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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 51 "Calculus Exploration 4:  Deriv
ative Rules (D-Rules)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT 258 28 "Constant Multiple Rule (CMR)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"DG\"\"\"7#*&%\"cGF&-%\"fG6#%\"xGF&F&*(F)F&F%F&7#F
*F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 259 26 
"Sum of Functions Rule (SR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"D
G\"\"\"7#,&-%\"fG6#%\"xGF&-%\"gGF+F&F&,&*&F%F&7#F)F&F&*&F%F&7#F-F&F&" 
}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 281 28 "Co
nstant Function Rule (CFR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%-If~for
~all~xG/*&%\"~G\"\"\"-%\"fG6#%\"xGF'%\"cG%>~where~c~is~a~constant~numb
erG/*(%'~then~GF'%\"DGF'7#F(F'*(F1F'7#F,F'%%~=~0GF'" }}{PARA 256 "" 0 
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 282 28 "Identity Function R
ule (IFR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\"
\"\"-%\"fG6#%\"xGF'F+/*(%'~then~GF'%\"DGF'7#F(F'*(F/F'7#F+F'%%~=~1GF'
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 283 20 "Po
wer Function Rule " }{TEXT -1 1 " " }{TEXT 284 5 "(PFR)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%%-If~for~all~xG/*&%\"~G\"\"\"-%\"fG6#%\"xGF')F+%
\"nG/*(%'~then~GF'%\"DGF'7#F(F'**F1F'7#F,F'%%~=~nGF')F+,&F-F'F'!\"\"F'
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 260 30 "Pr
oduct of Functions Rule (PR)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"
DG\"\"\"7#*&-%\"fG6#%\"xGF&-%\"gGF+F&F&,&*(F%F&7#F)F&F-F&F&*(F%F&7#F-F
&F)F&F&" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
261 31 "Quotient of Functions Rule (QR)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&%\"DG\"\"\"7#*&-%\"fG6#%\"xGF&-%\"gGF+!\"\"F&*&,&*(F%F&7#F)F&
F-F&F&*(F%F&7#F-F&F)F&F/F&F-!\"#" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT 286 24 "Power Chain Rule (PowCR)" }{TEXT -1 0 
"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"\"\"7#)-%\"fG6#%\"xG%\"
nGF&**F-F&)F),&F-F&F&!\"\"F&F%F&7#F)F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*(%9~Note~the~special~case:~G\"\"\"%\"DGF&7#)%\"xG%\"nGF&**F+F&
)F*,&F+F&F&!\"\"F&F'F&7#F*F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT 287 56 "Exponential Rule (ExpR) & Exponential Chain \+
Rule (ExpCR)" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"D
G\"\"\"7#-%$expG6#%\"xGF&*&F(F&%,~~~(~ExpR~)GF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$expG6#-%\"fG6#%\"xGF&**F)F&F'
F&7#F,F&%-~~~(~ExpCR~)GF&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT 289 54 "Natural Log Rule (LnR) & Natural Log Chain R
ule (LnCR)" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*&%\"DG
\"\"\"7#-%#lnG6#%\"xGF&*&F&F&F+!\"\"%+~~~(~LnR~)G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$/*(%\"~G\"\"\"%\"DGF&7#-%#lnG6#-%\"fG6#%\"xGF&*(F,!\"\"
F'F&7#F,F&%,~~~(~LnCR~)G" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT 291 42 "Sine Rule (SinR) & Sine Chain Rule (Si
nCR)" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"\"\"7
#-%$sinG6#%\"xGF&*&-%$cosGF*F&%,~~~(~SinR~)GF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$sinG6#-%\"fG6#%\"xGF&**-%$cos
GF+F&F'F&7#F,F&%-~~~(~SinCR~)GF&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT 293 46 "Cosine Rule (CosR) & Cosine Chain Rule
 (CosCR)" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"
\"\"7#-%$cosG6#%\"xGF&,$*&-%$sinGF*F&%,~~~(~CosR~)GF&!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$cosG6#-%\"fG6#%\"x
GF&,$**-%$sinGF+F&F'F&7#F,F&%-~~~(~CosCR~)GF&!\"\"" }}{PARA 256 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 298 48 "Tangent Rule (TanR) \+
& Tangent Chain Rule (TanCR)" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"DG\"\"\"7#-%$tanG6#%\"xGF&*&)-%$secGF*\"\"#F&%,~~
~(~TanR~)GF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7
#-%$tanG6#-%\"fG6#%\"xGF&**)-%$secGF+\"\"#F&F'F&7#F,F&%-~~~(~TanCR~)GF
&" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 299 52 
"Cotangent Rule (CotR) & Cotangent Chain Rule (CotCR)" }{TEXT -1 0 "" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"\"\"7#-%$cotG6#%\"xGF&,$*&
)-%$cscGF*\"\"#F&%,~~~(~CotR~)GF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$cotG6#-%\"fG6#%\"xGF&,$**)-%$cscGF+\"
\"#F&F'F&7#F,F&%-~~~(~CotCR~)GF&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 256 "" 0 "" {TEXT 300 46 "Secant Rule (SecR) & Secant Chain \+
Rule (SecCR)" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"D
G\"\"\"7#-%$secG6#%\"xGF&*(F(F&-%$tanGF*F&%,~~~(~SecR~)GF&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$secG6#-%\"fG6#%\"x
GF&*,F)F&-%$tanGF+F&F'F&7#F,F&%-~~~(~SecCR~)GF&" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 301 50 "Cosecant Rule (CscR) &
 Cosecant Chain Rule (CscCR)" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"DG\"\"\"7#-%$cscG6#%\"xGF&,$*(F(F&-%$cotGF*F&%,~~
~(~CscR~)GF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"
DGF&7#-%$cscG6#-%\"fG6#%\"xGF&,$*,F)F&-%$cotGF+F&F'F&7#F,F&%-~~~(~CscC
R~)GF&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 256 "" 
0 "" {TEXT 303 37 " Examples Using the Product Rule (PR)" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"\"\"7
#*&-%\"fG6#%\"xGF&-%\"gGF+F&F&,&*(F%F&7#F)F&F-F&F&*(F%F&7#F-F&F)F&F&" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 302 10 "Example 1." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"
\"\"7#*&-%\"fG6#%\"xGF&-%\"gGF+F&F&,&*(F%F&7#F)F&F-F&F&*(F%F&7#F-F&F)F
&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"fG6#%\"xG,&*&\"\"*\"\"\")F
'\"\"#F+F+*&\"\"(F+F'F+F+/*&%\"~GF+-%\"gGF&F+,&*$F,F+F+F+!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&%\"DG\"\"\"7#*&-%\"fG6#%\"xGF&-%\"gGF+F&F&%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"DG\"\"\"7#*&,&*&\"\"*F&)%\"xG\"\"#F&F&*&\"\"(F&F-
F&F&F&,&*$F,F&F&F&!\"\"F&F&%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*
&-%\"DG6#,&*&\"\"*\"\"\")%\"xG\"\"#F,F,*&\"\"(F,F.F,F,F,,&*$F-F,F,F,!
\"\"F,F,*&-F'6#F2F,F)F,F,%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,
&*&\"#=\"\"\"%\"xGF)F)\"\"(F)F),&*$)F*\"\"#F)F)F)!\"\"F)F)*(F/F)F*F),&
*&\"\"*F)F.F)F)*&F+F)F*F)F)F)F)%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,,*&\"#=\"\"\")%\"xG\"\"$F&F&*&F%F&F(F&!\"\"*&\"\"(F&)F(\"\"#F&F&F-F+%
!GF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"#=\"\"\")%\"xG\"\"$F'F'
*&\"#9F')F)\"\"#F'F'%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"#O\"
\"\")%\"xG\"\"$F&F&*&\"#@F&)F(\"\"#F&F&*&\"#=F&F(F&!\"\"\"\"(F0" }}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 10 "Example \+
2." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"\"\"7#*&-%\"fG6#%\"xGF
&-%\"gGF+F&F&,&*(F%F&7#F)F&F-F&F&*(F%F&7#F-F&F)F&F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$/-%\"fG6#%\"xG,(*$)F'\"\"#\"\"\"F,*&\"\"$F,F'F,F,\"\"
&!\"\"/*&%\"~GF,-%\"gGF&F,,**$)F'F.F,F,*&\"\"'F,F*F,F,*&F+F,F'F,F0F,F,
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&%\"DG\"\"\"7#*&-%\"fG6#%\"xGF&-%\"gGF+F&F&%!G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/*&%\"DG\"\"\"7#*&,(*$)%\"xG\"\"#F&F&*&\"\"$F&F,F&
F&\"\"&!\"\"F&,**$)F,F/F&F&*&\"\"'F&F+F&F&*&F-F&F,F&F1F&F&F&F&%!G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%\"DG6#,(*$)%\"xG\"\"#\"\"\"F.*&
\"\"$F.F,F.F.\"\"&!\"\"F.,**$)F,F0F.F.*&\"\"'F.F+F.F.*&F-F.F,F.F2F.F.F
.F.*&-F'6#F3F.F)F.F.%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&*&\"
\"#\"\"\"%\"xGF)F)\"\"$F)F),**$)F*F+F)F)*&\"\"'F))F*F(F)F)*&F(F)F*F)!
\"\"F)F)F)F)*&,(*&F+F)F1F)F)*&\"#7F)F*F)F)F(F3F),(*$F1F)F)*&F+F)F*F)F)
\"\"&F3F)F)%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"\"#\"\"\")%\"x
G\"\"%F&F&*&\"#:F&)F(\"\"$F&F&*&\"#9F&)F(F%F&F&*&F)F&F(F&!\"\"F-F&%!GF
&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&\"\"$\"\"\")%\"xG\"\"%F'F'*&
\"#@F')F)F&F'F'*&\"#>F')F)\"\"#F'F'*&\"#mF'F)F'!\"\"\"#5F'%!G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\"&\"\"\")%\"xG\"\"%F&F&*&\"#OF&
)F(\"\"$F&F&*&\"#LF&)F(\"\"#F&F&*&\"#qF&F(F&!\"\"\"#8F&" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 302 31 "Motivat
ion:  A Not Product Rule" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%@We~kn
ow~from~the~Sum~Rule~that~G\"\"\"-%\"DG6#,&%\"xGF&*$)F+\"\"#F&F&F&,&-F
(6#F+F&-F(6#F,F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%QSo,~maybe~pro
ducts~work~like~this~too,~possibly~G\"\"\"-%\"DG6#*&%#~xGF&)%\"xG\"\"#
F&F&*(-F(6#F-F&-F(6#*$F,F&F&%#??GF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
%%6But~by~the~Power~RuleG/,$**\"\"$\"\"\")%\"xG\"\"#F(%\"=GF(-%\"DG6#*
$)F*F'F(F(F(-F.6#*&%#~xGF(F)F(/*(%%and~GF(-F.6#F*F(-F.6#*$F)F(F(,$*(F+
F(F*F(%\".GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$0*&%%So,~G\"\"\"-%
\"DG6#*&%#~xGF&)%\"xG\"\"#F&F&*&-F(6#F-F&-F(6#*$F,F&F&%Tand~therefore~
the~simple~rule~above~can't~be~true~!G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coAnd~we~can~see~the~mor
e~complicated~Product~Rule~hiding~in~this~example:G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*(-%\"DG6#*&%#~xG\"\"\")%\"xG\"\"#F*F*%$=~3GF*F+F*,&
*&%#~1GF*F+F*F**(F-F*F,F*F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%.
~~~~~~~~~~~~~G,&*&-%\"DG6#%\"xG\"\"\")F*\"\"#F+F+*&-F(6#*$F,F+F+F*F+F+
" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT 305 38 " Examples Using the Quotient
 Rule (QR)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"DG6#*&-%\"fG6#%\"xG\"\"\"-%\"gGF*!\"\"*&,&*&-F%6#F
(F,F-F,F,*&-F%6#F-F,F(F,F/F,*$)F-\"\"#F,F/" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 304 10 "Example 1." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG
6#*&-%\"fG6#%\"xG\"\"\"-%\"gGF*!\"\"*&,&*&-F%6#F(F,F-F,F,*&-F%6#F-F,F(
F,F/F,*$)F-\"\"#F,F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6$/-%\"fG6#%\"xG,&*$)F'\"\"#\"\"\"\"\"*F'\"\"(/*&%\"~G
F,-%\"gGF&F,,&F)F,!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#*&-%\"fG6#%\"xG\"\"\"-%\"gGF*
!\"\"*&-F%6#*&,&*$)F+\"\"#F,\"\"*F+\"\"(F,,&F5F,F/F,F/F,%\"=GF," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*&-%\"DG6#,&*$)%\"xG\"\"#\"\"\"\"
\"*F-\"\"(F/,&F+F/!\"\"F/F/F/*&-F(6#F2F/F*F/F3F/*$)F2F.F/F3%!G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*&,&%\"xG\"#=\"\"(\"\"\"F+,&*$)F(
\"\"#F+F+!\"\"F+F+F+*&F(F+,&F-\"\"*F(F*F+!\"#F+*$)F,F/F+F0%!G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&*(,**$)%\"xG\"\"$\"\"\"\"#=F)!#=*$)
F)\"\"#F+\"\"(!\"(F+F+%\"+GF+,&F'F-F.!#9F+F+*$),&F.F+!\"\"F+F0F+F9%!G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$)%\"xG\"\"#\"\"\"!\"(F'!#=F*
F)F)*$),&F%F)!\"\"F)F(F)F/" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 304 10 "Example 2." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"DG6#*&-%\"fG6#%\"xG\"\"\"-%\"gGF*!\"\"*&,&*&-F%6#F
(F,F-F,F,*&-F%6#F-F,F(F,F/F,*$)F-\"\"#F,F/" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"fG6#%\"xG,(*$)F'\"\"#\"
\"\"F,F'\"\"$!\"&F,/*&%\"~GF,-%\"gGF&F,,**$)F'F-F,F,F)\"\"'F'!\"#F,F,
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"DG6#*&-%\"fG6#%\"xG\"\"\"-%\"gGF*!\"\"*&-F%6#*&,(*$)F+\"\"#
F,F,F+\"\"$!\"&F,F,,**$)F+F8F,F,F5\"\"'F+!\"#F,F,F/F,%\"=GF," }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*&,&*&-%\"DG6#,(*$)%\"xG\"\"#\"\"\"F/F-\"\"
$!\"&F/F/,**$)F-F0F/F/F+\"\"'F-!\"#F/F/F/F/*&-F(6#F2F/F*F/!\"\"F/*$)F2
F.F/F:%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*&,&%\"xG\"\"#\"\"$
\"\"\"F+,**$)F(F*F+F+*$)F(F)F+\"\"'F(!\"#F+F+F+F+*&,(F/F*F(\"#7F2F+F+,
(F/F+F(F*!\"&F+F+!\"\"F+*$)F,F)F+F8%!G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&*(,,*$)%\"xG\"\"%\"\"\"\"\"#*$)F)\"\"$F+\"#:*$)F)F,F+\"#9F)!
\"%F/F+F+%\"+GF+,,F'!\"$F-!#@F1!#>F)\"#m!#5F+F+F+*$),*F-F+F1\"\"'F)!\"
#F+F+F,F+!\"\"%!G" }}{PARA 256 "" 1 "" {XPPMATH 20 "6#*&,,*$)%\"xG\"\"
%\"\"\"!\"\"*$)F'\"\"$F)!\"'*$)F'\"\"#F)!\"&F'\"#i!\"(F)F)*$),*F+F)F/
\"\"'F'!\"#F)F)F1F)F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 304 
63 "Motivation for QR:  Try Some Algebra and Rules You Already Know" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?For~g(x)~nonzero,~we~know~thatG/*&-
%#~gG6#%\"xG\"\"\"-%\"gGF(!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-
%@So,~the~lefthand~side~is~just~gG6#%\"xG%'times~G*&\"\"\"F)-%\"gGF%!
\"\"%$andG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%8by~PR~and~CFR,~whate
verG\"\"\"-%\"DG6#*&F%F%-%\"gG6#%\"xG!\"\"F%%6is,~it~must~satisfy:~GF%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%\"DG6#-%\"gG6#%\"xG\"\"\"-%
!G6#*&F-F-F)!\"\"F-F-*&-F'F0F-F)F-F-*&-F'6#F-F-%$=~0GF-" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&-%\"DG6#*&\"\"\"F)-%\"gG6#%\"xG!\"\"F)F*F),$*
&-F&6#F*F)F*F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#*&\"\"\"F
(-%\"gG6#%\"xG!\"\",$*&-F%6#F)F(F)!\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%BNow,~use~this~and~PR~
again~to~getG\"\"\"-%\"DG6#*&-%\"fG6#%\"xGF%-%\"gGF,!\"\"F%%2in~a~simi
lar~way:GF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#*&-%\"fG6#%\"x
G\"\"\"-%!G6#*&F,F,-%\"gGF*!\"\"F,,&*&-F%6#F(F,F-F,F,*&-F%F/F,F(F,F," 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%0~~~~~~~~~~~~~~~G,&*&-%\"DG6#-%\"f
G6#%\"xG\"\"\"-%\"gGF,!\"\"F.*(-F(6#F/F.F/!\"#F*F.F1" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/%7~~~~~~~~~~~~~~~~~~~~~~G*&,&*&-%\"DG6#-%\"fG6#%\"x
G\"\"\"-%\"gGF-F/F/*&-F)6#F0F/F+F/!\"\"F/F0!\"#" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 261 "" 
0 "" {TEXT 308 44 " Examples Using the Power Chain Rule (PowCR)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"D
G6#)-%\"fG6#%\"xG%\"nG*(F,\"\"\")F(,&F,F.!\"\"F.F.-F%6#F(F." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 306 10 "Example 1." }{TEXT -1 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#)-%\"fG6#%\"xG%\"nG*(F,\"\"\")F(,&F
,F.!\"\"F.F.-F%6#F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"fG6#%\"x
G,(*$)F'\"\"$\"\"\"\"\"#F'!\"&F,F,/*&%\"~GF,%\"nGF,\"#5" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#
)-%\"fG6#%\"xG%\"nG*&-F%6#*$),(*$)F+\"\"$\"\"\"\"\"#F+!\"&F6F6\"#5F6F6
%\"=GF6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&),(*$)%\"xG\"\"$\"\"\"
\"\"#F*!\"&F,F,\"\"*F,-%\"DG6#F'F,\"#5%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&),(*$)%\"xG\"\"$\"\"\"\"\"#F)!\"&F+F+\"\"*F+,&*$)F)
F,F+\"\"'F-F+F+\"#5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 306 10 "Example 2." }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#)-%\"f
G6#%\"xG%\"nG*(F,\"\"\")F(,&F,F.!\"\"F.F.-F%6#F(F." }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$/-%\"fG6#%\"xG,(*$)F'\"\"#\"\"\"F,F'\"\"$!\"&F,/*&%\"
~GF,%\"nGF,#\"\"%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#)-%\"fG6#%\"xG%\"nG*&-F%6#*$),(*$)F
+\"\"#\"\"\"F6F+\"\"$!\"&F6#\"\"%F7F6F6%\"=GF6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,$*&),(*$)%\"xG\"\"#\"\"\"F,F*\"\"$!\"&F,#F,F-F,-%\"DG
6#F'F,#\"\"%F-%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&),(*$)%\"xG\"
\"#\"\"\"F+F)\"\"$!\"&F+#F+F,F+,&F)F*F,F+F+#\"\"%F," }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 306 10 "Example
 3." }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#)-%\"fG
6#%\"xG%\"nG*(F,\"\"\")F(,&F,F.!\"\"F.F.-F%6#F(F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$/-%\"fG6#%\"xG,,*$)F'\"\"'\"\"\"F,*$)F'\"\"&F,!\"(*$)F'
\"\"$F,\"\")*$)F'\"\"#F,\"\"*F,F,/*&%\"~GF,%\"nGF,#F,F3" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#
)-%\"fG6#%\"xG%\"nG*&-F%6#*$),,*$)F+\"\"'\"\"\"F6*$)F+\"\"&F6!\"(*$)F+
\"\"$F6\"\")*$)F+\"\"#F6\"\"*F6F6#F6F=F6F6%\"=GF6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&),,*$)%\"xG
\"\"'\"\"\"F,*$)F*\"\"&F,!\"(*$)F*\"\"$F,\"\")*$)F*\"\"#F,\"\"*F,F,-%
\"-G6##F7F3F,-%\"DG6#F'F,#F,F3%!G" }}{PARA 256 "" 1 "" {XPPMATH 20 "6#
,$*&),,*$)%\"xG\"\"'\"\"\"F+*$)F)\"\"&F+!\"(*$)F)\"\"$F+\"\")*$)F)\"\"
#F+\"\"*F+F+-%\"-G6##F6F2F+,*F,F**$)F)\"\"%F+!#NF4\"#CF)\"#=F+#F+F2" }
}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
0 "" }{TEXT 307 66 "Motivation for PowCR:  Try Some Algebra and Rules \+
You Already Know" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%<Suppose~you~want~t
o~find:~~G\"\"\"-%\"DG6#*$),(*$)%\"xG\"\"#F%F%*&\"\"$F%F.F%F%\"\"&!\"
\"\"#?F%F%%@.~~Based~upon~your~knowledge~ofGF%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%eoother~rules~you~could~first~multiply~out,~but~that~w
ould~be~tedious,~sinceG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*$),(*$)%
\"xG\"\"#\"\"\"F+*&\"\"$F+F)F+F+\"\"&!\"\"\"#?F+,^p*$)F)\"#SF+F+*&\"#g
F+)F)\"#RF+F+*&\"%5;F+)F)\"#QF+F+*&\"&!3DF+)F)\"#PF+F+*&\"'&HV#F+)F)\"
#OF+F+*&\"(swS\"F+)F)\"#NF+F+*&\"(!peKF+)F)\"#MF+F+*&\")g'Rc\"F+)F)\"#
LF+F/*&\"*00\\X\"F+)F)\"#KF+F/*&\"*?\\B&HF+)F)\"#JF+F/*&\"+WGd`8F+)F)
\"#IF+F+*&\"+?2p^yF+)F)\"#HF+F+*&\"*qNkY(F+)F)\"#GF+F+*&\",S98)>')F+)F
)\"#FF+F/*&\"-gGjjn8F+)F)\"#EF+F/*&\"-GN#om6'F+)F)\"#DF+F+*&\".XS;UYo
\"F+)F)\"#CF+F+*&\".!Q)*4R3LF+)F)\"#BF+F/*&\"/!\\fK9&)G\"F+)F)\"#AF+F/
*&\"/SSAza&f\"F+)F)\"#@F+F+*&\"/,3\")4;EtF+)F)F0F+F+*&\"/+-7'Rx(zF+)F)
\"#>F+F/*&\"0]s[\"eG@KF+)F)\"#=F+F/*&\"0+vzu)[NTF+)F)\"#<F+F+*&\"1D\"G
DN,H0\"F+)F)\"#;F+F+*&\"1+]_B)e9\">F+)F)\"#:F+F/*&\"1+vVj<$p8#F+)F)\"#
9F+F/*&\"1++v=!HUt'F+)F)\"#8F+F+*&\"0]7`Wwl\"HF+)F)\"#7F+F+*&\"2++]PML
N`\"F+)F)\"#6F+F/*&\"2+vo/[[=K\"F+)F)\"#5F+F+*&\"2+]7yvw:W\"F+)F)\"\"*
F+F+*&\"2DcEAG9?b$F+)F)\"\")F+F/*&\"2+v$f$=Q\"4>F+)F)\"\"(F+F+*&\"2]7.
K`Y*))>F+)F)\"\"'F+F+*&\"2+]PM-ueH%F+)F)F.F+F/*&\"2v$4'\\3(Q7PF+)F)\"
\"%F+F+*&\"2+]P%[@X8>F+)F)F-F+F/*&\"1]il(fi;9'F+F(F+F+*&\"1+voz\"4W9\"
F+F)F+F/\"/D1kJuO&*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*&%aoInstead,~you~might~try~to~use~PR~by~fir
st~considering~a~simpler~case:~G\"\"\"-%\"DG6#*$),(*$)%\"xG\"\"#F%F%*&
\"\"$F%F.F%F%\"\"&!\"\"F/F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%/
Then,~by~PR,~~G\"\"\"-%\"DG6#*$),(*$)%\"xG\"\"#F&F&*&\"\"$F&F/F&F&\"\"
&!\"\"F0F&F&-F(6#*&F,F&-%\"~G6#F,F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%.~~~~~~~~~~~~~G,&*(-%\"DG6#,(*$)%\"xG\"\"#\"\"\"F/*&\"\"$F/F-F/F/\"
\"&!\"\"F/F*F/%\"~GF/F/*&-%#~DGF)F/F*F/F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%!G*&-%\"2G6#,(*$)%\"xG\"\"#\"\"\"F.*&\"\"$F.F,F.F.\"
\"&!\"\"F.-%\"DGF(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&-%\"2G6#
,(*$)%\"xG\"\"#\"\"\"F.*&\"\"$F.F,F.F.\"\"&!\"\"F.,&*&F-F.F,F.F.F0F.F.
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%^pLet's~see~if~this~pattern~persists~when~we~try~n~=~3.~~By~PR~
and~the~above,~we~haveG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#*$
),(*$)%\"xG\"\"#\"\"\"F.*&\"\"$F.F,F.F.\"\"&!\"\"F0F.-F%6#*&F)F.)-%\"~
G6#F)F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%.~~~~~~~~~~~~~G,&*(-%\"
DG6#,(*$)%\"xG\"\"#\"\"\"F/*&\"\"$F/F-F/F/\"\"&!\"\"F/)F*F.F/%\"~GF/F/
*&-%#~DG6#*$F4F/F/F*F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%.~~~~~~~
~~~~~~G,&*(,&*&\"\"#\"\"\"%\"xGF*F*\"\"$F*F*),(*$)F+F)F*F**&F,F*F+F*F*
\"\"&!\"\"F)F*%\"~GF*F**(-%#~2G6#F.F*F'F*F.F*F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%.~~~~~~~~~~~~~G,&*(,&*&\"\"#\"\"\"%\"xGF*F*\"\"$F*F*)
,(*$)F+F)F*F**&F,F*F+F*F*\"\"&!\"\"F)F*%\"~GF*F**&-%#~2G6#F'F*F-F*F*" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&)-%#3~G6#,(*$)%\"xG\"\"#\"\"\"
F/*&\"\"$F/F-F/F/\"\"&!\"\"F.F/,&*&F.F/F-F/F/F1F/F/" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=Then~so~far~
the~pattern~is:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#*$),(*$)
%\"xG\"\"#\"\"\"F.*&\"\"$F.F,F.F.\"\"&!\"\"F-F.*&)-%#2~G6#F)%#~1GF.-F%
F7F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#*$),(*$)%\"xG\"\"#\"
\"\"F.*&\"\"$F.F,F.F.\"\"&!\"\"F0F.*&)-%#3~G6#F)F-F.-F%F7F." }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%boAp
parently,~we~can~extend~this~pattern~to~answer~our~original~question,G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<which~is~much~less~tedious!G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"DG6#*$),(*$)%\"xG\"\"#\"\"\"F.*&
\"\"$F.F,F.F.\"\"&!\"\"\"#?F.*&)-%$20~G6#F)\"#>F.-F%F8F." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%9~~~~~~~~~~~~~~~~~~~~~~~~G*&)-%$20~G6#,(*$)%\"
xG\"\"#\"\"\"F/*&\"\"$F/F-F/F/\"\"&!\"\"\"#>F/,&*&F.F/F-F/F/F1F/F/" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Dr. Joh
n Pais, Mathematics Department-MICDS" }}{PARA 0 "" 0 "" {TEXT -1 45 "E
-mail: pais@micds.org or pais@kinetigram.com" }}{PARA 0 "" 0 "" {TEXT 
-1 33 "URL:  http://kinetigram.com/micds" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Derivative Rules (D-Rules)
" }}{EXCHG {PARA 0 "" 0 "" {TEXT 266 28 "Constant Multiple Rule (CMR)
" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "restart:\nc:='c
':\n'D'*[c*'f'(x)]=c*('D'*['f'(x)]);\n#Diff([`c f`(x)],x)=c*Diff(f(x),
x);\n#[`c f`(x)]*`'`=c*[f(x)*`'`];\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"\"\"7#*&%\"cGF&-%\"
fG6#%\"xGF&F&*(F)F&F%F&7#F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 26 "Sum of Functions Rule (
SR)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "restart:\n'D'*['f'(x)+'g'(x)
]='D'*['f'(x)]+'D'*['g'(x)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"
DG\"\"\"7#,&-%\"fG6#%\"xGF&-%\"gGF+F&F&,&*&F%F&7#F)F&F&*&F%F&7#F-F&F&
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "
" 0 "" {TEXT -1 23 "Constant Function Rule " }{TEXT 280 5 "(CFR)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "restart:\nc:='c':\nf:=x->c:\n`If f
or all x`,` `*'f'(x)=f(x),` where c is a constant number`\n,` then `*'
D'*['f'(x)]='D'*['c']*` = 0`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%-If~
for~all~xG/*&%\"~G\"\"\"-%\"fG6#%\"xGF'%\"cG%>~where~c~is~a~constant~n
umberG/*(%'~then~GF'%\"DGF'7#F(F'*(F1F'7#F,F'%%~=~0GF'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
278 28 "Identity Function Rule (IFR)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 98 "restart:\nc:='c':\nf:=x->x:\n`If for all x`,` `*'f'(x)=f(x),\n` \+
then `*'D'*['f'(x)]='D'*['x']*` = 1`;\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%%-If~for~all~xG/*&%\"~G\"\"\"-%\"fG6#%\"xGF'F+/*(%'~then~GF'%\"D
GF'7#F(F'*(F/F'7#F+F'%%~=~1GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 25 "Power Function Rule (
PFR)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "restart:\nc:='c':\nn:='n':
\nf:=x->x^n:\n`If for all x`,` `*'f'(x)=f(x),\n` then `*'D'*['f'(x)]='
D'*[f(x)]*` = n`*x^(n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-If~for~
all~xG/*&%\"~G\"\"\"-%\"fG6#%\"xGF')F+%\"nG/*(%'~then~GF'%\"DGF'7#F(F'
**F1F'7#F,F'%%~=~nGF')F+,&F-F'F'!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 71 "Difference o
f Functions Rule (DR) (didn't include, since not on Maplet)" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 75 "restart:\nLimit(``*('f'(x)-'g'(x)),x=a)=L
imit('f'(x),x=a)-Limit('g'(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%&LimitG6$*&%!G\"\"\",&-%\"fG6#%\"xGF)-%\"gGF-!\"\"F)/F.%\"aG,&-F%6
$F+F2F)-F%6$F/F2F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 263 30 "Product of Functions Rule (PR)" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "restart:\n'D'*['f'(x)*'g'(x)]='D'*
['f'(x)]*'g'(x)+'D'*['g'(x)]*'f'(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/*&%\"DG\"\"\"7#*&-%\"fG6#%\"xGF&-%\"gGF+F&F&,&*(F%F&7#F)F&F-F&F&*(F
%F&7#F-F&F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 264 31 "Quotient of Functions Rule (QR)" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "restart:\n'D'*['f'(x)/'g'(x)]=('D
'*['f'(x)]*'g'(x)-'D'*['g'(x)]*'f'(x))/'g'(x)^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"DG\"\"\"7#*&-%\"fG6#%\"xGF&-%\"gGF+!\"\"F&*&,&*(F
%F&7#F)F&F-F&F&*(F%F&7#F-F&F)F&F/F&F-!\"#" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 24 "Power Chai
n Rule (PowCR)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "restart:\nc:='c'
:\nn:='n':\n'D'*['f'(x)^n]= n*f(x)^(n-1)*'D'*['f'(x)];\n` Note the spe
cial case: `*'D'*[x^n]=n*x^(n-1)*'D'*[x];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"DG\"\"\"7#)-%\"fG6#%\"xG%\"nGF&**F-F&)F),&F-F&F&!
\"\"F&F%F&7#F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%9~Note~the~spe
cial~case:~G\"\"\"%\"DGF&7#)%\"xG%\"nGF&**F+F&)F*,&F+F&F&!\"\"F&F'F&7#
F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 257 56 "Exponential Rule (ExpR) & Exponential Chain Rule
 (ExpCR)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "restart:\nc:='c':\nn:=
'n':\n'D'*[exp(x)]=exp(x)*`   ( ExpR )`;\n` `*'D'*[exp('f'(x))]= exp('
f'(x))*'D'*['f'(x)]*`   ( ExpCR )`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&%\"DG\"\"\"7#-%$expG6#%\"xGF&*&F(F&%,~~~(~ExpR~)GF&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$expG6#-%\"fG6#%\"xGF&*
*F)F&F'F&7#F,F&%-~~~(~ExpCR~)GF&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 54 "Natural Log \+
Rule (LnR) & Natural Log Chain Rule (LnCR)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 "restart:\nc:='c':\nn:='n':\n'D'*[ln(x)]=1/x,`   ( Ln
R )`;\n` `*'D'*[ln('f'(x))]= (1/'f'(x))*'D'*['f'(x)],`   ( LnCR )`;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*&%\"DG\"\"\"7#-%#lnG6#%\"xGF&*&F&F&
F+!\"\"%+~~~(~LnR~)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*(%\"~G\"\"\"
%\"DGF&7#-%#lnG6#-%\"fG6#%\"xGF&*(F,!\"\"F'F&7#F,F&%,~~~(~LnCR~)G" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 290 42 "Sine Rule (SinR) & Sine Chain Rule (SinCR)" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 123 "restart:\nc:='c':\nn:='n':\n'D'*[sin(x)]=cos
(x)*`   ( SinR )`;\n` `*'D'*[sin('f'(x))]= cos('f'(x))*'D'*['f'(x)]*` \+
  ( SinCR )`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"\"\"7#-%$si
nG6#%\"xGF&*&-%$cosGF*F&%,~~~(~SinR~)GF&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$sinG6#-%\"fG6#%\"xGF&**-%$cosGF+F&F'F&
7#F,F&%-~~~(~SinCR~)GF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 46 "Cosine Rule (CosR) & Cosine \+
Chain Rule (CosCR)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "restart:\nc:
='c':\nn:='n':\n'D'*[cos(x)]=-sin(x)*`   ( CosR )`;\n` `*'D'*[cos('f'(
x))]= -sin('f'(x))*'D'*['f'(x)]*`   ( CosCR )`;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"DG\"\"\"7#-%$cosG6#%\"xGF&,$*&-%$sinGF*F&%,~~~(~C
osR~)GF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&
7#-%$cosG6#-%\"fG6#%\"xGF&,$**-%$sinGF+F&F'F&7#F,F&%-~~~(~CosCR~)GF&!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 294 48 "Tangent Rule (TanR) & Tangent Chain Rule (TanCR)
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "restart:\nc:='c':\nn:='n':\n'D
'*[tan(x)]=sec(x)^2*`   ( TanR )`;\n` `*'D'*[tan('f'(x))]= sec('f'(x))
^2*'D'*['f'(x)]*`   ( TanCR )`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&
%\"DG\"\"\"7#-%$tanG6#%\"xGF&*&)-%$secGF*\"\"#F&%,~~~(~TanR~)GF&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$tanG6#-%\"fG
6#%\"xGF&**)-%$secGF+\"\"#F&F'F&7#F,F&%-~~~(~TanCR~)GF&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
295 52 "Cotangent Rule (CotR) & Cotangent Chain Rule (CotCR)" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 129 "restart:\nc:='c':\nn:='n':\n'D'*[cot(x)]
=-csc(x)^2*`   ( CotR )`;\n` `*'D'*[cot('f'(x))]= -csc('f'(x))^2*'D'*[
'f'(x)]*`   ( CotCR )`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"DG\"
\"\"7#-%$cotG6#%\"xGF&,$*&)-%$cscGF*\"\"#F&%,~~~(~CotR~)GF&!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$cotG6#-%\"fG
6#%\"xGF&,$**)-%$cscGF+\"\"#F&F'F&7#F,F&%-~~~(~CotCR~)GF&!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 296 46 "Secant Rule (SecR) & Secant Chain Rule (SecCR)" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 142 "restart:\nc:='c':\nn:='n':\n'D'*[sec(x)]
=sec(x)*tan(x)*`   ( SecR )`;\n` `*'D'*[sec('f'(x))]= sec('f'(x))*tan(
'f'(x))*'D'*['f'(x)]*`   ( SecCR )`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/*&%\"DG\"\"\"7#-%$secG6#%\"xGF&*(F(F&-%$tanGF*F&%,~~~(~SecR~)GF&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$secG6#-%\"f
G6#%\"xGF&*,F)F&-%$tanGF+F&F'F&7#F,F&%-~~~(~SecCR~)GF&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
297 50 "Cosecant Rule (CscR) & Cosecant Chain Rule (CscCR)" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 144 "restart:\nc:='c':\nn:='n':\n'D'*[csc(x)]=-
csc(x)*cot(x)*`   ( CscR )`;\n` `*'D'*[csc('f'(x))]= -csc('f'(x))*cot(
'f'(x))*'D'*['f'(x)]*`   ( CscCR )`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/*&%\"DG\"\"\"7#-%$cscG6#%\"xGF&,$*(F(F&-%$cotGF*F&%,~~~(~CscR~)GF&!
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"~G\"\"\"%\"DGF&7#-%$cscG
6#-%\"fG6#%\"xGF&,$*,F)F&-%$cotGF+F&F'F&7#F,F&%-~~~(~CscCR~)GF&!\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 31 "Algebraic Limit Rules: Examples" }}{EXCHG {PARA 258 
"" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 465 "r
estart:\nf:=x->x^2+3*x+4:\nLimit(``*f(x),x=1)=Limit(x^2,x=1)+Limit(3*x
,x=1)+Limit(4,x=1),` By SR`;\n``=Limit(x,x=1)^2+Limit(3*x,x=1)+Limit(4
,x=1),` By PFR 1`;\n``=1+Limit(3*x,x=1)+Limit(4,x=1),` By IFR`;\n``=5+
Limit(3*x,x=1),` By CFR`;\n``=5+3*Limit(x,x=1),` By CMR`;\n``=8,` By I
FR`;\n`So `,Limit(``*f(x),x=1)=limit(``*f(x),x=1);\n`Note that this is
 an example of the (by now) familiar fact that`;\n`if f is a continuou
s function at x `=a,` then `*Limit('f'(x),x=a)='f'(a);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$/-%&LimitG6$*&%!G\"\"\",(*$)%\"xG\"\"#F)F)*&\"\"$F
)F-F)F)\"\"%F)F)/F-F),(-F%6$F+F2F)-F%6$,$*&F0F)F-F)F)F2F)-F%6$F1F2F)%'
~By~SRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,(*$)-%&LimitG6$%\"xG/F
+\"\"\"\"\"#F-F--F)6$,$*&\"\"$F-F+F-F-F,F--F)6$\"\"%F,F-%*~By~PFR~1G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,(\"\"\"F&-%&LimitG6$,$*&\"\"$F&
%\"xGF&F&/F-F&F&-F(6$\"\"%F.F&%(~By~IFRG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$/%!G,&\"\"&\"\"\"-%&LimitG6$,$*&\"\"$F'%\"xGF'F'/F.F'F'%(~By~CFR
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G,&\"\"&\"\"\"*&\"\"$F'-%&Limi
tG6$%\"xG/F-F'F'F'%(~By~CMRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%!G\"
\")%(~By~IFRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$So~G/-%&LimitG6$*&%
!G\"\"\",(*$)%\"xG\"\"#F*F**&\"\"$F*F.F*F*\"\"%F*F*/F.F*,$*&\"\")F*F)F
*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnNote~that~this~is~an~example
~of~the~(by~now)~familiar~fact~thatG" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6$/%Dif~f~is~a~continuous~function~at~x~G%\"aG/*&%'~then~G\"\"\"-%&Lim
itG6$-%\"fG6#%\"xG/F0F%F)-F.6#F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 9 "Example 2" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 442 "restart:\nf:=x->(x^2-1)/(x-1):\nL
imit(f(x),x=2)=Limit(numer(f(x)),x=2)/Limit(denom(f(x)),x=2),` By QR`;
\n``=(Limit(x^2,x=2)+Limit(-1,x=2))/(Limit(x,x=2)+Limit(-1,x=2)),` By \+
SR`;\n\n``=3/(Limit(x,x=2)+Limit(-1,x=2)),` By PFR 1, IFR, CFR`;\n``=3
,` By IFR, CFR`;\n`So `,Limit(f(x),x=2)=limit(``*f(x),x=2);\n`Note tha
t this is another example of the (by now) familiar fact that`;\n`if f \+
is a continuous function at x `=a,` then `*Limit('f'(x),x=a)='f'(a);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F
-F-!\"\"F-,&F+F-F-F.F./F+F,*&-F%6$F(F0F--F%6$F/F0F.%'~By~QRG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$/%!G*&,&-%&LimitG6$*$)%\"xG\"\"#\"\"\"/F,F-F
.-F(6$!\"\"F/F.F.,&-F(6$F,F/F.F0F.F2%'~By~SRG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$/%!G,$*&\"\"$\"\"\",&-%&LimitG6$%\"xG/F-\"\"#F(-F+6$!\"
\"F.F(F2F(%4~By~PFR~1,~IFR,~CFRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%
!G\"\"$%-~By~IFR,~CFRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$So~G/-%&Li
mitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F.!\"\"F.,&F,F.F.F/F//F,F-,$*&\"\"$F.%!
GF.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%_oNote~that~this~is~another~
example~of~the~(by~now)~familiar~fact~thatG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$/%Dif~f~is~a~continuous~function~at~x~G%\"aG/*&%'~then~
G\"\"\"-%&LimitG6$-%\"fG6#%\"xG/F0F%F)-F.6#F%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 9 "Example
 3" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "restart:\nf:=x->(x^2-1)/(x-1
):\na:=1:\nLimit(f(x),x=a)=Limit(numer(f(x)),x=a)/Limit(denom(f(x)),x=
a),` By QR ??`;\nLimit(f(x),x=1)*` = `*Limit(factor(numer(f(x)))/denom
(f(` x`)),x=1)\n=Limit(factor(numer(f(x)))/denom(f(x)),x=1)*` = 2`,` B
y SR, IFR, CFR`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG6$*&,&*$
)%\"xG\"\"#\"\"\"F-F-!\"\"F-,&F+F-F-F.F./F+F-*&-F%6$F(F0F--F%6$F/F0F.%
*~By~QR~??G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*(-%&LimitG6$*&,&*$)%
\"xG\"\"#\"\"\"F.F.!\"\"F.,&F,F.F.F/F//F,F.F.%$~=~GF.-F&6$*(F0F.,&F,F.
F.F.F.,&%#~xGF.F.F/F/F1F.*&-F&6$F6F1F.%%~=~2GF.%1~By~SR,~IFR,~CFRG" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "
" {TEXT -1 9 "Example 4" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "restart
:\nf:=x->(x-1)/(x^2-1):\na:=-1:\nLimit(f(x),x=a)*` = `*Limit(numer(f(x
))/factor(denom(f(` x`))),x=a)\n=Limit(numer(f(x))/factor(denom(f(x)))
,x=a)*` which doesn't exist.`;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*
(-%&LimitG6$*&,&%\"xG\"\"\"F+!\"\"F+,&*$)F*\"\"#F+F+F+F,F,/F*F,F+%$~=~
GF+-F&6$*(F)F+,&%#~xGF+F+F,F,,&F7F+F+F+F,F1F+*&-F&6$*&F+F+,&F*F+F+F+F,
F1F+%6~which~doesn't~exist.GF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "Limit(f(x),x=a,
left)*` = `*Limit(numer(f(x))/factor(denom(f(` x`))),x=a,left)\n*`  = \+
`*Limit(numer(f(x))/factor(denom(f(x))),x=a,left)=limit(f(x),x=a,left)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,-%&LimitG6%*&,&%\"xG\"\"\"F+!
\"\"F+,&*$)F*\"\"#F+F+F+F,F,/F*F,%%leftGF+%$~=~GF+-F&6%*(F)F+,&%#~xGF+
F+F,F,,&F8F+F+F+F,F1F2F+%%~~=~GF+-F&6%*&F+F+,&F*F+F+F+F,F1F2F+,$%)infi
nityGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "Limit(f(x),x=a,right)*` = `*Limit(
numer(f(x))/factor(denom(f(` x`))),x=a,right)\n=Limit(numer(f(x))/fact
or(denom(f(x))),x=a,right)*` = `*limit(f(x),x=a,right);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*(-%&LimitG6%*&,&%\"xG\"\"\"F+!\"\"F+,&*$)F*\"
\"#F+F+F+F,F,/F*F,%&rightGF+%$~=~GF+-F&6%*(F)F+,&%#~xGF+F+F,F,,&F8F+F+
F+F,F1F2F+*(-F&6%*&F+F+,&F*F+F+F+F,F1F2F+F3F+%)infinityGF+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 22 "Algebraic Limit Rules:" }{TEXT 267 10 " Exercises" }{TEXT -1 0 
"" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 11 "Exercise 1." 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "n:=1:\nf:=x->2:\na:=4:\n``||n||`.
`,Limit(f(x),x=a)=``;``;\n``||n||`.`,Limit(f(x),x=a)=limit(f(x),x=a);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#1.G/-%&LimitG6$\"\"#/%\"xG\"\"%%
!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%#1.G/-%&LimitG6$\"\"#/%\"xG\"\"%F(" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 11 "Ex
ercise 2." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "n:=2:
\nf:=x->(5-4*x)^2:\na:=3:\n``||n||`.`,Limit(f(x),x=a)=``;``;\n``||n||`
.`,Limit(f(x),x=a)=limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
$%#2.G/-%&LimitG6$*$),&\"\"&\"\"\"*&\"\"%F,%\"xGF,!\"\"\"\"#F,/F/\"\"$
%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%#2.G/-%&LimitG6$*$),&\"\"&\"\"\"*&\"\"%F,%\"xGF,!\"\"
\"\"#F,/F/\"\"$\"#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 270 11 "Exercise 3." }{TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "n:=3:\nf:=x->x^2+3*x-7:\na:=-4:\n`
`||n||`.`,Limit(f(x),x=a)=``;``;\n``||n||`.`,Limit(f(x),x=a)=limit(f(x
),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#3.G/-%&LimitG6$,(*$)%\"xG
\"\"#\"\"\"F-*&\"\"$F-F+F-F-\"\"(!\"\"/F+!\"%%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#3.G/-%&LimitG6
$,(*$)%\"xG\"\"#\"\"\"F-*&\"\"$F-F+F-F-\"\"(!\"\"/F+!\"%!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 271 11 "Exercise 4." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 106 "n:=4:\nf:=x->3*x/(x+4):\na:=2:\n``||n||`.`,Limit(f(x
),x=a)=``;``;\n``||n||`.`,Limit(f(x),x=a)=limit(f(x),x=a);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$%#4.G/-%&LimitG6$,$*(\"\"$\"\"\"%\"xGF+,&F,F
+\"\"%F+!\"\"F+/F,\"\"#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$%#4.G/-%&LimitG6$,$*(\"\"$\"\"\"%\"xGF
+,&F,F+\"\"%F+!\"\"F+/F,\"\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 11 "Exercise 5." }{TEXT 
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "n:=5:\nf:=x->(x^2+1)/(3*
x^5+4):\na:=-1:\n``||n||`.`,Limit(f(x),x=a)=``;``;\n``||n||`.`,Limit(f
(x),x=a)=limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#5.G/-%&
LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F.F.F.,&*&\"\"$F.)F,\"\"&F.F.\"\"%F.!
\"\"/F,F5%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$%#5.G/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F.F.F.,&*&
\"\"$F.)F,\"\"&F.F.\"\"%F.!\"\"/F,F5F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 11 "Exercise 6.
" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "n:=6:\nf:=x->x^
2/(x^2+1):\na:=0:\n``||n||`.`,Limit(f(x),x=a)=``;``;\n``||n||`.`,Limit
(f(x),x=a)=limit(f(x),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#6.G/-
%&LimitG6$*&%\"xG\"\"#,&*$)F)F*\"\"\"F.F.F.!\"\"/F)\"\"!%!G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#6.G
/-%&LimitG6$*&%\"xG\"\"#,&*$)F)F*\"\"\"F.F.F.!\"\"/F)\"\"!F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 274 11 "Exercise 7." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 108 "n:=7:\nf:=x->(x^2+1)/x^2:\na:=0:\n``||n||`.`,Limit(f
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11 "" 1 "" {XPPMATH 20 "6$%#7.G/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F.
F.F.F,!\"#/F,\"\"!%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$%#7.G/-%&LimitG6$*&,&*$)%\"xG\"\"#\"\"\"F.F.
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0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 11 "Exercise 8." }{TEXT -1 
0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "n:=8:\nf:=x->x/(x^2-4):\na:
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 276 11 "Exercise 9." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
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11 "" 1 "" {XPPMATH 20 "6$%#9.G/-%&LimitG6$*&%\"hG\"\"\",&F*F**&F*F*F)
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11 "" 1 "" {XPPMATH 20 "6$%#9.G/-%&LimitG6$*&%\"hG\"\"\",&F*F**&F*F*F)
!\"\"F-F*/F)\"\"!F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 277 12 "Exercise 10." }{TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "n:=10:\nf:=h->(1-1/h)/(1-2/h):\na:
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{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
$%$10.G/-%&LimitG6$*&,&\"\"\"F**&F*F*%\"hG!\"\"F-F*,&F*F**&\"\"#F*F,F-
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 2755 "restart:\nwith(plottools):\nn:=2:                \+
          # Exercise number\nntxt:=convert(n,symbol):\n               \+
                # Define function f here below\n#f:=unapply(piecewise(
x<-2,2*x+9,-2<x and x<=1,x^2+1,1<x and x<3,3*x-1,3<x,x+6,`undefined`),
x):\nf:=unapply((5-4*x)^2):\nComputeLimits:=proc(fcn,real)  # Begin Sm
all program to compute limits\nlocal f,realtxt:\nf:=fcn:\nrealtxt:=con
vert(real,symbol):\nprint(`Limiting value that f(x) approaches as x ap
proaches `||realtxt||` from the left:`):\nprint(Limit('f'(x),x=real,le
ft)=limit(f(x),x=real,left),'f'(real)=f(real)):\nprint(`Limiting value
 that f(x) approaches as x approaches `||realtxt||` from the right:`):
\nprint(Limit('f'(x),x=real,right)=limit(f(x),x=real,right),'f'(real)=
f(real)):\nprint(`Limiting value that f(x) approaches as x approaches \+
`||realtxt||` from either side:`):\nprint(Limit('f'(x),x=real)=limit(f
(x),x=real),'f'(real)=f(real)):\nend:                           # End \+
Small program to compute limits\nComputeLimitsExercise:=proc(fcn,real)
  # Begin Small program to compute limits\nlocal f,realtxt:\nf:=fcn:\n
realtxt:=convert(real,symbol):\n#print(`Limiting value that f(x) appro
aches as x approaches `||realtxt||` from the left:`):\nprint(Limit('f'
(x),x=real,left)=`____ `,Limit('f'(x),x=real,right)=`____ `,\n      Li
mit('f'(x),x=real)=`____ `,'f'(real)=`____`):\nend:                   \+
        # End Small program to compute limits\n\n\n                   \+
            # Start creating individual plots here below\np[0]:=plot(f
(x),x=-3..4,color=blue,discont=true):\np[1]:=circle([-2,5],.125,color=
blue):\np[2]:=plot([[-2,0],[-2,5]],color=magenta,linestyle=2):\np[3]:=
disk([1,2],.125,color=blue):\np[4]:=plot([[1,0],[1,2]],color=magenta,l
inestyle=2):\np[5]:=circle([3,8],.125,color=blue):\np[6]:=plot([[3,0],
[3,9]],color=magenta,linestyle=2):\np[7]:=circle([3,9],.125,color=blue
):\n#p[3]:=circle([2,4],.,color=blue):\n#p[4]:=circle([2,1],.075,color
=blue):\n#p[6]:=circle([2,1],.075,color=blue):\n#p[7]:=plot([[2,0],[2,
4]],color=magenta,linestyle=2):\n#p[8]:=disk([3,1],.075,color=blue):\n
#p[9]:=plot([[3,0],[3,1]],color=magenta,linestyle=2):\nplotFunction:=p
lots[display](\n                   [seq(p[i],i=0..0)],\n              \+
      scaling=constrained,                   \n                   labe
ls=[``,``],\n                   #xtickmarks=4,ytickmarks=4,\n         \+
          view=[-3..4,-1..10]):\n                   #titlefont=[HELVET
ICA,DEFAULT,14],\n                   #title=`Limits of Piecewise Funct
ions: `||`Exercise `||ntxt):\nplotFunction;\n'f'(x)=f(x);``;\n`Compute
 the following limits and fill in the blanks:`;``;\nComputeLimitsExerc
ise(f,-2);``;\nComputeLimitsExercise(f,-1);``;\nComputeLimitsExercise(
f,0);``;\nComputeLimitsExercise(f,1);``;\nComputeLimitsExercise(f,2);`
`;\nComputeLimitsExercise(f,3);``;\n" }}{PARA 13 "" 1 "" {GLPLOT2D 
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%leftG%&____~G/-F%6%F'F+%&rightGF./-F%6$F'F+F./-F(6#F,%%____G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
&/-%&LimitG6%-%\"fG6#%\"xG/F*\"\"\"%%leftG%&____~G/-F%6%F'F+%&rightGF.
/-F%6$F'F+F./-F(6#F,%%____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%&LimitG6%-%\"fG6#%\"xG/F*\"\"#%%lef
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"" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%&Limi
tG6%-%\"fG6#%\"xG/F*\"\"$%%leftG%&____~G/-F%6%F'F+%&rightGF./-F%6$F'F+
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Filenam
e: ExploreCalc04TexPdf.mws" }}{PARA 0 "" 0 "" {TEXT -1 36 "Copyright 2
005, All Rights Reserved." }}{PARA 0 "" 0 "" {TEXT -1 44 "Permission i
s granted to use and modify for " }}{PARA 0 "" 0 "" {TEXT -1 37 "acade
mic and non-commercial purposes." }}{PARA 0 "" 0 "" {TEXT -1 13 "Dr. J
ohn Pais" }}{PARA 0 "" 0 "" {TEXT -1 28 "Mathematics Department-MICDS
" }}{PARA 0 "" 0 "" {TEXT -1 45 "E-mail: pais@micds.org or pais@kineti
gram.com" }}{PARA 0 "" 0 "" {TEXT -1 33 "URL:  http://kinetigram.com/m
icds" }}{PARA 0 "" 0 "" {TEXT -1 37 "_________________________________
____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "182 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }