00020001010008main.ACT000502000fbernoulli's.EAC010000003ce5DkxC[Bernoulli's inequality[If x>-1, n2 then we have[(1+x)n1+nx.[[This can be written as[if x>0, (1+x)n(1+x)-1n or[if x<0, (1+x)n(1+x)-1n.[We can view the above[as the slope of between[ two points:[ P=(1+x,(1+x)n) and[Q=(1,1).[[For n=2, and let t=x+1,[we have [case 1:t2-1t-1>2, [when t>1 or x>0.[case 2: t2-1t-1<2,[when t<1 or x<0.[We can explore this[by the following:[\&RBR`2F` 1RBPexQV&vrALCuxy5H  H  B@60)  x^20`&1WG7 @   ! @6 WDC7Ys() gp  H  A@6 @    1@  9[-525-4 -4.473684211 20.01385042 -3.473684211 -3.947368421 15.58171745 -2.947368421 -3.421052632 11.70360111 -2.421052632 -2.894736842 8.379501385 -1.894736842 -2.368421053 5.609418283 -1.368421053 -1.842105263 3.393351801 -0.8421052632 -1.315789474 1.731301939 -0.3157894737 -0.7894736842 0.6232686981 0.2105263158 -0.2631578947 0.06925207756 0.7368421053 0.2631578947 0.06925207756 1.263157895 0.7894736842 0.6232686981 1.789473684 1.315789474 1.731301939 2.315789474 1.842105263 3.393351801 2.842105263 2.368421053 5.609418283 3.368421053 2.894736842 8.379501385 3.894736842 3.421052632 11.70360111 4.421052632 3.947368421 15.58171745 4.947368421 4.473684211 20.01385042 5.4736842115256[Notice when t>1, the [slope is greater than 2.[[ When n=3, [ consider t3-1t-1 where[ t=(x+1). \ ``vrALCuxy 5H  H B@6    x^3 `&1WG7 @  H  A@6 ! @6B6EYi3DF 6&xUP @   1@  9  [-5-12521 -4.473684211 -89.5356466 16.54016621 -3.947368421 -61.50677941 12.63434903 -3.421052632 -40.03863537 9.282548476 -2.894736842 -24.25645138 6.484764543 -2.368421053 -13.28546435 4.24099723 -1.842105263 -6.250911212 2.551246537 -1.315789474 -2.278028867 1.415512465 -0.7894736842 -0.4920542353 0.8337950139 -0.2631578947-0.01822423094 0.8060941828 0.2631578947 0.01822423094 1.332409972 0.7894736842 0.4920542353 2.412742382 1.315789474 2.278028867 4.047091413 1.842105263 6.250911212 6.235457064 2.368421053 13.28546435 8.977839335 2.894736842 24.25645138 12.27423823 3.421052632 40.03863537 16.12465374 3.947368421 61.50677941 20.52908587 4.473684211 89.5356466 25.48753463512531[Notice if t>1 (or x>0),[then the slope is >3[if -2t<1, then[the slope is less than 3.[[ When n=7,[\``vrALCuxy5H  H  B@6  x^7 `VAVA @  H  A@6 ! @6gfYgf @  8='1@  9([(-5-7812513021 -4.743589744 -54044.29422 9409.671761 -4.487179487 -36628.07473 6675.392124 -4.230769231 -24262.48949 4638.608285 -3.974358974 -15662.8085 3148.909956 -3.717948718 -9820.277642 2081.683848 -3.461538462 -5955.032421 1334.972784 -3.205128205 -3474.704221 826.5394185 -2.948717949 -1938.35194 491.1345823 -2.692307692 -1025.351673 277.9702447 -2.435897436 -508.877194 148.3970938 -2.179487179 -233.6039763 73.78673447 -1.923076923 -97.26948009 33.61850635 -1.666666667 -35.72245085 13.77091907 -1.41025641 -11.09395953 5.017706614 -1.153846154 -2.722922918 1.728499926 -0.8974358974 -0.4688393566 0.7741180393 -0.641025641-0.04447621403 0.6364776929 -0.3846153846-1.245049345-3 0.7231214245 -0.1282051282-5.692955397-7 0.886364141 0.12820512825.692955395-7 1.147058171 0.38461538461.245049345-3 1.622976795 0.641025641 0.04447621403 2.661816261 0.8974358974 0.4688393566 5.178816273 1.153846154 2.722922918 11.19899897 1.41025641 11.09395953 24.60402635 1.666666667 35.72245085 52.08367627 1.923076923 97.26948009 104.2919368 2.179487179 233.6039763 197.207719 2.435897436 508.877194 353.7001887 2.692307692 1025.351673 605.2987158 2.948717949 1938.35194 994.167443 3.205128205 3474.704221 1575.284472 3.461538462 5955.032421 2418.825671 3.717948718 9820.277642 3612.753095 3.974358974 15662.8085 5265.608029 4.230769231 24262.48949 7509.508652 4.487179487 36628.07473 10503.35231 4.743589744 54044.29422 14436.2224357812519531[Notice that when t>1[the slope is >7.Rsolve(t^7-1t-1-7=0,t)Rt=- 1.491797988,t=- 0.8057864694+ 1.222904713,t=- 0.8057864694- 1.222904713,t= 0.5516854635- 1.25334886,t= 0.5516854635+ 1.25334886,t=1[[Example.[Consider the points[P=(t,t^2) and Q=(2,2)[\``vrALCuxy5H  H  B@6YP   x^2Y`&1WG7 !@  "H  A@6# ! @6tb&YV1fGYb V1fG`Y"$@  #!$#%1@  " 9 " "#[-525-3 -4.473684211 20.01385042 -2.473684211 -3.947368421 15.58171745 -1.947368421 -3.421052632 11.70360111 -1.421052632 -2.894736842 8.379501385 -0.8947368421 -2.368421053 5.609418283 -0.3684210526 -1.842105263 3.393351801 0.1578947368 -1.315789474 1.731301939 0.6842105263 -0.7894736842 0.6232686981 1.210526316 -0.2631578947 0.06925207756 1.736842105 0.2631578947 0.06925207756 2.263157895 0.7894736842 0.6232686981 2.789473684 1.315789474 1.731301939 3.315789474 1.842105263 3.393351801 3.842105263 2.368421053 5.609418283 4.368421053 2.894736842 8.379501385 4.894736842 3.421052632 11.70360111 5.421052632 3.947368421 15.58171745 5.947368421 4.473684211 20.01385042 6.4736842115257[ **Conjecture:[If t>0, [then t2-f(a)t-a>2,[where f(t)=t^2; the above[is true for a=2.[\&` uG` uGvrALCuxy'5H  (H  B@6Y )  x^2Y` *@  +H  A@6 , ! @6 E7)bYY%1 w IY(+-@  (,*-,(.1@  +)9)+)+(,('[(-10100-7 -9.487179487 90.00657462 -6.487179487 -8.974358974 80.539119 -5.974358974 -8.461538462 71.59763314 -5.461538462 -7.948717949 63.18211703 -4.948717949 -7.435897436 55.29257068 -4.435897436 -6.923076923 47.92899408 -3.923076923 -6.41025641 41.09138725 -3.41025641 -5.897435897 34.77975016 -2.897435897 -5.384615385 28.99408284 -2.384615385 -4.871794872 23.73438527 -1.871794872 -4.358974359 19.00065746 -1.358974359 -3.846153846 14.79289941 -0.8461538461 -3.333333333 11.11111111 -0.3333333333 -2.820512821 7.955292571 0.1794871795 -2.307692308 5.325443787 0.6923076923 -1.794871795 3.22156476 1.205128205 -1.282051282 1.64365549 1.717948718 -0.7692307692 0.5917159763 2.230769231 -0.2564102564 0.06574621959 2.743589744 0.2564102564 0.0657462196 3.256410256 0.7692307692 0.5917159763 3.769230769 1.282051282 1.64365549 4.282051282 1.794871795 3.22156476 4.794871795 2.307692308 5.325443787 5.307692308 2.820512821 7.955292571 5.820512821 3.333333333 11.11111111 6.333333333 3.846153846 14.79289941 6.846153846 4.358974359 19.00065746 7.358974359 4.871794872 23.73438527 7.871794872 5.384615385 28.99408284 8.384615385 5.897435897 34.77975016 8.897435897 6.41025641 41.09138725 9.41025641 6.923076923 47.92899408 9.923076923 7.435897436 55.29257068 10.43589744 7.948717949 63.18211703 10.94871795 8.461538462 71.59763314 11.46153846 8.974358974 80.539119 11.97435897 9.487179487 90.00657462 12.487179491010013[the conjecture is true[ when a=3.[(-401600-32 -37.94871795 1440.105194 -29.94871795 -35.8974359 1288.625904 -27.8974359 -33.84615385 1145.56213 -25.84615385 -31.79487179 1010.913872 -23.79487179 -29.74358974 884.6811308 -21.74358974 -27.69230769 766.8639053 -19.69230769 -25.64102564 657.4621959 -17.64102564 -23.58974359 556.4760026 -15.58974359 -21.53846154 463.9053254 -13.53846154 -19.48717949 379.7501644 -11.48717949 -17.43589744 304.0105194 -9.435897436 -15.38461538 236.6863905 -7.384615385 -13.33333333 177.7777778 -5.333333333 -11.28205128 127.2846811 -3.282051282 -9.230769231 85.20710059 -1.230769231 -7.17948718 51.54503616 0.8205128205 -5.128205128 26.29848784 2.871794872 -3.076923077 9.467455621 4.923076923 -1.025641026 1.051939514 6.974358974 1.025641026 1.051939513 9.025641026 3.076923077 9.467455621 11.07692308 5.128205128 26.29848784 13.12820513 7.179487179 51.54503616 15.17948718 9.230769231 85.20710059 17.23076923 11.28205128 127.2846811 19.28205128 13.33333333 177.7777778 21.33333333 15.38461538 236.6863905 23.38461538 17.43589744 304.0105194 25.43589744 19.48717949 379.7501644 27.48717949 21.53846154 463.9053254 29.53846154 23.58974359 556.4760026 31.58974359 25.64102564 657.4621959 33.64102564 27.69230769 766.8639053 35.69230769 29.74358974 884.6811308 37.74358974 31.79487179 1010.913872 39.79487179 33.84615385 1145.56213 41.84615385 35.8974359 1288.625904 43.8974359 37.94871795 1440.105194 45.9487179540160048[ **Conjecture:[tn-ant-a >n if t>0\/ i`V`7``V`7vrALCuxy05H  1H  B@62  x^2`3@  14 ! @6 74g8 C&IYE"YY5H  A@6@16@  54364171@  529252514(0[eAct02000fcsc/sec/cot.EAC0100000004a9Dkx*[csc(n)/sec(n)/cot(n)[[To use the function,[execute the line below.[R Define library\csc(n)=1/sin(n)RdoneR Define library\sec(n)=1/cos(n)RdoneR Define library\cot(n)=1/tan(n)Rdone[You will see the function[on the soft_key/cat[/User.[[Try to use it![In Rad,R csc(/2)R1R sec(/4)R2R cot(/3)R33R[In Deg,R csc(90)R1R sec(45)R2R cot(60)R33R define f1(x)=sec(x)xRdoneRf1(x)Rsinxcosx2Rsimplify(f1(x)-sec(x)*tan(x))R0Rf1(x)|x=2R UndefinedR f1(x)|x=0R0ReActcotH n1/T(n)cscH n1/L(n)f1H x:(sec(x),x)secH n1/P(n)020012eActivity Save.EAC010000000046 ( ~E [:eAct02000fruben-jan06.EAC01000000158aDkx[sin(x)R&define f()=piecewise(>1,^2,-1)RdoneRf(2)R4Rf(0)R-1Rf()Rx-1Rf(+2)Rx+1R\Graph2D@ Graph3DT LISTSYS`4NModify $ STATCALC NSTATSYS \NSequence, SheetH| Sheet3D| SolveEq@SolveLwrD SolveUprP StupFLG1\(StupListD StupPict$ ViewWind y1H , 8 D P \ h t           ! " # $ % &( '4 (@ )L *X +d ,p -| . 0 1 2 3 4 5 E F H I J K L$ M0 N< OH PT Q` Rl Sx T ] ^ _ ` a b      system]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 Dkx $  3xf(x)`P`P`p`p `(10qy`#YuY ` P[eActfH$x(x>1,x^2,x-1)02000dshiftings.EAC010000002c5aDkx[Shiftings and inverses[ a)If f(x)=sinx moves [down; look at y2,y3 [below.[=> f-1 moves ??[b) If f moves up[=> f-1 moves ??\graphsLGraph2D Graph3D LISTSYS4NModify $ STATCALC DNSTATSYS L\NSequence, Sheet| Sheet3DP| SolveEqSolveLwr SolveUpr StupFLG1(StupListD StupPictT$ ViewWindx y1,Hy2Hy3 Hy4 Hy5Hy64Hy7PHy8hH              ( 4  @ !L "X #d $p %| & ' ( ) * + , - . 0 1 2 3 4$ 50 E< FH HT I` Jl Kx L M N O P Q R S T ] ^ _ ` a b    , 8 system]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 Dkx $  3xL(x)|x/2|x(-)/2 3xM(x)  3xy2(x)+/2  3xy1(x+/2)  3x2*M(x)  3xL(x/2) 3xL(x) 3xQ(x)`P`P`` `(10qy`#YuY `P7#2Y` W[[About sec(x) and [csc(x)[\graphsGraph2D| Graph3D LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceX, Sheet| Sheet3D| SolveEq|SolveLwr SolveUpr StupFLG1(StupListD StupPict$ ViewWind( y1D$Hy2h$Hy3,Hy4,H      , 8 D P \ h t      ! " # $ % & ' ( ) * +( ,4 -@ .L 0X 1d 2p 3| 4 5 E F H I J K L M N O P Q$ R0 S< TH ]T ^X _\ `` ad bh l x    system]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 Dkx $  3xP(x)|x|x0 3xsec(x)|x|x0 3xL(x)|x/2|x(-)/2 3xcsc(x)|x/2|x(-)/2`P`P`p`p `(10qy`#YuY P`%0BG`` S[eActcotH n1/T(n)cscH n1/L(n)f1H x:(sec(x),x)secH n1/P(n)0301c8002000080000000011a9x_l\՝'{ )5ehm'i8!$ MbKcnZv0i: U UhF[]Zej -aZ塪`ϝsz [h)ݶt<|9wf{[.&/u$ }i+$H|B!G  {:0z0[._m+ xQcli~kY#fŬc,fb?ʘ}4fkb Ǭ/f_c+f7l"f7H̖l*fln٘}2fǬ?fcvs^٧bv1f쵘UbFn/bfV@̺c61򘭈@VluVlclG6l$fc'fbv8f_13f1ٙms1s1+fb5f/m1l{.lGތ7+1ك1+l$f嘍liٲƬ/[11-f?ٺų5[cWb;fl2f?ј,fǛYPoͥ齁֠9}M:W8=H#F5=z:tͣF'G io_[[Lf -xk:ch ⺥BmMh[ߝ G.-8ydU PxExexU8ƼzmmUms]EտB$W,tqYmy/ 4=͸:澟s5g.z6DhGClh/v!ڥp퓰}OI> 'a$lLC5/FrRjV?YOoK7gnyGs?-GI7*|s?u>({?t{ZF꙱UO=Vl~p<{vfm OKgLΦ٦wmټ~Kφ{K0,>>qglۡ};G5nnE,g{DkhtaQzFvhbϲ w,M[B۵o߽{u WbՏ~Z_\xJmRVTj++Uo}xp`J&}dwo~hoźǡJoZcov=XWF\˓;KO/֖̔'k=3=}X 3/kk'v-.,iݚ8ׅGB{9ڵB.K]ڛ- :'9hxҞXxҞLys[H>>Zg]0}س5tn[7w6oW;z^.-Zu~ڧC{=~`h mhzW#tV-pS^[k:օ oÝ \'wfK_w?vjG{x[mc~?EWڿmΔ:M}SJⰏB?rˇ}JW.h|aZt/Wuso]zUH:IghRRƉo\,$6ښSdR♾dehK5U:үՅ)*|:Uf-O93gT']VgOvwo+ g9oStBxgzU|料sT}I?~pxI맟 @?ybaDwguXw17}żp)3SgSŅ;f煫Sg;+S#3gM}y<*jg1O`6~nwu4_xz ׾>&LwRdEk!>}f鑰=sTZqu}Ooپyͣ6_}I;[>vɮ{p[[1dj[n)q:_ 3>]hxSnFwL?<*e{kl}鹽æ-q宥|]74~]oo䓴?7Iώ7K2>>Ozrh؁}Q6_<5r"-yר]鏾H~&uv3UYzvҿi}{5ǚ#$)BZre5,糾F|w峬?ϲZ|ՙgY6泬|>jYV1e,ϲZ,"5kW粬ߝϲZ|糬?ϲZ|7泬ֿ#mhd#,gYp>jO䳬?ϲZ|gY\>jYv|_gYϲZ|gC5{Zw>j|6,/gYY>j}+,ߖϲZ|泬>jYVgYh>jsYv?#_:=`=zOyN 瘎_Wϕw?m 6կm:ci?clY. -\w -\vk?Z֟wih/_3.NVceqvѿ9DhGClh/v!p zh}'aBmMh[C m"M'8p\Ԫ'jBkDk\ԪOwaG}BN_(j~_&)|@.\_.Z9XVt<{sSO^~xۺwu