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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       log(cos(x))
f(x) = -----------
             2    
            x
f(x)=log⁡(cos⁡(x))x2f{\left(x \right)} = \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}} 
f = log(cos(x))/x^2
Gráfico de la función
02468-8-6-4-2-10102-2
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:
log⁡(cos⁡(x))x2=0\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=2πx_{1} = 2 \pi
Solución numérica
x1=56.5486675910765x_{1} = 56.5486675910765
x2=75.3982238631513x_{2} = 75.3982238631513
x3=12.5663704222821x_{3} = 12.5663704222821
x4=37.6991114226912x_{4} = 37.6991114226912
x5=−75.3982212667289x_{5} = -75.3982212667289
x6=−31.4159267218315x_{6} = -31.4159267218315
x7=−6.28318500859429x_{7} = -6.28318500859429
x8=6.28318425041486x_{8} = 6.28318425041486
x9=−37.6991123409791x_{9} = -37.6991123409791
x10=−12.5663702114039x_{10} = -12.5663702114039
x11=−75.3982237850615x_{11} = -75.3982237850615
x12=56.5486697853444x_{12} = 56.5486697853444
x13=43.9822969520563x_{13} = 43.9822969520563
x14=43.9823007506628x_{14} = 43.9823007506628
x15=−62.8318536636829x_{15} = -62.8318536636829
x16=81.6814067097954x_{16} = 81.6814067097954
x17=−87.9645943582944x_{17} = -87.9645943582944
x18=−50.2654848480677x_{18} = -50.2654848480677
x19=62.8318527292118x_{19} = 62.8318527292118
x20=37.6991140183594x_{20} = 37.6991140183594
x21=69.1150378100591x_{21} = 69.1150378100591
x22=31.4159236692146x_{22} = 31.4159236692146
x23=−75.3982238842683x_{23} = -75.3982238842683
x24=81.6814092014976x_{24} = 81.6814092014976
x25=50.2654825592132x_{25} = 50.2654825592132
x26=6.28318536623994x_{26} = 6.28318536623994
x27=43.9822975717062x_{27} = 43.9822975717062
x28=−6.28318509127283x_{28} = -6.28318509127283
x29=62.8318534194081x_{29} = 62.8318534194081
x30=−12.5663711881532x_{30} = -12.5663711881532
x31=−37.699115831174x_{31} = -37.699115831174
x32=−62.8318552794767x_{32} = -62.8318552794767
x33=−6.28318703759797x_{33} = -6.28318703759797
x34=18.8495555495816x_{34} = 18.8495555495816
x35=87.9645944170753x_{35} = 87.9645944170753
x36=56.5486680400228x_{36} = 56.5486680400228
x37=100.530965170832x_{37} = 100.530965170832
x38=−18.8495552662988x_{38} = -18.8495552662988
x39=−94.2477799274769x_{39} = -94.2477799274769
x40=31.4159268946166x_{40} = 31.4159268946166
x41=−87.9645944082851x_{41} = -87.9645944082851
x42=25.1327406153308x_{42} = 25.1327406153308
x43=50.2654824463205x_{43} = 50.2654824463205
x44=−50.2654822745065x_{44} = -50.2654822745065
x45=6.28318528379982x_{45} = 6.28318528379982
x46=−43.9822971370899x_{46} = -43.9822971370899
x47=50.2654824267459x_{47} = 50.2654824267459
x48=75.3982240657869x_{48} = 75.3982240657869
x49=94.2477796642264x_{49} = 94.2477796642264
x50=−31.4159240473036x_{50} = -31.4159240473036
x51=31.4159254853307x_{51} = 31.4159254853307
x52=−81.6814088468767x_{52} = -81.6814088468767
x53=94.2477796093521x_{53} = 94.2477796093521
x54=69.1150385422669x_{54} = 69.1150385422669
x55=18.8495566653649x_{55} = 18.8495566653649
x56=−25.1327415699717x_{56} = -25.1327415699717
x57=12.566371085894x_{57} = 12.566371085894
x58=−43.9822973028755x_{58} = -43.9822973028755
x59=37.6991120390888x_{59} = 37.6991120390888
x60=18.8495568924594x_{60} = 18.8495568924594
x61=100.530964751937x_{61} = 100.530964751937
x62=−31.4159262422395x_{62} = -31.4159262422395
x63=−81.6814090383396x_{63} = -81.6814090383396
x64=−18.8495564609393x_{64} = -18.8495564609393
x65=−94.2477820859381x_{65} = -94.2477820859381
x66=−25.1327401011047x_{66} = -25.1327401011047
x67=−69.115038371213x_{67} = -69.115038371213
x68=25.1327418019752x_{68} = 25.1327418019752
x69=62.8318541663282x_{69} = 62.8318541663282
x70=−12.5663714894404x_{70} = -12.5663714894404
x71=−87.9645943472471x_{71} = -87.9645943472471
x72=94.2477796240335x_{72} = 94.2477796240335
x73=−56.5486674063126x_{73} = -56.5486674063126
x74=−56.5486687958913x_{74} = -56.5486687958913
x75=75.3982227157249x_{75} = 75.3982227157249
x76=−37.699111877239x_{76} = -37.699111877239
x77=12.5663724671728x_{77} = 12.5663724671728
x78=87.9645943359894x_{78} = 87.9645943359894
x79=87.9645940945969x_{79} = 87.9645940945969
x80=81.6814088355208x_{80} = 81.6814088355208
x81=−100.530965327794x_{81} = -100.530965327794
x82=37.6991093650365x_{82} = 37.6991093650365
x83=−81.681409137626x_{83} = -81.681409137626
x84=−100.530964572061x_{84} = -100.530964572061
x85=−37.6991117133998x_{85} = -37.6991117133998
x86=−69.1150387430457x_{86} = -69.1150387430457
x87=−56.5486680686757x_{87} = -56.5486680686757
x88=69.1150389965986x_{88} = 69.1150389965986
x89=43.9822971694655x_{89} = 43.9822971694655
x90=−25.1327411327427x_{90} = -25.1327411327427
x91=−69.1150373549062x_{91} = -69.1150373549062
x92=−94.2477794361694x_{92} = -94.2477794361694
x93=−50.2654827533158x_{93} = -50.2654827533158
x94=−62.8318524778695x_{94} = -62.8318524778695
x95=−43.9822971744421x_{95} = -43.9822971744421
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 log(cos(x))/x^2.
log⁡(cos⁡(0))02\frac{\log{\left(\cos{\left(0 \right)} \right)}}{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
−sin⁡(x)x2cos⁡(x)−2log⁡(cos⁡(x))x3=0- \frac{\sin{\left(x \right)}}{x^{2} \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−100.530964914873x_{1} = -100.530964914873
x2=−25.1327412287183x_{2} = -25.1327412287183
x3=−75.398223686155x_{3} = -75.398223686155
x4=−50.2654824574367x_{4} = -50.2654824574367
x5=81.6814089933346x_{5} = 81.6814089933346
x6=6.28318530717959x_{6} = 6.28318530717959
x7=50.2654824574367x_{7} = 50.2654824574367
x8=−43.9822971502571x_{8} = -43.9822971502571
x9=−37.6991118430775x_{9} = -37.6991118430775
x10=25.1327412287183x_{10} = 25.1327412287183
x11=−18.8495559215388x_{11} = -18.8495559215388
x12=−56.5486677646163x_{12} = -56.5486677646163
x13=18.8495559215388x_{13} = 18.8495559215388
x14=−6.28318530717959x_{14} = -6.28318530717959
x15=−62.8318530717959x_{15} = -62.8318530717959
x16=12.5663706143592x_{16} = 12.5663706143592
x17=56.5486677646163x_{17} = 56.5486677646163
x18=69.1150383789755x_{18} = 69.1150383789755
x19=37.6991118430775x_{19} = 37.6991118430775
x20=−31.4159265358979x_{20} = -31.4159265358979
x21=100.530964914873x_{21} = 100.530964914873
x22=94.2477796076938x_{22} = 94.2477796076938
x23=−12.5663706143592x_{23} = -12.5663706143592
x24=−94.2477796076938x_{24} = -94.2477796076938
x25=43.9822971502571x_{25} = 43.9822971502571
x26=75.398223686155x_{26} = 75.398223686155
x27=62.8318530717959x_{27} = 62.8318530717959
x28=87.9645943005142x_{28} = 87.9645943005142
x29=−81.6814089933346x_{29} = -81.6814089933346
x30=−87.9645943005142x_{30} = -87.9645943005142
x31=−69.1150383789755x_{31} = -69.1150383789755
x32=31.4159265358979x_{32} = 31.4159265358979
Signos de extremos en los puntos:
(-100.53096491487338, 0)

(-25.132741228718345, 0)

(-75.39822368615503, 0)

(-50.26548245743669, 0)

(81.68140899333463, 0)

(6.283185307179586, 0)

(50.26548245743669, 0)

(-43.982297150257104, 0)

(-37.69911184307752, 0)

(25.132741228718345, 0)

(-18.84955592153876, 0)

(-56.548667764616276, 0)

(18.84955592153876, 0)

(-6.283185307179586, 0)

(-62.83185307179586, 0)

(12.566370614359172, 0)

(56.548667764616276, 0)

(69.11503837897546, 0)

(37.69911184307752, 0)

(-31.41592653589793, 0)

(100.53096491487338, 0)

(94.2477796076938, 0)

(-12.566370614359172, 0)

(-94.2477796076938, 0)

(43.982297150257104, 0)

(75.39822368615503, 0)

(62.83185307179586, 0)

(87.96459430051421, 0)

(-81.68140899333463, 0)

(-87.96459430051421, 0)

(-69.11503837897546, 0)

(31.41592653589793, 0)


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
Puntos máximos de la función:
x32=−100.530964914873x_{32} = -100.530964914873
x32=−25.1327412287183x_{32} = -25.1327412287183
x32=−75.398223686155x_{32} = -75.398223686155
x32=−50.2654824574367x_{32} = -50.2654824574367
x32=81.6814089933346x_{32} = 81.6814089933346
x32=6.28318530717959x_{32} = 6.28318530717959
x32=50.2654824574367x_{32} = 50.2654824574367
x32=−43.9822971502571x_{32} = -43.9822971502571
x32=−37.6991118430775x_{32} = -37.6991118430775
x32=25.1327412287183x_{32} = 25.1327412287183
x32=−18.8495559215388x_{32} = -18.8495559215388
x32=−56.5486677646163x_{32} = -56.5486677646163
x32=18.8495559215388x_{32} = 18.8495559215388
x32=−6.28318530717959x_{32} = -6.28318530717959
x32=−62.8318530717959x_{32} = -62.8318530717959
x32=12.5663706143592x_{32} = 12.5663706143592
x32=56.5486677646163x_{32} = 56.5486677646163
x32=69.1150383789755x_{32} = 69.1150383789755
x32=37.6991118430775x_{32} = 37.6991118430775
x32=−31.4159265358979x_{32} = -31.4159265358979
x32=100.530964914873x_{32} = 100.530964914873
x32=94.2477796076938x_{32} = 94.2477796076938
x32=−12.5663706143592x_{32} = -12.5663706143592
x32=−94.2477796076938x_{32} = -94.2477796076938
x32=43.9822971502571x_{32} = 43.9822971502571
x32=75.398223686155x_{32} = 75.398223686155
x32=62.8318530717959x_{32} = 62.8318530717959
x32=87.9645943005142x_{32} = 87.9645943005142
x32=−81.6814089933346x_{32} = -81.6814089933346
x32=−87.9645943005142x_{32} = -87.9645943005142
x32=−69.1150383789755x_{32} = -69.1150383789755
x32=31.4159265358979x_{32} = 31.4159265358979
Decrece en los intervalos
(−∞,−100.530964914873]\left(-\infty, -100.530964914873\right]
Crece en los intervalos
[100.530964914873,∞)\left[100.530964914873, \infty\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
−sin⁡2(x)cos⁡2(x)−1+4sin⁡(x)xcos⁡(x)+6log⁡(cos⁡(x))x2x2=0\frac{- \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 1 + \frac{4 \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{6 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}}{x^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
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→−∞(log⁡(cos⁡(x))x2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(log⁡(cos⁡(x))x2)=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\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 log(cos(x))/x^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(cos⁡(x))xx2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(log⁡(cos⁡(x))xx2)=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x x^{2}}\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:
log⁡(cos⁡(x))x2=log⁡(cos⁡(x))x2\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}} = \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}
- Sí
log⁡(cos⁡(x))x2=−log⁡(cos⁡(x))x2\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}} = - \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}
- No
es decir, función
es
par
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