000100020100072nd.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.ACT000102000fshortestdis.EAC010000004df0^B[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 animation3 `3"d `7 9E(0vrALCuxy5H  H  A@6P %  (x-3)^2+3P cxsi @   ! @6 pY )H !`B@6 @    .@  3   ! @6 vh ' HQS`.@   .@    1@  9 [/x-value/length AB/slope AB/slope of the tangent[012infinity -5.999975267 0.2631578947 10.49360496 39.86315789 -5.4736592 0.5263157895 9.134289281 17.32631579 -4.947343393 0.7894736842 7.925843368 9.989473684 -4.421027605 1.052631579 6.873325879 6.452631579 -3.89471185 1.315789474 5.983042209 4.435789474 -3.368396077 1.578947368 5.261877669 3.178947368 -2.842080233 1.842105263 4.715421916 2.356390977 -2.315764478 2.105263158 4.34469145 1.805263158 -1.789448682 2.368421053 4.142690537 1.435087719 -1.263132927 2.631578947 4.093658013 1.191578947 -0.7368171267 2.894736842 4.176853619 1.040191388 -0.210501285 3.157894737 4.372928675 0.9578947368 0.3158144783 3.421052632 4.668912411 0.9287449393 0.8421302333 3.684210526 5.0597856 0.9413533835 1.368446052 3.947368421 5.547276601 0.9873684211 1.89476184 4.210526316 6.13743403 1.060526316 2.421077638 4.473684211 6.838186773 1.156037152 2.947393403 4.736842105 7.657505818 1.270175439 3.4737091825 8.6023252671.44.000025[[R(1.191578947)*(-0.7368171267)R- 0.877975776[ Method 1.[We use calculus approach[d=sqrt(x^2+y^2), where[y=(x-3)^2+3. To minimize[d is the same as to minimize[d^2. Therefore, we Rdefine f(x)=x^2+((x-3)^2+3)^2RdoneRdefine f1(x)=diff(f(x),x)RdoneRf1(x)R4x3-36x2+122x-144\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/x where[y=(x-3)^2+3; thus[the slope of AB isRdefine s(x)=((x-3)^2+3)/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-3)^2+3RdoneRdefine g1(x)=diff(g(x),x)RdoneRsolve(s(x)*g1(x)+1=0,x)Rx= 2.590979492R s(x)*g1(x)+1R2x-3x-32+3x+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,H4 @ L X d p |            ! " # $$ %0 &< 'H (T )` *l +x , - . 0 1 2 3 4 5 E F H I J K, L8 MD NP O\ Ph Qt R 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-3)*((x-3)^2+3)/x+1`P`P`p`p Pfcccc`rpcccc(10qy`#YuY Y yI(` "rrrsP[We get the x-intercept is[about x=2.5909794.eActfH$xx^2+((x-3)^2+3)^2f1H x:(f(x),x)gH x(x-3)^2+3g1H x:(g(x),x)sH x((x-3)^2+3)/x