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Trazado el gráfico de la función/ x/sin(x)^2

Gráfico de la función y = x/sin(x)^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          x   
f(x) = -------
          2   
       sin (x)
f(x)=xsin⁡2(x)f{\left(x \right)} = \frac{x}{\sin^{2}{\left(x \right)}} 
f = x/sin(x)^2
Gráfico de la función
02468-8-6-4-2-1010-50000005000000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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:
xsin⁡2(x)=0\frac{x}{\sin^{2}{\left(x \right)}} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 x/sin(x)^2.
0sin⁡2(0)\frac{0}{\sin^{2}{\left(0 \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
−2xcos⁡(x)sin⁡3(x)+1sin⁡2(x)=0- \frac{2 x \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{1}{\sin^{2}{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=54.9687756155963x_{1} = 54.9687756155963
x2=86.3880101981266x_{2} = 86.3880101981266
x3=80.1043708909521x_{3} = 80.1043708909521
x4=67.5368388204916x_{4} = 67.5368388204916
x5=−17.2497818346079x_{5} = -17.2497818346079
x6=−89.5298059530594x_{6} = -89.5298059530594
x7=−76.9625234358705x_{7} = -76.9625234358705
x8=−54.9687756155963x_{8} = -54.9687756155963
x9=4.60421677720058x_{9} = 4.60421677720058
x10=26.6848024909251x_{10} = 26.6848024909251
x11=64.3948849627586x_{11} = 64.3948849627586
x12=29.8283692130955x_{12} = 29.8283692130955
x13=−1.16556118520721x_{13} = -1.16556118520721
x14=23.5407082923052x_{14} = 23.5407082923052
x15=32.9715594404485x_{15} = 32.9715594404485
x16=−7.78988375114457x_{16} = -7.78988375114457
x17=45.5421150692309x_{17} = 45.5421150692309
x18=−64.3948849627586x_{18} = -64.3948849627586
x19=−67.5368388204916x_{19} = -67.5368388204916
x20=−29.8283692130955x_{20} = -29.8283692130955
x21=10.9499436485412x_{21} = 10.9499436485412
x22=−98.9551158352145x_{22} = -98.9551158352145
x23=92.6715879363332x_{23} = 92.6715879363332
x24=36.1144715353049x_{24} = 36.1144715353049
x25=1.16556118520721x_{25} = 1.16556118520721
x26=−58.1108600600615x_{26} = -58.1108600600615
x27=70.6787605627689x_{27} = 70.6787605627689
x28=39.2571723324086x_{28} = 39.2571723324086
x29=−83.2461991121237x_{29} = -83.2461991121237
x30=−32.9715594404485x_{30} = -32.9715594404485
x31=−23.5407082923052x_{31} = -23.5407082923052
x32=−48.6844162648433x_{32} = -48.6844162648433
x33=−39.2571723324086x_{33} = -39.2571723324086
x34=−70.6787605627689x_{34} = -70.6787605627689
x35=−14.1017251335659x_{35} = -14.1017251335659
x36=−4.60421677720058x_{36} = -4.60421677720058
x37=14.1017251335659x_{37} = 14.1017251335659
x38=95.8133575027966x_{38} = 95.8133575027966
x39=89.5298059530594x_{39} = 89.5298059530594
x40=42.3997088362447x_{40} = 42.3997088362447
x41=−73.8206542907788x_{41} = -73.8206542907788
x42=−20.3958423573092x_{42} = -20.3958423573092
x43=−42.3997088362447x_{43} = -42.3997088362447
x44=73.8206542907788x_{44} = 73.8206542907788
x45=7.78988375114457x_{45} = 7.78988375114457
x46=−61.2528940466862x_{46} = -61.2528940466862
x47=51.8266315338985x_{47} = 51.8266315338985
x48=58.1108600600615x_{48} = 58.1108600600615
x49=−80.1043708909521x_{49} = -80.1043708909521
x50=20.3958423573092x_{50} = 20.3958423573092
x51=−92.6715879363332x_{51} = -92.6715879363332
x52=76.9625234358705x_{52} = 76.9625234358705
x53=98.9551158352145x_{53} = 98.9551158352145
x54=−10.9499436485412x_{54} = -10.9499436485412
x55=−86.3880101981266x_{55} = -86.3880101981266
x56=−36.1144715353049x_{56} = -36.1144715353049
x57=17.2497818346079x_{57} = 17.2497818346079
x58=−45.5421150692309x_{58} = -45.5421150692309
x59=−95.8133575027966x_{59} = -95.8133575027966
x60=48.6844162648433x_{60} = 48.6844162648433
x61=−26.6848024909251x_{61} = -26.6848024909251
x62=−51.8266315338985x_{62} = -51.8266315338985
x63=83.2461991121237x_{63} = 83.2461991121237
x64=61.2528940466862x_{64} = 61.2528940466862
Signos de extremos en los puntos:
(54.96877561559635, 54.9733236521353)

(86.38801019812658, 86.3909041182369)

(80.1043708909521, 80.1074918192762)

(67.53683882049161, 67.5405405039634)

(-17.249781834607894, -17.2642747715272)

(-89.52980595305935, -89.5325983192143)

(-76.96252343587051, -76.9657717701096)

(-54.96877561559635, -54.9733236521353)

(4.604216777200577, 4.65851482876886)

(26.68480249092507, 26.6941711193826)

(64.39488496275855, 64.3987672586916)

(29.828369213095506, 29.836750495968)

(-1.1655611852072114, -1.3800501396893)

(23.54070829230515, 23.5513281936648)

(32.97155944044848, 32.9791417327101)

(-7.789883751144573, -7.821976656249)

(45.5421150692309, 45.5476044936817)

(-64.39488496275855, -64.3987672586916)

(-67.53683882049161, -67.5405405039634)

(-29.828369213095506, -29.836750495968)

(10.94994364854116, 10.9727748162644)

(-98.95511583521451, -98.9576422331465)

(92.67158793633321, 92.6742856347925)

(36.11447153530485, 36.1213939680409)

(1.1655611852072114, 1.3800501396893)

(-58.110860060061505, -58.115162181898)

(70.67876056276886, 70.6822976932733)

(39.25717233240859, 39.2635405954583)

(-83.24619911212368, -83.249202252239)

(-32.97155944044848, -32.9791417327101)

(-23.54070829230515, -23.5513281936648)

(-48.68441626484328, -48.6895513782775)

(-39.25717233240859, -39.2635405954583)

(-70.67876056276886, -70.6822976932733)

(-14.101725133565873, -14.1194534609607)

(-4.604216777200577, -4.65851482876886)

(14.101725133565873, 14.1194534609607)

(95.81335750279658, 95.8159667423276)

(89.52980595305935, 89.5325983192143)

(42.39970883624466, 42.4056051031498)

(-73.82065429077876, -73.8240408768555)

(-20.395842357309167, -20.4080997574018)

(-42.39970883624466, -42.4056051031498)

(73.82065429077876, 73.8240408768555)

(7.789883751144573, 7.821976656249)

(-61.252894046686194, -61.2569754864923)

(51.82663153389846, 51.8314553087146)

(58.110860060061505, 58.115162181898)

(-80.1043708909521, -80.1074918192762)

(20.395842357309167, 20.4080997574018)

(-92.67158793633321, -92.6742856347925)

(76.96252343587051, 76.9657717701096)

(98.95511583521451, 98.9576422331465)

(-10.94994364854116, -10.9727748162644)

(-86.38801019812658, -86.3909041182369)

(-36.11447153530485, -36.1213939680409)

(17.249781834607894, 17.2642747715272)

(-45.5421150692309, -45.5476044936817)

(-95.81335750279658, -95.8159667423276)

(48.68441626484328, 48.6895513782775)

(-26.68480249092507, -26.6941711193826)

(-51.82663153389846, -51.8314553087146)

(83.24619911212368, 83.249202252239)

(61.252894046686194, 61.2569754864923)


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=54.9687756155963x_{1} = 54.9687756155963
x2=86.3880101981266x_{2} = 86.3880101981266
x3=80.1043708909521x_{3} = 80.1043708909521
x4=67.5368388204916x_{4} = 67.5368388204916
x5=4.60421677720058x_{5} = 4.60421677720058
x6=26.6848024909251x_{6} = 26.6848024909251
x7=64.3948849627586x_{7} = 64.3948849627586
x8=29.8283692130955x_{8} = 29.8283692130955
x9=23.5407082923052x_{9} = 23.5407082923052
x10=32.9715594404485x_{10} = 32.9715594404485
x11=45.5421150692309x_{11} = 45.5421150692309
x12=10.9499436485412x_{12} = 10.9499436485412
x13=92.6715879363332x_{13} = 92.6715879363332
x14=36.1144715353049x_{14} = 36.1144715353049
x15=1.16556118520721x_{15} = 1.16556118520721
x16=70.6787605627689x_{16} = 70.6787605627689
x17=39.2571723324086x_{17} = 39.2571723324086
x18=14.1017251335659x_{18} = 14.1017251335659
x19=95.8133575027966x_{19} = 95.8133575027966
x20=89.5298059530594x_{20} = 89.5298059530594
x21=42.3997088362447x_{21} = 42.3997088362447
x22=73.8206542907788x_{22} = 73.8206542907788
x23=7.78988375114457x_{23} = 7.78988375114457
x24=51.8266315338985x_{24} = 51.8266315338985
x25=58.1108600600615x_{25} = 58.1108600600615
x26=20.3958423573092x_{26} = 20.3958423573092
x27=76.9625234358705x_{27} = 76.9625234358705
x28=98.9551158352145x_{28} = 98.9551158352145
x29=17.2497818346079x_{29} = 17.2497818346079
x30=48.6844162648433x_{30} = 48.6844162648433
x31=83.2461991121237x_{31} = 83.2461991121237
x32=61.2528940466862x_{32} = 61.2528940466862
Puntos máximos de la función:
x32=−17.2497818346079x_{32} = -17.2497818346079
x32=−89.5298059530594x_{32} = -89.5298059530594
x32=−76.9625234358705x_{32} = -76.9625234358705
x32=−54.9687756155963x_{32} = -54.9687756155963
x32=−1.16556118520721x_{32} = -1.16556118520721
x32=−7.78988375114457x_{32} = -7.78988375114457
x32=−64.3948849627586x_{32} = -64.3948849627586
x32=−67.5368388204916x_{32} = -67.5368388204916
x32=−29.8283692130955x_{32} = -29.8283692130955
x32=−98.9551158352145x_{32} = -98.9551158352145
x32=−58.1108600600615x_{32} = -58.1108600600615
x32=−83.2461991121237x_{32} = -83.2461991121237
x32=−32.9715594404485x_{32} = -32.9715594404485
x32=−23.5407082923052x_{32} = -23.5407082923052
x32=−48.6844162648433x_{32} = -48.6844162648433
x32=−39.2571723324086x_{32} = -39.2571723324086
x32=−70.6787605627689x_{32} = -70.6787605627689
x32=−14.1017251335659x_{32} = -14.1017251335659
x32=−4.60421677720058x_{32} = -4.60421677720058
x32=−73.8206542907788x_{32} = -73.8206542907788
x32=−20.3958423573092x_{32} = -20.3958423573092
x32=−42.3997088362447x_{32} = -42.3997088362447
x32=−61.2528940466862x_{32} = -61.2528940466862
x32=−80.1043708909521x_{32} = -80.1043708909521
x32=−92.6715879363332x_{32} = -92.6715879363332
x32=−10.9499436485412x_{32} = -10.9499436485412
x32=−86.3880101981266x_{32} = -86.3880101981266
x32=−36.1144715353049x_{32} = -36.1144715353049
x32=−45.5421150692309x_{32} = -45.5421150692309
x32=−95.8133575027966x_{32} = -95.8133575027966
x32=−26.6848024909251x_{32} = -26.6848024909251
x32=−51.8266315338985x_{32} = -51.8266315338985
Decrece en los intervalos
[98.9551158352145,∞)\left[98.9551158352145, \infty\right)
Crece en los intervalos
[−1.16556118520721,1.16556118520721]\left[-1.16556118520721, 1.16556118520721\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
2(x(1+3cos⁡2(x)sin⁡2(x))−2cos⁡(x)sin⁡(x))sin⁡2(x)=0\frac{2 \left(x \left(1 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)}{\sin^{2}{\left(x \right)}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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→−∞(xsin⁡2(x))y = \lim_{x \to -\infty}\left(\frac{x}{\sin^{2}{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(xsin⁡2(x))y = \lim_{x \to \infty}\left(\frac{x}{\sin^{2}{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x/sin(x)^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞1sin⁡2(x)=⟨0,∞⟩\lim_{x \to -\infty} \frac{1}{\sin^{2}{\left(x \right)}} = \left\langle 0, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨0,∞⟩xy = \left\langle 0, \infty\right\rangle x
lim⁡x→∞1sin⁡2(x)=⟨0,∞⟩\lim_{x \to \infty} \frac{1}{\sin^{2}{\left(x \right)}} = \left\langle 0, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨0,∞⟩xy = \left\langle 0, \infty\right\rangle x
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:
xsin⁡2(x)=−xsin⁡2(x)\frac{x}{\sin^{2}{\left(x \right)}} = - \frac{x}{\sin^{2}{\left(x \right)}}
- No
xsin⁡2(x)=xsin⁡2(x)\frac{x}{\sin^{2}{\left(x \right)}} = \frac{x}{\sin^{2}{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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