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Trazado el gráfico de la función/ x^2/(cos(x)-(3/2))

Gráfico de la función y = x^2/(cos(x)-(3/2))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
             2     
            x      
f(x) = ------------
       cos(x) - 3/2
f(x)=x2cos⁡(x)−32f{\left(x \right)} = \frac{x^{2}}{\cos{\left(x \right)} - \frac{3}{2}} 
f = x^2/(cos(x) - 3/2)
Gráfico de la función
02468-8-6-4-2-1010-100100
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:
x2cos⁡(x)−32=0\frac{x^{2}}{\cos{\left(x \right)} - \frac{3}{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^2/(cos(x) - 3/2).
02−32+cos⁡(0)\frac{0^{2}}{- \frac{3}{2} + \cos{\left(0 \right)}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
x2sin⁡(x)(cos⁡(x)−32)2+2xcos⁡(x)−32=0\frac{x^{2} \sin{\left(x \right)}}{\left(\cos{\left(x \right)} - \frac{3}{2}\right)^{2}} + \frac{2 x}{\cos{\left(x \right)} - \frac{3}{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=50.2853781381351x_{1} = 50.2853781381351
x2=0x_{2} = 0
x3=−31.4477629049505x_{3} = -31.4477629049505
x4=−94.2583901363381x_{4} = -94.2583901363381
x5=15.384078313763x_{5} = 15.384078313763
x6=−78.4761112993522x_{6} = -78.4761112993522
x7=−87.9759627556292x_{7} = -87.9759627556292
x8=21.7617891665261x_{8} = 21.7617891665261
x9=28.0965631409557x_{9} = 28.0965631409557
x10=−34.4123243298913x_{10} = -34.4123243298913
x11=−50.2853781381351x_{11} = -50.2853781381351
x12=−28.0965631409557x_{12} = -28.0965631409557
x13=44.0050355305737x_{13} = 44.0050355305737
x14=−44.0050355305737x_{14} = -44.0050355305737
x15=34.4123243298913x_{15} = 34.4123243298913
x16=−53.3133173740256x_{16} = -53.3133173740256
x17=8.86655987587145x_{17} = 8.86655987587145
x18=65.8975849587002x_{18} = 65.8975849587002
x19=53.3133173740256x_{19} = 53.3133173740256
x20=97.3380093822034x_{20} = 97.3380093822034
x21=−15.384078313763x_{21} = -15.384078313763
x22=−59.6063964574436x_{22} = -59.6063964574436
x23=−40.7179701727823x_{23} = -40.7179701727823
x24=47.0175866364576x_{24} = 47.0175866364576
x25=−81.6936519871376x_{25} = -81.6936519871376
x26=−72.187377914753x_{26} = -72.187377914753
x27=40.7179701727823x_{27} = 40.7179701727823
x28=91.0512783605725x_{28} = 91.0512783605725
x29=100.540912262872x_{29} = 100.540912262872
x30=−84.7640211937152x_{30} = -84.7640211937152
x31=−37.7256407796871x_{31} = -37.7256407796871
x32=94.2583901363381x_{32} = 94.2583901363381
x33=−91.0512783605725x_{33} = -91.0512783605725
x34=72.187377914753x_{34} = 72.187377914753
x35=81.6936519871376x_{35} = 81.6936519871376
x36=6.44302868842502x_{36} = 6.44302868842502
x37=−21.7617891665261x_{37} = -21.7617891665261
x38=−8.86655987587145x_{38} = -8.86655987587145
x39=−47.0175866364576x_{39} = -47.0175866364576
x40=−97.3380093822034x_{40} = -97.3380093822034
x41=87.9759627556292x_{41} = 87.9759627556292
x42=−65.8975849587002x_{42} = -65.8975849587002
x43=84.7640211937152x_{43} = 84.7640211937152
x44=37.7256407796871x_{44} = 37.7256407796871
x45=78.4761112993522x_{45} = 78.4761112993522
x46=59.6063964574436x_{46} = 59.6063964574436
Signos de extremos en los puntos:
(50.28537813813507, -5055.2375193299)

(0, 0)

(-31.447762904950526, -1975.92104913788)

(-94.2583901363381, -17767.2879407247)

(15.384078313762958, -96.6786225405606)

(-78.47611129935217, -2465.4004239752)

(-87.97596275562918, -15477.5397224483)

(21.761789166526114, -191.435494133993)

(28.096563140955716, -317.769921097195)

(-34.41232432989134, -475.685341898355)

(-50.28537813813507, -5055.2375193299)

(-28.096563140955716, -317.769921097195)

(44.00503553057369, -3870.88501139786)

(-44.00503553057369, -3870.88501139786)

(34.41232432989134, -475.685341898355)

(-53.31331737402556, -1138.92480411025)

(8.866559875871447, -33.4792071313582)

(65.89758495870016, -1738.99725739366)

(53.31331737402556, -1138.92480411025)

(97.33800938220337, -3791.87549212654)

(-15.384078313762958, -96.6786225405606)

(-59.606396457443566, -1423.16970359841)

(-40.71797017278228, -665.182748158072)

(47.017586636457565, -886.262513414773)

(-81.69365198713761, -13345.7051752545)

(-72.18737791475296, -2086.407492066)

(40.71797017278228, -665.182748158072)

(91.05127836057247, -3318.13441808471)

(100.54091226287183, -20214.9498299226)

(-84.76402119371517, -2875.97606364261)

(-37.72564077968713, -2844.44618480839)

(94.2583901363381, -17767.2879407247)

(-91.05127836057247, -3318.13441808471)

(72.18737791475296, -2086.407492066)

(81.69365198713761, -13345.7051752545)

(6.443028688425024, -80.9610896065579)

(-21.761789166526114, -191.435494133993)

(-8.866559875871447, -33.4792071313582)

(-47.017586636457565, -886.262513414773)

(-97.33800938220337, -3791.87549212654)

(87.97596275562918, -15477.5397224483)

(-65.89758495870016, -1738.99725739366)

(84.76402119371517, -2875.97606364261)

(37.72564077968713, -2844.44618480839)

(78.47611129935217, -2465.4004239752)

(59.606396457443566, -1423.16970359841)


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=50.2853781381351x_{1} = 50.2853781381351
x2=−31.4477629049505x_{2} = -31.4477629049505
x3=−94.2583901363381x_{3} = -94.2583901363381
x4=−87.9759627556292x_{4} = -87.9759627556292
x5=−50.2853781381351x_{5} = -50.2853781381351
x6=44.0050355305737x_{6} = 44.0050355305737
x7=−44.0050355305737x_{7} = -44.0050355305737
x8=−81.6936519871376x_{8} = -81.6936519871376
x9=100.540912262872x_{9} = 100.540912262872
x10=−37.7256407796871x_{10} = -37.7256407796871
x11=94.2583901363381x_{11} = 94.2583901363381
x12=81.6936519871376x_{12} = 81.6936519871376
x13=6.44302868842502x_{13} = 6.44302868842502
x14=87.9759627556292x_{14} = 87.9759627556292
x15=37.7256407796871x_{15} = 37.7256407796871
Puntos máximos de la función:
x15=0x_{15} = 0
x15=15.384078313763x_{15} = 15.384078313763
x15=−78.4761112993522x_{15} = -78.4761112993522
x15=21.7617891665261x_{15} = 21.7617891665261
x15=28.0965631409557x_{15} = 28.0965631409557
x15=−34.4123243298913x_{15} = -34.4123243298913
x15=−28.0965631409557x_{15} = -28.0965631409557
x15=34.4123243298913x_{15} = 34.4123243298913
x15=−53.3133173740256x_{15} = -53.3133173740256
x15=8.86655987587145x_{15} = 8.86655987587145
x15=65.8975849587002x_{15} = 65.8975849587002
x15=53.3133173740256x_{15} = 53.3133173740256
x15=97.3380093822034x_{15} = 97.3380093822034
x15=−15.384078313763x_{15} = -15.384078313763
x15=−59.6063964574436x_{15} = -59.6063964574436
x15=−40.7179701727823x_{15} = -40.7179701727823
x15=47.0175866364576x_{15} = 47.0175866364576
x15=−72.187377914753x_{15} = -72.187377914753
x15=40.7179701727823x_{15} = 40.7179701727823
x15=91.0512783605725x_{15} = 91.0512783605725
x15=−84.7640211937152x_{15} = -84.7640211937152
x15=−91.0512783605725x_{15} = -91.0512783605725
x15=72.187377914753x_{15} = 72.187377914753
x15=−21.7617891665261x_{15} = -21.7617891665261
x15=−8.86655987587145x_{15} = -8.86655987587145
x15=−47.0175866364576x_{15} = -47.0175866364576
x15=−97.3380093822034x_{15} = -97.3380093822034
x15=−65.8975849587002x_{15} = -65.8975849587002
x15=84.7640211937152x_{15} = 84.7640211937152
x15=78.4761112993522x_{15} = 78.4761112993522
x15=59.6063964574436x_{15} = 59.6063964574436
Decrece en los intervalos
[100.540912262872,∞)\left[100.540912262872, \infty\right)
Crece en los intervalos
(−∞,−94.2583901363381]\left(-\infty, -94.2583901363381\right]
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(x2(cos⁡(x)+4sin⁡2(x)2cos⁡(x)−3)2cos⁡(x)−3+4xsin⁡(x)2cos⁡(x)−3+1)2cos⁡(x)−3=0\frac{4 \left(\frac{x^{2} \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 3}\right)}{2 \cos{\left(x \right)} - 3} + \frac{4 x \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 3} + 1\right)}{2 \cos{\left(x \right)} - 3} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−93.7030340353641x_{1} = -93.7030340353641
x2=1.56807396891319x_{2} = 1.56807396891319
x3=−0.618359263229565x_{3} = -0.618359263229565
x4=5.85272096897357x_{4} = 5.85272096897357
x5=6.97038482489727x_{5} = 6.97038482489727
x6=−82.2446990693219x_{6} = -82.2446990693219
x7=−25.7186832137737x_{7} = -25.7186832137737
x8=94.8097374192358x_{8} = 94.8097374192358
x9=75.9623470137284x_{9} = 75.9623470137284
x10=24.6113242812875x_{10} = 24.6113242812875
x11=62.291378750181x_{11} = 62.291378750181
x12=−31.9952923414961x_{12} = -31.9952923414961
x13=56.0096143055116x_{13} = 56.0096143055116
x14=12.0760969579364x_{14} = 12.0760969579364
x15=−56.0096143055116x_{15} = -56.0096143055116
x16=44.5541791201343x_{16} = 44.5541791201343
x17=93.7030340353641x_{17} = 93.7030340353641
x18=0.618359263229565x_{18} = 0.618359263229565
x19=88.5271705195823x_{19} = 88.5271705195823
x20=18.3386078191145x_{20} = 18.3386078191145
x21=−12.0760969579364x_{21} = -12.0760969579364
x22=−1.56807396891318x_{22} = -1.56807396891318
x23=49.7282030327101x_{23} = 49.7282030327101
x24=−87.4204596988621x_{24} = -87.4204596988621
x25=25.7186832137737x_{25} = 25.7186832137737
x26=−5.85272096897357x_{26} = -5.85272096897357
x27=−75.9623470137284x_{27} = -75.9623470137284
x28=38.2741079005589x_{28} = 38.2741079005589
x29=82.2446990693219x_{29} = 82.2446990693219
x30=13.1855591616646x_{30} = 13.1855591616646
x31=99.9856845275761x_{31} = 99.9856845275761
x32=−69.6801470379161x_{32} = -69.6801470379161
x33=−99.9856845275761x_{33} = -99.9856845275761
x34=−1.56807396891319x_{34} = -1.56807396891319
x35=−19.4465140857371x_{35} = -19.4465140857371
x36=−13.1855591616646x_{36} = -13.1855591616646
x37=−49.7282030327101x_{37} = -49.7282030327101
x38=−38.2741079005589x_{38} = -38.2741079005589
x39=−43.4472952744091x_{39} = -43.4472952744091
x40=−6.97038482489727x_{40} = -6.97038482489727
x41=31.9952923414961x_{41} = 31.9952923414961

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:
Cóncava en los intervalos
[99.9856845275761,∞)\left[99.9856845275761, \infty\right)
Convexa en los intervalos
(−∞,−82.2446990693219]\left(-\infty, -82.2446990693219\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x2cos⁡(x)−32)=−∞\lim_{x \to -\infty}\left(\frac{x^{2}}{\cos{\left(x \right)} - \frac{3}{2}}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(x2cos⁡(x)−32)=−∞\lim_{x \to \infty}\left(\frac{x^{2}}{\cos{\left(x \right)} - \frac{3}{2}}\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^2/(cos(x) - 3/2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(xcos⁡(x)−32)=∞\lim_{x \to -\infty}\left(\frac{x}{\cos{\left(x \right)} - \frac{3}{2}}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞(xcos⁡(x)−32)=−∞\lim_{x \to \infty}\left(\frac{x}{\cos{\left(x \right)} - \frac{3}{2}}\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:
x2cos⁡(x)−32=x2cos⁡(x)−32\frac{x^{2}}{\cos{\left(x \right)} - \frac{3}{2}} = \frac{x^{2}}{\cos{\left(x \right)} - \frac{3}{2}}
- Sí
x2cos⁡(x)−32=−x2cos⁡(x)−32\frac{x^{2}}{\cos{\left(x \right)} - \frac{3}{2}} = - \frac{x^{2}}{\cos{\left(x \right)} - \frac{3}{2}}
- No
es decir, función
es
par
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