The.211
llertvatiYe
-What
f
'
(*)
tells
us about
f(x),
f
"
(x)
will
give
the
same
information
about
f
'
(x)
-Bevhp
f
'
(x)
>
0
-+
fincreases
(curve
rises
from
lett
to'ighQ
f
'
(x)
<
0 + f
decreases
(curve
falls
&om
lett to
right)
f
,
(4:0
-+
f neither
is increasing
nor
decreasing
(possible
max/min
points)
So, using
this analogy,
we
have
the tbllowing:
f
"
(x)
>
0
+
f
insreases
{concave
up)
(holds
water)
(.slopes
are
increasing)
f
"
(x)
<
0
-+
f
'
decreases
(eoncave
down)
(spills
water)
(slopes
are
decreasing)
f
.,
(x)
=
0
-+
f
,
is neither
increasing
nor decreasing
(possible
concavity
change)
(possible
inflection
pt)
-points
of
inflection
are
important
points in business
applications.
They
indicate
Trend
reversal
in
various
applications
&
often
indicate
arr
important
signal
for
the
market. See
diagram'
*For
distance-time
functions,
s:{t),
s
,,
-
f
,,
(t)
measures
how
velocity
is
changing
instantaneously.
This
is known
as
Acrele-ration.
This can
also
be symbolized
by dv/dt
or d'Sldt'
.
-EX34qple:
Show
with the
free-fall
functiorq
S
:
-16t2,
time
permitting.
-wiu
cover
noints
of
rnflection
in much
more
det*il
Iater
