{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" 
-1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 
1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time
s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 
2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 
1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 
0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 257 1 {CSTYLE "" -1 -1 "Ti
mes" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 257 "
" 0 "" {TEXT -1 39 "Lesson 4:  Definition of the Derivative" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 342 "In thi
s worksheet, you will see how  to solve two problems: the mathematical
 problem of finding the tangent line to the graph of a function, and t
he physical problem of defining the velocity of a moving object.  Alth
ough these problems might seem unrelated at first, it will turn out th
at they both lead to the same mathematical concept: the " }{TEXT 256 
10 "derivative" }{TEXT -1 15 " of a function." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "
Suppose we are given a function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }
{TEXT -1 13 " and a point " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }
{TEXT -1 18 " in the domain of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 
-1 23 ".  How can we find the " }{TEXT 259 12 "tangent line" }{TEXT 
-1 17 " to the graph of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 15 "
 at the point (" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 
", " }{XPPEDIT 18 0 "f(x[0]);" "6#-%\"fG6#&%\"xG6#\"\"!" }{TEXT -1 
160 ") on the graph?  Here is a picture of the sort of thing we are af
ter.  (Concentrate on the picture, and don't worry for now about the c
ommands that produce it.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "f := x-> x^2: \nf1 := D(f): \nt := x-> f(1) + f1(1)*(x - 1):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot([f(x),t(x)], x=-1..3, colour=[
black,red]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&
-%'CURVESG6$7S7$$!\"\"\"\"!$\"\"\"F*7$$!3PLLLLQ6G\"*!#=$\"3dO\"Ha@YAL)
F07$$!3immmT.\\p$)F0$\"3'=;Dz&o$[+(F07$$!3LLLL$))Qj^(F0$\"3ee3\"4-N&\\
cF07$$!3ULLL$=Kvl'F0$\"3!*4>@xMFKWF07$$!3hnmmTs!G!eF0$\"3u6RR)=dsO$F07
$$!3iLLL3yO5]F0$\"3>(Hyuby.^#F07$$!3i+++vE%)*=%F0$\"3/;^7k\"yav\"F07$$
!3)RLL$3WDTLF0$\"3#Qp?@5)R;6F07$$!3'4++]d(Q&\\#F0$\"3G&Qk%\\\"fpA'!#>7
$$!3:mmmm&4`i\"F0$\"3F'=)\\(=J;k#Fen7$$!3GKLLLQW*e)Fen$\"3m+))fOX&yP(!
#?7$$\"3HI#*******H,Q!#@$\"3#[T***o\"))\\W\"!#C7$$\"3Q(*******\\*3q)Fe
n$\"3#e\\-,Qd0d(F`o7$$\"3!********p=\\q\"F0$\"3l*o4OxZn!HFen7$$\"3_mmm
\"fBIY#F0$\"3oll58_[mgFen7$$\"3yKLLLO[kLF0$\"3S'y'*=,v>8\"F07$$\"3.KLL
L&Q\"GTF0$\"3ZZ\"R]x_Tq\"F07$$\"3+*****\\s]k,&F0$\"3s,`j(yxk^#F07$$\"3
WJLLLvv-eF0$\"3%Gnl!*\\*>nLF07$$\"3'3++]sgam'F0$\"3*Qv^wmOGW%F07$$\"3G
+++v\"ep[(F0$\"3-M*>9Faag&F07$$\"3#QLLLe/TM)F0$\"3cM/wH\"3C'pF07$$\"39
LLLeDBJ\"*F0$\"3)3nOM!3%zL)F07$$\"3Immm;kD!)**F0$\"3=<;C9=bg**F07$$\"3
Mmm;f`@'3\"!#<$\"3!*p&*[1Q')z6Fgs7$$\"3y****\\nZ)H;\"Fgs$\"3iGqVpN`_8F
gs7$$\"3YmmmJy*eC\"Fgs$\"37,<&pShAb\"Fgs7$$\"3')******R^bJ8Fgs$\"3A>/'
34RIx\"Fgs7$$\"3f*****\\5a`T\"Fgs$\"3!*\\.aUsA.?Fgs7$$\"3o****\\7RV'\\
\"Fgs$\"3^c+[aWJRAFgs7$$\"3k*****\\@fke\"Fgs$\"3*\\Te3%G&o^#Fgs7$$\"3/
LLL`4Nn;Fgs$\"3gu:e,#f+y#Fgs7$$\"3#*******\\,s`<Fgs$\"3-Ag^kV`vIFgs7$$
\"3[mm;zM)>$=Fgs$\"3'RgRzYjhN$Fgs7$$\"3$*******pfa<>Fgs$\"3oSK1ZD)pn$F
gs7$$\"3#HLLeg`!)*>Fgs$\"3#Qe$y6#=A*RFgs7$$\"3w****\\#G2A3#Fgs$\"3@MgH
nreNVFgs7$$\"3;LLL$)G[k@Fgs$\"3UzHC_h)\\o%Fgs7$$\"3#)****\\7yh]AFgs$\"
31%GUz`!Gl]Fgs7$$\"3xmmm')fdLBFgs$\"3Y2ta&)odXaFgs7$$\"3bmmm,FT=CFgs$
\"3?lEe&**>([eFgs7$$\"3FLL$e#pa-DFgs$\"3amy*f6TFE'Fgs7$$\"3!*******Rv&
)zDFgs$\"3K^[pE\\mbmFgs7$$\"3ILLLGUYoEFgs$\"3C7Y*yL,27(Fgs7$$\"3_mmm1^
rZFFgs$\"3e5US2$Q*\\vFgs7$$\"34++]sI@KGFgs$\"3-\"*)R!))3V@!)Fgs7$$\"34
++]2%)38HFgs$\"3%f)3\"*pS3'[)Fgs7$$\"\"$F*$\"\"*F*-%'COLOURG6&%$RGBGF*
F*F*-F$6$7S7$F($!\"$F*7$F.$!3ommmmFiDGFgs7$F4$!35LLLo!)*Qn#Fgs7$F9$!3n
mmmwxE.DFgs7$F>$!3YmmmOk]JBFgs7$FC$!3_LLL[9cg@Fgs7$FH$!3smmmhN2-?Fgs7$
FM$!37+++N&oz$=Fgs7$FR$!3!omm;)3Do;Fgs7$FW$!3?+++:v2*\\\"Fgs7$Fgn$!3BL
LL8>1D8Fgs7$F\\o$!3kmmmw))yr6Fgs7$Fbo$!3a,+++uR#***F07$Fio$!3_++++5#)f
#)F07$F^p$!3?++++E;!f'F07$Fcp$!3)pmmm\"G&R2&F07$Fhp$!3WMLLLF.rKF07$F]q
$!3%fLLL$HsV<F07$Fbq$\"3/+)*****\\9!H$F`o7$Fgq$\"3)Gmmm1:bg\"F07$F\\r$
\"3t,++]9#4L$F07$Far$\"3a+++]j\"R(\\F07$Ffr$\"3knmmm\"4#)o'F07$F[s$\"3
Gmmm;^Yi#)F07$F`s$\"3gKLLLG^g**F07$Fes$\"3oKLL=2Vs6Fgs7$F[t$\"3f*****
\\`pfK\"Fgs7$F`t$\"3!HLLLm&z\"\\\"Fgs7$Fet$\"3s******z-6j;Fgs7$Fjt$\"3
<******4#32$=Fgs7$F_u$\"3O*****\\#y'G*>Fgs7$Fdu$\"3G******H%=H<#Fgs7$F
iu$\"35mmm1>qMBFgs7$F^v$\"3%)*******HSu]#Fgs7$Fcv$\"3'HLL$ep'Rm#Fgs7$F
hv$\"3')******R>4NGFgs7$F]w$\"3#emm;@2h*HFgs7$Fbw$\"3]*****\\c9W;$Fgs7
$Fgw$\"3Lmmmmd'*GLFgs7$F\\x$\"3j*****\\iN7]$Fgs7$Fax$\"3aLLLt>:nOFgs7$
Ffx$\"35LLL.a#o$QFgs7$F[y$\"3ammm^Q40SFgs7$F`y$\"3y******z]rfTFgs7$Fey
$\"3gmmmc%GpL%Fgs7$Fjy$\"3/LLL8-V&\\%Fgs7$F_z$\"3=+++XhUkYFgs7$Fdz$\"3
=+++:o<E[Fgs7$Fiz$\"\"&F*-F^[l6&F`[l$\"*++++\"!\")$F*F*F\\el-%+AXESLAB
ELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fiz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 45 "The black curve is the graph of the function " 
}{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f
(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 55 ", and the red li
ne is the tangent line to the graph of " }{XPPEDIT 18 0 "f;" "6#%\"fG
" }{TEXT -1 7 " at at " }{XPPEDIT 18 0 "x[0] = .1;" "6#/&%\"xG6#\"\"!$
\"\"\"!\"\"" }{TEXT -1 108 ".  You can see on the picture what is mean
t by a tangent line: it is the line which intersects the graph of " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 242 " at the given point, and i
s \"parallel\" to the graph at that point.  It is important to realise
 that this is the only information we have about the tangent line.  In
 particular, we do not know that the tangent line will intersect the g
raph of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 15 " a second time.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
281 "The tangent line problem, then is to find the equation of a line \+
when we only know one point on the line.  Of course, if we also knew t
he slope of the line, we could write down its equation in point-slope \+
form, so  the problem becomes: how do we find the slope of the tangent
 line?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 
"The solution is to introduce a second point.  If we are given " }
{TEXT 257 3 "two" }{TEXT -1 24 " points on the graph of " }{XPPEDIT 
18 0 "f;" "6#%\"fG" }{TEXT -1 111 ", we can easily find the equation o
f the line joining them.  (A line joining two points on a graph is cal
led a " }{TEXT 258 12 "secant line." }{TEXT -1 67 ")  You should be ab
le to see what the following commands are doing." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "m := (f(3) - f(1))/(3 - 1);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"mG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "s := x-> f(1) + m*(x - 1) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"sGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%\"fG6#\"\"\"F0*&%\"mGF0,&9$
F0F0!\"\"F0F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot
([f(x),s(x)], x=-1..4, colour=[black,red]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"\"\"\"!$\"\"
\"F*7$$!3Fnmm\"HU,\"*)!#=$\"3I$pudcj!RzF07$$!3)HL$3FH'='zF0$\"3)yRmp7E
\"RjF07$$!35mm;/OU&*oF0$\"3Q\"))*3omoaZF07$$!3Plm;H_\">#eF0$\"3su-c$pp
%*Q$F07$$!3SLL3_!4Nv%F0$\"3A>Q#3$[efAF07$$!3YmmTg(fHw$F0$\"3a0:&eh')fT
\"F07$$!3^++vVLIPFF0$\"3YL\\q&fHG\\(!#>7$$!3#em;/,oln\"F0$\"3A(\\_NH!)
3\"GFQ7$$!3Y&***\\(oWB>'FQ$\"3qT4)GF8X$Q!#?7$$\"3C%ommTIOo%FQ$\"3wLD*z
QRO>#Ffn7$$\"3eML$3_>jU\"F0$\"3)HB5bP(QM?FQ7$$\"3E,++D;v/DF0$\"3SZ,%Hq
!ytiFQ7$$\"3y+++v=h(e$F0$\"3;-Tc'*e4(G\"F07$$\"3V+++v$[6j%F0$\"3Y9l7FN
vW@F07$$\"3EMLe*[z(ybF0$\"3!)o&R$f!yA6$F07$$\"3knmmTXg0nF0$\"3Ak?#pA8l
\\%F07$$\"3OommmJ<gwF0$\"3G0?L%HDy'eF07$$\"3U++D1Mcq()F0$\"3)3;0jCyAp(
F07$$\"3'fmmm\"pW`(*F0$\"3&*[MivE(H^*F07$$\"3K+]i!f#=$3\"!#<$\"3'p3LY_
%Gt6F\\r7$$\"3?+](=xpe=\"F\\r$\"31lo%e6(G19F\\r7$$\"37nm\"H28IH\"F\\r$
\"3Q-Mt1G)=n\"F\\r7$$\"3um;zpSS\"R\"F\\r$\"3O8GV&G0g$>F\\r7$$\"3GLL3_?
`(\\\"F\\r$\"38>p,ZAgUAF\\r7$$\"3fL$e*)>pxg\"F\\r$\"3%=7>rz@\\e#F\\r7$
$\"33+]Pf4t.<F\\r$\"3DeG$>=*p-HF\\r7$$\"3uLLe*Gst!=F\\r$\"3;-P:$f%fmKF
\\r7$$\"30+++DRW9>F\\r$\"3y0%p>a&4lOF\\r7$$\"3:++DJE>>?F\\r$\"3k)H%4#)
)Qr2%F\\r7$$\"3F+]i!RU07#F\\r$\"3#3fP/.+n\\%F\\r7$$\"3+++v=S2LAF\\r$\"
3[vi@t&>m)\\F\\r7$$\"3Jmmm\"p)=MBF\\r$\"37VXI[oV[aF\\r7$$\"3B++](=]@W#
F\\r$\"3U'G1$Qv4kfF\\r7$$\"35L$e*[$z*RDF\\r$\"33)zMJ4&\\^kF\\r7$$\"3e+
+]iC$pk#F\\r$\"3N<j.h9D1qF\\r7$$\"3[m;H2qcZFF\\r$\"3I%od&fW7\\vF\\r7$$
\"3O+]7.\"fF&GF\\r$\"3fbDY+XBQ\")F\\r7$$\"3Ymm;/OgbHF\\r$\"3D))H'\\m#f
N()F\\r7$$\"3w**\\ilAFjIF\\r$\"3Q=tMtpj$Q*F\\r7$$\"3yLLL$)*pp;$F\\r$\"
3kLM`())pH+\"!#;7$$\"3)RL$3xe,tKF\\r$\"3ieR;$Hj72\"Fcx7$$\"3Cn;HdO=yLF
\\r$\"3Z]#RA[779\"Fcx7$$\"3a+++D>#[Z$F\\r$\"3(32Y5uQu?\"Fcx7$$\"3SnmT&
G!e&e$F\\r$\"3'p'oJ)fQcG\"Fcx7$$\"3#RLLL)Qk%o$F\\r$\"3=v&)pa+md8Fcx7$$
\"37+]iSjE!z$F\\r$\"3+$[(G$*=hO9Fcx7$$\"3a+]P40O\"*QF\\r$\"3PEBRh'oU^
\"Fcx7$$\"\"%F*$\"#;F*-%'COLOURG6&%$RGBGF*F*F*-F$6$7S7$F($!\"(F*7$F.$!
3!pmmm\"p0klF\\r7$F4$!3uKL$3<XZ='F\\r7$F9$!3WmmmT%p\"edF\\r7$F>$!3:mmm
\"4m(G`F\\r7$FC$!3OLL$3i.9!\\F\\r7$FH$!3fmm;/R=0XF\\r7$FM$!3k++]P8#\\4
%F\\r7$FS$!3Kmm;/siqOF\\r7$FX$!3#)****\\(y$pZKF\\r7$Fhn$!3jKLL$yaE\"GF
\\r7$F]o$!3<mmm\">s%HCF\\r7$Fbo$!3]******\\$*4)*>F\\r7$Fgo$!3o******\\
_&\\c\"F\\r7$F\\p$!3%)******\\1aZ6F\\r7$Fap$!3+jmmT?)[o(F07$Ffp$!3WHLL
L=exJF07$F[q$\"3gMnmmm#pS'FQ7$F`q$\"3m,++DOD#3&F07$Feq$\"3(Qmmmm(y8!*F
07$Fjq$\"3K,+]i.tK8F\\r7$F`r$\"3!3++v3zMu\"F\\r7$Fer$\"3Yomm\"H_?<#F\\
r7$Fjr$\"3-nm;zihlDF\\r7$F_s$\"39LLL3#G,*HF\\r7$Fds$\"3OML$ezw5V$F\\r7
$Fis$\"3I++]PQ#\\\"QF\\r7$F^t$\"3%\\LL$e\"*[HUF\\r7$Fct$\"3=++++dxdYF
\\r7$Fht$\"3e+++D0xw]F\\r7$F]u$\"35,+]i&p@[&F\\r7$Fbu$\"3++++vgHKfF\\r
7$Fgu$\"3ElmmmZvOjF\\r7$F\\v$\"3%4+++v+'onF\\r7$Fav$\"3UKL$eR<*frF\\r7
$Ffv$\"3K-++])Hxe(F\\r7$F[w$\"3!fmm\"H!o-*zF\\r7$F`w$\"3X,+]7k.6%)F\\r
7$Few$\"3#emmmT9C#))F\\r7$Fjw$\"33****\\i!*3`#*F\\r7$F_x$\"3;NLLL*zym*
F\\r7$Fex$\"3fLL$3N1#45Fcx7$Fjx$\"3!pm;HYt70\"Fcx7$F_y$\"3A+++q(G**3\"
Fcx7$Fdy$\"3'pmmT6KU8\"Fcx7$Fiy$\"3eLLL`v&Q<\"Fcx7$F^z$\"30++DOl5;7Fcx
7$Fcz$\"3A++v.Uac7Fcx7$Fhz$\"#8F*-F][l6&F_[l$\"*++++\"!\")$F*F*F[el-%+
AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fhz%(DEFAULTG" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 130 "We chose x=2 as the second point.  Let's
 examine what happens as this point is chosen closer and closer to the
 starting point x=1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "c :
= 1;   left := c-1; right := c+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"cG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%leftG\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&rightG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "SecantLine := (a,b) -> f(a) + ( (f(b)-f(a)) / (b-a) )
 * (x-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+SecantLineGR6$%\"aG%\"
bG6\"6$%)operatorG%&arrowGF),&-%\"fG6#9$\"\"\"*&*&,&-F/6#9%F2F.!\"\"F2
,&%\"xGF2F1F9F2F2,&F8F2F1F9F9F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 146 "plots[display](plot(f(x), x= -1..4, color = black, t
hickness=2),\n               plot(\{SecantLine(c, c + 2/k) $ k =1..6 \+
\}, x= -1..4, color = red));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
300 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$!\"\"\"\"!$\"\"\"F*7$$!3Fnmm\"HU,
\"*)!#=$\"3I$pudcj!RzF07$$!3)HL$3FH'='zF0$\"3)yRmp7E\"RjF07$$!35mm;/OU
&*oF0$\"3Q\"))*3omoaZF07$$!3Plm;H_\">#eF0$\"3su-c$pp%*Q$F07$$!3SLL3_!4
Nv%F0$\"3A>Q#3$[efAF07$$!3YmmTg(fHw$F0$\"3a0:&eh')fT\"F07$$!3^++vVLIPF
F0$\"3YL\\q&fHG\\(!#>7$$!3#em;/,oln\"F0$\"3A(\\_NH!)3\"GFQ7$$!3Y&***\\
(oWB>'FQ$\"3qT4)GF8X$Q!#?7$$\"3C%ommTIOo%FQ$\"3wLD*zQRO>#Ffn7$$\"3eML$
3_>jU\"F0$\"3)HB5bP(QM?FQ7$$\"3E,++D;v/DF0$\"3SZ,%Hq!ytiFQ7$$\"3y+++v=
h(e$F0$\"3;-Tc'*e4(G\"F07$$\"3V+++v$[6j%F0$\"3Y9l7FNvW@F07$$\"3EMLe*[z
(ybF0$\"3!)o&R$f!yA6$F07$$\"3knmmTXg0nF0$\"3Ak?#pA8l\\%F07$$\"3OommmJ<
gwF0$\"3G0?L%HDy'eF07$$\"3U++D1Mcq()F0$\"3)3;0jCyAp(F07$$\"3'fmmm\"pW`
(*F0$\"3&*[MivE(H^*F07$$\"3K+]i!f#=$3\"!#<$\"3'p3LY_%Gt6F\\r7$$\"3?+](
=xpe=\"F\\r$\"31lo%e6(G19F\\r7$$\"37nm\"H28IH\"F\\r$\"3Q-Mt1G)=n\"F\\r
7$$\"3um;zpSS\"R\"F\\r$\"3O8GV&G0g$>F\\r7$$\"3GLL3_?`(\\\"F\\r$\"38>p,
ZAgUAF\\r7$$\"3fL$e*)>pxg\"F\\r$\"3%=7>rz@\\e#F\\r7$$\"33+]Pf4t.<F\\r$
\"3DeG$>=*p-HF\\r7$$\"3uLLe*Gst!=F\\r$\"3;-P:$f%fmKF\\r7$$\"30+++DRW9>
F\\r$\"3y0%p>a&4lOF\\r7$$\"3:++DJE>>?F\\r$\"3k)H%4#))Qr2%F\\r7$$\"3F+]
i!RU07#F\\r$\"3#3fP/.+n\\%F\\r7$$\"3+++v=S2LAF\\r$\"3[vi@t&>m)\\F\\r7$
$\"3Jmmm\"p)=MBF\\r$\"37VXI[oV[aF\\r7$$\"3B++](=]@W#F\\r$\"3U'G1$Qv4kf
F\\r7$$\"35L$e*[$z*RDF\\r$\"33)zMJ4&\\^kF\\r7$$\"3e++]iC$pk#F\\r$\"3N<
j.h9D1qF\\r7$$\"3[m;H2qcZFF\\r$\"3I%od&fW7\\vF\\r7$$\"3O+]7.\"fF&GF\\r
$\"3fbDY+XBQ\")F\\r7$$\"3Ymm;/OgbHF\\r$\"3D))H'\\m#fN()F\\r7$$\"3w**\\
ilAFjIF\\r$\"3Q=tMtpj$Q*F\\r7$$\"3yLLL$)*pp;$F\\r$\"3kLM`())pH+\"!#;7$
$\"3)RL$3xe,tKF\\r$\"3ieR;$Hj72\"Fcx7$$\"3Cn;HdO=yLF\\r$\"3Z]#RA[779\"
Fcx7$$\"3a+++D>#[Z$F\\r$\"3(32Y5uQu?\"Fcx7$$\"3SnmT&G!e&e$F\\r$\"3'p'o
J)fQcG\"Fcx7$$\"3#RLLL)Qk%o$F\\r$\"3=v&)pa+md8Fcx7$$\"37+]iSjE!z$F\\r$
\"3+$[(G$*=hO9Fcx7$$\"3a+]P40O\"*QF\\r$\"3PEBRh'oU^\"Fcx7$$\"\"%F*$\"#
;F*-%'COLOURG6&%$RGBGF*F*F*-%*THICKNESSG6#\"\"#-F$6$7S7$F($!\"(F*7$F.$
!3!pmmm\"p0klF\\r7$F4$!3uKL$3<XZ='F\\r7$F9$!3WmmmT%p\"edF\\r7$F>$!3:mm
m\"4m(G`F\\r7$FC$!3OLL$3i.9!\\F\\r7$FH$!3fmm;/R=0XF\\r7$FM$!3k++]P8#\\
4%F\\r7$FS$!3Kmm;/siqOF\\r7$FX$!3#)****\\(y$pZKF\\r7$Fhn$!3jKLL$yaE\"G
F\\r7$F]o$!3<mmm\">s%HCF\\r7$Fbo$!3]******\\$*4)*>F\\r7$Fgo$!3o******
\\_&\\c\"F\\r7$F\\p$!3%)******\\1aZ6F\\r7$Fap$!3+jmmT?)[o(F07$Ffp$!3WH
LLL=exJF07$F[q$\"3gMnmmm#pS'FQ7$F`q$\"3m,++DOD#3&F07$Feq$\"3(Qmmmm(y8!
*F07$Fjq$\"3K,+]i.tK8F\\r7$F`r$\"3!3++v3zMu\"F\\r7$Fer$\"3Yomm\"H_?<#F
\\r7$Fjr$\"3-nm;zihlDF\\r7$F_s$\"39LLL3#G,*HF\\r7$Fds$\"3OML$ezw5V$F\\
r7$Fis$\"3I++]PQ#\\\"QF\\r7$F^t$\"3%\\LL$e\"*[HUF\\r7$Fct$\"3=++++dxdY
F\\r7$Fht$\"3e+++D0xw]F\\r7$F]u$\"35,+]i&p@[&F\\r7$Fbu$\"3++++vgHKfF\\
r7$Fgu$\"3ElmmmZvOjF\\r7$F\\v$\"3%4+++v+'onF\\r7$Fav$\"3UKL$eR<*frF\\r
7$Ffv$\"3K-++])Hxe(F\\r7$F[w$\"3!fmm\"H!o-*zF\\r7$F`w$\"3X,+]7k.6%)F\\
r7$Few$\"3#emmmT9C#))F\\r7$Fjw$\"33****\\i!*3`#*F\\r7$F_x$\"3;NLLL*zym
*F\\r7$Fex$\"3fLL$3N1#45Fcx7$Fjx$\"3!pm;HYt70\"Fcx7$F_y$\"3A+++q(G**3
\"Fcx7$Fdy$\"3'pmmT6KU8\"Fcx7$Fiy$\"3eLLL`v&Q<\"Fcx7$F^z$\"30++DOl5;7F
cx7$Fcz$\"3A++v.Uac7Fcx7$Fhz$\"#8F*-F][l6&F_[l$\"*++++\"!\")$F*F*F_el-
F$6$7S7$F($!\"&F*7$F.$!3i++](oUIn%F\\r7$F4$!3++]7y)e&)Q%F\\r7$F9$!3%)*
**\\73F'oSF\\r7$F>$!3%)***\\(oXdYPF\\r7$FC$!3-+]i:F0EMF\\r7$FH$!3s**\\
7Gz))GJF\\r7$FM$!3/+]7.5>@GF\\r7$FS$!3_**\\7./(H]#F\\r7$FX$!3'***\\iS.
x&=#F\\r7$Fhn$!3[****\\(3\"\\f=F\\r7$F]o$!3i***\\P9/@d\"F\\r7$Fbo$!3i*
***\\7Xd[7F\\r7$Fgo$!3l(****\\PkrB*F07$F\\p$!3s)****\\([b1hF07$Fap$!3E
(**\\7`hOE$F07$Ffp$\"3)p+++DO\"o6FQ7$F[q$\"330+++&>0)HF07$F`q$\"3Z.+v=
-p6jF07$Feq$\"3o&*****\\2Mg#*F07$Fjq$\"3+,](=xZ&\\7F\\r7$F`r$\"3g+]i:$
4wb\"F\\r7$Fer$\"3M,+v=#R!z=F\\r7$Fjr$\"3!)**\\P4A@u@F\\r7$F_s$\"3I++D
chf#\\#F\\r7$Fds$\"3x+](of2L#GF\\r7$Fis$\"3B+]7yG>6JF\\r7$F^t$\"3w++vo
o6AMF\\r7$Fct$\"3q*****\\xJLu$F\\r7$Fht$\"3W++v$*yddSF\\r7$F]u$\"3#3+v
=<F;O%F\\r7$Fbu$\"3)****\\i0A#*p%F\\r7$Fgu$\"3&*)****\\2mD+&F\\r7$F\\v
$\"3r++]i0XE`F\\r7$Fav$\"3K**\\(o/Q*>cF\\r7$Ffv$\"3v,+](Q(zSfF\\r7$F[w
$\"3()**\\(=-,FC'F\\r7$F`w$\"3k+]P4tFelF\\r7$Few$\"3#*)***\\73\"o'oF\\
r7$Fjw$\"3=+](oz;)*=(F\\r7$F_x$\"3\\+++]*44](F\\r7$Fex$\"3-,+DJw/>yF\\
r7$Fjx$\"3(3+v=(4bM\")F\\r7$F_y$\"3g,++vdYC%)F\\r7$Fdy$\"3K,+Dc3uc()F
\\r7$Fiy$\"3m-++];$R0*F\\r7$F^z$\"3C,](=-*zq$*F\\r7$Fcz$\"3w+]7G:3u'*F
\\r7$Fhz$\"#5F*Fjdl-F$6$7S7$F($!3.LLLLLLLVF\\r7$F.$!3d6666YqUSF\\r7$F4
$!3!))))))Q6I)*y$F\\r7$F9$!3+WWW%HYa]$F\\r7$F>$!3RWWW%Rx\">KF\\r7$FC$!
3dbbb!3pU$HF\\r7$FH$!3-yxx-E7qEF\\r7$FM$!3U+++DUh'R#F\\r7$FS$!3OWWWp9v
8@F\\r7$FX$!3OLLLeezJ=F\\r7$Fhn$!3Bbbbb)p<a\"F\\r7$F]o$!3=WWW%z9jG\"F
\\r7$Fbo$!37)********Gt)**F07$Fgo$!3olmmm;q*4(F07$F\\p$!3,nmmmw$pJ%F07
$Fap$!3)3666O@**y\"F07$Ffp$\"3IYWWWa%\\@\"F07$F[q$\"3c\"yxxxh/w$F07$F`
q$\"3.nmm;C]@nF07$Feq$\"3]3666^_U$*F07$Fjq$\"3ZnmmT-#=A\"F\\r7$F`r$\"3
Y+++DFl&\\\"F\\r7$Fer$\"3GXWW%>o8y\"F\\r7$Fjr$\"3v566'=WP/#F\\r7$F_s$
\"3x))))))Q@vEBF\\r7$Fds$\"3G*))))Q'yr?EF\\r7$Fis$\"3)fmm;*ehwGF\\r7$F
^t$\"3SAAAsF*H:$F\\r7$Fct$\"3'GLLL8<&QMF\\r7$Fht$\"3<mmm;q%yr$F\\r7$F]
u$\"3!omm;/8\"))RF\\r7$Fbu$\"3WKLL$Q(>)G%F\\r7$Fgu$\"3))4666l$yb%F\\r7
$F\\v$\"33LLLLQtX[F\\r7$Fav$\"3-@AA(f6m5&F\\r7$Ffv$\"3m++++*>=R&F\\r7$
F[w$\"3I566'oy,m&F\\r7$F`w$\"3LnmmT4pSfF\\r7$Few$\"3GVWWWH%\\@'F\\r7$F
jw$\"34*****\\Pf?]'F\\r7$F_x$\"3)>AAAi'eynF\\r7$Fex$\"3iBAAscPhqF\\r7$
Fjx$\"3y666'3B=M(F\\r7$F_y$\"3YMLLL^_*f(F\\r7$Fdy$\"3e666h2)[*yF\\r7$F
iy$\"39dbbb.0f\")F\\r7$F^z$\"3RmmmT-rS%)F\\r7$Fcz$\"39omm\"p%H5()F\\r7
$Fhz$\"\"*F*Fjdl-F$6$7S7$F($!3'pmmmmmmm$F\\r7$F.$!3^AAsMlO7MF\\r7$F4$!
3exFl\\85\">$F\\r7$F9$!30*))Qw]lA%HF\\r7$F>$!3])))Q,A!y\"p#F\\r7$FC$!3
86h[Xa[UCF\\r7$FH$!3Vb0VxsN6AF\\r7$FM$!3%***\\(oWP?(>F\\r7$FS$!3w))QwN
D`C<F\\r7$FX$!3_m;/w8#yZ\"F\\r7$Fhn$!3b56hB'[SA\"F\\r7$F]o$!3`)))Q^WD0
+\"F\\r7$Fbo$!3e&****\\(G\"*)[(F07$Fgo$!3SILLe*QA'\\F07$F\\p$!3'3LL$e/
KFDF07$Fap$!3ky@s4>\"=;$FQ7$Ffp$\"3!>*)))QExIJ#F07$F[q$\"3[ibbbSSSXF07
$F`q$\"3EPLe9YJJrF07$Feq$\"3bBAAs%4ZU*F07$Fjq$\"3;M$e9r#4%>\"F\\r7$F`r
$\"3a+]PMhpL9F\\r7$Fer$\"3I!*)Q,<(p$o\"F\\r7$Fjr$\"3!GAZG;wK\">F\\r7$F
_s$\"37yx_@\"34;#F\\r7$Fds$\"3nyF!48G\"=CF\\r7$Fis$\"3^L$3_!*Q?k#F\\r7
$F^t$\"3#eW%pv'oQ)GF\\r7$Fct$\"3#pmm;\\-P8$F\\r7$Fht$\"3cMLeRh6yLF\\r7
$F]u$\"3cM$e9\"*)f9OF\\r7$Fbu$\"3cnmT5F<xQF\\r7$Fgu$\"3q@AAZp58TF\\r7$
F\\v$\"35om;/r,lVF\\r7$Fav$\"3]W%pv9&G$f%F\\r7$Ffv$\"3C-+]7C%G%[F\\r7$
F[w$\"3_AsM]jlx]F\\r7$F`w$\"3#\\LeRd/JK&F\\r7$Few$\"3U*)))Qw]2jbF\\r7$
Fjw$\"3k+]7`>I9eF\\r7$F_x$\"37YWW%Hji0'F\\r7$Fex$\"36ZW>8Pq.jF\\r7$Fjx
$\"3[CsM+_4\\lF\\r7$F_y$\"3?omm\"\\%eunF\\r7$Fdy$\"3`CA(fm?I.(F\\r7$Fi
y$\"3]766h!pTE(F\\r7$F^z$\"3?M$e9Y@1^(F\\r7$Fcz$\"3iM$3_&y]YxF\\r7$Fhz
$\"\")F*Fjdl-F$6$7S7$F($!\"%F*7$F.$!3gmm\"HdNvs$F\\r7$F4$!3?L3xJdY!\\$
F\\r7$F9$!3`m;/,f&QA$F\\r7$F>$!3Cm;H2)ya&HF\\r7$FC$!3cL3-jsP)o#F\\r7$F
H$!3tmT5S*R2W#F\\r7$FM$!3'**\\Pf$eK%=#F\\r7$FS$!3cmTg-?9>>F\\r7$FX$!3%
)*\\(=<'3[l\"F\\r7$Fhn$!3!HL$eR#4HQ\"F\\r7$F]o$!3Dm;z>,UV6F\\r7$Fbo$!3
'o***\\P47Q()F07$Fgo$!3/)***\\7.(4.'F07$F\\p$!3$*)***\\i!H@U$F07$Fap$!
3Ej;/w70`5F07$Ffp$\"3?qm;a8,k<F07$F[q$\"3#4nmm\"HV]TF07$F`q$\"3$***\\i
:&3k#pF07$Feq$\"3-mmm\"H<OQ*F07$Fjq$\"3#3]ilZcz?\"F\\r7$F`r$\"3G+voHWn
k9F\\r7$Fer$\"3dn;H#oKDt\"F\\r7$Fjr$\"37n\"zW<5&y>F\\r7$F_s$\"3+L$3-8I
QC#F\\r7$Fds$\"3(R$eR(*HU>DF\\r7$Fis$\"3k+vV)RF$fFF\\r7$F^t$\"3cM$eRsI
%=IF\\r7$Fct$\"3M++]7)4hG$F\\r7$Fht$\"3O+]7y:)za$F\\r7$F]u$\"38,DcwfN,
QF\\r7$Fbu$\"3c**\\(o/&o#3%F\\r7$Fgu$\"3zlm;H<ZNVF\\r7$F\\v$\"3.,+voaP
0YF\\r7$Fav$\"3wKeRs$[*\\[F\\r7$Ffv$\"3Y,+Dc6L<^F\\r7$F[w$\"3'f;H#=v\"
*o`F\\r7$F`w$\"38,D\"yv(*=j&F\\r7$Few$\"3OmmT5!4!*)eF\\r7$Fjw$\"3)*)\\
iSm!=ehF\\r7$F_x$\"3$\\LL$e\\U<kF\\r7$Fex$\"3ON$3FpRDo'F\\r7$Fjx$\"3eo
\"HK9fa%pF\\r7$F_y$\"3K,+]7[0(=(F\\r7$Fdy$\"3'*o;a82&RY(F\\r7$Fiy$\"3P
MLL3(4;r(F\\r7$F^z$\"3')*\\i:&emvzF\\r7$Fcz$\"3$=]PMF,%G#)F\\r7$Fhz$\"
3++++++++&)F\\rFjdl-F$6$7S7$F($!3#)*************z$F\\r7$F.$!31+++]TVQN
F\\r7$F4$!3Z****\\-r%3J$F\\r7$F9$!3/+++l;!\\0$F\\r7$F>$!3]*****\\lfsz#
F\\r7$FC$!3m****\\s@%3a#F\\r7$FH$!3p****\\U.6.BF\\r7$FM$!37++]-G&p0#F
\\r7$FS$!3z****\\AjP-=F\\r7$FX$!3!*****\\sih[:F\\r7$Fhn$!3W******pGf(G
\"F\\r7$F]o$!3c*****\\J$od5F\\r7$Fbo$!33'******4'f))zF07$Fgo$!3!z*****
*\\J(*Q&F07$F\\p$!3_)*******QC&)GF07$Fap$!3ok****\\AH4hFQ7$Ffp$\"3K-++
+4X$4#F07$F[q$\"3_/+++cT%Q%F07$F`q$\"3+,++v@N\\qF07$Feq$\"3K)******fs#
3%*F07$Fjq$\"3'3++v@Q'*>\"F\\r7$F`r$\"3[++]_u3Y9F\\r7$Fer$\"3*4++]PJKq
\"F\\r7$Fjr$\"37++]n(p$R>F\\r7$F_s$\"31+++Dp2%>#F\\r7$Fds$\"3_++]xgkeC
F\\r7$Fis$\"3W++]-V&*)o#F\\r7$F^t$\"3(3++]\\$pPHF\\r7$Fct$\"3=+++?am%>
$F\\r7$Fht$\"3E+++:B1YMF\\r7$F]u$\"3c++]P<I*o$F\\r7$Fbu$\"3c*****\\kx$
fRF\\r7$Fgu$\"3X)*****fG0-UF\\r7$F\\v$\"3G,++]/;hWF\\r7$Fav$\"3s)***\\
P/&fp%F\\r7$Ffv$\"3w,++5zj_\\F\\r7$F[w$\"3C++]<3;%>&F\\r7$F`w$\"3A,+]Z
=iYaF\\r7$Few$\"3w)*****\\'[Mp&F\\r7$Fjw$\"3u)***\\PM&=&fF\\r7$F_x$\"3
/+++gzs+iF\\r7$Fex$\"3;,++0\"Q_X'F\\r7$Fjx$\"3)*****\\x2k2nF\\r7$F_y$
\"3')******>EdRpF\\r7$Fdy$\"3T,++&o#R0sF\\r7$Fiy$\"31,++?`9VuF\\r7$F^z
$\"3c****\\<#Rmp(F\\r7$Fcz$\"3L,+]A_ERzF\\r7$Fhz$\"3G*************>)F
\\rFjdl-%+AXESLABELSG6%Q\"x6\"Q!6\"%(DEFAULTG-%%VIEWG6$;F(FhzFedo" 1 
2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 221 "Since we can draw the secant line throug
h any two points on the graph, we could now hold one of the points fix
ed and move the other one towards it along the graph.  (Try this by ed
iting the commands above: keep the value " }{XPPEDIT 18 0 "x = 1;" "6#
/%\"xG\"\"\"" }{TEXT -1 30 " fixed, and replace the value " }{XPPEDIT 
18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "x =
 1.5;" "6#/%\"xG$\"#:!\"\"" }{TEXT -1 224 ".  Experiment with other va
lues, too.)   We can let the two points come as close together as we l
ike, but of course we can never let them become equal, because then ou
r expression for the slope of the line becomes undefined." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Rather than experi
ment with just a few values, " }{TEXT 274 34 "we can animate the whole
 procedure" }{TEXT -1 42 ", and watch what happens as the two points" 
}}{PARA 0 "" 0 "" {TEXT -1 26 "come together.  Here is a " }{TEXT 260 
5 "Maple" }{TEXT -1 47 " procedure which will do this for any function
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
14 "The values of " }{TEXT 269 1 "a" }{TEXT 270 6 " and h" }{TEXT -1 
1 " " }{TEXT -1 15 "give the fixed " }{TEXT 271 1 "x" }{TEXT -1 23 "-v
alue and the initial " }{TEXT 272 1 "h" }{TEXT -1 8 "-value. " }{TEXT 
273 1 "h" }{TEXT -1 116 " can be any real number, positive or negative
, but gives the best results if its absolute value is relatively small
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
70 "The following commands create the various components of the animat
ion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 788 "animateSecant := p
roc( f, a, h )\n    local n, Background, Mover, Secants;\n\n    n := 3
0;\n\n    Background:=plots[display](plots[pointplot](\{[a,f(a)],[a,0]
\},symbol=circle),\n                        plot(f(x),x=(a-(h+signum(h
)*1))..(a+(h+signum(h)*1)))):\n    Mover:=plots[display](seq( plots[po
intplot](\{[a+(n-i)/(n/h), f(a+(n-i)/(n/h))],                         \+
                  [ a+(n-i)/(n/h),0]\},symbol=circle,color=blue),i=0..
n-1),\n                        insequence=true):\n    Secants:=plots[d
isplay](seq( plot((f(a+(n-i)/(n/h))-f(a))/(n-i)*(n/h)*(x-a)+f(a),     \+
                                                          x=(a-(h+sign
um(h)*1))..(a+(h+signum(h)*1)),color=blue),i=0..n-1),\n               \+
          insequence=true):\n\n    plots[display](Secants,Background,M
over);\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 149 "Now, we are ready to display the animation.  Reme
mber, to run the animation just select the plot and use the animation \+
toolbar or the Animation menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "animateSecant(f, 1, 3);" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%(ANIMATEG6@7(-
%'CURVESG6$7S7$$!\"$\"\"!$!#>F.7$$!3ommmmFiDG!#<$!3QLLL$Q6G\"=!#;7$$!3
5LLLo!)*Qn#F4$!3bmm;M!\\pt\"F77$$!3nmmmwxE.DF4$!3ILLL))Qj^;F77$$!3Ymmm
Ok]JBF4$!3CLLL=Kvl:F77$$!3_LLL[9cg@F4$!3omm;C2G![\"F77$$!3smmmhN2-?F4$
!3KLL$3yO5S\"F77$$!37+++N&oz$=F4$!38++]nU)*=8F77$$!3!omm;)3Do;F4$!3XLL
$3WDTB\"F77$$!3?+++:v2*\\\"F4$!31++]d(Q&\\6F77$$!3BLLL8>1D8F4$!3ammmc4
`i5F77$$!3kmmmw))yr6F4$!37MLL$QW*e)*F47$$!3a,+++uR#***!#=$!3w++++()>'*
*)F47$$!3_++++5#)f#)Fgo$!3:,+++0\"*H\")F47$$!3?++++E;!f'Fgo$!3m*******
H\"3&H(F47$$!3)pmmm\"G&R2&Fgo$!3\\LLL3k(p`'F47$$!3WMLLLF.rKFgo$!3nnmmm
j^NcF47$$!3%fLLL$HsV<Fgo$!3'zmmmYh=([F47$$\"3/+)*****\\9!H$!#?$!3+,++v
#\\N)RF47$$\"3)Gmmm1:bg\"Fgo$!3aommmCC(>$F47$$\"3t,++]9#4L$Fgo$!38****
*\\FRXL#F47$$\"3a+++]j\"R(\\Fgo$!3t*****\\#=/8:F47$$\"3knmmm\"4#)o'Fgo
$!3whmmmT&*elFgo7$$\"3Gmmm;^Yi#)Fgo$\"3QJLL$eDBJ\"Fgo7$$\"3gKLLLG^g**F
go$\"3(Hmmm;kD!)*Fgo7$$\"3oKLL=2Vs6F4$\"3Qjmm\"f`@'=F47$$\"3f*****\\`p
fK\"F4$\"3%z****\\nZ)HEF47$$\"3!HLLLm&z\"\\\"F4$\"3ckmm;$y*eMF47$$\"3s
******z-6j;F4$\"3f)******R^bJ%F47$$\"3<******4#32$=F4$\"3'e*****\\5a`^
F47$$\"3O*****\\#y'G*>F4$\"3'o****\\7RV'fF47$$\"3G******H%=H<#F4$\"3W'
*****\\@fkoF47$$\"35mmm1>qMBF4$\"3_ILLL&4Nn(F47$$\"3%)*******HSu]#F4$
\"3L)******\\,s`)F47$$\"3'HLL$ep'Rm#F4$\"3%[mm;zM)>$*F47$$\"3')******R
>4NGF4$\"3$*******pfa<5F77$$\"3#emm;@2h*HF4$\"3#GLLeg`!)4\"F77$$\"3]**
***\\c9W;$F4$\"3w****\\#G2A=\"F77$$\"3Lmmmmd'*GLF4$\"3;LLL$)G[k7F77$$
\"3j*****\\iN7]$F4$\"3#)****\\7yh]8F77$$\"3aLLLt>:nOF4$\"3ommm')fdL9F7
7$$\"35LLL.a#o$QF4$\"3Zmmm,FT=:F77$$\"3ammm^Q40SF4$\"34LL$e#pa-;F77$$
\"3y******z]rfTF4$\"3s******Rv&)z;F77$$\"3gmmmc%GpL%F4$\"3ALLLGUYo<F77
$$\"3/LLL8-V&\\%F4$\"3Wmmm1^rZ=F77$$\"3=+++XhUkYF4$\"35++]sI@K>F77$$\"
3=+++:o<E[F4$\"34++]2%)38?F77$$\"\"&F.$\"#@F.-%'COLOURG6&%$RGBG$F.F.Fc
[l$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F,F[[l%(DEFAULT
G-%'POINTSG6&7$$\"\"%F.$\"#;F.7$Fg\\lFc[lF_[l-%'SYMBOLG6#%'CIRCLEG-Fd
\\l6%7$$\"\"\"F.Fc[l7$Fc]lFc]lF\\]l-F(6$7S7$F,$\"\"*F.7$F2$\"3a()\\]>S
9%)zF47$F9$\"3]In$)z3t\\rF47$F>$\"3wwwph&\\jE'F47$FC$\"3*f4=UEAfV&F47$
FH$\"3=KU-sd-oYF47$FM$\"35_YKY&)H3SF47$FR$\"3mY+lN$G\"yLF47$FW$\"316;=
/51$y#F47$Ffn$\"3uv&yfRLsC#F47$F[o$\"3!QfmT2*yb<F47$F`o$\"3UG(z9<*3t8F
47$Feo$\"3czE&*z0![)**Fgo7$F[p$\"3O+T?&HkC#oFgo7$F`p$\"3SwQW4V-VVFgo7$
Fep$\"3]gfd=(*\\uDFgo7$Fjp$\"3[9QD9b'*p5Fgo7$F_q$\"3g8EBo'p0/$F07$Fdq$
\"3YoB57a]#3\"!#A7$Fjq$\"37fOHH'ywd#F07$F_r$\"3T6qgqP]46Fgo7$Fdr$\"3yK
(zcQ%)RZ#Fgo7$Fir$\"3-/%3d=9KZ%Fgo7$F^s$\"3'4N8/)H$o#oFgo7$Fcs$\"3H,)*
H!f\"=@**Fgo7$Fhs$\"3Y9;H*y$fu8F47$F]t$\"37:\"[x?&>e<F47$Fbt$\"3lQ,9,V
XDAF47$Fgt$\"3Wx;W.e$fw#F47$F\\u$\"3C,9;]D\\^LF47$Fau$\"3;G-#z;A:(RF47
$Ffu$\"3DiOV.Xd@ZF47$F[v$\"3?mH*H*H$3X&F47$F`v$\"3I*3k!eoD(G'F47$Fev$
\"3!*\\<4b*>n4(F47$Fjv$\"3*H'HD3juP!)F47$F_w$\"3m.5!QUem(*)F47$Fdw$\"3
h8%=R&>N,5F77$Fiw$\"3^eQc28?36F77$F^x$\"3)Q\"p<!4leA\"F77$Fcx$\"3GcAbf
.![M\"F77$Fhx$\"3Y*Rmv\"H7s9F77$F]y$\"3J8e1ww2/;F77$Fby$\"3eSznaHKI<F7
7$Fgy$\"3S6X#Q%[*3)=F77$F\\z$\"3W<]H!G*)3-#F77$Faz$\"3Qcf@Erov@F77$Ffz
$\"3qaV'\\E)>HBF77$F[[l$\"#DF.-F`[l6&Fb[l$\"#5!\"\"Fc[lFc[l7(-F(6$7S7$
F,$!39++++++g=F77$F2$!3cmmmb^bu<F77$F9$!3=LL[`+@+<F77$F>$!3ommc57g;;F7
7$FC$!3wmm'R:QC`\"F77$FH$!3gLLo4^n[9F77$FM$!3)om;_/;5P\"F77$FR$!37++:#
e/1H\"F77$FW$!3'om;?$HW27F77$Ffn$!3?++N#)zaC6F77$F[o$!3JLL`P.GR5F77$F`
o$!3Immm&\\l<k*F47$Feo$!3#=++gsuiz)F47$F[p$!31,++!H7t%zF47$F`p$!3T+++u
'z\"HrF47$Fep$!3!pmm,)oB'Q'F47$Fjp$!3XMLLRg!G]&F47$F_q$!3_MLLPUUaZF47$
Fdq$!3-,+]*GyQ)QF47$Fjq$!3)[LLth(H8JF47$F_r$!3i)***\\*[[yE#F47$Fdr$!3g
****\\))4yi9F47$Fir$!3`CLL$3vxA'Fgo7$F^s$\"3Iommr!zg[\"Fgo7$Fcs$\"3$fL
LL)G^1)*Fgo7$Fhs$\"3@IL$)>0\"\\%=F47$F]t$\"3o)***\\@2D(f#F47$Fbt$\"3CK
LL]()z4MF47$Fgt$\"3;*****>PS#\\UF47$F\\u$\"35'*****G-Zq]F47$Fau$\"3u'*
**\\UB0leF47$Ffu$\"3='****pI+tu'F47$F[v$\"3)\\mmEMR+a(F47$F`v$\"3')***
***puX'Q)F47$Fev$\"3!>LLe4QM:*F47$Fjv$\"3C+++10&>***F47$F_w$\"3EmmrLD4
y5F77$Fdw$\"3#****\\o8j0;\"F77$Fiw$\"3]mmmDK>T7F77$F^x$\"3$****\\iX0cK
\"F77$Fcx$\"3`LL$pY/pS\"F77$Fhx$\"3ALLjZW/!\\\"F77$F]y$\"3immJ()f\\s:F
77$Fby$\"3-++?*Qg#[;F77$Fgy$\"34nmwV\\4N<F77$F\\z$\"3dLL`/3w7=F77$Faz$
\"3C++06)ob*=F77$Ffz$\"3M++NRm#[(>F77$F[[l$\"39++++++g?F7F_[lFg[lF^\\l
-Fd\\l6&7$$\"+++++R!\"*$\"++++@:Ff[l7$F\\amFc[lF_[lF\\]lF`]lFf]l7(-F(6
$7S7$F,$!3$*************>=F77$F2$!33+++G*)HO<F77$F9$!3#)****zs5Zj;F77$
F>$!33++!G`o:e\"F77$FC$!3w****f*3B\"*\\\"F77$FH$!3;++?&\\pqT\"F77$FM$!
3*)****f4`*4M\"F77$FR$!35++!o*[Ai7F77$FW$!3#*****>B/w!=\"F77$Ffn$!3;++
?2sb*4\"F77$F[o$!3#*****R=(Hg,\"F77$F`o$!3[)****zg'eC%*F47$Feo$!3K****
*>v]jf)F47$F[p$!33+++!39Zw(F47$F`p$!3S*****z/yK'pF47$Fep$!3W*****>N(\\
NiF47$Fjp$!3O+++7d4q`F47$F_q$!33,++3q)pj%F47$Fdq$!3g+++/t?%y$F47$Fjq$!
3k,++oFNHIF47$F_r$!3**)****Rqd6?#F47$Fdr$!3Z*****>:?DT\"F47$Fir$!3;'**
*****ff'*eFgo7$F^s$\"3O'*****fD$)f;Fgo7$Fcs$\"3+++++;Y5)*Fgo7$Fhs$\"3.
(****zWnw#=F47$F]t$\"3m(****zw`Yc#F47$Fbt$\"39)****R=>1O$F47$Fgt$\"3#)
)****RMHH=%F47$F\\u$\"3[&****zS*R()\\F47$Fau$\"3_(*****fbwldF47$Ffu$\"
3x'****RY3+j'F47$F[v$\"3x'****>:plS(F47$F`v$\"3$y*****RMrN#)F47$Fev$\"
3'*)******RTq)*)F47$Fjv$\"3O*****>JT%3)*F47$F_w$\"3o****fh98e5F77$Fdw$
\"3c****>\"**=*Q6F77$Fiw$\"3%)*****zc.z@\"F77$F^x$\"3o*******4$f+8F77$
Fcx$\"3/++?ZHB!Q\"F77$Fhx$\"3i****f$>w;Y\"F77$F]y$\"3!)****z[]WU:F77$F
by$\"3i****RQKm;;F77$Fgy$\"3!*****>fcs,<F77$F\\z$\"3m****R-l!yx\"F77$F
az$\"3-++g\\X#*e=F77$Ffz$\"3()****>r[cO>F77$F[[l$\"3$*************>?F7
F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++QF^am$\"++++W9Ff[l7$F\\[nFc[lF_[lF\\]lF
`]lFf]l7(-F(6$7S7$F,$!33++++++!y\"F77$F2$!3CLLL+F/)p\"F77$F9$!3Xmm6#4K
ni\"F77$F>$!3YLL.be`Y:F77$FC$!3[LLBD!3eY\"F77$FH$!3#pm;2)QY&Q\"F77$FM$
!3ELL)Rdu4J\"F77$FR$!33++X6_%QB\"F77$FW$!3MLLQ9z2a6F77$Ffn$!39++0Kkcu5
F77$F[o$!3)pmmE*4zF**F47$F`o$!3WKLL?xS2#*F47$Feo$!39-++ynU'R)F47$F[p$!
3'3+++(e6#e(F47$F`p$!3-,++AkP(z'F47$Fep$!3uLL$Q#yv%3'F47$Fjp$!3:nmm%Q&
QP_F47$F_q$!3_ommy(\\&>XF47$Fdq$!31,+]=j`%o$F47$Fjq$!3Somm=zSXHF47$F_r
$!3P****\\=pYM@F47$Fdr$!3y****\\:$fAO\"F47$Fir$!3Qjmm;pTlbFgo7$F^s$\"3
GLLL[geL=Fgo7$Fcs$\"3kfmm;.T9)*Fgo7$Fhs$\"3Tjm;wVU5=F47$F]t$\"3)z***\\
9o0KDF47$Fbt$\"3]kmm<'R9J$F47$Fgt$\"3])****fJ=m6%F47$F\\u$\"3u&****peG
V!\\F47$Fau$\"3S(***\\x(ykm&F47$Ffu$\"3\\'****4i;F^'F47$F[v$\"3AJLLh*)
4tsF47$F`v$\"3O******4%p\\3)F47$Fev$\"3Ckm;/Zk?))F47$Fjv$\"3E+++=@$\\i
*F47$F_w$\"3\"HL$[*Qq\"Q5F77$Fdw$\"3!****\\b%[F<6F77$Fiw$\"3OLLL5Rh%>
\"F77$F^x$\"3)****\\Pu!ev7F77$Fcx$\"31nmYF9c`8F77$Fhx$\"3ammcRzIL9F77$
F]y$\"3^LLG5TR7:F77$Fby$\"3#*****f(3m]e\"F77$Fgy$\"3ULLjujNo;F77$F\\z$
\"3zmmE+A&Gu\"F77$Faz$\"3;++:)G!GA=F77$Ffz$\"37++0.JI)*=F77$F[[l$\"33+
+++++!)>F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++PF^am$\"++++p8Ff[l7$F[enFc[lF
_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3')************R<F77$F2$!3ymmmskyf;F77$
F9$!35LLV6J***e\"F77$F>$!3]mmExJ]6:F77$FC$!3Zmm'3'H\\K9F77$FH$!3ILLBm#
eQN\"F77$FM$!3kmmOQQ&4G\"F77$FR$!3))****4EbY07F77$FW$!3wmmc0aRF6F77$Ff
n$!35++!plv&\\5F77$F[o$!3-LLL,[G&p*F47$F`o$!3SmmmK)G-**)F47$Feo$!3i***
**R!G]'>)F47$F[p$!3))******fw^*R(F47$F`p$!3,+++'zu9j'F47$Fep$!3Fmmm&H=
S$fF47$Fjp$!3$RLLt0vY5&F47$F_q$!3=MLL\\D6-WF47$Fdq$!35,++L`'[e$F47$Fjq
$!3;NLLpIYhGF47$F_r$!3u*****H8wx1#F47$Fdr$!36+++z%)*>J\"F47$Fir$!3fILL
LyBM_Fgo7$F^s$\"3KhmmO&Rt+#Fgo7$Fcs$\"3sBLLL!f$=)*Fgo7$Fhs$\"3AILL/8=$
z\"F47$F]t$\"3%y****4')f%*\\#F47$Fbt$\"3UILL^+EiKF47$Fgt$\"3G(****zG2.
0%F47$F\\u$\"37&****fwd7#[F47$Fau$\"3S'****\\*>>nbF47$Ffu$\"33(****zxC
aR'F47$F[v$\"3-jmmq(G'RrF47$F`v$\"35******z`AMzF47$Fev$\"3IJLL3![Ul)F4
7$Fjv$\"3Q*****R#HUT%*F47$F_w$\"3;mmO<$4#=5F77$Fdw$\"3q*****)*pIc4\"F7
7$Fiw$\"3_mmm_UKr6F77$F^x$\"3s****\\(Qo0D\"F77$Fcx$\"3?LLt2**)oK\"F77$
Fhx$\"3GLL`&oR\\S\"F77$F]y$\"3KmmwrJM#[\"F77$Fby$\"3q****zO*oMb\"F77$F
gy$\"3Bmm1!4()\\j\"F77$F\\z$\"3'GLL\")*y*yq\"F77$Faz$\"3'*****pEgj&y\"
F77$Ffz$\"3m*****[LT+'=F77$F[[l$\"3')************R>F7F_[lFg[lF^\\l-Fd
\\l6&7$$\"+++++OF^am$\"++++'H\"Ff[l7$Fj^oFc[lF_[lF\\]lF`]lFf]l7(-F(6$7
S7$F,$F4F.7$F2$!3%******\\CI:i\"F77$F9$!3!****\\28aKb\"F77$F>$!33++]*
\\qkZ\"F77$FC$!3#)****\\'*y<*R\"F77$FH$!3/++v^EDA8F77$FM$!3+++v-J$4D\"
F77$FR$!31++vSe3x6F77$FW$!3<++v'*Gr+6F77$Ffn$!32++v\")[eC5F77$F[o$!35*
*****4'yFY*F47$F`o$!3O+++X*\\Ix)F47$Feo$!3q+++I)yl*zF47$F[p$!3o+++]%>p
@(F47$F`p$!3))******pJdlkF47$Fep$!3e++]n(yKy&F47$Fjp$!3s+++IZ'>(\\F47$
F_q$!3i,++?`n%G%F47$Fdq$!37,+]ZV>&[$F47$Fjq$!3#>+++A=vx#F47$F_r$!3A***
*\\Z`3,?F47$Fdr$!3)*****\\Uwth7F47$Fir$!3!y*****\\(eI!\\Fgo7$F^s$\"3C)
****\\-$4\"=#Fgo7$Fcs$\"3n'*****\\xIA)*Fgo7$Fhs$\"31(***\\K#Qfx\"F47$F
]t$\"3q(***\\2H'oY#F47$Fbt$\"35)****\\[!38KF47$Fgt$\"3u)*****fi*R)RF47
$F\\u$\"3Q&****\\%p=QZF47$Fau$\"3G'***\\7_!zY&F47$Ffu$\"3!o****\\$H8yi
F47$F[v$\"3Z(*****z&eh+(F47$F`v$\"3'))*****\\8[$y(F47$Fev$\"3O)***\\78
&y[)F47$Fjv$\"3_)*****HP\"zD*F47$F_w$\"3y&***\\_C[#)**F47$Fdw$\"3))***
\\Ub')R2\"F77$Fiw$\"3')*****\\fM![6F77$F^x$\"3%)***\\7.cbA\"F77$Fcx$\"
30+++)Q=-I\"F77$Fhx$\"3/++]J9dw8F77$F]y$\"3')***\\KB#H_9F77$Fby$\"3#)*
****fyr=_\"F77$Fgy$\"35++]0yh,;F77$F\\z$\"3++++'fVHn\"F77$Faz$\"35++Dl
<**[<F77$Ffz$\"3!****\\ncz<#=F77$F[[l$\"#>F.F_[lFg[lF^\\l-Fd\\l6&7$$\"
+++++NF^am$\"++++D7Ff[l7$FhhoFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!39+++
+++g;F77$F2$!3YLLL<SF$e\"F77$F9$!3rmm1]^^;:F77$F>$!3YLLt@yVT9F77$FC$!3
OLL8KG'eO\"F77$FH$!3ymmEPqk!H\"F77$FM$!3cLL8nB\"4A\"F77$FR$!3A++Sbhq[6
F77$FW$!3ULL$zQIS2\"F77$Ffn$!35-++m5%f***F47$F[o$!3#pmm'=CFI#*F47$F`o$
!3IMLLd5(eb)F47$Feo$!3u,++c[l'z(F47$F[p$!3f+++S7KMqF47$F`p$!3i+++W:n*H
'F47$Fep$!37LLLR#RDj&F47$Fjp$!3^nmm-WDR[F47$F_q$!3;omm!4Qs;%F47$Fdq$!3
q+++iL_&Q$F47$Fjq$!3CommqLd$p#F47$F_r$!3$*)****>c%RM>F47$Fdr$!3T*****f
!oZ67F47$Fir$!3egmmm'z=d%Fgo7$F^s$\"3=NLL8l%[N#Fgo7$Fcs$\"3jpmmmkDE)*F
go7$Fhs$\"3)Qmm1;&pe<F47$F]t$\"3X)****R&fEMCF47$Fbt$\"3ylmm=4!R;$F47$F
gt$\"3u*****>B&o<RF47$F\\u$\"3U(****R7;^l%F47$Fau$\"3%z*****H%='o`F47$
Ffu$\"3_'****>4T3;'F47$F[v$\"3#>LL$*Q)osoF47$F`v$\"3Q+++?ttKwF47$Fev$
\"3Ulmm;YX@$)F47$Fjv$\"3U*****f`/W2*F47$F_w$\"3+ILLJ<(Gy*F47$Fdw$\"3')
****f3CM_5F77$Fiw$\"3>LLLP\\uC6F77$F^x$\"3%******\\nV0?\"F77$Fcx$\"3!p
mm#ooat7F77$Fhx$\"3ymmYxJ?[8F77$F]y$\"3QLLt%HTAU\"F77$Fby$\"3%*****>NY
F!\\\"F77$Fgy$\"3YLL$4_[#o:F77$F\\z$\"3zmm'QH*)zj\"F77$Faz$\"3B++!Q]ZB
r\"F77$Ffz$\"3_++g)z<Ny\"F77$F[[l$\"39++++++g=F7F_[lFg[lF^\\l-Fd\\l6&7
$$\"+++++MF^am$\"++++c6Ff[l7$FgbpFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3
$*************>;F77$F2$!3jmmm*y<]a\"F77$F9$!3MLLQphxz9F77$F>$!3nmm'R90
kS\"F77$FC$!3`mmwnxaK8F77$FH$!3aLLyA9/f7F77$FM$!3wmm^J;*3>\"F77$FR$!3,
++0qkK?6F77$FW$!3mmm6zyMZ5F77$Ffn$!3+++]9L.Y(*F47$F[o$!3)HLLtAmx**)F47
$F`o$!3[mmmp@pQ$)F47$Feo$!37+++#)3t'f(F47$F[p$!3]+++IIs^oF47$F`p$!3g**
***z\"*pP8'F47$Fep$!3amm;6(*z\"[&F47$Fjp$!3TLLLvSa1ZF47$F_q$!3sMLLh3!)
\\SF47$Fdq$!3s++]wB&eG$F47$Fjq$!3cMLL@&G'4EF47$F_r$!33****\\wPqn=F47$F
dr$!3s****\\pf@h6F47$Fir$!3yFLL$e+2C%Fgo7$F^s$\"3@jmm,+gGDFgo7$Fcs$\"3
rLLL$=0-$)*Fgo7$Fhs$\"3pIL$))3_9u\"F47$F]t$\"3K)***\\+!p;S#F47$Fbt$\"3
pJLL_8s9JF47$Fgt$\"3'*)****R?u8&QF47$F\\u$\"3!f****HIX?d%F47$Fau$\"3%p
***\\Z;Lp_F47$Ffu$\"37(*****[#\\N/'F47$F[v$\"3sjmm)>=#RnF47$F`v$\"3C**
*****G$*>[(F47$Fev$\"3[KL$3#z0b\")F47$Fjv$\"3I+++U`*3*))F47$F_w$\"3Uim
;55E$e*F47$Fdw$\"3&)***\\HE)pI5F77$Fiw$\"3Mmmmz_X,6F77$F^x$\"3q***\\(=
8`v6F77$Fcx$\"3SLL`[`(oC\"F77$Fhx$\"3=LLVB\\$)>8F77$F]y$\"3cmm@c.>#R\"
F77$Fby$\"3s****R%[x'e9F77$Fgy$\"3WmmOO#z[`\"F77$F\\z$\"3@LLt\"*\\..;F
77$Faz$\"3-++NUKqv;F77$Ffz$\"3/++XIgDX<F77$F[[l$\"3$*************>=F7F
_[lFg[lF^\\l-Fd\\l6&7$$\"+++++LF^am$\"++++*3\"Ff[l7$Ff\\qFc[lF_[lF\\]l
F`]lFf]l7(-F(6$7S7$F,$!33++++++!e\"F77$F2$!3:+++i:w1:F77$F9$!3)*****p)
=PIW\"F77$F>$!31++?mCPr8F77$FC$!3))****R.FB*H\"F77$FH$!3G++I3eVF7F77$F
M$!37++!f*3(3;\"F77$FR$!3+++q%yY>4\"F77$FW$!31++Iq`m?5F77$Ffn$!3W,++jb
7'\\*F47$F[o$!3#3++g.g_w)F47$F`o$!3X+++#G8:7)F47$Feo$!3;,++3p!oR(F47$F
[p$!3S+++?[7pmF47$F`p$!3O+++#Hoy'fF47$Fep$!3&******H=g5L&F47$Fjp$!3?++
+[P$Qd%F47$F_q$!3F,++KOOKRF47$Fdq$!3=,++\"R\"='=$F47$Fjq$!3x,++sOoDDF4
7$F_r$!3X*****4*H,,=F47$Fdr$!3/+++L^&46\"F47$Fir$!3*\\******\\@&4RFgo7
$F^s$\"39+++!\\`Bq#Fgo7$Fcs$\"3M$*******Q:M)*Fgo7$Fhs$\"32(****p,4Us\"
F47$F]t$\"3i)****p/s!pBF47$Fbt$\"3/)****fyTb1$F47$Fgt$\"3k)****f<j]y$F
47$F\\u$\"31(****>[u*)[%F47$Fau$\"3#o****\\'[/q^F47$Ffu$\"3%o****fSdi#
fF47$F[v$\"3<)****z+[dg'F47$F`v$\"3**)*****f#\\7L(F47$Fev$\"3k)****\\A
h'))zF47$Fjv$\"3@,++[hQ2()F47$F_w$\"3U)*****)G]OQ*F47$Fdw$\"3$)****H<T
045F77$Fiw$\"3')*****>il\"y5F77$F^x$\"3+++]i*=0:\"F77$Fcx$\"32++!)GQ??
7F77$Fhx$\"35++SpmY\"H\"F77$F]y$\"3G++q<%R@O\"F77$Fby$\"3-++gL.3F9F77$
Fgy$\"3&*****z^*4:]\"F77$F\\z$\"3+++g*o!3o:F77$Faz$\"3:++!4)*e!R;F77$F
fz$\"3H++IiU*pq\"F77$F[[l$\"33++++++!y\"F7F_[lFg[lF^\\l-Fd\\l6&7$$\"++
+++KF^am$\"++++C5Ff[l7$FefqFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3')****
********R:F77$F2$!3;LLLM`]o9F77$F9$!3Vmm,3#)H19F77$F>$!3GLLV)yRjL\"F77
$FC$!31LL.Rw\"fE\"F77$FH$!3mmm\"Q>Ie>\"F77$FM$!3KLLGg,&38\"F77$FR$!3)*
***\\$*4nN1\"F77$FW$!3-LL$[hG)R**F47$Ffn$!35,+]6y@Y#*F47$F[o$!34lmmWQv
K&)F47$F`o$!3jKLL%RMV!zF47$Feo$!3c*****R$H)o>(F47$F[p$!3I+++5m_'['F47$
F`p$!3A+++mm'>!eF47$Fep$!3OLL$[l?.=&F47$Fjp$!3*pmm1UB6W%F47$F_q$!3$ymm
ESE\\\"QF47$Fdq$!3y++]0/^'3$F47$Fjq$!3`ommA)Q<W#F47$F_r$!3h****\\0AKM<
F47$Fdr$!3#*****\\'H%pg5F47$Fir$!3kmmm;CMyNFgo7$F^s$\"3=GLLyp5wGFgo7$F
cs$\"3Udmm;E5Q)*Fgo7$Fhs$\"3*Qmm^%f'pq\"F47$F]t$\"3g(***\\$4vkL#F47$Fb
t$\"3'Rmm'>AO;IF47$Fgt$\"3(y****z9_(=PF47$F\\u$\"3U'****4m.fS%F47$Fau$
\"3g(***\\#3e22&F47$Ffu$\"3X(****Hcl*3eF47$F[v$\"3'*HLL<yFskF47$F`v$\"
3i******H_]!=(F47$Fev$\"3#[mm\"HXEAyF47$Fjv$\"3c)****R&p(Q_)F47$F_w$\"
33HL$ycRS=*F47$Fdw$\"3S'***\\;(*4u)*F47$Fiw$\"3?LLLkf([0\"F77$F^x$\"3u
***\\ig1b7\"F77$Fcx$\"3umm14B`$>\"F77$Fhx$\"3]mmO:%)4j7F77$F]y$\"34LL=
z%)3K8F77$Fby$\"3!)****z#=$[&R\"F77$Fgy$\"3ILLBn19o9F77$F\\z$\"3DmmY(Q
EJ`\"F77$Faz$\"3%****\\%>ZT-;F77$Ffz$\"3#)***\\T\\K(o;F77$F[[l$\"3')**
**********R<F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++JF^am$\"++++5'*F^am7$Fd`r
Fc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!#:F.7$F2$!3nmmm1\"\\-V\"F77$F9$!3C
LLLF#f&p8F77$F>$!3nmmm5rI,8F77$FC$!3fmmmuDgK7F77$FH$!3ULLLzXAk6F77$FM$
!3pmmmC%H35\"F77$FR$!3'******RT(=N5F77$FW$!3=nmmEN+t'*F47$Ffn$!3y+++g+
J'**)F47$F[o$!3#HLLLlZ-I)F47$F`o$!3emmm1b:(o(F47$Feo$!3i+++g*ep*pF47$F
[p$!3@++++%GRI'F47$F`p$!33+++S]1OcF47$Fep$!3zmmmE6eH]F47$Fjp$!3yLLL$48
%3VF47$F_q$!3OMLLt\"*[(p$F47$Fdq$!3!3+++URo)HF47$Fjq$!3%[LLL(RzdBF47$F
_r$!3I******>9jn;F47$Fdr$!3y******fMV55F47$Fir$!3SHLLLL;ZKFgo7$F^s$\"3
5lmmm/')\\IFgo7$Fcs$\"3QILLL80U)*Fgo7$Fhs$\"3qILLtGs*o\"F47$F]t$\"3N)*
****R\"yQI#F47$Fbt$\"3kJLL`E=nHF47$Fgt$\"3)))*****>6W_OF47$F\\u$\"3o'*
****RG$GK%F47$Fau$\"3[(******Hr9(\\F47$Ffu$\"39(*****>Pn\"p&F47$F[v$\"
3UkmmEw!)QjF47$F`v$\"3Q*******>h(HqF47$Fev$\"3)=LLL$y'el(F47$Fjv$\"3Y*
*****fxOS$)F47$F_w$\"3IjmmY)GW)*)F47$Fdw$\"31)*****f#ewl*F47$Fiw$\"3`m
mm1jeJ5F77$F^x$\"3&)******\\U\\+6F77$Fcx$\"3ULLL*ygo;\"F77$Fhx$\"3DLLL
h,tM7F77$F]y$\"3immmSv.-8F77$Fby$\"3#******>.')QO\"F77$Fgy$\"3kmmm#QrZ
V\"F77$F\\z$\"3ALLL&3s\")\\\"F77$Faz$\"33+++e/xl:F77$Ffz$\"32+++E2ZI;F
77$F[[l$\"#<F.F_[lFg[lF^\\l-Fd\\l6&7$$\"\"$F.Fj]l7$FcjrFc[lF_[lF\\]lF`
]lFf]l7(-F(6$7S7$F,$!3)*************f9F77$F2$!3-+++zG*>R\"F77$F9$!3))*
**\\mC?GL\"F77$F>$!31++!HVuiE\"F77$FC$!3%*****H5vG*>\"F77$FH$!3;++&['*
=E8\"F77$FM$!31++0*o332\"F77$FR$!3&****\\'Gx!o+\"F77$FW$!3d****\\Q%yhS
*F47$Ffn$!3W++]3BSY()F47$F[o$!3)*)****>YTx1)F47$F`o$!3a+++>m(*puF47$Fe
o$!3))*****f)\\.(z'F47$F[p$!37+++!>I87'F47$F`p$!3'******RTj,Z&F47$Fep$
!3?++])fT)y[F47$Fjp$!3c+++mFqvTF47$F_q$!3#4++S%>0!e$F47$Fdq$!3#3++XVor
)GF47$Fjq$!3i,++C\"\\QF#F47$F_r$!3C****\\M1%4g\"F47$Fdr$!3c'****\\BE<g
*Fgo7$Fir$!3h'*****\\U)f\"HFgo7$F^s$\"3f(****\\&RhBKFgo7$Fcs$\"3!*)***
**\\++Y)*Fgo7$Fhs$\"3_(***\\,)zCn\"F47$F]t$\"3A)***\\'=\"GrAF47$Fbt$\"
3W)****p3.!=HF47$Fgt$\"3+*****>4Ihe$F47$F\\u$\"3%p*****=?wRUF47$Fau$\"
3O(***\\<X=s[F47$Ffu$\"3)o****p(=QubF47$F[v$\"35(****fVP`?'F47$F`v$\"3
7******pr,zoF47$Fev$\"3;(***\\P6Z*[(F47$Fjv$\"3g)****fceo:)F47$F_w$\"3
u&***\\D\"=[y)F47$Fdw$\"3#z***\\.o@T%*F47$Fiw$\"3()******[mH35F77$F^x$
\"3!)***\\P*=[v5F77$Fcx$\"34++gp#*=S6F77$Fhx$\"3\")****H2>O17F77$F]y$
\"3!)***\\@g')>F\"F77$Fby$\"3p****>\")))GK8F77$Fgy$\"3)*****4)4-9S\"F7
7$F\\z$\"3#)****>$y<KY\"F77$Faz$\"3/++b'>E\"H:F77$Ffz$\"39++&y&*3Af\"F
77$F[[l$\"39++++++g;F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++HF^am$\"++++5%)F^
am7$F`dsFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3$*************>9F77$F2$!3
?LLL^mt`8F77$F9$!3]mm'fE\"3'H\"F77$F>$!3FLL8b<CJ7F77$FC$!3ILL$fWsf;\"F
77$FH$!3ammO]L,,6F77$FM$!3ELLV`zyS5F77$FR$!3G*****HV!G%y*F47$FW$!3uLLL
]LNR\"*F47$Ffn$!37+++dX\\'\\)F47$F[o$!3#fmm1FN_$yF47$F`o$!3sKLLJxz_sF4
7$Feo$!3/+++756(f'F47$F[p$!3-+++!)>tQfF47$F`p$!3\")*****zyhUI&F47$Fep$
!3jLLLq?5GZF47$Fjp$!3NnmmQC*H/%F47$F_q$!3[nmm9ZhiMF47$Fdq$!3S+++\\u\\(
y#F47$Fjq$!3#zmmYF/**=#F47$F_r$!3<******[)\\U`\"F47$Fdr$!3](*****py6*4
*Fgo7$Fir$!3#Qmmm;0[e#Fgo7$F^s$\"31ILLVuO(R$Fgo7$Fcs$\"3)Hmmmw[*\\)*Fg
o7$Fhs$\"3OkmmHnBb;F47$F]t$\"33)****HB%oQAF47$Fbt$\"3Clmm?N#)oGF47$Fgt
$\"36*****R1>)>NF47$F\\u$\"3K'****z>\"pcTF47$Fau$\"3E(****\\t(*Gx%F47$
Ffu$\"3[(****R.!4daF47$F[v$\"3mILLXs'=2'F47$F`v$\"3w******RJFGnF47$Fev
$\"35lmmTW2BtF47$Fjv$\"3g)****>P\\L(zF47$F_w$\"3=GLL/u?&e)F47$Fdw$\"3z
(****pMvZA*F47$Fiw$\"3/KLL8*p+&)*F47$F^x$\"3s****\\P&p/0\"F77$Fcx$\"3w
mm')\\x^86F77$Fhx$\"3SmmE`O*z<\"F77$F]y$\"3KLLjjc$>C\"F77$Fby$\"3\")**
**RI<p+8F77$Fgy$\"3MLL`8G.o8F77$F\\z$\"3Vmm1\"[j#G9F77$Faz$\"3+++5N>[#
\\\"F77$Ffz$\"3')****p*=ZRb\"F77$F[[l$\"3$*************>;F7F_[lFg[lF^
\\l-Fd\\l6&7$$\"+++++GF^am$\"++++SyF^am7$F_^tFc[lF_[lF\\]lF`]lFf]l7(-F
(6$7S7$F,$!33++++++!Q\"F77$F2$!3rmmmB/[:8F77$F9$!39LLG&GU$f7F77$F>$!3m
mmOx!4i>\"F77$FC$!3lmmc\"QdE8\"F77$FH$!3[LL)et2%p5F77$FM$!3jmm\"y@n2,
\"F77$FR$!3*3++&zN[+&*F47$FW$!3!zmm@EGD())F47$Ffn$!3c,+]0oeY#)F47$F[o$
!3vLLLz!HFg(F47$F`o$!3ommmV)=c.(F47$Feo$!35,++Qq=(R'F47$F[p$!3%*******
pP8cdF47$F`p$!3d+++i,OQ^F47$Fep$!3/nm;UDOxXF47$Fjp$!39MLL6@G5RF47$F_q$
!3[MLL&[x^M$F47$Fdq$!3'3++NYEyo#F47$Fjq$!38NLLD%ff5#F47$F_r$!3a****\\j
!fvY\"F47$Fdr$!3W)****\\]4lf)Fgo7$Fir$!3/JLL$3EOD#Fgo7$F^s$\"3+nmmJ47r
NFgo7$Fcs$\"3]JLL$[(*Q&)*Fgo7$Fhs$\"3tIL$yl$*zj\"F47$F]t$\"3R)***\\zs3
1AF47$Fbt$\"3fJLLaRk>GF47$Fgt$\"3y)****f.3NX$F47$F\\u$\"3Y(****pP?O2%F
47$Fau$\"3-)***\\_4htYF47$Ffu$\"3>(****4>)zR`F47$F[v$\"37lmmaqRQfF47$F
`v$\"3_******4\"Hvd'F47$Fev$\"3;KL$euxm:(F47$Fjv$\"3]*****z<S)*y(F47$F
_w$\"3=km;$o'f&Q)F47$Fdw$\"3m(***\\!*QL3!*F47$Fiw$\"3=nmmOL<<'*F47$F^x
$\"3,++D\"=da-\"F77$Fcx$\"3WLL8Ii%o3\"F77$Fhx$\"3KLLB*RD'\\6F77$F]y$\"
3nmm6DZ)=@\"F77$Fby$\"37++gzX4p7F77$Fgy$\"3&omm*GNmM8F77$F\\z$\"3ALL$*
y\"4LR\"F77$Faz$\"39++ltw$eX\"F77$Ffz$\"35++b@ao::F77$F[[l$\"33++++++!
e\"F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++FF^am$\"++++!H(F^am7$F^htFc[lF_[lF
\\]lF`]lFf]l7(-F(6$7S7$F,$!3/++++++S8F77$F2$!31+++'>CsF\"F77$F9$!3&***
**f/LgA7F77$F>$!31++g*Rw6;\"F77$FC$!3+++?<BM*4\"F77$FH$!3/++S@@!y.\"F7
7$FM$!3.+++A[Y2)*F47$FR$!3s+++Eno;#*F47$FW$!3H+++uJq0')F47$Ffn$!3A,++a
!zm*zF47$F[o$!3q+++))GAqtF47$F`o$!3k+++c*R%=oF47$Feo$!3E,++kIE(>'F47$F
[p$!3%)******fb`tbF47$F`p$!3W+++O&eC(\\F47$Fep$!3Z+++9IiEWF47$Fjp$!3[+
++%yrvx$F47$F_q$!3.,++c-uFKF47$Fdq$!3!4++!ya:)e#F47$Fjq$!3X,++wX,A?F47
$F_r$!3Z*****zFo3S\"F47$Fdr$!3Q******R6!R4)Fgo7$Fir$!3C)*******pWA>Fgo
7$F^s$\"3[******>W([u$Fgo7$Fcs$\"3c&******>Yy&)*Fgo7$Fhs$\"3b(****fe]2
i\"F47$F]t$\"3C)****fK!\\t@F47$Fbt$\"3S)****zQk/x#F47$Fgt$\"3!*)****z+
(>(Q$F47$F\\u$\"3G(****fb\\0*RF47$Fau$\"3-(*****pTKuXF47$Ffu$\"3!y****
zM1DA&F47$F[v$\"3p)****R'o#\\!eF47$F`v$\"3:+++!3&yEkF47$Fev$\"35+++]5G
!*pF47$Fjv$\"3_*****R)4L1wF47$F_w$\"3Q)****>'f)f=)F47$Fdw$\"3I*****RV#
*=z)F47$Fiw$\"3w)*****fnF%Q*F47$F^x$\"3%******\\#[W+5F77$Fcx$\"37++S5Z
<g5F77$Fhx$\"31++?XrD@6F77$F]y$\"3/++g'yL==\"F77$Fby$\"32++!)Gu\\P7F77
$Fgy$\"3,++SWUH,8F77$F\\z$\"3#)****zw[Ne8F77$Faz$\"36++?7M>>9F77$Ffz$
\"3=++S`OUx9F77$F[[l$\"3/++++++S:F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++EF^a
m$\"++++gnF^am7$F]buFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!#8F.7$F2$!3TLL
Loz'*Q7F77$F9$!3gmm\"RKke=\"F77$F>$!3FLL$=sVh7\"F77$FC$!3OLL$GDFg1\"F7
7$FH$!3ymm\"p]'>15F77$FM$!3vLL$eYds]*F47$FR$!3b++]s)*)G$*)F47$FW$!3nKL
$e3y)Q$)F47$Ffn$!3!4++DIrnu(F47$F[o$!3vmmm'p;x8(F47$F`o$!3#GLL$o5E,mF4
7$Feo$!3a+++!4Rt*fF47$F[p$!3s******\\t$4R&F47$F`p$!3H+++5pb1[F47$Fep$!
3+LL$e[$)eF%F47$Fjp$!3$ommmXh[k$F47$F_q$!3enmmEII5JF47$Fdq$!3[++]#\\%[
)[#F47$Fjq$!3*zmmmsp!Q>F47$F_r$!3R****\\#\\xTL\"F47$Fdr$!35)****\\x#H
\"f(Fgo7$Fir$!3+hmm;zE\"f\"Fgo7$F^s$\"3(>LL$3zi=RFgo7$Fcs$\"33kmm;\\zh
)*Fgo7$Fhs$\"3Pkm;9v].;F47$F]t$\"3+****\\sL*39#F47$Fbt$\"3>lmm@[G@FF47
$Fgt$\"3-******zf)3K$F47$F\\u$\"34(****\\tyu!RF47$Fau$\"3!y***\\(QP]Z%
F47$Ffu$\"3_(****\\]9_5&F47$F[v$\"3OJLLtmXrcF47$F`v$\"3!*******\\5/wiF
47$Fev$\"3Rlm;aV)Q#oF47$Fjv$\"3U+++!z@GU(F47$F_w$\"3%3LL3Cvj)zF47$Fdw$
\"3S(***\\x4Xv&)F47$Fiw$\"39KLL$=!Q^\"*F47$F^x$\"3q)***\\(oCVv*F47$Fcx
$\"3zmmm!>.N.\"F77$Fhx$\"3kmm;\"*)))G4\"F77$F]y$\"3QLL3[Gy^6F77$Fby$\"
3-+++y-!f?\"F77$Fgy$\"3OLL$)f\\#zE\"F77$F\\z$\"3hmmmu0SB8F77$Faz$\"33+
+v]\"\\DQ\"F77$Ffz$\"3C++D&)=;R9F77$F[[l$\"#:F.F_[lFg[lF^\\l-Fd\\l6&7$
$\"+++++DF^am$\"++++]iF^am7$F\\\\vFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!
3)*************f7F77$F2$!3vmmmS<r+7F77$F9$!3ALLBV`7\\6F77$F>$!3mmm1W56
\"4\"F77$FC$!3`mmY)=7F.\"F77$FH$!3WLLLC*3fu*F47$FM$!3rlmm4,02#*F47$FR$
!3Q+++>I4\\')F47$FW$!3%ommw*H0s!)F47$Ffn$!3c+++^N'o\\(F47$F[o$!3!GLL`]
5_!pF47$F`o$!3wmmm!=#3%Q'F47$Feo$!3!)*****f6:uz&F47$F[p$!3k******R\"R$
3_F47$F`p$!3F*****RGb1k%F47$Fep$!3UmmmdR9DTF47$Fjp$!3iLLLH6:7NF47$F_q$
!39MLL(zlG*HF47$Fdq$!3]+++2N\"))Q#F47$Fjq$!3_MLLx[7a=F47$F_r$!3K*****p
q'[n7F47$Fdr$!3#o*****4Wo)3(Fgo7$Fir$!3AGLLL))3g7Fgo7$F^s$\"3Wkmm'R\"Q
#4%Fgo7$Fcs$\"3gKLLLOul)*Fgo7$Fhs$\"3?JLLUWE'e\"F47$F]t$\"3())*****=kH
3@F47$Fbt$\"3)>LL`D0@n#F47$Fgt$\"39*****>&\\daKF47$F\\u$\"3\"p****R\"z
SCQF47$Fau$\"3c)****\\g]dP%F47$Ffu$\"3C(****>mAz)\\F47$F[v$\"30kmm#[')
z`&F47$F`v$\"3l******>qHDhF47$Fev$\"3oILLew[dmF47$Fjv$\"3b*****ff7$RsF
47$F_w$\"3Gjmm>Xw'y(F47$Fdw$\"3G(****4_4!f$)F47$Fiw$\"3^lmm1O[=*)F47$F
^x$\"3.)****\\7,U]*F47$Fcx$\"3GLL$4nJo+\"F77$Fhx$\"3?LL8P1_k5F77$F]y$
\"3bmmc4>t@6F77$Fby$\"3!)****>FJIu6F77$Fgy$\"3_mmEvcbM7F77$F\\z$\"3ALL
`siW)G\"F77$Faz$\"3/++I*)[!fM\"F77$Ffz$\"3&*****4<,!4S\"F77$F[[l$\"3)*
************f9F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++CF^am$\"++++gdF^am7$F[f
vFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3$*************>7F77$F2$!35+++8bX
i6F77$F9$!3.++bijQ76F77$F>$!3))****Hm$yg0\"F77$FC$!3y)****4CrR***F47$F
H$!36****\\zF&)H%*F47$FM$!3V****\\`F%o!*)F47$FR$!3A++]lhHl$)F47$FW$!37
++]4zA0yF47$Ffn$!3C++]*zbpC(F47$F[o$!3'))****RJ/Fn'F47$F`o$!3%*)****HH
.p;'F47$Feo$!3'******>9\"\\(f&F47$F[p$!3b******H4uD]F47$F`p$!39*****zl
`ZZ%F47$Fep$!3%)****\\HWSuRF47$Fjp$!3S+++-3WzLF47$F_q$!3o+++o&Ga(GF47$
Fdq$!3_++]@D9*G#F47$Fjq$!33,++G+=q<F47$F_r$!3D****\\@fz+7F47$Fdr$!3y(*
***\\/wge'Fgo7$Fir$!3Ga*****\\(4*G*F07$F^s$\"3%p****\\)[8mUFgo7$Fcs$\"
3o'*****\\Bpp)*Fgo7$Fhs$\"3-)***\\q8-p:F47$F]t$\"3u)***\\l%*pv?F47$Fbt
$\"3z)*****)oDHi#F47$Fgt$\"3C*****R#RE)=$F47$F\\u$\"3t'****H4P8u$F47$F
au$\"3e(***\\AQYwUF47$Ffu$\"3%y*****=3jq[F47$F[v$\"3h(****>H;XS&F47$F`
v$\"3_)******)HbufF47$Fev$\"3i)***\\i44\"\\'F47$Fjv$\"3c*****>S.e0(F47
$F_w$\"3g'***\\)z`re(F47$Fdw$\"3:(***\\k!oD9)F47$Fiw$\"35(*****Hqe&o)F
47$F^x$\"3O(***\\iv2a#*F47$Fcx$\"3H,++7:g,)*F47$Fhx$\"3'*****4$Q_h.\"F
77$F]y$\"3s***\\5(4o\"4\"F77$Fby$\"3#*****RwfqU6F77$Fgy$\"3')****p!R'=
,7F77$F\\z$\"3+++Sq>\\`7F77$Faz$\"3*****\\yig#48F77$Ffz$\"3-++&*[$QEO
\"F77$F[[l$\"3$*************>9F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++BF^am$
\"++++!H&F^am7$Fj_wFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!33++++++!=\"F77
$F2$!3FLLL&G*>C6F77$F9$!3mmm'=QZc2\"F77$F>$!3ELL`)oX5-\"F77$FC$!35MLL(
f?3m*F47$FH$!3cmmmMmz8\"*F47$FM$!3%\\LLtRNmg)F47$FR$!3#=++?J*\\\"3)F47
$FW$!3GMLL@GSQvF47$Ffn$!3!3++![![q*pF47$F[o$!3pmmmA\")>SkF47$F`o$!3!QL
L`SC(\\fF47$Feo$!3-,++orc(R&F47$F[p$!3M+++?F9V[F47$F`p$!3y+++K?&)3VF47
$Fep$!3qLLL,\\mBQF47$Fjp$!3?nmmu/tYKF47$F_q$!3onmmQ8*zv#F47$Fdq$!3)4++
g`r%*=#F47$Fjq$!3/ommy^B'o\"F47$F_r$!3j*****f80T8\"F47$Fdr$!3q)*****zw
Y$3'Fgo7$Fir$!3OEmmmmIxfF07$F^s$\"3'QLLLP)))RWFgo7$Fcs$\"3?lmmm5kt)*Fg
o7$Fhs$\"3Skmm)Hy<b\"F47$F]t$\"3/*****>^-J/#F47$Fbt$\"39lmmAhutDF47$Fg
t$\"3#*)****f*G&>7$F47$F\\u$\"3)y****>Fm#eOF47$Fau$\"3M)*****Rq<xTF47$
Ffu$\"3c(****f(*QLv%F47$F[v$\"33KLL,h/r_F47$F`v$\"3/+++g*3Q#eF47$Fev$
\"3olmmmUpCjF47$Fjv$\"3Y+++3UHsoF47$F_w$\"3#3LLt2VvQ(F47$Fdw$\"3o*****
zgEh#zF47$Fiw$\"3CKLL`/p_%)F47$F^x$\"3C++++S&R+*F47$Fcx$\"3.omm9j)[`*F
47$Fhx$\"3_mm1HTy25F77$F]y$\"3WLL`K+jh5F77$Fby$\"3')****fD)366\"F77$Fg
y$\"3QLL81r\"y;\"F77$F\\z$\"3zmmEow`=7F77$Faz$\"39++Smjhs7F77$Ffz$\"3F
++!3ewVK\"F77$F[[l$\"33++++++!Q\"F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++AF^a
m$\"++++S[F^am7$FiiwFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3/++++++S6F77$
F2$!3immmdI%f3\"F77$F9$!3ILL=,%3*Q5F77$F>$!3ammm2,8g)*F47$FC$!3&emmO&*
pwK*F47$FH$!3+ML$)*[Sxz)F47$FM$!3*omm6/GkI)F47$FR$!3))****\\eCq(z(F47$
FW$!3cnm;LxdrsF47$Ffn$!3O,+]'HSru'F47$F[o$!3vKLLJ>p2iF47$F`o$!3(ommw^X
Dt&F47$Feo$!3=,++%>Vw>&F47$F[p$!3E+++5XagYF47$F`p$!3k+++1/&H9%F47$Fep$
!3mmm;t`#Hn$F47$Fjp$!3*RLLt9?S6$F47$F_q$!3BMLL4TbSEF47$Fdq$!3e++]]0!)*
3#F47$Fjq$!3gMLLH.H-;F47$F_r$!3c****\\]VTn5F47$Fdr$!3m*****\\Jf3e&Fgo7
$Fir$!3Y)HLL$e^lEF07$F^s$\"3Mmmmh=k8YFgo7$Fcs$\"3EHLL$y*ex)*Fgo7$Fhs$
\"3@JL$oANX`\"F47$F]t$\"3!*)***\\eb]5?F47$Fbt$\"3%>LLjbmX_#F47$Fgt$\"3
/*****z'=kbIF47$F\\u$\"3q(****4X&>vNF47$Fau$\"3A)***\\d-*y2%F47$Ffu$\"
3;)****H8Zgj%F47$F[v$\"3klmm5fdP^F47$F`v$\"3)))*****H\\1tcF47$Fev$\"3i
LL$3d(HehF47$Fjv$\"3Y+++9]y)o'F47$F_w$\"39km;cB$z=(F47$Fdw$\"3n)***\\^
^o4xF47$Fiw$\"3hlmmwQz>#)F47$F^x$\"3c****\\P/$Qv)F47$Fcx$\"3wMLL<6<o#*
F47$Fhx$\"3sKLL](eTz*F47$F]y$\"3zmm,%4z:.\"F77$Fby$\"3+++![n6&z5F77$Fg
y$\"3tmmc@yWM6F77$F\\z$\"3SLL8mLe$=\"F77$Faz$\"35++&\\5sfB\"F77$Ffz$\"
3;++l7[6'G\"F77$F[[l$\"3/++++++S8F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++@F^a
m$\"++++5WF^am7$FhcxFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!#6F.7$F2$!3'**
*****HooZ5F77$F9$!3$*****\\?%p@+\"F77$F>$!3W+++IL!)4&*F47$FC$!3Q******
4$>X**)F47$FH$!3Y,++XVo\"[)F47$FM$!3h+++&o?i+)F47$FR$!3g+++0c!R^(F47$F
W$!3%******\\k_Z+(F47$Ffn$!3-,++XDB(\\'F47$F[o$!3q******Rd=vfF47$F`o$!
3%*******HmO:bF47$Feo$!3Y+++?#>x*\\F47$F[p$!3;++++j%zZ%F47$F`p$!31+++!
y[q(RF47$Fep$!34+++Xe=ANF47$Fjp$!3M+++?)48)HF47$F_q$!3y+++!)o6BDF47$Fd
q$!3g+++l&H,*>F47$Fjq$!39,++![X$=:F47$F_r$!3[*****\\cB2+\"F47$Fdr$!3Q)
*****\\4Dy]Fgo7$Fir$\"3m%H+++]FY'Ffq7$F^s$\"3%))*****\\`R(y%Fgo7$Fcs$
\"3y(******\\Q:))*Fgo7$Fhs$\"3.)****\\:#H<:F47$F]t$\"3y)****\\g3z(>F47
$Fbt$\"3u)******)pQvCF47$Fgt$\"3;******R3L*)HF47$F\\u$\"3](*****HY7#\\
$F47$Fau$\"37)****\\Z.'yRF47$Ffu$\"3(y******Gb(=XF47$F[v$\"3J)*****>d5
/]F47$F`v$\"3`********3KAbF47$Fev$\"3!*)****\\(3!>*fF47$Fjv$\"3g******
>eF0lF47$F_w$\"3Q)****\\j@$))pF47$Fdw$\"3a)****\\pVK\\(F47$Fiw$\"3)*)*
*****H(*o)zF47$F^x$\"3*))****\\(oq.&)F47$Fcx$\"3],++?fX,!*F47$Fhx$\"3@
+++5iZ5&*F47$F]y$\"3'*****\\b\"G:+\"F77$Fby$\"3%******R_9z/\"F77$Fgy$
\"32+++P&y55\"F77$F\\z$\"3++++k!H'[6F77$Faz$\"31++]VyK*>\"F77$Ffz$\"31
++]WI&yC\"F77$F[[l$\"#8F.F_[lFg[lF^\\l-Fd\\l6&7$$\"\"#F.Fg\\l7$Fg]yFc[
lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3)*************f5F77$F2$!3JLLL-1V45F7
7$F9$!3klm;)R/Vl*F47$F>$!3eKLL_lZf\"*F47$FC$!3#HLLjmo8m)F47$FH$!37nm;+
#Gc;)F47$FM$!3dKL$)GL,1xF47$FR$!3a****\\^(3,B(F47$FW$!36ML$obFzt'F47$F
fn$!3q++]$zCtC'F47$F[o$!3kmmm[&zEu&F47$F`o$!3+LLLUx=)H&F47$Feo$!3u****
*fC&z(z%F47$F[p$!31+++!4[`H%F47$F`p$!3$******R:Z6\"QF47$Fep$!3]LL$oJY9
P$F47$Fjp$!3ommm#\\*f[GF47$F_q$!3Mnmm]'zcS#F47$Fdq$!3S++]z&e/*=F47$Fjq
$!3nnmmI1SM9F47$F_r$!35%****\\zF.M*Fgo7$Fdr$!35(****\\eUcd%Fgo7$Fir$\"
3QdLLLe1eRF07$F^s$\"3KJLLQ)[6'\\Fgo7$Fcs$\"3Immm;s[&))*Fgo7$Fhs$\"3&[m
mJ3\\+]\"F47$F]t$\"3i)***\\^;JX>F47$Fbt$\"3`lmmBu?ECF47$Fgt$\"3F*****>
\")>I#HF47$F\\u$\"3L(*****3Q04MF47$Fau$\"3W)***\\#p;$zQF47$Ffu$\"3f(**
**pWj9S%F47$F[v$\"3+JLLHbjq[F47$F`v$\"3;+++qodr`F47$Fev$\"3'fmm\"zT]De
F47$Fjv$\"3s)****fim<K'F47$F_w$\"3!3LLQ\"4r)y'F47$Fdw$\"3U)***\\QA!oF(
F47$Fiw$\"3MKLLB2+axF47$F^x$\"3A)***\\7Le`#)F47$Fcx$\"3YmmmA2uM()F47$F
hx$\"3#fmm'pOzE#*F47$F]y$\"3OJL$)p@x9(*F47$Fby$\"3*)****>ttJ;5F77$Fgy$
\"3CLLV_#4x1\"F77$F\\z$\"3Vmm'=wuO6\"F77$Faz$\"3-++0#e$oi6F77$Ffz$\"3&
****\\jF\"f47F77$F[[l$\"3)*************f7F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+
++++>F^am$\"++++5OF^am7$FdgyFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3$****
*********>5F77$F2$!3^mmmYPu6(*F47$F9$!3)>LL8f9pG*F47$F>$!3\\mmmu(\\\"4
))F47$FC$!3WmmmA!=#G$)F47$FH$!3oLLLb?d\\yF47$FM$!3Hmmmsf!eS(F47$FR$!3E
+++)*=JYpF47$FW$!3]mmmoC5rkF47$Ffn$!3O+++UqT(*fF47$F[o$!3pKLLdL<5bF47$
F`o$!33mmma)353&F47$Feo$!3!******>Fryf%F47$F[p$!3(*******z)\\F6%F47$F`
p$!3C+++GbCXOF47$Fep$!3$pmm')y12A$F47$Fjp$!3ZLLLl\"*)er#F47$F_q$!3)QLL
8UU#)G#F47$Fdq$!3k+++%f(y!z\"F47$Fjq$!3WMLL\"yb/N\"F47$F_r$!3s'*****R*
>Mn)Fgo7$Fdr$!3E+++?U.tSFgo7$Fir$\"3I&ommmc)psF07$F^s$\"3dhmmEB!\\8&Fg
o7$Fcs$\"3;GLLLfV*))*Fgo7$Fhs$\"3YJLL6g!G[\"F47$F]t$\"3r)****zp9F\">F4
7$Fbt$\"3KKLLdy-xBF47$Fgt$\"3Q*****Ry3n&GF47$F\\u$\"39(****z)H)fK$F47$
Fau$\"3)y*****4*H+y$F47$Ffu$\"3?)****RgrTG%F47$F[v$\"3ckmmQ`;PZF47$F`v
$\"3.******RG$3A&F47$Fev$\"39KLL$[2\"fcF47$Fjv$\"3j*****>Vd#QhF47$F_w$
\"39kmm#>+\"*e'F47$Fdw$\"3S(****>yg.1(F47$Fiw$\"3#[mmm9/6_(F47$F^x$\"3
a(*****\\(fM+)F47$Fcx$\"3?LLLDb-o%)F47$Fhx$\"3jJLLH66V*)F47$F]y$\"3*[m
mYyiUT*F47$Fby$\"3M)****RA-s%)*F47$Fgy$\"3emm'y'*RV.\"F77$F\\z$\"3/LLt
f/sy5F77$Faz$\"3******f?$Rg7\"F77$Ffz$\"3%)****>3&H8<\"F77$F[[l$\"3$**
***********>7F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++=F^am$\"++++SKF^am7$Fcaz
Fc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3q+++++++)*F47$F2$!3)*******p9=H$*
F47$F9$!33++]%yC&>*)F47$F>$!3S+++(*H#)e%)F47$FC$!3)*******yt1&*zF47$FH
$!38,+]5f^LvF47$FM$!3\"4++lh)f0rF47$FR$!3)4++X/:Dm'F47$FW$!3m++]!QxU?'
F47$Ffn$!3#4++0H4vu&F47$F[o$!3k*****f;nwF&F47$F`o$!3-+++n*HQ'[F47$Feo$
!3%4++!)HZzR%F47$F[p$!3()******p;:IRF47$F`p$!35+++-RMzMF47$Fep$!3O++]g
s'*pIF47$Fjp$!3E+++Q)yJe#F47$F_q$!3)3++?>03<#F47$Fdq$!3W++]3m6\"p\"F47
$Fjq$!3)4++?$4^m7F47$F_r$!3y$****\\37l+)Fgo7$Fdr$!3w'****\\&eUqNFgo7$F
ir$\"3a.++]Z;e5Fgo7$F^s$\"3%H++]\"el3`Fgo7$Fcs$\"37,++]YQ$*)*Fgo7$Fhs$
\"3G)***\\RHcl9F47$F]t$\"3-****\\Wx6!)=F47$Fbt$\"3o)****4H[yK#F47$Fgt$
\"31*****fv(R!z#F47$F\\u$\"3H)****p;7HC$F47$Fau$\"3m)***\\FJu!o$F47$Ff
u$\"3!z****4wzo;%F47$F[v$\"3-*****z9&p.YF47$F`v$\"3m******4))3q]F47$Fe
v$\"3?****\\(y5F\\&F47$Fjv$\"3k*****zB[Z&fF47$F_w$\"3M)***\\r%*[*Q'F47
$Fdw$\"3/****\\D$>R%oF47$Fiw$\"3'*******pv?)G(F47$F^x$\"3a****\\(=OLv(
F47$Fcx$\"3q,++G.J,#)F47$Fhx$\"3!4++!*eG%f')F47$F]y$\"3'>++&*R`P6*F47$
Fby$\"3Q,++;2BJ&*F47$Fgy$\"35++I$oq4+\"F77$F\\z$\"3+++gdhwV5F77$Faz$\"
37++:f]R*3\"F77$Ffz$\"3E++0Sx1L6F77$F[[l$\"33++++++!=\"F7F_[lFg[lF^\\l
-Fd\\l6&7$$\"+++++<F^am$\"++++!*GF^am7$Fb[[lFc[lF_[lF\\]lF`]lFf]l7(-F(
6$7S7$F,$!3M+++++++%*F47$F2$!3WLLL$>>m%*)F47$F9$!3Ummmx\\8_&)F47$F>$!3
JMLL>i\\3\")F47$FC$!3]LLLNn\">m(F47$FH$!3pnmml(fu@(F47$FM$!3vLLLg7R0oF
47$FR$!3#3++5>=(yjF47$FW$!30LLL#H_u$fF47$Ffn$!3g+++R:g(\\&F47$F[o$!3em
mmu4;X]F47$F`o$!35LLLz5lYYF47$Feo$!35,++CL-)>%F47$F[p$!3A+++gMbZPF47$F
`p$!3(******fFUMJ$F47$Fep$!3xLLLKxA>HF47$Fjp$!3/nmm5&o/X#F47$F_q$!3Wnm
mizO`?F47$Fdq$!3q+++BcW\"f\"F47$Fjq$!3unmm#3mD=\"F47$F_r$!3S'*****HUgR
tFgo7$Fdr$!3#********[<y1$Fgo7$Fir$\"3MOLLLQM*Q\"Fgo7$F^s$\"3?LLL.$4C[
&Fgo7$Fcs$\"3'HmmmOLt*)*Fgo7$Fhs$\"3)[mmw')>$[9F47$F]t$\"35*****4z?v%=
F47$Fbt$\"3[lmmC(o'yAF47$Fgt$\"3=*****zs'3CFF47$F\\u$\"35)****fMT)fJF4
7$Fau$\"35)****\\Mc9e$F47$Ffu$\"3_)****z\"ze\\SF47$F[v$\"3eKLLd\\AqWF4
7$F`v$\"3U******zZM>\\F47$Fev$\"3Olmm\"49jK&F47$Fjv$\"3m*****R/R7x&F47
$F_w$\"3oJLL](y)*='F47$Fdw$\"3!)******oyZFmF47$Fiw$\"3WKLL$*4JbqF47$F^
x$\"3v*****\\i7K](F47$Fcx$\"3WommI^fMzF47$Fhx$\"3gmmm[guv$)F47$F]y$\"3
rLLL9SC8))F47$Fby$\"3(3++!3#f_@*F47$Fgy$\"3SMLL()R,w'*F47$F\\z$\"3hmmY
b=\")35F77$Faz$\"35++q(z]F0\"F77$Ffz$\"3;++!>(f![4\"F77$F[[l$\"3/+++++
+S6F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++;F^am$\"++++gDF^am7$Fae[lFc[lF_[lF
\\]lF`]lFf]l7(-F(6$7S7$F,$F^amF.7$F2$!3!pmmm\"p0k&)F47$F9$!3uKL$3<XZ=)
F47$F>$!3WmmmT%p\"exF47$FC$!3:mmm\"4m(GtF47$FH$!3OLL$3i.9!pF47$FM$!3fm
m;/R=0lF47$FR$!3k++]P8#\\4'F47$FW$!3?nm;/siqcF47$Ffn$!3E++](y$pZ_F47$F
[o$!3jKLL$yaE\"[F47$F`o$!31nmm\">s%HWF47$Feo$!3Q+++]$*4)*RF47$F[p$!39+
++]_&\\c$F47$F`p$!3%)******\\1aZJF47$Fep$!3umm;/#)[oFF47$Fjp$!3$QLLL=e
xJ#F47$F_q$!3)RLLLtIf$>F47$Fdq$!3]++]PYx\"\\\"F47$Fjq$!3GMLLL7i)4\"F47
$F_r$!3o&****\\P'psmFgo7$Fdr$!3k)****\\74_c#Fgo7$Fir$\"37pmm;H_?<Fgo7$
F^s$\"3olmm\"zihl&Fgo7$Fcs$\"3\\JLL$3#G,**Fgo7$Fhs$\"3pJL$ezw5V\"F47$F
]t$\"3(*)***\\PQ#\\\"=F47$Fbt$\"3GKLLe\"*[HAF47$Fgt$\"3I*******pvxl#F4
7$F\\u$\"3#z****\\_qn2$F47$Fau$\"3W)***\\i&p@[$F47$Ffu$\"3A)****\\2'HK
RF47$F[v$\"3ElmmmZvOVF47$F`v$\"3;******\\2goZF47$Fev$\"3UKL$eR<*f^F47$
Fjv$\"3m******\\)Hxe&F47$F_w$\"37km;H!o-*fF47$Fdw$\"3y)***\\7k.6kF47$F
iw$\"3#emmmT9C#oF47$F^x$\"33****\\i!*3`sF47$Fcx$\"3SLLLL*zym(F47$Fhx$
\"3KKLL3N1#4)F47$F]y$\"3Ylm;HYt7&)F47$Fby$\"3c)******p(G**))F47$Fgy$\"
30mmmT6KU$*F47$F\\z$\"3:KLLLbdQ(*F47$Faz$\"30++DOl5;5F77$Ffz$\"31++v.U
ac5F77$F[[l$\"#6F.F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++:F^am$\"++++]AF^am7$F
__\\lFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3k*************f)F47$F2$!3O++
+SY\\\"=)F47$F9$!34*****RObt\"yF47$F>$!3N+++kE%yS(F47$FC$!3!))****zW:c
*pF47$FH$!3!3++gZZ`e'F47$FM$!3V*****zaw\\?'F47$FR$!3[+++%[C6\"eF47$FW$
!3g*****f6-QS&F47$Ffn$!3#3++g.'y(*\\F47$F[o$!3e*****>f[,e%F47$F`o$!3C*
****RI$H7UF47$Feo$!35+++w`<)z$F47$F[p$!3/+++SqN#Q$F47$F`p$!3q*****R-R;
)HF47$Fep$!3s*****fn[xh#F47$Fjp$!3=+++cy/&=#F47$F_q$!3a+++/N\\==F47$Fd
q$!3I+++_O5#R\"F47$Fjq$!3#3++SQwY,\"F47$F_r$!3(\\*****>&)y0gFgo7$Fdr$!
3Q(*****f2gi?Fgo7$Fir$\"3#>++++-<0#Fgo7$F^s$\"3=)*****zi\"*HeFgo7$Fcs$
\"3+++++3B0**Fgo7$Fhs$\"3_)****RsLQT\"F47$F]t$\"3%))****R)oK#y\"F47$Fb
t$\"31*****>f4.=#F47$Fgt$\"3S*****>nk9f#F47$F\\u$\"3u(****Rq*p$*HF47$F
au$\"3w)*****zF)GQ$F47$Ffu$\"3R)****>B/]\"QF47$F[v$\"3%z****fd%G.UF47$
F`v$\"3#*)*****>n&yh%F47$Fev$\"3[*******p?N*\\F47$Fjv$\"3o*****fl?US&F
47$F_w$\"3M)****zId1z&F47$Fdw$\"3w(****f&\\f%>'F47$Fiw$\"3=******Ry^*e
'F47$F^x$\"3S)******\\lH+(F47$Fcx$\"38+++OZ;,uF47$Fhx$\"3/)****z'4Q3yF
47$F]y$\"3)*)****RCDA@)F47$Fby$\"31)****>>;Le)F47$Fgy$\"3[*****fHG'3!*
F47$F\\z$\"3A)****>^K!*Q*F47$Faz$\"35+++[Fi%z*F47$Ffz$\"3%*****fNCG=5F
77$F[[l$\"3)*************f5F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++9F^am$\"++
++g>F^am7$F^i\\lFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3G*************>)F
47$F2$!3%HLLLOK*)z(F47$F9$!3Ulm;db'*\\uF47$F>$!3[KLL')e^dqF47$FC$!3KKL
L/[YimF47$FH$!3[mm;J8HpiF47$FM$!3:LL$=>pZ!fF47$FR$!3U****\\IwKFbF47$FW
$!3xLL$y-xp8&F47$Ffn$!3]++]%Gyyu%F47$F[o$!3_mmm+CkZVF47$F`o$!3?LLL;W6&
*RF47$Feo$!3#)*****>S^#)f$F47$F[p$!3%*******H)e(*>$F47$F`p$!3++++)RPd
\"GF47$Fep$!39LL$y94qY#F47$Fjp$!3'pmm'GvL_?F47$F_q$!3Jnmmui0,<F47$Fdq$
!3a++]mEV#H\"F47$Fjq$!3#enmmM:tI*Fgo7$F_r$!3e(****\\m!))Q`Fgo7$Fdr$!3a
+++&R#**f:Fgo7$Fir$\"3qMLL$3\")GQ#Fgo7$F^s$\"3mILLo(pO+'Fgo7$Fcs$\"3'=
mmm^z\"4**Fgo7$Fhs$\"35lm;_1f'R\"F47$F]t$\"3#*)***\\I*H(\\<F47$Fbt$\"3
*\\mmc-I68#F47$Fgt$\"3k)****Rk`^_#F47$F\\u$\"3c(****H))G1\"HF47$Fau$\"
3?)***\\(*ff$G$F47$Ffu$\"3a)*****)Q7xp$F47$F[v$\"3]JLL&Q9)pSF47$F`v$\"
3c*******o7rY%F47$Fev$\"3llm;/S7F[F47$Fjv$\"3p*****>Y62A&F47$F_w$\"3yI
L$oeY5f&F47$Fdw$\"3_)***\\*\\`\"yfF47$Fiw$\"3aKLLj7icjF47$F^x$\"3i)***
\\P>%Gv'F47$Fcx$\"3)fmm'Q&\\W8(F47$Fhx$\"3TmmmF%)pCvF47$F]y$\"3iJL$)ee
r6zF47$Fby$\"3c(****RoWtE)F47$Fgy$\"37JLL]a$\\n)F47$F\\z$\"3Hkmm!\\*[R
!*F47$Faz$\"3t****\\L,=G%*F47$Ffz$\"3F)***\\um?+)*F47$F[[l$\"3$*******
******>5F7F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++8F^am$\"++++!p\"F^am7$F]c]lFc
[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3q+++++++yF47$F2$!3Inmm'3qjT(F47$F9$
!3_LLL]dd#3(F47$F>$!3Hnmm3\"*=2nF47$FC$!3vmmmgTJHjF47$FH$!3#RLLj=NK&fF
47$FM$!3xnmmN=c/cF47$FR$!3.,++x2`V_F47$FW$!3/nmmR>:q[F47$Ffn$!3/,++L0(
z\\%F47$F[o$!3YLLL4i8:TF47$F`o$!3;nmmGb$zx$F47$Feo$!3'3++!GuK)R$F47$F[
p$!3H+++?1;<IF47$F`p$!3K+++sd$)\\EF47$Fep$!3cmmm>'piJ#F47$Fjp$!3vLLL,s
i>>F47$F_q$!3'QLL`/>Oe\"F47$Fdq$!3M+++\"ohF>\"F47$Fjq$!3ATLL`o'yY)Fgo7
$F_r$!3k%*****4G(>n%Fgo7$Fdr$!3/(*****HSQd5Fgo7$Fir$\"3spmmm,19FFgo7$F
^s$\"3enmmcKUxhFgo7$Fcs$\"3#[LLLBGJ\"**Fgo7$Fhs$\"3%>LL.eZ$z8F47$F]t$
\"3B*****p(H8<<F47$Fbt$\"37LLLf/&>3#F47$Fgt$\"35+++;E%)eCF47$F\\u$\"3r
)****>1ev#GF47$Fau$\"3'*)****\\@4V=$F47$Ffu$\"3E)****fa?/e$F47$F[v$\"3
'fmmY>Wj$RF47$F`v$\"3>+++g'ojJ%F47$Fev$\"3qKLL3tsgYF47$Fjv$\"3r*****zE
-s.&F47$F_w$\"3)\\mmc'eV\"R&F47$Fdw$\"3G*****H/7<w&F47$Fiw$\"3#fmmmoCP
7'F47$F^x$\"3s*****\\P=F]'F47$Fcx$\"3[MLLTVtnoF47$Fhx$\"3!RLLt)e,TsF47
$F]y$\"3#pmmOZ17h(F47$Fby$\"3q*****f<t8&zF47$Fgy$\"36omm/ECT$)F47$F\\z
$\"3#RLL$pk%**o)F47$Faz$\"37,++>vth!*F47$Ffz$\"3_-++$**)e<%*F47$F[[l$
\"3q+++++++)*F4F_[lFg[lF^\\l-Fd\\l6&7$$\"+++++7F^am$\"++++S9F^am7$F\\]
^lFc[lF_[lF\\]lF`]lFf]l7(-F(6$7S7$F,$!3M+++++++uF47$F2$!3x+++5y!Q.(F47
$F9$!3')****\\Vf=:nF47$F>$!3I+++JB'oN'F47$FC$!3Q*****p^jh*fF47$FH$!3Q,
+]T!zrj&F47$FM$!3h++]zWN/`F47$FR$!3)*****\\BRtf\\F47$FW$!3K++]^oK.YF47
$Ffn$!3s++]\"yi![UF47$F[o$!3'******z,IE)QF47$F`o$!3A+++TmvgNF47$Feo$!3
e+++aMS)>$F47$F[p$!3?+++5CcMGF47$F`p$!3=+++YT$R[#F47$Fep$!3)*****\\\"4
Ib;#F47$Fjp$!3K+++uo\"py\"F47$F_q$!3k+++;==m9F47$Fdq$!3g++]&p!4$4\"F47
$Fjq$!3$)3++g$=%GwFgo7$F_r$!3E(****\\&\\10SFgo7$Fdr$!3'>+++lcxa&F07$Fi
r$\"3]-++]#R_/$Fgo7$F^s$\"31+++Xn<^jFgo7$Fcs$\"3n'*****\\p2<**Fgo7$Fhs
$\"3a)***\\3X5i8F47$F]t$\"3I****\\Bg`%o\"F47$Fbt$\"3-*****H*3xK?F47$Fg
t$\"3K*****zeJDR#F47$F\\u$\"3`)****4C([WFF47$Fau$\"3U)***\\KC-&3$F47$F
fu$\"3U)****HqGJY$F47$F[v$\"34*****R+uG!QF47$F`v$\"3%*******HYilTF47$F
ev$\"3)))***\\71L%\\%F47$Fjv$\"3g+++uIp`[F47$F_w$\"3?****\\W^#=>&F47$F
dw$\"3;****\\'eq_a&F47$Fiw$\"3G******4\"G3*eF47$F^x$\"3%*****\\7[f_iF4
7$Fcx$\"3M+++W\">5g'F47$Fhx$\"3]+++ZLLdpF47$F]y$\"3L,+])3(p5tF47$Fby$
\"34+++o;SNwF47$Fgy$\"3x******e(\\v+)F47$F\\z$\"3*******zW./M)F47$Faz$
\"3u++]/\\H&p)F47$Ffz$\"3V,+]68(\\.*F47$F[[l$\"3M+++++++%*F4F_[lFg[lF^
\\l-Fd\\l6&7$$\"+++++6F^am$\"++++57F^am7$F[g^lFc[lF_[lF\\]lF`]lFf]lFg[
lF^\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve
 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 13 "" 1 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x-> 3*(1-x)*
x^2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operator
G%&arrowGF(,$*&,&\"\"\"F/9$!\"\"F/)F0\"\"#F/\"\"$F(F(F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 124 "You can see that as the moving point app
roaches the fixed one, the secant line swings round and approaches the
 tangent line." }}{PARA 0 "" 0 "" {TEXT -1 125 "Presumably, therefore,
 its slope approaches the slope of the tangent line.  We say that the \+
slope of the tangent line is the " }{TEXT 261 5 "limit" }{TEXT -1 96 "
 of the slope of the secant line.  In symbols, the slope of the secant
 line through the points (" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(x[0]);" "6#-%\"fG6#&%\"xG6#\"\"!" }
{TEXT -1 7 ") and (" }{XPPEDIT 18 0 "x[0]+h;" "6#,&&%\"xG6#\"\"!\"\"\"
%\"hGF(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(x[0]+h);" "6#-%\"fG6#,&&%
\"xG6#\"\"!\"\"\"%\"hGF+" }{TEXT -1 4 ") is" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "(f(x[0]+h)-f(x[0]))/h;" "6#*&,&-%\"fG6#,&&%\"xG6#\"\"!
\"\"\"%\"hGF-F--F&6#&F*6#F,!\"\"F-F.F3" }{TEXT -1 14 ", and we write" 
}}{PARA 256 "" 0 "" {TEXT -1 30 "Slope of the tangent line at (" }
{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 
0 "f(x[0]);" "6#-%\"fG6#&%\"xG6#\"\"!" }{TEXT -1 4 ") = " }{XPPEDIT 
18 0 "limit((f(x[0]+h)-f(x[0]))/h,h = 0);" "6#-%&limitG6$*&,&-%\"fG6#,
&&%\"xG6#\"\"!\"\"\"%\"hGF0F0-F)6#&F-6#F/!\"\"F0F1F6/F1F/" }{TEXT -1 
2 " ." }}{PARA 0 "" 0 "" {TEXT -1 68 "The symbol to the right of the e
quals sign is read as \"the limit as " }{XPPEDIT 18 0 "h;" "6#%\"hG" }
{TEXT -1 90 " goes to 0\", and indicates that we are to consider what \+
happens to the next expression as " }{XPPEDIT 18 0 "h;" "6#%\"hG" }
{TEXT -1 63 " becomes very small (close to 0).  Of course, we can neve
r put " }{XPPEDIT 18 0 "h = 0;" "6#/%\"hG\"\"!" }{TEXT -1 92 ", becase
 then we would be dividing by 0; the point is that the quotient behave
s nicely when " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 68 " is very s
mall (presumably because the numerator is also small when " }{XPPEDIT 
18 0 "h;" "6#%\"hG" }{TEXT -1 11 " is small.)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 
"Now let's look at our other problem: how do we define the instantaneo
us velocity of a moving object?  To be precise, let's suppose that we \+
have an object moving along the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 39 "-axis so that its position at any time " }{XPPEDIT 18 0 "
t;" "6#%\"tG" }{TEXT -1 28 " is given by the expression " }{XPPEDIT 
18 0 "s(t);" "6#-%\"sG6#%\"tG" }{TEXT -1 30 ".  For example, we might \+
have " }{XPPEDIT 18 0 "s(t) = t^2;" "6#/-%\"sG6#%\"tG*$F'\"\"#" }
{TEXT -1 5 ", or " }{XPPEDIT 18 0 "s(t) = sin(t);" "6#/-%\"sG6#%\"tG-%
$sinG6#F'" }{TEXT -1 49 ".  (How does the object move in these two cas
es?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "W
e can start by observing that there is no difficulty in defining the \+
" }{TEXT 262 7 "average" }{TEXT -1 50 " velocity of the object during \+
some time interval " }{XPPEDIT 18 0 "[t[0], t[1]];" "6#7$&%\"tG6#\"\"!
&F%6#\"\"\"" }{TEXT -1 105 ", using the elementary formula \"Distance \+
= Rate times Time\".  The distance travelled during the interval " }
{XPPEDIT 18 0 "[t[0], t[1]];" "6#7$&%\"tG6#\"\"!&F%6#\"\"\"" }{TEXT 
-1 4 " is " }{XPPEDIT 18 0 "s(t[1])-s(t[0]);" "6#,&-%\"sG6#&%\"tG6#\"
\"\"F*-F%6#&F(6#\"\"!!\"\"" }{TEXT -1 40 ", and dividing by the elapse
d time gives" }}{PARA 256 "" 0 "" {TEXT -1 42 "Average velocity during
 the time interval " }{XPPEDIT 18 0 "[t[0], t[1]];" "6#7$&%\"tG6#\"\"!
&F%6#\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(s(t[1])-s(t[0]))/(t[1]
-t[0]);" "6#*&,&-%\"sG6#&%\"tG6#\"\"\"F+-F&6#&F)6#\"\"!!\"\"F+,&&F)6#F
+F+&F)6#F0F1F1" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 85 "Here i
s a simple procedure to compute (decimal approximations to) average ve
locities:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "av := proc(s,t
0,t1) evalf((s(t1) - s(t0))/(t1 - t0)) end proc:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 69 "We can now define a position function and compute \+
average velocities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s :=
 t-> t^2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGR6#%\"tG6\"6$%)ope
ratorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "av(s,1,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"'
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "(You should check this \+
result by hand.)  Of course, our problem is not to compute the average
 velocity over some time interval " }{XPPEDIT 18 0 "[t[0], t[1]];" "6#
7$&%\"tG6#\"\"!&F%6#\"\"\"" }{TEXT -1 57 ", but to compute the instant
aneous velocity at some time " }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!
" }{TEXT -1 98 ".  The solution is to compute the average velocities o
ver shorter and shorter time intervals near " }{XPPEDIT 18 0 "t[0];" "
6#&%\"tG6#\"\"!" }{TEXT -1 58 ".  Here are some examples with our last
 position function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "av(s
,1,2); av(s,1,1.5); av(s,1,1.1); av(s,1,1.01);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++
D!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++@!\"*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"++++5?!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
118 "Experiment with a few more intervals, and convince yourself that \+
the average velocity over a short time interval near " }{XPPEDIT 18 0 
"t[0] = 1;" "6#/&%\"tG6#\"\"!\"\"\"" }{TEXT -1 59 " is very close to 2
.  Of course, we can never actually let " }{XPPEDIT 18 0 "t[1] = t[0];
" "6#/&%\"tG6#\"\"\"&F%6#\"\"!" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "av(s,1,1);" }}{PARA 8 "" 1 "" {TEXT -1 32 "Err
or, (in av) division by zero\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 
"but the point is that the average velocities behave nicely as " }
{XPPEDIT 18 0 "t[1];" "6#&%\"tG6#\"\"\"" }{TEXT -1 12 " approaches " }
{XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 45 ".  We say that t
he average velocities have a " }{TEXT 263 5 "limit" }{TEXT -1 4 " as \+
" }{XPPEDIT 18 0 "t[1];" "6#&%\"tG6#\"\"\"" }{TEXT -1 12 " approaches \+
" }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 61 ", and it make
s sense to define the instantaneous velocity at " }{XPPEDIT 18 0 "t[0]
;" "6#&%\"tG6#\"\"!" }{TEXT -1 18 " to be this limit:" }}{PARA 256 "" 
0 "" {TEXT -1 31 "Instantaneous velocity at time " }{XPPEDIT 18 0 "t[0
];" "6#&%\"tG6#\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "limit((s(t[1])
-s(t[0]))/(t[1]-t[0]),t[1] = t[0]);" "6#-%&limitG6$*&,&-%\"sG6#&%\"tG6
#\"\"\"F.-F)6#&F,6#\"\"!!\"\"F.,&&F,6#F.F.&F,6#F3F4F4/&F,6#F.&F,6#F3" 
}{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 140 "Now, this is exactly \+
the same limit we saw in the tangent line problem.  You can see this w
ith a simple substitution: define a new variable " }{XPPEDIT 18 0 "h;
" "6#%\"hG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "h = t[1]-t[0];" "6#/%\"
hG,&&%\"tG6#\"\"\"F)&F'6#\"\"!!\"\"" }{TEXT -1 10 ", so that " }
{XPPEDIT 18 0 "t[1] = t[0]+h;" "6#/&%\"tG6#\"\"\",&&F%6#\"\"!F'%\"hGF'
" }{TEXT -1 23 ", and the limit becomes" }}{PARA 256 "" 0 "" {TEXT -1 
31 "Instantaneous velocity at time " }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6
#\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "limit((s(t[0]+h)-s(t[0]))/h,
h = 0);" "6#-%&limitG6$*&,&-%\"sG6#,&&%\"tG6#\"\"!\"\"\"%\"hGF0F0-F)6#
&F-6#F/!\"\"F0F1F6/F1F/" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 
293 "This suggests that we might be able to interpret the instantaneou
s velocity as the slope of the tangent line to some graph.  In fact, c
omparing the limits from our two problems, it is clear that the instan
taneous velocity is the slope of the tangent line to the graph of the \+
position function " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 49 ".  Her
e is the picture for the position function " }{XPPEDIT 18 0 "s(t) = t^
2;" "6#/-%\"sG6#%\"tG*$F'\"\"#" }{TEXT -1 169 " that we were using abo
ve.  You should be able to see that the tangent line at the point (1,1
) has slope 2, as we would expect from our calculations of average vel
ocity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s := t-> t^2 ;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGR6#%\"tG6\"6$%)operatorG%&arrow
GF(*$)9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "plot([s(t),s(1)+2*(t-1)], t=-1..3, colour=[black,red]);" }}{PARA 
13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"\"
\"\"!$\"\"\"F*7$$!3PLLLLQ6G\"*!#=$\"3dO\"Ha@YAL)F07$$!3immmT.\\p$)F0$
\"3'=;Dz&o$[+(F07$$!3LLLL$))Qj^(F0$\"3ee3\"4-N&\\cF07$$!3ULLL$=Kvl'F0$
\"3!*4>@xMFKWF07$$!3hnmmTs!G!eF0$\"3u6RR)=dsO$F07$$!3iLLL3yO5]F0$\"3>(
Hyuby.^#F07$$!3i+++vE%)*=%F0$\"3/;^7k\"yav\"F07$$!3)RLL$3WDTLF0$\"3#Qp
?@5)R;6F07$$!3'4++]d(Q&\\#F0$\"3G&Qk%\\\"fpA'!#>7$$!3:mmmm&4`i\"F0$\"3
F'=)\\(=J;k#Fen7$$!3GKLLLQW*e)Fen$\"3m+))fOX&yP(!#?7$$\"3HI#*******H,Q
!#@$\"3#[T***o\"))\\W\"!#C7$$\"3Q(*******\\*3q)Fen$\"3#e\\-,Qd0d(F`o7$
$\"3!********p=\\q\"F0$\"3l*o4OxZn!HFen7$$\"3_mmm\"fBIY#F0$\"3oll58_[m
gFen7$$\"3yKLLLO[kLF0$\"3S'y'*=,v>8\"F07$$\"3.KLLL&Q\"GTF0$\"3ZZ\"R]x_
Tq\"F07$$\"3+*****\\s]k,&F0$\"3s,`j(yxk^#F07$$\"3WJLLLvv-eF0$\"3%Gnl!*
\\*>nLF07$$\"3'3++]sgam'F0$\"3*Qv^wmOGW%F07$$\"3G+++v\"ep[(F0$\"3-M*>9
Faag&F07$$\"3#QLLLe/TM)F0$\"3cM/wH\"3C'pF07$$\"39LLLeDBJ\"*F0$\"3)3nOM
!3%zL)F07$$\"3Immm;kD!)**F0$\"3=<;C9=bg**F07$$\"3Mmm;f`@'3\"!#<$\"3!*p
&*[1Q')z6Fgs7$$\"3y****\\nZ)H;\"Fgs$\"3iGqVpN`_8Fgs7$$\"3YmmmJy*eC\"Fg
s$\"37,<&pShAb\"Fgs7$$\"3')******R^bJ8Fgs$\"3A>/'34RIx\"Fgs7$$\"3f****
*\\5a`T\"Fgs$\"3!*\\.aUsA.?Fgs7$$\"3o****\\7RV'\\\"Fgs$\"3^c+[aWJRAFgs
7$$\"3k*****\\@fke\"Fgs$\"3*\\Te3%G&o^#Fgs7$$\"3/LLL`4Nn;Fgs$\"3gu:e,#
f+y#Fgs7$$\"3#*******\\,s`<Fgs$\"3-Ag^kV`vIFgs7$$\"3[mm;zM)>$=Fgs$\"3'
RgRzYjhN$Fgs7$$\"3$*******pfa<>Fgs$\"3oSK1ZD)pn$Fgs7$$\"3#HLLeg`!)*>Fg
s$\"3#Qe$y6#=A*RFgs7$$\"3w****\\#G2A3#Fgs$\"3@MgHnreNVFgs7$$\"3;LLL$)G
[k@Fgs$\"3UzHC_h)\\o%Fgs7$$\"3#)****\\7yh]AFgs$\"31%GUz`!Gl]Fgs7$$\"3x
mmm')fdLBFgs$\"3Y2ta&)odXaFgs7$$\"3bmmm,FT=CFgs$\"3?lEe&**>([eFgs7$$\"
3FLL$e#pa-DFgs$\"3amy*f6TFE'Fgs7$$\"3!*******Rv&)zDFgs$\"3K^[pE\\mbmFg
s7$$\"3ILLLGUYoEFgs$\"3C7Y*yL,27(Fgs7$$\"3_mmm1^rZFFgs$\"3e5US2$Q*\\vF
gs7$$\"34++]sI@KGFgs$\"3-\"*)R!))3V@!)Fgs7$$\"34++]2%)38HFgs$\"3%f)3\"
*pS3'[)Fgs7$$\"\"$F*$\"\"*F*-%'COLOURG6&%$RGBGF*F*F*-F$6$7S7$F($!\"$F*
7$F.$!3ommmmFiDGFgs7$F4$!35LLLo!)*Qn#Fgs7$F9$!3nmmmwxE.DFgs7$F>$!3Ymmm
Ok]JBFgs7$FC$!3_LLL[9cg@Fgs7$FH$!3smmmhN2-?Fgs7$FM$!37+++N&oz$=Fgs7$FR
$!3!omm;)3Do;Fgs7$FW$!3?+++:v2*\\\"Fgs7$Fgn$!3BLLL8>1D8Fgs7$F\\o$!3kmm
mw))yr6Fgs7$Fbo$!3a,+++uR#***F07$Fio$!3_++++5#)f#)F07$F^p$!3?++++E;!f'
F07$Fcp$!3)pmmm\"G&R2&F07$Fhp$!3WMLLLF.rKF07$F]q$!3%fLLL$HsV<F07$Fbq$
\"3/+)*****\\9!H$F`o7$Fgq$\"3)Gmmm1:bg\"F07$F\\r$\"3t,++]9#4L$F07$Far$
\"3a+++]j\"R(\\F07$Ffr$\"3knmmm\"4#)o'F07$F[s$\"3Gmmm;^Yi#)F07$F`s$\"3
gKLLLG^g**F07$Fes$\"3oKLL=2Vs6Fgs7$F[t$\"3f*****\\`pfK\"Fgs7$F`t$\"3!H
LLLm&z\"\\\"Fgs7$Fet$\"3s******z-6j;Fgs7$Fjt$\"3<******4#32$=Fgs7$F_u$
\"3O*****\\#y'G*>Fgs7$Fdu$\"3G******H%=H<#Fgs7$Fiu$\"35mmm1>qMBFgs7$F^
v$\"3%)*******HSu]#Fgs7$Fcv$\"3'HLL$ep'Rm#Fgs7$Fhv$\"3')******R>4NGFgs
7$F]w$\"3#emm;@2h*HFgs7$Fbw$\"3]*****\\c9W;$Fgs7$Fgw$\"3Lmmmmd'*GLFgs7
$F\\x$\"3j*****\\iN7]$Fgs7$Fax$\"3aLLLt>:nOFgs7$Ffx$\"35LLL.a#o$QFgs7$
F[y$\"3ammm^Q40SFgs7$F`y$\"3y******z]rfTFgs7$Fey$\"3gmmmc%GpL%Fgs7$Fjy
$\"3/LLL8-V&\\%Fgs7$F_z$\"3=+++XhUkYFgs7$Fdz$\"3=+++:o<E[Fgs7$Fiz$\"\"
&F*-F^[l6&F`[l$\"*++++\"!\")$F*F*F\\el-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%V
IEWG6$;F(Fiz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "To summarise:
 we have seen two uses for the limit " }{XPPEDIT 18 0 "limit((f(x[0]+h
)-f(x[0]))/h,h = 0);" "6#-%&limitG6$*&,&-%\"fG6#,&&%\"xG6#\"\"!\"\"\"%
\"hGF0F0-F)6#&F-6#F/!\"\"F0F1F6/F1F/" }{TEXT -1 78 " .  Its geometric \+
interpretation is the slope of the tangent line to graph of " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 15 " at the point (" }{XPPEDIT 
18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(x[0]
);" "6#-%\"fG6#&%\"xG6#\"\"!" }{TEXT -1 7 ").  If " }{XPPEDIT 18 0 "f;
" "6#%\"fG" }{TEXT -1 101 " represents the position of an object, then
 this limit represents the instantaneous velocity at time " }{XPPEDIT 
18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 70 ".  These are both example
s of the same idea: the limit represents the " }{TEXT 264 28 "instanta
neous rate of change" }{TEXT -1 17 " of the function " }{XPPEDIT 18 0 
"f;" "6#%\"fG" }{TEXT -1 74 ".  Something this useful should be given \+
a name.  Our limit is called the " }{TEXT 265 10 "derivative" }{TEXT 
-1 4 " of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 5 " (at " }
{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 18 "), and is denote
d " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 4 " ' (" }{XPPEDIT 18 0 "x
[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 6 ") .  (" }{TEXT 266 5 "Maple" }
{TEXT -1 17 " uses the symbol " }{TEXT 267 4 "D(f)" }{TEXT -1 12 " ins
tead of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 51 " ' .)  Since the
 value of the limit will depend on " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6
#\"\"!" }{TEXT -1 99 ", it is actually a function, and we will emphasi
se this by dropping the subscript and just writing " }}{PARA 256 "" 0 
"" {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 4 " ' (" }{XPPEDIT 18 0 "x;
" "6#%\"xG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "limit((f(x+h)-f(x))/h,h
 = 0);" "6#-%&limitG6$*&,&-%\"fG6#,&%\"xG\"\"\"%\"hGF-F--F)6#F,!\"\"F-
F.F1/F.\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 53 "For exam
ple, here are some derivatives worked out by " }{TEXT 268 5 "Maple" }
{TEXT -1 154 ".  Observe that they behave as functions should; in part
icular, they can be evaluated.  (Do any of these values relate to earl
ier parts of the worksheet?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "f := x-> 3*(1-x)*x^2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG
R6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&,&\"\"\"F/9$!\"\"F/)F0\"\"#F/\"
\"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f1 := D(f);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1GR6#%\"xG6\"6$%)operatorG%&arrow
GF(,&*$)9$\"\"#\"\"\"!\"$*(\"\"'F1,&F1F1F/!\"\"F1F/F1F1F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "D(f)(.1);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"#^!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "D(f)(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "g := x-> x^2 ;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arro
wGF(*$)9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "D(g)(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}}{MARK "19 1 \+
0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }