{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 }
{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 "Warning" -1 7 1 
{CSTYLE "" -1 -1 "Courier" 1 10 0 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 "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 "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 "Heading 2" -1 256 1 {CSTYLE "" 
-1 -1 "Times" 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 256 "
" 0 "" {TEXT -1 33 "Lesson 17: The Mean Value Theorem" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart: \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Recall Rolle's Theorem: " }}
{PARA 0 "" 0 "" {TEXT -1 30 "Let f be a function such that " }}{PARA 
0 "" 0 "" {TEXT -1 35 "       1) f is continuous on [a,b] " }}{PARA 0 
"" 0 "" {TEXT -1 39 "       2) f is differentiable on (a,b) " }}{PARA 
0 "" 0 "" {TEXT -1 23 "       3) f(a) = f(b). " }}{PARA 0 "" 0 "" 
{TEXT -1 53 "Then, there is some c in (a,b) such that f '(c) = 0. " }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "1) Use Rolle's Theorem to show th
at " }{XPPEDIT 18 0 "3*x - 2 + cos( Pi*x/2" "6#,(*&\"\"$\"\"\"%\"xGF&F
&\"\"#!\"\"-%$cosG6#*(%#PiGF&F'F&F(F)F&" }{TEXT -1 31 " has exactly on
e real root.    " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f1:= x \+
-> 3*x - 2 + cos( Pi * x / 2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#f1GR6#%\"xG6\"6$%)operatorG%&arrowGF(,(9$\"\"$\"\"#!
\"\"-%$cosG6#,$*&%#PiG\"\"\"F-F7#F7F/F7F(F(F(" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 46 "Let's plot f1 to see where roots are located. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(f1(x), x = -4 * Pi.. 4 \+
* Pi, color = red);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7S7$
$!3+++#*eqjc7!#;$!3cL-;,I%p!RF*7$$!3N[.^N[&=?\"F*$!3pKX%z%pg0PF*7$$!3C
n0%e6*=a6F*$!3+JY)37gte$F*7$$!3w,*p)GSe+6F*$!3fvS24Z$3]$F*7$$!3Aike7Oi
Y5F*$!3'*[12c0B9MF*7$$!3ZQQU2m>H**!#<$!3Cl6`]69yKF*7$$!3kBd)zA#HJ%*FD$
!3%4b$)*=-2#4$F*7$$!33_Z#G3Td\"*)FD$!3A<m$HdD:'GF*7$$!3%Hu&*4NdDQ)FD$!
3!3HTyg&GKEF*7$$!3Sbn:YN3^yFD$!3cx3cz%[!eCF*7$$!3xo_W6uR/tFD$!3Yc`>/eI
XBF*7$$!3ZERUmf(G#oFD$!39;k7*zFVF#F*7$$!3[iRPnoz!G'FD$!3#)G!*4*RoY<#F*
7$$!3n79YR>\\OdFD$!3q*zIe!G]7?F*7$$!3KWr;%G`>@&FD$!3aPZ:`yE'z\"F*7$$!3
>7(eM$>iNZFD$!3E;ab!fT.e\"F*7$$!3Wh<r&)y@pTFD$!3u8zuysFa8F*7$$!3wQ)G>O
*R*o$FD$!3/-L9Lu[=7F*7$$!3u[%f>Nc78$FD$!3WVVoo\\!*=6F*7$$!3AiIW9_?PEFD
$!3#*f\"\\&*Q9^/\"F*7$$!3q7Yp8G:&4#FD$!3U<)GTA4VF*FD7$$!3Ysa&Ru!**y:FD
$!3=\")*z#)eah_(FD7$$!3)>bRJxH//\"FD$!39&\\H&yNv%=&FD7$$!3-[s!R\"oiea!
#=$!3'z0NY&>@$)HFD7$$!3j+qiCf_S7!#>$!3jOGZEcSP5FD7$$\"3oul%)o22<aFir$
\"3a&G_Z?$>WGFir7$$\"3/,)*R%\\jS-\"FD$\"3I=G\"[j+W.\"FD7$$\"3XG?!*R;-X
:FD$\"3K%>u&H3wz=FD7$$\"3iWq**zBA$3#FD$\"3g\"f0\\a*>eKFD7$$\"3AXxe/ou4
EFD$\"3!z&)*)R\")4RD&FD7$$\"3^V2wei=>JFD$\"3++^?dVoVvFD7$$\"39g1d:>$[o
$FD$\"3/g[qROVM**FD7$$\"37*G@jp*3$>%FD$\"3O[T<zBO`6F*7$$\"3Y2PsiLwNZFD
$\"3))3\"R*pO0h7F*7$$\"3[W,lhj]F_FD$\"3>c=e26FL8F*7$$\"3/(\\BeM6^w&FD$
\"35/)>A>jiV\"F*7$$\"3o.XRRd&4F'FD$\"3ecRLw#3-f\"F*7$$\"3yGI6$))3(*z'F
D$\"3WD*3$yu'*3=F*7$$\"3X4MPG<m;tFD$\"3!RYLGr8F/#F*7$$\"3!4#[m[M'y&yFD
$\"3I5(e0&p([D#F*7$$\"3_)ob'G]5z$)FD$\"3%GPXH]=lR#F*7$$\"3$*eeyG)\\@\"
*)FD$\"3[4Iwm1S([#F*7$$\"3nSC%)[2yS%*FD$\"3!Qh#4ytRoDF*7$$\"3vM4*GmPl#
**FD$\"3i@xT`jiyEF*7$$\"3'R,./*pK[5F*$\"3cM.3wmVsGF*7$$\"3CFet&y@\")4
\"F*$\"3St<7$\\:94$F*7$$\"3Y)eqLU87:\"F*$\"3k2O%pz&oDLF*7$$\"3ftt#\\*)
G??\"F*$\"3W\"[][!f.1NF*7$$\"3+++#*eqjc7F*$\"3wm(fBNzGj$F*-%'COLOURG6&
%$RGBG$\"*++++\"!\")$\"\"!Fb[lFa[l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG
6$;$!+iqjc7F`[l$\"+iqjc7F`[l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 45 "There appears to be precisely one real root. " }}{PARA 0 
"" 0 "" {TEXT -1 43 "Let's have Maple numerically solve for it. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fsolve(f1(x) = 0, x = 0..5);
" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Z2ncR!#5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Hence, it appears that the real ro
ot of f1 is approximately  " }{XPPEDIT 18 0 ".3956670747;" "6#$\"+Z2nc
R!#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "How do we verify
 our numerical calculations with Rolle's Theorem? " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Suppose to the contrary t
hat f1 had 2 or more real roots; say a,b " }}{PARA 0 "" 0 "" {TEXT -1 
47 "are real roots of f1, i.e., f1(a) = f1(b) = 0. " }}{PARA 0 "" 0 "
" {TEXT -1 67 "Then by Rolle's Theorem there exists some c between a a
nd  b, with " }}{PARA 0 "" 0 "" {TEXT -1 31 "f1 '(c) = 0. But f1 '(x) \+
= 3 - " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 9 " /2 sin( " }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 21 " x/2 ) . Since |sin( " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 10 " )| < = 1 " }}{PARA 
0 "" 0 "" {TEXT -1 8 "for all " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }
{TEXT -1 18 " , we have that  |" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 9 " /2 sin( " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 13 " x
/2 )| < =  " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 9 " /2 < 3. " }}
{PARA 0 "" 0 "" {TEXT -1 68 "Hence, f1 '(x) > 0 for all x and therefor
e we have a contradiction. " }}{PARA 0 "" 0 "" {TEXT -1 38 "Thus, f1 h
as precisely one real root. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 53 "Let's plot f1 '(x) to see that indeed it is > 0
.     " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(D(f1)(x), x \+
= -4*Pi..4*Pi, color = red);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7c[l7$$!3+++#*eqjc7!#
;$\"3qlAuW\"y-A%!#<7$$!3&penI]THC\"F*$\"3upu:;?+\")RF-7$$!34u^@ZfCH7F*
$\"3+\"flbM*\\'p$F-7$$!3ln*)Gp\")RA7F*$\"3XUB-M9KTNF-7$$!3AhFO\"R]b@\"
F*$\"3q*)yL(Q'))zLF-7$$!3zalV8Eq37F*$\"3.W7UL-19KF-7$$!3N[.^N[&=?\"F*$
\"33ra*pvfd/$F-7$$!3)H!Hf0%Q**=\"F*$\"3UYe7iBx_FF-7$$!3!yXvc(>-y6F*$\"
3QzfxA>UoCF-7$$!3g7!eda0h;\"F*$\"3OVGC)GUE?#F-7$$!3Cn0%e6*=a6F*$\"3G)4
,C`=Z'>F-7$$!3#3!z4WyyS6F*$\"3o=/*[@A2u\"F-7$$!3TM_NslQF6F*$\"3Qa!RRfA
Bd\"F-7$$!3+oDh+`)R6\"F*$\"3gB/'pXapY\"F-7$$!3w,*p)GSe+6F*$\"3*)QcDs(p
#H9F-7$$!3i'=_G]6s4\"F*$\"3IJ(zBG52V\"F-7$$!3mrW$o(*QQ4\"F*$\"3*\\**4a
#QbO9F-7$$!3qcn\"3Xm/4\"F*$\"3_u>$e+%yY9F-7$$!3wT!*zCR4(3\"F*$\"35d\"G
.7s8Y\"F-7$$!3#=hjF()[.3\"F*$\"34a-x;jW.:F-7$$!3!>=G2#Qgt5F*$\"3k#fJua
/Bc\"F-7$$!31Atl;P6g5F*$\"3oL$[-(=bF<F-7$$!3Aike7OiY5F*$\"3mV#epz@(\\>
F-7$$!3eUaiCw>L5F*$\"3Q'*G/'[.v@#F-7$$!3%HUkmjr(>5F*$\"3GW6M7#f*>DF-7$
$!3@8RoU'eI,\"F*$\"3-&*)3+rX+o#F-7$$!3[.Mq[cM15F*$\"3#\\\"3tPloVGF-7$$
!3wN*GsaEj***F-$\"3a,66JQ14IF-7$$!3ZQQU2m>H**F-$\"3)4r&eN/MuJF-7$$!3))
3Vc70s/)*F-$\"3KIJX$=7VZ$F-7$$!31\"y/xTW-o*F-$\"3!R)='fO3iv$F-7$$!3C`_
%GKodb*F-$\"3Km/4+iG4SF-7$$!3kBd)zA#HJ%*F-$\"3&)\\\"*RT6!RA%F-7$$!3Uza
pTWS-$*F-$\"3?MM!)*R')oR%F-7$$!3'pB0al;N<*F-$\"3#yg]!>0\"G^%F-7$$!3v:,
EiF24\"*F-$\"3=(=K0#yzZXF-7$$!3`%*\\6p)GY/*F-$\"3Au8[P\"Qpc%F-7$$!3Kt)
pf(\\=!)*)F-$\"3C\\q`.b.qXF-7$$!33_Z#G3Td\"*)F-$\"3C?Rg1#eqb%F-7$$!3#*
)\\n)\\^W#y)F-$\"3o,n^6['*zWF-7$$!3_Z-\"p@\\\"\\')F-$\"3spJ\\,gAQVF-7$
$!37'*H&RG`e^)F-$\"3X6#=1)H.QTF-7$$!3%Hu&*4NdDQ)F-$\"3MXyqa-8))QF-7$$!
3Hq3RDJ7;$)F-$\"3Og91V.M[PF-7$$!3%e*fy**))o\\#)F-$\"3'3]%))z%3/g$F-7$$
!3S@6=uYD$=)F-$\"3?**zalT%fW$F-7$$!3u[id[/#o6)F-$\"39emctzi'G$F-7$$!33
w8(HA'Q]!)F-$\"3/+(RfE$>CJF-7$$!3k,lO(*>&R)zF-$\"3W#R_TL2/'HF-7$$!3'*G
;wrx^<zF-$\"315W-r@0(z#F-7$$!3Sbn:YN3^yF-$\"3/`ug#30fj#F-7$$!3q>yJzxu#
y(F-$\"3w]]S3dHuCF-7$$!33$))yC,7Wr(F-$\"38Wd9(zP(=BF-7$$!3OZ*RcCwgk(F-
$\"3?dw')**>-r@F-7$$!3k65!)y/uxvF-$\"3\"H#*Gep[G.#F-7$$!3@SJ7X*o5W(F-$
\"37GK3]NO\"z\"F-7$$!3xo_W6uR/tF-$\"3G:36_%p`g\"F-7$$!3'H$**=]q,%=(F-$
\"3m^8'e[pV\\\"F-7$$!3=(fM*)oOO1(F-$\"3Q&[CS)[/P9F-7$$!3FHpI3lW.qF-$\"
3!p\\YApE#H9F-7$$!3Ph#zwKcK%pF-$\"37q\">[DRaV\"F-7$$!3[$f^q9mI)oF-$\"3
)\\?K(yqib9F-7$$!3ZERUmf(G#oF-$\"33@!R&Q)4'*[\"F-7$$!3YN9m\">cto'F-$\"
3vn<$)>'[[h\"F-7$$!3[W*)*oTO=b'F-$\"3v(Q))=J>E!=F-7$$!3Z`k8UmJ;kF-$\"3
dgWD3^WW?F-7$$!3[iRPnoz!G'F-$\"3tdM*R\"*3%HBF-7$$!3(H*[Q^(eF@'F-$\"3W$
=*oalv%[#F-7$$!3OCeRN1sWhF-$\"3GydnNP)fk#F-7$$!3ubnS>DowgF-$\"3uCI8j1D
6GF-7$$!38(o<MSW'3gF-$\"3[a0$\\Wr'yHF-7$$!3^='GuG11%fF-$\"33L#**3hNj9$
F-7$$!3!*\\&R9<oD(eF-$\"3Lrrc8*HBJ$F-7$$!3G\"[]a0IX!eF-$\"3qi!)eg,wuMF
-7$$!3n79YR>\\OdF-$\"3%)))ysHGxJOF-7$$!3eXyjvsN0cF-$\"36e7B?Vc7RF-7$$!
3]yU\"=hAUZ&F-$\"3y?\\AnAxaTF-7$$!3S62*z%z3V`F-$\"39uAqKg:[VF-7$$!3KWr
;%G`>@&F-$\"3kH')Q#GRX[%F-7$$!3)**eG`'=T_^F-$\"38-&[')*[*f_%F-7$$!3]O+
\\Y/(G4&F-$\"3y'*=[\"y6Tb%F-7$$!3Mf22P(*4j]F-$\"3Q4f^un3jXF-7$$!30$[^w
-HL.&F-$\"3iXX*4;W'oXF-7$$!3(e?K#=$eN+&F-$\"3-@i)**yr2d%F-7$$!3qGH\")3
wyt\\F-$\"3?$oU&4]YpXF-7$$!3+?e8rZqa[F-$\"3`$=)))GL1IXF-7$$!3>7(eM$>iN
ZF-$\"3O[ziq8GPWF-7$$!3#Q(>_@4-%f%F-$\"3[Gp=MN2iUF-7$$!3OO_e4*>CX%F-$
\"3MCYGENoCSF-7$$!3knoh.%>;Q%F-$\"3](ROhU?i)QF-7$$!3!*)\\[w*)=3J%F-$\"
3oOtd;v!ot$F-7$$!3<I,o\"R=+C%F-$\"3El>&y)3HyNF-7$$!3Wh<r&)y@pTF-$\"3k1
rO7\"HET$F-7$$!3MIgwaKE\\SF-$\"3)4\"*f]$4V@JF-7$$!35+.#Qi3$HRF-$\"3$4O
q)[U$f#GF-7$$!3WpX(G*RN4QF-$\"3f9Gn!G*fODF-7$$!3wQ)G>O*R*o$F-$\"3cLU%*
R\"oOE#F-7$$!3E\"\\O%4O')\\NF-$\"3u/71:Nfz>F-7$$!3vVT%p&yK5MF-$\"3$Ri/
^:WVu\"F-7$$!3C'z^W5#zqKF-$\"3Aldmok<p:F-7$$!3u[%f>Nc78$F-$\"33$H`g-sC
Y\"F-7$$!3euY'e:y.5$F-$\"3i`BO>!*o[9F-7$$!3U+*p(f**\\pIF-$\"3Zh&fO([bQ
9F-7$$!3EE^nj<iQIF-$\"3`bbhDM4K9F-7$$!36_.enNu2IF-$\"3?1dHt)>$H9F-7$$!
3'zd&[r`'o(HF-$\"3eGDTS2CI9F-7$$!3!Q!3Rvr)f%HF-$\"3Ii6)4'Q&[V\"F-7$$!3
jHgHz*3^\"HF-$\"3$e,9WQ[JW\"F-7$$!3[b7?$yIU)GF-$\"3w\"fU5![5b9F-7$$!3'
)e@#))*zrgFF-$\"3h541u#f)Q:F-7$$!3AiIW9_?PEF-$\"3'*f!e#\\0Wx;F-7$$!3%)
\\fD9@p,DF-$\"3#zzON$4L')=F-7$$!3YP)oS,zhO#F-$\"3Wp6nsB\\X@F-7$$!3E\"G
vRYA%)H#F-$\"3s;r)GuV.H#F-7$$!33D<)Q\"fmIAF-$\"3[UPd6jAVCF-7$$!3))o\")
yj$4H;#F-$\"3ZiAp-*4Cg#F-7$$!3q7Yp8G:&4#F-$\"3%\\!yOJI4mFF-7$$!3R?s(\\
bK1.#F-$\"3gI%RGOYW#HF-7$$!3kF)fiH7h'>F-$\"35%zQg2vN3$F-7$$!36NCaP?f,>
F-$\"3PAS;'3Y=C$F-7$$!3eU]#)y<2P=F-$\"3'eCL26NwR$F-7$$!3_d-Rh7.3<F-$\"
3?]tFSQT&p$F-7$$!3Ysa&Ru!**y:F-$\"3ook38'=Z'RF-7$$!3L#\\^7]]VW\"F-$\"3
v!o5'z([M?%F-7$$!3A7vae-r48F-$\"3)ylZ')=]&)Q%F-7$$!35KN%e,q]<\"F-$\"3w
H*)pZUx6XF-7$$!3)>bRJxH//\"F-$\"3q*)zLx(Hwc%F-7$$!3$RQ^EFi9$z!#=$\"3!e
?>-#Qg)[%F-7$$!3-[s!R\"oieaFacm$\"3%HG&3zyy(=%F-7$$!3.6haeK)\\7%Facm$
\"3[(*)pI;g![RF-7$$!3/u\\=.(R8z#Facm$\"3=n)))fXyom$F-7$$!3b0W]Dz^C@Fac
m$\"3YiY<**y_9NF-7$$!3/PQ#y9'pd9Facm$\"3-H'3fWPlN$F-7$$!3d&oK9qV(3z!#>
$\"3'R(f\\x)QY>$F-7$$!3j+qiCf_S7F`em$\"3fg!yK\"ogIIF-7$$\"3#Gf4qFyeo&F
`em$\"3SCL`?I*)fGF-7$$\"3i=Y'yCG7E\"Facm$\"3%)4FJ\"4O3p#F-7$$\"3)zFG!o
'oQ&>Facm$\"3y;tkU_VDDF-7$$\"3IP>>)34lk#Facm$\"3YRb%eXYcO#F-7$$\"3*fD>
&G**yJSFacm$\"3-H/;9[Pq?F-7$$\"3oul%)o22<aFacm$\"3<\\zD_q%*==F-7$$\"3c
#HAk&G&)GyFacm$\"3YJO$Hgr'>:F-7$$\"3/,)*R%\\jS-\"F-$\"3jkVbzcKI9F-7$$
\"3!zNv2..V:\"F-$\"3%[R4EL=^Z\"F-7$$\"3w94:nDa%G\"F-$\"3U1KT3%4Ne\"F-7
$$\"3grk_.@y99F-$\"3?wfNF#y4v\"F-7$$\"3XG?!*R;-X:F-$\"3-(3.t9S0(>F-7$$
\"3]#yD\\#=dz;F-$\"3b4]NpuLUAF-7$$\"3aO&\\*4?79=F-$\"39eB.*=`ya#F-7$$
\"3c89Y-rR\")=F-$\"3CA![z(*[!4FF-7$$\"3e!Ht\\>s'[>F-$\"3#RH$*H)3\\tGF-
7$$\"3$y;&[(GZf,#F-$\"33*RP'RUMRIF-7$$\"3iWq**zBA$3#F-$\"3QgD!fjeZ?$F-
7$$\"3?#)32Lz.\\@F-$\"3Ej2j&['QkLF-7$$\"3x>Z9'[`[@#F-$\"3N7nsOK7?NF-7$
$\"3Md&=#R!p1G#F-$\"3a\"*GMbeIqOF-7$$\"3#\\R#H#f%[YBF-$\"3Mc!)R<1L8QF-
7$$\"33q+W)p:\"yCF-$\"3;w0Eg,*=2%F-7$$\"3AXxe/ou4EF-$\"31U$oNC(y%G%F-7
$$\"3z%*48om5PFF-$\"33]>HtavQWF-7$$\"3OWUnJlYkGF-$\"3)y!>O7GLNXF-7$$\"
3uc+c(\\1j*GF-$\"3c0J)>Z0+b%F-7$$\"3:peWjk9GHF-$\"3iZ0-S<!3c%F-7$$\"3a
\"oJ$Hk)*fHF-$\"359m?;YpnXF-7$$\"3%R\\<_RE=*HF-$\"3)**HBA'omqXF-7$$\"3
L1L5hjmBIF-$\"37NL\"[/6(pXF-7$$\"3s=\"*)pK1b0$F-$\"3UM*pVbH[c%F-7$$\"3
7J\\(GHYt3$F-$\"37!pe*)fMgb%F-7$$\"3^V2wei=>JF-$\"3kclyt\"[La%F-7$$\"3
oAK'Hn(fgKF-$\"3i(fMhG>5W%F-7$$\"3$=qlr34?S$F-$\"3'3?il\"3)yE%F-7$$\"3
*4=o8]?Ma$F-$\"3UaAXO0WKSF-7$$\"39g1d:>$[o$F-$\"3c_T6PwEYPF-7$$\"3R<$e
2O'*=\"QF-$\"3)RIid=.uX$F-7$$\"3kuf%f!3'*QRF-$\"3/M![P_x.:$F-7$$\"3!>j
L6DDg1%F-$\"3O3z];6QPGF-7$$\"37*G@jp*3$>%F-$\"3BP=9)\\T3`#F-7$$\"3s$*=
#H6e(GVF-$\"39*o)\\7&)RCAF-7$$\"3I)\\A&HlUkWF-$\"3AP]&p>YI&>F-7$$\"3)G
5Bh%\\4+YF-$\"3#\\x/0Rh!H<F-7$$\"3Y2PsiLwNZF-$\"3y`j'e(ydi:F-7$$\"3s;`
X7\"*pe[F-$\"3kS\\yLptn9F-7$$\"3)f#p=i[j\")\\F-$\"33P6(oBd)H9F-7$$\"3^
G)>'*zoB,&F-$\"3!=r-m8+&H9F-7$$\"3/JF0PF5V]F-$\"3BjQT.E!GV\"F-7$$\"3eL
c[um$Q2&F-$\"3lN5#>%pvR9F-7$$\"3BN&=>hqX5&F-$\"3=dk<ZpM]9F-7$$\"3SRVy'
[Qg;&F-$\"3)GrCm$oK#[\"F-7$$\"3[W,lhj]F_F-$\"3$R`-#oVWG:F-7$$\"3i#[$p2
w!>O&F-$\"3'Q5%y$zgin\"F-7$$\"3w?ot`)3j\\&F-$\"33oV0pz&G)=F-7$$\"3\"*e
,y*452j&F-$\"3P:2CH>1R@F-7$$\"3/(\\BeM6^w&F-$\"3IS_:;f\\LCF-7$$\"3q[i@
WCd\"*eF-$\"32z%RxHdPt#F-7$$\"3O+!4EaL!=gF-$\"3/j!oB.!\\WIF-7$$\"3-_<+
TY\\WhF-$\"3%\\]<%pHZ`LF-7$$\"3o.XRRd&4F'F-$\"3ucU)zx`&[OF-7$$\"3'\\8C
`-WJS'F-$\"3<SqVPcbHRF-7$$\"3CmPD6BLNlF-$\"3q>ryTPiqTF-7$$\"3](R$=(f?v
m'F-$\"3#GhAOz,9O%F-7$$\"3yGI6$))3(*z'F-$\"3=/yheRp$\\%F-7$$\"3#fdq())
zKkoF-$\"3YRp8[0ENXF-7$$\"3&R7GW4Z*GpF-$\"3)HlZ$**H-hXF-7$$\"3)>n&3+ic
$*pF-$\"3#p#yy5hrqXF-7$$\"37>Ku0`=eqF-$\"39,)=05SUc%F-7$$\"3G9$eq^Bu=(
F-$\"3h`?HMJ@.XF-7$$\"3X4MPG<m;tF-$\"3C(o7l][/Q%F-7$$\"3IiiWe@'>X(F-$
\"3#R)prqHH\">%F-7$$\"3;:\">&)eise(F-$\"3WZok_/`[RF-7$$\"3gTbb.G\"\\l(
F-$\"3FDeT3*y.\"QF-7$$\"3.o>f=IcAxF-$\"3%eq\"z?]3jOF-7$$\"3Y%RGOB8-z(F
-$\"3Cl]`&\\5$3NF-7$$\"3!4#[m[M'y&yF-$\"3)p)frJ9!yM$F-7$$\"3zzO;Y'=I#z
F-$\"3+&41zJ\"[*=$F-7$$\"3!y`iO%Q<))zF-$\"3'z`GH>y\"HIF-7$$\"3!eRh6/HL
0)F-$\"31@)HWqp&oGF-7$$\"3ra-mQU[=\")F-$\"3F;%HepO$4FF-7$$\"3]szlLYz[#
)F-$\"3=ADhw5j,CF-7$$\"3_)ob'G]5z$)F-$\"387E(*p(3*=@F-7$$\"37J#)oGiO7&
)F-$\"3y'yQ(3Y\"z'=F-7$$\"3st2sGuiX')F-$\"3_$e-ooWjm\"F-7$$\"3K;LvG'))
)y()F-$\"3C8*QC3**H_\"F-7$$\"3$*eeyG)\\@\"*)F-$\"3C*4&RMf8W9F-7$$\"3S'
R*Gh#*=X*)F-$\"3r>n&p#>-N9F-7$$\"35KHz$pG#y*)F-$\"3.Zq\"[7A,V\"F-7$$\"
3ynkHE\"o7,*F-$\"3pZXEA(\\%H9F-7$$\"3[.+!)evIW!*F-$\"396(e*Hl+L9F-7$$
\"3kwq!QU'Q5\"*F-$\"3!zh#*>5eFX\"F-7$$\"3!)\\T\"))Glk<*F-$\"3Y%[xVBk\"
*[\"F-7$$\"38'HG)=Ii3$*F-$\"3^tp:@w>5;F-7$$\"3nSC%)[2yS%*F-$\"3;\"ee][
44z\"F-7$$\"3?kXNx*>Ac*F-$\"3QF2?hL'H+#F-7$$\"3q(ome?fOo*F-$\"3?&Q(R!
\\(=^AF-7$$\"3B6)yVV)40)*F-$\"3Sm,(z&odEDF-7$$\"3vM4*GmPl#**F-$\"3H#))
>`&49>GF-7$$\"3oOL.$>Hh***F-$\"3Q^&zaA\\/*HF-7$$\"3ptvJs?d15F*$\"3!*)3
](4:(=;$F-7$$\"3q8=LD7`85F*$\"3H\\>QB9OJLF-7$$\"3s`gMy.\\?5F*$\"3]R@l
\\c*o\\$F-7$$\"3vLXP%o3W.\"F*$\"3F*>->Zj#3QF-7$$\"3'R,./*pK[5F*$\"3?EL
c38:\"3%F-7$$\"3?<iB*ov21\"F*$\"3!z90%\\h&>G%F-7$$\"3g?%p!)QCK2\"F*$\"
34&owHR)*QV%F-7$$\"3-CE!p3tc3\"F*$\"3qjfEhm=JXF-7$$\"3CFet&y@\")4\"F*$
\"3MrEG[F6qXF-7$$\"3')**z38+W,6F*$\"3y>!y_\\%RqXF-7$$\"3Is,WS#eZ5\"F*$
\"3u`Pp:3TmXF-7$$\"3#\\M#znk236F*$\"39)3+)HD<eXF-7$$\"3a<X9&p%R66F*$\"
3u.],9?qXXF-7$$\"3yi)[)\\6.=6F*$\"3ou>H$=6#3XF-7$$\"3&y?`XgnY7\"F*$\"3
OVdxp`MaWF-7$$\"3:)*='R^Sz8\"F*$\"3-%4xPc****H%F-7$$\"3Y)eqLU87:\"F*$
\"3)pXD#o%\\$*3%F-7$$\"3m%yf7H<R;\"F*$\"3wx58=rRVQF-7$$\"3-\")*[\"f6iw
6F*$\"3;D3X)=rRc$F-7$$\"3Sx\"Qq-D$*=\"F*$\"3_>n/7=;iKF-7$$\"3ftt#\\*)G
??\"F*$\"3C-'o\")\\Z*\\HF-7$$\"3)>XEa\"\\&)37F*$\"3=oY**Qv@#y#F-7$$\"3
>Ib#f$4o:7F*$\"3-Ifbr*))ph#F-7$$\"3T3YUcp]A7F*$\"3'HhoucfhX#F-7$$\"3!o
oBp(HLH7F*$\"3:PCsqld,BF-7$$\"3SV=#z,&)HC\"F*$\"3L:WQf'f\"=?F-7$$\"3++
+#*eqjc7F*$\"3IMxDb=sz<F--%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F`^pF_^
p-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+iqjc7F^^p$\"+iqjc7F^^p%(DEF
AULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+
1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "2)  Suppose an object moves
 on a line with position function given by: " }{XPPEDIT 18 0 "s(t) = t
/2 - sin(t)" "6#/-%\"sG6#%\"tG,&*&F'\"\"\"\"\"#!\"\"F*-%$sinG6#F'F," }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 86 " Find where the instant
aneous velocity equals the average velocity on the interval [0," }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 4 " ]. " }}{PARA 0 "" 0 "" 
{TEXT -1 17 " Plot a picture. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 
"First let's plot the position function. " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "s:= t -> t/2 - sin(t);" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGR6#%\"tG6
\"6$%)operatorG%&arrowGF(,&9$#\"\"\"\"\"#-%$sinG6#F-!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(s(t), t = 0..Pi, color \+
= red);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6&-%'CURVESG6#7S7$$\"\"!F)F(7$$\"3%)eD2LzxZo!#>$!3LNRJZ%Q
&=MF-7$$\"3)\\$px*G*f!G\"!#=$!3)>_Zx[B!ojF-7$$\"3+5@exGm]>F3$!3X!)p#3@
U)H'*F-7$$\"3[99!=3o^i#F3$!3!3o7A\\NDG\"F37$$\"35!\\D0[nkH$F3$!3!G`r&G
W&))e\"F37$$\"37\"=Za&z%)=RF3$!3+<<$\\0'))f=F37$$\"3edXa()oGjXF3$!3H\"
)z([!4\"\\7#F37$$\"3W%3**Hbm(H_F3$!39-xVO!G(zBF37$$\"37PRr4)3T*eF3$!3-
T.WH'f;h#F37$$\"3y\"[)yykYxlF3$!3<K#3HU=Y#GF37$$\"3[s'ocGo$zrF3$!3Q#f<
H^P'))HF37$$\"3UQ0;gr'p&yF3$!3=<.V-]pWJF37$$\"3vFMt?$[t`)F3$!3OX:1)pE(
oKF37$$\"3u\"p30k@I>*F3$!3OP=U7eFbLF37$$\"3#*R7\\HeV)y*F3$!3Co+yM(3VS$
F37$$\"3#G[))*)3W'\\5!#<$!3L&f\"\\^.CCMF37$$\"30'[@XS@'46Fdp$!3T:M\"Rs
&>2MF37$$\"3G9w$3G*Qz6Fdp$!3q%3_[h&yYLF37$$\"3>%3*3tc9T7Fdp$!3'*e#Q[_E
eD$F37$$\"3Ey0DBA!*38Fdp$!3n(*R#4'>]9JF37$$\"3qjE.#[AMP\"Fdp$!3\"fdC7$
ptQHF37$$\"3*R:_MgU2W\"Fdp$!3%f<$yhi$=r#F37$$\"3\"Qu#)**[jD]\"Fdp$!3)G
KT,)H\"RY#F37$$\"3=hw\"ymX#p:Fdp$!3E&y(>Q'fP:#F37$$\"3mwM!*4(4&Q;Fdp$!
3#=*\\`(pMXy\"F37$$\"3CDL:iU!))p\"Fdp$!3BUiTY.;C9F37$$\"3o(R1/.CRw\"Fd
p$!3z5*H!*HsY%**F-7$$\"32Id)H7*>J=Fdp$!3mG\\:=(y'o]F-7$$\"3Gfb7wY,(*=F
dp$\"3I4IsJXMY7!#?7$$\"3cM5'zg%pg>Fdp$\"3\"fZ#few&*RbF-7$$\"3**oJ8:.SJ
?Fdp$\"3s!*)\\%H'f\"*>\"F37$$\"3)**p!zPD$\\4#Fdp$\"3u0tmyq4<=F37$$\"3k
<!fhumF;#Fdp$\"3CA#y)3dS:DF37$$\"3wak3@YBCAFdp$\"3AU^_C->\"=$F37$$\"39
/b<W_V\"H#Fdp$\"34I'\\pQLL%RF37$$\"3#*z_V$zlYN#Fdp$\"3ooVK.*f9p%F37$$
\"3cmjYO*f2U#Fdp$\"3c#RWW$[p.bF37$$\"3)eq)=U!z`[#Fdp$\"3#=')[fu#oDjF37
$$\"3#Q'HHd#HIb#Fdp$\"3t,/(*=\"zM@(F37$$\"3z^r&[X%==EFdp$\"3E_\\94Gd#4
)F37$$\"3'=BS\\0:[o#Fdp$\"3GoPL7;\\8!*F37$$\"3[gN,?R*3v#Fdp$\"3W\"\\0t
sBh%**F37$$\"3@/0LMNh6GFdp$\"3O,f!H\\$y\"3\"Fdp7$$\"3w#oUX107)GFdp$\"3
[*zFvVZJ=\"Fdp7$$\"3W46xe&[M%HFdp$\"336GL+U([F\"Fdp7$$\"3H.6)e58)4IFdp
$\"3I\">=[33NP\"Fdp7$$\"3Kt2RXCLtIFdp$\"31MxQ6!f%o9Fdp7$$\"3!)***\\/l#
fTJFdp$\"3O1_3Ajzq:Fdp-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%+AXESLABEL
SG6$Q\"t6\"Q!6\"-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG" 1 2 0 1 10 0 2 
6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 80 "From the plot, we see that the object first moves
 to the left of the origin and " }}{PARA 0 "" 0 "" {TEXT -1 38 "then c
ontinuously moves to the right. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 54 "The average velocity of the object on an \+
interval [0, " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 15 "] is given
 by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "
         ( s(" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 15 ") - s(0) )
 / ( " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 9 " - 0 ) . " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The instantaneo
usl velocity at time c is given by s '(c). Thus, " }}{PARA 0 "" 0 "" 
{TEXT -1 23 "we are asked to solve: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 13 "         ( s(" }{XPPEDIT 18 0 "Pi;" "6#
%#PiG" }{TEXT -1 15 ") - s(0) ) / ( " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 17 " - 0 ) = s '(c); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 19 "precisely the MVT. " }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s(Pi);" }
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"#" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s(0);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "D(s);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#R6#%\"tG6\"6$%)operatorG%&arrowGF&,&#\"\"\"\"\"#F,-%$cosG6#9$!\"\"F&
F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "We need to solve: " }
{XPPEDIT 18 0 "1/2 = 1/2 - cos(t)" "6#/*&\"\"\"F%\"\"#!\"\",&*&F%F%F&F
'F%-%$cosG6#%\"tGF'" }{TEXT -1 14 " for t in [0, " }{XPPEDIT 18 0 "Pi;
" "6#%#PiG" }{TEXT -1 30 " ], i.e.,  we want cos(t) = 0 " }}{PARA 0 "
" 0 "" {TEXT -1 13 "for t in [0, " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 19 "]. Solution is t = " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 46 " /2. Let's plot a figure with the secant line " }}{PARA 
0 "" 0 "" {TEXT -1 32 "between the points (0, 0) and ( " }{XPPEDIT 18 
0 "Pi;" "6#%#PiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 41 " /2 ), and with the tangent line for t = " }{XPPEDIT 18 
0 "Pi;" "6#%#PiG" }{TEXT -1 6 " /2 . " }}{PARA 0 "" 0 "" {TEXT -1 32 "
These lines should be parallel. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "D(s)(Pi/2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "s(Pi/2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#PiG#
\"\"\"\"\"%F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(
plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords
 has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "da
ta:= ([0,0], [Pi, Pi/2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
48 "a1:= plot( [data], color=magenta, thickness=2 ):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 55 "a2:= plot(s(t), t = 0..5, color = green, \+
thickness=2): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "a3:= plot
((Pi/4 - 1) + 1/2 * ( x - Pi/2), x = 0..Pi, color = magenta, thickness
=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a4:= textplot([1/5,
1,`Secant Line`], align=\{RIGHT\}, color = blue, thickness=2):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a5:= textplot([1.5,-.8, `Tan
gent Line`], color = blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "display(\{a1,a2,a3,a4,a5\});" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3G
LLL3x&)*3\"!#=$!3zg$4m)fsFa!#>7$$\"3umm\"H2P\"Q?F-$!3#Hv`Q5()\\+\"F-7$
$\"3MLL$eRwX5$F-$!3V!e)H-hl-:F-7$$\"33ML$3x%3yTF-$!3g)H?f\"=ao>F-7$$\"
3emm\"z%4\\Y_F-$!3Ax)yuzZeQ#F-7$$\"3`LLeR-/PiF-$!3!R038\"y$>s#F-7$$\"3
]***\\il'pisF-$!38.)z2>/&4IF-7$$\"3>MLe*)>VB$)F-$!36n]ED6RLKF-7$$\"3Y+
+DJbw!Q*F-$!3e[goZ!RQP$F-7$$\"3%ommTIOo/\"!#<$!3\\R;#orlUU$F-7$$\"3YLL
3_>jU6Ffn$!3!yM;SgMTQ$F-7$$\"37++]i^Z]7Ffn$!3]@QJ.w'*QKF-7$$\"33++](=h
(e8Ffn$!3=9$Q$40C#)HF-7$$\"3/++]P[6j9Ffn$!3i22W'H0li#F-7$$\"3UL$e*[z(y
b\"Ffn$!3.uHQWex4AF-7$$\"3wmm;a/cq;Ffn$!3a$pe[euuf\"F-7$$\"3%ommmJ<gw
\"Ffn$!3U/*R>o;'*z*F07$$\"3/+]iSj0x=Ffn$!3A)\\E@;lR\\\"F07$$\"3gmmm\"p
W`(>Ffn$\"3]\\qop*3#RoF07$$\"3K+]i!f#=$3#Ffn$\"3ca%=3\">9+<F-7$$\"3?+]
(=xpe=#Ffn$\"3?iV/g\"Q?w#F-7$$\"37nm\"H28IH#Ffn$\"3)zAoqTV;'RF-7$$\"3u
m;zpSS\"R#Ffn$\"3joAGBIDR^F-7$$\"3GLL3_?`(\\#Ffn$\"3IW5/e_=$['F-7$$\"3
fL$e*)>pxg#Ffn$\"37qTkx0c]zF-7$$\"33+]Pf4t.FFfn$\"3Km'\\jA;'y#*F-7$$\"
3uLLe*Gst!GFfn$\"3s&[J5P`c2\"Ffn7$$\"30+++DRW9HFfn$\"3n3x0\\:-K7Ffn7$$
\"3:++DJE>>IFfn$\"39'z&\\#p,vQ\"Ffn7$$\"3F+]i!RU07$Ffn$\"3&fzAp[A#R:Ff
n7$$\"3+++v=S2LKFfn$\"3dem4H3*yq\"Ffn7$$\"3Jmmm\"p)=MLFfn$\"333O$>#>]e
=Ffn7$$\"3B++](=]@W$Ffn$\"3sT<%p)y7<?Ffn7$$\"35L$e*[$z*RNFfn$\"3Q'Q&oe
:#z:#Ffn7$$\"3e++]iC$pk$Ffn$\"3n()*>d.rvI#Ffn7$$\"3[m;H2qcZPFfn$\"3dL,
w<nMVCFfn7$$\"3O+]7.\"fF&QFfn$\"3srszpu4zDFfn7$$\"3Ymm;/OgbRFfn$\"3s.s
6(o[[q#Ffn7$$\"3w**\\ilAFjSFfn$\"3tfC'H<a#GGFfn7$$\"3yLLL$)*pp;%Ffn$\"
3))=)R<*\\RQHFfn7$$\"3)RL$3xe,tUFfn$\"3Pc^+Wk_TIFfn7$$\"3Cn;HdO=yVFfn$
\"3^'3:`5jP8$Ffn7$$\"3a+++D>#[Z%Ffn$\"3l'eE$zVK4KFfn7$$\"3SnmT&G!e&e%F
fn$\"3a%er&)ogZG$Ffn7$$\"3#RLLL)Qk%o%Ffn$\"3ee::rq$>M$Ffn7$$\"37+]iSjE
!z%Ffn$\"3_C2+hA5#R$Ffn7$$\"3a+]P40O\"*[Ffn$\"3=Kx$Qb2(HMFfn7$$\"\"&F)
$\"3cQJmuU#*eMFfn-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-%*THICKNESSG6#\"
\"#-F$6%7S7$F($!\"\"F)7$$\"3%)eD2LzxZoF0$!3'=PYL56wl*F-7$$\"3)\\$px*G*
f!G\"F-$!33K:6b.qf$*F-7$$\"3+5@exGm]>F-$!3,X*37coY-*F-7$$\"3[99!=3o^i#
F-$!3w#H*4ffT(o)F-7$$\"35!\\D0[nkH$F-$!3Sastfiw^$)F-7$$\"37\"=Za&z%)=R
F-$!3))3kFAgdS!)F-7$$\"3edXa()oGjXF-$!3@@xAclN=xF-7$$\"3W%3**Hbm(H_F-$
!3yd/]Bn6&Q(F-7$$\"37PRr4)3T*eF-$!3+KI9&fXH0(F-7$$\"3y\"[)yykYxlF-$!3b
edggnE6nF-7$$\"3[s'ocGo$zrF-$!3xjc;deJ5kF-7$$\"3UQ0;gr'p&yF-$!3MJ(>*>k
^rgF-7$$\"3vFMt?$[t`)F-$!3d&GL'ReKJdF-7$$\"3u\"p30k@I>*F-$!38acuz\"*[.
aF-7$$\"3#*R7\\HeV)y*F-$!3/!Qa_3#y0^F-7$$\"3#G[))*)3W'\\5Ffn$!3'eed]bz
<v%F-7$$\"30'[@XS@'46Ffn$!3upDRxH*=X%F-7$$\"3G9w$3G*Qz6Ffn$!3_G>\"ef`I
5%F-7$$\"3>%3*3tc9T7Ffn$!3/zXbM;F%z$F-7$$\"3Ey0DBA!*38Ffn$!3s3ru$)))[b
MF-7$$\"3qjE.#[AMP\"Ffn$!3[\"oO)*e()G8$F-7$$\"3*R:_MgU2W\"Ffn$!3/I#RF)
pG'z#F-7$$\"3\"Qu#)**[jD]\"Ffn$!3)4G'3]D=([#F-7$$\"3=hw\"ymX#p:Ffn$!39
%p64mrP:#F-7$$\"3mwM!*4(4&Q;Ffn$!3o;E[]9X2=F-7$$\"3CDL:iU!))p\"Ffn$!3v
tLB*oyf]\"F-7$$\"3o(R1/.CRw\"Ffn$!3k6!oz%)z.=\"F-7$$\"32Id)H7*>J=Ffn$!
3s'\\82&Q/S%)F07$$\"3Gfb7wY,(*=Ffn$!3?N?s$>m#\\^F07$$\"3cM5'zg%pg>Ffn$
!3%=F[>gp_'>F07$$\"3**oJ8:.SJ?Ffn$\"3S\\%emv:+d\"F07$$\"3)**p!zPD$\\4#
Ffn$\"3Y**\\`*)oiYZF07$$\"3k<!fhumF;#Ffn$\"3^#)3&zIP$Q\")F07$$\"3wak3@
YBCAFfn$\"3xtAV0J<@6F-7$$\"39/b<W_V\"H#Ffn$\"3u?v(3AwrX\"F-7$$\"3#*z_V
$zlYN#Ffn$\"3k*Rwr'*GLx\"F-7$$\"3cmjYO*f2U#Ffn$\"3zK=L#o*z.@F-7$$\"3)e
q)=U!z`[#Ffn$\"3MHN%4@&*oU#F-7$$\"3#Q'HHd#HIb#Ffn$\"33>[Y'GY^w#F-7$$\"
3z^r&[X%==EFfn$\"3%*edGuA#44$F-7$$\"3'=BS\\0:[o#Ffn$\"3Nf6qu_2CMF-7$$
\"3[gN,?R*3v#Ffn$\"3S-y1+'pWv$F-7$$\"3@/0LMNh6GFfn$\"3-@Dlrw1eSF-7$$\"
3w#oUX107)GFfn$\"3&QT8FKDgS%F-7$$\"3W46xe&[M%HFfn$\"3BZb&QzUsr%F-7$$\"
3H.6)e58)4IFfn$\"3Y;bSHb1\\]F-7$$\"3Kt2RXCLtIFfn$\"3kmQ&pAimO&F-7$$\"3
!)***\\/l#fTJFfn$\"3,***\\ADjzq&F--Fhz6&FjzF[[lF(F[[lF^[l-%%TEXTG6'7$$
\"+++++?!#5$\"\"\"F)Q,Secant~Line6\"%+ALIGNRIGHTG-Fhz6&FjzF(F(F[[lF^[l
-F$6%7$F'7$$\"37$z*e`EfTJFfn$\"3c'*[zEjzq:FfnFhjlF^[l-F[[m6%7$$\"#:Fg[
l$F][lFg[lQ-Tangent~LineFd[mFf[m-%+AXESLABELSG6%Q\"tFd[mQ!6\"%(DEFAULT
G-%%VIEWG6$;F(FczF]]m" 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" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "30 0 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }