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sqrt((-1-cos(2*x))/(-1+x^2))
Trazado el gráfico de la función/ sqrt((-1-cos(2*x))/(-1+x^2))

Gráfico de la función y = sqrt((-1-cos(2*x))/(-1+x^2))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
            _______________
           / -1 - cos(2*x) 
f(x) =    /  ------------- 
         /            2    
       \/       -1 + x
f(x)=−cos⁡(2x)−1x2−1f{\left(x \right)} = \sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}} 
f = sqrt((-cos(2*x) - 1)/(x^2 - 1))
Gráfico de la función
02468-8-6-4-2-101013
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:
−cos⁡(2x)−1x2−1=0\sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
Solución numérica
x1=32.9867228626928x_{1} = 32.9867228626928
x2=86.3937979737193x_{2} = 86.3937979737193
x3=−36.1283155162826x_{3} = -36.1283155162826
x4=73.8274273593601x_{4} = 73.8274273593601
x5=4.71238898038469x_{5} = 4.71238898038469
x6=39.2699081698724x_{6} = 39.2699081698724
x7=95.8185759344887x_{7} = 95.8185759344887
x8=45.553093477052x_{8} = 45.553093477052
x9=221.48228207808x_{9} = 221.48228207808
x10=70.6858347057703x_{10} = 70.6858347057703
x11=26.7035375555132x_{11} = 26.7035375555132
x12=−89.5353906273091x_{12} = -89.5353906273091
x13=−26.7035375555132x_{13} = -26.7035375555132
x14=−17.2787595947439x_{14} = -17.2787595947439
x15=−61.261056745001x_{15} = -61.261056745001
x16=4.71238898038469x_{16} = 4.71238898038469
x17=356.570766182442x_{17} = 356.570766182442
x18=−92.6769832808989x_{18} = -92.6769832808989
x19=61.261056745001x_{19} = 61.261056745001
x20=−7.85398163397448x_{20} = -7.85398163397448
x21=67.5442420521806x_{21} = 67.5442420521806
x22=80.1106126665397x_{22} = 80.1106126665397
x23=−14.1371669411541x_{23} = -14.1371669411541
x24=14.1371669411541x_{24} = 14.1371669411541
x25=29.845130209103x_{25} = 29.845130209103
x26=39.2699081698724x_{26} = 39.2699081698724
x27=10.9955742875643x_{27} = 10.9955742875643
x28=17.2787595947439x_{28} = 17.2787595947439
x29=−51.8362787842316x_{29} = -51.8362787842316
x30=−29.845130209103x_{30} = -29.845130209103
x31=−23.5619449019234x_{31} = -23.5619449019234
x32=−48.6946861306418x_{32} = -48.6946861306418
x33=42.4115008234622x_{33} = 42.4115008234622
x34=−73.8274273593601x_{34} = -73.8274273593601
x35=23.5619449019235x_{35} = 23.5619449019235
x36=−86.3937979737193x_{36} = -86.3937979737193
x37=54.9778714378214x_{37} = 54.9778714378214
x38=20.4203522483337x_{38} = 20.4203522483337
x39=−48.6946861306418x_{39} = -48.6946861306418
x40=51.8362787842316x_{40} = 51.8362787842316
x41=−67.5442420521806x_{41} = -67.5442420521806
x42=−7.85398163397448x_{42} = -7.85398163397448
x43=−4.71238898038469x_{43} = -4.71238898038469
x44=−45.553093477052x_{44} = -45.553093477052
x45=−70.6858347057703x_{45} = -70.6858347057703
x46=64.4026493985908x_{46} = 64.4026493985908
x47=−2206.96883914683x_{47} = -2206.96883914683
x48=−504.225620901162x_{48} = -504.225620901162
x49=−42.4115008234622x_{49} = -42.4115008234622
x50=−83.2522053201295x_{50} = -83.2522053201295
x51=92.6769832808989x_{51} = 92.6769832808989
x52=−95.8185759344887x_{52} = -95.8185759344887
x53=89.5353906273091x_{53} = 89.5353906273091
x54=−39.2699081698724x_{54} = -39.2699081698724
x55=−362.853951489621x_{55} = -362.853951489621
x56=76.9690200129499x_{56} = 76.9690200129499
x57=48.6946861306418x_{57} = 48.6946861306418
x58=−20.4203522483337x_{58} = -20.4203522483337
x59=−64.4026493985908x_{59} = -64.4026493985908
x60=2461.4378440876x_{60} = 2461.4378440876
x61=7.85398163397448x_{61} = 7.85398163397448
x62=−80.1106126665397x_{62} = -80.1106126665397
x63=−1.5707963267949x_{63} = -1.5707963267949
x64=98.9601685880785x_{64} = 98.9601685880785
x65=205.774318810131x_{65} = 205.774318810131
x66=−58.1194640914112x_{66} = -58.1194640914112
x67=83.2522053201295x_{67} = 83.2522053201295
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞−cos⁡(2x)−1x2−1=0\lim_{x \to -\infty} \sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}} = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞−cos⁡(2x)−1x2−1=0\lim_{x \to \infty} \sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}} = 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 sqrt((-1 - cos(2*x))/(-1 + x^2)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−cos⁡(2x)−1x2−1x)=0\lim_{x \to -\infty}\left(\frac{\sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−cos⁡(2x)−1x2−1x)=0\lim_{x \to \infty}\left(\frac{\sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}}}{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:
−cos⁡(2x)−1x2−1=−cos⁡(2x)−1x2−1\sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}} = \sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}}
- Sí
−cos⁡(2x)−1x2−1=−−cos⁡(2x)−1x2−1\sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}} = - \sqrt{\frac{- \cos{\left(2 x \right)} - 1}{x^{2} - 1}}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = sqrt((-1-cos(2*x))/(-1+x^2))
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