,,,EXAMPLES F
         CALCULUS3 A F/ C\RSE,'









            ,TRANSCRIB$ ,US+
         ,,UNIFI$ ,,5GLI% ,,BRL









            ,EXAMPLES #A-BJ
      ,BRL PAGES P#A-P#B & #A-#AF
P#A   
























P#B
    ,,EXAMPLE #A
  ^1,EXAMPLE #C ,FACTOR
,P"<X"> "7 #BX9#C"-#EX9#B"-#DX"6#C4
  ^1,SOLU;N
  ,P"<#A">
    "7 #B"<#A">9#C"-#E"<#A">9#B
    "-#D"<#A">"6#C
    "7 "-#D "7@: #J
,"!=E1 BY ! ,FACTOR ,!OREM1 X"6#A IS A
FACTOR4 ,WE F9D ANO!O FACTOR BY L;G
DIVI.N3

          #BX9#C"-#GX"6#C
        "333333333333333333333333
  X"6#A _ #BX9#C"-#EX9#B"-#DX"6#C
          #BX9#C"6#BX9#B
          "3333333333333
                "-#GX9#B"-#DX
                "-#GX9#B"-#GX
                "333333333333
                          #CX"6#C
                          #CX"6#C
                          "333333

#A            ,,EXAMPLE #A
    ,,EXAMPLE #B
  ^1,EXAMPLE #D ,FACTOR
#BX9#C/B"6#DX9#A/B"-#FX9<"-#A/B>4
  ^1,SOLU;N ,! T]M ) ! SMALLE/ EXPON5T
IS "-#FX9<"-#A/B> & WE USE #BX9<"-#A/B>
Z A COMMON FACTOR4
  #BX9#C/B"6#DX9#A/B"-#FX9<"-#A/B>
    "7 #BX9<"-#A/B>"<X9#B"6#BX"-#C">
    "7 #BX9<"-#A/B>"<X"-#A">"<X"6#C">















#B            ,,EXAMPLE #B
    ,,EXAMPLE #C
  ^1,EXAMPLE ,RA;NALIZE ! NUM]ATOR 9 !
EXPRES.N
  ;;(%X"6#D+"-#B./X)
  ^1,SOLU;N ,WE MULTIPLY ! NUM]ATOR & !
DENOM9ATOR BY ! 3JUGATE RADICAL
;;%X"6#D+"6#B3
      ,D N EXP& ! DENOM9ATOR4
  ;;;(%X"6#D+"-#B./X)
    "7 "<(%X"6#D+"-#B./X)">
    "<(%X"6#D+"6#B./%X"6#D+"6#B)">
    "7 ("<X"6#D">"-#D
    ./X"<%X"6#D+"6#B">)
    "7 (X./X"<%X"6#D+"6#B">)
      "<X "7@: #J">
    "7 (#A./%X"6#D+"6#B);'








#C            ,,EXAMPLE #C
    ,,EXAMPLE #D
  ^1,EXAMPLE #B ,A L9E> FUNC;N IS GIV5
BY ;Y "7 #F-#EX4 ,IF ;X 9CR1SES BY #B1
H[ DOES ;Y *ANGE8
  ^1,SOLU;N ,! RATE ( *ANGE IS
  (,.DY./,.DX) "7 SLOPE "7 "-#E
& WE >E GIV5 T ,.DX "7 #B4 ,?US
  ,.DY "7 "<"-#E">,.DX
    "7 "<"-#E">"<#B">
    "7 "-#AJ
& S ;Y DECR1SES BY #AJ4

    ,,EXAMPLE #E
  ,? SU7E/S T ! SLOPE ( ! TANG5T L9E #A
%D 2 ;M "7 #B4 ,WE SAY T ! SLOPE ( !
TANG5T L9E IS ! .1LIMIT ( ! SLOPES ( !
SECANT L9ES1 & WE EXPRESS ? SYMBOLICALLY
BY WRIT+
  ;;;LIM.5<,Q\O,P> M5<,,PQ> "7 M;'
&
  ;;;LIM.5<X\O#A> (X9#B"-#A./X"-#A)
    "7 #B;'


#D         ,,EXAMPLES #D & #E
    ,,EXAMPLE #F
#AA4 ,Z DRY AIR MOVES UPW>D1 X EXP&S & 9
  S DO+ COOLS AT A RATE ( AB #A^J,C =
  EA* #AJJ ;M RISE1 UP TO AB #AB KM4




















#E            ,,EXAMPLE #F
    ,,EXAMPLE #G
"333333333333333333333333333333333333333
          ,PROP]TIES ( ,LIMITS

  ,SUPPOSE T ! LIMITS
  ;;;LIM.5<X\OA> F"<X"> &
    LIM.5<X\OA> G"<X">
BO? EXI/ & LET ;C 2 A 3/ANT4 ,!N
""=;;;
#A4 LIM.5<X\OA> .<F"<X">"6G"<X">.>
  "7 LIM.5<X\OA> F"<X">"
  "6LIM.5<X\OA> G"<X">
#B4 LIM.5<X\OA> .<F"<X">"-G"<X">.>
  "7 LIM.5<X\OA> F"<X">"
  "-LIM.5<X$OA> G"<X">
#C4 LIM.5<X\OA> .<CF"<X">.>
  "7 CLIM.5<X\OA> F"<X">
#D4 LIM.5<X\OA> .<F"<X">G"<X">.>
  "7 LIM.5<X\OA> F"<X">"
  LIM.5<X\OA> G"<X">
#E4 LIM.5<X\OA> (F"<X">./G"<X">)
  "7 (LIM.5<X\OA> F"<X">"
  ./LIM.5<X\OA> G"<X">) IF
  LIM.5<X\OA> G"<X"> "7@: #J
#F            ,,EXAMPLE #G
#F4 LIM.5<X\OA> .<F"<X">.>9N
  "7 .<LIM.5<X\OA> F"<X">.>9N IF N IS A
  POSITIVE INTEGER 
#G4 LIM.5<X\OA> %9<A>F"<X">+
  "7 %9<A>LIM.5<X$OA> F"<X">+ IF THE
  ROOT ON THE RIGHT SIDE EXISTS
""=;'
"333333333333333333333333333333333333333
















#G         ,,EXAMPLE #G (3T4)
    ,,EXAMPLE #H
  ^1,EXAMPLE #B ,F9D
;;;LIM.5X\O#E> "<X9#B"6#BX"-#C">;' US+ !
PROP]TIES ( LIMITS4
  ^1,SOLU;N
""=;;;
  LIM.5<X\O#E> "<X9#B"6#BX"-#C">
    "7 LIM.5<X\O#E> X9#B
    "6LIM.5<X\O#E> #BX
    "-LIM.5<X\O#E> #C
      "<,PROPERTIES #A & #B">
    "7 LIM.5<X\O#E> X9#B
    "6#BLIM.5<X\O#E> X
    "-LIM.5<X\O#E> #C
      "<,PROPERTY #C">
    "7 #E9#B"6#B"<#E">"-#C
    "7 #CB
""=;'






#H            ,,EXAMPLE #H
    ,,EXAMPLE #I
  ^1,EXAMPLE #C
    ,EVALUATE US+ ! PROP]TIES ( LIMITS4
""=;;;
"<A">
  LIM.5<X\O#A> (X9#D"-#EX9#B"6#A./X"6#B)
"<B"> LIM.5<X\O#C> %X9#B"6X+
""=;'
  ^1,SOLU;N
"<A">
  ;;;LIM.5<X\O#A> (X9#D"-#EX9#B"6#A
    ./X"6#B)
    "7 (LIM.5<X\O#A> (X9#D"-#EX9#B"6#A
    ./X"6#B)
      "<,PROPERTY #E">
    "7 (LIM.5<X\O#A> X9#D
    "-#ELIM.5<X\O#A> X9#B
    "6LIM.5<X\O#A> #A
    ./LIM.5<X\O#A> X"6LIM.5<X\O#A> #B)
      "<,PROPERTIES #B1 #C1 & #A">
    "7 (#A9#D"-#E"<#A">9#B"6#A./#A"6#B)
    "7 "-#A;'


#I            ,,EXAMPLE #I
    ,,EXAMPLE #AJ
"<;B"> ;;;LIM.5<X\O#C> %X9#B"6X+
  "7 %LIM.5<X\O#C> "<X9#B"6X">+
    "<,PROPERTY #G">
  "7 %LIM.5<X\O#C> X9#B"
  "6LIM.5<X\O#C> X+
    "<,PROPERTY #A">
  "7 %#C9#B"6#C+
  "7 %#AB+
  "7 #B%#C+
""=;'













#AJ          ,,EXAMPLE #AJ
    ,,EXAMPLE #AA
  ^1,EXAMPLE #G ,EVALUATE
  ;;;LIM.5<X\O#J> (%X"6#A+"-#A./X)4;'
  ^1,SOLU;N ,"H ! ALGEBRAIC SIMPLIFICA;N
3SI/S ( RA;NALIZ+ ! NUM]ATOR1 T IS1
MULTIPLY+ NUM]ATOR & DENOM9ATOR BY !
3JUGATE RADICAL ;;%X"6#A+"6#A3
      ,D N EXP& ! DENOM9ATOR4
  ;;;LIM.5<X\O#J> (%X"6#A+"-#A./X)
    "7 LIM.5<X\O#J> "<(%X"6#A+"-#A
      ./X)"> "<(%X"6#A+"6#A
      ./%X"6#A+"6#A)">
    "7 LIM.5<X\O#J> ("<X"6#A">"-#A
      ./X"<%X"6#A+"6#A">)
    "7 LIM.5<X\O#J> (X
      ./X"<%X"6#A+"6#A">)
    "7 LIM.5<X\O#J> (#A./%X"6#A+"6#A)
    "7 (LIM.5<X\O#J> #A
      ./%LIM.5<X\O#J> "<X"6#A">+
      "6LIM.5<X\O#J> #A)
    "7 (#A./%#J"6#A+"6#A)
    "7 #A/B;'


#AA          ,,EXAMPLE #AA
    ,,EXAMPLE #AB
  ,3SID] ! FUNC;N ;F DESCRIB$ BY

;;;F"<X"> "7 ,_<X9#B  IF X _@< #A
             ,_<#C"-X IF X @> #A;'

    ,EXAMPLE #AC
  ,= 9/.E1 WE -PUTE F"<#J">1 F"<#A">1 &
F"<#B"> Z FOLL[S3
,S9CE #J _@< #A1 WE H
  F"<#J"> "7 #J9#B "7 #J4
,S9CE #A _@< #A1 WE H
  F"<#A"> "7 #A9#B "7 #A4
,S9CE #B "> #A1 WE H
  F"<#B"> "7 #C"-#B "7 #A4









#AB       ,,EXAMPLES #AB & #AC
    ,,EXAMPLE #AD
  ^1,EXAMPLE #B ,%[ T
;;;LIM.5<X\O#J> _\X_\ "7 #J4;'
  ^1,SOLU;N ,RECALL T

  ;;;_\X_\ "7 ,_<X   IF X _@> #J
              ,_<"-X IF X "< #J;'

,"!=E
  ;;;LIM.5<X\O#J9"6> _\X_\
    "7 LIM.5<X\O#J9"6> X "7 #J;'
&
  ;;;LIM.5<X\O#J9"-> _\X_\
    "7 LIM.5<X\O#J9"-> "<"-X"> "7 #J;'
  ,S9CE ! LEFT & "R LIMITS >E EQUAL1 WE
H
  ;;;LIM.5<X\O#J> _\X_\ "7 #J







#AC          ,,EXAMPLE #AD
    ,,EXAMPLE #AE
  ^1,EXAMPLE #C ,! ^1,H1VISIDE ^1FUNC;N
;,H IS DEF9$ BY

;;;,H"<T"> "7 ,"<#J IF T "< #J
              ,"<#A IF T _"> #J

    ,,EXAMPLE #AF
  ^1,EXAMPLE #D ,IF
  ;;;F"<X">

    "7 ,.<"-X"-#B   IF X _"< "-#A
       ,.<X         IF "-#A "< X "< #A
       ,.<X9#B"-#BX IF X _"> #A;'

DET]M9E :E!R OR N
;;;LIM.5<X\O"-#A> F"<X"> AND
LIM.5<X\O#A> F"<X">;' EXI/4






#AD       ,,EXAMPLES #AE & #AF
    ,,EXAMPLE #AG
  ^1,EXAMSLE #E
,": >E ! FOLL[+ FUNC;NS 4CONT9U\S8
""=;;;
"<A"> F"<X"> "7 ,_<X9#B"6#A IF X "< #J
                ,_<#J       IF X "7 #J
                ,_<X9#B"-#A IF X "> #J

"<B"> G"<X"> "7 ,_<X"6#A IF X "7@: #B
                ,_<#A    IF X "7 #B
""=;'

    ,,EXAMPLE #AH
#AC4 ,LET

F"<X">

  "7 ,_<#A"-_\X_\ IF _\X_\ _@< #A
     ,_<_\X_\"-#A
     ,_<  IF #A "< _\X_\ _"< #B
     ,_<"<X"-#C">9#B IF ;X @> #B
     ,_<"<X"6#C">9#B IF ;X "< "-#B


#AE       ,,EXAMPLES #AG & #AH
    ,,EXAMPLE #AI
AV]AGE VELOC;Y
  "7 ;(4T.E TRAVELL$_/"T ELAPS$;)
  "7 ;(,.DS./,.T;)
  "7 ;(F"<#C4A">"-F"<#C">_/#J4A;)
  "7 (#D4I"<#C4A">9#B"-#D4I"<#C">9#B"
    ./#J4A)
  "7 #BI4HI M_/S

    ,,EXAMPLE #BJ
"<II"> ;;;(,.D,T./,.DT)
  "7 (,T"<#C4E">"-,T"<#B">./#C4E"-#B)
  "7 (#F"-#H4C./#A4E) "7 ("-#B4C./#A4E)
  "7 "-#A4E^J_/MIN;'
"<III"> ;;;(,.D,T./,.DT)
  "7 (,T"<#C4J">"-,T"<#B">./#C"-#B)
  "7 (#F4E"-#H4C./#A) "7 "-#A4H^J_/MIN;'
"<IV"> ;;;(,.D,T./,.DT)
  "7 (,T"<#B4E">"-,T"<#B">./#B4E"-#B)
  "7 (#G4B"-#H4C./#J4E)
  "7 "-#B4B^J_/MIN;'

          ,,5D ,,( ,,EXAMPLES

#AF       ,,EXAMPLES #AI & #BJ