微分方程与几何变换演示

微分方程与几何变换演示
几何画板及绘图内容加载中,请稍等……
2017-07-01 19:13
①本范例演示微分方程在圆柱体上的几何变换与切线动态演示;②涉及核心数学概念包括微分方程、切线、几何变换、数列级数与分段函数;③构造要点:使用 GeoGebra 命令如 Function, Textfield, Slider, Locus 等创建函数图像、动态切线与参数变换;④适合高中及以上数学教学与函数几何性质研究。

GeoGebra 命令及参数

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  2. Function({0, 10.000000000000007, 0.7999999999999999, 0.8752, 0.9583510016, 1.0501921165211063, 1.1514982233981628, 1.263069454555173, 1.3857153663916875, 1.5202320567937317, 1.6673700100965332, 1.8277898615658303, 2.0020025772719054, 2.190289760992533, 2.3925989895045614, 2.6084083834365264, 2.8365543271746514, 3.075016883946109, 3.3206599257894736, 3.568928812531464, 3.8135199012510497, 4.046056410943917, 4.25583783303534, 4.429777619763094, 4.552703457739886, 4.608249911396379, 4.580583646111448, 4.45709291399128, 4.231854805636671, 3.909133129956639, 3.5055265020977577, 3.0491637365600748, 2.57513378738342, 2.11818827840561, 1.7055615946682963, 1.352892706250512, 1.0643516067442036, 0.8357583727568898, 0.6585245134342721, 0.5228261566833096, 0.41949473536419357, 0.34082759143067537, 0.2807343836656168, 0.23456804205780568, 0.19885338932534985, 0.1710178419573574, 0.14916385146209857, 0.13189087758100787, 0.11816174580275705, 0.1072045590576779, 0.09844153572474389, 0.09143763354596164, 0.08586348120696945, 0.08146857795164501, 0.07806184582032603, 0.07549745396861147, 0.07366443772631025, 0.07247906451364376, 0.07187920231923771, 0.07182016055994969, 0.07227162429436992, 0.07321540980615962, 0.07464384575286731, 0.07655863863219126, 0.07897012068650712, 0.08189680703220767, 0.08536520989550472, 0.0894098735573291, 0.09407360550421448, 0.09940788844524814, 0.10547346506710684, 0.11234109321722148, 0.12009247402843674, 0.1288213596123233, 0.13863485055370955, 0.1496548966731337, 0.16202001747250178, 0.1758872613836666, 0.1914344253994446, 0.20886255883776694, 0.22839877677893342, 0.2502994099704455, 0.2748535184855003, 0.30238679582399625, 0.33326588800362184, 0.36790314787535267, 0.40676183755211404, 0.4503617802867135, 0.49928544577344075, 0.5541844274923833, 0.6157862343844756, 0.6849012677614057, 0.7624297823319031, 0.849368529894804, 0.946816645054974, 1.0559801397075836, 1.1781741069153335, 1.3148213674157503, 1.4674457850453027, 1.6376577782014474, 1.827128591215014, 2.0375485653525263, 2.270562836605849, 2.5276754251252616, 2.810109379752678, 3.118606312263284, 3.453143195113333, 3.8125378820381135, 4.193908340518526, 4.59194657258632, 4.997972498477316, 5.398757918743226, 5.775179310994545, 6.100909968303936, 6.341651173506961, 6.455871154189063, 6.3986038151959574, 6.130163355363517, 5.630649650952778, 4.917404749431624, 4.056111633816887, 3.152205038905274, 2.3178134351158235, 1.6311501318060313, 1.1168232918781755, 0.756748681757512, 0.514670004106935, 0.3548021373924212, 0.2493997501863131, 0.17930222620011588, 0.1320001499075696, 0.09951632771335805, 0.07678916971282522, 0.06058945402736665, 0.048833640092713404, 0.04015801875901577, 0.033655879232542416, 0.02871455018443249, 0.024913380561623573, 0.021959070544805045, 0.01964410679583877, 0.01781965998503759, 0.016377655473242516, 0.015238743418813207, 0.01434411647047296, 0.013649872077709813, 0.013123080977713404, 0.012739015229556802, 0.012479174849631689, 0.012329871780013943, 0.012281208041044727, 0.012326336552525479, 0.012460927657235908, 0.012682787782368922, 0.012991592721366955, 0.013388709161673264, 0.013877085925554406, 0.014461201990982733, 0.015147062426151336, 0.015942236383812692, 0.016855933588361967, 0.017899117536888425, 0.019084655085272226, 0.020427503316225167, 0.02194493567147016, 0.023656810337525015, 0.025585884852055932, 0.027758181884177814, 0.030203412169784283, 0.03295546168039075, 0.03605295129715737, 0.039539878575715594, 0.043466352646939405, 0.04788943492911771, 0.0528741001542532, 0.05849433426280995, 0.06483438802589168, 0.07199020784160909, 0.08007106805439923, 0.08920143239391325, 0.09952307575498079, 0.11119750157138109, 0.12440869449887992, 0.1393662530350599, 0.1563089520706804, 0.17550879117674179, 0.19727559064193984, 0.22196220380387516, 0.24997042092038702, 0.28175764647273094, 0.31784443802115053, 0.3588230000046898, 0.4053667293884296, 0.45824091063473216, 0.5183146533916113, 0.5865741550499938, 0.6641373482852715, 0.7522699555703611, 0.8524029106905859, 0.9661510101554472, 1.0953325083495467, 1.241989144450885, 1.4084057494385365}) → Rblue
  3. Function({0, 10.000000000000007, 2, 2.14, 2.292368, 2.45564016860032, 2.627425793304048, 2.8043280921716884, 2.9819804566819763, 3.1552363235686913, 3.3185237109045955, 3.4663323497715304, 3.5937551229903404, 3.6969763018489727, 3.773603916723309, 3.8227846184721708, 3.8451000492138423, 3.842298033606764, 3.8169398603707934, 3.7720424428296266, 3.7107711021692444, 3.6362097201092465, 3.551211070326843, 3.458315687960833, 3.3597219789776487, 3.25729047127178, 3.1525681740032825, 3.046822840440097, 2.941080399912699, 2.8361615174171764, 2.7327151276512365, 2.6312480126847726, 2.532150226272885, 2.435716570971093, 2.3421645275885528, 2.2516491046584304, 2.1642750740110808, 2.0801070217968904, 1.9991775929457996, 1.921494252584976, 1.8470448363460938, 1.7758021154964108, 1.7077275633029219, 1.6427744758950666, 1.5808905735762808, 1.5220201863225862, 1.4661061093540262, 1.4130912004881924, 1.3629197798931465, 1.3155388843609659, 1.2708994219344034, 1.2289572683424057, 1.1896743440250448, 1.1530197094216288, 1.1189707165957112, 1.0875142571831562, 1.0586481501453924, 1.032382718024729, 1.0087426075305694, 0.9877689195933428, 0.969521725817893, 0.9540830628940996, 0.9415605143177554, 0.9320915099967088, 0.9258484990075403, 0.9230451785163897, 0.9239439914218198, 0.9288651338404469, 0.9381973357681412, 0.9524106844379077, 0.9720717334253957, 0.997861053818144, 1.030593192369523, 1.0712386363503337, 1.120946741970982, 1.1810675141734257, 1.2531684318131036, 1.3390399576520977, 1.4406797378167957, 1.560240718404249, 1.6999228748522723, 1.8617833098895586, 2.0474381649392734, 2.2576375321304867, 2.491719033672806, 2.746994340765915, 3.0181957749091564, 3.2971887948096117, 3.573194408522225, 3.8336989603578346, 4.066019249399235, 4.259192055032767, 4.405632747383816, 4.502034997507946, 4.549288161212706, 4.551587854028264, 4.515165342197581, 4.447061026394608, 4.354198584974381, 4.242825319026365, 4.118261558482537, 3.984859082423165, 3.846076049506505, 3.704602290318623, 3.5624951813827916, 3.421305784700086, 3.2821870577489713, 3.1459825500904715, 3.013297126088622, 2.8845524128030764, 2.7600298383759343, 2.6399038545582254, 2.524267518478837, 2.4131521820478317, 2.3065426597056584, 2.2043889332589166, 2.1066152046212867, 2.0131269147173256, 1.9238161992846714, 1.838566140191369, 1.7572540860323298, 1.6797542516613917, 1.6059397578837202, 1.5356842359086604, 1.468863093426683, 1.4053545181387794, 1.3450402786012012, 1.2878063701394962, 1.2335435444335152, 1.1821477545147898, 1.1335205418620415, 1.087569388677317, 1.044208056026854, 1.0033569271733676, 0.964943375015572, 0.9289021730532784, 0.8951759707370328, 0.8637158565239328, 0.8344820355949364, 0.8074446542186843, 0.7825848094904441, 0.759895792067013, 0.739384621143897, 0.7210739460585137, 0.7050044085834295, 0.6912375855547729, 0.6798596647422378, 0.6709860501226894, 0.6647671489467143, 0.6613956659538958, 0.6611158244406965, 0.6642350550896093, 0.6711388474912405, 0.6823096515897804, 0.6983509495305099, 0.7200178877186384, 0.7482561425416335, 0.7842509348162474, 0.8294881845134614, 0.8858294625479514, 0.9556011871983708, 1.0416956025879083, 1.1476750744317383, 1.2778599774149182, 1.4373609540497192, 1.6319854724436134, 1.8679056138380372, 2.1509266891317207, 2.4851729590987484, 2.8710715625359295, 3.302769989229881, 3.7656466126437906, 4.235257188137062, 4.679379026911385, 5.063932915536749, 5.36124158506847, 5.556832017393484, 5.651292443715768, 5.656874082555752, 5.591625364492454, 5.474222350384324, 5.3209543370172465, 5.144677122366229, 4.95493043156397, 4.7585241238835625, 4.560197598449798, 4.363188577501065, 4.169671565876159, 3.9810773251375466, 3.7983189937911317, 3.6219499281143914, 3.4522733359439246, 3.289418456079505, 3.1333936803672198, 2.9841237947461003, 2.8414762473476123, 2.7052797929990735, 2.57533780448469, 2.451437824202693, 2.3333584443949196, 2.220874274059486, 2.1137595249751744, 2.0117905938987186}) → Fred
  4. Function({0, 10.000000000000007, 3, 3.0300000000000002, 3.030606, 2.99888840494944, 2.933228307707805, 2.833733773931501, 2.7025234940632, 2.5437809611612554, 2.363511999185351, 2.1690028563418657, 1.9680593510552193, 1.7681776174492405, 1.5758205076247531, 1.3959370129946684, 1.2317819070296283, 1.0850064660902485, 0.9559331612261428, 0.8439113783246616, 0.7476692174814377, 0.6656095472688406, 0.5960302659688136, 0.5372707563510544, 0.4877977378277122, 0.4462468638265951, 0.41143499738417455, 0.38235486550208764, 0.35816036771183946, 0.33814793224341827, 0.32173717139632285, 0.30845262462759976, 0.29790744343262554, 0.2897893119267784, 0.28384858467526963, 0.2798884663402758, 0.27775699326656594, 0.27734056465541435, 0.27855878543590756, 0.28136040949997726, 0.28572020224183253, 0.29163657088476974, 0.29912983775015467, 0.30824105458741896, 0.3190312751925885, 0.3315812189775179, 0.34599127021589937, 0.362381766695016, 0.3808935376978819, 0.4016886547932299, 0.4249513598771144, 0.4508891332254171, 0.47973385978465083, 0.5117430442061349, 0.5472010137080183, 0.5864200320340678, 0.6297412266517384, 0.6775352037229384, 0.7302021898189944, 0.7881714940230655, 0.851900026769145, 0.9218695398920896, 0.9985821628812718, 1.0825536998724907, 1.1743040169596972, 1.2743436867000055, 1.3831558640003194, 1.5011721451183608, 1.6287409143016107, 1.7660864243950487, 1.913256617437885, 2.070057522691536, 2.2359720664212657, 2.4100614450857036, 2.5908480963317704, 2.7761811195047987, 2.9630882740709072, 3.147624107179657, 3.3247321116738258, 3.4881507353378867, 3.630408508994157, 3.7429707724784556, 3.816614352397887, 3.84210677324076, 3.8112368774663974, 3.7181652062270456, 3.5609254725934125, 3.3427343783520764, 3.072630691561974, 2.7649878188382586, 2.437726826810236, 2.1095567566506577, 1.797039753718554, 1.5123989136630267, 1.2626376519472047, 1.0499461232059475, 0.8729135001630619, 0.7279561104004364, 0.6105409596363705, 0.5160329428055951, 0.44017724976332595, 0.37931028396450384, 0.33040196445130704, 0.2910096833659781, 0.25919587610569983, 0.23343837334529555, 0.21254764535028323, 0.19559631300216546, 0.18186177882281102, 0.1707807953631828, 0.16191408379488942, 0.1549190571872191, 0.14952891663038737, 0.1455366850123063, 0.1427830359337704, 0.14114702988138905, 0.14053907781929442, 0.140895616419758, 0.1421751058521005, 0.14435505770578572, 0.14742987387322756, 0.1514093325755245, 0.1563175995524066, 0.16219267412679667, 0.16908620394726873, 0.17706362061835582, 0.186204562564728, 0.19660356237981585, 0.2083709843350491, 0.2216342042051435, 0.23653902845818023, 0.25325135338724664, 0.27195906702437866, 0.29287419766928724, 0.31623531246815284, 0.3423101674585271, 0.37139860648857415, 0.40383569989529494, 0.4399951040594887, 0.48029260896229625, 0.5251898213369262, 0.5751979041980556, 0.6308812571373532, 0.6928609727572861, 0.7618178389721565, 0.838494569367079, 0.9236968274580346, 1.0182924564960036, 1.1232081226426254, 1.2394223107436142, 1.3679532592322892, 1.5098399598429741, 1.6661137497962248, 1.8377572559377304, 2.0256464784180075, 2.230470600267564, 2.452622679192076, 2.6920527821179205, 2.9480735556690627, 3.219107136314137, 3.502362610650078, 3.7934366714360417, 4.085839731198304, 4.370470541644708, 4.635101665756707, 4.864004924951548, 5.037946687415222, 5.134910067552731, 5.132009519944317, 5.009035757999757, 4.753700113263411, 4.367728665220968, 3.871575379284556, 3.3045134643601823, 2.7177755140650772, 2.162057539384862, 1.674974275030326, 1.2746918686989481, 0.9616396922086896, 0.7250944204271372, 0.5500466387873444, 0.42177820233920427, 0.32794655384035604, 0.25903599761331675, 0.2080460229341814, 0.16995348504358515, 0.14119500459604778, 0.11925002509769692, 0.10233118814075949, 0.08916368026856408, 0.07883118528642025, 0.07066928066769074, 0.06419179563548591, 0.05903980757035121, 0.054946131510731425, 0.05171042714441159, 0.04918161468310019, 0.0472453548983172, 0.04581506573414953, 0.04482543064014008, 0.044227679438614415, 0.04398614327174678, 0.04407573571097838}) → Rred
  5. Textfield(R01) → IBoxR1
  6. Textfield(F01) → IBoxF1
  7. Textfield(R02) → IBoxR2
  8. Textfield(F02) → FBox2