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
{yotle
d;r
i4
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$kJffii:f+e
U43Y3
- l,Kx]+[K
-Lt (olr)
(;.tt:)*
fu&ra)'
rLtq r^Q[*" h*r. I o ,a\* ,
Se-f +'t :s).S-,\\rQ, -[t(x]-D-o
?oss
Kz-[:
Cxtl)Cn-,):Q
)<= -( Ft
Cotlc.
p(c,ulu(
ntf
-\
+'tcl =
6tr)(3)
Ct ta)3
<o
Crr)e0
>o
l'r{r) = ------>
t)
(;
tt
lp
@, \'tP(..[-r* p-tV{r:
Ft/
t)
Gtt+c,),
(r, t)
ftt$ ;
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
[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.
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
+tA
Co,r,
I
$t[A:
'tx3
v
cqvi[g
Sr{ -Pt( =o - S-\$Q:
+ t t-f
{"
$"6^1- lLxL
lzP- o
K=o ?
-l-+ ++
-L
+"c,): []>c
-Q"til
- lt
csel cq tr{.
WrfwtnJ,nt.
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-
or
"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---
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)
Inn
*fLx
L+^-.-{ = h,
ax fr+r)
F--[ t+X-'( K){ trtfx=.t)
tsL
-5
-1(
{
\
(
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
Koo
>&-fx-{
XSoo
,\
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
'(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)
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+
f,= -t
|
,
+- 0 R ,- ^\
^a=
t-2)
[,
J*
n
f -
9'(>)
<o
tt)t
-_
f-t{)(r)
=!'/ \-1
Ccrv\Cct v
,,[).)
., tI
GO(*g)
__
-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 +
.-
-\
- ,-l
+t.f+
-O
+'fro)
{'rt3)-- Se>d
v
tq)(
_cnGt <O
( r)'(
l.wQtt"[t*,. forur{
qalrA)
Y:o
l"t+
lzto
q9'(*p,
n'2^
d,2cr
.o*
.t
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-
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.
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 thggrg)maximur
a'
"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
.-3
Cn1f-
-p
-\
r -
loutk: L,t/
-\
s)
(" I t)
7
,, h/Qr
l",,,ib:
(j; t) (aral)
.Grr
Absolute Extrema on a crosed interval
Example
Find the absolute maximum and absolute minimum (if any) of
f(f)
-ltsz'
- ::,
f-1' -.2:'
Absolute Extrema on a generat interval
Example
Find the absolute maximum and absolute minimum (if any) of
f(u)
u+
16
u> o.
""G.il
