00010001010008main.ACT0001020011converseOfMVT.EAC0100000084fd^t[Converse of MVT.[Mean Value Theorem[If f is continuous on[[a,b] and differentiable[on (a,b), then there[is a point p in (a,b)[ such that[f'(p)= f(b)-f(a)b-a.[ Conversely,[under what condition, for[all x in (a,b) there[is a p in (a,b) such[that[f'(x)= f(p)-f(a)p-a or[f'(x)= f(b)-f(p)b-p.[[ Example 1:[Consider the function Rdefine f(x)=x-x^3/3 Rdone[[ in [-1,1].\ graph of f``vrALCuxy.@ 3    (-x^3)/3+xH! A@6 @6gwXYgYq .@   .@!   1@  9R diff(f(x),x)R-x2+1Rdiff(f(x),x)|x=0R1[*drag the function into[the graph strip.[Consider the derivative[at x=0.[Case 1.[We can not find[a point p so that[ f(p)-f(-1)p+1=1.Rsolve( f(p)-f(-1)p+1=1,p)Rp=12-32,p=12+32[\f' and left end point Graph2D| Graph3D LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceX, Sheet| Sheet3D| SolveEq|SolveLwr SolveUpr StupFLG1(StupListD StupPict$ ViewWind( y1DHy2\ Hy3|(Hy4H         ( 4 @ L X d p  | ! " # $ % & ' ( ) * + , - .$ 00 1< 2H 3T 4` 5l Ex F H I J K L M N O P Q R S T ], ^0 _4 `8 a< b@ D P \ h t system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D 0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D i ^t $  3xf(x) 3x:(f(x),x,1) 3x(f(x)-f(-1))/(x+1) 3x0.5`P`P`f`ff)) f`fff(10qy`f#YuYf  P `cxsh S[Remark: the graph of[y=f'(x) is higher than[that of the y=fx-f-1x+1,[in particular, at x=0.[Case 2.[We can not find[a point p so that[ f(1)-f(p)1-p=1.Rsolve( f(1)-f(p)1-p=1,p)Rp=-12-32,p=-12+32[\f' and right end point XGraph2Dh Graph3D| LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceD, Sheetp| Sheet3D| SolveEqhSolveLwrl SolveUprx StupFLG1(StupListD StupPict$ ViewWind y10Hy2H Hy3h$H             ( 4 @  L !X "d #p $| % & ' ( ) * + , - . 0 1 2 3$ 40 5< EH FT H` Il Jx K L M N O P Q R S T ] ^ _ ` a  b   , 8 D system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D 0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D i ^t $  3xf(x) 3x:(f(x),x,1) 3x(f(1)-f(x))/(1-x)`P`P``fHH `ff(10qy`f#YuYf YP `&1WH R[Remark:[The graph of y=f'(x) is[higher than that of [y=f1-fx1-x. In particular[at x=0; this implies that[the converse of MVT is[ false there.[[ Example 2.R#define g(x)=cos(x)|x/2|x-/2RdoneR/20R20[-sin(x)Rdiff(g(x),x)|x=1R-sin1R\Geometry Window % `(eY'BeCvTvrALCuxy5@  H  B@6$Ye29Ab0   cos(x)%Y% `'Be!rA8 @   `@6W2g  @6cT!) $1)6b GsuY@  .H  3  H  A@6@0#   @6C C e( rbTp`.@  .H! 1@  9[ -1.258809198 -0.1084781746 -1.136979026 -0.1552335397 -1.015148853 -0.2039847874 -0.8933186809 -0.254383302 -0.7714885084 -0.3060574276 -0.649658336 -0.3586159789 -0.5278281636 -0.4116520008 -0.4059979912 -0.4647467386 -0.2841678188 -0.5174737725 -0.1623376464 -0.5694032729-0.04050747397 -0.6201063288 0.08132269845 -0.6691593005 0.2031528709 -0.7161481486 0.3249830433 -0.7606726893 0.4468132157 -0.8023507294 0.5686433881 -0.8408220332 0.6904735605 -0.8757520755 0.8123037329 -0.9068355395 0.9341339053 -0.9337995184 1.055964078 -0.956406384 1.17779425 -0.9744562901 1.299624423 -0.9877892816 1.421454595 -0.9962869842 1.543284767 -0.9998738571 1.66511494 -0.9985179926 1.786945112 -0.9922314534 1.908775285 -0.9810701458 2.030605457 -0.9651332267 2.152435629 -0.9445620555 2.274265802 -0.9195386985Rsolve( g(/2)-g(x)(/2)-x=-sin1Rx= 0.5707963268[[Remark:[The slope of the secant[line connecting x=a and[x=p or x=p and x=b[will 'assumes' ALL the[slopes of the tangent [lines IF f is concave[up or concave down in[(a,b).[**[a) Suppose x0=0 refer[to yang/2006/roles/[converse-MVT.tex.[b)Consider the following[animation, we fix the point[B and let A be moving[and we collect the [slopes of AB, we drag[the x values of A and[the slopes of AB,[we get the derivative[of the function.\animation-fix leftV(h`E(0`Cq22TqtvrALCuxyH" 22P`FuP`CuQ`@c&P`7uQ`4i8w``1c&S`(W(``%Q`"Dy``8wU`2e0``&S` @``(W`2p` ` yY6pY HY 6sFYuQYW(WY&S Yg0Ye0a"@Y4i8w`Y2pYsFYBqBY$HYc&SYQ Y @c0YyY@YY6sPY(W(`YpYg4i8Y6sFY$IYuQYDyY Y(W0Yg4i@YS$PY"Dy`Y 6sFY$HY$IY22 II@ %X )yP Cg 8aaP uYP gyiP Yc PXI @c'P )1 !P Tba hX wp b4p Fpp 0'i p aSP ufGP U5I0 5Y90 QQh vh p6p G46Gp #q0 IG tqbq I90 #Vu' $QQp pEh9 C"vp X2"0 T7P Y#p 0A4 7UPP re0 BFX fG eGg RGf1 "V"p  &p rU qp $@ 5H  H  B@6`@0#   cos(x)`P`$H !.@  "3  #  @6CGrH`Yewh re`$.@  "%.@  "!#%$"#&1@   9'  22P`CuQ`7uQ`1c&S`%Q`8wU`&S`(W` ` HYuQY&S Ye0a"@Y2pYBqBYc&SY @c0YY6sPYpY6sFYuQY(W0YS$PY 6sFY$IY$I 6sF S$P (W0 uQ 6sF p Y6sP  @c0 c&S BqB 2p e0a"@ &S uQ H  (W&S8wU%Q1c&S7uQCuQP22 HV vP xvR $$ G vHe@ h10 X7xp HC 6yt $b  % 1P yBF bTp0 Dri %XP @A` 2P@ d(@ BBW@  2P!@ A s0t` HC $v@ hp s1$A G8W !T0 yY$` h$ AXy  qS aVy 577 pf3P W Vq0 1d0@ p #v6` V$` 3Ep  TP qAp Q6B(0 79etp 05E (5H  )H  A@6YwX%a YP`P$H *@  )+ " @63cF%YsCCf6v i#Y,#`@6W2g`)-@  ,+*[,]8-+), ,)+'#2([-2g(-2)2g(2)R diff(g(x),x)R-sinxR g(p)-g(-/2)p+/2Rcospp+2Rsolve(-sinx=cospp+2,p)R-sinx-cospp+2=0[Remark:[1. In the geometry above[collect the x value of A.[2. Collect the slope[of the secant lines.[3. drag the x and slope[back to the graph.[4. Construct a tangent[to the curve->animate[5. Collect the x value[and the slope.[6. Drag it back to the[graph.[7. Study the relationship[between the graphs from[ step 3 and 6.[[case 1. let's try x=-1Rsolve( -sin(-1)=cospp+2,p)Rp=- 0.5707963268R[case 2. try x=-0.2Rsolve( -sin(-0.2)=cospp+2,p)Rp= 1.028219229R[case 3. try x=-0.01Rsolve( -sin(-0.01)=cospp+2,p)Rp= 1.539686994Rcos 1.539686994 1.539686994+2R9.999833448-3R -sin(-0.01)R9.999833334-3[[Case 1.[Fix left end x=-/2R(g(x)-g(-/2))/(x+/2)Rcosxx+2R\left end is fixed.Graph2D| Graph3D LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceX, Sheet| Sheet3D| SolveEq|SolveLwr SolveUpr StupFLG1(StupListD StupPict$ ViewWind( y1D0Hy2t$Hy3Hy4H        ( 4 @ L X d p |   ! " # $ % & ' ( ) * + , -$ .0 0< 1H 2T 3` 4l 5x E F H I J K L M N O P Q R S T, ]8 ^< _@ `D aH bL P \ h t  system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D 0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D i ^t $  3x:(g(x),x,1)|x(-)/2|x0 3xP(x)/(x+/2) 3x0.3 3xg(x)`P`P`W2g`W2g" `(10qy`#YuY i&TTY `cxsh S[Remark:[(1)The graph of y=g'(x)[is lower than that of[y=cosxx+2.[(2) If we pick x0=0, [then all x in (-2,0),[we can find a p in (-2,2)[so that the converse[of Mean Value Theorem[holds.[[Case 2. Fix right end[x=2.R(g(2 )-g(x))/(2-x)Rcosxx-2R\Fix the right end /lGraph2Dh Graph3D| LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceD, Sheetp| Sheet3D| SolveEqhSolveLwrl SolveUprx StupFLG1(StupListD StupPict$ ViewWind y104Hy2d$Hy3H           $ 0 < H T  ` !l "x # $ % & ' ( ) * + , - . 0 1 2, 38 4D 5P E\ Fh Ht I J K L M N O P Q R S T ] ^ _ ` a  b$ ( 4 @ L X system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D 0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D i ^t $ " 3x:(g(x),x,1)|x/2|x(-)/2 3xP(x)/(x-/2) 3xP(x)`P`P`W2g=d`W2g=d=d" =d`=d=d=d(10qy`=d#YuY=d `cxsh R[[eActFH@-xf(x)-f(-2)-(((f(1)-f(-2))/(1-(-2))))*(x+2)GH@-xg(x)-g(-2)-(((g(1)-g(-2))/(1-(-2))))*(x+2)fH xx-x^3/3gH,xP(x)|x/2|x-/2table1 = $0<HT`lx ,8DP\ht(4@LXdp|`````u`p`e```U`P`E`@`5`0`%` ````` PY YPYYPYYPYYPYYPYYPYYPYYPYYY  P  P  P  P  P  P  P  P P