Example:
@
Class
Meeting#74 Derivatives
&
Sraphs
.
-Bsea[:
f
'(x)
is to f(x)
as
f
"(x)
is
to
f
'(x)
[what
f
'(x)
tells
you
about
f(x), f
"(x)
will
give you
the
same
information
about
f
'(xlJ.
-f
'(x)
>
0
-+
fincreases
(curve
rises from
left to
rightx
t
)
*f
'(*)
<
0
+
f
decreases
(curve
falls from left
to
righ$(
J,
)
@
"-f
i*i=
0
->
f neither is increasing
nor
decreasing
(possible
maxlmin
points) (called
critical
points)
*f
"(x)
>
0
-+
f increases
(slopes
increase)
(curve
is
concave up)
(holds
water)
-f
"(x)
<
0'*+ f
'
decreases
(slopes
decrease)
(curve
is
concave down)
(spills
water)
-f
"(x)
=
o
->
f
is neither increases nor
decreases
(possible
concavity change)(possible
point
of
lnflection)
f
'(x):0
is not
a sufficient
condition
for
a mrur/min
point.
Give example
6,:*rat
)F0)
f
"(x):
0
is not
a
sufficient condition for
an
inflection
point.
Give eample
(y=*+
at
x=O)
-f
'(x)>0
tbr x<a
and
x>c
*f
'(x)<0 for
a<x<c
*f
'(x)=0
tbr:<=a
and
>rc
*f
"(x)>0
for
x>b
-f
"(x)<0
for
x<b
-f
"(x)=g
for x=b
Example:
Given f(x)
=
*'+6*
+i,
find intervals
on the x-axis for
which f
increases,
decreases, concave
up, &
concave
down.
Find
all
critical
points
& sketch.
@
%d*
**
f,rx)
=
a3+dp*
f
a'rt-
f*"
t*d*adif7'*
f,/{x}=r:
-*
&,x:*3
*F
K**.,
fraJ=I
(
C,s
j
t-e-
o'tt-
**'W'*
P"#
kv|ru-a
"'
';:;
+,
,{rr'J*
,
'
l-tr"}
I
,{""*'"-r.
c.r
d
r1f.P'i.',rt
a^t{_
g
?
P#
+'g<s
-Consider
the
following
curve:
*1*r
c/tu:t,
e^C
1v*+*f's
t
(
j
1
-x
"\
ua+
4--*"
n
*J
