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Trazado el gráfico de la función/ xsinx/(4+tg^2(x))

Gráfico de la función y = xsinx/(4+tg^2(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         x*sin(x) 
f(x) = -----------
              2   
       4 + tan (x)
f(x)=xsin⁡(x)tan⁡2(x)+4f{\left(x \right)} = \frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4} 
f = (x*sin(x))/(tan(x)^2 + 4)
Gráfico de la función
02468-8-6-4-2-10102.5-2.5
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⁡(x)tan⁡2(x)+4=0\frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=65.9734457253857x_{1} = 65.9734457253857
x2=−89.5353907014139x_{2} = -89.5353907014139
x3=−21.9911485751286x_{3} = -21.9911485751286
x4=21.9911485751286x_{4} = 21.9911485751286
x5=−3.14159265358979x_{5} = -3.14159265358979
x6=−15.707963267949x_{6} = -15.707963267949
x7=9.42477796076938x_{7} = 9.42477796076938
x8=6.28318530717959x_{8} = 6.28318530717959
x9=−34.5575191894877x_{9} = -34.5575191894877
x10=−73.8274272831926x_{10} = -73.8274272831926
x11=−67.544242126936x_{11} = -67.544242126936
x12=64.4026493290031x_{12} = 64.4026493290031
x13=28.2743338823081x_{13} = 28.2743338823081
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=−7.85398152870528x_{15} = -7.85398152870528
x16=−53.4070751110265x_{16} = -53.4070751110265
x17=84.8230016469244x_{17} = 84.8230016469244
x18=−45.5530935521522x_{18} = -45.5530935521522
x19=86.3937979058448x_{19} = 86.3937979058448
x20=29.8451302874799x_{20} = 29.8451302874799
x21=42.4115006809338x_{21} = 42.4115006809338
x22=42.4115007525711x_{22} = 42.4115007525711
x23=−1.5707964157513x_{23} = -1.5707964157513
x24=59.6902604182061x_{24} = 59.6902604182061
x25=56.5486677646163x_{25} = 56.5486677646163
x26=−23.5619449773569x_{26} = -23.5619449773569
x27=72.2566310325652x_{27} = 72.2566310325652
x28=−50.2654824574367x_{28} = -50.2654824574367
x29=95.8185759189139x_{29} = 95.8185759189139
x30=−95.818575869486x_{30} = -95.818575869486
x31=78.5398163397448x_{31} = 78.5398163397448
x32=−87.9645943005142x_{32} = -87.9645943005142
x33=37.6991118430775x_{33} = 37.6991118430775
x34=−6.28318530717959x_{34} = -6.28318530717959
x35=−37.6991118430775x_{35} = -37.6991118430775
x36=−43.9822971502571x_{36} = -43.9822971502571
x37=47.1238898038469x_{37} = 47.1238898038469
x38=−14.1371668678372x_{38} = -14.1371668678372
x39=20.42035217694x_{39} = 20.42035217694
x40=−100.530964914873x_{40} = -100.530964914873
x41=3.14159265358979x_{41} = 3.14159265358979
x42=−72.2566310325652x_{42} = -72.2566310325652
x43=−81.6814089933346x_{43} = -81.6814089933346
x44=−65.9734457253857x_{44} = -65.9734457253857
x45=0x_{45} = 0
x46=−28.2743338823081x_{46} = -28.2743338823081
x47=−56.5486677646163x_{47} = -56.5486677646163
x48=1.57079626923918x_{48} = 1.57079626923918
x49=7.85398171361397x_{49} = 7.85398171361397
x50=31.4159265358979x_{50} = 31.4159265358979
x51=43.9822971502571x_{51} = 43.9822971502571
x52=−47.1238898038469x_{52} = -47.1238898038469
x53=100.530964914873x_{53} = 100.530964914873
x54=−97.3893722612836x_{54} = -97.3893722612836
x55=−51.8362786963082x_{55} = -51.8362786963082
x56=81.6814089933346x_{56} = 81.6814089933346
x57=−75.398223686155x_{57} = -75.398223686155
x58=−78.5398163397448x_{58} = -78.5398163397448
x59=95.8185760136054x_{59} = 95.8185760136054
x60=50.2654824574367x_{60} = 50.2654824574367
x61=94.2477796076938x_{61} = 94.2477796076938
x62=73.8274274384211x_{62} = 73.8274274384211
x63=−23.5619448833633x_{63} = -23.5619448833633
x64=−59.6902604182061x_{64} = -59.6902604182061
x65=12.5663706143592x_{65} = 12.5663706143592
x66=14.1371670350454x_{66} = 14.1371670350454
x67=34.5575191894877x_{67} = 34.5575191894877
x68=−29.8451301097218x_{68} = -29.8451301097218
x69=−58.1194640203505x_{69} = -58.1194640203505
x70=51.836278862996x_{70} = 51.836278862996
x71=15.707963267949x_{71} = 15.707963267949
x72=87.9645943005142x_{72} = 87.9645943005142
x73=−80.1106125976626x_{73} = -80.1106125976626
x74=−9.42477796076938x_{74} = -9.42477796076938
x75=−36.128315443513x_{75} = -36.128315443513
x76=−31.4159265358979x_{76} = -31.4159265358979
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))/(4 + tan(x)^2).
0sin⁡(0)tan⁡2(0)+4\frac{0 \sin{\left(0 \right)}}{\tan^{2}{\left(0 \right)} + 4}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
−x(2tan⁡2(x)+2)sin⁡(x)tan⁡(x)(tan⁡2(x)+4)2+xcos⁡(x)+sin⁡(x)tan⁡2(x)+4=0- \frac{x \left(2 \tan^{2}{\left(x \right)} + 2\right) \sin{\left(x \right)} \tan{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 4\right)^{2}} + \frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=87.138945811754x_{1} = 87.138945811754
x2=−90.2804556957995x_{2} = -90.2804556957995
x3=24.3132699931952x_{3} = 24.3132699931952
x4=18.0330900698586x_{4} = 18.0330900698586
x5=−29.845130209103x_{5} = -29.845130209103
x6=−98.2194935076171x_{6} = -98.2194935076171
x7=1.5707963267949x_{7} = 1.5707963267949
x8=4.71238898038469x_{8} = 4.71238898038469
x9=36.1283155162826x_{9} = 36.1283155162826
x10=−85.65343015591x_{10} = -85.65343015591
x11=2.40467079662226x_{11} = 2.40467079662226
x12=83.997442131108x_{12} = 83.997442131108
x13=5.49385370718343x_{13} = 5.49385370718343
x14=−32.2503175803598x_{14} = -32.2503175803598
x15=60.5216851226385x_{15} = 60.5216851226385
x16=−80.1106126665397x_{16} = -80.1106126665397
x17=−95.8185759344887x_{17} = -95.8185759344887
x18=38.5324692900873x_{18} = 38.5324692900873
x19=454.703366375065x_{19} = 454.703366375065
x20=11.7561606649435x_{20} = 11.7561606649435
x21=99.7050166950322x_{21} = 99.7050166950322
x22=−76.2289489691309x_{22} = -76.2289489691309
x23=67.5442420521806x_{23} = 67.5442420521806
x24=10.2725607549115x_{24} = 10.2725607549115
x25=69.9460058299699x_{25} = 69.9460058299699
x26=70.6858347057703x_{26} = 70.6858347057703
x27=−36.8767140943411x_{27} = -36.8767140943411
x28=−46.3003430565376x_{28} = -46.3003430565376
x29=54.238892641524x_{29} = 54.238892641524
x30=−5.49385370718343x_{30} = -5.49385370718343
x31=0.997817540674201x_{31} = 0.997817540674201
x32=−51.8362787842316x_{32} = -51.8362787842316
x33=58.1194640914112x_{33} = 58.1194640914112
x34=73.0874722075759x_{34} = 73.0874722075759
x35=−73.8274273593601x_{35} = -73.8274273593601
x36=4.01872293357174x_{36} = 4.01872293357174
x37=16.5483332246309x_{37} = 16.5483332246309
x38=−10.2725607549115x_{38} = -10.2725607549115
x39=−99.7050166950322x_{39} = -99.7050166950322
x40=−4.01872293357174x_{40} = -4.01872293357174
x41=27.453878484238x_{41} = 27.453878484238
x42=80.1106126665397x_{42} = 80.1106126665397
x43=46.3003430565376x_{43} = 46.3003430565376
x44=−16.5483332246309x_{44} = -16.5483332246309
x45=−47.9562027081837x_{45} = -47.9562027081837
x46=90.2804556957995x_{46} = 90.2804556957995
x47=98.2194935076171x_{47} = 98.2194935076171
x48=76.2289489691309x_{48} = 76.2289489691309
x49=0x_{49} = 0
x50=−91.9364513523705x_{50} = -91.9364513523705
x51=33.735647763708x_{51} = 33.735647763708
x52=−7.85398163397448x_{52} = -7.85398163397448
x53=−27.453878484238x_{53} = -27.453878484238
x54=−11.7561606649435x_{54} = -11.7561606649435
x55=−58.865755516954x_{55} = -58.865755516954
x56=−93.4219711563462x_{56} = -93.4219711563462
x57=−83.997442131108x_{57} = -83.997442131108
x58=−54.238892641524x_{58} = -54.238892641524
x59=−62.0071694687546x_{59} = -62.0071694687546
x60=−68.2900468139916x_{60} = -68.2900468139916
x61=32.2503175803598x_{61} = 32.2503175803598
x62=−69.9460058299699x_{62} = -69.9460058299699
x63=−33.735647763708x_{63} = -33.735647763708
x64=−55.7243617726486x_{64} = -55.7243617726486
x65=−19.6879744764593x_{65} = -19.6879744764593
x66=14.1371669411541x_{66} = 14.1371669411541
x67=−71.4315058489604x_{67} = -71.4315058489604
x68=55.7243617726486x_{68} = 55.7243617726486
x69=−25.9686624297459x_{69} = -25.9686624297459
x70=62.0071694687546x_{70} = 62.0071694687546
x71=−63.663110358029x_{71} = -63.663110358029
x72=77.7144563954775x_{72} = 77.7144563954775
x73=45.553093477052x_{73} = 45.553093477052
x74=25.9686624297459x_{74} = 25.9686624297459
x75=−18.0330900698586x_{75} = -18.0330900698586
x76=−40.0178633797347x_{76} = -40.0178633797347
x77=−41.6736614988808x_{77} = -41.6736614988808
x78=92.6769832808989x_{78} = 92.6769832808989
x79=82.5119289102023x_{79} = 82.5119289102023
x80=40.0178633797347x_{80} = 40.0178633797347
x81=−7.13941996437343x_{81} = -7.13941996437343
x82=−49.4416503630997x_{82} = -49.4416503630997
x83=91.9364513523705x_{83} = 91.9364513523705
x84=47.9562027081837x_{84} = 47.9562027081837
x85=−77.7144563954775x_{85} = -77.7144563954775
x86=68.2900468139916x_{86} = 68.2900468139916
Signos de extremos en los puntos:
(87.13894581175403, -12.3763087646776)

(-90.28045569579949, 12.8225080937804)

(24.313269993195227, -3.45264222691292)

(18.033090069858616, -2.56044421510836)

(-29.845130209103036, -1.12127665170554e-29)

(-98.21949350761713, -13.9501156336138)

(1.5707963267948966, 5.8895428941999e-33)

(4.71238898038469, -1.59017658143397e-31)

(36.12831551628262, -3.66424875021481e-28)

(-85.65343015590999, -12.1653166830211)

(2.4046707966222645, 0.335020556538555)

(83.99744213110796, 11.9301098752818)

(5.493853707183429, -0.777532226748693)

(-32.2503175803598, 4.58011040647008)

(60.52168512263849, -8.59574594780144)

(-80.11061266653972, -1.92283264304371e-27)

(-95.81857593448869, 3.676520165044e-28)

(38.53246929008733, 5.47244715183819)

(454.7033663750651, 64.582180890724)

(11.756160664943506, -1.66847511356391)

(99.70501669503224, -14.1611083032433)

(-76.2289489691309, 10.8267221900235)

(67.54424205218055, -1.3132184568469e-27)

(10.27256075491152, -1.45763299397885)

(69.9460058299699, 9.93432901594521)

(70.68583470577035, 6.77618297499813e-29)

(-36.87671409434108, -5.23725594007668)

(-46.30034305653763, 6.57579015556219)

(54.238892641523954, -7.70336350275712)

(-5.493853707183429, -0.777532226748693)

(0.9978175406742009, 0.130960306067674)

(-51.83627878423159, 3.09398107171563e-30)

(58.119464091411174, 1.39112146798308e-29)

(73.08747220757594, -10.3805252333763)

(-73.82742735936014, -4.43565443427593e-28)

(4.018722933571741, -0.567362924067202)

(16.54833322463088, -2.34951143571411)

(-10.27256075491152, -1.45763299397885)

(-99.70501669503224, -14.1611083032433)

(-4.018722933571741, -0.567362924067202)

(27.453878484238032, 3.89877781647259)

(80.11061266653972, -1.92283264304371e-27)

(46.30034305653763, 6.57579015556219)

(-16.54833322463088, -2.34951143571411)

(-47.956202708183724, -6.81098836459494)

(90.28045569579949, 12.8225080937804)

(98.21949350761713, -13.9501156336138)

(76.2289489691309, 10.8267221900235)

(0, 0)

(-91.93645135237047, -13.0577154126865)

(33.73564776370796, 4.7910880047921)

(-7.853981633974483, 7.36192861774987e-31)

(-27.453878484238032, 3.89877781647259)

(-11.756160664943506, -1.66847511356391)

(-58.86575551695403, 8.36054305299913)

(-93.42197115634625, -13.2687078181799)

(-83.99744213110796, 11.9301098752818)

(-54.238892641523954, -7.70336350275712)

(-62.0071694687546, -8.80673558504463)

(-68.29004681399161, -9.69912415120835)

(32.2503175803598, 4.58011040647008)

(-69.9460058299699, 9.93432901594521)

(-33.73564776370796, 4.7910880047921)

(-55.72436177264862, -7.91435195205804)

(-19.687974476459267, 2.79557707373584)

(14.137166941154069, 4.29347676987172e-30)

(-71.43150584896037, 10.1453198768279)

(55.72436177264862, -7.91435195205804)

(-25.968662429745905, 3.68780913408032)

(62.0071694687546, -8.80673558504463)

(-63.66311035802902, 9.04193923563947)

(77.71445639547751, 11.0377136292074)

(45.553093477052, 1.74530768724744e-35)

(25.968662429745905, 3.68780913408032)

(-18.033090069858616, -2.56044421510836)

(-40.01786337973469, 5.68342974137886)

(-41.67366149888085, -5.91862382854916)

(92.6769832808989, -2.69152684487792e-27)

(82.51192891020227, 11.7191179837181)

(40.01786337973469, 5.68342974137886)

(-7.139419964373433, 1.012038154624)

(-49.441650363099725, -7.02197513700988)

(91.93645135237047, -13.0577154126865)

(47.956202708183724, -6.81098836459494)

(-77.71445639547751, 11.0377136292074)

(68.29004681399161, -9.69912415120835)


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=87.138945811754x_{1} = 87.138945811754
x2=24.3132699931952x_{2} = 24.3132699931952
x3=18.0330900698586x_{3} = 18.0330900698586
x4=−98.2194935076171x_{4} = -98.2194935076171
x5=1.5707963267949x_{5} = 1.5707963267949
x6=−85.65343015591x_{6} = -85.65343015591
x7=5.49385370718343x_{7} = 5.49385370718343
x8=60.5216851226385x_{8} = 60.5216851226385
x9=−95.8185759344887x_{9} = -95.8185759344887
x10=11.7561606649435x_{10} = 11.7561606649435
x11=99.7050166950322x_{11} = 99.7050166950322
x12=10.2725607549115x_{12} = 10.2725607549115
x13=70.6858347057703x_{13} = 70.6858347057703
x14=−36.8767140943411x_{14} = -36.8767140943411
x15=54.238892641524x_{15} = 54.238892641524
x16=−5.49385370718343x_{16} = -5.49385370718343
x17=−51.8362787842316x_{17} = -51.8362787842316
x18=58.1194640914112x_{18} = 58.1194640914112
x19=73.0874722075759x_{19} = 73.0874722075759
x20=4.01872293357174x_{20} = 4.01872293357174
x21=16.5483332246309x_{21} = 16.5483332246309
x22=−10.2725607549115x_{22} = -10.2725607549115
x23=−99.7050166950322x_{23} = -99.7050166950322
x24=−4.01872293357174x_{24} = -4.01872293357174
x25=−16.5483332246309x_{25} = -16.5483332246309
x26=−47.9562027081837x_{26} = -47.9562027081837
x27=98.2194935076171x_{27} = 98.2194935076171
x28=0x_{28} = 0
x29=−91.9364513523705x_{29} = -91.9364513523705
x30=−7.85398163397448x_{30} = -7.85398163397448
x31=−11.7561606649435x_{31} = -11.7561606649435
x32=−93.4219711563462x_{32} = -93.4219711563462
x33=−54.238892641524x_{33} = -54.238892641524
x34=−62.0071694687546x_{34} = -62.0071694687546
x35=−68.2900468139916x_{35} = -68.2900468139916
x36=−55.7243617726486x_{36} = -55.7243617726486
x37=14.1371669411541x_{37} = 14.1371669411541
x38=55.7243617726486x_{38} = 55.7243617726486
x39=62.0071694687546x_{39} = 62.0071694687546
x40=45.553093477052x_{40} = 45.553093477052
x41=−18.0330900698586x_{41} = -18.0330900698586
x42=−41.6736614988808x_{42} = -41.6736614988808
x43=−49.4416503630997x_{43} = -49.4416503630997
x44=91.9364513523705x_{44} = 91.9364513523705
x45=47.9562027081837x_{45} = 47.9562027081837
x46=68.2900468139916x_{46} = 68.2900468139916
Puntos máximos de la función:
x46=−90.2804556957995x_{46} = -90.2804556957995
x46=−29.845130209103x_{46} = -29.845130209103
x46=4.71238898038469x_{46} = 4.71238898038469
x46=36.1283155162826x_{46} = 36.1283155162826
x46=2.40467079662226x_{46} = 2.40467079662226
x46=83.997442131108x_{46} = 83.997442131108
x46=−32.2503175803598x_{46} = -32.2503175803598
x46=−80.1106126665397x_{46} = -80.1106126665397
x46=38.5324692900873x_{46} = 38.5324692900873
x46=454.703366375065x_{46} = 454.703366375065
x46=−76.2289489691309x_{46} = -76.2289489691309
x46=67.5442420521806x_{46} = 67.5442420521806
x46=69.9460058299699x_{46} = 69.9460058299699
x46=−46.3003430565376x_{46} = -46.3003430565376
x46=0.997817540674201x_{46} = 0.997817540674201
x46=−73.8274273593601x_{46} = -73.8274273593601
x46=27.453878484238x_{46} = 27.453878484238
x46=80.1106126665397x_{46} = 80.1106126665397
x46=46.3003430565376x_{46} = 46.3003430565376
x46=90.2804556957995x_{46} = 90.2804556957995
x46=76.2289489691309x_{46} = 76.2289489691309
x46=33.735647763708x_{46} = 33.735647763708
x46=−27.453878484238x_{46} = -27.453878484238
x46=−58.865755516954x_{46} = -58.865755516954
x46=−83.997442131108x_{46} = -83.997442131108
x46=32.2503175803598x_{46} = 32.2503175803598
x46=−69.9460058299699x_{46} = -69.9460058299699
x46=−33.735647763708x_{46} = -33.735647763708
x46=−19.6879744764593x_{46} = -19.6879744764593
x46=−71.4315058489604x_{46} = -71.4315058489604
x46=−25.9686624297459x_{46} = -25.9686624297459
x46=−63.663110358029x_{46} = -63.663110358029
x46=77.7144563954775x_{46} = 77.7144563954775
x46=25.9686624297459x_{46} = 25.9686624297459
x46=−40.0178633797347x_{46} = -40.0178633797347
x46=92.6769832808989x_{46} = 92.6769832808989
x46=82.5119289102023x_{46} = 82.5119289102023
x46=40.0178633797347x_{46} = 40.0178633797347
x46=−7.13941996437343x_{46} = -7.13941996437343
x46=−77.7144563954775x_{46} = -77.7144563954775
Decrece en los intervalos
[99.7050166950322,∞)\left[99.7050166950322, \infty\right)
Crece en los intervalos
(−∞,−99.7050166950322]\left(-\infty, -99.7050166950322\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→−∞(xsin⁡(x)tan⁡2(x)+4)y = \lim_{x \to -\infty}\left(\frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}\right)
True

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

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(sin⁡(x)tan⁡2(x)+4)y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}\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:
xsin⁡(x)tan⁡2(x)+4=xsin⁡(x)tan⁡2(x)+4\frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4} = \frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}
- Sí
xsin⁡(x)tan⁡2(x)+4=−xsin⁡(x)tan⁡2(x)+4\frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4} = - \frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор