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Trazado el gráfico de la función/ log(tan(sin(3*x))^(2))

Gráfico de la función y = log(tan(sin(3*x))^(2))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /   2          \
f(x) = log\tan (sin(3*x))/
f(x)=log⁡(tan⁡2(sin⁡(3x)))f{\left(x \right)} = \log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} \right)} 
f = log(tan(sin(3*x))^2)
Gráfico de la función
02468-8-6-4-2-1010-2010
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:
log⁡(tan⁡2(sin⁡(3x)))=0\log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−asin⁡(π4)3+π3x_{1} = - \frac{\operatorname{asin}{\left(\frac{\pi}{4} \right)}}{3} + \frac{\pi}{3}
x2=asin⁡(π4)3+π3x_{2} = \frac{\operatorname{asin}{\left(\frac{\pi}{4} \right)}}{3} + \frac{\pi}{3}
x3=−asin⁡(π4)3x_{3} = - \frac{\operatorname{asin}{\left(\frac{\pi}{4} \right)}}{3}
x4=asin⁡(π4)3x_{4} = \frac{\operatorname{asin}{\left(\frac{\pi}{4} \right)}}{3}
Solución numérica
x1=−8.67869344649495x_{1} = -8.67869344649495
x2=−97.6904852982058x_{2} = -97.6904852982058
x3=98.135456775558x_{3} = 98.135456775558
x4=−47.8699743181213x_{4} = -47.8699743181213
x5=−43.6811841133349x_{5} = -43.6811841133349
x6=25.8788257429928x_{6} = 25.8788257429928
x7=−43.2362126359827x_{7} = -43.2362126359827
x8=−40.0946199823929x_{8} = -40.0946199823929
x9=−5.98207227025742x_{9} = -5.98207227025742
x10=−62.0857685575214x_{10} = -62.0857685575214
x11=12.265257577437x_{11} = 12.265257577437
x12=62.0857685575214x_{12} = 62.0857685575214
x13=−25.8788257429928x_{13} = -25.8788257429928
x14=−74.0499130980363x_{14} = -74.0499130980363
x15=88.2657073374364x_{15} = 88.2657073374364
x16=56.2475547276941x_{16} = 56.2475547276941
x17=3.88767716786422x_{17} = 3.88767716786422
x18=27.973220845386x_{18} = 27.973220845386
x19=−35.9058297776065x_{19} = -35.9058297776065
x20=−11.8202861000847x_{20} = -11.8202861000847
x21=42.1890150847861x_{21} = 42.1890150847861
x22=−49.9643694205145x_{22} = -49.9643694205145
x23=−96.0410616731648x_{23} = -96.0410616731648
x24=86.1713122350432x_{24} = 86.1713122350432
x25=40.0946199823929x_{25} = 40.0946199823929
x26=33.209208601369x_{26} = 33.209208601369
x27=−27.973220845386x_{27} = -27.973220845386
x28=−77.7937318254704x_{28} = -77.7937318254704
x29=84.07691713265x_{29} = 84.07691713265
x30=−53.7081881479487x_{30} = -53.7081881479487
x31=−19.1506689584609x_{31} = -19.1506689584609
x32=−21.6900355382064x_{32} = -21.6900355382064
x33=96.0410616731648x_{33} = 96.0410616731648
x34=−84.07691713265x_{34} = -84.07691713265
x35=−91.8522714683784x_{35} = -91.8522714683784
x36=−18.1034714072643x_{36} = -18.1034714072643
x37=−99.784880400599x_{37} = -99.784880400599
x38=−3.88767716786422x_{38} = -3.88767716786422
x39=46.3778052895725x_{39} = 46.3778052895725
x40=−13.9146812024779x_{40} = -13.9146812024779
x41=−59.9913734551282x_{41} = -59.9913734551282
x42=−69.8611228932499x_{42} = -69.8611228932499
x43=−23.7844306405996x_{43} = -23.7844306405996
x44=24.3866567144439x_{44} = 24.3866567144439
x45=−67.7667277908567x_{45} = -67.7667277908567
x46=−65.6723326884635x_{46} = -65.6723326884635
x47=−1.79328206547102x_{47} = -1.79328206547102
x48=90.3601024398296x_{48} = 90.3601024398296
x49=−16.0090763048711x_{49} = -16.0090763048711
x50=16.0090763048711x_{50} = 16.0090763048711
x51=−45.7755792157281x_{51} = -45.7755792157281
x52=−89.7578763659852x_{52} = -89.7578763659852
x53=59.9913734551282x_{53} = 59.9913734551282
x54=−55.8025832503419x_{54} = -55.8025832503419
x55=5.98207227025742x_{55} = 5.98207227025742
x56=34.2564061525656x_{56} = 34.2564061525656
x57=81.9825220302568x_{57} = 81.9825220302568
x58=91.8522714683784x_{58} = 91.8522714683784
x59=−30.0676159477792x_{59} = -30.0676159477792
x60=10.1708624750438x_{60} = 10.1708624750438
x61=64.1801636599146x_{61} = 64.1801636599146
x62=54.1531596253009x_{62} = 54.1531596253009
x63=61.0385710063248x_{63} = 61.0385710063248
x64=20.1978665096575x_{64} = 20.1978665096575
x65=−87.663481263592x_{65} = -87.663481263592
x66=32.1620110501724x_{66} = 32.1620110501724
x67=−81.9825220302568x_{67} = -81.9825220302568
x68=52.0587645229077x_{68} = 52.0587645229077
x69=74.0499130980363x_{69} = 74.0499130980363
x70=−52.0587645229077x_{70} = -52.0587645229077
x71=47.8699743181213x_{71} = 47.8699743181213
x72=66.2745587623078x_{72} = 66.2745587623078
x73=100.229851877951x_{73} = 100.229851877951
x74=2.39550813931537x_{74} = 2.39550813931537
x75=8.07646737265061x_{75} = 8.07646737265061
x76=30.0676159477792x_{76} = 30.0676159477792
x77=−93.9466665707716x_{77} = -93.9466665707716
x78=−31.7170395728201x_{78} = -31.7170395728201
x79=38.0002248799997x_{79} = 38.0002248799997
x80=−71.9555179956431x_{80} = -71.9555179956431
x81=−9.72589099769155x_{81} = -9.72589099769155
x82=76.1443082004295x_{82} = 76.1443082004295
x83=−100.229851877951x_{83} = -100.229851877951
x84=−75.6993367230772x_{84} = -75.6993367230772
x85=18.1034714072643x_{85} = 18.1034714072643
x86=−38.0002248799997x_{86} = -38.0002248799997
x87=71.9555179956431x_{87} = 71.9555179956431
x88=69.8611228932499x_{88} = 69.8611228932499
x89=93.9466665707716x_{89} = 93.9466665707716
x90=22.2922616120507x_{90} = 22.2922616120507
x91=78.2387033028227x_{91} = 78.2387033028227
x92=−57.896978352735x_{92} = -57.896978352735
x93=44.2834101871793x_{93} = 44.2834101871793
x94=−33.8114346752133x_{94} = -33.8114346752133
x95=49.9643694205145x_{95} = 49.9643694205145
x96=68.368953864701x_{96} = 68.368953864701
x97=−79.8881269278636x_{97} = -79.8881269278636
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 log(tan(sin(3*x))^2).
log⁡(tan⁡2(sin⁡(0⋅3)))\log{\left(\tan^{2}{\left(\sin{\left(0 \cdot 3 \right)} \right)} \right)}
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
6(tan⁡2(sin⁡(3x))+1)cos⁡(3x)tan⁡(sin⁡(3x))=0\frac{6 \left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} + 1\right) \cos{\left(3 x \right)}}{\tan{\left(\sin{\left(3 x \right)} \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−π6x_{1} = - \frac{\pi}{6}
x2=π6x_{2} = \frac{\pi}{6}
Signos de extremos en los puntos:
 -pi      /   2   \ 
(----, log\tan (1)/)
  6                 

 pi     /   2   \ 
(--, log\tan (1)/)
 6


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:
La función no tiene puntos mínimos
Puntos máximos de la función:
x2=−π6x_{2} = - \frac{\pi}{6}
x2=π6x_{2} = \frac{\pi}{6}
Decrece en los intervalos
(−∞,−π6]\left(-\infty, - \frac{\pi}{6}\right]
Crece en los intervalos
[π6,∞)\left[\frac{\pi}{6}, \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
18(tan⁡2(sin⁡(3x))+1)(−(tan⁡2(sin⁡(3x))+1)cos⁡2(3x)tan⁡2(sin⁡(3x))−sin⁡(3x)tan⁡(sin⁡(3x))+2cos⁡2(3x))=018 \left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} + 1\right) \left(- \frac{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} + 1\right) \cos^{2}{\left(3 x \right)}}{\tan^{2}{\left(\sin{\left(3 x \right)} \right)}} - \frac{\sin{\left(3 x \right)}}{\tan{\left(\sin{\left(3 x \right)} \right)}} + 2 \cos^{2}{\left(3 x \right)}\right) = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞log⁡(tan⁡2(sin⁡(3x)))=log⁡(⟨0,tan⁡2(1)⟩)\lim_{x \to -\infty} \log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} \right)} = \log{\left(\left\langle 0, \tan^{2}{\left(1 \right)}\right\rangle \right)}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=log⁡(⟨0,tan⁡2(1)⟩)y = \log{\left(\left\langle 0, \tan^{2}{\left(1 \right)}\right\rangle \right)}
lim⁡x→∞log⁡(tan⁡2(sin⁡(3x)))=log⁡(⟨0,tan⁡2(1)⟩)\lim_{x \to \infty} \log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} \right)} = \log{\left(\left\langle 0, \tan^{2}{\left(1 \right)}\right\rangle \right)}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=log⁡(⟨0,tan⁡2(1)⟩)y = \log{\left(\left\langle 0, \tan^{2}{\left(1 \right)}\right\rangle \right)}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función log(tan(sin(3*x))^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(tan⁡2(sin⁡(3x)))x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(log⁡(tan⁡2(sin⁡(3x)))x)=0\lim_{x \to \infty}\left(\frac{\log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \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:
log⁡(tan⁡2(sin⁡(3x)))=log⁡(tan⁡2(sin⁡(3x)))\log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} \right)} = \log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} \right)}
- Sí
log⁡(tan⁡2(sin⁡(3x)))=−log⁡(tan⁡2(sin⁡(3x)))\log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} \right)} = - \log{\left(\tan^{2}{\left(\sin{\left(3 x \right)} \right)} \right)}
- No
es decir, función
es
par
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