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tan(x)-2/x
Trazado el gráfico de la función/ tan(x)-2/x

Gráfico de la función y = tan(x)-2/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                2
f(x) = tan(x) - -
                x
f(x)=tan⁡(x)−2xf{\left(x \right)} = \tan{\left(x \right)} - \frac{2}{x} 
f = tan(x) - 2/x
Gráfico de la función
02468-8-6-4-2-1010-200200
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:
tan⁡(x)−2x=0\tan{\left(x \right)} - \frac{2}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−40.8895777660408x_{1} = -40.8895777660408
x2=−84.8465692433091x_{2} = -84.8465692433091
x3=72.2842925036825x_{3} = 72.2842925036825
x4=−50.3052188363296x_{4} = -50.3052188363296
x5=15.8336114149477x_{5} = 15.8336114149477
x6=−6.57833373272234x_{6} = -6.57833373272234
x7=−25.2119030642106x_{7} = -25.2119030642106
x8=40.8895777660408x_{8} = 40.8895777660408
x9=69.1439554764926x_{9} = 69.1439554764926
x10=−91.1281305511393x_{10} = -91.1281305511393
x11=−9.62956034329743x_{11} = -9.62956034329743
x12=−56.5839987378634x_{12} = -56.5839987378634
x13=25.2119030642106x_{13} = 25.2119030642106
x14=−81.7058821480364x_{14} = -81.7058821480364
x15=53.4444796697636x_{15} = 53.4444796697636
x16=66.0037377708277x_{16} = 66.0037377708277
x17=−75.4247339745236x_{17} = -75.4247339745236
x18=31.479374920314x_{18} = 31.479374920314
x19=75.4247339745236x_{19} = 75.4247339745236
x20=91.1281305511393x_{20} = 91.1281305511393
x21=−37.7520396346102x_{21} = -37.7520396346102
x22=−12.7222987717666x_{22} = -12.7222987717666
x23=−3.6435971674254x_{23} = -3.6435971674254
x24=−18.954681766529x_{24} = -18.954681766529
x25=50.3052188363296x_{25} = 50.3052188363296
x26=−69.1439554764926x_{26} = -69.1439554764926
x27=34.6152330552306x_{27} = 34.6152330552306
x28=−94.2689923093066x_{28} = -94.2689923093066
x29=22.0814757672807x_{29} = 22.0814757672807
x30=−62.863657228703x_{30} = -62.863657228703
x31=37.7520396346102x_{31} = 37.7520396346102
x32=56.5839987378634x_{32} = 56.5839987378634
x33=−34.6152330552306x_{33} = -34.6152330552306
x34=−97.4099011706723x_{34} = -97.4099011706723
x35=94.2689923093066x_{35} = 94.2689923093066
x36=59.7237354324305x_{36} = 59.7237354324305
x37=−44.0276918992479x_{37} = -44.0276918992479
x38=28.3447768697864x_{38} = 28.3447768697864
x39=100.550852725424x_{39} = 100.550852725424
x40=6.57833373272234x_{40} = 6.57833373272234
x41=−59.7237354324305x_{41} = -59.7237354324305
x42=84.8465692433091x_{42} = 84.8465692433091
x43=−47.1662676027767x_{43} = -47.1662676027767
x44=47.1662676027767x_{44} = 47.1662676027767
x45=−78.5652673845995x_{45} = -78.5652673845995
x46=−66.0037377708277x_{46} = -66.0037377708277
x47=3.6435971674254x_{47} = 3.6435971674254
x48=9.62956034329743x_{48} = 9.62956034329743
x49=−100.550852725424x_{49} = -100.550852725424
x50=−28.3447768697864x_{50} = -28.3447768697864
x51=−15.8336114149477x_{51} = -15.8336114149477
x52=78.5652673845995x_{52} = 78.5652673845995
x53=87.9873209346887x_{53} = 87.9873209346887
x54=81.7058821480364x_{54} = 81.7058821480364
x55=97.4099011706723x_{55} = 97.4099011706723
x56=−31.479374920314x_{56} = -31.479374920314
x57=−87.9873209346887x_{57} = -87.9873209346887
x58=−22.0814757672807x_{58} = -22.0814757672807
x59=−72.2842925036825x_{59} = -72.2842925036825
x60=−53.4444796697636x_{60} = -53.4444796697636
x61=44.0276918992479x_{61} = 44.0276918992479
x62=18.954681766529x_{62} = 18.954681766529
x63=62.863657228703x_{63} = 62.863657228703
x64=12.7222987717666x_{64} = 12.7222987717666
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 tan(x) - 2/x.
tan⁡(0)−20\tan{\left(0 \right)} - \frac{2}{0}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
tan⁡2(x)+1+2x2=0\tan^{2}{\left(x \right)} + 1 + \frac{2}{x^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
2((tan⁡2(x)+1)tan⁡(x)−2x3)=02 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2}{x^{3}}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=53.4070882400762x_{1} = 53.4070882400762
x2=−53.4070882400762x_{2} = -53.4070882400762
x3=−59.6902698223522x_{3} = -59.6902698223522
x4=91.1061895988645x_{4} = 91.1061895988645
x5=31.4159910385691x_{5} = 31.4159910385691
x6=100.530966883351x_{6} = 100.530966883351
x7=−56.5486788248068x_{7} = -56.5486788248068
x8=−40.8407338562166x_{8} = -40.8407338562166
x9=−21.9913366258898x_{9} = -21.9913366258898
x10=−12.5673782310434x_{10} = -12.5673782310434
x11=18.8498545326302x_{11} = 18.8498545326302
x12=−97.3893744264699x_{12} = -97.3893744264699
x13=12.5673782310434x_{13} = 12.5673782310434
x14=59.6902698223522x_{14} = 59.6902698223522
x15=−69.1150444367376x_{15} = -69.1150444367376
x16=40.8407338562166x_{16} = 40.8407338562166
x17=−47.123908915844x_{17} = -47.123908915844
x18=28.2744223630525x_{18} = 28.2744223630525
x19=56.5486788248068x_{19} = 56.5486788248068
x20=−72.2566363340409x_{20} = -72.2566363340409
x21=−84.8230049240194x_{21} = -84.8230049240194
x22=−37.6991491711314x_{22} = -37.6991491711314
x23=37.6991491711314x_{23} = 37.6991491711314
x24=−65.9734526904055x_{24} = -65.9734526904055
x25=87.9645972388826x_{25} = 87.9645972388826
x26=−25.1328672093776x_{26} = -25.1328672093776
x27=78.5398204679406x_{27} = 78.5398204679406
x28=−100.530966883351x_{28} = -100.530966883351
x29=84.8230049240194x_{29} = 84.8230049240194
x30=9.42716513079408x_{30} = 9.42716513079408
x31=72.2566363340409x_{31} = 72.2566363340409
x32=−91.1061895988645x_{32} = -91.1061895988645
x33=34.557567651394x_{33} = 34.557567651394
x34=75.3982283521748x_{34} = 75.3982283521748
x35=3.20220404348268x_{35} = 3.20220404348268
x36=97.3893744264699x_{36} = 97.3893744264699
x37=−18.8498545326302x_{37} = -18.8498545326302
x38=−81.6814126632857x_{38} = -81.6814126632857
x39=−75.3982283521748x_{39} = -75.3982283521748
x40=−9.42716513079408x_{40} = -9.42716513079408
x41=25.1328672093776x_{41} = 25.1328672093776
x42=−78.5398204679406x_{42} = -78.5398204679406
x43=−31.4159910385691x_{43} = -31.4159910385691
x44=81.6814126632857x_{44} = 81.6814126632857
x45=6.29121666025161x_{45} = 6.29121666025161
x46=−62.8318611346764x_{46} = -62.8318611346764
x47=62.8318611346764x_{47} = 62.8318611346764
x48=−50.2654982052414x_{48} = -50.2654982052414
x49=−43.9823206571687x_{49} = -43.9823206571687
x50=−3.20220404348268x_{50} = -3.20220404348268
x51=−87.9645972388826x_{51} = -87.9645972388826
x52=−28.2744223630525x_{52} = -28.2744223630525
x53=50.2654982052414x_{53} = 50.2654982052414
x54=−15.7084792414691x_{54} = -15.7084792414691
x55=69.1150444367376x_{55} = 69.1150444367376
x56=47.123908915844x_{56} = 47.123908915844
x57=−34.557567651394x_{57} = -34.557567651394
x58=21.9913366258898x_{58} = 21.9913366258898
x59=43.9823206571687x_{59} = 43.9823206571687
x60=−94.2477819966962x_{60} = -94.2477819966962
x61=65.9734526904055x_{61} = 65.9734526904055
x62=15.7084792414691x_{62} = 15.7084792414691
x63=94.2477819966962x_{63} = 94.2477819966962
x64=−6.29121666025161x_{64} = -6.29121666025161
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(2((tan⁡2(x)+1)tan⁡(x)−2x3))=∞\lim_{x \to 0^-}\left(2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2}{x^{3}}\right)\right) = \infty
lim⁡x→0+(2((tan⁡2(x)+1)tan⁡(x)−2x3))=−∞\lim_{x \to 0^+}\left(2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2}{x^{3}}\right)\right) = -\infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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
[100.530966883351,∞)\left[100.530966883351, \infty\right)
Convexa en los intervalos
(−∞,−100.530966883351]\left(-\infty, -100.530966883351\right]
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
True

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

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(x)−2x)y = \lim_{x \to \infty}\left(\tan{\left(x \right)} - \frac{2}{x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x) - 2/x, 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→−∞(tan⁡(x)−2xx)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} - \frac{2}{x}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡(x)−2xx)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} - \frac{2}{x}}{x}\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:
tan⁡(x)−2x=−tan⁡(x)+2x\tan{\left(x \right)} - \frac{2}{x} = - \tan{\left(x \right)} + \frac{2}{x}
- No
tan⁡(x)−2x=tan⁡(x)−2x\tan{\left(x \right)} - \frac{2}{x} = \tan{\left(x \right)} - \frac{2}{x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = tan(x)-2/x
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