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Trazado el gráfico de la función/ tgx+ctg(x/2)+tg(2x/3)

Gráfico de la función y = tgx+ctg(x/2)+tg(2x/3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   /x\      /2*x\
f(x) = tan(x) + cot|-| + tan|---|
                   \2/      \ 3 /
f(x)=(tan⁡(x)+cot⁡(x2))+tan⁡(2x3)f{\left(x \right)} = \left(\tan{\left(x \right)} + \cot{\left(\frac{x}{2} \right)}\right) + \tan{\left(\frac{2 x}{3} \right)} 
f = tan(x) + cot(x/2) + tan((2*x)/3)
Gráfico de la función
02468-8-6-4-2-1010-25002500
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:
(tan⁡(x)+cot⁡(x2))+tan⁡(2x3)=0\left(\tan{\left(x \right)} + \cot{\left(\frac{x}{2} \right)}\right) + \tan{\left(\frac{2 x}{3} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=49.0991017443071x_{1} = 49.0991017443071
x2=−79.3215426106732x_{2} = -79.3215426106732
x3=−39.5429310220932x_{3} = -39.5429310220932
x4=−33.7757929185592x_{4} = -33.7757929185592
x5=67.9486576658459x_{5} = 67.9486576658459
x6=98.1710985322121x_{6} = 98.1710985322121
x7=33.7757929185592x_{7} = 33.7757929185592
x8=−7.44956602030916x_{8} = -7.44956602030916
x9=−96.0915987867095x_{9} = -96.0915987867095
x10=47.1238898038469x_{10} = 47.1238898038469
x11=92.4039604286781x_{11} = 92.4039604286781
x12=−3.92331892451831x_{12} = -3.92331892451831
x13=82.8477897064642x_{13} = 82.8477897064642
x14=65.9734457253857x_{14} = 65.9734457253857
x15=26.2991219418479x_{15} = 26.2991219418479
x16=−60.4719866891346x_{16} = -60.4719866891346
x17=−45.1486778633867x_{17} = -45.1486778633867
x18=1.84381917901569x_{18} = 1.84381917901569
x19=−9.42477796076938x_{19} = -9.42477796076938
x20=−41.6224307675958x_{20} = -41.6224307675958
x21=−67.9486576658459x_{21} = -67.9486576658459
x22=11.3999899012296x_{22} = 11.3999899012296
x23=−35.8552926640618x_{23} = -35.8552926640618
x24=52.625348840098x_{24} = 52.625348840098
x25=58.392486943632x_{25} = 58.392486943632
x26=−14.9262369970204x_{26} = -14.9262369970204
x27=22.7728748460571x_{27} = 22.7728748460571
x28=518.362787842316x_{28} = 518.362787842316
x29=−47.1238898038469x_{29} = -47.1238898038469
x30=63.9982337849254x_{30} = 63.9982337849254
x31=28.2743338823081x_{31} = 28.2743338823081
x32=−90.3244606831755x_{32} = -90.3244606831755
x33=79.3215426106733x_{33} = 79.3215426106733
x34=71.4749047616367x_{34} = 71.4749047616367
x35=−26.2991219418479x_{35} = -26.2991219418479
x36=−30.2495458227684x_{36} = -30.2495458227684
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=−52.625348840098x_{38} = -52.625348840098
x39=−22.7728748460571x_{39} = -22.7728748460571
x40=96.0915987867095x_{40} = 96.0915987867095
x41=−98.1710985322121x_{41} = -98.1710985322121
x42=103.672557568463x_{42} = 103.672557568463
x43=−63.9982337849254x_{43} = -63.9982337849254
x44=−79.3215426106733x_{44} = -79.3215426106733
x45=−71.4749047616367x_{45} = -71.4749047616367
x46=3.92331892451831x_{46} = 3.92331892451831
x47=84.8230016469244x_{47} = 84.8230016469244
x48=−1.84381917901569x_{48} = -1.84381917901569
x49=60.4719866891346x_{49} = 60.4719866891346
x50=−128.023572526253x_{50} = -128.023572526253
x51=−86.7982135873846x_{51} = -86.7982135873846
x52=9.42477796076938x_{52} = 9.42477796076938
x53=−84.8230016469244x_{53} = -84.8230016469244
x54=−11.3999899012296x_{54} = -11.3999899012296
x55=−20.6933751005544x_{55} = -20.6933751005544
x56=41.6224307675958x_{56} = 41.6224307675958
x57=90.3244606831755x_{57} = 90.3244606831755
x58=−28.2743338823081x_{58} = -28.2743338823081
x59=30.2495458227684x_{59} = 30.2495458227684
x60=14.9262369970204x_{60} = 14.9262369970204
x61=54.7048485856006x_{61} = 54.7048485856006
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
tan⁡2(x)+2tan⁡2(2x3)3−cot⁡2(x2)2+76=0\tan^{2}{\left(x \right)} + \frac{2 \tan^{2}{\left(\frac{2 x}{3} \right)}}{3} - \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{7}{6} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=88.7964096300637x_{1} = 88.7964096300637
x2=76.224965172799x_{2} = 76.224965172799
x3=−55.7219262779724x_{3} = -55.7219262779724
x4=−113.924077015876x_{4} = -113.924077015876
x5=−13.3981859439086x_{5} = -13.3981859439086
x6=−24.3009258991689x_{6} = -24.3009258991689
x7=−88.7964096300637x_{7} = -88.7964096300637
x8=1375.19084078569x_{8} = 1375.19084078569
x9=19.6762974081827x_{9} = 19.6762974081827
x10=−5.45136997763013x_{10} = -5.45136997763013
x11=−25.5428083266919x_{11} = -25.5428083266919
x12=−201.893745159296x_{12} = -201.893745159296
x13=−57.3754092512602x_{13} = -57.3754092512602
x14=−80.8495936637852x_{14} = -80.8495936637852
x15=5.45136997763013x_{15} = 5.45136997763013
x16=−38.5258533297214x_{16} = -38.5258533297214
x17=−82.0914760913082x_{17} = -82.0914760913082
x18=63.2419201697694x_{18} = 63.2419201697694
x19=82.0914760913082x_{19} = 82.0914760913082
x20=95.0745210943377x_{20} = 95.0745210943377
x21=32.2477418654474x_{21} = 32.2477418654474
x22=62.0000377422464x_{22} = 62.0000377422464
x23=−69.9468537085249x_{23} = -69.9468537085249
x24=−32.2477418654474x_{24} = -32.2477418654474
x25=−95.0745210943377x_{25} = -95.0745210943377
x26=93.4210381210499x_{26} = 93.4210381210499
x27=−74.5714821995111x_{27} = -74.5714821995111
x28=74.5714821995111x_{28} = 74.5714821995111
x29=−19.6762974081827x_{29} = -19.6762974081827
x30=−36.8723703564336x_{30} = -36.8723703564336
x31=−51.0972977869861x_{31} = -51.0972977869861
x32=49.8554153594631x_{32} = 49.8554153594631
x33=24.3009258991689x_{33} = 24.3009258991689
x34=−49.8554153594631x_{34} = -49.8554153594631
x35=36.8723703564336x_{35} = 36.8723703564336
x36=−62.0000377422464x_{36} = -62.0000377422464
x37=−99.6991495853239x_{37} = -99.6991495853239
x38=−87.5545272025406x_{38} = -87.5545272025406
x39=18.0228144348948x_{39} = 18.0228144348948
x40=−93.4210381210499x_{40} = -93.4210381210499
x41=13.3981859439086x_{41} = 13.3981859439086
x42=−76.224965172799x_{42} = -76.224965172799
x43=44.3923642482307x_{43} = 44.3923642482307
x44=−68.7049712810019x_{44} = -68.7049712810019
x45=12.1563035163856x_{45} = 12.1563035163856
x46=−43.1504818207076x_{46} = -43.1504818207076
x47=99.6991495853239x_{47} = 99.6991495853239
x48=−12.1563035163856x_{48} = -12.1563035163856
x49=126.495521473141x_{49} = 126.495521473141
x50=38.5258533297214x_{50} = 38.5258533297214
x51=80.8495936637852x_{51} = 80.8495936637852
x52=−18.0228144348948x_{52} = -18.0228144348948
x53=55.7219262779724x_{53} = 55.7219262779724
x54=69.9468537085249x_{54} = 69.9468537085249
x55=−0.826741486643925x_{55} = -0.826741486643925
x56=51.0972977869861x_{56} = 51.0972977869861
Signos de extremos en los puntos:
(88.79640963006366, 2.82474292020604)

(76.22496517279896, 3.98075765927311)

(-55.72192627797235, 3.98075765927311)

(-113.92407701587648, -3.9807576592731)

(-13.39818594390863, -2.82474292020604)

(-24.300925899168888, 2.82474292020604)

(-88.79640963006366, -2.82474292020604)

(1375.1908407856856, -3.98075765927311)

(19.676297408182684, 3.98075765927311)

(-5.451369977630129, 2.82474292020604)

(-25.54280832669192, -9.15613936847922)

(-201.89374515929623, -2.82474292020601)

(-57.3754092512602, -3.98075765927311)

(-80.84959366378517, 2.82474292020604)

(5.451369977630129, -2.82474292020604)

(-38.52585332972144, -3.98075765927311)

(-82.0914760913082, -9.15613936847922)

(63.241920169769436, 9.15613936847922)

(82.0914760913082, 9.15613936847922)

(95.07452109433773, 3.98075765927311)

(32.24774186544739, 2.82474292020604)

(62.000037742246406, -2.82474292020604)

(-69.94685370852491, -2.82474292020604)

(-32.24774186544739, -2.82474292020604)

(-95.07452109433773, -3.98075765927311)

(93.42103812104988, -3.98075765927311)

(-74.57148219951111, 3.98075765927312)

(74.57148219951111, -3.98075765927312)

(-19.676297408182684, -3.98075765927311)

(-36.87237035643359, 3.98075765927311)

(-51.09729778698615, -2.82474292020604)

(49.85541535946312, -9.1561393684793)

(24.300925899168888, -2.82474292020604)

(-49.85541535946312, 9.1561393684793)

(36.87237035643359, -3.98075765927311)

(-62.000037742246406, 2.82474292020604)

(-99.69914958532392, 2.82474292020604)

(-87.55452720254064, 9.1561393684793)

(18.022814434894833, -3.98075765927311)

(-93.42103812104988, 3.98075765927311)

(13.39818594390863, 2.82474292020604)

(-76.22496517279896, -3.98075765927311)

(44.39236424823068, 9.15613936847922)

(-68.70497128100187, 9.15613936847934)

(12.1563035163856, -9.15613936847927)

(-43.15048182070765, 2.82474292020604)

(99.69914958532392, -2.82474292020604)

(-12.1563035163856, 9.15613936847927)

(126.49552147314118, 2.82474292020603)

(38.52585332972144, 3.98075765927311)

(80.84959366378517, -2.82474292020604)

(-18.022814434894833, 3.98075765927311)

(55.72192627797235, -3.98075765927311)

(69.94685370852491, 2.82474292020604)

(-0.826741486643925, -3.98075765927311)

(51.09729778698615, 2.82474292020604)


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=88.7964096300637x_{1} = 88.7964096300637
x2=76.224965172799x_{2} = 76.224965172799
x3=−55.7219262779724x_{3} = -55.7219262779724
x4=−24.3009258991689x_{4} = -24.3009258991689
x5=19.6762974081827x_{5} = 19.6762974081827
x6=−5.45136997763013x_{6} = -5.45136997763013
x7=−80.8495936637852x_{7} = -80.8495936637852
x8=63.2419201697694x_{8} = 63.2419201697694
x9=82.0914760913082x_{9} = 82.0914760913082
x10=95.0745210943377x_{10} = 95.0745210943377
x11=32.2477418654474x_{11} = 32.2477418654474
x12=−74.5714821995111x_{12} = -74.5714821995111
x13=−36.8723703564336x_{13} = -36.8723703564336
x14=−49.8554153594631x_{14} = -49.8554153594631
x15=−62.0000377422464x_{15} = -62.0000377422464
x16=−99.6991495853239x_{16} = -99.6991495853239
x17=−87.5545272025406x_{17} = -87.5545272025406
x18=−93.4210381210499x_{18} = -93.4210381210499
x19=13.3981859439086x_{19} = 13.3981859439086
x20=44.3923642482307x_{20} = 44.3923642482307
x21=−68.7049712810019x_{21} = -68.7049712810019
x22=−43.1504818207076x_{22} = -43.1504818207076
x23=−12.1563035163856x_{23} = -12.1563035163856
x24=126.495521473141x_{24} = 126.495521473141
x25=38.5258533297214x_{25} = 38.5258533297214
x26=−18.0228144348948x_{26} = -18.0228144348948
x27=69.9468537085249x_{27} = 69.9468537085249
x28=51.0972977869861x_{28} = 51.0972977869861
Puntos máximos de la función:
x28=−113.924077015876x_{28} = -113.924077015876
x28=−13.3981859439086x_{28} = -13.3981859439086
x28=−88.7964096300637x_{28} = -88.7964096300637
x28=1375.19084078569x_{28} = 1375.19084078569
x28=−25.5428083266919x_{28} = -25.5428083266919
x28=−201.893745159296x_{28} = -201.893745159296
x28=−57.3754092512602x_{28} = -57.3754092512602
x28=5.45136997763013x_{28} = 5.45136997763013
x28=−38.5258533297214x_{28} = -38.5258533297214
x28=−82.0914760913082x_{28} = -82.0914760913082
x28=62.0000377422464x_{28} = 62.0000377422464
x28=−69.9468537085249x_{28} = -69.9468537085249
x28=−32.2477418654474x_{28} = -32.2477418654474
x28=−95.0745210943377x_{28} = -95.0745210943377
x28=93.4210381210499x_{28} = 93.4210381210499
x28=74.5714821995111x_{28} = 74.5714821995111
x28=−19.6762974081827x_{28} = -19.6762974081827
x28=−51.0972977869861x_{28} = -51.0972977869861
x28=49.8554153594631x_{28} = 49.8554153594631
x28=24.3009258991689x_{28} = 24.3009258991689
x28=36.8723703564336x_{28} = 36.8723703564336
x28=18.0228144348948x_{28} = 18.0228144348948
x28=−76.224965172799x_{28} = -76.224965172799
x28=12.1563035163856x_{28} = 12.1563035163856
x28=99.6991495853239x_{28} = 99.6991495853239
x28=80.8495936637852x_{28} = 80.8495936637852
x28=55.7219262779724x_{28} = 55.7219262779724
x28=−0.826741486643925x_{28} = -0.826741486643925
Decrece en los intervalos
[126.495521473141,∞)\left[126.495521473141, \infty\right)
Crece en los intervalos
(−∞,−99.6991495853239]\left(-\infty, -99.6991495853239\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→−∞((tan⁡(x)+cot⁡(x2))+tan⁡(2x3))y = \lim_{x \to -\infty}\left(\left(\tan{\left(x \right)} + \cot{\left(\frac{x}{2} \right)}\right) + \tan{\left(\frac{2 x}{3} \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((tan⁡(x)+cot⁡(x2))+tan⁡(2x3))y = \lim_{x \to \infty}\left(\left(\tan{\left(x \right)} + \cot{\left(\frac{x}{2} \right)}\right) + \tan{\left(\frac{2 x}{3} \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x) + cot(x/2) + tan((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→−∞((tan⁡(x)+cot⁡(x2))+tan⁡(2x3)x)y = x \lim_{x \to -\infty}\left(\frac{\left(\tan{\left(x \right)} + \cot{\left(\frac{x}{2} \right)}\right) + \tan{\left(\frac{2 x}{3} \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((tan⁡(x)+cot⁡(x2))+tan⁡(2x3)x)y = x \lim_{x \to \infty}\left(\frac{\left(\tan{\left(x \right)} + \cot{\left(\frac{x}{2} \right)}\right) + \tan{\left(\frac{2 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:
(tan⁡(x)+cot⁡(x2))+tan⁡(2x3)=−tan⁡(2x3)−tan⁡(x)−cot⁡(x2)\left(\tan{\left(x \right)} + \cot{\left(\frac{x}{2} \right)}\right) + \tan{\left(\frac{2 x}{3} \right)} = - \tan{\left(\frac{2 x}{3} \right)} - \tan{\left(x \right)} - \cot{\left(\frac{x}{2} \right)}
- No
(tan⁡(x)+cot⁡(x2))+tan⁡(2x3)=tan⁡(2x3)+tan⁡(x)+cot⁡(x2)\left(\tan{\left(x \right)} + \cot{\left(\frac{x}{2} \right)}\right) + \tan{\left(\frac{2 x}{3} \right)} = \tan{\left(\frac{2 x}{3} \right)} + \tan{\left(x \right)} + \cot{\left(\frac{x}{2} \right)}
- No
es decir, función
no es
par ni impar
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