000200020100072nd.ACT000102000f2ndexample-.EAC010000004915^E[Finding the shortest[distance from a point[to a given curve.[\AuthorProfessor,Dr. Wei-Chi Yang Department of Math/Stats Radford University Radford, VA 24142 e-mail:wyang@radford.edu URL:www.radford.edu/ ~wyang[ Objective.[We would like to add[an animation to a [common Calculus [ problem. \ an animation0f5% ` B5``r"f5%0hSR@vrALCuxy5H  H  A@6P Cu  4-x^2P `xshB @   ! @6 Uwr YXHu XHuY !B@6 @    .@  3   ! @62`'3 Tpeg S8Dea`.@   .@    1@  9 [/x-value/length AB/slope AB/slope of the tangent[[[ Method 1.[We use calculus approach[d=sqrt(x^2+(y-1)^2), where[y=-x^2+4. To minimize[d is the same as to minimize[d^2. Therefore, we Rdefine f(x)=x^2+(3-x^2)^2RdoneRdefine f1(x)=diff(f(x),x)RdoneRf1(x)R4x3-10xRrfactor(f1(x))R22x-52x+5xRsolve(f1(x)=0,x)Rx=0,x=-102,x=102R\drawing f and f'Graph2D, Graph3D@ LISTSYSL4NModify $ STATCALC NSTATSYS \NSequence, Sheet4| Sheet3D| SolveEq,SolveLwr0 SolveUpr< StupFLG1H(StupListpD StupPict$ ViewWind     $ 0 < H T ` l x       ! " # $ % & ' ( ) *, +8 ,D -P .\ 0h 1t 2 3 4 5 E F H I J K L M N O P( Q4 R@ SL TX ]d ^h _l `p at bx |     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@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i ^ $ `P`P`T`T Qia```(10qy`#YuY YgD2`B#yO[By using the G-solve,[we found the min of f is[at x=2.591 which is roughly[what we found from the[ animation.[Method 2 Using the[slope.[Intuitively, the shortest[distance happens when[ the slope of AB is perpendicular[to the slope of the tangent[ at point A.[But slope AB = (y-1)/x [where[y=-x^2+4; thus[the slope of AB isRdefine s(x)=(3-x^2)/xRdoneR[We need to solve x where[s(x)*g'(x)=-1, where[g(x) is the original function[ (x-3)^2+3.Rdefine g(x)=-x^2+4RdoneRdefine g1(x)=diff(g(x),x)RdoneRsolve(s(x)*g1(x)+1=0,x)Rx=-102,x=102R s(x)*g1(x)+1R2x2-3+1R[\"dragging the function to the graphGraph2D, Graph3D@ LISTSYSL4NModify $ STATCALC NSTATSYS \NSequence, Sheet4| Sheet3D| SolveEq,SolveLwr0 SolveUpr< StupFLG1H(StupListpD StupPict$ ViewWind     $ 0 < H T ` l x       ! " # $ % & ' ( ) *, +8 ,D -P .\ 0h 1t 2 3 4 5 E F H I J K L M N O P( Q4 R@ SL TX ]d ^h _l `p at bx |     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@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i ^ $ `P`P`p`p `x`(10qy`#YuY Y yI(`shBO\or use graph editorGraph2D@ Graph3DT LISTSYS`4NModify $ STATCALC NSTATSYS \NSequence, SheetH| Sheet3D| SolveEq@SolveLwrD SolveUprP StupFLG1\(StupListD StupPict$ ViewWind y1 H( 4 @ L X d p |           ! " # $ %$ &0 '< (H )T *` +l ,x - . 0 1 2 3 4 5 E F H I J K L, M8 ND OP P\ Qh Rt S T ] ^ _ ` a b      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 ^ $  3x2*(x^2-3)+1`P`P`p`p `X(10qy`#YuY X8`VcxP[We get the x-intercept is[about x=2.5909794.eActfH xx^2+(3-x^2)^2f1H x:(f(x),x)gH x-x^2+4g1H x:(g(x),x)sH x(3-x^2)/x010008main.ACT0003020013Inverse_ax+b_dw.EAC0100000012d0 Inverse_ax+b_dw.EAC main.ACT$( K `E '.[DFinding the ip of a function about any lin, form y=a+bLT General idea:iVTo findi(jy=x), w&terchange y and xsolv yǐ{8Ϟ( ) is similarH(Fory=xhave (x,y) -> (y,x)[H(4For the line y=a+b, we have (x,y) -> (y-b a,5)[" Example: 28Def]a function, and set a- b-values f.@Rx dMf(x)=x+1R+done<a:= -b:=2=2[rInterchange x with y'M:`=fO)Ďx+2-2 solve(ans,y)Ry=x 2 +2x+3R:'f_inv:=getrightK[1]LKIsym_line:=a+bx+2s\"Drag&drop f(x),  and L̈26<N(FinaForm$NGraph2D& 3 LISTSYS@4< Modify 0P<STATCALC Td< \\x Sequence,xSheetO|`o lveEq~`wr(UptupFLG1(<Lis{ DPicdViewWin_osvev4xȐԐP   ^ !<Pd(u42@rLrXrd p!| " # $, %@ &T 'hĒІ)܆ 0*荸+  ,-( .< 0P$ 1d%%2x<3H 4T 5` El F xZI J K L M N̆ O|؆ P Q0Rц SO][ A ^_(,a0b4u8D ͆PΆ\hhׇt؆نچۆKh FinancialForman   E system] ^_` a bV6 @@6C@x6:Y& a MatDatabUSB%.16   H <,<   3ߊ5< range of f(x)y|@y-bJaR9Rcsolve(2k ,y)R%y2[5Rx1dx|a+b$,xk-1/x\#"View graphs with restrictedЈ26DNH%FinaForm$NGraph2D 3, LISTSYS8@4< Modify lP<STATCALC d< \x S:equence,xSheetO | olveEq`wr(Up(tupFLG14(<Lis{\DPicViewWind_osvev xy^  H2 P(0(<7P T-  n)`lx#̒ؒ  !,"@2 #T,$h8 %|D &P !'\(h  )itm*+W-.01Ȇ2Ԇ345E F H, IJJT K@L| MXNd-OpƆ P|wQ{RST]p } ^_`Ćab̒В܆͆臺FDב@ ؑT ّh$ ڑ|0 6ۑ< FinancialFormat  ! system]"^_` a bx6 @c60` `à6 @x !( a. MatDatab:USB%.16EJ   F<a   < ` 0(10qy`@#YuY b  $ʇo0%aߞج` e5y`   $Y[eActFHxa b fH x(x)+1f_inv@@  xxsym_line x020012eActivity Save.EAC010000001306 Inverse_ax+b_dw.EAC main.ACT$( K `E '.[DFinding the ip of a function about any lin, form y=a+bLT General idea:iVTo findi(jy=x), w&terchange y and xsolv yǐ{8Ϟ( ) is similarH(Fory=xhave (x,y) -> (y,x)[H(4For the line y=a+b, we have (x,y) -> (y-b a,5)[" Example: 28Def]a function, and set a- b-values f.@Rx dMf(x)=x+1R+done<a:= -b:=2=2[rInterchange x with y'M:`=fO)Ďx+2-2 solve(ans,y)Ry=x 2 +2x+3R:'f_inv:=getrightK[1]LKIsym_line:=a+bx+2s\"Drag&drop f(x),  and L̈26<N(FinaForm$NGraph2D& 3 LISTSYS@4< Modify 0P<STATCALC Td< \\x Sequence,xSheetO|`o lveEq~`wr(UptupFLG1(<Lis{ DPicdViewWin_osvev4xȐԐP   ^ !<Pd(u42@rLrXrd p!| " # $, %@ &T 'hĒІ)܆ 0*荸+  ,-( .< 0P$ 1d%%2x<3H 4T 5` El F xZI J K L M N̆ O|؆ P Q0Rц SO][ A ^_(,a0b4u8D ͆PΆ\hhׇt؆نچۆKh FinancialForman   E system] ^_` a bV6 @@6C@x6:Y& a MatDatabUSB%.16   H <,<   3ߊ5<r{y| 0}R  solve(y-ba!2 ,y)R%y2[@ 5Range is {y|1} -> domain of f(x) x|a+b$,xkx-1/x\"View graphs with restrictedЈ26DNH%FinaForm$NGraph2D 3, LISTSYS8@4< Modify lP<STATCALC d< \x S:equence,xSheetO | olveEq`wr(Up(tupFLG14(<Lis{\DPicViewWind_osvev xy^  H2 P(0(<7P T-  n)`lx#̒ؒ  !,"@2 #T,$h8 %|D &P !'\(h  )itm*+W-.01Ȇ2Ԇ345E F H, IJJT K@L| MXNd-OpƆ P|wQ{RST]p } ^_`Ćab̒В܆͆臺FDב@ ؑT ّh$ ڑ|0 6ۑ< FinancialFormat  ! system]"^_` a bx6 @c60` `à6 @x !( a. MatDatab:USB%.16EJ   F<a   < ` 0(10qy`@#YuY b  $ʇo0%aߞج` e5y`   $Y[eActFHxa b fH x(x)+1f_inv@@  xxsym_line x02000fshortestdis.EAC010000002051 shortestdis.EAC main.ACT",/3  ^E '=[48.602327>1. 4.000025[ R  (1.191576)*(-0 171267)RH 877975Q Method 1.bWe use calculus approachd=sqrt(x^2+y^2), where/y=(x-3)^2+3. To minimize[d is the same a $d^2<herefore, we R=define f(x)=xi(rwR)done991:diff(D,x)5.!4x3-362 +122/-144\bdrawing f and f'̉%S Graph2D, 3@ LISTSYSL4NModify; ($< STATCALC U(< \< Sequence<,xSheet4|olveEqLwr0 (Up< tupFLG1H<(<Lis{pdDPic ViewWind؆,@x ( <P$dx <  HT`lx !"̒#ؒ$%&'@(T)h*|,+8D-P  .\0h1t2345EFHȒIԒJKLM,N@OT^k(Q|R@STX ]d ^h_l`patbx|͒2ΒВ! systemlist]=~[ @ @@x @@x @@x  aF seq_histb NewFolde]3 system]l0^]= (10qy& #YuYU T`H$HY gD2#ʪ!Hxƨ   `B#yhOq[;By using the G-solve, we found min of f is=at x=2.591 which roughlyCwh%HfromjA animation. Method 2 Uslope!* Intuitively,shortestdistance happensenPAB perpendicular(to+Vgenh +at point A.[But slope AB = y/x where y=(x-3)^2+3; thus9the9of<isRQWdefine s(x)=(B)/xR%done5We need to solve V*g'\-1,g(x)e original function .Bg1diff(~,x)(È9+1=03x = 2.590979492R s(x)*g1 +1R2$x-A3.&(V+B[[\"dragging the function tograpḧ26S Graph2D, 3@ LISTSYSL4NModify; ($< STATCALC U(< \< Sequence<,xSheet4|olveEqLwr0 (Up< tupFLG1H<(<Lis{pdDPic ViewWind؆,@x ( <P$dx <  HT`lx !"̒#ؒ$%&'@(T)h*|,+8D-P  .\0h1t2345EFHȒIԒJKLM,N@OT^k(Q|R@STX ]d ^h_l`patbx|͒2ΒВ! systemlist]=~[ @ @@x @@x @@x  aF seq_histb NewFolde]3 system]l0^]= (10qy`H#YuYU 0 l$HY yI(U   `shBOq\;or use graph editorЈ''ES Graph2D@ 3T LISTSYS`4NModify; ($< STATCALC U(< \< Sequence<,xSheetH|ĒolveEqLwr (UpP tupFLG1\<(<Lis{dDPicȐ ViewWind,yJ,4hLX|x p  |ĒВܒ !" #$ %,0&@<'TH(hT)|`*l+x,  -.0123̒4ؒ5EFHIJ K"L,8M@DNTPOh\hQtRS T]- ^_`abȒ͒ԒΒВ! systemlist]=~o  8p0p @@x @@x  aW seq_histb NewFolden system]l0^]= \x.UEtt--dQnu :kٍ:kYΚuLg:Ig:Yg:[}Ag#:Φu%YLg9ݢ:['uv.Igte=%}UgyuvUg+un+dl΢:K,:KlYg]:kYΒ:ۡV MgS:kam٬6κuvVguv^g_S:[gGg/lήl^=:{Sg솀YXgt\g}:_gm:{Hg#~GYv[g:YJg:{Xg:[zYA=qBguvYgY3:{Ygti:{jlw{6{8Lv-%N Z#7jPy x7 ܅C2=Lr-6%޷RS)Kmjl\2u;c<]h~,6jQ,Q KIk)ݿn^;Oseo$5=Ze]:Z6V6^6n>aTqx)˶m+fbo̲6Gm4g]_OiƥvFyi]vEՀ4ߖmߖmߖmߖ^O|*Ѐ1?4a/N_ߜ੊~MUl~+o}B=Icye+yEjbikUJ? =ߐ폨-w>ٺeV;g$ mUT߁Iu'ʶݝtYͯKO>⵪WmT>N7Jy]V8e7>([%n?fmKeBgֻr)-5:87v_jd_: Twbބ<s>wϘ%qwhe"CU|s"ߒȷ&m|{"e敉ב/рxkGL&lYMo~Zݛ8OzUOzLuo:5{V-C>|(,+M){jl"oJ>\/O=<oOQݷaltxdlBV8"ߕoL7%݉D/O7ߛFW]uL?khxR^xaWTw8r6SYnڭ.fv(dK)=В +f&n7e;BqvM6vT5,Ù;t߼̌,s\3N8>d.'Pp"' s:|tU':؍C6SLx5J{I2iyi7I*fioI[./ȓF/ʓFb2%yHb>̿:ܒ{ՉtU'+hkt"C2幄u:2չ~'2+!'rRVt"geඹ9/:}ݟ TXg3nji^Fi^"MiܠVY 4,vYyk퍹 vC697ɫ)QPC ljx@.+0(w 5ozvA>!Y ljX{d@P9WV ljxF6j 3~xfi&PGYjc_]4 i)-ߴܬ͜=>kOV.yy[IgˉEYhȺEeF;zl%lT~)ktKL4VޒYPW[:vX~;(?"=U9^D;>ww:qЋ;ߙ>zǰֻWR_ 8TAy@Ef*څjO흳\¶Y-:_w<7m{*o4?4guo_x;*o?4@Ʈʄ{a~]5ÒŤ5Kt:}}lNr>ۤ?Sjn\&Pjy҇OLqO_3KUm9<Czq͹'K3C5^fkfB][R!BgM٦fǮ>rBRXT)|b`>t9W}PfN+pgM~n~GWlɺˮzmχMb*۸ Ɠ2vc;TZ'~t2%wLmo۲qJ}ߖh[6qݲYaO=lW-JMOyxO}]{f7mr]-LP| #l7>N NT"}AR0mڼU߹XuA cG< 3Qg&j B6~).âϣ'wǼ?ꎩؕp9) зCGLo%:VUo[zrT0;<ߗoMlLk]xN\( kw^k?K*SǪ{no]e;lOZv:*& $[oO6Y׼nYK3Wm?q)츇c[d勇~3lڹ`bGHKŎ:4l;U1 Jܫ~ WիceY9=ԟ/;Og nsW+ʹP FYֈ7:/:I*߬*60&gD:ݒG*Jy։LU:n#PN-yYpKtθ%T:ܒG*n#l]:n#Ϋn#k^JW:oJ'J"jP5R j,+ФʞܠLYUH>Rt;nsip )wFU#δߘ_ሪ{Ts)wNuOixRzTs{ܹ=R<=R=RݺGUl˝t;QUeeO,X4eO/{~=m쑶v^UUj T#MՠuV%ҦW/KS+TKigJJyf;p.񪊠[ m<7 uU^-.q v?ܟ)B*;˫;kF:w_T/TB.VBG}b}yMصK(Ke:+~6R,BHʠwJȩ ,1zV;W[!_xYJ nWyy+)yTn[eLȱF9c^8zc`WUvTx0S]Qa9k n#PO.5]h?P.T=Yk|ve%ϩS'Xf;ZUV=o&N{w v+B㴕7}l;_ևjKO\>O]7{*ԤsxV"؋>87[n?A/q@;\9T|>Qb꼪@y @Q??&MzϕΕ;5l /s}}5-;ciz@ 瀴qiI8#K.K"zl[,SZiK|=*f ypvCMc6NjQ sTx~<2ϕsNczs|Ej#s}Tke% }|6r7| XűX0N?8 B F~٧اQاa}iاa}iاaOs)ܖ?Rkݚ޿ild3|kGFdzZֲq;֮k>|ֲq㹕k=o;di9w'&V`M[ӫ[Wwov&W}N}Z)pz2 TL?o΃ZV[111pc(_;1'=996X?=zp#u˺㩉ѩ؞ ݮzV&hؖ361>:Xzbrx?kĿ'o},Җ+<]x꽩ᑯ? 9M&=o1ݟ;縫շY>eML/ְ`|Ov]4W#nbZwPzyåQw_yY6ʼ€dm!,h^b~.I3GoRJݒ43Yg̿BgXz]aomx=^3;v?7pDs=φV>wj越>h/>of_k<`f̿>8hff_`k13sf_࢙>xk\63,eo-/}k̿Ak43}mf03#f_`k1393}p/'L^03̿sf_%3}73Wyk4탨>탄>Xkf̿A>af~A0dfzٔ>8lff̿I3U6xW43lΛYhf2ſf_3}pknd_^G}$p-rngO¼N?8F/1hyv9~ѽ04d5