00020001010008main.ACT0002020007CMV.EAC010000002a9f  CMV.EAC main.ACT$'+~E '=[485Geometric Interpretation of Cauchy Mean Value Theorem\= AuthorTuProfessor Wei-Chi Yang Radford University 9 e- l: wyang@r2.eduE!URL: http://www.#/5C Objectns:[T(1) would like to g a g#i#how'() ?Theorem is stated.[9(2) We would briefly describe how CMV8 proved and usAI[Tm.bN (B!) Suppose the function f:[a,b]eRYg re continuouswthat e]G ir restriLs to (N)<differeAable. Mr, assumat g(x)0 for all x in 0N. =nre@ a point t)pt whichT   f(b)-f(a) gg(a)=f(t) g .[# - Remark: [>P(1) Simply put, if we apthe Mean Value Theorem ongraph of a parametric X Xequation,FnT will obtai@ CMV. In o!r words, assume func> s f and g sNsfy VcondidCauchyholkcbe interpreted Ias any numbt fohichg curve P definAb5WP(t)=[g,f $] for atb has slope equal to theof ecant that runs from 0.poi (g(aca))Gbb)). [9 T(2) We use+following example demonstrat% motivationproCMV.JTtechniqubd here be appliarbi_ryse w'K holds[ An E:3cider>parametric curve N~=Y[x(t),y]=[g f t*cost*sin%*] with t in [0,2] We let A=(-,0), B=(0 Xand C /2). FiE] on the curve of P whereslop%tangent _!line is same as0AC. [[WKey: claim tha#f we rotate AC to AB (a horizontal jsegm~),t whichSmake(for)ebᒽlllK,same t that makes the horizontal tangent forthin curve Q(t). So3key i>o find %MequationC\.Motiv+ => play anim<ΈPTZ kpWx4 `9"( Q3S59 5#! DvKr  A 'L'C uxy qq @  tcos( )v0-0.5tcos( )+sin-1.570796326795 H" (.(32`4dP 6`82)y@b0qAt0cCF'i`UEY`TGF9H`$1P1W0#R0St"HUETHW`Xp`YRub0gd0YesI0QgDHCi5pG0'rXyt0vD0wrw0 W@@|YxT#DY(gH`YEs 8bX$#$x I 03t0<1FPH9H`$TYesIp`lrv(%aPt eP T( &SQ$@ӆ0Crh & %"q#TU#<` QbI YfA &bRT 479$5FY 4Y<(9x 5  w6!rx$P)d$30#0 bS` &XyV#$EEpIp0DTW@<Ib`H6r T@%`y0$8CUHB `hgHgpHebs"P+H gH"yE6YAfYvU Y)W$'>"Bg `r%hFTB4 w"3'$Z 1@ XXm tcos( )~--0.5+%sin% -1.570796327] (10qH     tcos( )00-0.5+%sin%-1.570796326795JlWw  s"s((32`4dP 6`82)y@b0qAt0cCF'm`UEY`TGF9H`$1P1W0#R0St"HUETHW`Xp`YRub0gd0YesI0QgDHCi5pG0'rXyt00`vD wrw W@@$YxT#P (gH`Es<Y8b0X$_<x I H3txY1FT9H`$TYesIpl(%Q dS S T ( i$" Dq2 SH#<VgT6y 5FB Q#x 'wi` 3!71 6sl 5!! )s$T  Y @Hp r3P2@$$5a0P<bT$<&4`TBTEDlC$ $I%<@"yyg 4&0BS<hbX2YY@X 5hH  .@ 3  $ tcos( )sinN(10qՈH"hMJ6 TW&3& `W4#$7D5pXsCBe`HY 'Es` Y&V!.@ ;.@!1 "@m1ĘL2PH 1@ T4H D@6cdw YQPw$YÐa tcos( )0-0.5+%sin%-1.570796326795&{_ oo H  E@>6Eb5hP Y $U u1H!Y o tcos( )--0.5+%sin% -1.570796327(10qHt@H Gw"3'&!fXUK !U)G!5PxYBqx @Ibb `3uH ~Ale5+ ! `%C@,67W2g N"_@   #yH o2-[, $8#8%<5<&0#hFw"3'RY'  #' 9(PH B4w)4*N(' v0v   T%d&3 +<@  #,Z!Ƙ'lW2gf@-Ę'EI#"*)G!(5."B(`0 `v$vi2`sqaP<efSHW`F$IU8$AP0$3E#x%@憄5p .%z spwgidaaS`RXF DU8P6R0$(I#0 uF@<pCpHeT fi0YS8` i $Rv0$DTYcRYa QS`T"#xYAPY؎l`(57 61YW 1U 7v t (FQr0y9V<2( P4Tw`1H  &c  4IPC 6Yd 5%Gp1u@$#bv 0Q`<3A0Hv'WPTS2x 'C($iswxffRH1@VIECy92xS'H2B<C&QB`5yEfDfX4lhdaT`$b/*1H!7H 6RedXH Y rETx%u31~"T((` `^`tp`gyv s`iq eSYe@$WFIY`0I99S<A2)HH3%B0T% 6P`00 %wxyli0YvuIhh8`a(RTDGx69s(2g %xahUX` `DYys`Y&,Y1i,Y7`YV Yv YG8PYvyY4y#lS4xex&7vDY"PP +((vH "u 4b'p)Q'V`$8'@0rP<'B xR0Tar#q P0lvypw$a5 (6 &Pa0 4 9 7aE  2gY $!!Fl Q5@ˆx@TV(0DRpriRuR 3q VR"  \VXBFEXV `$8bUFvpAe u60  G$A H#123 `<H@C6A0B& `X66I BsqS rLTP`. (!'&R   n(q% [Part I: $/ The followings are to describe howget t*E scatter plot>rcurv"Q(t):[fTStep 1. We construct drag and (op the parametric equation [[t*cos(t)], sin ] into=Geo<y Strip above.[&IIn obr words, we have [xY,y^=j=[g~,f%QT2. Doints A=[-,0],B=[0C (/2)] graph. )&noteatline for AC is1Ky-0=m(x+) or y=(1a*/vwriQ5inhMm asUllows:1-[x(t),y]=[t*cos (1/2)*(.)+/2][;Step 3. Select the point D on [H], Tsind ]]. [DVD 4. ConstruGa perpendicular linXas@gdroughband)to AB;PersI segments AC,/Gthin graph resp ively at G,FjEU 5. Animatealo e r=t by proȆFs e range of. arameter foraTion)U)Step 6. Collect and drag the x-value of Ddistance DE into*polar plot we getGthin curve Q(t).[ [+ Part II.:Gettin equation for0H alytically:t [$~+To graph Q, key is notiʎDG=EFI siG=f-GF or b  +y2(m*x+b1)&2A+g.7  =t*sin(t)-((1/2)*(t*cos)+)[L**In other words,  heights (y values ofin graph) should be/ same as[Wf{yc. T_efore, we write Q4 8[x3,y1]=[,]|Note.$ Ifrdragpartric equation 8BR[v],@]], with t in [0,2], backtoFc4Jgeometry strip above, we will get the graph of Q(t).[ Part III.6 To find where0 has a horizontal tangenthnote ?dy xQ=t(dt a~Q=0 impliesatUf-(m*g +b))(((t*cos-1.=0. T)fo-vneed K)to solve for t  that f(t)=m*g  or[[  f'6)g ,=m=$ f(b)-f(a( ggbkPThis is exactly wthe Cauchy mean valueorem states. Now it=andard 8exercise . We demonstratTasllows:Rdefin=-t*sin6R done0g0cos0f(?kt g(t) t=1&2. ,t,0,2,3)R1t="( 2.425497143BT(QmR&q[MWe find t to be about Q (when m=(1/2),a=-, a3b=0 respectively.)]Note. MSince we are coll: ng the 'dista ' f-yve dur# animation,DILchose this example inRifiedtervalcau/b is non-negWve.\Trace three graphs.3 NFinaForm0$NGraph2DT 3h LISTSYSt4< Modify P<STATCALC d< \x S;equence,xShee] \|ؒ olveEq_wrX(UpdtupFLG1p(<LisDPicܐViewWinL_osvevdxt_((H2m` y<|<c ,Hyt3 7 Ć$K( ;O  $0<FÆT`lx !"#%̒"؍%  &(,'(< )P *dg+x, 8-D .P 0\ 1h 2t3 4 5, E@ FT Hh I|Ȇ JԆ K/L͆ MyN O( PQQ( R4 S@L ] X ^_dahblp,|͑@ ΑTFБhב| ؑ ّć   ۆfRܐ FinancialFormat  E system]Setu^a _LIST`,bT@ @@ @xĮP(1) Simply put, if we apthe Mean Value Theorem ongraph of a parametric X Xequation,FnT will obtai@ CMV. In o!r words, assume func> s f and g sNsfy VcondidCauchyholkcbe interpreted Ias any numbt fohichg curve P definAb5WP(t)=[g,f $] for atb has slope equal to theof ecant that runs from 0.poi (g(aca))Gbb)). [9 T(2) We use+following example demonstrat% motivationproCMV.JTtechniqubd here be appliarbi_ryse w'K holds[ An E:3cider>parametric curve N~=Y[x(t),y]=[g f t*cost*sin%*] with t in [0,2] We let A=(-,0), B=(0 Xand C /2). FiE] on the curve of P whereslop%tangent _!line is same as0AC. [[WKey: claim tha#f we rotate AC to AB (a horizontal jsegm~),t whichSmake(for)ebᒽlllK,same t that makes the horizontal tangent forthin curve Q(t). So3key i>o find %MequationC\.Motiv+ => play anim<ΈPTZ kpWx4 `9"( Q3S59 5#! DvKr  A 'L'C uxy qq @  tcos( )v0-0.5tcos( )+sin-1.570796326795 H" (.(32`4dP 6`82)y@b0qAt0cCF'i`UEY`TGF9H`$1P1W0#R0St"HUETHW`Xp`YRub0gd0YesI0QgDHCi5pG0'rXyt0vD0wrw0 W@@|YxT#DY(gH`YEs 8bX$#$x I 03t0<1FPH9H`$TYesIp`lrv(%aPt eP T( &SQ$@ӆ0Crh & %"q#TU#<` QbI YfA &bRT 479$5FY 4Y<(9x 5  w6!rx$P)d$30#0 bS` &XyV#$EEpIp0DTW@<Ib`H6r T@%`y0$8CUHB `hgHgpHebs"P+H gH"yE6YAfYvU Y)W$'>"Bg `r%hFTB4 w"3'$Z 1@ XXm tcos( )~--0.5+%sin% -1.570796327] (10qH     tcos( )00-0.5+%sin%-1.570796326795JlWw  s"s((32`4dP 6`82)y@b0qAt0cCF'm`UEY`TGF9H`$1P1W0#R0St"HUETHW`Xp`YRub0gd0YesI0QgDHCi5pG0'rXyt00`vD wrw W@@$YxT#P (gH`Es<Y8b0X$_<x I H3txY1FT9H`$TYesIpl(%Q dS S T ( i$" Dq2 SH#<VgT6y 5FB Q#x 'wi` 3!71 6sl 5!! )s$T  Y @Hp r3P2@$$5a0P<bT$<&4`TBTEDlC$ $I%<@"yyg 4&0BS<hbX2YY@X 5hH  .@ 3  $ tcos( )sinN(10qՈH"hMJ6 TW&3& `W4#$7D5pXsCBe`HY 'Es` Y&V!.@ ;.@!1 "@m1ĘL2PH 1@ T4H D@6cdw YQPw$YÐa tcos( )0-0.5+%sin%-1.570796326795&{_ oo H  E@>6Eb5hP Y $U u1H!Y o tcos( )--0.5+%sin% -1.570796327(10qHt@H Gw"3'&!fXUK !U)G!5PxYBqx @Ibb `3uH ~Ale5+ ! `%C@,67W2g N"_@   #yH o2-[, $8#8%<5<&0#hFw"3'RY'  #' 9(PH B4w)4*N(' v0v   T%d&3 +<@  #,Z!Ƙ'lW2gf@-Ę'EI#"*)G!(5."B(`0 `v$vi2`sqaP<efSHW`F$IU8$AP0$3E#x%@憄5p .%z spwgidaaS`RXF DU8P6R0$(I#0 uF@<pCpHeT fi0YS8` i $Rv0$DTYcRYa QS`T"#xYAPY؎l`(57 61YW 1U 7v t (FQr0y9V<2( P4Tw`1H  &c  4IPC 6Yd 5%Gp1u@$#bv 0Q`<3A0Hv'WPTS2x 'C($iswxffRH1@VIECy92xS'H2B<C&QB`5yEfDfX4lhdaT`$b/*1H!7H 6RedXH Y rETx%u31~"T((` `^`tp`gyv s`iq eSYe@$WFIY`0I99S<A2)HH3%B0T% 6P`00 %wxyli0YvuIhh8`a(RTDGx69s(2g %xahUX` `DYys`Y&,Y1i,Y7`YV Yv YG8PYvyY4y#lS4xex&7vDY"PP +((vH "u 4b'p)Q'V`$8'@0rP<'B xR0Tar#q P0lvypw$a5 (6 &Pa0 4 9 7aE  2gY $!!Fl Q5@ˆx@TV(0DRpriRuR 3q VR"  \VXBFEXV `$8bUFvpAe u60  G$A H#123 `<H@C6A0B& `X66I BsqS rLTP`. (!'&R   n(q% [Part I: $/ The followings are to describe howget t*E scatter plot>rcurv"Q(t):[fTStep 1. We construct drag and (op the parametric equation [[t*cos(t)], sin ] into=Geo<y Strip above.[&IIn obr words, we have [xY,y^=j=[g~,f%QT2. Doints A=[-,0],B=[0C (/2)] graph. )&noteatline for AC is1Ky-0=m(x+) or y=(1a*/vwriQ5inhMm asUllows:1-[x(t),y]=[t*cos (1/2)*(.)+/2][;Step 3. Select the point D on [H], Tsind ]]. [DVD 4. ConstruGa perpendicular linXas@gdroughband)to AB;PersI segments AC,/Gthin graph resp ively at G,FjEU 5. Animatealo e r=t by proȆFs e range of. arameter foraTion)U)Step 6. Collect and drag the x-value of Ddistance DE into*polar plot we getGthin curve Q(t).[ [+ Part II.:Gettin equation for0H alytically:t [$~+To graph Q, key is notiʎDG=EFI siG=f-GF or b  +y2(m*x+b1)&2A+g.7  =t*sin(t)-((1/2)*(t*cos)+)[L**In other words,  heights (y values ofin graph) should be/ same as[Wf{yc. T_efore, we write Q4 8[x3,y1]=[,]|Note.$ Ifrdragpartric equation 8BR[v],@]], with t in [0,2], backtoFc4Jgeometry strip above, we will get the graph of Q(t).[ Part III.6 To find where0 has a horizontal tangenthnote ?dy xQ=t(dt a~Q=0 impliesatUf-(m*g +b))(((t*cos-1.=0. T)fo-vneed K)to solve for t  that f(t)=m*g  or[[  f'6)g ,=m=$ f(b)-f(a( ggbkPThis is exactly wthe Cauchy mean valueorem states. Now it=andard 8exercise . We demonstratTasllows:Rdefin=-t*sin6R done0g0cos0f(?kt g(t) t=1&2. ,t,0,2,3)R1t="( 2.425497143BT(QmR&q[MWe find t to be about Q (when m=(1/2),a=-, a3b=0 respectively.)]Note. MSince we are coll: ng the 'dista ' f-yve dur# animation,DILchose this example inRifiedtervalcau/b is non-negWve.\Trace three graphs.3 NFinaForm0$NGraph2DT 3h LISTSYSt4< Modify P<STATCALC d< \x S;equence,xShee] \|ؒ olveEq_wrX(UpdtupFLG1p(<LisDPicܐViewWinL_osvevdxt_((H2m` y<|<c ,Hyt3 7 Ć$K( ;O  $0<FÆT`lx !"#%̒"؍%  &(,'(< )P *dg+x, 8-D .P 0\ 1h 2t3 4 5, E@ FT Hh I|Ȇ JԆ K/L͆ MyN O( PQQ( R4 S@L ] X ^_dahblp,|͑@ ΑTFБhב| ؑ ّć   ۆfRܐ FinancialFormat  E system]Setu^a _LIST`,bT@ @@ @xĮ8`GF{GxK_o߶-/Ǜug7wn]qxmk o-`|uxkxKxsxSx5_s5_s5_s5_s5_s5_S5_S5_S5_S5_SV)`Յߟ`u{G$Vi=#;4BbbWJl}u Dv۾>ܻc`Ob%Dc=W}]㛽ثD^2A VbA,ضL4ۖ bA VĮ buA65A-]:5A;- b=A 6Į bA v0łX:- bǃز B! buAt1n bgA,A\bXMY[ĢA,bA!% &5A)u @k bA5A@Ħ v,uئ v2~6ׂXgy; b[Alk(uecW=A,Ķ%Al{  b_;F<XG bA7uGXw b  b#A ?RA 6ĂA bE _W2۫φvD^vVX"u~!+{&X%6~ۓ{q,t9h`׏T^m'?O~)Z;wҪv况|Z佟- [mQJʜ$6"%&QJ.}d?Wrd>o.o)o-_]0.HE{- [0Lnx3mi u0?emXM?c%)y)v^})GʈRRNH9)崔r.$(Ee~[e~[e~[e~;;o'PEPOprk\H38LсJMm|l_ƋoО9>6o9ŗ_i.KK\gw#Z3\=nR寽gڭk',t|e\׹qǿԳ'V XN3[sXLסy罽:)gb}x"*SQ9=#{CH}smLVECv߾w{RQ'ݸSaVFT|ͱݭ3ѩ EJR'J)R6MmzٮvVaD\-G5R,2!Z)\'%-z)ǥĤ eSRI9-)oF'K̓o7TdUg [M29)7KXJ
Duˮ۬HO8фuSNtD'zX>Dtү9d)՝|Ή@ˑ= Gj >pd`Ñv' lGtԩ VVDdKFw=m5q&V*ej3HO;52l[}_;r zX6 -}D6F?9i{5ݷߔ I[Gla #[X -yEޕ-yL~С6{ثRy -+kErH[q?jfˎ_MuH.wE!{_~+>^?jfmĶ^^3mw>ľԒ↦Ug7dUͲs}Gu9ZםSu;Oϙ {dF=\EsuVQu⦲T}(|\\R{GG7Q Rݥ&m9h[G+_Y fWvS{p¼t鱅'mzҶg+K,}\)9yhc=2&CΥΥӪsBK9ܩ'ug^v%τ%%{6P3 ?yo+v߲w_$ Sm[F]ֹiaYzN woxw뽃{Ḅ^ӶʟP7zeIաzbvVOhqM2{4]0Rƃzϗ6n&f?[?i姦 oJ -)';R4])g}ª#f/9lW//IKR~n<o_o_>xƙmL=5ןt6u?UΣ^\տؙjhS:GJUG{fj`>>vbM{ _6_ {Ǽ5ּ>ȬHdD%iMdV'⻚v556'~*ZkY/KlNĿM~s۳{wp|Ƚ_L/\DDD<{=^B uD,Ndy¿џfcp.}d0άJdLC\뿶nxh`XlcX6/gLd6$2Df["=Nd6='j9Yy,cw9 s4NT[\08+RKq:kEVG{Q'iO^Ľ.K,r6MsϝYUHMZ9);2/͉)YC6ϑpADNRsyοq"}v.yʩL߫ȻĿKs~S9( Ldu;&Y&hzhFH9[hU.;-wNd !1J$+Luy*m_+oZuUZ/_/oZ$SҚy4j^&m*o6pVuFiWi&iWiU|4&pVw4]!M?JYt=|K=m`cM^ΚDhYjmTkUZӺղ=՚$mHbjM:d5$T]A՚m s6W9՚Z-T /*T^lPxUkM6Z~Tvu$=̼6䒙L2s$3d#6332agd4=''=#=#)Gj28eR<]]RFPtzxnay.Q F/(z.bhW*_|˞i.eq_R:=e~DRyۿnKM퍉yJ GǤ],ɥ0%5Udə'.8]#O>%)q!-(vN%J͵P,.ug%\^*-e6^5?cIdt.Y]1Bvŏfd%@o~ױzf*OIsTTcMեljѝM-*BzeSSSl#F6puǼBuguou<}5/~TA:qoNd wsdzux]ɱ]CCށWoN %z/ؽC'G-4K.ٰw`ؠ6־[c#C9,{u}ժ{fCc:ټxXa(n~uHu!zrp\~pwWn;û:ϝ~`.Ib~z_yd/_׹)o#޾12Kִ=7yo4%񍛶/]_>nY"?w͏hmLRsY|[Ncq3!w\O^JFo+oPXY@rmKgv&2}Od$⛆Gz %_zJǮa~}w$z|eW\'3sL1 YZϛd7݇q`p˰ =Oz?fK/*|(c`F5:Ƹq\x\8aoLkýU)=V?"z5l%m@? =ޭ|yG2sxhl/9OFzG{ ;>iS36{Q>11!/$Gux?<|𘎏/Xlxl<&Ǩo4&zP`r|#PBQ?# 5w/Nj_cFYF!o5Vvor4GٕoWrhܬdl`,6&?u+cVw?Y1=d[lcOd P,KL$dj33q2o`A[nu&;Np:RoSoSo󬷿j#$ByyDX-U^a?I%R+%&E IV]J.)IPH*Ј )3* #r'#t<_&?<}r>}'?/맹G.T!EO4|ڽy7[~y-ߨW+xz٭(1FL>hxUf oxgWPoj{3.5G8<]twcaL넔RNKy[Y)tٶ/%&QJ.)=RF+w_Փ7݇nG>4VnyÑB3ʓW݊U?v_N%k2{]Y.=} G._9:kq_hswX:/Ĵs%٘UیأfV]D*~6/7LqtU2yzW?wdz˹Rib>'yai#Ⳡxrh#~h#FH>楍xYڈ9q~Q.RF4Y}})yE\2v?1V֑[Uڕu96OC}rhoGGGGuˡrQRRR^.xK0}Xwvу:Rڿd6 .KSRܶpy{cɴ[k.]񃪰KB7M5|}C?+3/(ٓ׋l Ϝ5/ǑZ^7ϥ܁"7ٱRNy}ޟzTl]hUSMUگv֧ݺi?ګەܫLqܠotwtjxϯvSj7ϯvS]LKML7M>txS_S]u߮>vuvS{ =iwڥvW= jOڡoz7=0=ig[%kW[$kG[vazdػYA=v#"^?{=7:=G;{!~}Dk}1wr04Vt7mhN;0Y jɠY} j;#kOAX>{2p^'ŇjAA'Œ%VOwk}OU| 5*^UtH^?Ll?Fz : M߹Oz/m-.5OXH7M'K&\$SS衞}˲ޱ'3'{~}tc] VS%s h.MTIWb>jY1s% 5hAԛԛޤ\u]z4=w[|WS|Aoh.}FG?y݉j.m%xEi%zVAvVPA+Fv>i+6m7L[i+fk+oڊ9Vд?ۊ,7}'-}'{GĺFWZ5~H]5Ob}k4#Zf6SK^:[j|~oOek,jU=(ϟ7W5oko AsgEWtT,t gˬ6P2y)hloeń&n("iMNZc%&n^#{{MQC)ucQCr޽=ԼlsӍ/ee˭-kZni?ܐ{[4?7?ίjnv'RՖ5ML5 R_o]r9vjI+MCJœ5TUY*umERSvojxt=~̿$_,ڌ