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Trazado el gráfico de la función/ x^(3)/(1-cos^(3)x)

Gráfico de la función y = x^(3)/(1-cos^(3)x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
             3    
            x     
f(x) = -----------
              3   
       1 - cos (x)
f(x)=x31−cos⁡3(x)f{\left(x \right)} = \frac{x^{3}}{1 - \cos^{3}{\left(x \right)}} 
f = x^3/(1 - cos(x)^3)
Gráfico de la función
02468-8-6-4-2-1010-100000000100000000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=6.28318530717959x_{2} = 6.28318530717959
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:
x31−cos⁡3(x)=0\frac{x^{3}}{1 - \cos^{3}{\left(x \right)}} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 x^3/(1 - cos(x)^3).
031−cos⁡3(0)\frac{0^{3}}{1 - \cos^{3}{\left(0 \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
−3x3sin⁡(x)cos⁡2(x)(1−cos⁡3(x))2+3x21−cos⁡3(x)=0- \frac{3 x^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(1 - \cos^{3}{\left(x \right)}\right)^{2}} + \frac{3 x^{2}}{1 - \cos^{3}{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=9.20276242945215x_{1} = 9.20276242945215
x2=−34.4994655954632x_{2} = -34.4994655954632
x3=76.8544103932078x_{3} = 76.8544103932078
x4=−97.3688281929125x_{4} = -97.3688281929125
x5=70.5661807617724x_{5} = 70.5661807617724
x6=−70.5661807617724x_{6} = -70.5661807617724
x7=14.4117611458066x_{7} = 14.4117611458066
x8=70.8054896337473x_{8} = 70.8054896337473
x9=58.2515959829935x_{9} = 58.2515959829935
x10=−89.6415715237089x_{10} = -89.6415715237089
x11=−40.7916254984257x_{11} = -40.7916254984257
x12=83.1420514692039x_{12} = 83.1420514692039
x13=39.1085892225111x_{13} = 39.1085892225111
x14=−32.8103351863957x_{14} = -32.8103351863957
x15=28.2032701505156x_{15} = 28.2032701505156
x16=65.9431049059039x_{16} = 65.9431049059039
x17=72.2289324391492x_{17} = 72.2289324391492
x18=−9.20276242945215x_{18} = -9.20276242945215
x19=8.23376307408263x_{19} = 8.23376307408263
x20=−51.9763160124177x_{20} = -51.9763160124177
x21=−33.1631207107052x_{21} = -33.1631207107052
x22=−65.9431049059039x_{22} = -65.9431049059039
x23=34.4994655954632x_{23} = 34.4994655954632
x24=−89.4292102107962x_{24} = -89.4292102107962
x25=40.7916254984257x_{25} = 40.7916254984257
x26=−57.9873339863584x_{26} = -57.9873339863584
x27=−84.7994111055208x_{27} = -84.7994111055208
x28=−72.2289324391492x_{28} = -72.2289324391492
x29=−13.862720236882x_{29} = -13.862720236882
x30=51.9763160124177x_{30} = 51.9763160124177
x31=−59.6567195436983x_{31} = -59.6567195436983
x32=−20.1942897936321x_{32} = -20.1942897936321
x33=−26.5068737665397x_{33} = -26.5068737665397
x34=−39.1085892225111x_{34} = -39.1085892225111
x35=−15.5786734174273x_{35} = -15.5786734174273
x36=−78.5143363896437x_{36} = -78.5143363896437
x37=−95.7159678012889x_{37} = -95.7159678012889
x38=89.4292102107962x_{38} = 89.4292102107962
x39=−45.7026495977655x_{39} = -45.7026495977655
x40=−91.0842248400038x_{40} = -91.0842248400038
x41=91.0842248400038x_{41} = 91.0842248400038
x42=−21.899499977872x_{42} = -21.899499977872
x43=95.9211844583754x_{43} = 95.9211844583754
x44=97.3688281929125x_{44} = 97.3688281929125
x45=59.6567195436983x_{45} = 59.6567195436983
x46=−8.23376307408263x_{46} = -8.23376307408263
x47=−64.2772204232222x_{47} = -64.2772204232222
x48=−53.3695785942227x_{48} = -53.3695785942227
x49=26.5068737665397x_{49} = 26.5068737665397
x50=103.653259473329x_{50} = 103.653259473329
x51=−28.2032701505156x_{51} = -28.2032701505156
x52=78.5143363896437x_{52} = 78.5143363896437
x53=45.4035411205358x_{53} = 45.4035411205358
x54=20.1942897936321x_{54} = 20.1942897936321
x55=84.7994111055208x_{55} = 84.7994111055208
x56=−14.4117611458066x_{56} = -14.4117611458066
x57=−47.0813781180829x_{57} = -47.0813781180829
x58=64.2772204232222x_{58} = 64.2772204232222
x59=21.899499977872x_{59} = 21.899499977872
x60=−83.1420514692039x_{60} = -83.1420514692039
x61=53.3695785942227x_{61} = 53.3695785942227
x62=32.8103351863957x_{62} = 32.8103351863957
x63=15.5786734174273x_{63} = 15.5786734174273
x64=−76.8544103932078x_{64} = -76.8544103932078
Signos de extremos en los puntos:
(9.202762429452154, 404.214333583657)

(-34.49946559546322, -20582.7825962644)

(76.85441039320779, 454628.22382234)

(-97.36882819291249, -461707.889679986)

(70.56618076177244, 351989.043397167)

(-70.56618076177244, -351989.043397167)

(14.411761145806569, 2934.79538767312)

(70.80548963374734, 354374.711601576)

(58.25159598299348, 197211.152934428)

(-89.64157152370889, -719468.37965707)

(-40.791625498425724, -33999.0846754401)

(83.14205146920388, 575492.349444912)

(39.108589222511085, 60064.7707475362)

(-32.81033518639566, -35512.7981459512)

(28.203270150515582, 11259.3049147937)

(65.94310490590392, 143475.573629862)

(72.22893243914923, 188518.271517505)

(-9.202762429452154, -404.214333583657)

(8.233763074082635, 531.145820700731)

(-51.976316012417705, -140035.154640052)

(-33.16312071070516, -36276.5153310003)

(-65.94310490590392, -143475.573629862)

(34.49946559546322, 20582.7825962644)

(-89.42921021079624, -716069.980967163)

(40.791625498425724, 33999.0846754401)

(-57.987333986358415, -195431.097783069)

(-84.7994111055208, -305021.013842594)

(-72.22893243914923, -188518.271517505)

(-13.862720236882016, -2718.17557022929)

(51.976316012417705, 140035.154640052)

(-59.65671954369825, -106246.4569046)

(-20.19428979363209, -8329.21359855978)

(-26.50687376653973, -18764.0982949452)

(-39.108589222511085, -60064.7707475362)

(-15.578673417427316, -1914.19882928462)

(-78.51433638964372, -242118.70134329)

(-95.71596780128891, -877849.651267687)

(89.42921021079624, 716069.980967163)

(-45.70264959776547, -95145.8622034015)

(-91.08422484000383, -377969.359724425)

(91.08422484000383, 377969.359724425)

(-21.899499977872033, -5284.4972845503)

(95.92118445837536, 881611.281085131)

(97.36882819291249, 461707.889679986)

(59.65671954369825, 106246.4569046)

(-8.233763074082635, -531.145820700731)

(-64.27722042322225, -266086.213577417)

(-53.36957859422269, -76086.7705015048)

(26.50687376653973, 18764.0982949452)

(103.65325947332902, 556980.754388093)

(-28.203270150515582, -11259.3049147937)

(78.51433638964372, 242118.70134329)

(45.40354112053584, 93909.180795307)

(20.19428979363209, 8329.21359855978)

(84.7994111055208, 305021.013842594)

(-14.411761145806569, -2934.79538767312)

(-47.081378118082874, -52252.3634722407)

(64.27722042322225, 266086.213577417)

(21.899499977872033, 5284.4972845503)

(-83.14205146920388, -575492.349444912)

(53.36957859422269, 76086.7705015048)

(32.81033518639566, 35512.7981459512)

(15.578673417427316, 1914.19882928462)

(-76.85441039320779, -454628.22382234)


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:
Puntos mínimos de la función:
x1=9.20276242945215x_{1} = 9.20276242945215
x2=76.8544103932078x_{2} = 76.8544103932078
x3=70.5661807617724x_{3} = 70.5661807617724
x4=−89.6415715237089x_{4} = -89.6415715237089
x5=83.1420514692039x_{5} = 83.1420514692039
x6=39.1085892225111x_{6} = 39.1085892225111
x7=28.2032701505156x_{7} = 28.2032701505156
x8=65.9431049059039x_{8} = 65.9431049059039
x9=72.2289324391492x_{9} = 72.2289324391492
x10=−51.9763160124177x_{10} = -51.9763160124177
x11=−33.1631207107052x_{11} = -33.1631207107052
x12=34.4994655954632x_{12} = 34.4994655954632
x13=40.7916254984257x_{13} = 40.7916254984257
x14=89.4292102107962x_{14} = 89.4292102107962
x15=−45.7026495977655x_{15} = -45.7026495977655
x16=91.0842248400038x_{16} = 91.0842248400038
x17=97.3688281929125x_{17} = 97.3688281929125
x18=59.6567195436983x_{18} = 59.6567195436983
x19=−8.23376307408263x_{19} = -8.23376307408263
x20=26.5068737665397x_{20} = 26.5068737665397
x21=103.653259473329x_{21} = 103.653259473329
x22=78.5143363896437x_{22} = 78.5143363896437
x23=45.4035411205358x_{23} = 45.4035411205358
x24=20.1942897936321x_{24} = 20.1942897936321
x25=84.7994111055208x_{25} = 84.7994111055208
x26=−14.4117611458066x_{26} = -14.4117611458066
x27=64.2772204232222x_{27} = 64.2772204232222
x28=21.899499977872x_{28} = 21.899499977872
x29=53.3695785942227x_{29} = 53.3695785942227
x30=32.8103351863957x_{30} = 32.8103351863957
x31=15.5786734174273x_{31} = 15.5786734174273
Puntos máximos de la función:
x31=−34.4994655954632x_{31} = -34.4994655954632
x31=−97.3688281929125x_{31} = -97.3688281929125
x31=−70.5661807617724x_{31} = -70.5661807617724
x31=14.4117611458066x_{31} = 14.4117611458066
x31=70.8054896337473x_{31} = 70.8054896337473
x31=58.2515959829935x_{31} = 58.2515959829935
x31=−40.7916254984257x_{31} = -40.7916254984257
x31=−32.8103351863957x_{31} = -32.8103351863957
x31=−9.20276242945215x_{31} = -9.20276242945215
x31=8.23376307408263x_{31} = 8.23376307408263
x31=−65.9431049059039x_{31} = -65.9431049059039
x31=−89.4292102107962x_{31} = -89.4292102107962
x31=−57.9873339863584x_{31} = -57.9873339863584
x31=−84.7994111055208x_{31} = -84.7994111055208
x31=−72.2289324391492x_{31} = -72.2289324391492
x31=−13.862720236882x_{31} = -13.862720236882
x31=51.9763160124177x_{31} = 51.9763160124177
x31=−59.6567195436983x_{31} = -59.6567195436983
x31=−20.1942897936321x_{31} = -20.1942897936321
x31=−26.5068737665397x_{31} = -26.5068737665397
x31=−39.1085892225111x_{31} = -39.1085892225111
x31=−15.5786734174273x_{31} = -15.5786734174273
x31=−78.5143363896437x_{31} = -78.5143363896437
x31=−95.7159678012889x_{31} = -95.7159678012889
x31=−91.0842248400038x_{31} = -91.0842248400038
x31=−21.899499977872x_{31} = -21.899499977872
x31=95.9211844583754x_{31} = 95.9211844583754
x31=−64.2772204232222x_{31} = -64.2772204232222
x31=−53.3695785942227x_{31} = -53.3695785942227
x31=−28.2032701505156x_{31} = -28.2032701505156
x31=−47.0813781180829x_{31} = -47.0813781180829
x31=−83.1420514692039x_{31} = -83.1420514692039
x31=−76.8544103932078x_{31} = -76.8544103932078
Decrece en los intervalos
[103.653259473329,∞)\left[103.653259473329, \infty\right)
Crece en los intervalos
(−∞,−89.6415715237089]\left(-\infty, -89.6415715237089\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=6.28318530717959x_{2} = 6.28318530717959
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x31−cos⁡3(x))=−∞\lim_{x \to -\infty}\left(\frac{x^{3}}{1 - \cos^{3}{\left(x \right)}}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(x31−cos⁡3(x))=∞\lim_{x \to \infty}\left(\frac{x^{3}}{1 - \cos^{3}{\left(x \right)}}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x^3/(1 - cos(x)^3), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(x21−cos⁡3(x))=∞\lim_{x \to -\infty}\left(\frac{x^{2}}{1 - \cos^{3}{\left(x \right)}}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞(x21−cos⁡3(x))=∞\lim_{x \to \infty}\left(\frac{x^{2}}{1 - \cos^{3}{\left(x \right)}}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
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:
x31−cos⁡3(x)=−x31−cos⁡3(x)\frac{x^{3}}{1 - \cos^{3}{\left(x \right)}} = - \frac{x^{3}}{1 - \cos^{3}{\left(x \right)}}
- No
x31−cos⁡3(x)=x31−cos⁡3(x)\frac{x^{3}}{1 - \cos^{3}{\left(x \right)}} = \frac{x^{3}}{1 - \cos^{3}{\left(x \right)}}
- Sí
es decir, función
es
impar
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