00010001010008main.ACT0001020010introducinge.EAC010000005356"^\: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 functionsXGraph2D| Graph3D LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceX, Sheet| Sheet3D| SolveEq|SolveLwr SolveUpr StupFLG1(StupListD StupPict$ ViewWind( y1DHy2\ Hy3|H            ( 4 @ L  X !d "p #| $ % & ' ( ) * + , - . /  0 1 2 3$ 40 5< EH FT H` Il Jx K L M N O P Q R S T ] ^ _ ` a  b   , 8 D 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)  3x_(2)*2^x`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(` S[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?[[ 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      @