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Trazado el gráfico de la función/ x^5/sin(x)

Gráfico de la función y = x^5/sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          5  
         x   
f(x) = ------
       sin(x)
f(x)=x5sin⁡(x)f{\left(x \right)} = \frac{x^{5}}{\sin{\left(x \right)}} 
f = x^5/sin(x)
Gráfico de la función
02468-8-6-4-2-1010-1000000010000000
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:
x5sin⁡(x)=0\frac{x^{5}}{\sin{\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^5/sin(x).
05sin⁡(0)\frac{0^{5}}{\sin{\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
−x5cos⁡(x)sin⁡2(x)+5x4sin⁡(x)=0- \frac{x^{5} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{5 x^{4}}{\sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=10.5531104299019x_{1} = 10.5531104299019
x2=−29.6782238599607x_{2} = -29.6782238599607
x3=42.2938264412084x_{3} = 42.2938264412084
x4=−32.8356099874983x_{4} = -32.8356099874983
x5=−51.7399407908695x_{5} = -51.7399407908695
x6=26.5171686179843x_{6} = 26.5171686179843
x7=−61.1795112699548x_{7} = -61.1795112699548
x8=−45.4435075515668x_{8} = -45.4435075515668
x9=−89.4795700005049x_{9} = -89.4795700005049
x10=39.1428589802905x_{10} = 39.1428589802905
x11=−13.7893118576574x_{11} = -13.7893118576574
x12=45.4435075515668x_{12} = 45.4435075515668
x13=83.1921757248865x_{13} = 83.1921757248865
x14=−64.3250751965547x_{14} = -64.3250751965547
x15=−58.0335192248369x_{15} = -58.0335192248369
x16=−42.2938264412084x_{16} = -42.2938264412084
x17=58.0335192248369x_{17} = 58.0335192248369
x18=20.1774438696657x_{18} = 20.1774438696657
x19=54.8870259692204x_{19} = 54.8870259692204
x20=−80.0482313681549x_{20} = -80.0482313681549
x21=32.8356099874983x_{21} = 32.8356099874983
x22=98.909660402511x_{22} = 98.909660402511
x23=−39.1428589802905x_{23} = -39.1428589802905
x24=−3.79022237829253x_{24} = -3.79022237829253
x25=−23.3510066366273x_{25} = -23.3510066366273
x26=−73.7597432512514x_{26} = -73.7597432512514
x27=92.6230533845309x_{27} = 92.6230533845309
x28=16.9925901138555x_{28} = 16.9925901138555
x29=51.7399407908695x_{29} = 51.7399407908695
x30=−67.4702705534315x_{30} = -67.4702705534315
x31=73.7597432512514x_{31} = 73.7597432512514
x32=−92.6230533845309x_{32} = -92.6230533845309
x33=67.4702705534315x_{33} = 67.4702705534315
x34=76.9040953473809x_{34} = 76.9040953473809
x35=−54.8870259692204x_{35} = -54.8870259692204
x36=35.9902726647882x_{36} = 35.9902726647882
x37=−26.5171686179843x_{37} = -26.5171686179843
x38=29.6782238599607x_{38} = 29.6782238599607
x39=61.1795112699548x_{39} = 61.1795112699548
x40=7.25024830711031x_{40} = 7.25024830711031
x41=95.7664129259896x_{41} = 95.7664129259896
x42=70.6151463420358x_{42} = 70.6151463420358
x43=64.3250751965547x_{43} = 64.3250751965547
x44=48.5921497153139x_{44} = 48.5921497153139
x45=−7.25024830711031x_{45} = -7.25024830711031
x46=−95.7664129259896x_{46} = -95.7664129259896
x47=−16.9925901138555x_{47} = -16.9925901138555
x48=23.3510066366273x_{48} = 23.3510066366273
x49=−10.5531104299019x_{49} = -10.5531104299019
x50=89.4795700005049x_{50} = 89.4795700005049
x51=86.335949286752x_{51} = 86.335949286752
x52=−76.9040953473809x_{52} = -76.9040953473809
x53=−70.6151463420358x_{53} = -70.6151463420358
x54=13.7893118576574x_{54} = 13.7893118576574
x55=3.79022237829253x_{55} = 3.79022237829253
x56=80.0482313681549x_{56} = 80.0482313681549
x57=−20.1774438696657x_{57} = -20.1774438696657
x58=−83.1921757248865x_{58} = -83.1921757248865
x59=−35.9902726647882x_{59} = -35.9902726647882
x60=−98.909660402511x_{60} = -98.909660402511
x61=−48.5921497153139x_{61} = -48.5921497153139
x62=−86.335949286752x_{62} = -86.335949286752
Signos de extremos en los puntos:
(10.553110429901942, -144836.622542368)

(-29.67822385996075, -23348933.9988476)

(42.29382644120838, -136269528.568455)

(-32.835609987498266, 38610285.1882944)

(-51.739940790869504, 372518724.77898)

(26.517168617984314, 13342035.1057279)

(-61.17951126995476, -859954703.185496)

(-45.44350755156676, 194971978.395094)

(-89.47957000050494, 5745084383.58212)

(39.142858980290455, 92635461.1324094)

(-13.789311857657383, 530317.704370101)

(45.44350755156676, 194971978.395094)

(83.19217572488648, 3992044599.32892)

(-64.32507519655475, 1104611511.64742)

(-58.03351922483685, 660694176.070092)

(-42.29382644120838, -136269528.568455)

(58.03351922483685, 660694176.070092)

(20.177443869665733, 3445652.04683287)

(54.88702596922039, -500199279.101362)

(-80.04823136815494, -3293095042.76767)

(32.835609987498266, 38610285.1882944)

(98.90966040251095, -9478677533.92323)

(-39.142858980290455, 92635461.1324094)

(-3.7902223782925293, -1294.84632241203)

(-23.351006636627282, -7100067.9578743)

(-73.75974325125135, -2188227762.21252)

(92.6230533845309, -6826959607.90923)

(16.992590113855474, -1476824.51218319)

(51.739940790869504, 372518724.77898)

(-67.47027055343149, -1402011338.71759)

(73.75974325125135, -2188227762.21252)

(-92.6230533845309, -6826959607.90923)

(67.47027055343149, -1402011338.71759)

(76.9040953473809, 2695648764.73051)

(-54.88702596922039, -500199279.101362)

(35.99027266478819, -60964471.6454713)

(-26.517168617984314, 13342035.1057279)

(29.67822385996075, -23348933.9988476)

(61.17951126995476, -859954703.185496)

(7.250248307110308, 24335.9050167958)

(95.76641292598956, 8065981548.75513)

(70.61514634203581, 1760253730.08186)

(64.32507519655475, 1104611511.64742)

(48.592149715313944, -272343866.939746)

(-7.250248307110308, 24335.9050167958)

(-95.76641292598956, 8065981548.75513)

(-16.992590113855474, -1476824.51218319)

(23.351006636627282, -7100067.9578743)

(-10.553110429901942, -144836.622542368)

(89.47957000050494, 5745084383.58212)

(86.33594928675204, -4804911857.65449)

(-76.9040953473809, 2695648764.73051)

(-70.61514634203581, 1760253730.08186)

(13.789311857657383, 530317.704370101)

(3.7902223782925293, -1294.84632241203)

(80.04823136815494, -3293095042.76767)

(-20.177443869665733, 3445652.04683287)

(-83.19217572488648, 3992044599.32892)

(-35.99027266478819, -60964471.6454713)

(-98.90966040251095, -9478677533.92323)

(-48.592149715313944, -272343866.939746)

(-86.33594928675204, -4804911857.65449)


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=−32.8356099874983x_{1} = -32.8356099874983
x2=−51.7399407908695x_{2} = -51.7399407908695
x3=26.5171686179843x_{3} = 26.5171686179843
x4=−45.4435075515668x_{4} = -45.4435075515668
x5=−89.4795700005049x_{5} = -89.4795700005049
x6=39.1428589802905x_{6} = 39.1428589802905
x7=−13.7893118576574x_{7} = -13.7893118576574
x8=45.4435075515668x_{8} = 45.4435075515668
x9=83.1921757248865x_{9} = 83.1921757248865
x10=−64.3250751965547x_{10} = -64.3250751965547
x11=−58.0335192248369x_{11} = -58.0335192248369
x12=58.0335192248369x_{12} = 58.0335192248369
x13=20.1774438696657x_{13} = 20.1774438696657
x14=32.8356099874983x_{14} = 32.8356099874983
x15=−39.1428589802905x_{15} = -39.1428589802905
x16=51.7399407908695x_{16} = 51.7399407908695
x17=76.9040953473809x_{17} = 76.9040953473809
x18=−26.5171686179843x_{18} = -26.5171686179843
x19=7.25024830711031x_{19} = 7.25024830711031
x20=95.7664129259896x_{20} = 95.7664129259896
x21=70.6151463420358x_{21} = 70.6151463420358
x22=64.3250751965547x_{22} = 64.3250751965547
x23=−7.25024830711031x_{23} = -7.25024830711031
x24=−95.7664129259896x_{24} = -95.7664129259896
x25=89.4795700005049x_{25} = 89.4795700005049
x26=−76.9040953473809x_{26} = -76.9040953473809
x27=−70.6151463420358x_{27} = -70.6151463420358
x28=13.7893118576574x_{28} = 13.7893118576574
x29=−20.1774438696657x_{29} = -20.1774438696657
x30=−83.1921757248865x_{30} = -83.1921757248865
Puntos máximos de la función:
x30=10.5531104299019x_{30} = 10.5531104299019
x30=−29.6782238599607x_{30} = -29.6782238599607
x30=42.2938264412084x_{30} = 42.2938264412084
x30=−61.1795112699548x_{30} = -61.1795112699548
x30=−42.2938264412084x_{30} = -42.2938264412084
x30=54.8870259692204x_{30} = 54.8870259692204
x30=−80.0482313681549x_{30} = -80.0482313681549
x30=98.909660402511x_{30} = 98.909660402511
x30=−3.79022237829253x_{30} = -3.79022237829253
x30=−23.3510066366273x_{30} = -23.3510066366273
x30=−73.7597432512514x_{30} = -73.7597432512514
x30=92.6230533845309x_{30} = 92.6230533845309
x30=16.9925901138555x_{30} = 16.9925901138555
x30=−67.4702705534315x_{30} = -67.4702705534315
x30=73.7597432512514x_{30} = 73.7597432512514
x30=−92.6230533845309x_{30} = -92.6230533845309
x30=67.4702705534315x_{30} = 67.4702705534315
x30=−54.8870259692204x_{30} = -54.8870259692204
x30=35.9902726647882x_{30} = 35.9902726647882
x30=29.6782238599607x_{30} = 29.6782238599607
x30=61.1795112699548x_{30} = 61.1795112699548
x30=48.5921497153139x_{30} = 48.5921497153139
x30=−16.9925901138555x_{30} = -16.9925901138555
x30=23.3510066366273x_{30} = 23.3510066366273
x30=−10.5531104299019x_{30} = -10.5531104299019
x30=86.335949286752x_{30} = 86.335949286752
x30=3.79022237829253x_{30} = 3.79022237829253
x30=80.0482313681549x_{30} = 80.0482313681549
x30=−35.9902726647882x_{30} = -35.9902726647882
x30=−98.909660402511x_{30} = -98.909660402511
x30=−48.5921497153139x_{30} = -48.5921497153139
x30=−86.335949286752x_{30} = -86.335949286752
Decrece en los intervalos
[95.7664129259896,∞)\left[95.7664129259896, \infty\right)
Crece en los intervalos
(−∞,−95.7664129259896]\left(-\infty, -95.7664129259896\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
x3(x2(1+2cos⁡2(x)sin⁡2(x))−10xcos⁡(x)sin⁡(x)+20)sin⁡(x)=0\frac{x^{3} \left(x^{2} \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{10 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 20\right)}{\sin{\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→−∞(x5sin⁡(x))y = \lim_{x \to -\infty}\left(\frac{x^{5}}{\sin{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(x5sin⁡(x))y = \lim_{x \to \infty}\left(\frac{x^{5}}{\sin{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x^5/sin(x), 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→−∞(x4sin⁡(x))y = x \lim_{x \to -\infty}\left(\frac{x^{4}}{\sin{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(x4sin⁡(x))y = x \lim_{x \to \infty}\left(\frac{x^{4}}{\sin{\left(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:
x5sin⁡(x)=x5sin⁡(x)\frac{x^{5}}{\sin{\left(x \right)}} = \frac{x^{5}}{\sin{\left(x \right)}}
- Sí
x5sin⁡(x)=−x5sin⁡(x)\frac{x^{5}}{\sin{\left(x \right)}} = - \frac{x^{5}}{\sin{\left(x \right)}}
- No
es decir, función
es
par
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