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x^3*cot(4*x)
Trazado el gráfico de la función/ x^3*cot(4*x)

Gráfico de la función y = x^3*cot(4*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        3         
f(x) = x *cot(4*x)
f(x)=x3cot⁡(4x)f{\left(x \right)} = x^{3} \cot{\left(4 x \right)} 
f = x^3*cot(4*x)
Gráfico de la función
02468-8-6-4-2-1010-5000050000
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:
x3cot⁡(4x)=0x^{3} \cot{\left(4 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−7π8x_{1} = - \frac{7 \pi}{8}
x2=−3π8x_{2} = - \frac{3 \pi}{8}
x3=π8x_{3} = \frac{\pi}{8}
x4=5π8x_{4} = \frac{5 \pi}{8}
Solución numérica
x1=67.9369411338793x_{1} = 67.9369411338793
x2=97.7820713429823x_{2} = 97.7820713429823
x3=52.2289778659303x_{3} = 52.2289778659303
x4=−13.7444678594553x_{4} = -13.7444678594553
x5=82.0741080750334x_{5} = 82.0741080750334
x6=44.3749962319558x_{6} = 44.3749962319558
x7=1.96349540849362x_{7} = 1.96349540849362
x8=80.5033117482384x_{8} = 80.5033117482384
x9=27.8816348006094x_{9} = 27.8816348006094
x10=−64.009950316892x_{10} = -64.009950316892
x11=−20.0276531666349x_{11} = -20.0276531666349
x12=−49.872783375738x_{12} = -49.872783375738
x13=−30.2378292908018x_{13} = -30.2378292908018
x14=22.3838476568273x_{14} = 22.3838476568273
x15=23.9546439836222x_{15} = 23.9546439836222
x16=60.0829594999048x_{16} = 60.0829594999048
x17=−47.5165888855456x_{17} = -47.5165888855456
x18=−52.2289778659303x_{18} = -52.2289778659303
x19=−9.8174770424681x_{19} = -9.8174770424681
x20=96.2112750161874x_{20} = 96.2112750161874
x21=−56.1559686829176x_{21} = -56.1559686829176
x22=−7.46128255227576x_{22} = -7.46128255227576
x23=42.0188017417635x_{23} = 42.0188017417635
x24=4.31968989868597x_{24} = 4.31968989868597
x25=56.1559686829176x_{25} = 56.1559686829176
x26=−3.53429173528852x_{26} = -3.53429173528852
x27=−67.9369411338793x_{27} = -67.9369411338793
x28=93.8550805259951x_{28} = 93.8550805259951
x29=−100.138265833175x_{29} = -100.138265833175
x30=74.2201264410589x_{30} = 74.2201264410589
x31=−12.1736715326604x_{31} = -12.1736715326604
x32=−89.9280897090078x_{32} = -89.9280897090078
x33=12.1736715326604x_{33} = 12.1736715326604
x34=−79.717913584841x_{34} = -79.717913584841
x35=−65.5807466436869x_{35} = -65.5807466436869
x36=92.2842841992002x_{36} = 92.2842841992002
x37=−87.5718952188155x_{37} = -87.5718952188155
x38=70.2931356240716x_{38} = 70.2931356240716
x39=45.9457925587507x_{39} = 45.9457925587507
x40=−34.164820107789x_{40} = -34.164820107789
x41=−96.2112750161874x_{41} = -96.2112750161874
x42=−5.89048622548086x_{42} = -5.89048622548086
x43=−43.5895980685584x_{43} = -43.5895980685584
x44=−69.5077374606742x_{44} = -69.5077374606742
x45=−1.96349540849362x_{45} = -1.96349540849362
x46=−91.4988860358027x_{46} = -91.4988860358027
x47=48.3019870489431x_{47} = 48.3019870489431
x48=71.8639319508665x_{48} = 71.8639319508665
x49=−74.2201264410589x_{49} = -74.2201264410589
x50=−57.7267650097125x_{50} = -57.7267650097125
x51=49.872783375738x_{51} = 49.872783375738
x52=−17.6714586764426x_{52} = -17.6714586764426
x53=−35.7356164345839x_{53} = -35.7356164345839
x54=−86.0010988920206x_{54} = -86.0010988920206
x55=64.009950316892x_{55} = 64.009950316892
x56=53.7997741927252x_{56} = 53.7997741927252
x57=−78.1471172580461x_{57} = -78.1471172580461
x58=−82.0741080750334x_{58} = -82.0741080750334
x59=30.2378292908018x_{59} = 30.2378292908018
x60=89.9280897090078x_{60} = 89.9280897090078
x61=−21.5984494934298x_{61} = -21.5984494934298
x62=−16.1006623496477x_{62} = -16.1006623496477
x63=62.4391539900971x_{63} = 62.4391539900971
x64=−27.8816348006094x_{64} = -27.8816348006094
x65=−60.0829594999048x_{65} = -60.0829594999048
x66=−93.8550805259951x_{66} = -93.8550805259951
x67=8.24668071567321x_{67} = 8.24668071567321
x68=84.4303025652257x_{68} = 84.4303025652257
x69=9.8174770424681x_{69} = 9.8174770424681
x70=88.3572933822129x_{70} = 88.3572933822129
x71=−97.7820713429823x_{71} = -97.7820713429823
x72=38.0918109247762x_{72} = 38.0918109247762
x73=36.5210145979813x_{73} = 36.5210145979813
x74=−23.9546439836222x_{74} = -23.9546439836222
x75=26.3108384738145x_{75} = 26.3108384738145
x76=−75.7909227678538x_{76} = -75.7909227678538
x77=40.4480054149686x_{77} = 40.4480054149686
x78=−31.8086256175967x_{78} = -31.8086256175967
x79=−53.7997741927252x_{79} = -53.7997741927252
x80=66.3661448070844x_{80} = 66.3661448070844
x81=−38.0918109247762x_{81} = -38.0918109247762
x82=16.1006623496477x_{82} = 16.1006623496477
x83=−39.6626072515711x_{83} = -39.6626072515711
x84=−25.5254403104171x_{84} = -25.5254403104171
x85=34.164820107789x_{85} = 34.164820107789
x86=18.45685683984x_{86} = 18.45685683984
x87=14.5298660228528x_{87} = 14.5298660228528
x88=−83.6449044018282x_{88} = -83.6449044018282
x89=−42.0188017417635x_{89} = -42.0188017417635
x90=86.0010988920206x_{90} = 86.0010988920206
x91=−45.9457925587507x_{91} = -45.9457925587507
x92=20.0276531666349x_{92} = 20.0276531666349
x93=−71.8639319508665x_{93} = -71.8639319508665
x94=5.89048622548086x_{94} = 5.89048622548086
x95=75.7909227678538x_{95} = 75.7909227678538
x96=31.8086256175967x_{96} = 31.8086256175967
x97=−61.6537558266997x_{97} = -61.6537558266997
x98=100.138265833175x_{98} = 100.138265833175
x99=58.5121631731099x_{99} = 58.5121631731099
x100=78.1471172580461x_{100} = 78.1471172580461
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^3*cot(4*x).
03cot⁡(0⋅4)0^{3} \cot{\left(0 \cdot 4 \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
x3(−4cot⁡2(4x)−4)+3x2cot⁡(4x)=0x^{3} \left(- 4 \cot^{2}{\left(4 x \right)} - 4\right) + 3 x^{2} \cot{\left(4 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−0.284857832509479x_{1} = -0.284857832509479
x2=0.284857832509479x_{2} = 0.284857832509479
Signos de extremos en los puntos:
(-0.28485783250947855, 0.0106390069022123)

(0.28485783250947855, 0.0106390069022123)


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:
La función no tiene puntos mínimos
Puntos máximos de la función:
x2=−0.284857832509479x_{2} = -0.284857832509479
x2=0.284857832509479x_{2} = 0.284857832509479
Decrece en los intervalos
(−∞,−0.284857832509479]\left(-\infty, -0.284857832509479\right]
Crece en los intervalos
[0.284857832509479,∞)\left[0.284857832509479, \infty\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
2x(16x2(cot⁡2(4x)+1)cot⁡(4x)−12x(cot⁡2(4x)+1)+3cot⁡(4x))=02 x \left(16 x^{2} \left(\cot^{2}{\left(4 x \right)} + 1\right) \cot{\left(4 x \right)} - 12 x \left(\cot^{2}{\left(4 x \right)} + 1\right) + 3 \cot{\left(4 x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−87.5697541732718x_{1} = -87.5697541732718
x2=84.4280818569863x_{2} = 84.4280818569863
x3=−53.7962892737124x_{3} = -53.7962892737124
x4=−75.7884489375654x_{4} = -75.7884489375654
x5=8.22400650125734x_{5} = 8.22400650125734
x6=−71.8613229481552x_{6} = -71.8613229481552
x7=20.0182954817936x_{7} = 20.0182954817936
x8=5.85882469188875x_{8} = 5.85882469188875
x9=82.0718236179034x_{9} = 82.0718236179034
x10=80.5009827188895x_{10} = 80.5009827188895
x11=4.27670831820519x_{11} = 4.27670831820519
x12=89.9260047580913x_{12} = 89.9260047580913
x13=27.8749115647534x_{13} = 27.8749115647534
x14=−83.6426628429806x_{14} = -83.6426628429806
x15=−52.2253881515166x_{15} = -52.2253881515166
x16=−39.6578804402903x_{16} = -39.6578804402903
x17=31.8027320818734x_{17} = 31.8027320818734
x18=−89.9260047580913x_{18} = -89.9260047580913
x19=−9.79841540209022x_{19} = -9.79841540209022
x20=−67.934181333679x_{20} = -67.934181333679
x21=−27.8749115647534x_{21} = -27.8749115647534
x22=−23.9468192466062x_{22} = -23.9468192466062
x23=62.4361512110445x_{23} = 62.4361512110445
x24=64.0070212188413x_{24} = 64.0070212188413
x25=42.014339927382x_{25} = 42.014339927382
x26=96.2093262196225x_{26} = 96.2093262196225
x27=−85.9989187425869x_{27} = -85.9989187425869
x28=−61.6507147994337x_{28} = -61.6507147994337
x29=−91.4968368764179x_{29} = -91.4968368764179
x30=−20.0182954817936x_{30} = -20.0182954817936
x31=48.2981055329588x_{31} = 48.2981055329588
x32=56.1526299665193x_{32} = 56.1526299665193
x33=−45.9417120251482x_{33} = -45.9417120251482
x34=26.3037140617996x_{34} = 26.3037140617996
x35=−35.7303703383804x_{35} = -35.7303703383804
x36=−65.5778876974823x_{36} = -65.5778876974823
x37=45.9417120251482x_{37} = 45.9417120251482
x38=−96.2093262196225x_{38} = -96.2093262196225
x39=−16.08902527208x_{39} = -16.08902527208
x40=18.4467035965976x_{40} = 18.4467035965976
x41=30.2316297197341x_{41} = 30.2316297197341
x42=44.3707712817119x_{42} = 44.3707712817119
x43=92.2822524786986x_{43} = 92.2822524786986
x44=16.08902527208x_{44} = 16.08902527208
x45=34.1593328874937x_{45} = 34.1593328874937
x46=49.8690240935011x_{46} = 49.8690240935011
x47=1.87210673494537x_{47} = 1.87210673494537
x48=−8.22400650125734x_{48} = -8.22400650125734
x49=93.8530828077241x_{49} = 93.8530828077241
x50=−17.6608547068894x_{50} = -17.6608547068894
x51=−82.0718236179034x_{51} = -82.0718236179034
x52=75.7884489375654x_{52} = 75.7884489375654
x53=−79.7155616107374x_{53} = -79.7155616107374
x54=9.79841540209022x_{54} = 9.79841540209022
x55=−5.85882469188875x_{55} = -5.85882469188875
x56=−30.2316297197341x_{56} = -30.2316297197341
x57=97.7801538511486x_{57} = 97.7801538511486
x58=23.9468192466062x_{58} = 23.9468192466062
x59=−64.0070212188413x_{59} = -64.0070212188413
x60=70.2904683239994x_{60} = 70.2904683239994
x61=36.5158812890039x_{61} = 36.5158812890039
x62=78.1447180109551x_{62} = 78.1447180109551
x63=−1.87210673494537x_{63} = -1.87210673494537
x64=−12.1582888593509x_{64} = -12.1582888593509
x65=−21.5897718000664x_{65} = -21.5897718000664
x66=74.2176002581006x_{66} = 74.2176002581006
x67=60.0798389768097x_{67} = 60.0798389768097
x68=12.1582888593509x_{68} = 12.1582888593509
x69=66.3633196916654x_{69} = 66.3633196916654
x70=−34.1593328874937x_{70} = -34.1593328874937
x71=−57.7235171323762x_{71} = -57.7235171323762
x72=40.4433703651964x_{72} = 40.4433703651964
x73=14.5169730004709x_{73} = 14.5169730004709
x74=85.9989187425869x_{74} = 85.9989187425869
x75=−31.8027320818734x_{75} = -31.8027320818734
x76=52.2253881515166x_{76} = 52.2253881515166
x77=71.8613229481552x_{77} = 71.8613229481552
x78=−42.014339927382x_{78} = -42.014339927382
x79=−93.8530828077241x_{79} = -93.8530828077241
x80=−78.1447180109551x_{80} = -78.1447180109551
x81=−74.2176002581006x_{81} = -74.2176002581006
x82=53.7962892737124x_{82} = 53.7962892737124
x83=38.0868892427927x_{83} = 38.0868892427927
x84=−38.0868892427927x_{84} = -38.0868892427927
x85=−60.0798389768097x_{85} = -60.0798389768097
x86=22.3754742123117x_{86} = 22.3754742123117
x87=−43.5852970079695x_{87} = -43.5852970079695
x88=−97.7801538511486x_{88} = -97.7801538511486
x89=−49.8690240935011x_{89} = -49.8690240935011
x90=−100.136393457088x_{90} = -100.136393457088
x91=88.355171367267x_{91} = 88.355171367267
x92=−56.1526299665193x_{92} = -56.1526299665193
x93=100.136393457088x_{93} = 100.136393457088
x94=−47.5126432228376x_{94} = -47.5126432228376
x95=67.934181333679x_{95} = 67.934181333679
x96=−3.48200465376189x_{96} = -3.48200465376189
x97=58.5089588866363x_{97} = 58.5089588866363
x98=−25.5180968098495x_{98} = -25.5180968098495
x99=−13.7308395096978x_{99} = -13.7308395096978
x100=−69.5050400239439x_{100} = -69.5050400239439

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[−1.87210673494537,1.87210673494537]\left[-1.87210673494537, 1.87210673494537\right]
Convexa en los intervalos
(−∞,−100.136393457088]\left(-\infty, -100.136393457088\right]
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→−∞(x3cot⁡(4x))y = \lim_{x \to -\infty}\left(x^{3} \cot{\left(4 x \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(x3cot⁡(4x))y = \lim_{x \to \infty}\left(x^{3} \cot{\left(4 x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x^3*cot(4*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→−∞(x2cot⁡(4x))y = x \lim_{x \to -\infty}\left(x^{2} \cot{\left(4 x \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(x2cot⁡(4x))y = x \lim_{x \to \infty}\left(x^{2} \cot{\left(4 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:
x3cot⁡(4x)=x3cot⁡(4x)x^{3} \cot{\left(4 x \right)} = x^{3} \cot{\left(4 x \right)}
- Sí
x3cot⁡(4x)=−x3cot⁡(4x)x^{3} \cot{\left(4 x \right)} = - x^{3} \cot{\left(4 x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = x^3*cot(4*x)
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