,derivatives & ,graph+

  ,"o (! tradi;nal big uses =
derivatives is z a tool 9 ond]/&+ &
graph+ func;ns4,x wd 2 easy 6imag9e t 9
? "d ( -put] algebra sy/ems & graph+
calculators1 ? tool (! #17th ,c5tury _h
2come obsolete4 ,i ?9k ? is false4 ,let
me try 6illu/rate j h[ -pletely "o c 2
misl$ 0me*anical graph+ tools4 ,my
"picul> example has be5 run on ,maple1 b
x's easy 5 6f9d curves t def1t !
me*anical graph+ tool ( yr *oice4
  ,we'll />t )a func;n t is sli<tly
messy1 b re,y n all t -plicat$4
  f(x) .k e^-x^^2^
    _/25,000,000"@(sin (?x_/50#)
    -?1_/(2000x-250,000)^2"#@)_4
,"h e ".k<*] #2.718281828459 is a
special numb]1 ! base (! natural logs1 t
we w meet m c>e;lly d[n ! road a bit4 ,=
"r n[1 all t matt]s is t x's "s numb] 2t
#2 & #3_4

                                      #j
  ,! "q is1 :at does ! graph ( y .k f(x)
look l8 ,9 "picul>1 :at >e ! maximum &
m9imum values atta9$ 0! func;n8
  ,! atta*$ ,maple ,"w%eet looks at ?
func;n 9 a bun* ( di6]5t plot w9d[s1 :
,i hope %d %[ y j h[ m* "o's view (a
func;n dep5ds on ! w9d[ y *oose1 h[ h>d
x is 6pick ! "r w9d[4 ,"o al (t5 ne$s m
?an "o w9d[ 9 ord] 6illu/rate all !
important prop]ties (a func;n4 ,=!
record1 ! w9d[s *os5 9 ! ,maple "w%eet 7
  ,',key 6items 9 ! foll[+ ta#
#a -#20 "k x "k #20
#b ,diagonal1 y .k -,= to +,=
#c -#90 "k x "k -#70
#d ,p>abola1 y .k -#1 to +,=
#e -#90 "k x "k -#70, -10 "k y "k #10
#f ,horiz4 l9e ne> y .k -#1
#g -#100 "k x "k #100
#h -#1000 "k x "k #1000
#i ,s9e wave1 y .k -#1 to #1
#aj -#10,000 "k x "k #10,000
#aa -#20,000 "k x "k #20,000
#ab #100 "k x "k #200
                                      #a
#ac #120 "k x "k #130
#ad #124 "k x "k #126
#ae #124.9 "k x "k #125.1
#af ,v]t4 asymp4 to y .k -,=
#ag #124.99 "k x "k #125.01, -5 "k y
  "k #1,'
 

















                                      #b

--------------------------------------
,figure   ,plot w9d[  ,app>5t %ape
--------------------------------------
#1        #a          #b
--------------------------------------
#2        #c          #d
--------------------------------------
#3        #e          #f
--------------------------------------
#4        #g          ,curve
--------------------------------------
#5        #h          #i
--------------------------------------
#6        #aj         ,wave packet
--------------------------------------
#7        #aa         ,wave packet
--------------------------------------
#8        #ab         ,diagonal l9e
--------------------------------------
#9        #ac         ,l9e )a blip
--------------------------------------
#10       #ad         ,l9e )a l;g blip
--------------------------------------
#11       #ae         #af
--------------------------------------
#12       #ag         #af
--------------------------------------
                                      #d
  ,:at ? example ultimately %[s is !
import.e ( 2+ a# 6look at a func;n
analytic,y 9 ord] 6det]m9e ": xs 9t]e/+
f1tures >e4 ,ev5 9 ! _w ( ,maple &
graph+ calculators1 calculus / has a
role 6play 9 graph+4
  ,9cid5t,y1 y mi<t 2 9t]e/$ 6"k h[ ,i
came up ) ? weird func;n4 ,y "k :at !
graph ( y .k sin x looks l4 ,divid+ ;x
by #50 6get y .k sin (x_/50) /ret*es !
graph 0a factor ( #50 9 ! horizontal
direc;n1 yield+ v sl[ waves4 ,! func;n
y .k e^-x^^2 is a bell curve1 relat$ 6!
bell curve us$ 9 /atisics1 & %[n 2l 9
,figure #1_4
  ,divid+ ;x by #5000 6get
e^-x^^2^_/25,000,000 /ret*es ? curve 0a
factor ( #5000 9 ! horizontal direc;n4
,:5 we multiply ? 0! s9e curve
y .k sin (x_/50), we get a wave
oscillat+ 2t ! bottom & top (a wide
5velope made 0? /ret*$ bell curve & xs
neg,n4 ,f9,y1 ! func;n
y .k -?1_/(x-125)^2"# has a v]tical
                                      #e
asymptote at x .k #125_4 ,0replac+ ? )
-?1_/(2000x-250,000)^2"#, we get a
func;n t / has a v]tical asymptote at ?
po9t1 b an exce$+ly n>r[ "o t ,maple
misses un.s we zoom way 94

    ,derivatives1 slopes1 & extrema3 a
    preview
  ,we alr "k a gd bit ab ! 3nec;n 2t
derivatives & graph+4 ,a positive
derivative & an 9cr1s+ func;n g tgr1 z d
a negative derivative &a decr1s+ func;n4
,a derivative ( z]o m1ns a horizontal
tang5t l9e1 : happ5s at maxima & m9ima (
;f_4 ,"! >e "s details t >5't q "r 9 ?
r\< summ>y1 ?\<4 ,= 9/.e1 ! func;n
f(x) .k \x\ has a m9imum at x .k #0, :
is n a po9t ": f'(x) .k #0, b a po9t ":
f'(x) does n exi/4 ,simil>ly1
g(x) .k x^3 has g'(0) .k #0, b x doesn't
h a maximum or a m9imum at (0, 0)_4 ,x
has a horizontal tang5t l9e at (0, 0), b
x if 9cr1s+ "ey": 9 ! s5se t :5"e a "k b
, x foll[s t g(a) "k g(b)_4 ,a gd bit (
                                      #f
\r "w =! next ll :ile w 2 devot$ 6gett+
all ! details "r 9 ? associ,n 2t1 on "o
h&1 po9ts ": ! derivative is positive1
negative1 or z]o1 &1 on ! o!r h&1 po9ts
": ! func;n is 9cr1s+1 decr1s+1 or at an
extremum4

    ,global & local extrema
  ,if we >e plott+ a func;n y .k f(x) on
an 9t]val ,i, "o ?+ t natur,y 9t]e/s u
is is ": ! graph r1*es xs maximum &
m9imum values4 ,a po9t at : ;f r1*es !
l[e/ value x atta9s any": on ;,i is
call$ a m9imum po9t ( ;f on ,i, or1 6be
m precise1 a global or absolute m9imum4
,simil>ly1 a global maximum is a po9t ":
;f atta9s ! l>ge/ value x atta9s any":
on ! 9t]val ;,i_4 ,a func;n c h m ?an "o
global maximum or m9imum3 ?9k ab
f(x) .k sin x on ! 9t]val
#0 "k: x "k: #20_4 ,maxima & m9ima >e
collectively call$ extrema4
  ,a local or relative m9imum is a po9t
o!r ?an an 5dpo9t ( ;,i at : ! graph is
                                      #g
l[] ?an x is at any o!r ne>by po9t4 ,x
ne$ n 2 a global maximum1 h["e4 ,local
maxima >e def9$ ! same way z local
m9ima4 ,on ! surface (! e>?1 ! elev,n
func;n has a global maximum at ! top (
,m.t ,"ee/2 ! top ( e m.ta9 or hill is a
local maximum4 ,z an example1 3sid] !
func;n y .k x^4"-8x^2 on ! 9t]val
,i .k @(-2, 3@), : is %[n 9 ,figure #2_4
  ,? func;n has a local m9imum at
(2, -16) &a local maximum at (0, 0)_4 ,!
global maximum is at (3, 9)_2 "! >e
global m9ima at (-2, -16) & at (2, 16)_4
  ,cle>ly ! only po9ts t cd 2 global
extrema ( ;f on an 9t]val ;,i >e local
extrema ( ;f &! 5dpo9ts (! 9t]val ;,i_2
b :at c we say ab ! local extrema ( ;f_8
  ,let x0 2 a local maximum ( ;f_4 ,!n
"! exi/s an 9t]val @(x0-.@e, x0+.@e@) 9
: e po9t ;x satisfies f(x) "k: f(x0)_4
,let #0 "k h "k .@e_4 ,!n
  ?f(x0+h)-f(x0)_/h# .1: #0_4
,s9ce ? holds = e small1 positive h, x
foll[s t
                                      #h
  "lim%h $o #0^+]  ?f(x0+h)-f(x0)_/h#
    .1: #0,
if ! limit exi/s4 ,simil>ly1 if
-.@e "k h "k #0, !n
  ?f(x0+h)-f(x0)_/h# "k: #0_4
,s9ce ? holds = e small1 negative h, x
foll[s t
  "lim%h $o #0^-]  ?f(x0+h)-f(x0)_/h#
    "k: #0,
if x exi/s4 ,s9ce ! limit f "o side is
negative &! limit f ! o!r side is
positive1 x foll[s t if
  f'(x0)
    .k "lim%h $o #0] ?f(x0+h)-f(x0)_/h#
    .k "lim%h $o #0^-]  ?f(x0+h)-f(x0)
      _/h#
    .k "lim%h $o #0^+]  ?f(x0+h)-f(x0)
      _/h#
exi/s1 !n f'(x0) .k #0_4
  ,we h "!=e %[n t e local m9imum o3urs
ei at a po9t x0 ": f'(x0) .k #0, or at a
po9t x0 ": f'(x0) is undef9$4 ,! same ?+
holds ( local m9ima1 ei 0us+ ! same
pro(1 or 0us+ ! fact t a local m9imum (
                                      #i
y .k f(x) is a local maximum (
y .k -f(x)_4
  ,a po9t at : f'(x) is ei z]o or
undef9$ is call$ a critical po9t ( ;f_4
,:at we h "!=e %[n is t e local extremum
( ;f on ! 9t]val ;,i is a critical po9t
( f, & t e global extremum is ei a
critical po9t ( ;f or an 5dpo9t ( ;,i_4

    ,examples

,example #1
  ,suppose we >e ask$ 6f9d all local &
global extrema (! func;n
f(x) .k #3x^5"-5x^3 on ! 9t]val
,i .k @(-2, #2@)_4 ,we "k t
  f'(x) .k #15x^4"-15x^2
    .k #15x^2"(x-1)(x+1)_4
,! derivative is def9$ at e po9t ;x 9 !
9t]val ;,i_2 s ! only critical po9ts >e
! po9ts at x .k #0, x .k #1, & x .k -#1,
": f'(x) .k #0_4
  ,is "! a way 6tell :e!r ea* ( ^! po9ts
is a local maximum or m9umum or nei )\t
                                     #aj
hav+ ! graph (! func;n8 ,yes4 ,0look+ at
": ea* (! factors #15, x^2, x+1, & x-1 (
f'(x) is positive1 negative1 or z]o1 we
c see t
  f'(x) .1 #0 if x "k -#1 or x .1 #1
  f'(x) "k: #0 if -#1 "k: x "k: #1_4
,? m1ns t ! po9t at x .k -#1 m/ 2 a
local maximum1 t ! po9t at x .k #1 m/ 2
a mocal m9imum1 & t ! po9t at x .k #0 m/
2 nei4
  ,6get ! global extrema1 recall t a
global extremum m/ 2 ei a local extremum
or an 5dpo9t (! 9t]val 9 "q4 ,! #4
possi# global extrema >e "!=e ! #4 po9ts
li/$ 9 ! foll[+ ta#4









                                     #aa
-----------------------
;x   f(x)  ,nature
-----------------------
-#2  -#56  ,global m94
-----------------------
-#1  #2
-----------------------
#1   -#2
-----------------------
#2   #56   ,global max4
-----------------------

  ,d y n[ h a cle> picture 9 yr m9d (!
graph ( y .k f(x)_8 ,if n1 !n try 6pull
tgr "ey?+ we've j le>n$ & 6develop s* an
image4 ,! actual plot is 9 ,figure #3_4

,example #2
  ,suppose we >e ask$ 6f9d all local &
global extrema (! func;n
f(x) .k x-6x^2_/3 on ! 9t]val
,i .k @(-4,125@)_4 ,we "k t
  f'(x) .k #1-4x^-1_/3,
: is z]o :5
                                     #ab
  #1-4x^-1_/3 .k #0
  #4x^-1_/3 .k #1
  ?1_/<3>x]# .k ?1_/4#
  <3>x] .k #4
  x .k #4^3 .k #64_4
  ,"! is a second critical po9t1 ?\<1 :
wd 2 easy 6miss3 ! derivative
  f'(x) .k #1-4x^-1_/3
is undef9$ :5 x .k #0_4
  ,look+ at ! =mula =! derivative1 x
isn't h>d 6see t
  f'(x) .1 #0 if x "k #0 or if x .1 #64
  f'(x) "k #0 if #0 "k x "k #64_4
,! critical po9t at x .k #0 is "!=e a
local maximum1 &! critical po9t at
x .k #64 is a local m9imum4 ,! global
extrema m/ 2 ei ^! local extrema or else
! 5dpo9ts (! 9t]val4 ,e possi# global
extremum is "!=e li/$ 9 ! foll[+ ta#4





                                     #ac
-------------------------------------
;x    f(x)               ,nature
-------------------------------------
-#4   -#4-12>2] ".k<*]
        -#20.97
-------------------------------------
#0    #0                 ,global max4
-------------------------------------
#64   -#32               ,global m94
-------------------------------------
#125  -#25
-------------------------------------

  ,! plot (! func;n is 9 ,figure #4_4

,example #3
  ,f9,y1 let's try ! same ?+ )
f(x) .k x^-2 on ! 9t]val (-2, 2), i4e41
on ! op5 9t]val -#2 "k x "k #2_4 ,!
derivative is
  f'(x) .k -?2_/x^3"#,
: is n"e z]o1 b : is undef9$ at x .k #0
_4 ,"! is "!=e a critical po9t at
x .k #0_4 ,? critical po9t is nei a
                                     #ad
maximum nor a m9imum1 b a v]tical
asymptote1 z %[n 9 ,figure #5_4
  ,! func;n f(x) .k x^-2 "!=e has no
global maximum on ! 9t]val (-2, 2)_4 ,x
c 2 made z l>ge z we l 0tak+ po9ts close
to x .k #0_4
  ,:at may 2 m subtle is t f(x) .k x^-2
al has no global m9imum on ! 9t]val
(-2, 2)_4 ,! value ( f(x) gets small] &
small] z ;x gets clos] & clos] to #2 or
to -#2_4 ,! global m9imum "!=e c't 2 at
any po9t except x .k +-2_4 ,b ! global
m9imum al c't 2 at x .k #2 or x .k -#2,
s9ce nei ( ^! po9ts lies 9 ! 9t]val
-#2 "k x "k #2_4 ,all we c "!=e say is t
f(x) c 2 made z close z we l to ?1_/4#
0tak+ ;x close 5 6"o (! 5dpo9ts (!
9t]val (-2, 2)_2 b t ;f has no m9imum on
? 9t]val4

    ,rolle's ,!orem
  ,rolle's ,!orem &! ,m1n ,value ,!orem
>e two results : >e geometric,y obvi\s1
b : play a c5tral role 9 ! =mal
                                     #ae
develop;t ( calculus if "o is 2+ logic,y
precise4 ,s9ce we're n try+ 6prove e la/
?+ we d1 ^! !orems >e less c5tralto u2 b
!y >e / "p (! basic corpus ( mat]ial p
expect y 6h le>n$ 9 calculus4
  ,rolle's ,!orem says t if ;f is 3t9u\s
on ! clos$ 9t]val @(a, b@) & di6]5tia#
on ! op5 9t]val (a, b), & if
f(a) .k f(b), !n "! is "s po9t ;c,
a "k c "k b, at : f'(c) .k #0_4
  ,! pro( is easy3 ,if ;f is a 3/ant1 !n
f'(c) .k #0 = e a "k c "k b_4 ,if n1 !n
;f has 6h a global maximum &a global
m9imum on @(a, b@)_4 ,n bo? ( ^! c 2
5dpo9ts (! 9t]val1 or else ;f wd 2 a
3/ant4 ,at l1/ "o m/ "!=e 2 a critical
po9t---a po9t at : ei f'(c) .k #0 or
f'(c) is undef9$4 ,s9ce ;f is di6]5tia#
"ey": on (a, b), ! only critical po9ts
>e ^? ": f'(c) .k #0, s we're d"o4
  ,a typical func;n = : ,rolle's ,!orem
applies is %[n 9 ,figure #6_4 ,! po9ts
": f'(c) .k #0 >e %[n 0small seg;ts (!
tang5t l9es at ^? po9ts4
                                     #af
  ,! ,m1n ,value ,!orem is analog\s
6,rolle's ,!orem1 b x drops ! 3di;n t
f(a) .k f(b)_4 ,x ass]ts t if ;f is
3t9u\s on ! clos$ 9t]val @(a, b@) &
di6]5tia# on ! op5 9t]val (a, b), !n "!
is "s po9t ;c, a "k c "k b, at :
  f'(c) .k ?f(b)-f(a)_/b-a#_4
  ,? is actu,y m 9tuitive ?an ! not,n f/
makes y ?9k4 ,! "r h& side1
?f(b)-f(a)_/b-a#, is ! av]age rate (
*ange (! func;n ;f 2t x .k a & x .k b_4
,! left h& side1 f'(c), is ! 9/antane\s
rate ( *ange ( ;f at ;c_4 ,s :at ! ,m1n
,value ,!orem is say+ is t "! is alw a
po9t ": ! 9/antane\s rate ( *ange is
equal 6! av]age rate ( *ange4 ,if y
drive f "h 6,"dton av]ag+ #60 mph1 !n at
"s po9t al;g ! way1 y m/ h be5 go+
exactly #60_4 ,2lieva#8
  ,ano!r way 6?9k ab x is 6say t ?f(b)
-f(a)_/b-a# is ! slope (! secant l9e
jo9+ ! po9ts (a, f(a)) & (b, f(b))_4
,:at ! ,m1n ,value ,!orem says is t "!
is a po9t ;c 2t ;a & ;b at : ! tang5t
                                     #ag
l9e is p>allel 6? secant l9e4
  ,! pro( is "o ( ^? cl"e ma!matical
>gu;ts t uses :at we alr "k 9 a slick
way4 ,3sid] ! func;n
  h(x) .k f(x)-@((?f(b)-f(a)_/b-a#)(x-a)
    +f(a)@)_4
;h is j ;f m9us ! secant l9e jo9+
(a, f(a)) to (b, f(b))_4 ,?us ;h is a
di6];e ( #2 3t9u\s func;ns1 s ;h is
3t9u\s on @(a, b@)_4 ,al1
  h'(x) .k f'(x)-?f(b)-f(a)_/b-a#
is def9$ at e po9t on (a, b)_4 ,f9,y1
  h(a) .k f(a)-@((?f(b)-f(a)_/b-a#)(a-a)
    +f(a)@) .k #0
  h(b) .k f(b)-@((?f(b)-f(a)_/b-a#)(b-a)
    +f(a)@) .k #0_4
  ,we c "!=e apply ,rolle's ,!orem to ;h
6get a po9t ;c ) a "k c "k b at :
  #0 .k h'(c) .k f'(c)-?f(b)-f(a)_/b-a#
  f'(c) .k ?f(b)-f(a)_/b-a#_4
  ,figure #7 %[s a func;n = : ! ,m1n
,value ,!orem applies4 ,! secant l9e
jo9+ (a, f(a)) to (b, f(b)) is al %[n1
al;g )! #2 po9ts at : ! tang5t l9es >e
                                     #ah
p>allel 6? secant l9e4

    ,! ,f/ ,derivative ,te/
  ,a func;n ;f is 9cr1s+ 9 ! 9t]val
@(a, b@) if = any po9ts
a "k: s "k t "k: b, we h f(s) "k f(t)_4
;f is decr1s+ 9 ! 9t]val @(a, b@) if =
any po9ts a "k: s "k t "k: b, we h
f(s) .1 f(t)_4
  ,\r 9tuitive no;n has be5 t 9cr1s+
func;ns & positive derivatives g tgr1 &
t decr1s+ func;ns & negative derivatives
g tgr4 ,we al "k we h 6be a ll m
delicate ?an t3 if f(x) .k x^3, !n ;f is
9cr1s+ "ey":1 b
f'(0) .k #3(0^2") .k #0_4
  ,! correct !orem turns \ 6be3
  ,if f'(x) .1 #0 = all x @e (a, b), !n
;f is 9cr1s+ on @(a, b@)_4
  ,if f'(x) "k #0 = all x @e (a, b), !n
;f is decr1s+ on @(a, b@)_4
  ,if f'(x) .k #0 = all x @e (a, b), !n
;f is 3/ant on @(a, b@)_4

                                     #ai
  ,6prove ! f/ case1 we prove xs
3trapositive1 : is logic,y equival5t3
  ,if ;f is n 9cr1s+ on @(a, b@), !n =
"s c @e (a, b), f'(c) "k: #0_4
  ,if ;f is n 9cr1s+ on @(a, b@), !n "!
exi/ po9ts a "k: s "k t "k: b )
f(s) .1: f(t)_4 ,0! ,m1n ,value ,!orem1
we c "!=e f9d a po9t ;c 2t ;s & ;t s* t
  f'(c) .k ?f(t)-f(s)_/t-s# "k: #0,
z claim$4 ,see if y c d ! o!r #2 cases
yrf4
  ,! ,f/ ,derivative ,te/ h>ve/s ? idea
9 ord] 6classify ! 2havior ( ;f at
critical po9ts4 ,x says1
  ,let ;f 2 di6]5tia# 9 an 9t]val >.d
;c, & let f'(c) .k #0_4
  ,if = all x "k c ne> ;c, f'(x) "k #0,
&= all x .1 c ne> ;c, f'(x) .1 #0, !n ;c
is a local m9imum = ;f_4
  ,if = all x "k c ne> ;c, f'(x) .1 #0,
&= all x .1 c ne> ;c, f'(x) "k #0, !n ;c
is a local maximum = ;f_4
  ,if = all x /.k c ne> ;c, f'(x) .1 #0,
or = all x /.k c ne> ;c, f'(x) "k #0, !n
                                     #bj
;c is nei a local maximum nor a local
m9imum = ;f_4
  ,? makes s5se 2c :at we >e re,y say+
is1
  ,if ;f is decr1s+ 6! left ( ;c &
9cr1s+ 6! "r ( ;c, !n ;c is a local
m9imum4
  ,if ;f is 9cr1s+ 6! left ( ;c &
decr1s+ 6! "r ( ;c, !n ;c is a local
maximum4
  ,if ;f is decr1s+ on bo? sides ( ;c or
9cr1s+ on bo? sides ( ;c, !n ;c is nei a
local m9imum nor a local maximum4 ,x's
wor? tak+ a m9ute & draw+ pictures ( all
^! situ,ns4 ,graphs %[+ ?ree (! ?+s t cd
happ5 >e %[n 9 ,figure #8_4

    ,examples4

,example #1_4
  ,let f(x) .k x^3"-5x^2"+x-1_4 ,f9d all
critical po9ts ( ;f_4 ,det]m9e ": ;f is
9cr1s+ & decr1s+1 & locate xs local
maxima & m9ima4 ,sket* ! graph ( ;f_4
                                     #ba
  ,if f(x) .k x^3"-5x^2"+x-1, !n
f'(x) .k #3x^2"-10x+1_4 ,critical po9ts
o3ur ": f'(x) is undef9$---: n"e happ5s
---or ": f'(x) .k #0, i4e41 at
  x .k ?10+->88]_/6# .k ?5+->22]_/3#_4
,x is easy 6see t f'(x) is an
upw>d-po9t+ p>abola1 : m1ns t
  f'(x) "k #0 if ?5-.>22.]_/3# "k x "k ?
    5+.>22.]_/3#
  f'(x) .1 #0 if x "k ?5-.>22.]_/3# or
    x .1 ?5+.>22.]_/3#_4
  ;f "!=e has a local maximum at
  (?5->22]_/3#, f(?5->22]_/3#))
    .k (?5->22]_/3#, ?-232+44>22]_/27#)
    ".k<*] (0.103194747, -0.948952093)
&a local m9imum at
  (?5+>22]_/3#, f(?5+>22]_/3#))
    .k (?5+>22]_/3#, ?-232-44>22]_/27#)
    ".k<*] (3.230138587, -16.23623309)_4
  ,a sket* is 9 ,figure #9_4

,example #2_4
  ,let f(x) .k ?x^2"+2x+2_/x+2#_4 ,f9d
all critical po9ts ( ;f_4 ,det]m9e ": ;f
                                     #bb
is 9cr1s+ & decr1s+1 & locate xs local
maxima & m9ima4 ,sket* ! graph ( ;f_4
  ,if f(x) .k ?x^2"+2x+2_/x+2#, !n
  f'(x)
    .k ?(x+2)(2x+2)-(x^2"+2x+2)(1)_/(x+
      2)^2"#
    .k ?x^2"+4x+2_/(x+2)^2"#_4
,critical po9ts o3ur ": f'(x) is undef9$
(at x .k -#2) or ": f'(x) .k #0, i4e41
at
  x .k ?-4+->8]_/2# .k -#2+->2]_4
,x is easy 6see t ! num]ator ( f'(x) is
an upw>d-po9t+ p>abola1 negative 2t !
two roots & positive else":4 ,!
denom9ator ( f'(x) is alw non-negative4
,? m1ns t
  f'(x) ^"k #0 if -#2-.>2^.] ^"k x ^"k
    -#2 ^or ^if -2 ^"k x ^"k -#2+.>2^.]
    ^_4
  f'(x) ^.1 #0 ^if x ^"k -#2-.>2^.] ^
    or x ^.1 -#2+.>2^.]^_4
  ,! critical po9t at x .k -#2 is a
v]tical asymptote4 ,2c ! derivative is
negative on bo? sides ( ? asymptote1 x
                                     #bc
m/ 2 t
  "lim%x $o -#2^-]  f(x) .k -,=
  "lim%x $o -#2^+]  f(x) .k +,=_4
  ;f has a local maximum at
  (-2->2], f(-2->2]))
    .k (-2->2], -2-2>2])
    ".k<*] (-3.414213562, -4.828427123)
&a local m9imum at
  (-2+>2], f(-2+>2]))
    .k (-2+>2], -2+2>2])
    ".k<*] (-0.585786438, 0.8284271251)
  ,a sket* is 9 ,figure #10_4

,example #3_4
  ,let f(x) .k #3x^4"-8x^3"-1_4 ,f9d all
critical po9ts ( ;f_4 ,det]m9e ": ;f is
9cr1s+ & decr1s+1 & locate xs local
maxima & m9ima4 ,sket* ! graph ( ;f_4
  ,if f(x) .k #3x^4"-8x^3"-1, !n
f'(x) .k #12x^3"-24x^2 .k #12x^2"(x-2)_4
,critical po9ts o3ur ": f'(x) is undef9$
---: n"e happ5s---or ": f'(x) .k #0, i4
e41 at
  x .k #0 or x .k #2_4
                                     #bd
,x is easy 6see t f'(x) .1 #0 if
x .1 #2, & t f'(x) "k #0 if x "k #2_4 ,?
m1ns t ;f is 9cr1s+ 6! "r ( #2, &
decr1s+ 6! left ( #2_4 ;f "!=e has a
local m9imum at (2, f(2)) .k (2, 15)_2 !
critical po9t at x .k #0 is nei a local
maximum nor a local m9imum4
  ,a sket* is 9 ,figure #11_4

    ,a ,cau;n>y hi/ory4
  ,we n[ "k t
  ,if = all ;x 9 ! 9t]val a "k x "k b,
f'(x) .1 #0, !n ;f is 9cr1s+ 9 ! 9t]val
a "k x "k b_4
  ,if = all ;x 9 ! 9t]val a "k x "k b,
f'(x) "k #0, !n ;f is decr1s+ 9 ! 9t]val
a "k x "k b_4
  ,if = all ;x 9 ! 9t]val a "k x "k b,
f'(x) .k #0, !n ;f is a 3/ant 9 ! 9t]val
a "k x "k b_4
  ,s ! f/ derivative gives u a lot (
9=m,n ab ! graph ( ;f_4
  ,= 9/.e1 let
f(x) .k #9x^4"-11x^3"-3x_4 ,! derivative
                                     #be
( ;f is
  f'(x) .k #36x^3"-33x^2"-3
    .k #3(x-1)(12x^2"+x+1)_4
,! only root ( f' is x .k #1_4 ,x is
easy 6see t f'(x) .1 #0 if x .1 #1, & t
f'(x) "k #0 if x "k #1_4 ,?us1 ;f
decr1ses 6! po9t (1, -5), & ;f !n
9cr1ses 6! "r ( ? po9t4 ,? / l1ves a
l>ge degree ( unc]ta9ty ab ! %ape (!
fraph ( y .k f(x), ?\<4 ,s"eal
possibilities >e %[n 9 ,figures #12, #13
, #14, & #15_4
  ,all ^! graphs satisfy ! 3di;ns impos$
on f(x) .k #9x^4"-11x^3"-3x 0! sign ( xs
f/ derivative4 ,c we use calculus 6tell
: is ! "r graph8

    ,3cav;y & ,second ,derivatives4
  ,! answ] turns \ 6be yes1 & 6rely n on
! 9=m,n ! derivative gives u1 b on !
9=m,n ! second derivative gives u4 ,6see
?1 let's apply ! f/ derivative te/ n 6!
func;n ;f, ^: derivative is f', b 6!
func;n f', ^: derivative is f''_4 ,we
                                     #bf
le>n t
  ,if = all ;x 9 ! 9t]val a "k x "k b,
f''(x) .1 #0, !n f' is 9cr1s+ 9 ! 9t]val
a "k x "k b_4
  ,if = all ;x 9 ! 9t]val a "k x "k b,
f''(x) "k #0, !n f' is decr1s+ 9 !
9t]val a "k x "k b_4
  ,if = all ;x 9 ! 9t]val a "k x "k b,
f''(x) .k #0, !n f' is a 3/ant 9 !
9t]val a "k x "k b_4

  ,:at does x m1n = f' 6be 9cr1s+1 ?\<8
,x m1ns t ! slope (! tang5t l9e to ;f is
gett+ l>g]1 : m1ns t ;f is ei 9cr1s+ at
a fa/] & fa/] rate z y g 6! "r1 or t ;f
is decr1s+1 b do+ s m sl[ly z y g 6! "r4
,func;ns ^: derivatives >e 9cr1s+ >e sd
6be .3cave .up4 ,?ree func;ns t >e 3cave
up >e %[n 9 ,figure #16_4
  ,simil>ly1 = f' 6be decr1s+ m1ns t !
slope (! tang5t l9e to ;f is gett+
small]1 : m1ns t ;f is ei decr1s+ at a
fa/] & fa/] rate z y g 6! "r1 or t ;f is
9cr1s+1 b do+ s m sl[ly z y g 6! "r4
                                     #bg
,func;ns ^: derivatives >e decr1s+ >e sd
6be .3cave .d[n4 ,?ree 3cave d[n func;ns
>e %[n 9 ,figure #17_4
  ,s :at we've n[ le>n$ is
  ,if = all ;x 9 ! 9t]val a "k x "k b,
f''(x) .1 #0, !n ;f is 3cave up 9 !
9t]val a "k x "k b_4
  ,if = all ;x 9 ! 9t]val a "k x "k b,
f''(x) "k #0, !n ;f is 3cave d[n 9 !
9t]val a "k x "k b_4
  ,if = all ;x 9 ! 9t]val a "k x "k b,
f''(x) .k #0, !n ;f is a /rai<t l9e 9 !
9t]val a "k x "k b_4

  ,"o o!r bit ( t]m9ology is use;l3 ,an
.9flec;n .po9t is a po9t at : ! second
derivative ( ;f *anges sign1 i4e41 a
po9t ;c s* t ;f is 3cave up on "o side (
;c & 3cave d[n on ! o!r side ( ;c_4 ,!
?ree func;ns %[n 9 ,figure #18 all h
9flec;n po9ts at x .k #0_4
  ,! only way =! second derivative ( ;f
6*ange sign at x .k c is if f''(c) is ei
#0 or undef9$2 s all 9flec;n po9ts
                                     #bh
satisfy ? 3di;n4 ,x is n ! case1 h["e1 t
e po9t at : f''(x) is z]o or undef9$ is
an 9flec;n po9t4 ,3sid]1 = 9/.e1 !
func;n f(x) .k x^4_4 ,we h
f'(x) .k #4x^3 & f''(x) .k #12x^2, s
f''(0) .k #0_4 ,b a look at ! graph (
y .k f(x) .k x^4 (,figure #19) %[s t ;f
is 3cave up "ey":1 s t "! is no 9flec;n
po9t at x .k #0_4 ,"o cd al see ?
algebraic,y1 s9ce cle>ly
f''(x) .k #12x^2 .1: #0 = all ;x_4

    ,! cau;n>y example resolv$4
  ,n[ let's return 6\r 9itial "q ab !
graph ( y .k #9x^4"-11x^3"-3x_4 ,we h
  y .k #9x^4"-11x^3"-3x
  y' .k #36x^3"-33x^2"-3 .k #3(x-1)(
    12x^2"+x+1)
  y'' .k #108x^2"-66x .k #6x(18x-11)_4
  ,? m1ns t "! >e possi# 9flec;n po9ts =
;y at x .k #0 & at x .k ?11_/18#, ":
y'' .k #0_4 ,9 fact1 x is easy 6see
0look+ at ! sign ( ea* (! factors ( y''
t
                                     #bi
  y'' ^.1 #0 if x ^.1 ?11_/18#
  y'' ^"k #0 ^if #0 ^"k y ^"k ?11_/18#
  y'' ^.1 #0 ^if x ^"k #0^_4
,the ^function y ^.k #9x^^4^-11x^^3^-3x
 ^must ^therefore ^be ^concave ^down ^
between x ^.k #0 ^and x ^.k ?11_/18#[
 ^and ^it ^must ^be ^concave ^up ^
everywhere ^else_4 ^,a ^look ^at ^the ^
graphs ^in ^,figures #12--#15 ^shows ^
that ^only ^the ^graph ^in ^,figure #13
 ^satisfies ^this ^condition_4 ^,it ^
is ^in ^fact ^the ^correct ^graph_4

    ,an example ( \r graph+ tools4
  ,3sid] ! func;n
  f(x) .k #4x+?1_/x-1#+?1_/x+1#
  f'(x) .k #4-?1_/(x-1)^2"#-?1_/(x+
    1)^2"# .k ?2(2x^4"-5x^2"+1)_/(x-1)^2
    "(x+1)^2"#
  f''(x) .k ?2_/(x-1)^3"#+?2_/(x+1)^3"#
    .k ?4x(x^2"+3)_/(x-1)^3"(x+1)^3"#_4
,suppose we want 6det]m9e all critical
po9ts ( ;f, all 9flec;n po9ts ( ;f, &
all extrema ( ;f_4 ,we want 6"k ": ;f is
                                     #cj
9cr1s+ & decr1s+1 & ": x is 3cave up &
3cave d[n4 ,f9,y1 we want ! graph ( ;f_4
  ,s9ce f'' is a bit simpl] ?an f', or
at l1/1 s9ce x seems 6factor fur!r1
let's />t ) x4 ,! possi# posi;ns (
9flec;n po9ts >e at x .k +-1, ": f''(x)
is undef9$1 & at x .k #0, ":
f''(x) .k #0_4 ,! po9ts at x .k +-1 >e n
9flec;n po9ts1 b v]tical asymptotes1 z c
qkly 2 se5 0look+ at ! equ,n = f(x)_4
,on ! o!r h&1 x is easy 6see 0look+ at !
signs ( ea* (! factors t make up f''(x)
t if -#1 "k x "k #0, !n f''(x) .1 #0, &
if #0 "k x "k #1, !n f''(x) "k #0_4 ,!
po9t (0, 0) is "!=e an 9flec;n po9t =
;f_4 ,x is equ,y cle> t
  f''(x) "k #0 9 (-,=, -1)
  f''(x) .1 #0 9 (-1, 0)
  f''(x) "k #0 9 (0, 1)
  f''(x) .1 #0 9 (1, ,=)_4
,?us1 ;f is 3cave d[n 9
(-,=, -1).+(0, 1), & 3cave up 9
(-1, 0).+(1, ,=)_4 ,9cid5t,y1 x is use;l
6draw a numb] l9e 6keep track ( ^!
                                     #ca
9t]vals1 : lets y keep all y 9=m,n 9 "o
n1t picture4 ,a second numb] l9e c !n
hold ! 9=m,n obta9$ f f'(x)_4
  ,n[ let's look at
  f'(x) .k ?2(2x^4"-5x^2"+1)_/(x-1)^2"(x
    +1)^2"#_4
,"! >e critical po9ts at x .k +-1, ":
f'(x) is undef9$4 ,^! po9ts h alr be5
%[n 6be v]tical asymptotes4 ,"! >e al
critical po9ts at ! po9ts ":
f'(x) .k #0, i4e41 at ! roots (
  #2x^4"-5x^2"+1 .k #0_4
,? equ,n looks na/y1 b 9 fact1 x's j a
quadratic equ,n 9 x^2_4 ,! quadratic
=mula says t xs roots >e
  x^2 .k ?5+->17]_/4#
  x .k +->?5+-.>17.]_/4#]
  x ".k<*] +-1.51022, +-0.468213_4
  ,a bit ( ?"\ or a bit ( plu7+ 9 numb]s
lets u "w \ ! sign (! num]ator 9 f'(x)_3
  2(2x^4"-5x^2"+1) "k #0 if #0.468213
    "k \x\ "k #1.51022,
& #2(2x^4"-5x^2"+1) .1: #0 o!rwise4 ,!
denom9ator ( f'(x) is alw non-negative4
                                     #cb
,s except at ! #2 v]tical asymptotes1 we
h
  f'(x)
    .1 #0 if x "k -#1.51
    f'(x) "k #0 if -#1.51 "k x "k -#0.47
    f'(x) .1 #0 if -#0.47 "k x "k #0.47
    f'(x) "k #0 if #0.47 "k x "k #1.51
    f'(x) .1 #0 if #1.51 "k x_4
,! critical po9ts at -#1.51 & #0.47 >e
"!=e local maxima1 :ile ! "os at -#0.47
& #1.51 >e local m9ima4 ,:5 we take all
? 9=m,n tgr1 we get ! graph ( y .k f(x),
: is %[n 9 ,figure #20_4

    ,! second derivative te/4
  ,al? we c use ! second derivative ( ;f
l ? 6get -plicat$ 9=m,n ab ! graph ( f,
"! is al a simple ?+ we c d )! second
derivative4 ,suppose f'(c) .k #0_4 ,!n
x .k c mi<t 2 a local maximum or a local
m9imum or nei4 ,"o way 6tell 9 mo/ cases
is 6take ! second derivative ( ;f_4 ,if
f'(c) .k #0, & if f''(x) exi/s & is
3t9u\s 9 a nei<borhood ( ;c, !n
                                     #cc
  ,if f''(c) .1 #0, !n (c, f(c)) is a
    local m9imum4
  ,if f''(c) "k #0, !n (c, f(c)) is a
    local maximum4
  ,if f''(c) .k #0, !n any?+ c happ5 at
    (c, f(c))_4
  ,calculus books typic,y prove ?1 b x
al j makes s5se4 ,try 6draw sket*es1 &
y'll see t ! only way = ;f 6h a
horizontal tang5t l9e at (c, f(c)) &= ;f
6be 3cave up at (c, f(c)) is if
(c, f(c)) is a local m9imum4 ,simil>ly1
! only way = ;f 6h a horizontal tang5t
l9e at (c, f(c)) &= ;f 6be 3cave d[n at
(c, f(c)) is if (c, f(c)) is a local
maximum4
  ,6see t any?+ c happ5 if f'(c) .k #0 &
f''(c) .k #0, 3sid] ! func;ns
  f(x) .k x^4
  g(x) .k -x^4
  h(x) .k x^3
  k(x) .k -x^3_4
  ,all ^! func;ns h y' .k y'' .k #0 at
x .k #0_4 ,b ;f has a local & global
                                     #cd
m9imum at x .k #0, ;g has a local &
global maximum at x .k #0, ;h has an
9flec;n po9t at x .k #0 & is alw 9cr1s+1
& ;k has an 9flec;n po9t at x .k #0 & is
alw decr1s+4

,two qk examples4
  ,if f(x) .k #3x^4"+4x^3"-12x^2"+13, !n
f'(x) .k #12x^3"+12x^2"-24x
.k #12x(x-1)(x+2)_4 ,! func;n ;f "!=e
has critical po9ts at #0, #1, & -#2_4 ,!
second derivative is
f''(x) .k #36x^2"+24x-24, s
f''(0) .k -#24 "k #0,
f''(1) .k #36 .1 #0,
f''(-2) .k #72 .1 #0_4 ,"!=e ;f has a
local maximum at (0, 13), a local m9imum
at (1, 8), &a local m9imum at
(-2, -19)_4,! plot is %[n 9 ,figure #21
_4
  ,if g(x) .k #4x^5"+5x^4"+11, !n
g'(x) .k #20x^4"+20x^3 .k #20x^3"(x+1)_4
,! func;n ;g "!=e has critical po9ts at
#0 & -#1_4 ,! second derivative is
                                     #ce
g''(x) .k #80x^3"+60x^2, s g''(0) .k #0
& g''(-1) .k -#20 "k #0_4 ,"!=e ;g has a
local maximum at (-1, 12)_4 ,! second
drivative te/ doesn't tell u ! nature (!
critical po9t at x .k #0, b 0look+ at
g'(x) .k #20x^3"(x+1), x is cle> t
g'(x) "k #0 if ;x is sli<tly less ?an
#0, & t g'(x) .1 #0 if ;x is sli<tly
grt] ?an #0_4 ,?us1 (0, 11) m/ 2 a local
m9imum = ;g_4,! plot is %[n 9 ,figure
#22_4

    ,limits at ,9f9;y4
  ,0us+ ! 9=m,n ! f/ & second
derivatives ( ;f give u1 we c -e v close
6-pletely plott+ ! graphs ( mo/ func;ns4
,"! is only "o bit (! picture we hav5't
yet ?"\ ab---! 2havior ( func;ns at l>ge
(or l>ge negative) values ( ;x_4
  ,3sid]1 = 9/.e1 ! func;n
  f(x) .k ?x^2"-1_/x#
  f'(x) .k ?x^2"+1_/x^2"#
  f''(x) .k -?2_/x^3_4#
,? func;n has no critical po9ts or
                                     #cf
9flec;n po9ts1 s9ce nei ! f/ nor second
derivatives c equal #0_4 ,x has a
v]tical asymptote at x .k #0, &
  "lim%x $o #0^+]  f(x) .k
    "lim%x $o #0^+]  ?x^2"-1_/x# .k -,=
  "lim%x $o #0^-]  f(x) .k
    "lim%x $o #0^-]  ?x^2"-1_/x# .k
    +,=_4
  ,bo? ! num]ator & denom9ator (!
derivative >e alw non-negative2 s except
at ! po9t x .k #0, we h f'(x) .1 #0_4 ,!
func;n ;f is "!=e 9cr1s+ except at !
4cont9u;y at ! v]tical asymptote4 ,j
look+ at ! second derivative1 we c see t
  f''(x) "k #0 if x .1 #0,
  f''(x) .1 #0 if x "k #0_4
,?us ;f is 3cave d[n = positive x, &
3cave up = negative ;x_4
  ,ev5 ) all ? 9=m,n1 ?\<1 "! >e / a
v>iety ( di6]5t %apes ! graph ( ;f cd
take4 ,?ree possibilities >e %[n 9
,figures #23, #24, & #25_4,^! plots di6]
9 ! 2havior ( ;f = l>ge values ( ;x &=
l>ge negative values ( ;x_4 ,h[ c we
                                     #cg
tell : ( ^! graphs is "r8
  ,"p (! answ] is t ! graphs 9 ,figures
#23--#25 di6] 9 _! limits at 9f9;y4 ,=!
func;n 9 ,figure #23,
  "lim%x $o ,=] f(x) .k #0,
:ile =! func;ns 9 ,figures #24 & #25,
  "lim%x $o ,=] f(x) .k ,=_4
  ,s h[ wd we "w \
  "lim%x $o ,=] ?x^2"-1_/x#=
,well1 ! natural ?+ 6d wd 2 6br1k !
func;n up z
  f(x) .k ?x^2"-1_/x# .k x-?1_/x#_4
,:5 ;x gets v l>ge1 ?1_/x# $o #0_2 s
f(x) .k x-?1_/x# ".k<*] x $o ,=_4 ,?us1
  "lim%x $o ,=] f(x) .k ,=_4
,? elim9ates ! graph 9 ,figure #23 f
3sid],n1 & l1ves u ) "s?+ l ! graphs 9
,figures #24 & #25_4
  ,n[1 ? example reli$ on \r see+ h[
6br1k f(x) up 96a piece t w5t to #0 &a
piece t obvi\sly w5t to ,=_4 ,we may n
alw f9d ? trivial4 ,h[1 9 g5]al1 c we
-pute limits ( r,nal func;ns at +-,=_8
,6f9d \1 let's j look at #3 typical
                                     #ch
examples4
  ,f/1 "h's a case ": ! degree (!
denom9ator is l>g] ?an ! degree (!
num]ator4 ,9 ? case1 we divide bo? top &
bottom 0! hi<e/ p[] ( ;x f.d 9 !
denom9ator4
  "lim%x $o ,=] ?2x^3"-11x^2"+5_/4x^5"
      +x^2"-3x#
    .k "lim%x $o ,=] ,??2x^3"_/x^5"#-?
      11x^2"_/x^5"#+?5_/x^5"#,_/?4x^5"
      _/x^5"#+?x^2"_/x^5"#-?3x_/x^5"#,#
    .k "lim%x $o ,=] ,??2_/x^2"#-?11
      _/x^3"#+?5_/x^5"#,_/4+?1_/x^3"#-?3
      _/x^4"#,#_4
,z x $o ,=, ! num]ator approa*es #0 &!
denom9ator approa*es #4_2 s ! limit m/ 2
?0_/4# .k #0_4 ,x isn't h>d 6see t !
same ?+ wd happ5 :5"e ! denom9ator has
hi<] degree ?an ! num]ator4
  ,next1 let's look at a case ": !
num]ator & denom9ator h ! same degree4
,ag1 we divide top & bottom 0! hi<e/ p[]
( ;x f.d 9 ! denom9ator4
  "lim%x $o ,=] ?2x^5"-11x^2"+5_/4x^5"
                                     #ci
      +x^2"-3x#
    .k "lim%x $o ,=] ,??2x^5"_/x^5"#-?
      11x^2"_/x^5"#+?5_/x^5"#,_/?4x^5"
      _/x^5"#+?x^2"_/x^5"#-?3x_/x^5"#,#
    .k "lim%x $o ,=] ,?2-?11_/x^3"#+?5
      _/x^5"#,_/4+?1_/x^3"#-?3
      _/x^4"#,#_4
,z x $o ,=, ! num]ator approa*es #2 &!
denom9ator approa*es #4_2 s ! limit m/ 2
?2_/4# .k ?1_/2#_4 ,x isn't h>d 6see t ?
is al typical---if ! num]ator &
denom9ator h ! same degree1 !n ! limit
is ! ratio ( _! l1d+ coe6ici5ts4
  ,f9,y1 "h's an example ": ! num]ator
has hi<] degree ?an ! denom9ator4 ,?
case takes a bit m ?"\4 ,let
  f(x) .k ?2x^5"-4x^4"+x^3"+4x^2"-3x
    +5_/x^3"-2x^2"+8#_4
,if we ag divide top & bottom 0! hi<e/
p[] ( ;x 9 ! denom9ator1 we get "s?+ l
?3
  "lim%x $o ,=] ?2x^5"-4x^4"+x^3"+4x^2"-
      3x+5_/x^3"-2x^2"+8#
    .k "lim%x $o ,=] ,??2x^5"_/x^3"#-?
                                     #dj
      4x^4"_/x^3"#+?x^3"_/x^3"#+?4x^2"
      _/x^3"#-?3x_/x^3"#+?5_/x^3"#,_/
      ?x^3"_/x^3"#-?2x^2"_/x^3"#+?8
      _/x^3"#,#
    .k "lim%x $o ,=] ,?2x^2"-4x+1+?4_/x#
      -?3_/x^2"#+?5_/x^3"#,_/1-?2_/x#+?8
      _/x^3"#,#
    .k "lim%x $o ,=] ,?x^2"(2-?4_/x#+?1
      _/x^2"#+?4_/x^3"#-?3_/x^4"#+?5
      _/x^5"#),_/1-?2_/x#+?8_/x^3"#,#_4
  ,n[ "o r1sons l ?3 ,:5 ;x is v l>ge 9
absolute value1 ! denom9ator "h is close
to #1_4 ,! num]ator is x^2 "ts "s?+ v
close to #2_4 ,s r\<ly1 ! :ole func;n is
?2x^2"_/1#, : approa*es +,= :5
x $o +-,=_4
  ,n[1 calculus books seem n 6want 6d ?1
b x's v easy at ? po9t 6be a bit m
sophi/icat$ ?an ?4 ,if we take ! orig9al
r,nal func;n & j divide ! denom9ator 96!
num]ator ) l;g divi.n1 we get
  ?2x^5"-4x^4"+x^3"+4x^2"-3x+5_/x^3"
    -2x^2"+8#
    .k #2x^2"+1+?-3x+2_/x^3"-2x^2"+8#_4
                                     #da
,:5 ;x is l>ge 9 absolute value1 !
frac;n ?-3x+2_/x^3"-2x^2"+8# is close to
#0, s9ce xs denom9ator has a bi7] degree
?an xs num]ator4 ,?us1 = l>ge x,
  f(x) ".k<*] #2x^2"+1_4
,x is "!=e ! case t
  "lim%x $o +-,=] ?2x^5"-4x^4"+x^3"
    +4x^2"-3x+5_/x^3"-2x^2"+8#
    .k "lim%x $o ,=] (2x^2"+1) .k +,=_2
b "o c say m ?an t4 ,! func;n f(x) is
asymptotic to #2x^2"+1_4 ,figure #26 %[s
! graphs ( f(x) & #2x^2"+1 tgr4
  ,let's f9i% ? sec;n )! example we />t$
)3 f(x) .k ?x^2"-1_/x#_4 ,if we d j :at
we 7 ask$ 6d 9 ? la/ example & divide "?
by x, we get f(x) .k x-?1_/x#_4 ,:5 ;x
is l>ge 9 absolute value1 ?1_/x# is
close to #0_2 s f(x) ".k<*] x_4 ,figure
#27 %[s ;x & f(x) tgr1 3firm+ t ! two
curves >e asymptotic1 && %[+ t ! plot 9
,figure #25 0 correct4

    ,a sket*y summ>y4

                                     #db
  ,= me1 a use;l way 63ceptualize m* (
:at we h d"o s f> ) derivatives & graph+
is z foll[s3
  ,let ;f & f' & f'' 2 3t9u\s 9 an
9t]val4 ,? implies t if f'(c) .1 #0, !n
f'(x) .1 #0 = e ;x 9 a nei<borhood (
;c_4 ,! same sort ( ?+ holds = ;f & f'',
& if ! func;n is less ?an #0 9/1d (
grt]4 ,func;ns l ? >e ! normal sort (
unsurpris+ func;ns4 ,"! >e !n ?ree
possi# k9ds ( 2havior at a po9t ;x_4

  ,if f'(x) .1 #0, !n ;f is 9cr1s+ ne>
;x_4
  ,if f'(x) "k #0, !n ;f is decr1s+ ne>
;x_4
  ,if f'(x) .k #0, !n we don't "k :at's
happ5+ at ;x_4 ,"! cd 2 a local maximum
or m9imum ! way -x^2 & x^2, resp41 h at
x .k #0_4 ,! func;n ;f cd al 2 9cr1s+ or
decr1s+ ne> ;x, ! way x^3 & -x^3, resp41
>e at x .k #0_4 ,x's 9t]e/+ t ^! po9ts
": "! >e lots ( possi# k9ds ( 2havior1 &
": we don't "k :at's go+ on1 >e j !
                                     #dc
po9ts t 9t]e/ u mo/4

  ,! second derivative has a simil> set
( possibilities4
  ,if f''(x) .1 #0, !n ;f is 3cave up
ne> ;x_4
  ,if f''(x) "k #0, !n ;f is 3cave d[n
ne> ;x_4
  ,if f''(x) .k #0, !n we don't "k :at's
happ5+ at ;x_4 ,"! cd 2 an 9flec;n po9t
l x^3"+x has at x .k #0_4 ,"! cd al 2 a
maximum or m9imum at ;x, z -x^4 & x^4,
resp41 h at x .k #0_4

  ,f9,y1 if f'(x) .k #0, !n "! >e ag
?ree possibilities dep5d+ on ! sign (!
second derivative at ;x_4
  ,if f''(x) .1 #0, !n ;f has a local
m9imum at ;x_4
  ,if f''(x) "k #0, !n ;f has a local
maximum at ;x_4
  ,if f''(x) .k #0, !n we don't "k :at's
happ5+ at ;x_4 ,"! cd 2 an 9flec;n po9t
l x^3 has at x .k #0_4 ,"! cd al 2 a
                                     #dd
maximum or m9imum at ;x, z -x^4 & x^4,
resp41 h at x .k #0_4

  ,x seems 6me 6be a use;l po9t ( view1
z well z "o ! calculus books don't (t5
l1d "o 6"u/&1 t 3di;ns l f'(x) .k #0 or
f''(x) .k #0 >e n tell+ y t ;x is an
extremum or an 9flec;n po9t4 ,9/1d1 !y
>e tell+ y t ;x is "o (! few po9ts ^:
/atus we don't "k1 & : we "\ "!=e
69ve/igate m fully4

    ,max-,m9 ,pro#ms4
  ,"o (! big ?+s calculus is gd = is
solv+ pro#ms 9 : "o looks =! maximum or
m9imum value ( "s quant;y4 ,! tools t
let u plot graphs >e1 af all1 exactly !
tools "o ne$s 6f9d global or local
maxima & m9ima4 ,9 mo/ ( ^! pro#ms1 "!
is a -mon me?od3 ,f9d ! quant;y 6be
optimiz$4 ,write t quant;y z a func;n (a
s+le v>ia#4 ,f9d ! maximum or m9imum ( t
func;n4 ,"h >e a few examples1 tak5 f !
appall+ly /&>diz$ li/ ( /&>d optimiz,n
                                     #de
pro#ms "o meets 9 ? busi;s4

,maximiz+ >ea4
  ,a typical v].n ( ? pro#m is "s?+ l ?3
,a f>m] wants 6f;e 9 a corral1 : is
suppos$ 6be a rectangle divid$ d[n !
c5t] 0ano!r f;e1 z %[n 9 ,figure #28_4 (
,presumably ! f>m] has an 9f9ite am.t (
flat l&1 & he is rais+ #2 types (
animals t don't play well tgr1 l %eep &
jagu>s_4) ,! total am.t ( f5c+ availa# =
? project is #200 y>ds4 ,:at >e !
dim5.ns (! corral )! maximum >ea8
  ,if we let ;x & ;y 2 ! dim5.ns (!
corral1 !n we "k ! quant;y we >e try+
6maximize is ! >ea1 ,a .k xy_4 ,? isn't
yet a func;n ( #1 v>ia#1 ?\<---we ne$
6get rid ei ( ;x or ( ;y 0f9d+ "s equ,n
relat+ ! two4
  ,? equ,n is ! 3/ra9t t #2x+3y .k #200,
: says t ! total am.t ( f5c+ is #200
y>ds4 ,us+ ?1 we c write
  y .k ?200-2x_/3#,
: m1ns t ! func;n we want 6maximize is
                                     #df
  ,a(x) .k x(?200-2x_/3#)
    .k ?200_/3#x-?2_/3#x^2_4
,we want ! maximum value ( ? func;n on !
9t]val #0 "k: x "k: #100, s9ce cle>ly ;x
c't 2 less ?an #0 & s9ce if x .1 #100,
!n y .k ?200-2x_/3# "k #0_4
  ,at ? po9t1 ! pro#m is easy3 we -pute
  ,a'(x) .k ?200_/3#-?4_/3#x_4
,! only critical po9t happ5s :5
,a'(x) .k #0, at x .k #50_4 ,! global
maximum >ea m/ "!=e 2 ei at ? po9t or at
"o (! 5dpo9ts (! 9t]val @(0,100@)_4
,s9ce ! >ea is cle>ly #0 at ! 5dpo9ts1
?\<1 ! maximum m/ 2 :5 x .k #50,
y .k ?100_/3#, ,a .k ?5000_/3#_4

,m9imiz+ l5g?4
  ,a v>i,n on ! corral pro#m wd 2 ?3
,suppose ! f>m] ne$s ! same %ape corral
z 9 ! previ\s pro#m4 ,he "ks1 h["e1 t !
corral ne$s a total >ea ( #4000 squ>e
y>ds4 ,:at's ! m9imum am.t ( f5c+ ne$$
6d ! job1 & :at >e ! dim5.ns (! result+
corral8
                                     #dg
  ,? "t1 ! quant;y 6be m9imiz$ is !
total f;e l5g?1 #2x+3y_4 ,z 2f1 ? isn't
a func;n ( #1 v>ia#1 s we c't yet use
calculus on x4
  ,we c get rid ( "o (! two v>ia#s1 ?\<1
0rememb]+ t ! >ea is suppos$ 6be
xy .k #4000 squ>e y>ds1 : m1ns t
y .k ?4000_/x#_4 ,plu7+ ? 9 = ;y m1ns t
! quant;y 6be m9imiz$ is
  l(x) .k #2x+?12000_/x#_4
,we ne$ 6m9imize ? on ! 9t]val (0, +,=),
s9ce ! wid? c't 2 less ?an #0, b ! l5g?
c 2 z l;g z desir$4 ,9 r1l life1 "o's
nei<bors mi<t n 2 s oblig+4
  ,z 2f1 ! pro#m is re,y pretty easy
0n[4 ,! func;n l(x) has a s+le critical
po9t ":
  #0 .k l'(x) .k #2-?12000_/x^2"#,
i4e41 ": x^2 .k #6000, or
x .k >6000] .k #20>15] ".k<*] #77.5
y>ds4 ,x is cle> t if ;x is close
69f9;y1 !n ! am.t ( f5c+ ne$$ is v l>ge4
,? is al true :5 ;x is close to #0, s9ce
9 t case1 ;y is v l>ge4 ,! global m9imum
                                     #dh
m/ "!=e 2 at ! critical po9t1 ":
x .k #20>15],
y .k #4000_/x .k ?40_/3#>15],
l .k #80>15]_4

,m9imiz+ surface >ea4
  ,a m obvi\sly use;l pro#m is ! foll[+3
,we want 63/ruct a cyl9drical b>rel
design$ 6hold #8 cubic feet ( fluid4
,:at >e ! dim5.ns t m9imize ! am.t (
/eel us$ =! top1 bottom1 & sides (!
b>rel8
  ,if we imag9e t ! b>rel is ;h feet hi<
& has a radius ( ;r feet1 !n ! am.t (
/eel us$ = ea* (! top & bottom is
.pr^2_4 ,! am.t us$ =! side is #2.prh_4
,s we >e try+ 6m9imize ! surface >ea
  ,s .k .pr^2"+2.prh_4
  ,z usual1 ? isn't a func;n ( "o v>ia#1
b ( two1 ;r & ;h_4 ,we ne$ an equ,n
relat+ ^! two v>ia#s 9 ord] 6get rid (
"o4 ,? equ,n is ! 3/ra9t t ! volume (!
b>rel 2 #8 cubic feet1 : says 9 algebra
t
                                     #di
  .pr^2"h .k #8
  h .k ?8_/.pr^2"#_4
,? m1ns we ne$ 6m9imize
  ,s(r) .k .pr^2"+2.pr?8_/.pr^2"#
    .k .pr^2"+?16_/r#_4
,we presumably want 6d ? on ! 9t]val
#0 "k r "k +,=, ?\< ! b>rels built ne> !
5ds ( ? 9t]val wd n 2 easy 6h&le4
  ,once we're d[n 6calculus1 x's n h>d4
,x's cle> t if ;r is close ei to #0 or
to ,=, !n ,s(r) is v l>ge4 ,! m9imum
"!=e has 6happ5 at a critical po9t (
;,s_4 ,n[1
  ,s'(r) .k #2.pr-?16_/r^2"#
    .k ?2.pr^3"-16_/r^2"#,
s ! only critical po9t = ;,s 9 ! 9t]val
(0, ,=) o3urs ":
  #2.pr^3"-16 .k #0
  r^3 .k ?8_/.p#
  r .k ?2_/.p^1_/3"#_4
,at ? po9t1
  h .k ?8_/.pr^2"# .k ?2_/.p^1_/3"#_4
  ,x's wor?:ile 63template :y x mi<t 2 t
r1l b>rels >e n built 6dim5.ns l ^!4
                                     #ej
,m9imiz+ "t4
  ,f9,y1 let's d a pro#m : is pos$ 9 a
frivol\s way1 b : l1ds u 6"s v important
pr9ciples4
  ,a fi%]man is #1 mile \ 9 a lake )a
/rai<t ea/-we/ %orel9e :5 he cat*es a
/up5d\s fi%4 ,he wants 6%[ ? fi% 6his
frs z soon z possi#4 ,! frs >e at a b>
#1 mile 9l& & #2 miles we/ ( 8 posi;n1 z
%[n 9 ,figure #29_4 ,! fi%]man c r[ at
#2 mph1 & c run at #5 mph4 ,6:at po9t %d
he r[ 9 ord] 6m9imize 8 travel "t 6! b>8
  ,! optimal pa? w probably n 2 a /rai<t
l9e1 s9ce ! fi%]man c run fa/] ?an he c
r[4 ,he w "!=e probably want 6h1d m
ne>ly t[>d ! %ore1 9 ord] 6trade r[+
4t.e = runn+ 4t.e4
  ,z %[n 9 ! figure1 if ! fi%]man h1ds
=a po9t a 4t.e ;x f ! ne>e/ po9t on
%ore1 !n 0! ,py?agor1n ,!orem1 he ne$s
6r[ a 4t.e ( >1+x^2"] & 6run a 4t.e
>1+(2-x)^2"]_4 ,! "t ? w take is
  ,t(x)
    .k ?>1+x^2"]_/2#+?>1+(2-x)^2"]_/5#_4
                                     #ea
  ,6m9imize ! "t1 we f/ take xs
derivative & look = critical po9ts4 ,we
f9d t
  ,t'(x)
    .k ?2x_/4>1+x^2"]#+?-2(2-x)_/10>1+(2
      -x)^2"]#
    .k ?5x>1+(2-x)^2"]+(2x-4)>1+x^2"]_/
      10>1+x^2"]>1+(2-x)^2"]#_4
  ,? looks bad1 b at l1/ ! denom9ator is
alw positive1 s t ,t'(x) is alw def9$4
,we "!=e ne$ only f9d ": ,t'(x) .k #0_4
,? happ5s :5 ! num]ator is #0, i4e41 :5
  #5x>1+(2-x)^2"] .k -(2x-4)>1+x^2"]
  #25x^2"(1+(2-x)^2") .k (2x-4)^2"(1+x^2
    ")
  #21x^4"-84x^3"+105x^2"+16x-16 .k #0_4
,y probably c't solve ? equ,n exactly---
,i c't1 anyway---b y c use ,maple 6get
approximate solu;ns1 : >e r\<ly
  x .k -#0.3972
  x .k #0.3632_4
,cle>ly ! second ( ^! is ! "o we want4
,! fi%]man obvi\sly %dn't r[ away f !
b>4
                                     #eb

  ,n[1 "! is ano!r & deep] way 6?9k ab ?
pro#m4 ,let .?1 & .?2 2 ! angles m>k$ 9
,figure #30 2l1 & let v1 & v2 2 !
velocities at : ! fi%]man c move 9 wat]
& on l&1 resp4
  ,!n ,t'(x) .k x is equival5t to
  ?2x_/4>1+x^2"]# .k ?2(2-x)_/10>1+(2
    -x)^2"]#
  ?x_/2>1+x^2"]# .k ?(2-x)_/5>1+(2
    -x)^2"]#
  ?sin .?1_/v1# .k ?sin .?2_/v2#_4
  ,? la/ equ,n is call$ ,snell's law1 &
applies n j 6fi%]m51 b 6! mo;n ( li<t "?
two di6]5t m$ia 9 : li<t travels at
di6]5t spe$s4 ,:5 a ray ( li<t travel+
"? air 5t]s wat]1 x b5ds at an angle
dictat$ 0,snell's ,law1 ": v1 & v2 >e !
spe$s ( li<t 9 air & 9 wat]1 resp4 ,?us1
li<t travels j l ! fi%]man1 alw tak+ !
pa? t m9imizes xs travel "t4 ,? is true
ev5 if ! ray ( li<t is travel+ "? a
m$ium 9 : xs veloc;y---call$ ! 9dex (
refrac;n---v>ies 3t9u\sly4 ,"! is a :ole
                                     #ec
bran* ( ma!matics call$ ! calculus (
v>i,ns : is aim$ at solv+ pro#ms (
det]m9+ ! optimum pa? 9 situ,ns l ?1 & x
turns \ t :at >e call$ v>i,nal pr9ciples
l ,snell's ,law play a big role 9
physics 9 a v>iety ( situ,ns4 ,if y want
6le>n m ab ?1 ,*apt] #19 ( ,volume #2 (!
,feynman ,lectures is devot$ 6? idea4
,x's an alm v]batim lecture ( ,feynman's
& x's a lovely ?+ 6r1d = anybody :o ?9ks
? sort ( ma? & physics is excit+4 ,y saw
x "h f/1 ?\<4












                                     #ed