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  • Gráfico de la función y =:
  • 6*x-x^3 6*x-x^3
  • -(4-x^2)^(1/2) -(4-x^2)^(1/2)
  • 4/(x^2+1) 4/(x^2+1)
  • 3-(x+2)/(x^2+2*x) 3-(x+2)/(x^2+2*x)

  • Expresiones idénticas

  • (sinx+tgx)/(dos *x*ctgx)
  • ( seno de x más tgx) dividir por (2 multiplicar por x multiplicar por ctgx)
  • ( seno de x más tgx) dividir por (dos multiplicar por x multiplicar por ctgx)
  • (sinx+tgx)/(2xctgx)
  • sinx+tgx/2xctgx
  • (sinx+tgx) dividir por (2*x*ctgx)

  • Expresiones semejantes

  • (sinx-tgx)/(2*x*ctgx)
Trazado el gráfico de la función/ (sinx+tgx)/(2*x*ctgx)

Gráfico de la función y = (sinx+tgx)/(2*x*ctgx)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(x) + tan(x)
f(x) = ---------------
          2*x*cot(x)
f(x)=sin⁡(x)+tan⁡(x)2xcot⁡(x)f{\left(x \right)} = \frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}} 
f = (sin(x) + tan(x))/(((2*x)*cot(x)))
Gráfico de la función
02468-8-6-4-2-1010-50005000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=1.5707963267949x_{2} = 1.5707963267949
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)+tan⁡(x)2xcot⁡(x)=0\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−πx_{1} = - \pi
x2=πx_{2} = \pi
x3=2πx_{3} = 2 \pi
Solución numérica
x1=−91.1080860717611x_{1} = -91.1080860717611
x2=−25.1327416534355x_{2} = -25.1327416534355
x3=−31.41592673575x_{3} = -31.41592673575
x4=−18.849554936845x_{4} = -18.849554936845
x5=−81.6814090386147x_{5} = -81.6814090386147
x6=−62.8318521658186x_{6} = -62.8318521658186
x7=31.4159239820201x_{7} = 31.4159239820201
x8=−62.8318540332717x_{8} = -62.8318540332717
x9=12.5663700981581x_{9} = 12.5663700981581
x10=47.1209763081767x_{10} = 47.1209763081767
x11=40.8389774408875x_{11} = 40.8389774408875
x12=−84.8261831073356x_{12} = -84.8261831073356
x13=34.5566546899828x_{13} = 34.5566546899828
x14=43.9822971695215x_{14} = 43.9822971695215
x15=−9.42567821453527x_{15} = -9.42567821453527
x16=−40.8376695273069x_{16} = -40.8376695273069
x17=−21.9911796982577x_{17} = -21.9911796982577
x18=47.1270880348955x_{18} = 47.1270880348955
x19=81.6814092182089x_{19} = 81.6814092182089
x20=−65.9735393758622x_{20} = -65.9735393758622
x21=−75.3982238987855x_{21} = -75.3982238987855
x22=25.132742193978x_{22} = 25.132742193978
x23=94.247779609352x_{23} = 94.247779609352
x24=−15.7080841076436x_{24} = -15.7080841076436
x25=−47.1257220786859x_{25} = -47.1257220786859
x26=−56.5486706799278x_{26} = -56.5486706799278
x27=−59.6904319845233x_{27} = -59.6904319845233
x28=9.42658199496347x_{28} = 9.42658199496347
x29=−3.14326707194023x_{29} = -3.14326707194023
x30=62.8318526701909x_{30} = 62.8318526701909
x31=91.1095039335036x_{31} = 91.1095039335036
x32=−75.3982155408245x_{32} = -75.3982155408245
x33=−87.9645943582203x_{33} = -87.9645943582203
x34=−18.8495568079986x_{34} = -18.8495568079986
x35=15.7088922765864x_{35} = 15.7088922765864
x36=65.9735389843344x_{36} = 65.9735389843344
x37=75.3982213788012x_{37} = 75.3982213788012
x38=−53.4080340967238x_{38} = -53.4080340967238
x39=25.1327403244751x_{39} = 25.1327403244751
x40=75.3982241638922x_{40} = 75.3982241638922
x41=100.530964745217x_{41} = 100.530964745217
x42=97.3913416584044x_{42} = 97.3913416584044
x43=−72.2558000971762x_{43} = -72.2558000971762
x44=21.9911796938987x_{44} = 21.9911796938987
x45=18.849558891133x_{45} = 18.849558891133
x46=84.8213388084006x_{46} = 84.8213388084006
x47=28.2742637865287x_{47} = 28.2742637865287
x48=59.6912473597501x_{48} = 59.6912473597501
x49=−94.247779428311x_{49} = -94.247779428311
x50=−37.6991118774179x_{50} = -37.6991118774179
x51=−84.8200753771042x_{51} = -84.8200753771042
x52=56.5486675835781x_{52} = 56.5486675835781
x53=−12.5663701515109x_{53} = -12.5663701515109
x54=−43.9822971744317x_{54} = -43.9822971744317
x55=62.8318564649026x_{55} = 62.8318564649026
x56=37.6991120547296x_{56} = 37.6991120547296
x57=−28.2734476034419x_{57} = -28.2734476034419
x58=−34.5557272431001x_{58} = -34.5557272431001
x59=34.5611202441731x_{59} = 34.5611202441731
x60=−6.28318509237865x_{60} = -6.28318509237865
x61=18.849555492068x_{61} = 18.849555492068
x62=−69.1150388295625x_{62} = -69.1150388295625
x63=−12.5663731336943x_{63} = -12.5663731336943
x64=−56.5486673390634x_{64} = -56.5486673390634
x65=12.5663704177689x_{65} = 12.5663704177689
x66=69.1150375370464x_{66} = 69.1150375370464
x67=78.5390028929786x_{67} = 78.5390028929786
x68=−69.1150357551256x_{68} = -69.1150357551256
x69=87.9645943361683x_{69} = 87.9645943361683
x70=6.28318528394097x_{70} = 6.28318528394097
x71=91.1033742827819x_{71} = 91.1033742827819
x72=53.4089756107496x_{72} = 53.4089756107496
x73=72.2566119315779x_{73} = 72.2566119315779
x74=31.4159269876038x_{74} = 31.4159269876038
x75=−50.2654822658225x_{75} = -50.2654822658225
x76=−25.1327382847962x_{76} = -25.1327382847962
x77=50.2654824463177x_{77} = 50.2654824463177
x78=−40.8437719191275x_{78} = -40.8437719191275
x79=−31.4159219217159x_{79} = -31.4159219217159
x80=81.68140841851x_{80} = 81.68140841851
x81=−97.3903823516316x_{81} = -97.3903823516316
x82=69.1150394191749x_{82} = 69.1150394191749
x83=−100.530964509946x_{83} = -100.530964509946
x84=−78.5380926204531x_{84} = -78.5380926204531
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) + tan(x))/(((2*x)*cot(x))).
sin⁡(0)+tan⁡(0)0⋅2cot⁡(0)\frac{\sin{\left(0 \right)} + \tan{\left(0 \right)}}{0 \cdot 2 \cot{\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
12xcot⁡(x)(cos⁡(x)+tan⁡2(x)+1)+(−2x(−cot⁡2(x)−1)−2cot⁡(x))(sin⁡(x)+tan⁡(x))4x2cot⁡2(x)=0\frac{1}{2 x \cot{\left(x \right)}} \left(\cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) + \frac{\left(- 2 x \left(- \cot^{2}{\left(x \right)} - 1\right) - 2 \cot{\left(x \right)}\right) \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)}{4 x^{2} \cot^{2}{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=34.5574417217695x_{1} = 34.5574417217695
x2=−100.530964914873x_{2} = -100.530964914873
x3=−40.8403379519905x_{3} = -40.8403379519905
x4=−3.14175093489092x_{4} = -3.14175093489092
x5=−25.1327412287183x_{5} = -25.1327412287183
x6=−75.398223686155x_{6} = -75.398223686155
x7=−50.2654824574367x_{7} = -50.2654824574367
x8=81.6814089933346x_{8} = 81.6814089933346
x9=50.2654824574367x_{9} = 50.2654824574367
x10=−43.9822971502571x_{10} = -43.9822971502571
x11=21.991151641644x_{11} = 21.991151641644
x12=−37.6991118430775x_{12} = -37.6991118430775
x13=−72.2565554808085x_{13} = -72.2565554808085
x14=25.1327412287183x_{14} = 25.1327412287183
x15=−9.42485781925051x_{15} = -9.42485781925051
x16=72.2566292955242x_{16} = 72.2566292955242
x17=−78.5396461911396x_{17} = -78.5396461911396
x18=−18.8495559215388x_{18} = -18.8495559215388
x19=59.6903504740077x_{19} = 59.6903504740077
x20=84.8228398927363x_{20} = 84.8228398927363
x21=−15.7079741591662x_{21} = -15.7079741591662
x22=−56.5486677646163x_{22} = -56.5486677646163
x23=18.8495559215388x_{23} = 18.8495559215388
x24=−6.28318530717959x_{24} = -6.28318530717959
x25=−21.99115164051x_{25} = -21.99115164051
x26=−47.1240716887076x_{26} = -47.1240716887076
x27=12.5663706143592x_{27} = 12.5663706143592
x28=−97.3894633954265x_{28} = -97.3894633954265
x29=−40.8410747688973x_{29} = -40.8410747688973
x30=56.5486677646163x_{30} = 56.5486677646163
x31=−62.8318530717959x_{31} = -62.8318530717959
x32=15.7080473943606x_{32} = 15.7080473943606
x33=78.5397437779814x_{33} = 78.5397437779814
x34=69.1150383789755x_{34} = 69.1150383789755
x35=6.28318530717959x_{35} = 6.28318530717959
x36=53.4072667518135x_{36} = 53.4072667518135
x37=37.6991118430775x_{37} = 37.6991118430775
x38=−31.4159265358979x_{38} = -31.4159265358979
x39=100.530964914873x_{39} = 100.530964914873
x40=−84.8226545167245x_{40} = -84.8226545167245
x41=−53.4071612202693x_{41} = -53.4071612202693
x42=94.2477796076938x_{42} = 94.2477796076938
x43=−34.5573403476478x_{43} = -34.5573403476478
x44=9.42495587951317x_{44} = 9.42495587951317
x45=−91.1063775106345x_{45} = -91.1063775106345
x46=−12.5663706143592x_{46} = -12.5663706143592
x47=65.97345482794x_{47} = 65.97345482794
x48=−94.2477796076938x_{48} = -94.2477796076938
x49=28.274327526925x_{49} = 28.274327526925
x50=40.8405347827156x_{50} = 40.8405347827156
x51=43.9822971502571x_{51} = 43.9822971502571
x52=75.398223686155x_{52} = 75.398223686155
x53=−59.6902758155797x_{53} = -59.6902758155797
x54=91.1058606234583x_{54} = 91.1058606234583
x55=62.8318530717959x_{55} = 62.8318530717959
x56=−28.2742529777451x_{56} = -28.2742529777451
x57=97.389573015133x_{57} = 97.389573015133
x58=−84.8233930037709x_{58} = -84.8233930037709
x59=87.9645943005142x_{59} = 87.9645943005142
x60=91.1066053663186x_{60} = 91.1066053663186
x61=−65.9734547229338x_{61} = -65.9734547229338
x62=47.1235460529235x_{62} = 47.1235460529235
x63=−81.6814089933346x_{63} = -81.6814089933346
x64=−87.9645943005142x_{64} = -87.9645943005142
x65=−69.1150383789755x_{65} = -69.1150383789755
x66=31.4159265358979x_{66} = 31.4159265358979
Signos de extremos en los puntos:
(34.557441721769464, 2.60544241909504e-19)

(-100.53096491487338, -1.52764277031079e-31)

(-40.8403379519905, -1.10499008809096e-16)

(-3.141750934890923, -4.99443894134616e-17)

(-25.132741228718345, -3.81910692577698e-32)

(-75.39822368615503, -1.14573207773309e-31)

(-50.26548245743669, -7.63821385155396e-32)

(81.68140899333463, 1.88255223925939e-31)

(50.26548245743669, 7.63821385155396e-32)

(-43.982297150257104, -6.68343712010972e-32)

(21.991151641643985, 1.00525924381744e-24)

(-37.69911184307752, -5.72866038866547e-32)

(-72.25655548080847, -1.1273039336493e-19)

(25.132741228718345, 3.81910692577698e-32)

(-9.424857819250505, -1.07882095276346e-18)

(72.25662929552416, 3.15009033747855e-26)

(-78.53964619113955, -2.66786990270092e-18)

(-18.84955592153876, -2.86433019433274e-32)

(59.690350474007744, 2.75475304481601e-19)

(84.82283989273634, 2.01766630804503e-18)

(-15.707974159166167, -2.23936882398876e-22)

(-56.548667764616276, -8.59299058299821e-32)

(18.84955592153876, 2.86433019433274e-32)

(-6.283185307179586, -9.54776731444245e-33)

(-21.991151640509973, -1.00376583622348e-24)

(-47.12407168870756, -5.806085316048e-18)

(12.566370614359172, 1.90955346288849e-32)

(-97.38946339542655, -1.77073058979625e-19)

(-40.84107476889726, -1.15060815318006e-16)

(56.548667764616276, 8.59299058299821e-32)

(-62.83185307179586, -9.54776731444245e-32)

(15.708047394360637, 7.97163235164562e-19)

(78.53974377798144, 8.82433229843939e-20)

(69.11503837897546, 2.81541104661861e-31)

(6.283185307179586, 9.54776731444245e-33)

(53.40726675181354, 6.3138091250563e-18)

(37.69911184307752, 5.72866038866547e-32)

(-31.41592653589793, -4.77388365722123e-32)

(100.53096491487338, 1.52764277031079e-31)

(-84.82265451672453, -4.27954673616227e-17)

(-53.40716122026933, -2.57359145580604e-19)

(94.2477796076938, 1.24937720620631e-31)

(-34.55734034764782, -7.40075829164777e-18)

(9.424955879513174, 2.65795427218555e-17)

(-91.10637751063449, -3.61815017326727e-18)

(-12.566370614359172, -1.90955346288849e-32)

(65.97345482794003, 2.60149754277779e-23)

(-94.2477796076938, -1.24937720620631e-31)

(28.274327526924953, 1.44249900139147e-23)

(40.840534782715636, 5.07830496843494e-18)

(43.982297150257104, 6.68343712010972e-32)

(75.39822368615503, 1.14573207773309e-31)

(-59.69027581557969, -2.35408833715755e-22)

(91.10586062345827, 3.11189662741211e-17)

(62.83185307179586, 9.54776731444245e-32)

(-28.274252977745054, -3.78827011247992e-19)

(97.38957301513302, 4.169491411353e-18)

(-84.82339300377086, -6.91378946352496e-17)

(87.96459430051421, 1.33668742402194e-31)

(91.10660536631859, 8.41022481934631e-17)

(-65.97345472293381, -2.48351882236391e-23)

(47.12354605292351, 7.40760077486107e-17)

(-81.68140899333463, -1.88255223925939e-31)

(-87.96459430051421, -1.33668742402194e-31)

(-69.11503837897546, -2.81541104661861e-31)

(31.41592653589793, 4.77388365722123e-32)


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=81.6814089933346x_{1} = 81.6814089933346
x2=50.2654824574367x_{2} = 50.2654824574367
x3=25.1327412287183x_{3} = 25.1327412287183
x4=18.8495559215388x_{4} = 18.8495559215388
x5=12.5663706143592x_{5} = 12.5663706143592
x6=56.5486677646163x_{6} = 56.5486677646163
x7=69.1150383789755x_{7} = 69.1150383789755
x8=6.28318530717959x_{8} = 6.28318530717959
x9=37.6991118430775x_{9} = 37.6991118430775
x10=100.530964914873x_{10} = 100.530964914873
x11=94.2477796076938x_{11} = 94.2477796076938
x12=43.9822971502571x_{12} = 43.9822971502571
x13=75.398223686155x_{13} = 75.398223686155
x14=62.8318530717959x_{14} = 62.8318530717959
x15=87.9645943005142x_{15} = 87.9645943005142
x16=31.4159265358979x_{16} = 31.4159265358979
Puntos máximos de la función:
x16=−100.530964914873x_{16} = -100.530964914873
x16=−25.1327412287183x_{16} = -25.1327412287183
x16=−75.398223686155x_{16} = -75.398223686155
x16=−50.2654824574367x_{16} = -50.2654824574367
x16=−43.9822971502571x_{16} = -43.9822971502571
x16=−37.6991118430775x_{16} = -37.6991118430775
x16=−18.8495559215388x_{16} = -18.8495559215388
x16=−56.5486677646163x_{16} = -56.5486677646163
x16=−6.28318530717959x_{16} = -6.28318530717959
x16=−62.8318530717959x_{16} = -62.8318530717959
x16=−31.4159265358979x_{16} = -31.4159265358979
x16=−12.5663706143592x_{16} = -12.5663706143592
x16=−94.2477796076938x_{16} = -94.2477796076938
x16=−81.6814089933346x_{16} = -81.6814089933346
x16=−87.9645943005142x_{16} = -87.9645943005142
x16=−69.1150383789755x_{16} = -69.1150383789755
Decrece en los intervalos
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Crece en los intervalos
[−6.28318530717959,6.28318530717959]\left[-6.28318530717959, 6.28318530717959\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=1.5707963267949x_{2} = 1.5707963267949
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)+tan⁡(x)2xcot⁡(x))y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(sin⁡(x)+tan⁡(x)2xcot⁡(x))y = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (sin(x) + tan(x))/(((2*x)*cot(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→−∞(12xcot⁡(x)(sin⁡(x)+tan⁡(x))x)y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{2 x \cot{\left(x \right)}} \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(12xcot⁡(x)(sin⁡(x)+tan⁡(x))x)y = x \lim_{x \to \infty}\left(\frac{\frac{1}{2 x \cot{\left(x \right)}} \left(\sin{\left(x \right)} + \tan{\left(x \right)}\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)+tan⁡(x)2xcot⁡(x)=−sin⁡(x)−tan⁡(x)2xcot⁡(x)\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}} = \frac{- \sin{\left(x \right)} - \tan{\left(x \right)}}{2 x \cot{\left(x \right)}}
- No
sin⁡(x)+tan⁡(x)2xcot⁡(x)=−−sin⁡(x)−tan⁡(x)2xcot⁡(x)\frac{\sin{\left(x \right)} + \tan{\left(x \right)}}{2 x \cot{\left(x \right)}} = - \frac{- \sin{\left(x \right)} - \tan{\left(x \right)}}{2 x \cot{\left(x \right)}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор