00010001010008main.ACT00020200072^x.EAC010000003fe8"^\:IR[Introducing a special[number.[[ Example One:[Compare some values of[2xand 2xxfor[corresponding values of[x.[\tangent at a point``vrALCuxyH# `G6!`shB`BRc `G6 `6!0`!&0`1WG@`G6 YcxpYcxp G6 1WG@!&06!0G6 BRc shBG6!`P 2P I11 GWP 2Hp 4#1P 2 xDP 8P wW0 1Pp v3rU%1HQWtEpP7d BE%%fTvq ! 5H  H  A@6%`%VYS6   2^x%``&1WG7 1@  9[-50.03125 0.02166084935 -4.473684211 0.0450077043 0.03119696325 -3.947368421 0.06482219028 0.0449313183 -3.421052632 0.0933599351 0.06471217575 -2.894736842 0.1344613233 0.093201487 -2.368421053 0.1936574553 0.1342331185 -1.842105263 0.2789144794 0.193328785 -1.315789474 0.4017056133 0.2784411125 -0.7894736842 0.5785551186 0.40102385 -0.2631578947 0.8332620064 0.5775782123 0.2631578947 1.20010272 0.8318550167 0.7894736842 1.728443787 1.198076338 1.315789474 2.489385179 1.72552531 1.842105263 3.585328385 2.48518183 2.368421053 5.163756792 3.579274457 2.894736842 7.43708284 5.155037642 3.421052632 10.71123281 7.42452525 3.947368421 15.42681597 10.6931466 4.473684211 22.21841828 15.40076712532 22.180902\ List EditorGraph2D@ Graph3DT LISTCAL`, LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceH, Sheett| Sheet3D| SolveEqlSolveLwrp SolveUpr| StupFLG1(StupListD StupPict$ ViewWind 4 @ 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 ]D ^D _,D `pD a b       list3/list2system]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 "^\:I $ `P`P`p`p `(10qy`#YuY  $0<HT`lx G6!`shB`BRc `G6 `6!0`!&0`1WG@`G6 YcxpYcxp G6 1WG@!&06!0G6 BRc shBG6!  $0<HT`lxP PpC H" 3YQ 4F#0 etU0 xDy@ pV0 xUQ` 3& @  rrCxpHXS(8P7Vy Cp#(T&Yp"A   $0<HT`lx`P 2P I11 GWP 2Hp 4#1P 2 xDP 8P wW0 1Pp v3rU%1HQWtEpP7d BE%%fTvq !  $0<HT`lxqy qxX  qxT q  qyR qw'x qQ qxc2 qW 14 1E 1ix 2 14$ 1P 1P 1P 1D3 1rUy 1P ` P[[**What is list3/list2?R2^xxR2xln2Rln2R 0.6931471806R[ Exercise One.[Insert a Graph Editor[strip and explore the[ functions axand axx[where a=2.2, 2.4,[2.6, 2.8 and 3.[For each function[compare the value of[the function and the[derivative at the same[ value of x.[What do you notice?[Now insert a List Editor[and investigate in a[similar way as shown in[ the example.[[ Exercise Two.[Insert another Graph[Editor strip or List[Editor and experiment[to find the value of a[in axand axx for[which ax=axx.[[Exercise Three.[ Now consider axx.[You should be confident[from the previous[exercise that this[function is an[exponential function.[But exactly what[ function?[Use your previous[findings to make a[conjecture about[axx.[[Once you have made a[conjecture, you can[check your result using[the computer algebra[capabilities of this unit.\Check your conj. $RaxxGraph2D, 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^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system   0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D 0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D i "^\:I $ `P`P`p`p `(10qy`#YuY ` O[[Exercise Four.[Prove that your[conjecture from exercise[three is correct using[traditional calculus[ techniques.[[End.[[[ [ [[[[[eActdybydx P 8Tp       @    x  $0<HT`l y  $0<HTdt      @ 020010introduction.EAC0100000075bc"^\:I[Introducing a special[number.[\AuthorAnthony Harradine Noel Baker Centre for School Mathematics, Prince Alfred Collge Adelaide, SA 5074 Australia e-mail: aharra@pac.edu.au[[ Objective:[To be introduced to a[special number by[comparing the value of[numerous exponential[functions to the value[of the corresponding[derivative function.[[ Example One:[Compare some values of[2xand 2xxfor[corresponding values of[x.[[To do this we can use[the table maker in[the Classpad. We can[define two functions -[tap the strip below to[see this in the Graph[Editor.[\Defined functions(Graph2Dh Graph3D| LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceD, Sheetp| Sheet3D| SolveEqhSolveLwrl SolveUprx StupFLG1(StupListD StupPict$ ViewWind y10Hy2H Hh t                ( !4 "@ #L $X %d &p '| ( ) * + , - . / 0 1 2 3 4 5 E F$ H0 I< JH KT L` Ml Nx O P Q R S T ] ^ _ ` a b      system]listsystem^]=system_^system`_systema`systemb~  aseq_histbNewFolde system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system   0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D 0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D i "^\:I $  3x2^x 3x:(2^x,x,1)`P`P`p`p `(10qy`#YuY     $0<HT`lx ,8DP\ht(4@LXdp|qU 8b6 w%r"@TQwDDy`5H !pw@CaAU("gVwDVx"3 TVDg@ x'3AeBW  `0s0 gap1G@cVR4)P'h'hY`U6T& 6%p1  RrP5b@pCtP$(c@Iv&t(&t(` R[With the Graph Editor[active, tap the right[arrow and then the[first icon to set the[table limits. Set the[start value to 0, the[end to 20 with steps of[1. Then tap the right[arrow again and then[the second icon to[produce a table of[values.[If you study this table[closely you should find[that the derivative[values are always less[than the corresponding[values of the function.[If you tap the graph[icon when the Graph[Editor is active you will[see this graphically.[[Further analysis can be[carried out using the[List Editor. First go to[the V shaped icon in the[top left of this window.[Choose Settings, then[Setup, then Basic Format[and ensure that the[Decimal Calculation option[is unchecked. Tap the[icon below to open the[ List Editor.[\ List EditorGraph2D@ Graph3DT LISTCAL`$ LISTSYS4NModify $ STATCALC NSTATSYS \NSequence@, Sheetl| Sheet3D| SolveEqdSolveLwrh SolveUprt StupFLG1(StupListD StupPict$ ViewWind , 8 D P \ h t            ! " # $ %( &4 '@ (L )X *d +p ,| - . 0 1 2 3 4 5 E F H I J K$ L0 M< NH OT P` Ql Rx S T ] ^ _ ` a b      N2^xdiff(2^x,x)eActxIʮfeActyIʮfeActdybydxsystem`_systema`systemb~++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ aseq_histbNewFolde system]listsystem^]=system+system]listsystem^]=system+system]listsystem^]=system+system]listsystem^]=system+system]listsystem^]=system+system]listsystem^]=system+system]listsystem^]=system+system]listsystem^]=system+system]listsystem^]=system+system]listsystem^]=system+  0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D 0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D i "^\:I $ `P`P`p`p `(10qy`#YuY ` P[[Each list can be given a[title by simply typing it[into the list header. I[named the first list x[and then entered the[numbers 1 to 10. The[second list was named[y and in the calulation[(Cal) box at the bottom[I entered 2^x (without[the " "). The ClassPad[calculates the values.[List3 was then named[dybydx and in the cal.[box the command[diff(2^x,x) was entered.[[What do you notice[about the values of the[derivate of 2^x?[\ **Animation``vrALCuxy@ `G6!`shB`BRc `G6 `6!0`!&0`1WG@`G6 YcxpYcxp G6 1WG@!&06!0G6 BRc shBG6!R8"P cP CpG4P V IVP Vwg7P 6t EX X0tP aPB "y`9p Fi'F aS!U0f'Cep10H Id#S2q 7Fp9&rIt3pfdR 5H  H  A@60`DU   3^x0``&1WG7 .@  3   !! @6 uDAbY $$0f "! Y .@  .@    1@   9  [-54.115226337-34.521038225-3 -4.4736842117.336859057-3 8.0603635-3 -3.947368421 0.0130805687 0.01437047345 -3.421052632 0.02332078021 0.02562049565 -2.894736842 0.04157761044 0.04567767375 -2.368421053 0.07412692346 0.081436749 -1.842105263 0.1321576859 0.145190058 -1.315789474 0.2356182225 0.2588530745 -0.7894736842 0.4200735386 0.461504292 -0.2631578947 0.7489309442 0.8227960397 0.2631578947 1.33523659 1.466927462 0.7894736842 2.380535568 2.615321553 1.315789474 4.244153909 4.662743657 1.842105263 7.566718448 8.313003988 2.368421053 13.49037507 14.82089498 2.894736842 24.05140626 26.42353328 3.421052632 42.88021202 47.10937467 3.947368421 76.44927548 83.9892672 4.473684211 136.2981069 149.74083375243 266.9664522[\ list editorGraph2D@ Graph3DT LISTCAL`, LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceH, Sheett| Sheet3D| SolveEqlSolveLwrp SolveUpr| StupFLG1(StupListD StupPict$ ViewWind 4 @ 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 ]D ^D _,D `pD a b       list3/list2system]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 "^\:I $ `P`P`p`p `(10qy`#YuY  $0<HT`lx`G6!`shB`BRc `G6 `6!0`!&0`1WG@`G6 YcxpYcxp G6 1WG@!&06!0G6 BRc shBG6! $0<HT`lxR&3p 3hYp 0V 3 x wa@ A&4` 2v 5a"P  58` H D 3R6Y85V$ASVgD47Pp@Q@b`(! dI'T6)C $0<HT`lxR8"P cP CpG4P V IVP Vwg7P 6t EX X0tP aPB "y`9p Fi'F aS!U0f'Cep10H Id#S2q 7Fp9&rIt3pfdR  $0<HT`lx ( (Y  (5x (Ve (Y ( ) (y5 '8  '7(  '6  '7Q  '7#u  '6  '4u '71Q '7&E '7 '9g '7S ` P[**What is list3/list2?R3^xxR3xln3Rln3R 1.098612289R[ Exercise One.[Insert a Graph Editor[strip and explore the[ functions axand axx[where a=2.2, 2.4,[2.6, 2.8 and 3.[For each function[compare the value of[the function and the[derivative at the same[ value of x.[What do you notice?[Now insert a List Editor[and investigate in a[similar way as shown in[ the example.[[ Exercise Two.[Insert another Graph[Editor strip or List[Editor and experiment[to find the value of a[in axand axx for[which ax=axx.[[Exercise Three.[ Now consider axx.[You should be confident[from the previous[exercise that this[function is an[exponential function.[But exactly what[ function?[Use your previous[findings to make a[conjecture about[axx.[[Once you have made a[conjecture, you can[check your result using[the computer algebra[capabilities of this unit.\Check your conj.$RaxxGraph2D, 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^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system system]listsystem^]=system   0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D 0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D i "^\:I $ `P`P`p`p `(10qy`#YuY ` O[[Exercise Four.[Prove that your[conjecture from exercise[three is correct using[traditional calculus[ techniques.[[End.[[[ [ [[[[[eActdybydx P 8Tp       @    x  $0<HT`l y  $0<HTdt      @