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

Gráfico de la función y = sin(x)^tan(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          tan(x)   
f(x) = sin      (x)
f(x)=sin⁡tan⁡(x)(x)f{\left(x \right)} = \sin^{\tan{\left(x \right)}}{\left(x \right)} 
f = sin(x)^tan(x)
Gráfico de la función
02468-8-6-4-2-10100.02.0
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:
sin⁡tan⁡(x)(x)=0\sin^{\tan{\left(x \right)}}{\left(x \right)} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 sin(x)^tan(x).
sin⁡tan⁡(0)(0)\sin^{\tan{\left(0 \right)}}{\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
((tan⁡2(x)+1)log⁡(sin⁡(x))+cos⁡(x)tan⁡(x)sin⁡(x))sin⁡tan⁡(x)(x)=0\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sin{\left(x \right)} \right)} + \frac{\cos{\left(x \right)} \tan{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin^{\tan{\left(x \right)}}{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−72.7242516821245x_{1} = -72.7242516821245
x2=27.8067132327489x_{2} = 27.8067132327489
x3=78.0721956901856x_{3} = 78.0721956901856
x4=−37.2314911935183x_{4} = -37.2314911935183
x5=65.5058250758264x_{5} = 65.5058250758264
x6=−49.7978618078774x_{6} = -49.7978618078774
x7=−81.2137883437754x_{7} = -81.2137883437754
x8=71.789010383006x_{8} = 71.789010383006
x9=15.2403426183897x_{9} = 15.2403426183897
x10=−60.1578810677653x_{10} = -60.1578810677653
x11=−100.063344265314x_{11} = -100.063344265314
x12=82.1490296428939x_{12} = 82.1490296428939
x13=−97.8569929108428x_{13} = -97.8569929108428
x14=−12.0987499647999x_{14} = -12.0987499647999
x15=38.1667324926368x_{15} = 38.1667324926368
x16=−62.3642324222366x_{16} = -62.3642324222366
x17=−24.6651205791591x_{17} = -24.6651205791591
x18=69.5826590285347x_{18} = 69.5826590285347
x19=−3.60921330314904x_{19} = -3.60921330314904
x20=−16.1755839175082x_{20} = -16.1755839175082
x21=−18.3819352719795x_{21} = -18.3819352719795
x22=84.3553809973652x_{22} = 84.3553809973652
x23=−9.89239861032863x_{23} = -9.89239861032863
x24=96.9217516117243x_{24} = 96.9217516117243
x25=19.317176571098x_{25} = 19.317176571098
x26=−93.7801589581345x_{26} = -93.7801589581345
x27=−66.4410663749449x_{27} = -66.4410663749449
x28=44.4499177998164x_{28} = 44.4499177998164
x29=−22.4587692246878x_{29} = -22.4587692246878
x30=−35.025139839047x_{30} = -35.025139839047
x31=40.3730838471081x_{31} = 40.3730838471081
x32=−87.496973650955x_{32} = -87.496973650955
x33=25.6003618782776x_{33} = 25.6003618782776
x34=75.8658443357143x_{34} = 75.8658443357143
x35=−43.5146765006979x_{35} = -43.5146765006979
x36=−68.6474177294162x_{36} = -68.6474177294162
x37=59.2226397686468x_{37} = 59.2226397686468
x38=−5.81556465762034x_{38} = -5.81556465762034
x39=21.5235279255693x_{39} = 21.5235279255693
x40=88.4322149500735x_{40} = 88.4322149500735
x41=34.0898985399285x_{41} = 34.0898985399285
x42=90.6385663045448x_{42} = 90.6385663045448
x43=−41.3083251462266x_{43} = -41.3083251462266
x44=94.7154002572531x_{44} = 94.7154002572531
x45=31.8835471854572x_{45} = 31.8835471854572
x46=−79.0074369893041x_{46} = -79.0074369893041
x47=−91.5738076036633x_{47} = -91.5738076036633
x48=63.2994737213551x_{48} = 63.2994737213551
x49=2.67397200403055x_{49} = 2.67397200403055
x50=−47.5915104534061x_{50} = -47.5915104534061
x51=−53.8746957605857x_{51} = -53.8746957605857
x52=−56.081047115057x_{52} = -56.081047115057
Signos de extremos en los puntos:
(-72.72425168212449, 1.49536626668525)

(27.80671323274889, 1.49536626668525)

(78.07219569018558, 1.49536626668525)

(-37.23149119351827, 0.668732485330622)

(65.50582507582641, 1.49536626668525)

(-49.797861807877446, 0.668732485330622)

(-81.21378834377538, 0.668732485330622)

(71.789010383006, 1.49536626668525)

(15.240342618389718, 1.49536626668525)

(-60.15788106776532, 1.49536626668525)

(-100.06334426531414, 0.668732485330622)

(82.14902964289386, 0.668732485330622)

(-97.85699291084283, 1.49536626668525)

(-12.098749964799925, 0.668732485330622)

(38.16673249263677, 0.668732485330622)

(-62.36423242223662, 0.668732485330622)

(-24.665120579159098, 0.668732485330622)

(69.58265902853469, 0.668732485330622)

(-3.609213303149041, 1.49536626668525)

(-16.175583917508213, 1.49536626668525)

(-18.38193527197951, 0.668732485330622)

(84.35538099736517, 1.49536626668525)

(-9.892398610328627, 1.49536626668525)

(96.92175161172435, 1.49536626668525)

(19.317176571098006, 0.668732485330622)

(-93.78015895813455, 0.668732485330622)

(-66.44106637494491, 1.49536626668525)

(44.449917799816355, 0.668732485330622)

(-22.4587692246878, 1.49536626668525)

(-35.02513983904697, 1.49536626668525)

(40.37308384710806, 1.49536626668525)

(-87.49697365095496, 0.668732485330622)

(25.600361878277592, 0.668732485330622)

(75.86584433571429, 0.668732485330622)

(-43.51467650069786, 0.668732485330622)

(-68.6474177294162, 0.668732485330622)

(59.22263976864682, 1.49536626668525)

(-5.815564657620339, 0.668732485330622)

(21.523527925569304, 1.49536626668525)

(88.43221495007346, 0.668732485330622)

(34.08989853992848, 1.49536626668525)

(90.63856630454475, 1.49536626668525)

(-41.30832514622656, 1.49536626668525)

(94.71540025725305, 0.668732485330622)

(31.88354718545718, 0.668732485330622)

(-79.00743698930408, 1.49536626668525)

(-91.57380760366325, 1.49536626668525)

(63.29947372135511, 0.668732485330622)

(2.6739720040305457, 1.49536626668525)

(-47.591510453406144, 1.49536626668525)

(-53.87469576058573, 1.49536626668525)

(-56.08104711505703, 0.668732485330622)


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=−37.2314911935183x_{1} = -37.2314911935183
x2=−49.7978618078774x_{2} = -49.7978618078774
x3=−81.2137883437754x_{3} = -81.2137883437754
x4=−100.063344265314x_{4} = -100.063344265314
x5=82.1490296428939x_{5} = 82.1490296428939
x6=−12.0987499647999x_{6} = -12.0987499647999
x7=38.1667324926368x_{7} = 38.1667324926368
x8=−62.3642324222366x_{8} = -62.3642324222366
x9=−24.6651205791591x_{9} = -24.6651205791591
x10=69.5826590285347x_{10} = 69.5826590285347
x11=−18.3819352719795x_{11} = -18.3819352719795
x12=19.317176571098x_{12} = 19.317176571098
x13=−93.7801589581345x_{13} = -93.7801589581345
x14=44.4499177998164x_{14} = 44.4499177998164
x15=−87.496973650955x_{15} = -87.496973650955
x16=25.6003618782776x_{16} = 25.6003618782776
x17=75.8658443357143x_{17} = 75.8658443357143
x18=−43.5146765006979x_{18} = -43.5146765006979
x19=−68.6474177294162x_{19} = -68.6474177294162
x20=−5.81556465762034x_{20} = -5.81556465762034
x21=88.4322149500735x_{21} = 88.4322149500735
x22=94.7154002572531x_{22} = 94.7154002572531
x23=31.8835471854572x_{23} = 31.8835471854572
x24=63.2994737213551x_{24} = 63.2994737213551
x25=−56.081047115057x_{25} = -56.081047115057
Puntos máximos de la función:
x25=−72.7242516821245x_{25} = -72.7242516821245
x25=27.8067132327489x_{25} = 27.8067132327489
x25=78.0721956901856x_{25} = 78.0721956901856
x25=65.5058250758264x_{25} = 65.5058250758264
x25=71.789010383006x_{25} = 71.789010383006
x25=15.2403426183897x_{25} = 15.2403426183897
x25=−60.1578810677653x_{25} = -60.1578810677653
x25=−97.8569929108428x_{25} = -97.8569929108428
x25=−3.60921330314904x_{25} = -3.60921330314904
x25=−16.1755839175082x_{25} = -16.1755839175082
x25=84.3553809973652x_{25} = 84.3553809973652
x25=−9.89239861032863x_{25} = -9.89239861032863
x25=96.9217516117243x_{25} = 96.9217516117243
x25=−66.4410663749449x_{25} = -66.4410663749449
x25=−22.4587692246878x_{25} = -22.4587692246878
x25=−35.025139839047x_{25} = -35.025139839047
x25=40.3730838471081x_{25} = 40.3730838471081
x25=59.2226397686468x_{25} = 59.2226397686468
x25=21.5235279255693x_{25} = 21.5235279255693
x25=34.0898985399285x_{25} = 34.0898985399285
x25=90.6385663045448x_{25} = 90.6385663045448
x25=−41.3083251462266x_{25} = -41.3083251462266
x25=−79.0074369893041x_{25} = -79.0074369893041
x25=−91.5738076036633x_{25} = -91.5738076036633
x25=2.67397200403055x_{25} = 2.67397200403055
x25=−47.5915104534061x_{25} = -47.5915104534061
x25=−53.8746957605857x_{25} = -53.8746957605857
Decrece en los intervalos
[94.7154002572531,∞)\left[94.7154002572531, \infty\right)
Crece en los intervalos
(−∞,−100.063344265314]\left(-\infty, -100.063344265314\right]
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→−∞sin⁡tan⁡(x)(x)y = \lim_{x \to -\infty} \sin^{\tan{\left(x \right)}}{\left(x \right)}
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞sin⁡tan⁡(x)(x)y = \lim_{x \to \infty} \sin^{\tan{\left(x \right)}}{\left(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:
sin⁡tan⁡(x)(x)=(−sin⁡(x))−tan⁡(x)\sin^{\tan{\left(x \right)}}{\left(x \right)} = \left(- \sin{\left(x \right)}\right)^{- \tan{\left(x \right)}}
- No
sin⁡tan⁡(x)(x)=−(−sin⁡(x))−tan⁡(x)\sin^{\tan{\left(x \right)}}{\left(x \right)} = - \left(- \sin{\left(x \right)}\right)^{- \tan{\left(x \right)}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sin(x)^tan(x)
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