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y=sin(x+3)/(x^5*(\cos(2*x^3)/\sin(2*x^3)))
Trazado el gráfico de la función/ y=sin(x+3)/(x^5*(\cos(2*x^3)/\sin(2*x^3)))

Gráfico de la función y = y=sin(x+3)/(x^5*(\cos(2*x^3)/\sin(2*x^3)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        sin(x + 3) 
f(x) = ------------
             /   3\
        5 cos\2*x /
       x *---------
             /   3\
          sin\2*x /
f(x)=sin⁡(x+3)x5cos⁡(2x3)sin⁡(2x3)f{\left(x \right)} = \frac{\sin{\left(x + 3 \right)}}{x^{5} \frac{\cos{\left(2 x^{3} \right)}}{\sin{\left(2 x^{3} \right)}}} 
f = sin(x + 3)/((x^5*(cos(2*x^3)/sin(2*x^3))))
Gráfico de la función
02468-8-6-4-2-1010-200200
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=0.922635074322014x_{2} = 0.922635074322014
x3=1.16244735150963x_{3} = 1.16244735150963
x4=1.33067003949147x_{4} = 1.33067003949147
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+3)x5cos⁡(2x3)sin⁡(2x3)=0\frac{\sin{\left(x + 3 \right)}}{x^{5} \frac{\cos{\left(2 x^{3} \right)}}{\sin{\left(2 x^{3} \right)}}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−3x_{1} = -3
Solución numérica
x1=−99.8289214865311x_{1} = -99.8289214865311
x2=−63.7770458019178x_{2} = -63.7770458019178
x3=50.25101358432x_{3} = 50.25101358432
x4=−33.331207869056x_{4} = -33.331207869056
x5=90.241640218312x_{5} = 90.241640218312
x6=67.8102976311186x_{6} = 67.8102976311186
x7=88.2910891360752x_{7} = 88.2910891360752
x8=−46.9563995070422x_{8} = -46.9563995070422
x9=42.1346569805983x_{9} = 42.1346569805983
x10=−91.8902389182422x_{10} = -91.8902389182422
x11=78.1694152762756x_{11} = 78.1694152762756
x12=12.2210541486568x_{12} = 12.2210541486568
x13=−78.2990190577827x_{13} = -78.2990190577827
x14=−51.7486381796021x_{14} = -51.7486381796021
x15=78.6706454041684x_{15} = 78.6706454041684
x16=−70.1154732412612x_{16} = -70.1154732412612
x17=86.4801475540085x_{17} = 86.4801475540085
x18=−49.8026692147941x_{18} = -49.8026692147941
x19=−33.2565756420669x_{19} = -33.2565756420669
x20=9.81534244285994x_{20} = 9.81534244285994
x21=98.121239553254x_{21} = 98.121239553254
x22=−4.1038440723062x_{22} = -4.1038440723062
x23=23.8282112344034x_{23} = 23.8282112344034
x24=80.0812125011867x_{24} = 80.0812125011867
x25=64.7211175002429x_{25} = 64.7211175002429
x26=64.1668893183998x_{26} = 64.1668893183998
x27=−73.6316107083779x_{27} = -73.6316107083779
x28=−62.5815582983129x_{28} = -62.5815582983129
x29=−89.8374028224263x_{29} = -89.8374028224263
x30=−39.990617058652x_{30} = -39.990617058652
x31=82.2118220636342x_{31} = 82.2118220636342
x32=−72.0318970311207x_{32} = -72.0318970311207
x33=6.24430141399867x_{33} = 6.24430141399867
x34=40.112692684082x_{34} = 40.112692684082
x35=19.2807289177624x_{35} = 19.2807289177624
x36=−50.5896141765949x_{36} = -50.5896141765949
x37=13.8080164927834x_{37} = 13.8080164927834
x38=94.8862747523076x_{38} = 94.8862747523076
x39=−15.5959680699004x_{39} = -15.5959680699004
x40=48.036775115204x_{40} = 48.036775115204
x41=−93.9710617850561x_{41} = -93.9710617850561
x42=−45.7684574304806x_{42} = -45.7684574304806
x43=−53.7275457333903x_{43} = -53.7275457333903
x44=−1.98775701035949x_{44} = -1.98775701035949
x45=36.0294154207901x_{45} = 36.0294154207901
x46=−21.700911641093x_{46} = -21.700911641093
x47=22.0907922786994x_{47} = 22.0907922786994
x48=72.2472075205006x_{48} = 72.2472075205006
x49=52.2538134968796x_{49} = 52.2538134968796
x50=75.8319427728261x_{50} = 75.8319427728261
x51=−40.7781264583823x_{51} = -40.7781264583823
x52=5.56977332281327x_{52} = 5.56977332281327
x53=−25.9555971457474x_{53} = -25.9555971457474
x54=−28.1599857440451x_{54} = -28.1599857440451
x55=1.84527014864403x_{55} = 1.84527014864403
x56=5.3593894570072x_{56} = 5.3593894570072
x57=−87.8010679213519x_{57} = -87.8010679213519
x58=19.9000725328619x_{58} = 19.9000725328619
x59=−37.6989818623476x_{59} = -37.6989818623476
x60=−23.9566231820435x_{60} = -23.9566231820435
x61=−30.092950771979x_{61} = -30.092950771979
x62=−43.8970357910736x_{62} = -43.8970357910736
x63=−9.74968601332573x_{63} = -9.74968601332573
x64=69.9225947215812x_{64} = 69.9225947215812
x65=−95.8790933366488x_{65} = -95.8790933366488
x66=−78.1187402540623x_{66} = -78.1187402540623
x67=47.7564992483677x_{67} = 47.7564992483677
x68=−97.9667629521309x_{68} = -97.9667629521309
x69=91.9435363694188x_{69} = 91.9435363694188
x70=26.4699515729987x_{70} = 26.4699515729987
x71=−55.790072052417x_{71} = -55.790072052417
x72=31.9939873525322x_{72} = 31.9939873525322
x73=−5.74956624873947x_{73} = -5.74956624873947
x74=−59.84091785628x_{74} = -59.84091785628
x75=−17.9028916416884x_{75} = -17.9028916416884
x76=100.05715623053x_{76} = 100.05715623053
x77=−18.8735914016239x_{77} = -18.8735914016239
x78=54.0337908636453x_{78} = 54.0337908636453
x79=−81.732597339168x_{79} = -81.732597339168
x80=−65.8203340659954x_{80} = -65.8203340659954
x81=59.9716437873313x_{81} = 59.9716437873313
x82=37.8073529799311x_{82} = 37.8073529799311
x83=−75.7531912259686x_{83} = -75.7531912259686
x84=28.227831870277x_{84} = 28.227831870277
x85=14.2496162144817x_{85} = 14.2496162144817
x86=−13.4355585030886x_{86} = -13.4355585030886
x87=−11.4233994450921x_{87} = -11.4233994450921
x88=30.2579840116705x_{88} = 30.2579840116705
x89=96.3006589853465x_{89} = 96.3006589853465
x90=−68.0429310597671x_{90} = -68.0429310597671
x91=56.479802516484x_{91} = 56.479802516484
x92=−86.6318051354324x_{92} = -86.6318051354324
x93=62.266196337338x_{93} = 62.266196337338
x94=74.6624521612035x_{94} = 74.6624521612035
x95=−7.9675582928489x_{95} = -7.9675582928489
x96=−84.6805019659695x_{96} = -84.6805019659695
x97=34.6973044545644x_{97} = 34.6973044545644
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 + 3)/((x^5*(cos(2*x^3)/sin(2*x^3)))).
sin⁡(3)05cos⁡(2⋅03)sin⁡(2⋅03)\frac{\sin{\left(3 \right)}}{0^{5} \frac{\cos{\left(2 \cdot 0^{3} \right)}}{\sin{\left(2 \cdot 0^{3} \right)}}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=0.922635074322014x_{2} = 0.922635074322014
x3=1.16244735150963x_{3} = 1.16244735150963
x4=1.33067003949147x_{4} = 1.33067003949147
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⁡(x+3)x5cos⁡(2x3)sin⁡(2x3))y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x + 3 \right)}}{x^{5} \frac{\cos{\left(2 x^{3} \right)}}{\sin{\left(2 x^{3} \right)}}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(sin⁡(x+3)x5cos⁡(2x3)sin⁡(2x3))y = \lim_{x \to \infty}\left(\frac{\sin{\left(x + 3 \right)}}{x^{5} \frac{\cos{\left(2 x^{3} \right)}}{\sin{\left(2 x^{3} \right)}}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x + 3)/((x^5*(cos(2*x^3)/sin(2*x^3)))), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(sin⁡(2x3)x5cos⁡(2x3)sin⁡(x+3)x)y = x \lim_{x \to -\infty}\left(\frac{\frac{\sin{\left(2 x^{3} \right)}}{x^{5} \cos{\left(2 x^{3} \right)}} \sin{\left(x + 3 \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(sin⁡(2x3)x5cos⁡(2x3)sin⁡(x+3)x)y = x \lim_{x \to \infty}\left(\frac{\frac{\sin{\left(2 x^{3} \right)}}{x^{5} \cos{\left(2 x^{3} \right)}} \sin{\left(x + 3 \right)}}{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⁡(x+3)x5cos⁡(2x3)sin⁡(2x3)=−sin⁡(2x3)sin⁡(x−3)x5cos⁡(2x3)\frac{\sin{\left(x + 3 \right)}}{x^{5} \frac{\cos{\left(2 x^{3} \right)}}{\sin{\left(2 x^{3} \right)}}} = - \frac{\sin{\left(2 x^{3} \right)} \sin{\left(x - 3 \right)}}{x^{5} \cos{\left(2 x^{3} \right)}}
- No
sin⁡(x+3)x5cos⁡(2x3)sin⁡(2x3)=sin⁡(2x3)sin⁡(x−3)x5cos⁡(2x3)\frac{\sin{\left(x + 3 \right)}}{x^{5} \frac{\cos{\left(2 x^{3} \right)}}{\sin{\left(2 x^{3} \right)}}} = \frac{\sin{\left(2 x^{3} \right)} \sin{\left(x - 3 \right)}}{x^{5} \cos{\left(2 x^{3} \right)}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = y=sin(x+3)/(x^5*(\cos(2*x^3)/\sin(2*x^3)))
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