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

Gráfico de la función y = tan(sin(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=31.4159265358979x_{1} = 31.4159265358979
x2=3.14159265358979x_{2} = 3.14159265358979
x3=−47.1238898038469x_{3} = -47.1238898038469
x4=−12.5663706143592x_{4} = -12.5663706143592
x5=−34.5575191894877x_{5} = -34.5575191894877
x6=−69.1150383789755x_{6} = -69.1150383789755
x7=75.398223686155x_{7} = 75.398223686155
x8=−65.9734457253857x_{8} = -65.9734457253857
x9=−50.2654824574367x_{9} = -50.2654824574367
x10=−56.5486677646163x_{10} = -56.5486677646163
x11=59.6902604182061x_{11} = 59.6902604182061
x12=72.2566310325652x_{12} = 72.2566310325652
x13=91.106186954104x_{13} = 91.106186954104
x14=−91.106186954104x_{14} = -91.106186954104
x15=−62.8318530717959x_{15} = -62.8318530717959
x16=−6.28318530717959x_{16} = -6.28318530717959
x17=6.28318530717959x_{17} = 6.28318530717959
x18=62.8318530717959x_{18} = 62.8318530717959
x19=−25.1327412287183x_{19} = -25.1327412287183
x20=94.2477796076938x_{20} = 94.2477796076938
x21=−9.42477796076938x_{21} = -9.42477796076938
x22=−37.6991118430775x_{22} = -37.6991118430775
x23=65.9734457253857x_{23} = 65.9734457253857
x24=−100.530964914873x_{24} = -100.530964914873
x25=−43.9822971502571x_{25} = -43.9822971502571
x26=25.1327412287183x_{26} = 25.1327412287183
x27=21.9911485751286x_{27} = 21.9911485751286
x28=87.9645943005142x_{28} = 87.9645943005142
x29=−40.8407044966673x_{29} = -40.8407044966673
x30=−97.3893722612836x_{30} = -97.3893722612836
x31=43.9822971502571x_{31} = 43.9822971502571
x32=−53.4070751110265x_{32} = -53.4070751110265
x33=97.3893722612836x_{33} = 97.3893722612836
x34=100.530964914873x_{34} = 100.530964914873
x35=−94.2477796076938x_{35} = -94.2477796076938
x36=−31.4159265358979x_{36} = -31.4159265358979
x37=18.8495559215388x_{37} = 18.8495559215388
x38=78.5398163397448x_{38} = 78.5398163397448
x39=−103.672557568463x_{39} = -103.672557568463
x40=−18.8495559215388x_{40} = -18.8495559215388
x41=53.4070751110265x_{41} = 53.4070751110265
x42=47.1238898038469x_{42} = 47.1238898038469
x43=12.5663706143592x_{43} = 12.5663706143592
x44=81.6814089933346x_{44} = 81.6814089933346
x45=34.5575191894877x_{45} = 34.5575191894877
x46=−75.398223686155x_{46} = -75.398223686155
x47=−15.707963267949x_{47} = -15.707963267949
x48=50.2654824574367x_{48} = 50.2654824574367
x49=−81.6814089933346x_{49} = -81.6814089933346
x50=−3.14159265358979x_{50} = -3.14159265358979
x51=−59.6902604182061x_{51} = -59.6902604182061
x52=−172.787595947439x_{52} = -172.787595947439
x53=194.778744522567x_{53} = 194.778744522567
x54=−87.9645943005142x_{54} = -87.9645943005142
x55=−28.2743338823081x_{55} = -28.2743338823081
x56=−21.9911485751286x_{56} = -21.9911485751286
x57=9.42477796076938x_{57} = 9.42477796076938
x58=56.5486677646163x_{58} = 56.5486677646163
x59=15.707963267949x_{59} = 15.707963267949
x60=84.8230016469244x_{60} = 84.8230016469244
x61=−78.5398163397448x_{61} = -78.5398163397448
x62=37.6991118430775x_{62} = 37.6991118430775
x63=−72.2566310325652x_{63} = -72.2566310325652
x64=−84.8230016469244x_{64} = -84.8230016469244
x65=69.1150383789755x_{65} = 69.1150383789755
x66=0x_{66} = 0
x67=28.2743338823081x_{67} = 28.2743338823081
x68=40.8407044966673x_{68} = 40.8407044966673
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(sin(x)).
tan⁡(sin⁡(0))\tan{\left(\sin{\left(0 \right)} \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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(sin⁡(x))+1)cos⁡(x)=0\left(\tan^{2}{\left(\sin{\left(x \right)} \right)} + 1\right) \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
Signos de extremos en los puntos:
 -pi           
(----, -tan(1))
  2            

 pi         
(--, tan(1))
 2


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=−π2x_{1} = - \frac{\pi}{2}
Puntos máximos de la función:
x1=π2x_{1} = \frac{\pi}{2}
Decrece en los intervalos
[−π2,π2]\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]
Crece en los intervalos
(−∞,−π2]∪[π2,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\pi}{2}, \infty\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⁡(x)+2cos⁡2(x)tan⁡(sin⁡(x)))(tan⁡2(sin⁡(x))+1)=0\left(- \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} \tan{\left(\sin{\left(x \right)} \right)}\right) \left(\tan^{2}{\left(\sin{\left(x \right)} \right)} + 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=65.9734457253857x_{1} = 65.9734457253857
x2=−54.2984478147474x_{2} = -54.2984478147474
x3=48.0152625075678x_{3} = 48.0152625075678
x4=−21.9911485751286x_{4} = -21.9911485751286
x5=21.9911485751286x_{5} = 21.9911485751286
x6=96.4979995575627x_{6} = 96.4979995575627
x7=−15.707963267949x_{7} = -15.707963267949
x8=−4.03296535731068x_{8} = -4.03296535731068
x9=−11.6749979106383x_{9} = -11.6749979106383
x10=97.3893722612836x_{10} = 97.3893722612836
x11=−24.2413685249975x_{11} = -24.2413685249975
x12=−61.940480368075x_{12} = -61.940480368075
x13=−76.2895963898759x_{13} = -76.2895963898759
x14=6.28318530717959x_{14} = 6.28318530717959
x15=68.2236656752546x_{15} = 68.2236656752546
x16=7.17455801090047x_{16} = 7.17455801090047
x17=41.7320772003882x_{17} = 41.7320772003882
x18=28.2743338823081x_{18} = 28.2743338823081
x19=−94.2477796076938x_{19} = -94.2477796076938
x20=−12.5663706143592x_{20} = -12.5663706143592
x21=−83.9316289432035x_{21} = -83.9316289432035
x22=17.9581832178179x_{22} = 17.9581832178179
x23=−70.0064110826963x_{23} = -70.0064110826963
x24=85.7143743506453x_{24} = 85.7143743506453
x25=10.3161506644903x_{25} = 10.3161506644903
x26=−53.4070751110265x_{26} = -53.4070751110265
x27=−39.9493317929464x_{27} = -39.9493317929464
x28=−90.2148142503831x_{28} = -90.2148142503831
x29=−33.6661464857668x_{29} = -33.6661464857668
x30=90.2148142503831x_{30} = 90.2148142503831
x31=59.6902604182061x_{31} = 59.6902604182061
x32=32.3072992396188x_{32} = 32.3072992396188
x33=−79.4311890434657x_{33} = -79.4311890434657
x34=−91.9975596578249x_{34} = -91.9975596578249
x35=56.5486677646163x_{35} = 56.5486677646163
x36=26.0241139324392x_{36} = 26.0241139324392
x37=−2.25021994986891x_{37} = -2.25021994986891
x38=72.2566310325652x_{38} = 72.2566310325652
x39=−50.2654824574367x_{39} = -50.2654824574367
x40=−26.0241139324392x_{40} = -26.0241139324392
x41=78.5398163397448x_{41} = 78.5398163397448
x42=61.940480368075x_{42} = 61.940480368075
x43=−87.9645943005142x_{43} = -87.9645943005142
x44=37.6991118430775x_{44} = 37.6991118430775
x45=−6.28318530717959x_{45} = -6.28318530717959
x46=−13.4577433180801x_{46} = -13.4577433180801
x47=−37.6991118430775x_{47} = -37.6991118430775
x48=−43.9822971502571x_{48} = -43.9822971502571
x49=−48.0152625075678x_{49} = -48.0152625075678
x50=74.5068509824341x_{50} = 74.5068509824341
x51=24.2413685249975x_{51} = 24.2413685249975
x52=63.7232257755168x_{52} = 63.7232257755168
x53=70.0064110826963x_{53} = 70.0064110826963
x54=19.7409286252596x_{54} = 19.7409286252596
x55=−41.7320772003882x_{55} = -41.7320772003882
x56=−72.2566310325652x_{56} = -72.2566310325652
x57=−81.6814089933346x_{57} = -81.6814089933346
x58=98.2807449650045x_{58} = 98.2807449650045
x59=−65.9734457253857x_{59} = -65.9734457253857
x60=−85.7143743506453x_{60} = -85.7143743506453
x61=0x_{61} = 0
x62=−17.9581832178179x_{62} = -17.9581832178179
x63=−63.7232257755168x_{63} = -63.7232257755168
x64=2.25021994986891x_{64} = 2.25021994986891
x65=−28.2743338823081x_{65} = -28.2743338823081
x66=−19.7409286252596x_{66} = -19.7409286252596
x67=−68.2236656752546x_{67} = -68.2236656752546
x68=43.9822971502571x_{68} = 43.9822971502571
x69=8.53340525704849x_{69} = 8.53340525704849
x70=100.530964914873x_{70} = 100.530964914873
x71=−97.3893722612836x_{71} = -97.3893722612836
x72=−35.4488918932086x_{72} = -35.4488918932086
x73=81.6814089933346x_{73} = 81.6814089933346
x74=−75.398223686155x_{74} = -75.398223686155
x75=4.03296535731068x_{75} = 4.03296535731068
x76=−57.4400404683372x_{76} = -57.4400404683372
x77=−77.6484436360239x_{77} = -77.6484436360239
x78=50.2654824574367x_{78} = 50.2654824574367
x79=94.2477796076938x_{79} = 94.2477796076938
x80=−59.6902604182061x_{80} = -59.6902604182061
x81=12.5663706143592x_{81} = 12.5663706143592
x82=52.5157024073056x_{82} = 52.5157024073056
x83=46.232517100126x_{83} = 46.232517100126
x84=34.5575191894877x_{84} = 34.5575191894877
x85=39.9493317929464x_{85} = 39.9493317929464
x86=30.524553832177x_{86} = 30.524553832177
x87=15.707963267949x_{87} = 15.707963267949
x88=87.9645943005142x_{88} = 87.9645943005142
x89=−99.6395922111525x_{89} = -99.6395922111525
x90=76.2895963898759x_{90} = 76.2895963898759
x91=83.9316289432035x_{91} = 83.9316289432035
x92=−9.42477796076938x_{92} = -9.42477796076938
x93=−55.6572950608954x_{93} = -55.6572950608954
x94=−46.232517100126x_{94} = -46.232517100126
x95=54.2984478147474x_{95} = 54.2984478147474
x96=−31.4159265358979x_{96} = -31.4159265358979
x97=75.398223686155x_{97} = 75.398223686155
x98=91.9975596578249x_{98} = 91.9975596578249

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
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Convexa en los intervalos
(−∞,−94.2477796076938]\left(-\infty, -94.2477796076938\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞tan⁡(sin⁡(x))=⟨−tan⁡(1),tan⁡(1)⟩\lim_{x \to -\infty} \tan{\left(\sin{\left(x \right)} \right)} = \left\langle - \tan{\left(1 \right)}, \tan{\left(1 \right)}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−tan⁡(1),tan⁡(1)⟩y = \left\langle - \tan{\left(1 \right)}, \tan{\left(1 \right)}\right\rangle
lim⁡x→∞tan⁡(sin⁡(x))=⟨−tan⁡(1),tan⁡(1)⟩\lim_{x \to \infty} \tan{\left(\sin{\left(x \right)} \right)} = \left\langle - \tan{\left(1 \right)}, \tan{\left(1 \right)}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−tan⁡(1),tan⁡(1)⟩y = \left\langle - \tan{\left(1 \right)}, \tan{\left(1 \right)}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(sin(x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(tan⁡(sin⁡(x))x)=0\lim_{x \to -\infty}\left(\frac{\tan{\left(\sin{\left(x \right)} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(tan⁡(sin⁡(x))x)=0\lim_{x \to \infty}\left(\frac{\tan{\left(\sin{\left(x \right)} \right)}}{x}\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:
tan⁡(sin⁡(x))=−tan⁡(sin⁡(x))\tan{\left(\sin{\left(x \right)} \right)} = - \tan{\left(\sin{\left(x \right)} \right)}
- No
tan⁡(sin⁡(x))=tan⁡(sin⁡(x))\tan{\left(\sin{\left(x \right)} \right)} = \tan{\left(\sin{\left(x \right)} \right)}
- Sí
es decir, función
es
impar
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