00010001010008main.ACT00030200072^x.EAC010000002187^[The derivative of[2^x\``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`Vq   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\Graph2D@ 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`gsystema`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 ^ $ `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 !  $0<HT`lxqy T" qxX qxT q  qyR qw'x qQ qxc2 qW 14 1E 1ix 2 14$ 1P 1P 1P 1D3 1rUy 1P ` P[eActconstant 02000eCircleArea.EAC01000000214d^ [\Circle Animation``vr ALCuxy5@  H! I@6iEW#h"i@ ! @61AFYG%dT H! E@6G%dT 1AF H# B@6 1 Yd 0%$iH!  @#  &1WG7 H#    @6 #A@6 H! C@6Q(vtqTY%@  @   H  H  H@6iEW`#h"i@` 2H# H# F@6G%dTY1AFY@  # @6 H  D@6Gs!9U aY%@  H! @  H!    [Open Geometry and select[Edit/Animation/Go (once)[Data from animation[Radius Area[1 3.141592654 1.526315789 7.318779561 2.052631579 13.23646101 2.578947368 20.89463701 3.105263158 30.29330755 3.631578947 41.43247264 4.157894737 54.31213227 4.684210526 68.93228645 5.210526316 85.29293517 5.736842105 103.3940784 6.263157895 123.2357163 6.789473684 144.8178486 7.315789474 168.1404755 7.842105263 193.203597 8.368421053 220.007213 8.894736842 248.5513235 9.421052632 278.8359286 9.947368421 310.8610282 10.47368421 344.626622411 380.1327111[\Graph2D@ Graph3DT LISTCAL`( LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceD, Sheetp| Sheet3D| SolveEqhSolveLwrl SolveUprx StupFLG1(StupListD StupPict$ ViewWind 0 < H T ` l x            ! " # $ %, &8 'D (P )\ *h +t , - . 0 1 2 3 4 5 E F H I J K( L4 M@ NL OX Pd Qp R| S T ]D ^D _(D `l ap bt x     list2/(list1)^2system]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@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i ^ $ `P`P`p`p `(10qy`#YuY  $0<HT`lx Rcx&1WWG6RccxpxsphBR`!&1`shBP&1WPxsh@1WG@!&06!0G6  BRc  shBshB   $0<HT`lxe@1yV26Fcp0uP2G&@C"p2(dPRQp9@x@##W0DxH`huP 5 rHU#PxY(`( Dbf"@' $0<HT`lxe@eSce$BeCBe)e@e0e93e1e&$ eE e5V e0 eCv e@ e8e6Pe6`eAe7` P[eActa b 020015DynamicPolarGraph.EAC0100000020e8^8[*When I first graphed[these functions, I was[given an error. Ignore[ the error!\Two Polar CurvesHGraph2D Graph3D LISTSYS4NModify $ STATCALC 0NSTATSYS 8\NSequence, Sheet| Sheet3D<| SolveEqSolveLwr SolveUpr StupFLG1(StupListD StupPict@$ ViewWindd r1Hr2 Hr3 Hxt4$Hxt5,Hyt4,$Hyt5P,H|              $ 0  < !H "T #` $l %x & ' ( ) * + , - . 0 1 2 3 4 5, E8 FD HP I\ Jh Kt L M N O P Q R S T ] ^ _ ` a b    ( 4 system]listsystem^]=system_^system`_systema`systemb~////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// a7b  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@PSheet1Jl^Sheet2Jl^Sheet3Jl^Sheet4Jl^Sheet5Jl^SheetSheet3D 0@PSheet1Jl^Sheet2Jl^Sheet3Jl^Sheet4Jl^Sheet5Jl^SheetSheet3D i ^ $   5na*P(n) 5na*P(b*n) 5na*L(b*n) 4t(2+P(t))*P(t/a) 4t(1+P(t))*P(t/5+a*P(t)) 4t(2+P(t))*L(t/a) 4t(1+P(t))*L(t/5+a*P(t))`P`P``0 Yp00`0#YuY0 ((( ((($&!7!D !UGY$H9P`` V[-Graph the curves[-Open the diamond menu[and select Graph Controller[-Select Manual mode [ -Tap Modify[-Press the right arrow in[the graph window to [advance a; down arrow[ to advance b[ Example 1.[For equation 4, we choose[the variable a from 1 to[6; step 0.2. The variable['b' from 1 to 10 will not[affect the graph in this[case.[[ Exercise.[Oberve the animation[from Example 1. When['a' gets larger and larger,[(1) How is the graph[being affected?[(2) Prove or disprove[that the total area of [the parametric equation[gets closer and closer[to the area of a circle, [when 'a' gets larger and[larger.R define x(t)=1+costcost5+ccostRdoneR define y(t)=1+costsint5+ccostRdoneRaR0Rx(t)tR-sincostc+t5+costsincostc+t5+5sintcoscostc+t5-5sintsincostc+t5c-5sintcostsincostc+t5c5R define z(t)=-5cost5sint+sint5cost+sint55RdoneR y(t)*z(t)3tRERROR :Insufficient MemoryR[ Example 2.[For equation 5, we choose[the variable a from 0 to[6; step 0.2. Zoom in[the picture if necessary.[The variable 'b' from 1[to 10 will not affect the[graph in this case.eActa (b (xH4"t(1+P(t))*P(((t)/(5))+c*P(t))yH4"t(1+P(t))*L(((t)/(5))+c*P(t))zHXEt((-(5*P(((t)/(5)))*L(t)+L(((t)/(5)))*P(t)+L(((t)/(5)))))/(5))