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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        x*tan(x)
f(x) = 2
f(x)=2xtan⁡(x)f{\left(x \right)} = 2^{x \tan{\left(x \right)}} 
f = 2^(x*tan(x))
Gráfico de la función
02468-8-6-4-2-101002e36
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:
2xtan⁡(x)=02^{x \tan{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−74x_{1} = -74
x2=93.7392360779319x_{2} = 93.7392360779319
x3=−87.4573040965086x_{3} = -87.4573040965086
x4=7.94906547193951x_{4} = 7.94906547193951
x5=−65.3758026134456x_{5} = -65.3758026134456
x6=−33.460843974514x_{6} = -33.460843974514
x7=−8x_{7} = -8
x8=−52x_{8} = -52
x9=58.25x_{9} = 58.25
x10=−67.75x_{10} = -67.75
x11=23.8512071242679x_{11} = 23.8512071242679
x12=39.7658127296485x_{12} = 39.7658127296485
x13=−93.5372433556428x_{13} = -93.5372433556428
x14=33.5767464220834x_{14} = 33.5767464220834
x15=83.8655762391978x_{15} = 83.8655762391978
x16=52x_{16} = 52
x17=−39.6274617482893x_{17} = -39.6274617482893
x18=77.7704361051493x_{18} = 77.7704361051493
x19=67.9299821346181x_{19} = 67.9299821346181
x20=−30x_{20} = -30
x21=−83.6797619147589x_{21} = -83.6797619147589
x22=96x_{22} = 96
x23=89.9967672341884x_{23} = 89.9967672341884
x24=36.25x_{24} = 36.25
x25=−49.4007444461329x_{25} = -49.4007444461329
x26=−27.2366341005014x_{26} = -27.2366341005014
x27=−11.2490636164206x_{27} = -11.2490636164206
x28=61.8299628020166x_{28} = 61.8299628020166
x29=−17.5424646489314x_{29} = -17.5424646489314
x30=14.25x_{30} = 14.25
x31=−1.60959069459541x_{31} = -1.60959069459541
x32=45.9059949386518x_{32} = 45.9059949386518
x33=80.25x_{33} = 80.25
x34=−55.5463465490928x_{34} = -55.5463465490928
x35=−89.75x_{35} = -89.75
x36=55.7024946175115x_{36} = 55.7024946175115
x37=−77.59184479844x_{37} = -77.59184479844
x38=49.5594301423322x_{38} = 49.5594301423322
x39=74x_{39} = 74
x40=−61.6615017614917x_{40} = -61.6615017614917
x41=17.6184584305389x_{41} = 17.6184584305389
x42=99.8126587555757x_{42} = 99.8126587555757
x43=30x_{43} = 30
x44=−71.4858341979056x_{44} = -71.4858341979056
x45=−45.75x_{45} = -45.75
x46=−23.7478604283627x_{46} = -23.7478604283627
x47=−96x_{47} = -96
x48=−43.2347994312404x_{48} = -43.2347994312404
x49=−99.6199761477733x_{49} = -99.6199761477733
x50=71.6720262218199x_{50} = 71.6720262218199
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 2^(x*tan(x)).
20tan⁡(0)2^{0 \tan{\left(0 \right)}}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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
2xtan⁡(x)(x(tan⁡2(x)+1)+tan⁡(x))log⁡(2)=02^{x \tan{\left(x \right)}} \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \log{\left(2 \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−99.6183394336316x_{1} = -99.6183394336316
x2=−74x_{2} = -74
x3=33.5650208576075x_{3} = 33.5650208576075
x4=−49.3933311095239x_{4} = -49.3933311095239
x5=−8x_{5} = -8
x6=7.94758041818416x_{6} = 7.94758041818416
x7=−27.2213857692459x_{7} = -27.2213857692459
x8=−52x_{8} = -52
x9=58.25x_{9} = 58.25
x10=−67.75x_{10} = -67.75
x11=−61.6601354030739x_{11} = -61.6601354030739
x12=−77.589402614256x_{12} = -77.589402614256
x13=61.8268379697226x_{13} = 61.8268379697226
x14=52x_{14} = 52
x15=99.8104085167801x_{15} = 99.8104085167801
x16=−30x_{16} = -30
x17=−17.5358490903254x_{17} = -17.5358490903254
x18=96x_{18} = 96
x19=36.25x_{19} = 36.25
x20=−93.5345960280675x_{20} = -93.5345960280675
x21=−83.678905101525x_{21} = -83.678905101525
x22=0x_{22} = 0
x23=77.7669801721682x_{23} = 77.7669801721682
x24=39.7600082125317x_{24} = 39.7600082125317
x25=14.25x_{25} = 14.25
x26=55.6965973793536x_{26} = 55.6965973793536
x27=−87.454551029532x_{27} = -87.454551029532
x28=80.25x_{28} = 80.25
x29=71.6679170716591x_{29} = 71.6679170716591
x30=−89.75x_{30} = -89.75
x31=−39.624881783627x_{31} = -39.624881783627
x32=89.9967250645693x_{32} = 89.9967250645693
x33=−23.7137089835621x_{33} = -23.7137089835621
x34=−11.2316472982516x_{34} = -11.2316472982516
x35=−33.4530096122582x_{35} = -33.4530096122582
x36=93.7368246905657x_{36} = 93.7368246905657
x37=67.9290126385135x_{37} = 67.9290126385135
x38=45.9042534928273x_{38} = 45.9042534928273
x39=74x_{39} = 74
x40=17.6050336016714x_{40} = 17.6050336016714
x41=83.8636278678182x_{41} = 83.8636278678182
x42=23.8470753746242x_{42} = 23.8470753746242
x43=−71.4816515446886x_{43} = -71.4816515446886
x44=30x_{44} = 30
x45=−45.75x_{45} = -45.75
x46=−55.5423054709898x_{46} = -55.5423054709898
x47=−96x_{47} = -96
Signos de extremos en los puntos:
(-99.61833943363156, 1.63866513219626e-39)

(-74, 1.58790900237183e-128)

(33.565020857607536, 3.31499297221772e-16)

(-49.393331109523935, 1.98804236404477e-18)

(-8, 4.21369523372168e-17)

(7.947580418184165, 3.26500808864899e-26)

(-27.221385769245916, 4.13428753741543e-15)

(-52, 1.75686715812741e-95)

(58.25, 2.71030452962077e-134)

(-67.75, 1.91644199750828e-98)

(-61.66013540307387, 9.6557612599462e-45)

(-77.58940261425604, 2.04065126683741e-33)

(61.82683796972256, 4.91684909643655e-30)

(52, 1.75686715812741e-95)

(99.8104085167801, 4.1504023602923e-27)

(-30, 1.42592055465052e-58)

(-17.535849090325392, 8.34430119408943e-21)

(96, 2.89987025510949e-158)

(36.25, 5.83105080724914e-90)

(-93.53459602806753, 4.38733746221933e-25)

(-83.67890510152503, 3.91687381553293e-56)

(0, 1)

(77.76698017216819, 1.48148484626021e-23)

(39.76000821253173, 3.68061644829314e-23)

(14.25, 1.39175579250676e-38)

(55.696597379353555, 6.83026052982068e-20)

(-87.45455102953201, 1.87317760503026e-15)

(80.25, 6.46748163225783e-173)

(71.66791707165906, 3.93574062215682e-15)

(-89.75, 1.11543182910651e-124)

(-39.62488178362696, 6.60887864629277e-33)

(89.99672506456929, 3.17524060405033e-55)

(-23.71370898356214, 2.11159384470216e-47)

(-11.23164729825155, 8.8094089506234e-15)

(-33.45300961225823, 9.80549163005508e-21)

(93.7368246905657, 1.51651476719276e-16)

(67.9290126385135, 3.19036755545125e-51)

(45.90425349282726, 1.90455903467066e-38)

(74, 1.58790900237183e-128)

(17.605033601671398, 2.17565107655221e-16)

(83.86362786781818, 9.73795404054003e-37)

(23.84707537462418, 3.22977033664645e-25)

(-71.48165154468862, 8.42659109795534e-22)

(30, 1.42592055465052e-58)

(-45.75, 9.20125931926392e-70)

(-55.542305470989824, 3.9106654834059e-27)

(-96, 2.89987025510949e-158)


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=0x_{1} = 0
La función no tiene puntos máximos
Decrece en los intervalos
[0,∞)\left[0, \infty\right)
Crece en los intervalos
(−∞,0]\left(-\infty, 0\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
2xtan⁡(x)(2x(tan⁡2(x)+1)tan⁡(x)+(x(tan⁡2(x)+1)+tan⁡(x))2log⁡(2)+2tan⁡2(x)+2)log⁡(2)=02^{x \tan{\left(x \right)}} \left(2 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)^{2} \log{\left(2 \right)} + 2 \tan^{2}{\left(x \right)} + 2\right) \log{\left(2 \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−74x_{1} = -74
x2=−17.528816187949x_{2} = -17.528816187949
x3=39.7539051792938x_{3} = 39.7539051792938
x4=−52x_{4} = -52
x5=−55.5380972509922x_{5} = -55.5380972509922
x6=58.25x_{6} = 58.25
x7=−67.75x_{7} = -67.75
x8=−39.6221778860703x_{8} = -39.6221778860703
x9=67.9280114788194x_{9} = 67.9280114788194
x10=93.7344311561396x_{10} = 93.7344311561396
x11=52x_{11} = 52
x12=23.8426919598751x_{12} = 23.8426919598751
x13=−30x_{13} = -30
x14=−83.6780238178204x_{14} = -83.6780238178204
x15=89.9966815516549x_{15} = 89.9966815516549
x16=17.5908632803025x_{16} = 17.5908632803025
x17=96x_{17} = 96
x18=36.25x_{18} = 36.25
x19=−93.5319016536416x_{19} = -93.5319016536416
x20=−49.3856619215542x_{20} = -49.3856619215542
x21=14.25x_{21} = 14.25
x22=−77.5868814914086x_{22} = -77.5868814914086
x23=−71.4773667290766x_{23} = -71.4773667290766
x24=80.25x_{24} = 80.25
x25=−89.75x_{25} = -89.75
x26=83.8616192237483x_{26} = 83.8616192237483
x27=−23.7130790737585x_{27} = -23.7130790737585
x28=7.94599865464823x_{28} = 7.94599865464823
x29=99.8081171947765x_{29} = 99.8081171947765
x30=−33.44475782823x_{30} = -33.44475782823
x31=33.5527647140669x_{31} = 33.5527647140669
x32=61.823590620763x_{32} = 61.823590620763
x33=−99.6166598668128x_{33} = -99.6166598668128
x34=74x_{34} = 74
x35=55.6905023170195x_{35} = 55.6905023170195
x36=30x_{36} = 30
x37=−61.6587199007142x_{37} = -61.6587199007142
x38=−45.75x_{38} = -45.75
x39=−96x_{39} = -96
x40=77.7634403159374x_{40} = 77.7634403159374
x41=45.9024368161541x_{41} = 45.9024368161541

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:
No tiene corvaduras en todo el eje numérico
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 2^(x*tan(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→−∞(2xtan⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{2^{x \tan{\left(x \right)}}}{x}\right)
True

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