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  • Gráfico de la función y =:
  • y^2+1 y^2+1
  • x/(x^2-1) x/(x^2-1)
  • 1-x^2 1-x^2
  • x/5 x/5

  • Expresiones idénticas

  • tan(tres *x)/((dos *x))
  • tangente de (3 multiplicar por x) dividir por ((2 multiplicar por x))
  • tangente de (tres multiplicar por x) dividir por ((dos multiplicar por x))
  • tan(3x)/((2x))
  • tan3x/2x
  • tan(3*x) dividir por ((2*x))

  • Expresiones con funciones

  • Tangente tan
  • tan(x)/x
  • tanh(x+3)
  • tan(x^2)
  • tan(x)+x
  • tan(5*x^3-3*x^2)
Trazado el gráfico de la función/ tan(3*x)/((2*x))

Gráfico de la función y = tan(3*x)/((2*x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       tan(3*x)
f(x) = --------
         2*x
f(x)=tan⁡(3x)2xf{\left(x \right)} = \frac{\tan{\left(3 x \right)}}{2 x} 
f = tan(3*x)/((2*x))
Gráfico de la función
02468-8-6-4-2-1010-2525
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⁡(3x)2x=0\frac{\tan{\left(3 x \right)}}{2 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=30.3687289847013x_{1} = 30.3687289847013
x2=−83.7758040957278x_{2} = -83.7758040957278
x3=−15.707963267949x_{3} = -15.707963267949
x4=36.6519142918809x_{4} = 36.6519142918809
x5=76.4454212373516x_{5} = 76.4454212373516
x6=−31.4159265358979x_{6} = -31.4159265358979
x7=32.4631240870945x_{7} = 32.4631240870945
x8=74.3510261349584x_{8} = 74.3510261349584
x9=21.9911485751286x_{9} = 21.9911485751286
x10=−21.9911485751286x_{10} = -21.9911485751286
x11=4.18879020478639x_{11} = 4.18879020478639
x12=28.2743338823081x_{12} = 28.2743338823081
x13=80.634211442138x_{13} = 80.634211442138
x14=14.6607657167524x_{14} = 14.6607657167524
x15=−96.342174710087x_{15} = -96.342174710087
x16=72.2566310325652x_{16} = 72.2566310325652
x17=24.0855436775217x_{17} = 24.0855436775217
x18=−94.2477796076938x_{18} = -94.2477796076938
x19=87.9645943005142x_{19} = 87.9645943005142
x20=−50.2654824574367x_{20} = -50.2654824574367
x21=−55.5014702134197x_{21} = -55.5014702134197
x22=−43.9822971502571x_{22} = -43.9822971502571
x23=−39.7935069454707x_{23} = -39.7935069454707
x24=−97.3893722612836x_{24} = -97.3893722612836
x25=50.2654824574367x_{25} = 50.2654824574367
x26=2.0943951023932x_{26} = 2.0943951023932
x27=59.6902604182061x_{27} = 59.6902604182061
x28=−7.33038285837618x_{28} = -7.33038285837618
x29=83.7758040957278x_{29} = 83.7758040957278
x30=−61.7846555205993x_{30} = -61.7846555205993
x31=68.0678408277789x_{31} = 68.0678408277789
x32=−35.6047167406843x_{32} = -35.6047167406843
x33=−41.8879020478639x_{33} = -41.8879020478639
x34=−53.4070751110265x_{34} = -53.4070751110265
x35=−85.870199198121x_{35} = -85.870199198121
x36=−79.5870138909414x_{36} = -79.5870138909414
x37=12.5663706143592x_{37} = 12.5663706143592
x38=−81.6814089933346x_{38} = -81.6814089933346
x39=94.2477796076938x_{39} = 94.2477796076938
x40=52.3598775598299x_{40} = 52.3598775598299
x41=81.6814089933346x_{41} = 81.6814089933346
x42=−30.3687289847013x_{42} = -30.3687289847013
x43=54.4542726622231x_{43} = 54.4542726622231
x44=85.870199198121x_{44} = 85.870199198121
x45=−28.2743338823081x_{45} = -28.2743338823081
x46=−24.0855436775217x_{46} = -24.0855436775217
x47=−48.1710873550435x_{47} = -48.1710873550435
x48=−72.2566310325652x_{48} = -72.2566310325652
x49=−68.0678408277789x_{49} = -68.0678408277789
x50=96.342174710087x_{50} = 96.342174710087
x51=37.6991118430775x_{51} = 37.6991118430775
x52=−11.5191730631626x_{52} = -11.5191730631626
x53=65.9734457253857x_{53} = 65.9734457253857
x54=39.7935069454707x_{54} = 39.7935069454707
x55=−52.3598775598299x_{55} = -52.3598775598299
x56=−17.8023583703422x_{56} = -17.8023583703422
x57=−87.9645943005142x_{57} = -87.9645943005142
x58=92.1533845053006x_{58} = 92.1533845053006
x59=−19.8967534727354x_{59} = -19.8967534727354
x60=61.7846555205993x_{60} = 61.7846555205993
x61=78.5398163397448x_{61} = 78.5398163397448
x62=−6.28318530717959x_{62} = -6.28318530717959
x63=15.707963267949x_{63} = 15.707963267949
x64=−57.5958653158129x_{64} = -57.5958653158129
x65=−2.0943951023932x_{65} = -2.0943951023932
x66=−26.1799387799149x_{66} = -26.1799387799149
x67=−70.162235930172x_{67} = -70.162235930172
x68=56.5486677646163x_{68} = 56.5486677646163
x69=46.0766922526503x_{69} = 46.0766922526503
x70=17.8023583703422x_{70} = 17.8023583703422
x71=−90.0589894029074x_{71} = -90.0589894029074
x72=48.1710873550435x_{72} = 48.1710873550435
x73=41.8879020478639x_{73} = 41.8879020478639
x74=−65.9734457253857x_{74} = -65.9734457253857
x75=10.471975511966x_{75} = 10.471975511966
x76=43.9822971502571x_{76} = 43.9822971502571
x77=70.162235930172x_{77} = 70.162235930172
x78=−37.6991118430775x_{78} = -37.6991118430775
x79=−59.6902604182061x_{79} = -59.6902604182061
x80=−92.1533845053006x_{80} = -92.1533845053006
x81=−75.398223686155x_{81} = -75.398223686155
x82=100.530964914873x_{82} = 100.530964914873
x83=−13.6135681655558x_{83} = -13.6135681655558
x84=34.5575191894877x_{84} = 34.5575191894877
x85=6.28318530717959x_{85} = 6.28318530717959
x86=98.4365698124802x_{86} = 98.4365698124802
x87=−99.4837673636768x_{87} = -99.4837673636768
x88=−63.8790506229925x_{88} = -63.8790506229925
x89=58.6430628670095x_{89} = 58.6430628670095
x90=19.8967534727354x_{90} = 19.8967534727354
x91=−9.42477796076938x_{91} = -9.42477796076938
x92=−33.5103216382911x_{92} = -33.5103216382911
x93=−4.18879020478639x_{93} = -4.18879020478639
x94=63.8790506229925x_{94} = 63.8790506229925
x95=−74.3510261349584x_{95} = -74.3510261349584
x96=−46.0766922526503x_{96} = -46.0766922526503
x97=8.37758040957278x_{97} = 8.37758040957278
x98=90.0589894029074x_{98} = 90.0589894029074
x99=26.1799387799149x_{99} = 26.1799387799149
x100=−77.4926187885482x_{100} = -77.4926187885482
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(3*x)/((2*x)).
tan⁡(0⋅3)0⋅2\frac{\tan{\left(0 \cdot 3 \right)}}{0 \cdot 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
12x(3tan⁡2(3x)+3)−tan⁡(3x)2x2=0\frac{1}{2 x} \left(3 \tan^{2}{\left(3 x \right)} + 3\right) - \frac{\tan{\left(3 x \right)}}{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
9(tan⁡2(3x)+1)tan⁡(3x)−3(tan⁡2(3x)+1)x+tan⁡(3x)x2x=0\frac{9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−37.7020586193797x_{1} = -37.7020586193797
x2=59.6921217440971x_{2} = 59.6921217440971
x3=−65.9751298043699x_{3} = -65.9751298043699
x4=−28.2782623603624x_{4} = -28.2782623603624
x5=−77.4940525549667x_{5} = -77.4940525549667
x6=76.4468746426753x_{6} = 76.4468746426753
x7=19.9023342041602x_{7} = 19.9023342041602
x8=−79.5884099297947x_{8} = -79.5884099297947
x9=100.532070129183x_{9} = 100.532070129183
x10=30.3723866910662x_{10} = 30.3723866910662
x11=61.7864537594165x_{11} = 61.7864537594165
x12=−9.43653299924975x_{12} = -9.43653299924975
x13=41.890554238415x_{13} = 41.890554238415
x14=26.1841813074948x_{14} = 26.1841813074948
x15=−26.1841813074948x_{15} = -26.1841813074948
x16=−39.7962986804533x_{16} = -39.7962986804533
x17=21.9961984071905x_{17} = 21.9961984071905
x18=−99.4848842112145x_{18} = -99.4848842112145
x19=−92.1545901877593x_{19} = -92.1545901877593
x20=−41.890554238415x_{20} = -41.890554238415
x21=48.1733936907667x_{21} = 48.1733936907667
x22=−55.5034719939252x_{22} = -55.5034719939252
x23=−52.3619994250983x_{23} = -52.3619994250983
x24=46.0791033967769x_{24} = 46.0791033967769
x25=−57.59779431589x_{25} = -57.59779431589
x26=52.3619994250983x_{26} = 52.3619994250983
x27=15.7150294035936x_{27} = 15.7150294035936
x28=24.0901548043733x_{28} = 24.0901548043733
x29=−72.2581686851326x_{29} = -72.2581686851326
x30=−33.513636601095x_{30} = -33.513636601095
x31=10.4825608710591x_{31} = 10.4825608710591
x32=−85.8714930952558x_{32} = -85.8714930952558
x33=−8.39079467531438x_{33} = -8.39079467531438
x34=−50.267692715982x_{34} = -50.267692715982
x35=−46.0791033967769x_{35} = -46.0791033967769
x36=6.30075451006221x_{36} = 6.30075451006221
x37=17.8085946429789x_{37} = 17.8085946429789
x38=−75.3996972758354x_{38} = -75.3996972758354
x39=80.6355893520718x_{39} = 80.6355893520718
x40=68.0694730948384x_{40} = 68.0694730948384
x41=−24.0901548043733x_{41} = -24.0901548043733
x42=83.7771303379305x_{42} = 83.7771303379305
x43=56.5506324811787x_{43} = 56.5506324811787
x44=37.7020586193797x_{44} = 37.7020586193797
x45=−4.21493490101902x_{45} = -4.21493490101902
x46=−31.4194623838264x_{46} = -31.4194623838264
x47=28.2782623603624x_{47} = 28.2782623603624
x48=−96.343327974564x_{48} = -96.343327974564
x49=39.7962986804533x_{49} = 39.7962986804533
x50=94.2489584987906x_{50} = 94.2489584987906
x51=−13.6217185706789x_{51} = -13.6217185706789
x52=4.21493490101902x_{52} = 4.21493490101902
x53=12.5751980850599x_{53} = 12.5751980850599
x54=−21.9961984071905x_{54} = -21.9961984071905
x55=−94.2489584987906x_{55} = -94.2489584987906
x56=−70.1638194780521x_{56} = -70.1638194780521
x57=81.6827692391381x_{57} = 81.6827692391381
x58=−68.0694730948384x_{58} = -68.0694730948384
x59=8.39079467531438x_{59} = 8.39079467531438
x60=−74.3525204774901x_{60} = -74.3525204774901
x61=−35.6078367878772x_{61} = -35.6078367878772
x62=−2.14462535489162x_{62} = -2.14462535489162
x63=−53.4091553787305x_{63} = -53.4091553787305
x64=72.2581686851326x_{64} = 72.2581686851326
x65=−30.3723866910662x_{65} = -30.3723866910662
x66=43.9848230807054x_{66} = 43.9848230807054
x67=−83.7771303379305x_{67} = -83.7771303379305
x68=92.1545901877593x_{68} = 92.1545901877593
x69=65.9751298043699x_{69} = 65.9751298043699
x70=63.8807899107774x_{70} = 63.8807899107774
x71=−97.3905131257488x_{71} = -97.3905131257488
x72=70.1638194780521x_{72} = 70.1638194780521
x73=−48.1733936907667x_{73} = -48.1733936907667
x74=2.14462535489162x_{74} = 2.14462535489162
x75=98.4376985407423x_{75} = 98.4376985407423
x76=14.6683354032458x_{76} = 14.6683354032458
x77=78.5412309908953x_{77} = 78.5412309908953
x78=34.5607337431424x_{78} = 34.5607337431424
x79=−87.9658573926638x_{79} = -87.9658573926638
x80=−81.6827692391381x_{80} = -81.6827692391381
x81=90.060223122721x_{81} = 90.060223122721
x82=87.9658573926638x_{82} = 87.9658573926638
x83=74.3525204774901x_{83} = 74.3525204774901
x84=96.343327974564x_{84} = 96.343327974564
x85=−19.9023342041602x_{85} = -19.9023342041602
x86=−43.9848230807054x_{86} = -43.9848230807054
x87=−17.8085946429789x_{87} = -17.8085946429789
x88=58.6449574258964x_{88} = 58.6449574258964
x89=−63.8807899107774x_{89} = -63.8807899107774
x90=−59.6921217440971x_{90} = -59.6921217440971
x91=−15.7150294035936x_{91} = -15.7150294035936
x92=85.8714930952558x_{92} = 85.8714930952558
x93=−61.7864537594165x_{93} = -61.7864537594165
x94=−11.5288000418182x_{94} = -11.5288000418182
x95=36.654945229763x_{95} = 36.654945229763
x96=54.4563129318412x_{96} = 54.4563129318412
x97=−6.30075451006221x_{97} = -6.30075451006221
x98=50.267692715982x_{98} = 50.267692715982
x99=32.4665459324263x_{99} = 32.4665459324263
x100=−90.060223122721x_{100} = -90.060223122721
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−(9(tan⁡2(3x)+1)tan⁡(3x)−3(tan⁡2(3x)+1)x+tan⁡(3x)x2x)=9\lim_{x \to 0^-}\left(\frac{9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}}{x}\right) = 9
lim⁡x→0+(9(tan⁡2(3x)+1)tan⁡(3x)−3(tan⁡2(3x)+1)x+tan⁡(3x)x2x)=9\lim_{x \to 0^+}\left(\frac{9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}}{x}\right) = 9
- los límites son iguales, es decir omitimos el punto correspondiente

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
[100.532070129183,∞)\left[100.532070129183, \infty\right)
Convexa en los intervalos
[−2.14462535489162,2.14462535489162]\left[-2.14462535489162, 2.14462535489162\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⁡(3x)2x)y = \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{2 x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(3x)2x)y = \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{2 x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(3*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→−∞(12xtan⁡(3x)x)y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{2 x} \tan{\left(3 x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(12xtan⁡(3x)x)y = x \lim_{x \to \infty}\left(\frac{\frac{1}{2 x} \tan{\left(3 x \right)}}{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⁡(3x)2x=tan⁡(3x)2x\frac{\tan{\left(3 x \right)}}{2 x} = \frac{\tan{\left(3 x \right)}}{2 x}
- No
tan⁡(3x)2x=−tan⁡(3x)2x\frac{\tan{\left(3 x \right)}}{2 x} = - \frac{\tan{\left(3 x \right)}}{2 x}
- No
es decir, función
no es
par ni impar
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