00020001010008main.ACT000202000ederivslope.EAC01000000194e  x^(2)-3x-5  `bu  v: +D Dr  A 'L' C uxy q9  K1@6 "G& `ts7 ڎ Slope:  jjiVfD Y yB9a  69eu F.@v 3X $ x^2-3 -5 H A@6  `  7.@      <"@1@ 2PH 7 derivslope.EAC main.ACT!+ 2  ^E 2i[4#D_ ative As Sh of Tangent Line+ by Ravin KumarAlcorn State University , MS 39096VrkK@aD.eduo Objec[Objective of this e-Aity to exhibit why deriva0a fun>on atpoint6slopRangelineL the graphoeA. Recallat@0Dhcur= ed  `l,atd. Tfollowing anim߆showɐj at differ֊s.[h\|AE:Έ $/2A`3R .` 1B Dv r  A  L' C uxy h q9  K1@6 w$qY9$sCJSlope:  fj"jtangentp aRBeptr0Gd Ph39$|(5HM5"PPR }  0.5x^2-3F `^ cxsi  .@  304J4d<"@1Ġ2PH [[ 3A line PQ through a point P of curve is called;F seca(@W of the curve if it passes rough ano" r point Q2. Difference between a seca,line?d a chord is similar toxd?F? segment. ORwonders what happenoxPQ ae draws nearΈ P. Obse followingimation?swer.[\S e TangentΈ!%3 8 ` ## Y d@!5D0vr  A  "L' C uxy c l5H" u~ Q@67 &  U 0.1x^3-+1JV"bshBSY@V "8@&yx yEf!XHP&56Y #.@ $3Ȗ"% tangent@R6Y!5d Y #%f3 aETD&uHcy'y"@(1Ơ2PH Ȉ2!%[Let us analyze. The abov5nimation shows that as Q approaches P, secant line PQ'rough P'V%at Pi refore, [    slope of secant line PQP. = 1tange2at P[SLNow let P(a,f(a)) and Q(a+h+h)), where equationthe curve is yrf(x)[SPQ P-gf-a5*h"QePuivalto he0. Thus, [# ht05nBut the right hand side ofabove equation is nothing b9f'(a),@valu6deriv2;Hf at x = a.[Hence,Pslopz angent line=}l tosnx Example 1: qGrapha funcgive nfollow strip. Use i7 o approximateits1.  Solu\:|% In order to approximate the value of $ivative we draw a tangent lin3o6cur$ at x = 1 andYits slope.[Exap't strip. Notic at between x-s -3V 3, run is 6frise-10. Therefore,nǚ;ly equal D/6 -1.67, whichkX an1-+4Ffunction .\A}ing S"Ή )$* x)8`cYI@ rH1 4Q( 1v 9>r  A 'L' C uxy qq*H  u+H"B62cSIb"4>,ȐUBQ@ x $Q60 Wi0 -=.@.3ȒX/ 0.2x^3- sin()-w0 A߉~H dpH` , 1.H  /.20-<"@3e1@2PH 7+[ ZLet us now2e the slope calculation utility of ClassPad and fiout7 actual value*jS?`tangent line.\<V7S~Έ4ڈlCC}29ep!9W4g<@vr  A  L & C uxy R 5h9  K1@x6  "G& `2XBif CJSlope:  f6ɐjiQ@ xv $Q60 Wi  N7F.H%83‡79 0.2x^3- sin()-1":nH AlH dpH;y.@!M98<.H  :76";=B1Ġ2PH 75y[UTActual value of slope shown in the above strip is pretty close to our approximation.\d?What if we were give[at`func8 withr graph is[ f(x)=0.2x3-x2sinx*1R diff(.2*x^3-x^29m-1,x,1,1)R - 5cos1" +10sin;-3D5RKQ approx(ans)Ri"> 1.623244275b2D'T[-`These calculations and the slope obtained ug wim,+ClassPad confirm<reN.p Example 2:/ Firequ of tangent line to curve y=x  -4x-1 at x=1po Solu؎n E delvar xY"[resto x as a variable][R y=x3! -4x-1R&-4(Z1x81s[1 in xansh>[J5[saves to h thevaluex-coordinate ofpointdiff(,x)搋-[calculnsu derivative at= 1mi[+*[saves derivative value at x=1 to slope m]2R?GxQ3Y -4x-1Rj-42x /[calculat}y-coordinate of the point when]CanskcW&ϴRkˍ delvar xdoneZ[resto x as&riableEqu@onangeline iso y=m*(x-h)+y=-x-31F!Expand the following strip to see+equation calculated by ClassPad.[\  Verific4Έ> & < `(s0 ` p Dv<>r KA 'L'C uxy qq?9  0@6% "Gts7G! Eq: Wgfb`!l 2we R% A.@  B3 C$ x^3-4 -1DaH HB@q6mw} `3E}FHsÈ<"@G1ĘL2PH  `ATPX~?b9[  Exercises: [Given below is a function and.graph with tangent line at x = 2. Actual slopeClso displayed. You may expSthe strip to seef , approximatePof1{", (verify your ^answer using)cdetermƈuvaluud]vat![f[p6Feel fro chH!ftering, hitEnterJQ while you are on the line of *expression. This will automatically update;Graph , window, and gener.anolr problem to do.[4]E \UΈ*eAct020012eActivity Save.EAC01000000194e  x^(2)-3x-5  `bu  v: +D Dr  A 'L' C uxy q9  K1@6 "G& `ts7 ڎ Slope:  jjiVfD Y yB9a  69eu F.@v 3X $ x^2-3 -5 H A@6  `  7.@      <"@1@ 2PH 7 derivslope.EAC main.ACT!+ 2  ^E 2i[4#D_ ative As Sh of Tangent Line+ by Ravin KumarAlcorn State University , MS 39096VrkK@aD.eduo Objec[Objective of this e-Aity to exhibit why deriva0a fun>on atpoint6slopRangelineL the graphoeA. Recallat@0Dhcur= ed  `l,atd. Tfollowing anim߆showɐj at differ֊s.[h\|AE:Έ $/2A`3R .` 1B Dv r  A  L' C uxy h q9  K1@6 w$qY9$sCJSlope:  fj"jtangentp aRBeptr0Gd Ph39$|(5HM5"PPR }  0.5x^2-3F `^ cxsi  .@  304J4d<"@1Ġ2PH [[ 3A line PQ through a point P of curve is called;F seca(@W of the curve if it passes rough ano" r point Q2. Difference between a seca,line?d a chord is similar toxd?F? segment. ORwonders what happenoxPQ ae draws nearΈ P. Obse followingimation?swer.[\S e TangentΈ!%3 8 ` ## Y d@!5D0vr  A  "L' C uxy c l5H" u~ Q@67 &  U 0.1x^3-+1JV"bshBSY@V "8@&yx yEf!XHP&56Y #.@ $3Ȗ"% tangent@R6Y!5d Y #%f3 aETD&uHcy'y"@(1Ơ2PH Ȉ2!%[Let us analyze. The abov5nimation shows that as Q approaches P, secant line PQ'rough P'V%at Pi refore, [    slope of secant line PQP. = 1tange2at P[SLNow let P(a,f(a)) and Q(a+h+h)), where equationthe curve is yrf(x)[SPQ P-gf-a5*h"QePuivalto he0. Thus, [# ht05nBut the right hand side ofabove equation is nothing b9f'(a),@valu6deriv2;Hf at x = a.[Hence,Pslopz angent line=}l tosnx Example 1: qGrapha funcgive nfollow strip. Use i7 o approximateits1.  Solu\:|% In order to approximate the value of $ivative we draw a tangent lin3o6cur$ at x = 1 andYits slope.[Exap't strip. Notic at between x-s -3V 3, run is 6frise-10. Therefore,nǚ;ly equal D/6 -1.67, whichkX an1-+4Ffunction .\A}ing S"Ή )$* x)8`cYI@ rH1 4Q( 1v 9>r  A 'L' C uxy qq*H  u+H"B62cSIb"4>,ȐUBQ@ x $Q60 Wi0 -=.@.3ȒX/ 0.2x^3- sin()-w0 A߉~H dpH` , 1.H  /.20-<"@3e1@2PH 7+[ ZLet us now2e the slope calculation utility of ClassPad and fiout7 actual value*jS?`tangent line.\<V7S~Έ4ڈlCC}29ep!9W4g<@vr  A  L & C uxy R 5h9  K1@x6  "G& `2XBif CJSlope:  f6ɐjiQ@ xv $Q60 Wi  N7F.H%83‡79 0.2x^3- sin()-1":nH AlH dpH;y.@!M98<.H  :76";=B1Ġ2PH 75y[UTActual value of slope shown in the above strip is pretty close to our approximation.\d?What if we were give[at`func8 withr graph is[ f(x)=0.2x3-x2sinx*1R diff(.2*x^3-x^29m-1,x,1,1)R - 5cos1" +10sin;-3D5RKQ approx(ans)Ri"> 1.623244275b2D'T[-`These calculations and the slope obtained ug wim,+ClassPad confirm<reN.p Example 2:/ Firequ of tangent line to curve y=x  -4x-1 at x=1po Solu؎n E delvar xY"[resto x as a variable][R y=x3! -4x-1R&-4(Z1x81s[1 in xansh>[J5[saves to h thevaluex-coordinate ofpointdiff(,x)搋-[calculnsu derivative at= 1mi[+*[saves derivative value at x=1 to slope m]2R?GxQ3Y -4x-1Rj-42x /[calculat}y-coordinate of the point when]CanskcW&ϴRkˍ delvar xdoneZ[resto x as&riableEqu@onangeline iso y=m*(x-h)+y=-x-31F!Expand the following strip to see+equation calculated by ClassPad.[\  Verific4Έ> & < `(s0 ` p Dv<>r KA 'L'C uxy qq?9  0@6% "Gts7G! Eq: Wgfb`!l 2we R% A.@  B3 C$ x^3-4 -1DaH HB@q6mw} `3E}FHsÈ<"@G1ĘL2PH  `ATPX~?b9[  Exercises: [Given below is a function and.graph with tangent line at x = 2. Actual slopeClso displayed. You may expSthe strip to seef , approximatePof1{", (verify your ^answer using)cdetermƈuvaluud]vat![f[p6Feel fro chH!ftering, hitEnterJQ while you are on the line of *expression. This will automatically update;Graph , window, and gener.anolr problem to do.[4]E \UΈ*eAct0301d700200008000000001901xlq'Әpb'ʩ6iIC7v롞e]e2U8; vMh&NX'@1M1?mV6ƴ}7ˀSMJ 568x(e7G?88xy?>r#C_hiΦ;Ʒ656ƓMMK5/TTSRMK5/TTSRMK5/TdSMK6/TdSMK6/TdSMBiΦ;Ʒ6?MM㩦083Ig;FG}xK ;O. 5g Pͤvh%'e*[:U6wl|1Ee!5TvzA)&QӍN2w3gl |άlPuIf,lh{1uuVWsF拒Jf%YynvȽ򘴱SxdKCp*^ZeTdUXt*o|[G).ggH{Wz{uoF.nb^m]N]M:Vg=:ۤ]m:^g{tvuYg:r:Qg}VgOٲnYݬtvGgt9Mݪ:^Fgy:ۢ:3u֧uvtKgI,Gt6ΆtVVٰΖt[ggtGgl^ٗt=ݧWu6l~uF욐,:۬1u6!v'd_#.==:و:alUg_uΦtc':;u7t7uvNg?ً: e}?Ofbki9:@8` Jv yKPGmwnoGL?dz|xa'S_Nuyߗ.Juޱ3Qo}W~yg[ـa3GEow'.x$EsbkƚӴ4Ə~)eBʬR R^K!iߐ ߐ ߐ ߐN|ZЄo8Q>l֧G8~o/?_[<&TzO=j9g|׭a9k4nmU'+gcCGkL&.wz9y}sBIF/'%^j~5UsV-cg꽲wZv^]:ԛlnvrG[olu#lTM_6Stn %r:q΄>a_Ӿ]=ԛ}ږDm QK&jP5NM'zqE ^nywSɾ' T曛I3P\I7ի|v[qzayqiUj'k%jo<rٝ,ZOBuݞ%lg!,ÏkxDvo6N$jDm9 lƿQe] MRޔr\rH,jJdƍ.S>++Jo<$i`.yRkTn9 v="=622S`WY>j{ zj<ص$S`=u, vj9 l9|{x{_ ^vHdk4?/m)RޕT;$Ҟ I{7ViOZcxѽZe+voh4{DJ,_uO}ҠYi#' ^ҠU_}NFA/iP2pP }euS~ں}^P _CRީCrX=1w'MHN[]^2bO.ߒ:%쬘?z{NoHbq m푁dĞYO% T~!NڏX5nuSәg˿S'>t8+Eo_ {o>_n1qC!WUCta.zqNwXRlaT#2km1mUd. c6`ВmW?mj2Bv]Jg5jDq+덶̔2 e{u{pt,s1j^ϙƐϙ꫚5uEx"xن9VuՕ`ΰWjq]|[W=6[⿝ o76U7=+[Vu6Z=͐/۞HFNv;!Aey}V 9|{~= [{>\"֝t?8g+F=M<̄>=/8:Ϛ)cysOm1Y8{l|xh}{ŒUV;|gr5[n|orw河 )ywlhJ'A{~ZKs]fxpsŸ^,}QήÚQ4;j3cOQȠ~VϡaZgI=~w*d6dm.ߛ/”5aqkP}T CX%RhcǗNspV#h?(>mG*J|ۗLTD4wؠjQu:'B̗3mx͟gn??s \gn?3@վ'gn?ӟ'p?s gI'gn?GO ~e8~K'pƟgn?O{'?s ԟgn?wO=C ]8Yԟ63@ܟNx5{??]'ǟM8و?sgn?q.n?)gn?c'~'pΟ^gn?O_v%wI!?a5_e)r ~n$;`?yߖgvx6tksURAhy \?~f}]i?ww5G_3֚}Vtn7n )RKYrNKR.HR.e2SʀRFLHr\ʲۉOW7o8Z`Diigpφ}d/?}O 8ӺFIMOj,km+_>pl>rlԇk?[_oxi_1|׷9$ރ5\6G´6'G-3[2'ӹ5%KMdMe3[|=jr6es6]*Y%#3/Esy4oC*> a?"3dgl!/]S;?1=eG_\IfI5.N.|-e (~vXHOfd&Qdeg"+*ZRa+aF 2N- ډ v>5'{L=Ç/̛9ih<{{{ ޒQ[2 ]Rz|inW2drGƼI31 ũzgI}jN*}i jלd;".ZjRtN5EEk@mR"M4pW_ϕG2ꠕSNYVpmvlSaqVYEž5%g=j'Ӫ`Z=\ӹ[!N[{ gfs[!%c}񙖻ה#7Pr1;o>PFHjM\LmWdvbͽr{75s޲RnHV\| V7wD6άXrۻU슔̾A{i 8go;bd}ɁSe 5}U-,;d@w1;Qӗڗ4_{>Hp;B^fi_I+a^s%gw8푻\9e_9 -2 ,zF˺̶Aozfnii΂5YIjP䁭,uJ1R+64CΖ:d;}4ll9[Dʖ"ŕJ{Q4^rW0ܸ/|3FohqjyozqePt-g5>ngfQ^j054sk$"GtaL}8{pYied߀֞Tt8\a ]Rw;ܭњ=oCP/i$} 332eʃN%JRrW>-[1D. !!'/nDKN" ~qCF>G-9*A.UM*)_-sv7eGҫԌ3S+tnMj32[FikVsԳG^-SY=jCx;pwb}u,i,;qx/ӻ2W]B$nv-Q)>eKNg'­?JwraLk_~[2^Ịۏuǹ]!oIgߨEJyz/ TpaΜIˍx7^Kf-[uŒU-\0 ^ȣZU3٬.Wpk͌Dv[VΜ.Z`2jpŖ-1-w>:$̌<8u8/u+ڔofmsk Jll.guS̹)]坕<rzVQM rΨL}'T5r;Gsrp>t!VI04fa