00020002010008main.ACT0001020012eActivity Save.EAC0100000010ed shrink_circle.EACingACT,/9<@  ^I '<[4SX Cp[KIOrginated from the textbook by Stewart, James, Essential Calculus: Early P +Transcend'als Thomson, BrS s/Cole, 2007.\ AuthorG FProfessor,Dr. Wei-Chi Yang Department of Math/Stats Radford University, VA 24142 USA e-mail: wyP@r7.edu URL: http://www. /~0 [ Objectx.[/VSome problems on finding limits can be done usCASpabil. Howe , it will/] >more excitR to solve themgether with getry animations.jNote. 6a) A complete generalization to 3D case can be found ih llowing article by Mead  a,Yang:[=Open.\Anim ! tap-->Έň  '4Xh `8#TsY0 YaF&e$ eBt DvKr  A 'L'C uxey  H! ,H F'@=68c%UO t%x[Y<^p VB%5ZYH#FwTD@Ysulu" @  9"4Ei0fY {ŘX ȩ!r%hFTBW`  !@   1 W2gf@FŘ' d9E126{c!C5&75wsuP Coord: nfH!LjxDh3$jTA4tp Y0sWw7Ty@EqggfZ9  0@6c!C5% 5sWw7T26 Shrinking Circle: +qy ut]m `@R$ !A  BCzC=%@=N]ĘotU 8"0 2iA2f rhUah xGpXUV Y(,0!H B !@Q6X$`.H!XEG<'GFG@ Yo !Ȑr%hFTB$YW2g"ei#@ E$ %&@  '8(8")5VSU3Y3   4H'XbY*JH#J'GFԈ8A)19% !b KU[[ A problem.[N1We have a fixed circle of C1: (x-1)^2+y^2=1 and w0nother0 2: #x3, which will be shrinking.[+ 1) A is U cent[2)&2) B!point2 on4y-axisW*3) F.top iRsecti0݊Δ`'2Lx-0cepf line BFU@would!kk see wheradius(r) goeo 0at|pp,oL?[IZThe following matrix is t coordinates$r point L a#ircle C2 shrink9o 0. So (ourInjecture[limit ofJexists.[6 3.414213562H060441189 5037317044 544332125N 582449523h 618259366651911k68353599713i48h74125N76729436 7917976680 3.8147122363609755256006478474486057N91578181 3.90732015 921745144 934882619M467586357396163 3.9668153033P9820698792439h9099883 3.9997965550[ 5We can conjecture thate limit (L) approaches to 4.=OProve anlytically[hNLet B bV e point (0,rS nd (x1,y1) coordinatgof F$z,0")L. Step 1)First, we needfixx1 y1 in terms)Nr. TF on both circlso[x1U2]+y1=r( (1)N-1)QG +y1 2 =1 (2)[It follows from that x1>, -2x1+1Y Yand =x1xfN=r.[Hence [+x1=r֐ڎ4y1=4-P4$4(11W9qStep 2.Y,We need to find z in terms of r. Let F' be ohe x-axis so tha perpendicularN AL.[ By@simi$riangles ==== [K length ALB@=* *F'+F,uzrez-x1y .[Th$implies)zr:r-<. Substituting x1 ay1?."finally gives z as a function of r follows:[CAS capability:R deTe z(r)=r2'2-24-RbJdone[ [TQn014 Exercises.'S(1) What if the fixed circle is replacby (x-2)^2+y^2=4, try to see wBhappen[ hint:->o)se 2v(2) [\"What if the fixed circle is replacby a straight line orcubic function passingL rough (0,0), try to see wu happens.[' hint:->o Or case   1.eActzH,r((r^(2))/(2-(4-r^(2))))010013shrinkingcircle.ACT0001020011shrink_circle.EAC0100000010ed shrink_circle.EACingACT,/9<@  ^I '<[4SX Cp[KIOrginated from the textbook by Stewart, James, Essential Calculus: Early P +Transcend'als Thomson, BrS s/Cole, 2007.\ AuthorG FProfessor,Dr. Wei-Chi Yang Department of Math/Stats Radford University, VA 24142 USA e-mail: wyP@r7.edu URL: http://www. /~0 [ Objectx.[/VSome problems on finding limits can be done usCASpabil. Howe , it will/] >more excitR to solve themgether with getry animations.jNote. 6a) A complete generalization to 3D case can be found ih llowing article by Mead  a,Yang:[=Open.\Anim ! tap-->Έň  '4Xh `8#TsY0 YaF&e$ eBt DvKr  A 'L'C uxey  H! ,H F'@=68c%UO t%x[Y<^p VB%5ZYH#FwTD@Ysulu" @  9"4Ei0fY {ŘX ȩ!r%hFTBW`  !@   1 W2gf@FŘ' d9E126{c!C5&75wsuP Coord: nfH!LjxDh3$jTA4tp Y0sWw7Ty@EqggfZ9  0@6c!C5% 5sWw7T26 Shrinking Circle: +qy ut]m `@R$ !A  BCzC=%@=N]ĘotU 8"0 2iA2f rhUah xGpXUV Y(,0!H B !@Q6X$`.H!XEG<'GFG@ Yo !Ȑr%hFTB$YW2g"ei#@ E$ %&@  '8(8")5VSU3Y3   4H'XbY*JH#J'GFԈ8A)19% !b KU[[ A problem.[N1We have a fixed circle of C1: (x-1)^2+y^2=1 and w0nother0 2: #x3, which will be shrinking.[+ 1) A is U cent[2)&2) B!point2 on4y-axisW*3) F.top iRsecti0݊Δ`'2Lx-0cepf line BFU@would!kk see wheradius(r) goeo 0at|pp,oL?[IZThe following matrix is t coordinates$r point L a#ircle C2 shrink9o 0. So (ourInjecture[limit ofJexists.[6 3.414213562H060441189 5037317044 544332125N 582449523h 618259366651911k68353599713i48h74125N76729436 7917976680 3.8147122363609755256006478474486057N91578181 3.90732015 921745144 934882619M467586357396163 3.9668153033P9820698792439h9099883 3.9997965550[ 5We can conjecture thate limit (L) approaches to 4.=OProve anlytically[hNLet B bV e point (0,rS nd (x1,y1) coordinatgof F$z,0")L. Step 1)First, we needfixx1 y1 in terms)Nr. TF on both circlso[x1U2]+y1=r( (1)N-1)QG +y1 2 =1 (2)[It follows from that x1>, -2x1+1Y Yand =x1xfN=r.[Hence [+x1=r֐ڎ4y1=4-P4$4(11W9qStep 2.Y,We need to find z in terms of r. Let F' be ohe x-axis so tha perpendicularN AL.[ By@simi$riangles ==== [K length ALB@=* *F'+F,uzrez-x1y .[Th$implies)zr:r-<. 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