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Trazado el gráfico de la función/ (x^4/cox(x))/sqrt(x)/(1-x^2)

Gráfico de la función y = (x^4/cox(x))/sqrt(x)/(1-x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /   4    \
       |  x     |
       |------*x|
       |cos(x)  |
       |--------|
       |   ___  |
       \ \/ x   /
f(x) = ----------
              2  
         1 - x
f(x)=xx4cos⁡(x)1x1−x2f{\left(x \right)} = \frac{x \frac{x^{4}}{\cos{\left(x \right)}} \frac{1}{\sqrt{x}}}{1 - x^{2}} 
f = ((x*(x^4/cos(x)))/sqrt(x))/(1 - x^2)
Gráfico de la función
02468-8-6-4-2-1010-1000010000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−1x_{1} = -1
x2=0x_{2} = 0
x3=1x_{3} = 1
x4=1.5707963267949x_{4} = 1.5707963267949
x5=4.71238898038469x_{5} = 4.71238898038469
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:
xx4cos⁡(x)1x1−x2=0\frac{x \frac{x^{4}}{\cos{\left(x \right)}} \frac{1}{\sqrt{x}}}{1 - x^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=0x_{1} = 0
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^4/cos(x))*x)/sqrt(x))/(1 - x^2).
004cos⁡(0)101−02\frac{0 \frac{0^{4}}{\cos{\left(0 \right)}} \frac{1}{\sqrt{0}}}{1 - 0^{2}}
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
−x722cos⁡(x)+x4cos⁡(x)+x(x4sin⁡(x)cos⁡2(x)+4x3cos⁡(x))x1−x2+2x6x(1−x2)2cos⁡(x)=0\frac{- \frac{x^{\frac{7}{2}}}{2 \cos{\left(x \right)}} + \frac{\frac{x^{4}}{\cos{\left(x \right)}} + x \left(\frac{x^{4} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{4 x^{3}}{\cos{\left(x \right)}}\right)}{\sqrt{x}}}{1 - x^{2}} + \frac{2 x^{6}}{\sqrt{x} \left(1 - x^{2}\right)^{2} \cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−12.3679451429993x_{1} = -12.3679451429993
x2=−18.7170744794989x_{2} = -18.7170744794989
x3=72.2220346776721x_{3} = 72.2220346776721
x4=50.2157541172622x_{4} = 50.2157541172622
x5=−21.8775586988855x_{5} = -21.8775586988855
x6=−84.7935300820509x_{6} = -84.7935300820509
x7=−47.0708473605521x_{7} = -47.0708473605521
x8=15.5490670493603x_{8} = 15.5490670493603
x9=−43.9254674286543x_{9} = -43.9254674286543
x10=62.7920682105289x_{10} = 62.7920682105289
x11=56.5044633765815x_{11} = 56.5044633765815
x12=31.3363803263524x_{12} = 31.3363803263524
x13=−65.9355550373727x_{13} = -65.9355550373727
x14=−5.8903740979634x_{14} = -5.8903740979634
x15=53.3602711495463x_{15} = 53.3602711495463
x16=−72.2220346776721x_{16} = -72.2220346776721
x17=−100.506097899296x_{17} = -100.506097899296
x18=97.3637031478906x_{18} = 97.3637031478906
x19=81.6508040350002x_{19} = 81.6508040350002
x20=78.5079873328373x_{20} = 78.5079873328373
x21=−59.6483820588707x_{21} = -59.6483820588707
x22=−37.6328153120862x_{22} = -37.6328153120862
x23=−78.5079873328373x_{23} = -78.5079873328373
x24=−2.43083372960275x_{24} = -2.43083372960275
x25=−94.2212549300287x_{25} = -94.2212549300287
x26=−28.1859574673931x_{26} = -28.1859574673931
x27=87.9361751849824x_{27} = 87.9361751849824
x28=−40.7795052204397x_{28} = -40.7795052204397
x29=−9.16081582863083x_{29} = -9.16081582863083
x30=25.0333308393354x_{30} = 25.0333308393354
x31=47.0708473605521x_{31} = 47.0708473605521
x32=−31.3363803263524x_{32} = -31.3363803263524
x33=−56.5044633765815x_{33} = -56.5044633765815
x34=−53.3602711495463x_{34} = -53.3602711495463
x35=34.4851994748049x_{35} = 34.4851994748049
x36=94.2212549300287x_{36} = 94.2212549300287
x37=21.8775586988855x_{37} = 21.8775586988855
x38=91.0787477156772x_{38} = 91.0787477156772
x39=40.7795052204397x_{39} = 40.7795052204397
x40=2.43083372960275x_{40} = 2.43083372960275
x41=28.1859574673931x_{41} = 28.1859574673931
x42=−50.2157541172622x_{42} = -50.2157541172622
x43=65.9355550373727x_{43} = 65.9355550373727
x44=69.078869710533x_{44} = 69.078869710533
x45=75.3650686465816x_{45} = 75.3650686465816
x46=−97.3637031478906x_{46} = -97.3637031478906
x47=−15.5490670493603x_{47} = -15.5490670493603
x48=43.9254674286543x_{48} = 43.9254674286543
x49=5.8903740979634x_{49} = 5.8903740979634
x50=−34.4851994748049x_{50} = -34.4851994748049
x51=59.6483820588707x_{51} = 59.6483820588707
x52=−62.7920682105289x_{52} = -62.7920682105289
x53=18.7170744794989x_{53} = 18.7170744794989
x54=−69.078869710533x_{54} = -69.078869710533
x55=−25.0333308393354x_{55} = -25.0333308393354
x56=−87.9361751849824x_{56} = -87.9361751849824
x57=100.506097899296x_{57} = 100.506097899296
x58=−81.6508040350002x_{58} = -81.6508040350002
x59=−75.3650686465816x_{59} = -75.3650686465816
x60=84.7935300820509x_{60} = 84.7935300820509
x61=−91.0787477156772x_{61} = -91.0787477156772
x62=37.6328153120862x_{62} = 37.6328153120862
x63=9.16081582863083x_{63} = 9.16081582863083
x64=12.3679451429993x_{64} = 12.3679451429993
Signos de extremos en los puntos:
(-12.367945142999284, -552.330109283776*I)

(-18.717074479498912, -1533.41177912104*I)

(72.22203467767214, 44362.6545437911)

(50.21575411726218, -17898.2035043917)

(-21.877558698885462, 2257.94464677945*I)

(-84.79353008205095, 66245.4161656403*I)

(-47.0708473605521, 15229.55897351*I)

(15.549067049360316, 969.541972375597)

(-43.92546742865425, -12814.9762445222*I)

(62.79206821052886, -31276.3403229883)

(56.50446337658148, -24030.7532068533)

(31.336380326352398, -5520.01223905214)

(-65.93555503737265, 35335.4727892044*I)

(-5.890374097963395, -93.8561841891924*I)

(53.360271149546314, 20829.2428093217)

(-72.22203467767214, 44362.6545437911*I)

(-100.5060978992964, -101311.399868171*I)

(97.36370314789059, 93579.6936788192)

(81.65080403500025, -60279.5132968612)

(78.50798733283726, 54648.1045527362)

(-59.64838205887072, 27510.5767126856*I)

(-37.6328153120862, -8713.21937956262*I)

(-78.50798733283726, 54648.1045527362*I)

(-2.4308337296027474, 14.6324424567505*I)

(-94.22125493002875, -86213.2321573915*I)

(-28.18595746739309, 4239.62265200371*I)

(87.93617518498237, -72552.1852821653)

(-40.779505220439674, 10645.839543525*I)

(-9.16081582863083, 266.287337179446*I)

(25.033330839335424, -3156.01935814965)

(47.0708473605521, 15229.55897351)

(-31.336380326352398, -5520.01223905214*I)

(-56.50446337658148, -24030.7532068533*I)

(-53.360271149546314, 20829.2428093217*I)

(34.48519947480494, 7007.83554874258)

(94.22125493002875, -86213.2321573915)

(21.877558698885462, 2257.94464677945)

(91.07874771567724, 79206.0758080783)

(40.779505220439674, 10645.839543525)

(2.4308337296027474, 14.6324424567505)

(28.18595746739309, 4239.62265200371)

(-50.21575411726218, -17898.2035043917*I)

(65.93555503737265, 35335.4727892044)

(69.07886971053296, -39695.219318744)

(75.36506864658162, -49344.694031099)

(-97.36370314789059, 93579.6936788192*I)

(-15.549067049360316, 969.541972375597*I)

(43.92546742865425, -12814.9762445222)

(5.890374097963395, -93.8561841891924)

(-34.48519947480494, 7007.83554874258*I)

(59.64838205887072, 27510.5767126856)

(-62.79206821052886, -31276.3403229883*I)

(18.717074479498912, -1533.41177912104)

(-69.07886971053296, -39695.219318744*I)

(-25.033330839335424, -3156.01935814965*I)

(-87.93617518498237, -72552.1852821653*I)

(100.5060978992964, -101311.399868171)

(-81.65080403500025, -60279.5132968612*I)

(-75.36506864658162, -49344.694031099*I)

(84.79353008205095, 66245.4161656403)

(-91.07874771567724, 79206.0758080783*I)

(37.6328153120862, -8713.21937956262)

(9.16081582863083, 266.287337179446)

(12.367945142999284, -552.330109283776)


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=72.2220346776721x_{1} = 72.2220346776721
x2=15.5490670493603x_{2} = 15.5490670493603
x3=53.3602711495463x_{3} = 53.3602711495463
x4=97.3637031478906x_{4} = 97.3637031478906
x5=78.5079873328373x_{5} = 78.5079873328373
x6=47.0708473605521x_{6} = 47.0708473605521
x7=34.4851994748049x_{7} = 34.4851994748049
x8=21.8775586988855x_{8} = 21.8775586988855
x9=91.0787477156772x_{9} = 91.0787477156772
x10=40.7795052204397x_{10} = 40.7795052204397
x11=2.43083372960275x_{11} = 2.43083372960275
x12=28.1859574673931x_{12} = 28.1859574673931
x13=65.9355550373727x_{13} = 65.9355550373727
x14=59.6483820588707x_{14} = 59.6483820588707
x15=84.7935300820509x_{15} = 84.7935300820509
x16=9.16081582863083x_{16} = 9.16081582863083
Puntos máximos de la función:
x16=50.2157541172622x_{16} = 50.2157541172622
x16=62.7920682105289x_{16} = 62.7920682105289
x16=56.5044633765815x_{16} = 56.5044633765815
x16=31.3363803263524x_{16} = 31.3363803263524
x16=81.6508040350002x_{16} = 81.6508040350002
x16=87.9361751849824x_{16} = 87.9361751849824
x16=25.0333308393354x_{16} = 25.0333308393354
x16=94.2212549300287x_{16} = 94.2212549300287
x16=69.078869710533x_{16} = 69.078869710533
x16=75.3650686465816x_{16} = 75.3650686465816
x16=43.9254674286543x_{16} = 43.9254674286543
x16=5.8903740979634x_{16} = 5.8903740979634
x16=18.7170744794989x_{16} = 18.7170744794989
x16=100.506097899296x_{16} = 100.506097899296
x16=37.6328153120862x_{16} = 37.6328153120862
x16=12.3679451429993x_{16} = 12.3679451429993
Decrece en los intervalos
[97.3637031478906,∞)\left[97.3637031478906, \infty\right)
Crece en los intervalos
(−∞,2.43083372960275]\left(-\infty, 2.43083372960275\right]
Asíntotas verticales
Hay:
x1=−1x_{1} = -1
x2=0x_{2} = 0
x3=1x_{3} = 1
x4=1.5707963267949x_{4} = 1.5707963267949
x5=4.71238898038469x_{5} = 4.71238898038469
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(xx4cos⁡(x)1x1−x2)y = \lim_{x \to -\infty}\left(\frac{x \frac{x^{4}}{\cos{\left(x \right)}} \frac{1}{\sqrt{x}}}{1 - x^{2}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(xx4cos⁡(x)1x1−x2)y = \lim_{x \to \infty}\left(\frac{x \frac{x^{4}}{\cos{\left(x \right)}} \frac{1}{\sqrt{x}}}{1 - x^{2}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (((x^4/cos(x))*x)/sqrt(x))/(1 - x^2), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(x4x(1−x2)cos⁡(x))y = x \lim_{x \to -\infty}\left(\frac{x^{4}}{\sqrt{x} \left(1 - x^{2}\right) \cos{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(x4x(1−x2)cos⁡(x))y = x \lim_{x \to \infty}\left(\frac{x^{4}}{\sqrt{x} \left(1 - x^{2}\right) \cos{\left(x \right)}}\right)
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:
xx4cos⁡(x)1x1−x2=−x5−x(1−x2)cos⁡(x)\frac{x \frac{x^{4}}{\cos{\left(x \right)}} \frac{1}{\sqrt{x}}}{1 - x^{2}} = - \frac{x^{5}}{\sqrt{- x} \left(1 - x^{2}\right) \cos{\left(x \right)}}
- No
xx4cos⁡(x)1x1−x2=x5−x(1−x2)cos⁡(x)\frac{x \frac{x^{4}}{\cos{\left(x \right)}} \frac{1}{\sqrt{x}}}{1 - x^{2}} = \frac{x^{5}}{\sqrt{- x} \left(1 - x^{2}\right) \cos{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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