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Trazado el gráfico de la función/ acos(2*x)/(3*x)^7

Gráfico de la función y = acos(2*x)/(3*x)^7

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       acos(2*x)
f(x) = ---------
              7 
         (3*x)
f(x)=acos⁡(2x)(3x)7f{\left(x \right)} = \frac{\operatorname{acos}{\left(2 x \right)}}{\left(3 x\right)^{7}} 
f = acos(2*x)/(3*x)^7
Gráfico de la función
02468-8-6-4-2-1010-20000002000000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
acos⁡(2x)(3x)7=0\frac{\operatorname{acos}{\left(2 x \right)}}{\left(3 x\right)^{7}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=12x_{1} = \frac{1}{2}
Solución numérica
x1=1471.32973958607x_{1} = 1471.32973958607
x2=1158.89542606175x_{2} = 1158.89542606175
x3=602.056473706186x_{3} = 602.056473706186
x4=985.114787855564x_{4} = 985.114787855564
x5=1332.52222531459x_{5} = 1332.52222531459
x6=1297.8078768074x_{6} = 1297.8078768074
x7=111.361225874841x_{7} = 111.361225874841
x8=636.934404724423x_{8} = 636.934404724423
x9=1610.0643229548x_{9} = 1610.0643229548
x10=950.337370059771x_{10} = 950.337370059771
x11=497.334885342624x_{11} = 497.334885342624
x12=741.493806795019x_{12} = 741.493806795019
x13=1124.15254357883x_{13} = 1124.15254357883
x14=1436.63503680835x_{14} = 1436.63503680835
x15=567.164656154555x_{15} = 567.164656154555
x16=811.145364530444x_{16} = 811.145364530444
x17=1193.63218636955x_{17} = 1193.63218636955
x18=1714.07210895198x_{18} = 1714.07210895198
x19=706.65228172375x_{19} = 706.65228172375
x20=392.453015240084x_{20} = 392.453015240084
x21=880.758634212624x_{21} = 880.758634212624
x22=1228.36304569086x_{22} = 1228.36304569086
x23=322.416357480814x_{23} = 322.416357480814
x24=1506.01989310111x_{24} = 1506.01989310111
x25=671.799395430489x_{25} = 671.799395430489
x26=915.552110207711x_{26} = 915.552110207711
x27=75.9361790041621x_{27} = 75.9361790041621
x28=1401.93564791926x_{28} = 1401.93564791926
x29=462.394254611521x_{29} = 462.394254611521
x30=181.93284963984x_{30} = 181.93284963984
x31=1540.70562708945x_{31} = 1540.70562708945
x32=1679.40674150116x_{32} = 1679.40674150116
x33=532.257873291891x_{33} = 532.257873291891
x34=427.434296752946x_{34} = 427.434296752946
x35=1367.2314283789x_{35} = 1367.2314283789
x36=1644.73751320616x_{36} = 1644.73751320616
x37=1263.08821109672x_{37} = 1263.08821109672
x38=252.257554492062x_{38} = 252.257554492062
x39=1089.40330217957x_{39} = 1089.40330217957
x40=1575.3870646973x_{40} = 1575.3870646973
x41=357.448011350186x_{41} = 357.448011350186
x42=1019.88470788229x_{42} = 1019.88470788229
x43=217.119743191886x_{43} = 217.119743191886
x44=287.354413827215x_{44} = 287.354413827215
x45=146.685508638801x_{45} = 146.685508638801
x46=845.956534315827x_{46} = 845.956534315827
x47=776.324635246017x_{47} = 776.324635246017
x48=1748.73371205935x_{48} = 1748.73371205935
x49=40.3960598031214x_{49} = 40.3960598031214
x50=1054.64744793835x_{50} = 1054.64744793835
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en acos(2*x)/(3*x)^7.
acos⁡(0⋅2)(0⋅3)7\frac{\operatorname{acos}{\left(0 \cdot 2 \right)}}{\left(0 \cdot 3\right)^{7}}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
−212187x71−4x2−7acos⁡(2x)2187x8=0- \frac{2 \frac{1}{2187 x^{7}}}{\sqrt{1 - 4 x^{2}}} - \frac{7 \operatorname{acos}{\left(2 x \right)}}{2187 x^{8}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=78.7940795564258x_{1} = 78.7940795564258
x2=278.394976367891x_{2} = 278.394976367891
x3=1021.08313134483x_{3} = 1021.08313134483
x4=749.282527123547x_{4} = 749.282527123547
x5=178.797838279845x_{5} = 178.797838279845
x6=476.974609689871x_{6} = 476.974609689871
x7=501.75849223042x_{7} = 501.75849223042
x8=153.84650574719x_{8} = 153.84650574719
x9=1119.82984114028x_{9} = 1119.82984114028
x10=452.183587057101x_{10} = 452.183587057101
x11=551.306544013764x_{11} = 551.306544013764
x12=897.586840972451x_{12} = 897.586840972451
x13=526.535672975249x_{13} = 526.535672975249
x14=128.866678098607x_{14} = 128.866678098607
x15=971.693523872324x_{15} = 971.693523872324
x16=650.333498726326x_{16} = 650.333498726326
x17=1243.20824116267x_{17} = 1243.20824116267
x18=946.994361535753x_{18} = 946.994361535753
x19=823.45094976312x_{19} = 823.45094976312
x20=798.731940538824x_{20} = 798.731940538824
x21=625.584659076191x_{21} = 625.584659076191
x22=848.166350522206x_{22} = 848.166350522206
x23=1095.14703237356x_{23} = 1095.14703237356
x24=699.81686287529x_{24} = 699.81686287529
x25=1144.5101890313x_{25} = 1144.5101890313
x26=1070.46169361927x_{26} = 1070.46169361927
x27=103.852087939105x_{27} = 103.852087939105
x28=402.578085767997x_{28} = 402.578085767997
x29=996.389750376227x_{29} = 996.389750376227
x30=303.254548429237x_{30} = 303.254548429237
x31=28.5243286418337x_{31} = 28.5243286418337
x32=872.878273483271x_{32} = 872.878273483271
x33=1218.53710952328x_{33} = 1218.53710952328
x34=328.101562506083x_{34} = 328.101562506083
x35=1169.18814179761x_{35} = 1169.18814179761
x36=675.077493012357x_{36} = 675.077493012357
x37=1045.77375215041x_{37} = 1045.77375215041
x38=576.071457923845x_{38} = 576.071457923845
x39=53.6819515431366x_{39} = 53.6819515431366
x40=600.830733257726x_{40} = 600.830733257726
x41=922.292167722582x_{41} = 922.292167722582
x42=774.00918299421x_{42} = 774.00918299421
x43=377.762414470033x_{43} = 377.762414470033
x44=228.632245079739x_{44} = 228.632245079739
x45=253.521444050014x_{45} = 253.521444050014
x46=1193.86376206063x_{46} = 1193.86376206063
x47=203.72526650764x_{47} = 203.72526650764
x48=352.937188345093x_{48} = 352.937188345093
x49=427.384931578109x_{49} = 427.384931578109
x50=724.551811671774x_{50} = 724.551811671774
Signos de extremos en los puntos:
(78.79407955642583, 1.39508190624688e-16*I)

(278.39497636789145, 2.47496942120498e-20*I)

(1021.0831313448284, 3.28534697691978e-24*I)

(749.2825271235473, 2.76067689126431e-23*I)

(178.7978382798453, 5.14457080803649e-19*I)

(476.97460968987076, 6.14960778081261e-22*I)

(501.7584922304195, 4.3426396431877e-22*I)

(153.84650574718955, 1.43954342991918e-18*I)

(1119.8298411402766, 1.74076321958067e-24*I)

(452.18358705710114, 8.87219453195338e-22*I)

(551.3065440137643, 2.27410284513643e-22*I)

(897.5868409724508, 7.97421312594656e-24*I)

(526.5356729752486, 3.11864814962527e-22*I)

(128.86667809860663, 4.83842508513076e-18*I)

(971.6935238723241, 4.62068652511177e-24*I)

(650.3334987263257, 7.30852386735321e-23*I)

(1243.2082411626664, 8.47935181714575e-25*I)

(946.9943615357529, 5.51600855436775e-24*I)

(823.45094976312, 1.44265098023491e-23*I)

(798.731940538824, 1.77901976976562e-23*I)

(625.5846590761911, 9.54183429154065e-23*I)

(848.166350522206, 1.17717462983356e-23*I)

(1095.1470323735552, 2.02928978590909e-24*I)

(699.81686287529, 4.414921172857e-23*I)

(1144.5101890313044, 1.49827311050176e-24*I)

(1070.4616936192724, 2.37394535043899e-24*I)

(103.8520879391047, 2.11596427127981e-17*I)

(402.5780857679967, 1.97016571846197e-21*I)

(996.3897503762267, 3.88797675942522e-24*I)

(303.2545484292369, 1.37661115355481e-20*I)

(28.524328641833748, 1.40975706451349e-13*I)

(872.8782734832705, 9.66193126601589e-24*I)

(1218.5371095232783, 9.73362418679613e-25*I)

(328.1015625060831, 8.02036998113639e-21*I)

(1169.1881417976088, 1.29370594339493e-24*I)

(675.0774930123573, 5.65411407648308e-23*I)

(1045.7737521504127, 2.78734649267313e-24*I)

(576.0714579238447, 1.6815096951713e-22*I)

(53.681951543136584, 1.91106220009712e-15*I)

(600.830733257726, 1.25928189379296e-22*I)

(922.2921677225821, 6.61578926902286e-24*I)

(774.0091829942099, 2.20835980556204e-23*I)

(377.7624144700329, 3.04904803536308e-21*I)

(228.63224507973877, 9.54710793918443e-20*I)

(253.5214440500136, 4.70179524499972e-20*I)

(1193.8637620606335, 1.12050821131225e-24*I)

(203.72526650763982, 2.10431329479303e-19*I)

(352.9371883450929, 4.86140755112575e-21*I)

(427.38493157810893, 1.30683849833981e-21*I)

(724.5518116717735, 3.47715518010723e-23*I)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
La función no tiene puntos mínimos
La función no tiene puntos máximos
No cambia el valor en todo el eje numérico
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
4(−2(1−4x2)32+7x21−4x2+14acos⁡(2x)x3)2187x6=0\frac{4 \left(- \frac{2}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}} + \frac{7}{x^{2} \sqrt{1 - 4 x^{2}}} + \frac{14 \operatorname{acos}{\left(2 x \right)}}{x^{3}}\right)}{2187 x^{6}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=818.258632133597x_{1} = 818.258632133597
x2=496.863369528893x_{2} = 496.863369528893
x3=723.795042308353x_{3} = 723.795042308353
x4=21.6980023586667x_{4} = 21.6980023586667
x5=421.129053394141x_{5} = 421.129053394141
x6=440.067913304901x_{6} = 440.067913304901
x7=307.406137090723x_{7} = 307.406137090723
x8=648.189215854146x_{8} = 648.189215854146
x9=874.916194984461x_{9} = 874.916194984461
x10=856.031943603464x_{10} = 856.031943603464
x11=250.472195645206x_{11} = 250.472195645206
x12=136.378693804879x_{12} = 136.378693804879
x13=950.438141397837x_{13} = 950.438141397837
x14=931.559836364419x_{14} = 931.559836364419
x15=383.239562161987x_{15} = 383.239562161987
x16=799.369480308063x_{16} = 799.369480308063
x17=893.798900886767x_{17} = 893.798900886767
x18=704.896690561853x_{18} = 704.896690561853
x19=591.461497214716x_{19} = 591.461497214716
x20=402.186345531058x_{20} = 402.186345531058
x21=231.480349028689x_{21} = 231.480349028689
x22=742.691436996212x_{22} = 742.691436996212
x23=610.373127285493x_{23} = 610.373127285493
x24=60.0202467019942x_{24} = 60.0202467019942
x25=288.434502043211x_{25} = 288.434502043211
x26=837.146104219462x_{26} = 837.146104219462
x27=780.47859918509x_{27} = 780.47859918509
x28=193.471499659078x_{28} = 193.471499659078
x29=98.2433681495375x_{29} = 98.2433681495375
x30=79.1450115184696x_{30} = 79.1450115184696
x31=553.63057978937x_{31} = 553.63057978937
x32=155.422142702756x_{32} = 155.422142702756
x33=515.788716218373x_{33} = 515.788716218373
x34=685.996316133469x_{34} = 685.996316133469
x35=912.680101809778x_{35} = 912.680101809778
x36=477.934892966081x_{36} = 477.934892966081
x37=629.282336684525x_{37} = 629.282336684525
x38=761.585936487191x_{38} = 761.585936487191
x39=40.8648530328833x_{39} = 40.8648530328833
x40=345.332727475288x_{40} = 345.332727475288
x41=269.456702265464x_{41} = 269.456702265464
x42=667.093849299232x_{42} = 667.093849299232
x43=459.003130933128x_{43} = 459.003130933128
x44=174.452520284407x_{44} = 174.452520284407
x45=326.372076040896x_{45} = 326.372076040896
x46=364.288450344229x_{46} = 364.288450344229
x47=117.320033193332x_{47} = 117.320033193332
x48=572.547349471644x_{48} = 572.547349471644
x49=534.711075823417x_{49} = 534.711075823417
x50=212.480414874531x_{50} = 212.480414874531
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(4(−2(1−4x2)32+7x21−4x2+14acos⁡(2x)x3)2187x6)=−∞\lim_{x \to 0^-}\left(\frac{4 \left(- \frac{2}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}} + \frac{7}{x^{2} \sqrt{1 - 4 x^{2}}} + \frac{14 \operatorname{acos}{\left(2 x \right)}}{x^{3}}\right)}{2187 x^{6}}\right) = -\infty
lim⁡x→0+(4(−2(1−4x2)32+7x21−4x2+14acos⁡(2x)x3)2187x6)=∞\lim_{x \to 0^+}\left(\frac{4 \left(- \frac{2}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}} + \frac{7}{x^{2} \sqrt{1 - 4 x^{2}}} + \frac{14 \operatorname{acos}{\left(2 x \right)}}{x^{3}}\right)}{2187 x^{6}}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
No tiene corvaduras en todo el eje numérico
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(acos⁡(2x)(3x)7)=0\lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(2 x \right)}}{\left(3 x\right)^{7}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(acos⁡(2x)(3x)7)=0\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(2 x \right)}}{\left(3 x\right)^{7}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función acos(2*x)/(3*x)^7, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(12187x7acos⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{2187 x^{7}} \operatorname{acos}{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(12187x7acos⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{2187 x^{7}} \operatorname{acos}{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
acos⁡(2x)(3x)7=−acos⁡(−2x)2187x7\frac{\operatorname{acos}{\left(2 x \right)}}{\left(3 x\right)^{7}} = - \frac{\operatorname{acos}{\left(- 2 x \right)}}{2187 x^{7}}
- No
acos⁡(2x)(3x)7=acos⁡(−2x)2187x7\frac{\operatorname{acos}{\left(2 x \right)}}{\left(3 x\right)^{7}} = \frac{\operatorname{acos}{\left(- 2 x \right)}}{2187 x^{7}}
- No
es decir, función
no es
par ni impar
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