-D.ppepdent
varlsble
-notatiqr,r
-ff
v:f(x).
then dv/dx=f
'fx)
or v'
;-ff
s:g(t), then ds/dFg'(t)
or s'
-If
R:k(w), then dRldw:l{(w)
or
R'
-[n
our
previous
example for
f(x)
:x2
,the
derivative could
be
symboiize as dy/dx
=
2x, ifwe
wanted to
use
dependent
variable
notation.
(also,
y'=2x)
-Also,
the derivative
gives
us important
information
about the
shape
ofthe
graph
off.
1)
between
a
and
b,
the
tangent
lines
have
a
poritve
slope which
causes
the
graph
to
increase (
t
),
2)
between b and d,
the tangent lines have
a
negativo
stope
which causes the
graph
to decrease
(
+
)
3)
beyond
d, the slopes become
positive
again
&
the
function increases.
4)
the slopes
are decreasing from a
to c which causes
the
graph
to
be
coilcave down
(spills
water)
5)
the slopes are increasing from
c
and
beyond which causes the
graph
to
be concave
up
(holds
water)
6)
at
the
point
where
>rq
the
concavity
changes
which
is
called a
point
of
inflection.
7)
at the
points
where :r=b
and x=4 the slopes
arezero.
These
are
higMow
points.
These
are
the
possible
maximurn/minimum
points.
*Summarv
f'(x)>0-+fincreases
f'
(x)
<
0
-+
fdecreases
f
'
(x):0
-)
possible
m&(
or
min
point
f'
(x)
increases -+
concave
up
f
'
(x)
decreases
-+
concave
down
f"(*)
:0
-)
possible
inflection
pont
-You
can sketch the
graph
of
f
' on a separate axis
system when
desired. Do:
1-*2
