Curve Sketching with the Second Derivative
Example
Determine where the function
r(x)
u2
_;*3
is increasing and decreasing, and where its graph is concave
up and concave down. Find all relative extrema and points of
inflection, and sketch the graph.
&^r*1 : 3qa[*"ns 3. I r 9.2'
_xl
{f>): x>+3
ry irnc vua^p/f,e ( *
\^ $ e^v */s
c
{yotle
d;r
i4
H
4
nlr r
+'rx\ =
(,.?t:)z:.---
6ar-:)2-
9zT" +'ffl
Crrr
=d .
bF: o
F:O F
r,.^) *>\ 4
L - ' ,IJ
Cnr{
#
i6Q p.r,,t$: (rt+ra)3 @,4
7
fucv\€^$\
4g_
+(
s"[*
n
(PX
*L
{'(-r) =
-f
0+ f\^+",w +
alz
_
<o
{'ct)
G11213,p
6 Gt)
60) )o
(rt tr2-
q4tFt
C"wcaurfur
{'(*)=
-,c\
(Ktt3)L
fu)"ca)
+'(t0=
- (ar*)
a (x'*t
a,)
(nq*17
t$kJffii:f+e
U43Y3
- l,Kx]+[K
-Lt (olr)
(;.tt:)*
fu&ra)'
rLtq r^Q[*" h*r. I o ,a\* ,
Se-f +'t :s).S-,\\rQ, -[t(x]-D-o
?oss
Kz-[:
o
Cxtl)Cn-,):Q
)<= -( Ft
Cotlc.
A
0
p(c,ulu(
ntf
t
-\
+'tcl =
6tr)(3)
Ct ta)3
<o
Crr)e0
>o
l'r{r) = ------>
t)
(;
tt
lp
@, \'tP(..[-r* p-tV{r:
Ft/
t)
Gtt+c,),
(r, t)
ftt€$ ;
il'v L>d
^l-"-{ ^$y "rtt
frKr
t(w ,*+3 rf== r.
fr
[l
*n A=
X€ -$o
)&f:
((, {cr))
#*n*avity nnd lntl*ct[*n
F*Er-rts
Example
The first derivative of a certain function f(x) is
f'(x):x2-2x-8.
(a) Find intervats on wnic@ increasing and decreasing.
(b) Find intervals on which the graph of f is concave up and
concave down.
(c) Find ,n. 9Aordinate of the relative extrema and inflection
points of f.
C) 5'rt 4l= o - S-\w ' /&tK-s
-l
|
+La=C-t)Gr)
d for = CaC-tt)
.
)e
+0
r+
ttiJeA
t\
+
r..-)-L
o
r
[s)= ta tl) >o
,
,
Q<t=Dk {)"- o
:
f,=-a X:4
^-?E-cPr -
A
-.
I"d-
-
t5
''.
t-.
:
ti
n\
\v'}J
: 22,<.-r--'
9a-+ +t\ :d - S-[*.
+tl (-x\
{tlfo)
f,
-ir
t{
=-\<o
G t/c
ltta=lzo
PoWN
rrl
r> ( c-
,j6 l''p{.
-0 I t.*&,,
(c) Lo. *\ r^rtotF o*
t[\I
Loc,q_\ l/A[tn 4{
t^P(e.[-,r,r
ln=L ^d
f,,*f
[,t/e cft^ dt^tp
Gzr7)
trt? )
it (r t7 )
?*+ X-co.,noL,*{es.
The Second Derivative Test
Suppose f't(x) exists on an open interval containing x
that f'(c) 0.
:
I
-
c and
However,if f"(c) 0 or af f"(c) does not exist, the test is
inconclusive and f may have a relative maximum, a relative
minimuffi, or no relative extremum at all at x c.
-
-
The Second Derivative Test
Example
Find the critical points of
f(x):x3 +3x2+1
and use the second derivative test to classify each critical point
as a relative maximum or minimum.
l;E
g
+tA
Co,r,
I
$t[A:
'tx3
v
cqvi[g
Sr{ -Pt( =o - S-\$Q:
+ t t-f
{"
O
$"6^1- lLxL
lzP- o
K=o ?
-l-+ ++
-L
+"c,): []>c
-Q"til
- lt
csel cq tr{.
Y
I
€WrfwtnJ,nt.
I
3.3. Curve Sketching
Vertical Asymptotes
The vertical line x - c is a vertical asymptote of the graph of
f (x)
if either
lim f (x): *oo (or -
oc)
X---+C-
A
or
f
t
"U_
C\,u-rb
8): *oo (or - oo)
s# K u[1"4
Aeno4/.r tr^q
k^ -- O
.
i.@"a
Vertical Asymptotes
Example
Determine all vertical asymptotes of the graph of
g(x):##i
Vet,.
lF,C
^\W
tfns
K}-lr-L{ - o
Cx+1)G-'t) - e
K=-\
*
4l
I
\
It(o v
tr='t'
r---
t
C\nBcb v\q,rrun ,l-'*
- {d + o
act)z+z{*t) = }-3-- 0
?,(.{l
'1,
uqv^['
4Sy r,ttf ,
{
Wo{\i.
A\PoTA
vEraT-AsyKP,
(tt" t= {N qrrft' N)
t+) ('t)
1
Inn
*fLx
L+^-.-{ = h,
}
ax fr+r)
F--[ t+X-'( K){ trtfx=.t)
tsL
-5
-1(
{
I
{
\
(
I
\I
I
Horizontal Asymptotes
The horizontal line y
of f(x) if
or
-
b is a horizontar asymptote of the graph
"IT""f(x)-F
,IT." r(x) -
bL
lt
Horizontal Asymptotes
Example
Determine all horizontal asymptotes of the graph of
, \ 2x2*2x
glx):
*24*_4
K€oo
>&-fx-{
XSoo
P
I
,\
lI
I
I
X= tf
General Procedure for sketching the Graph
1. Find the domain of f (x). /
Step 2. Find and plot all intercepts. ,./
Step 3. Determine all vertical and horizontal asymptotes and
Step
draw them.
step
4. Find
f
'(x) and determine the critical numbers and
intervals of increase and decrease.
Determine all relative extrema. plot qach relative
maximum with a "cap" and each relative minimum with
a "cup".
Step
5.
Step
6. Find f" (x) and determine intervals of concavity and
Step
7.
points of inflection. Plot inflection points with a "twist"
Complete the sketch by joining the ptotted points.
Curve Sketching
Example
:
#rF
,tf
Vqn h"q\ oty
X: il\
'n
t-+r" rlo^yr t-*! as y ur,tp of y
sketch the graph of r(x)
I
l-t*
I i{-c^,iJ':lu
IncNl-ss
sr(A
=
=
G
F =M I =.Q
[]-wrtru,
r))
=-
xt D't
@*fr{ttx+A-Erl ,- -rlx
(
(x+ t)H'3
-4 f x-r\
t
\'t
l)
(o+ ()3
+
L{
(x+ r)3
\
TvFEil
C"
f"
1t JIS
{-
+f+
t
f,= -t
|
,
+- 0 R ,- ^\
*
^a=
t-2)
[,
J*
n
f -
9'(>)
i
+
F
<o
tt)t
-_
f-t{)(r)
=!'/ \-1
Ccrv\Cct v
,,[).)
., tI
GO(*g)
__
r
z
t
-0
lro)
=
Loro[
rf -o- -{
(n-r)
GL\) (-l)
t\)3
"..nn
(^
5C>
*$ (l I t)
Ff)2)
pa
[xt
[xtt)s4
-tn-t
f Lzts -LL
(F+
& (x-D
[r+t)t
(
r)'1
bA-[t
Crt l)+
Pos*,U
lf
ltt
t\r.
+
[q \,n $tq.[1-* J 4 : /
R +
0
-
-
.-
-\
K
- ,-l
+t.f+
-O
L
+'fro)
{'rt3)-- Se>d
v
tq)(
3
_cnGt <O
( r)'(
l.wQtt"[t*,. forur{
qalrA)
Y:o
l"t+
lzto
q9'(*p,
n'2^
d,2cr
.o*
.t
D
E|4?{ I
Curve Sketching
Example
Sketch the graph of f (x)
- "l:
x-5
.glakltr
t.*
Curve Sketching
Example
Sketch the graph of f (x)
- : !-.
t\
-f/1
-T-
|
1
|
q|IrtF{I
*
3.4. Opti mization
Absolute Maxima and Minima of a function
Let f be a function defined on an interval / containing the
number c. Then
x in l.
in /.
Collectively, absolute maxima and minima are called absolute
extrema.
| 'E!IE
Absolute Extrema on a Closed interval
How to Find the Absolute Extrema of a Continuous
Function f on a::;, x ::;, b
1. Find all critical numbers of f in a < x < b.
Step 2. Compute f (x) at the critical numbers found in step
Step
Step
3.
-
1
and at the endpoints x
aand X b.
The largest and smallest values found in step 2 are,
respectively, the absolute maximum and absolute
minimum values of f(x) on a < x < b.
-
il't
nE
Absolute Extrema on a closed interval
Example,-*
7l
/
Find th€ggrg)maximur
a'
I
"n@inimum
(if any) of
f(x): x3 *3x2 + 1; -3< x{2.
{lCD = 3&+(aK
zx>+6r =O
3x(xt
X=
=Q
o'
+ + + o- +- -o++
{- +-
+ tz+ I o
}t rI7+
I +rzrt=Jt
I
L
.-3
Cn1f-
-p
-\
r -
loutk: L,t/
-\
s)
(" I t)
7
,, h/Qr
g
l",,,ib:
(j; t) (aral)
t
.Grr
Absolute Extrema on a crosed interval
Example
Find the absolute maximum and absolute minimum (if any) of
f(f)
-ltsz'
- ::,
f-1' -.2:'
1
@
Absolute Extrema on a generat interval
Example
Find the absolute maximum and absolute minimum (if any) of
f(u)
-
u+
16
u
:
u> o.
""G.il