( x ) < 0 
for
each
 x 
in
the
interval,
then
 f 
is
 decreasing
on
that
interval;
 
 c. If
 f ' ( x ) = 0 
for
each
 x 
in
the
interval,
then
 f 
is

 constant
on
that
interval.
 So,
determining
the
intervals
on
which
a
function
is
increasing
 (or
decreasing)
amounts
to
solving
an
inequality.

Recall
that
 solving
inequalities
was
covered
in
the
Chapter
R
prerequisite
 skills
(Section
R.5).
 
 Definition:
Critical
numbers
 The
critical
numbers
for
a
function
 f 
are
those
numbers
 c 
 in
the
domain
of
 f 
for
which
 f ' ( c ) = 0 
or
 f ' ( c ) 
does
not
 exist.


 
 A
critical
point
is
the
coordinate
pair
( c, f ( c )) .
 
 To
find
the
intervals
where
a
function
is
 increasing/decreasing:
 
 1. Draw
a
number
line
and
indicate
all
values
of
 x 
that
are
 NOT
in
the
domain
of
 f .
 
 2. To
that
number
line,
add
all
critical
numbers
of
 f .

This
 will
divide
the
number
line
into
several
open
intervals.
 
 3. Choose
a
specific
value
of
 x 
inside
each
interval
and
plug
 it
into
the
derivative
to
determine
if
 f ' ( x ) > 0 
or
 f ' ( x ) < 0 .
 
 4. Use
the
test
in
Step
3
to
determine
whether
the
function
 is
increasing
or
decreasing
on
each
interval.
 Example:

Determine
where
the
function
is
increasing

 
and
where
the
function
is
decreasing.
 
 1.

 f ( x ) = x 2 − 2 x − 15 
 
 
 
 
 
 
 
 
 2.

 h ( x ) = xe−3 x 
 Clicker
Check­out:

Choose
any
letter
to
check
out.
 
 
 In
recitation
tomorrow:
Quiz
on
Sections
4.1‐4.3
(Rules
of

 Differentiation)
 
 Next
time
–
Section
5.2
 
 ...
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This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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