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Trazado el gráfico de la función/ (1-2*abs(sin(x))*sin(x))/sqrt(x*(12-x))

Gráfico de la función y = (1-2*abs(sin(x))*sin(x))/sqrt(x*(12-x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       1 - 2*|sin(x)|*sin(x)
f(x) = ---------------------
             ____________   
           \/ x*(12 - x)
f(x)=−sin⁡(x)2∣sin⁡(x)∣+1x(12−x)f{\left(x \right)} = \frac{- \sin{\left(x \right)} 2 \left|{\sin{\left(x \right)}}\right| + 1}{\sqrt{x \left(12 - x\right)}} 
f = (-sin(x)*2*Abs(sin(x)) + 1)/sqrt(x*(12 - x))
Gráfico de la función
02468-8-6-4-2-10102-2
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=12x_{2} = 12
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⁡(x)2∣sin⁡(x)∣+1x(12−x)=0\frac{- \sin{\left(x \right)} 2 \left|{\sin{\left(x \right)}}\right| + 1}{\sqrt{x \left(12 - x\right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π4x_{1} = \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
Solución numérica
x1=−79.3252145031423x_{1} = -79.3252145031423
x2=8.63937979737193x_{2} = 8.63937979737193
x3=−85.6083998103219x_{3} = -85.6083998103219
x4=58.9048622548086x_{4} = 58.9048622548086
x5=−5.49778714378214x_{5} = -5.49778714378214
x6=−74.6128255227576x_{6} = -74.6128255227576
x7=266.249977391735x_{7} = 266.249977391735
x8=40.0553063332699x_{8} = 40.0553063332699
x9=−55.7632696012188x_{9} = -55.7632696012188
x10=57.3340659280137x_{10} = 57.3340659280137
x11=−73.0420291959627x_{11} = -73.0420291959627
x12=−93.4623814442964x_{12} = -93.4623814442964
x13=32.2013246992954x_{13} = 32.2013246992954
x14=−54.1924732744239x_{14} = -54.1924732744239
x15=84.037603483527x_{15} = 84.037603483527
x16=−87.1791961371168x_{16} = -87.1791961371168
x17=951.11717587431x_{17} = 951.11717587431
x18=7.06858347057703x_{18} = 7.06858347057703
x19=19.6349540849362x_{19} = 19.6349540849362
x20=14.9225651045515x_{20} = 14.9225651045515
x21=69.9004365423729x_{21} = 69.9004365423729
x22=−30.6305283725005x_{22} = -30.6305283725005
x23=51.0508806208341x_{23} = 51.0508806208341
x24=2.35619449019234x_{24} = 2.35619449019234
x25=82.4668071567321x_{25} = 82.4668071567321
x26=52.621676947629x_{26} = 52.621676947629
x27=90.3207887907066x_{27} = 90.3207887907066
x28=−80.8960108299372x_{28} = -80.8960108299372
x29=46.3384916404494x_{29} = 46.3384916404494
x30=−60.4756585816035x_{30} = -60.4756585816035
x31=−10.2101761241668x_{31} = -10.2101761241668
x32=−29.0597320457056x_{32} = -29.0597320457056
x33=−18.0641577581413x_{33} = -18.0641577581413
x34=77.7544181763474x_{34} = 77.7544181763474
x35=−36.9137136796801x_{35} = -36.9137136796801
x36=−47.9092879672443x_{36} = -47.9092879672443
x37=101.316363078271x_{37} = 101.316363078271
x38=−68.329640215578x_{38} = -68.329640215578
x39=44.7676953136546x_{39} = 44.7676953136546
x40=−24.3473430653209x_{40} = -24.3473430653209
x41=27.4889357189107x_{41} = 27.4889357189107
x42=−41.6261026600648x_{42} = -41.6261026600648
x43=33.7721210260903x_{43} = 33.7721210260903
x44=−35.3429173528852x_{44} = -35.3429173528852
x45=63.6172512351933x_{45} = 63.6172512351933
x46=−49.4800842940392x_{46} = -49.4800842940392
x47=−98.174770424681x_{47} = -98.174770424681
x48=−62.0464549083984x_{48} = -62.0464549083984
x49=38.484510006475x_{49} = 38.484510006475
x50=25.9181393921158x_{50} = 25.9181393921158
x51=96.6039740978861x_{51} = 96.6039740978861
x52=88.7499924639117x_{52} = 88.7499924639117
x53=−3.92699081698724x_{53} = -3.92699081698724
x54=−11.7809724509617x_{54} = -11.7809724509617
x55=76.1836218495525x_{55} = 76.1836218495525
x56=65.1880475619882x_{56} = 65.1880475619882
x57=−16.4933614313464x_{57} = -16.4933614313464
x58=−91.8915851175014x_{58} = -91.8915851175014
x59=−99.7455667514759x_{59} = -99.7455667514759
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 (1 - 2*Abs(sin(x))*sin(x))/sqrt(x*(12 - x)).
−sin⁡(0)2∣sin⁡(0)∣+10(12−0)\frac{- \sin{\left(0 \right)} 2 \left|{\sin{\left(0 \right)}}\right| + 1}{\sqrt{0 \left(12 - 0\right)}}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=12x_{2} = 12
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−sin⁡(x)2∣sin⁡(x)∣+1x(12−x))=0\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} 2 \left|{\sin{\left(x \right)}}\right| + 1}{\sqrt{x \left(12 - x\right)}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(−sin⁡(x)2∣sin⁡(x)∣+1x(12−x))=0\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} 2 \left|{\sin{\left(x \right)}}\right| + 1}{\sqrt{x \left(12 - x\right)}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (1 - 2*Abs(sin(x))*sin(x))/sqrt(x*(12 - x)), dividida por x con x->+oo y x ->-oo
No se ha logrado calcular el límite a la izquierda
lim⁡x→−∞(−sin⁡(x)2∣sin⁡(x)∣+1xx(12−x))\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} 2 \left|{\sin{\left(x \right)}}\right| + 1}{x \sqrt{x \left(12 - x\right)}}\right)
No se ha logrado calcular el límite a la derecha
lim⁡x→∞(−sin⁡(x)2∣sin⁡(x)∣+1xx(12−x))\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} 2 \left|{\sin{\left(x \right)}}\right| + 1}{x \sqrt{x \left(12 - x\right)}}\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⁡(x)2∣sin⁡(x)∣+1x(12−x)=2sin⁡(x)∣sin⁡(x)∣+1−x(x+12)\frac{- \sin{\left(x \right)} 2 \left|{\sin{\left(x \right)}}\right| + 1}{\sqrt{x \left(12 - x\right)}} = \frac{2 \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + 1}{\sqrt{- x \left(x + 12\right)}}
- No
−sin⁡(x)2∣sin⁡(x)∣+1x(12−x)=−2sin⁡(x)∣sin⁡(x)∣+1−x(x+12)\frac{- \sin{\left(x \right)} 2 \left|{\sin{\left(x \right)}}\right| + 1}{\sqrt{x \left(12 - x\right)}} = - \frac{2 \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + 1}{\sqrt{- x \left(x + 12\right)}}
- No
es decir, función
no es
par ni impar
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