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Trazado el gráfico de la función/ lncosx/x

Gráfico de la función y = lncosx/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       log(cos(x))
f(x) = -----------
            x
f(x)=log⁡(cos⁡(x))xf{\left(x \right)} = \frac{\log{\left(\cos{\left(x \right)} \right)}}{x} 
f = log(cos(x))/x
Gráfico de la función
02468-8-6-4-2-10105-5
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))x=0\frac{\log{\left(\cos{\left(x \right)} \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=2πx_{1} = 2 \pi
Solución numérica
x1=−94.2477821181699x_{1} = -94.2477821181699
x2=94.2477796464198x_{2} = 94.2477796464198
x3=−37.6991118773063x_{3} = -37.6991118773063
x4=−62.831853342621x_{4} = -62.831853342621
x5=37.6991120412136x_{5} = 37.6991120412136
x6=−6.28318565828478x_{6} = -6.28318565828478
x7=31.41592690231x_{7} = 31.41592690231
x8=81.6814086046101x_{8} = 81.6814086046101
x9=25.1327418222459x_{9} = 25.1327418222459
x10=−100.530965520471x_{10} = -100.530965520471
x11=50.2654824463258x_{11} = 50.2654824463258
x12=62.8318541847386x_{12} = 62.8318541847386
x13=6.28318518794684x_{13} = 6.28318518794684
x14=−69.115038338702x_{14} = -69.115038338702
x15=−37.6991119922772x_{15} = -37.6991119922772
x16=−18.8495564858153x_{16} = -18.8495564858153
x17=−62.831854107628x_{17} = -62.831854107628
x18=56.5486680287307x_{18} = 56.5486680287307
x19=25.1327414643316x_{19} = 25.1327414643316
x20=−31.4159241420874x_{20} = -31.4159241420874
x21=87.9645941066782x_{21} = 87.9645941066782
x22=50.2654825255164x_{22} = 50.2654825255164
x23=−43.9822972719791x_{23} = -43.9822972719791
x24=75.3982227288473x_{24} = 75.3982227288473
x25=12.5663704278981x_{25} = 12.5663704278981
x26=12.5663726458004x_{26} = 12.5663726458004
x27=69.1150378552832x_{27} = 69.1150378552832
x28=−43.9822972155883x_{28} = -43.9822972155883
x29=−69.1150373702266x_{29} = -69.1150373702266
x30=−62.8318524860201x_{30} = -62.8318524860201
x31=31.4159255197709x_{31} = 31.4159255197709
x32=−50.2654827892748x_{32} = -50.2654827892748
x33=−75.398221304309x_{33} = -75.398221304309
x34=−81.6814090383932x_{34} = -81.6814090383932
x35=25.1327406361642x_{35} = 25.1327406361642
x36=12.5663709619672x_{36} = 12.5663709619672
x37=−56.5486688149424x_{37} = -56.5486688149424
x38=87.9645943080185x_{38} = 87.9645943080185
x39=75.3982240692465x_{39} = 75.3982240692465
x40=−37.6991117193748x_{40} = -37.6991117193748
x41=94.2477796093521x_{41} = 94.2477796093521
x42=−87.9645943784668x_{42} = -87.9645943784668
x43=81.6814067419791x_{43} = 81.6814067419791
x44=−56.5486674103626x_{44} = -56.5486674103626
x45=−31.4159267241418x_{45} = -31.4159267241418
x46=18.8495569463446x_{46} = 18.8495569463446
x47=69.1150378169902x_{47} = 69.1150378169902
x48=43.9822972608986x_{48} = 43.9822972608986
x49=81.681409202583x_{49} = 81.681409202583
x50=100.53096475248x_{50} = 100.53096475248
x51=−6.28318510435776x_{51} = -6.28318510435776
x52=−25.1327401479732x_{52} = -25.1327401479732
x53=−94.2477794368085x_{53} = -94.2477794368085
x54=18.8495555620182x_{54} = 18.8495555620182
x55=69.1150387526606x_{55} = 69.1150387526606
x56=−87.9645943583769x_{56} = -87.9645943583769
x57=43.9822971694838x_{57} = 43.9822971694838
x58=−6.28318735811168x_{58} = -6.28318735811168
x59=−25.1327416014258x_{59} = -25.1327416014258
x60=−56.5486681966342x_{60} = -56.5486681966342
x61=50.2654824911769x_{61} = 50.2654824911769
x62=31.4159268762441x_{62} = 31.4159268762441
x63=94.2477796395327x_{63} = 94.2477796395327
x64=75.3982237987167x_{64} = 75.3982237987167
x65=6.28318528398235x_{65} = 6.28318528398235
x66=56.5486698298894x_{66} = 56.5486698298894
x67=87.9645943360203x_{67} = 87.9645943360203
x68=−18.8495546630502x_{68} = -18.8495546630502
x69=−81.6814090343752x_{69} = -81.6814090343752
x70=−50.265482275851x_{70} = -50.265482275851
x71=62.8318534985437x_{71} = 62.8318534985437
x72=−94.2477799278511x_{72} = -94.2477799278511
x73=−50.2654849051414x_{73} = -50.2654849051414
x74=−69.11503874655x_{74} = -69.11503874655
x75=100.530965148878x_{75} = 100.530965148878
x76=−62.8318536719726x_{76} = -62.8318536719726
x77=−12.5663702322309x_{77} = -12.5663702322309
x78=−81.6814088542447x_{78} = -81.6814088542447
x79=43.9822969697753x_{79} = 43.9822969697753
x80=−18.8495552966503x_{80} = -18.8495552966503
x81=−43.9822971744709x_{81} = -43.9822971744709
x82=56.5486675921582x_{82} = 56.5486675921582
x83=−31.4159265657376x_{83} = -31.4159265657376
x84=37.6991094417897x_{84} = 37.6991094417897
x85=−75.3982238853375x_{85} = -75.3982238853375
x86=37.6991117778923x_{86} = 37.6991117778923
x87=−100.53096457419x_{87} = -100.53096457419
x88=18.8495564016278x_{88} = 18.8495564016278
x89=−75.3982234907138x_{89} = -75.3982234907138
x90=6.28318553619608x_{90} = 6.28318553619608
x91=69.1150390046369x_{91} = 69.1150390046369
x92=−25.1327415788287x_{92} = -25.1327415788287
x93=62.8318527325798x_{93} = 62.8318527325798
x94=−12.5663715613679x_{94} = -12.5663715613679
x95=−12.5663711678178x_{95} = -12.5663711678178
x96=−87.9645943759507x_{96} = -87.9645943759507
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.
log⁡(cos⁡(0))0\frac{\log{\left(\cos{\left(0 \right)} \right)}}{0}
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)xcos⁡(x)−log⁡(cos⁡(x))x2=0- \frac{\sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=31.4159265358979x_{1} = 31.4159265358979
x2=−12.5663706143592x_{2} = -12.5663706143592
x3=75.398223686155x_{3} = 75.398223686155
x4=−69.1150383789755x_{4} = -69.1150383789755
x5=−50.2654824574367x_{5} = -50.2654824574367
x6=−56.5486677646163x_{6} = -56.5486677646163
x7=−75.398223686155x_{7} = -75.398223686155
x8=−62.8318530717959x_{8} = -62.8318530717959
x9=−6.28318530717959x_{9} = -6.28318530717959
x10=6.28318530717959x_{10} = 6.28318530717959
x11=62.8318530717959x_{11} = 62.8318530717959
x12=−25.1327412287183x_{12} = -25.1327412287183
x13=94.2477796076938x_{13} = 94.2477796076938
x14=−37.6991118430775x_{14} = -37.6991118430775
x15=−100.530964914873x_{15} = -100.530964914873
x16=−43.9822971502571x_{16} = -43.9822971502571
x17=25.1327412287183x_{17} = 25.1327412287183
x18=87.9645943005142x_{18} = 87.9645943005142
x19=43.9822971502571x_{19} = 43.9822971502571
x20=−31.4159265358979x_{20} = -31.4159265358979
x21=−94.2477796076938x_{21} = -94.2477796076938
x22=18.8495559215388x_{22} = 18.8495559215388
x23=−18.8495559215388x_{23} = -18.8495559215388
x24=12.5663706143592x_{24} = 12.5663706143592
x25=81.6814089933346x_{25} = 81.6814089933346
x26=−81.6814089933346x_{26} = -81.6814089933346
x27=50.2654824574367x_{27} = 50.2654824574367
x28=−87.9645943005142x_{28} = -87.9645943005142
x29=56.5486677646163x_{29} = 56.5486677646163
x30=37.6991118430775x_{30} = 37.6991118430775
x31=100.530964914873x_{31} = 100.530964914873
x32=69.1150383789755x_{32} = 69.1150383789755
Signos de extremos en los puntos:
(31.41592653589793, 0)

(-12.566370614359172, 0)

(75.39822368615503, 0)

(-69.11503837897546, 0)

(-50.26548245743669, 0)

(-56.548667764616276, 0)

(-75.39822368615503, 0)

(-62.83185307179586, 0)

(-6.283185307179586, 0)

(6.283185307179586, 0)

(62.83185307179586, 0)

(-25.132741228718345, 0)

(94.2477796076938, 0)

(-37.69911184307752, 0)

(-100.53096491487338, 0)

(-43.982297150257104, 0)

(25.132741228718345, 0)

(87.96459430051421, 0)

(43.982297150257104, 0)

(-31.41592653589793, 0)

(-94.2477796076938, 0)

(18.84955592153876, 0)

(-18.84955592153876, 0)

(12.566370614359172, 0)

(81.68140899333463, 0)

(-81.68140899333463, 0)

(50.26548245743669, 0)

(-87.96459430051421, 0)

(56.548667764616276, 0)

(37.69911184307752, 0)

(100.53096491487338, 0)

(69.11503837897546, 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:
Puntos mínimos de la función:
x1=−12.5663706143592x_{1} = -12.5663706143592
x2=−69.1150383789755x_{2} = -69.1150383789755
x3=−50.2654824574367x_{3} = -50.2654824574367
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=−75.398223686155x_{5} = -75.398223686155
x6=−62.8318530717959x_{6} = -62.8318530717959
x7=−6.28318530717959x_{7} = -6.28318530717959
x8=−25.1327412287183x_{8} = -25.1327412287183
x9=−37.6991118430775x_{9} = -37.6991118430775
x10=−100.530964914873x_{10} = -100.530964914873
x11=−43.9822971502571x_{11} = -43.9822971502571
x12=−31.4159265358979x_{12} = -31.4159265358979
x13=−94.2477796076938x_{13} = -94.2477796076938
x14=−18.8495559215388x_{14} = -18.8495559215388
x15=−81.6814089933346x_{15} = -81.6814089933346
x16=−87.9645943005142x_{16} = -87.9645943005142
Puntos máximos de la función:
x16=31.4159265358979x_{16} = 31.4159265358979
x16=75.398223686155x_{16} = 75.398223686155
x16=6.28318530717959x_{16} = 6.28318530717959
x16=62.8318530717959x_{16} = 62.8318530717959
x16=94.2477796076938x_{16} = 94.2477796076938
x16=25.1327412287183x_{16} = 25.1327412287183
x16=87.9645943005142x_{16} = 87.9645943005142
x16=43.9822971502571x_{16} = 43.9822971502571
x16=18.8495559215388x_{16} = 18.8495559215388
x16=12.5663706143592x_{16} = 12.5663706143592
x16=81.6814089933346x_{16} = 81.6814089933346
x16=50.2654824574367x_{16} = 50.2654824574367
x16=56.5486677646163x_{16} = 56.5486677646163
x16=37.6991118430775x_{16} = 37.6991118430775
x16=100.530964914873x_{16} = 100.530964914873
x16=69.1150383789755x_{16} = 69.1150383789755
Decrece en los intervalos
[−6.28318530717959,6.28318530717959]\left[-6.28318530717959, 6.28318530717959\right]
Crece en los intervalos
(−∞,−100.530964914873]\left(-\infty, -100.530964914873\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+2sin⁡(x)xcos⁡(x)+2log⁡(cos⁡(x))x2x=0\frac{- \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 1 + \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}}{x} = 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))x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(log⁡(cos⁡(x))x)=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\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, dividida por x 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,
la inclinada coincide con la asíntota horizontal a la derecha
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,
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))x=−log⁡(cos⁡(x))x\frac{\log{\left(\cos{\left(x \right)} \right)}}{x} = - \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}
- No
log⁡(cos⁡(x))x=log⁡(cos⁡(x))x\frac{\log{\left(\cos{\left(x \right)} \right)}}{x} = \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}
- No
es decir, función
no es
par ni impar
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