3.4 - MATH
1081
 Monday,
February
7
 


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Unformatted text preview: MATH
1081
 Monday,
February
7
 
 Chapter
3
–
Section
4
 
 DEFINITION
OF
THE
DERIVATIVE
 
 Homework
#4
(due
2/14):
 
 Section
3.4
#18,
48,
56
 
 
 
 
 
 
 
 
 
 Section
4.1
#34,
56,
62,
72
 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 
 Last
time,
we
defined
the
average
rate
of
change
of
 f ( x ) 
 between
 x = a 
and
 x = b 
and
the
instantaneous
rate
of
change
of
 f ( x ) 
at
 x = a .
 
 
 The
average
rate
of
change
is
simply
the
slope
of
the
line
 between
two
different
points
on
the
curve.
 
 The
instantaneous
rate
of
change
is
the
limit
of
the
slopes
of
 these
lines
as
the
two
points
get
closer
and
closer
together.
 
 Let’s
start
with
 P = ( 3, f ( 3)) 
and
Q = ( 4, f ( 4 )) 
that
are
two
 different
points
on
the
graph
of
a
function,
which
we
will
see
 on
the
next
slide.
 
 We
will
call
the
line
between
these
two
points
a
secant
line.
 
 We
then
move
the
point
Q 
closer
and
closer
to
the
point
 P .
 The
“limiting”
line,
which
we
call
the
tangent
line
at
P,
will
be
 THE
line
that
passes
through
the
graph
at
that
single
point
P
 and
has
the
same
direction
of
the
curve
there.
 
 The
slope
of
this
tangent
line
is
defined
to
be
the
limit
of
the
 slopes
of
the
secant
lines
and
is
often
referred
to
as
the
slope
of
 the
curve
at
the
point
P. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 So,
the
instantaneous
rate
of
change
of
 f ( x ) 
at
 x = a ,
which
on
 a
graph
can
be
thought
of
as
the
slope
of
the
tangent
line
to
 f ( x ) 
at
 x = a ,
is
computed
as
 
 f (a + h) − f (a) 
 
 
 
 
 
 lim 
 h→ 0 h 
 as
long
as
the
(two‐sided)
limit
exists.
 
 We
will
use
the
notation

 
 







 f '(a)
 
 to
notate
this
very
important
limit.
 Now,
if
this
limit
exists,
then
we
have
 x = a 
being
assigned
 exactly
one
value
called
 f ' ( a ) .



That
is,
we
can
think
of
this
 f ' 
 as
a
function
that
assigns
to
each
input
value
 x = a 
in
the
 domain
of
the
original
function
 f ( x ) ,
the
output
value
 f ' ( a ) 
 which
equals
the
slope
of
the
original
curve
 f ( x ) 
at
that
point
 x = a .
 
 
 So,
for
a
function
 f ( x ) 
at
 x = a ,
the
value
of
 f ( a ) 
tells
us
where
 the
point
is
on
the
curve
and
the
value
 f ' ( a ) 
tells
us
the
slope
 of
the
curve
at
that
point.
 
 Definition:

Derivative
 
 The
derivative
of
the
function
 f 
at
 x 
is
defined
as

 
 f ( x + h) − f ( x) 




 
 f ' ( x ) = lim h→ 0 h 
 
 provided
the
limit
exists.
 
 
 In
essence,
all
that
we
have
done
here
is
to
rename
the
specific
 value
 a 
to
the
general
variable
 x 
to
emphasize
its
identification
 as
a
function
that
can
be
evaluated
at
many
different
points.
 
 
 The
definition
of
derivative
includes
the
phrase
“provided
the
 limit
exists”.

For
what
kinds
of
situations
will
the
derivative
 NOT
exist
at
 x = a ?
 
 
 • when
the
function
is
not
defined
at
 x = a 
 • when
the
function
is
not
continuous
at
 x = a 
 • when
the
function
has
a
sharp
corner
at
 x = a 
 • when
the
tangent
line
to
the
curve
at
 x = a 
is
vertical
 
 
 
 
 
 
 
 From
page
209
of
your
book,
 
 
 
 
 
 
 
 
 
 
 
 
 Example:

Find
the
derivative
for
the
function.
 
 
 1.

 f ( x ) = 3
 
 
 2.

 g ( x ) = x 2 
 
 
 3.

 h ( x ) = 4 
 x 

 Example:

Find
the
equation
of
the
tangent
line
to
the
curve
at

 
the
indicated
point.
 
 1.

 f ( x ) = x 2 
at
the
point
( 2, 4 ) 
 
 
 
 2.

 f ( x ) = x 2 
at
the
point
( −3, 9 ) 
 
 
 
 4 3.

 h ( x ) = 
at
the
point
 x = 1
 x As
introduced
in
Workshop
1,
the
marginal
cost
for
the
 production
of
 a 
units
is
the
rate
of
change
of
the
cost
function
 C ( x ) 
at
 x = a .


 
 This
marginal
cost
can
be
used
to
approximate
the
cost
to
 produce
the
 a + 1st
unit.
 
 That
is,
for
a
cost
function
 C ( x ) ,
the
derivative
 C ' ( x ) 
(called
 the
marginal
cost
function)
measures
the
rate
of
change
of
cost
 and
can
be
used
to
approximate
the
cost
to
increase
production
 by
one
unit.
 
 We
can
similarly
define
the
marginal
revenue
and
marginal
 profit
functions.
 Example:

Suppose
that
the
revenue
(in
dollars)
generated

 
from
the
sale
of
 x 
picnic
tables
is
given
by

 x2 .


 R ( x ) = 20 x − 500 
 Approximate
the
revenue
from
the
sale
of
the
1001st
 picnic
table.
 
 Clicker
Check­Out:

Choose
any
letter
to
check
out
now.
 
 Next
time
–
Section
4.1
 
 
 ...
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