00020002010012Shortest_Dist1.ACT0002020012Shortest_Dist1.EAC0100000011d5 shortestdis.EAC main.ACT",/3F `I '*[4 6E!B `Qp9  `Ticb 'FsDzvKr  A 'L'C u x y ' 09! 13@@6%e2F& PRS De  : = ːgf7UbqA p@7s%  u4.` Bta8Td6Y2'V@ BOԠ  .A.XbeR.$s$a. @ q      !9*2@168BvdTV&SaGfO>^ Length OA: $ z1zYUz(z Coord xx5H" . (x-3)^2+3Du!: $   /.@s3s+0.@   ">1Ġ.L9T6  )R[[ Problem: >@We would like to find the shortest distance between O=(0,0) a*X f(x)=(x-3)u2}+3.R+dede 2+3R0doneA͌ Method 1.Calculus Approach+]V We note thatPe distance between a point A=(x,y) on y=f(x) is d=sqrt(x^2+y^2), where U*. To minimize d9equivaleYt^2-DforeO e substituy-3)^2f+3 p 'and define squaDlow:R/D(x)=(dшR)done9hD1:diff(D,x)5.!$4x39-36 +122/-144\+Sketch D and D'fi the root of D'=D1.Ј3?EN&%FinaForm$NGraph2D% 3 LISTSYS$@4< Modify XP<STATCALC |d< \x Sequence,xSheetO | olveEq`wr(UptupFLG1 (<Lis{HDPicViewWind_osvev-̐xy^Uy2(< P |  f,O8DP\ht Ȇ !Ԇ"#$1%h &| '(  )4  *E@I+L,X-d.p0|123  5EĆFІH܆IJKT]6k M NO6P<† QbRVTS `?l ]Z xi ^_ Z ab ͆Ά׆̒Aؒ ْڒے FinanciaUFormat   system]"^_` a "bxR @R @xX[[LBy using the G-solve, we found min of D ($root*'=D1) is at about Vxc=2.59097949183Phich, roughly wh6ffromanimation. Now shortest ] "distance5O=(0,0) to y=f(xRD|xR" 4.092059375 Y7Q0D[ 8Method 2 UAslope-N gent line% point.P /Intuitively, happens when83OA is perpendicular to the tangent line of f at poiA.[NBut slope K$= y/x where y=(x-3)^2+3; thus we defN_5Xrbe as U follows:Rci;s(x)=([)/xR%done5[MWe needsolve x,P*f'V-1. )f1(x)- =diff(f(x),x)(zf+1=03Yx="b 2.590979492Y yI)4This is consistent with wha e found from Method 1.eActDH$xx^2+((x-3)^2+3)^2D1H x:(D(x),x)fH x(x-3)^2+3f1H x:(f(x),x)gH x(x-3)^2+3g1H x:(g(x),x)sH x((x-3)^2+3)/x020012eActivity Save.EAC0100000011d7 shortestdis.EAC main.ACT",/3F `I '*[4 6E!B `Qp9  `Ticb 'FsDzvKr  A 'L'Cu x y ' 09! 13@@6%e2F& PRS De  : = ːgf7UbqA p@7s%  u4.` Bta8Td6Y2'V@ BOԠ  .A.XbeR.$s$a. @ q      !9*2@168BvdTV&SaGfO>^ Length OA: $ z1zYUz(z Coord xx5H" . (x-3)^2+3Du!: $   /.@s3s+0.@   ">1Ġ.L9T6  )R[[ Problem: >@We would like to find the shortest distance between O=(0,0) a*X f(x)=(x-3)u2}+3.R+dede 2+3R0doneA͌ Method 1.Calculus Approach+]V We note thatPe distance between a point A=(x,y) on y=f(x) is d=sqrt(x^2+y^2), where U*. To minimize d9equivaleYt^2-DforeO e substituy-3)^2f+3 p 'and define squaDlow:R/D(x)=(dшR)done9hD1:diff(D,x)5.!$4x39-36 +122/-144\+Sketch D and D'fi the root of D'=D1.Ј3?EN&%FinaForm$NGraph2D% 3 LISTSYS$@4< Modify XP<STATCALC |d< \x Sequence,xSheetO | olveEq`wr(UptupFLG1 (<Lis{HDPicViewWind_osvev-̐xy^Uy2(< P |  f,O8DP\ht Ȇ !Ԇ"#$1%h &| '(  )4  *E@I+L,X-d.p0|123  5EĆFІH܆IJKT]6k M NO6P<† QbRVTS `?l ]Z xi ^_ Z ab ͆Ά׆̒Aؒ ْڒے FinanciaUFormat   system]"^_` a "bxR @R @xX[[LBy using the G-solve, we found min of D ($root*'=D1) is at about Vxc=2.59097949183Phich, roughly wh6ffromanimation. Now shortest ] "distance5O=(0,0) to y=f(xRD|xR" 4.092059375 Y7Q0D[ 8Method 2 UAslope-N gent line% point.P /Intuitively, happens when83OA is perpendicular to the tangent line of f at poiA.[NBut slope K$= y/x where y=(x-3)^2+3; thus we defN_5Xrbe as U follows:Rci;s(x)=([)/xR%done5[MWe needsolve x,P*f'V-1. )f1(x)- =diff(f(x),x)(zf+1=03Yx="b 2.590979492Y yI)4This is consistent with wha e found from Method 1.eActDH$xx^2+((x-3)^2+3)^2D1H x:(D(x),x)fH x(x-3)^2+3f1H x:(f(x),x)gH x(x-3)^2+3g1H x:(g(x),x)sH x((x-3)^2+3)/x010008main.ACT0001020012eActivity Save.EAC0100000011d9 Shortest_Dist1.EACACT,/9<@S `I '*[4 6E!B `Qp9  `Ticb 'FsDzvKr  A 'L'Cu xy / 89! A3@H6%e2F& PRS Dr  : = ːgf7UbqA p@7s%  u4.`3 Bta8Td6Y2'V@ B,OԠ  .A.XbeR.$s$a. @ q @     .9*2@>6EBvdTV&SaGfO>^ Length OA: $ z1zYUz(z Coord xx5H" . (x-3)^2+3Du!: $  /.ș3s  .@   "H1Ġ.L9T6r  )[[ Problem: >@We would like to find the shortest distance between O=(0,0) a*X f(x)=(x-3)u2}+3.Rdede 2+3R0doneA͌ Method 1.Calculus Approach+V We note thatPe distance between a point A=(x,y) on y=f(x) is d=sqrt(x^2+y^2), where U*. To minimize d9equivaleYt^2-DforeO e substituy-3)^2f+3 p 'and define squaDlow:R/D(x)=(dшR)done9hD1:diff(D,x)5.!$4x39-36 +122/-144\+Sketch D and D'fi the root of D'=D1.Ј3?EN&%FinaForm$NGraph2D% 3 LISTSYS$@4< Modify XP<STATCALC |d< \x Sequence,xSheetO | olveEq`wr(UptupFLG1 (<Lis{HDPicViewWind_osvev-̐xy^Uy2(< P |  f,O8DP\ht Ȇ !Ԇ"#$1%h &| '(  )4  *E@I+L,X-d.p0|123  5EĆFІH܆IJKT]6k M NO6P<† QbRVTS `?l ]Z xi ^_ Z ab ͆Ά׆̒Aؒ ْڒے FinanciaUFormat   system]"^_` a "bxR @R @xX[[LBy using the G-solve, we found min of D ($root*'=D1) is at about Vxc=2.59097949183Phich, roughly wh6ffromanimation. 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