00020002010008main.ACT0001020012eActivity Save.EAC010000001e0e slopeoftangent.EACline.ACT),69=P^E '*[5Sk of the Tq Ll at a point 5\= AuthorUvGProfessor, Dr.Wei-Chi Y Departm鈇Math/Stats Radford University, VA 24142 USA e-mail: wy2@r8 .edu URL:www./ ~([<  ObjectkWe shall investigate( 5relationship between the slopes of secant lin and0 ('= =tange4 at a point. _ f'(a)=  f(a+h)-)h0DL ExamplegGiv graphfunclow, Rdef f(x)=-(x-1)^2+2R"done2 expand(2)(K-xV2^ +2TQs diff?,xA-.x-1R [ find the ' slope of tangent -line at0 point B=(-0.1,0.79) byPusing various approaches.\qqΈŊ  Y `F3 47D vKr  A 'L'?C uxy  q 5H  !C@6&   $ -(x+-1)^2+2; `K Y"iI%e n@Si !%@6 q!Y Rd3 H BCY   GGY". ;.Ș 3DD L6HA/yG!xrun .@      "61Ġ29 7d<q[Method 1. Animation. The lengths BC, 4 slope of and Ot2 4tangent line atgpoi B is captured8- below: Part 1derivve froms+ileft:I column 1=xC2=펀#column 3=slope BC <-2  8.0183614. -1.9677983058 7.824042579/ 4.0$7355966S66317652 4.0#l0339491l 441527443l$8711932 7.2533283H 3.9#8389j2؆1670l6$88, 6.8837687$F 7745881367700954603.8$ -1.742386441 6.52090163 3.8#6101847466 6.3428829077$m67798305m716689745I3.7$6457813% 5.992944277$637966ۆ7 8210223157$58136n 5.65113.6#54917627m4832$3.6$516731743287I6$ -1.484772881 5.1536247183.5$ 7525711867 4.991842046#620369492% 4.832083577m$m36779% 4.674348014$35596610n4.518634!32376440k364940123.4$2915627121326F3$25931n063606913.3$2j593 3.915964509327322 -1.194957627% 77033610572$725K\6267200\2$n3055423n 4851145212$0983529% 3.34551781"06615084207927981$153072343141$H00174745 2.93876124$ -0.969542.807180214 3.069545763 -0.9373440678& 2.6775978678%$8 051423729855001194p%p p82o42442919$ 840738983& 3008199159% 808537288^17920882p%7%355932& 2.05958418$74413 1.934327I8%78322031.826283209 2.811932203 -0.6797305085& 1.71260101887%8 4752881368 600893597p7%4p15327118& 1.491157717$ 583125423] 1.38338998]6%o5509528 1.2775866#n187'3;1737447L6$m 48656K071859825$ 454318644 0.9719279931 2.554318644 -0.4221169492& 0.8739451299&9 3899152549 7779068626r4&r 3577135593r 6838086299&32551 59164565754&293310&0.50145 2.3%7 261108474 0.4131052R3$228779KR326716^o&805084 0.2422429413 2.296705085 -0.1645033898&915'651769 9&9832301694 0.0790116285Lr'r*s001*2.410428-4 2%0000  Part 2. \derivative from the rightΈ   `@Gs . Dv$Kr  A kL'C ux'y 5H  #H C@36 & ]fȖU -(x+-1)^2+2G6 YURT#r @ sF#D64 Y# Q SQdd3AY:Y m"..Ș.3ɔDB 6 Y  ,  @;B 6HAByG!xBruBB.@ Bz"@1Ġ2P9 ""!Ȗy=-(x+-1)^2+2708HbA<%N[]column 1=the x of C  2=length BC 3=slope<W2 2.110473880.1 1.964408475- 2.08329918? . 355915254728816949758817[771183050xn8932G2I 2.03539034n 2067745763838914367 24236610182204237I910032779l278644I977008653475\0457 0.349147X9 1.944980342 0.3847322 1.679676271930495207422372887440M46I91K49[n45591525 1.65990370839 4915067797 57290169589106913$ 527098305537318I 878716103562688501718#8664940I5982813%4661F1n54056338F14305HZ 841853548 0.6694644068 1.394982915*27 70505593275254 816034309n7i474576n237610178023638977623128^9478995438118385277967729217772203321698'75693738883013181M1 1.7399792Ȇ9186L84n$1458~7219m040.9541966102 1.1 11864 702781663$ 7 8978813567 074620339I682375977[02537om03902Jm66066333$60970034378H6j7209656271و 67845762761303862I132154322586998201677\#8W1559393591Ɇ8 1.530170Y23895 0.825479661 1.49927764927452033 0.7898881356766667693I 3101118647542l02n3229587I 1.3457m 187050847$961 38129491568315581020716846475282042385247661193663924988060.57633L523269Y52301 0.5407474576 1.186900309 559252542$ 705132739161254I 594844068746956n 089390793I63043sn 433972881%03756390I666027113983465709070836278*05 927648238 1.73721017$3~61JJ516^7Ɇr9\2806779K 809242503o8083p 0.2560152542 0.7468076365 1.843984% 22042372888 6821946073879576271%18483=34p6153870)p 915167797% 0.14924067o5495589950759_\11364N2̆ 47512755298635084o 0.078 401647291 2.09 424661016,325791?95753389>6.874+64-O 247920777/t931A-0.02871694916 0.1676506412 2.1'%: 6430847458: N 850943990N;(5`*v9#000001*2.416517994-4199v  By inspection, when the )distance BC gets closerIto 0,3 slope of*^1 )&_/o c gent line at  poiB.hMethod 2.Numerically,RDefOf()=- -1)^2+2R$ done[ We let x=-0.1 R'". 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