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Trazado el gráfico de la función/ sin(x)^3*((-atan(x))/3+x/3)

Gráfico de la función y = sin(x)^3*((-atan(x))/3+x/3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          3    /-atan(x)    x\
f(x) = sin (x)*|--------- + -|
               \    3       3/
f(x)=(x3+(−1)atan⁡(x)3)sin⁡3(x)f{\left(x \right)} = \left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} 
f = (x/3 + (-atan(x))/3)*sin(x)^3
Gráfico de la función
02468-8-6-4-2-10105-5
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:
(x3+(−1)atan⁡(x)3)sin⁡3(x)=0\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=28.274327539771x_{1} = 28.274327539771
x2=21.9911516425052x_{2} = 21.9911516425052
x3=43.9823032542865x_{3} = 43.9823032542865
x4=−12.5664316443175x_{4} = -12.5664316443175
x5=−59.690275753337x_{5} = -59.690275753337
x6=−56.5486761233879x_{6} = -56.5486761233879
x7=−81.6813767358288x_{7} = -81.6813767358288
x8=72.2566292958367x_{8} = 72.2566292958367
x9=−9.42485171603722x_{9} = -9.42485171603722
x10=−78.5398074796132x_{10} = -78.5398074796132
x11=−59.6902445763065x_{11} = -59.6902445763065
x12=−15.7079741677484x_{12} = -15.7079741677484
x13=−34.557546390117x_{13} = -34.557546390117
x14=94.2477801894889x_{14} = 94.2477801894889
x15=9.42482919322591x_{15} = 9.42482919322591
x16=65.9734548151346x_{16} = 65.9734548151346
x17=87.9646063132303x_{17} = 87.9646063132303
x18=6.28325837390066x_{18} = 6.28325837390066
x19=−37.6991249695058x_{19} = -37.6991249695058
x20=31.4158973414617x_{20} = 31.4158973414617
x21=−3.14147637996428x_{21} = -3.14147637996428
x22=−6.283115394968x_{22} = -6.283115394968
x23=−43.9823032321748x_{23} = -43.9823032321748
x24=−21.9911516410072x_{24} = -21.9911516410072
x25=3.14163928523582x_{25} = 3.14163928523582
x26=0x_{26} = 0
x27=12.5663013873734x_{27} = 12.5663013873734
x28=−81.6814265139205x_{28} = -81.6814265139205
x29=−65.9734547046097x_{29} = -65.9734547046097
x30=50.2654784098211x_{30} = 50.2654784098211
x31=3.14157419664107x_{31} = 3.14157419664107
x32=6.28317671486296x_{32} = 6.28317671486296
x33=−87.9646059653208x_{33} = -87.9646059653208
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)^3*((-atan(x))/3 + x/3).
((−1)atan⁡(0)3+03)sin⁡3(0)\left(\frac{\left(-1\right) \operatorname{atan}{\left(0 \right)}}{3} + \frac{0}{3}\right) \sin^{3}{\left(0 \right)}
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
(13−13(x2+1))sin⁡3(x)+3(x3+(−1)atan⁡(x)3)sin⁡2(x)cos⁡(x)=0\left(\frac{1}{3} - \frac{1}{3 \left(x^{2} + 1\right)}\right) \sin^{3}{\left(x \right)} + 3 \left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−47.1238897398428x_{1} = -47.1238897398428
x2=72.2566310277228x_{2} = 72.2566310277228
x3=15.7079634238123x_{3} = 15.7079634238123
x4=97.3893722386395x_{4} = 97.3893722386395
x5=−42.4196515709562x_{5} = -42.4196515709562
x6=14.1633536976303x_{6} = 14.1633536976303
x7=−43.982297174703x_{7} = -43.982297174703
x8=95.8221117869534x_{8} = 95.8221117869534
x9=34.5575190521566x_{9} = 34.5575190521566
x10=−36.1379432026438x_{10} = -36.1379432026438
x11=37.6991119965993x_{11} = 37.6991119965993
x12=40.8407044980859x_{12} = 40.8407044980859
x13=−34.5575191251925x_{13} = -34.5575191251925
x14=12.5663704809964x_{14} = 12.5663704809964
x15=−51.8429042612973x_{15} = -51.8429042612973
x16=36.1379432026438x_{16} = 36.1379432026438
x17=−7.9047266518689x_{17} = -7.9047266518689
x18=−78.5398162431523x_{18} = -78.5398162431523
x19=56.5486676281888x_{19} = 56.5486676281888
x20=−59.6902604573859x_{20} = -59.6902604573859
x21=−100.530964808226x_{21} = -100.530964808226
x22=−53.407075259837x_{22} = -53.407075259837
x23=18.8495559923432x_{23} = 18.8495559923432
x24=−29.8568869013277x_{24} = -29.8568869013277
x25=−31.4159266845618x_{25} = -31.4159266845618
x26=−12.5663706164231x_{26} = -12.5663706164231
x27=−65.9734457652762x_{27} = -65.9734457652762
x28=20.4379300963978x_{28} = 20.4379300963978
x29=58.1253545010966x_{29} = 58.1253545010966
x30=6.28318528466365x_{30} = 6.28318528466365
x31=−23.577034707719x_{31} = -23.577034707719
x32=−9.42477811595977x_{32} = -9.42477811595977
x33=−3.14159285481255x_{33} = -3.14159285481255
x34=4.80501573129414x_{34} = 4.80501573129414
x35=−95.8221117869534x_{35} = -95.8221117869534
x36=−87.9645943592141x_{36} = -87.9645943592141
x37=−45.5606634099038x_{37} = -45.5606634099038
x38=94.2477796093528x_{38} = 94.2477796093528
x39=−80.1148552083624x_{39} = -80.1148552083624
x40=−43.9822972046548x_{40} = -43.9822972046548
x41=73.8320385087492x_{41} = 73.8320385087492
x42=21.9911485851914x_{42} = 21.9911485851914
x43=−15.7079632965462x_{43} = -15.7079632965462
x44=51.8429042612973x_{44} = 51.8429042612973
x45=−6.28318518053504x_{45} = -6.28318518053504
x46=−21.9911485864908x_{46} = -21.9911485864908
x47=−89.5391788958977x_{47} = -89.5391788958977
x48=−73.8320385087492x_{48} = -73.8320385087492
x49=62.8318530430108x_{49} = 62.8318530430108
x50=64.4079514776374x_{50} = 64.4079514776374
x51=75.398223692586x_{51} = 75.398223692586
x52=53.4070751439253x_{52} = 53.4070751439253
x53=7.9047266518689x_{53} = 7.9047266518689
x54=0x_{54} = 0
x55=−25.1327412189989x_{55} = -25.1327412189989
x56=80.1148552083624x_{56} = 80.1148552083624
x57=28.2743338652787x_{57} = 28.2743338652787
x58=−67.549291920442x_{58} = -67.549291920442
x59=−14.1633536976303x_{59} = -14.1633536976303
x60=65.9734457527599x_{60} = 65.9734457527599
x61=−69.1150382652533x_{61} = -69.1150382652533
x62=−1.88455017054346x_{62} = -1.88455017054346
x63=1.88455017054346x_{63} = 1.88455017054346
x64=9.42477807889789x_{64} = 9.42477807889789
x65=92.680640992997x_{65} = 92.680640992997
x66=−87.9645942994217x_{66} = -87.9645942994217
x67=−28.2743337405708x_{67} = -28.2743337405708
x68=84.8230015969263x_{68} = 84.8230015969263
x69=−37.6991118770157x_{69} = -37.6991118770157
x70=−81.6814090376207x_{70} = -81.6814090376207
x71=−58.1253545010966x_{71} = -58.1253545010966
x72=87.9645943354964x_{72} = 87.9645943354964
x73=−50.2654823161497x_{73} = -50.2654823161497
x74=43.9822971693753x_{74} = 43.9822971693753
x75=29.8568869013277x_{75} = 29.8568869013277
x76=31.41592659636x_{76} = 31.41592659636
x77=86.397726460829x_{77} = 86.397726460829
x78=42.4196515709562x_{78} = 42.4196515709562
x79=−56.5486676809069x_{79} = -56.5486676809069
x80=50.2654824463717x_{80} = 50.2654824463717
x81=21.9911485108066x_{81} = 21.9911485108066
Signos de extremos en los puntos:
(-47.12388973984283, 3.98310346097366e-21)

(72.25663102772283, 2.67597069160318e-24)

(15.707963423812345, -1.79234694014091e-20)

(97.38937223863952, 3.70884132285647e-22)

(-42.419651570956205, -13.622783998703)

(14.163353697630292, 4.21667496713117)

(-43.982297174703014, 2.06639646233012e-22)

(95.82211178695336, 31.4199944574926)

(34.557519052156564, 2.85040058445483e-20)

(-36.13794320264376, -11.5300006037631)

(37.699111996599285, 4.36068792687663e-20)

(40.84070449808586, -3.73890091284254e-26)

(-34.55751912519251, 2.92505830313435e-21)

(12.566370480996397, -8.75640298167247e-21)

(-51.842904261297335, 16.7626944041327)

(36.13794320264376, -11.5300006037631)

(-7.904726651868903, 2.14495149966393)

(-78.53981624315229, 2.31258304446127e-20)

(56.54866762818882, -4.65491461069023e-20)

(-59.69026045738589, -1.16550125376709e-21)

(-100.5309648082264, -4.00155274285572e-20)

(-53.40707525983695, -5.69599789226226e-20)

(18.849555992343227, 2.05070614248904e-21)

(-29.85688690132773, -9.43790005122283)

(-31.41592668456182, 3.27213702463331e-20)

(-12.566370616423054, 3.24545686181302e-26)

(-65.97344576527618, -1.36299383969224e-21)

(20.43793009639784, 6.30241932128866)

(58.12535450109656, 18.8562721445462)

(6.283185284663652, -1.8530931632127e-23)

(-23.577034707718983, -7.34703244897591)

(-9.424778115959775, -9.91675067455694e-21)

(-3.1415928548125533, -5.10303885725562e-21)

(4.805015731294137, -1.13178788432727)

(-95.82211178695336, 31.4199944574926)

(-87.96459435921413, 5.82547059105447e-21)

(-45.560663409903775, 14.6693431242966)

(94.24777960935283, 1.41080617543143e-25)

(-80.1148552083624, -26.1848064691713)

(-43.98229720465481, 2.276863022088e-21)

(73.83203850874915, -24.0908268341739)

(21.991148585191436, -6.95143729038979e-24)

(-15.707963296546192, -1.1070354004941e-22)

(51.842904261297335, 16.7626944041327)

(-6.28318518053504, -3.29751523230015e-21)

(-21.991148586490784, -1.00068687626426e-23)

(-89.53917889589773, 29.3258855090146)

(-73.83203850874915, -24.0908268341739)

(62.83185304301077, -4.87168450060204e-22)

(64.40795147763735, 20.9500098681499)

(75.39822369258601, 6.54644327525439e-24)

(53.40707514392534, -6.15476868236516e-22)

(7.904726651868903, 2.14495149966393)

(0, 0)

(-25.132741218998916, -7.22344924354435e-24)

(80.1148552083624, -26.1848064691713)

(28.27433386527867, 4.40175054211864e-23)

(-67.54929192044199, -21.996924731538)

(-14.163353697630292, 4.21667496713117)

(65.97344575275994, -4.40466653095097e-22)

(-69.11503826525328, -3.31203997990776e-20)

(-1.8845501705434633, 0.22994789474801)

(1.8845501705434633, 0.22994789474801)

(9.424778078897885, -4.37359944257925e-21)

(92.68064099299698, -30.3729351221931)

(-87.96459429942173, -3.75539371192506e-26)

(-28.27433374057076, 2.53790472959541e-20)

(84.82300159692633, 3.46893455991417e-21)

(-37.699111877015746, 4.71099422940883e-22)

(-81.68140903762065, 2.31971907254417e-21)

(-58.12535450109656, 18.8562721445462)

(87.96459433549641, 1.23299113518815e-21)

(-50.26548231614965, -4.57978063086762e-20)

(43.98229716937529, 9.88405807000468e-23)

(29.85688690132773, -9.43790005122283)

(31.415926596359974, 2.20121989284558e-21)

(86.39772646082896, -28.2788466810656)

(42.419651570956205, -13.622783998703)

(-56.54866768090694, -1.07529612384756e-20)

(50.26548244637171, -2.19983144722745e-23)

(21.99114851080661, 1.81545208169053e-21)


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=−42.4196515709562x_{1} = -42.4196515709562
x2=−36.1379432026438x_{2} = -36.1379432026438
x3=36.1379432026438x_{3} = 36.1379432026438
x4=−29.8568869013277x_{4} = -29.8568869013277
x5=−23.577034707719x_{5} = -23.577034707719
x6=4.80501573129414x_{6} = 4.80501573129414
x7=−80.1148552083624x_{7} = -80.1148552083624
x8=73.8320385087492x_{8} = 73.8320385087492
x9=−73.8320385087492x_{9} = -73.8320385087492
x10=0x_{10} = 0
x11=80.1148552083624x_{11} = 80.1148552083624
x12=−67.549291920442x_{12} = -67.549291920442
x13=92.680640992997x_{13} = 92.680640992997
x14=29.8568869013277x_{14} = 29.8568869013277
x15=86.397726460829x_{15} = 86.397726460829
x16=42.4196515709562x_{16} = 42.4196515709562
Puntos máximos de la función:
x16=14.1633536976303x_{16} = 14.1633536976303
x16=95.8221117869534x_{16} = 95.8221117869534
x16=−51.8429042612973x_{16} = -51.8429042612973
x16=−7.9047266518689x_{16} = -7.9047266518689
x16=20.4379300963978x_{16} = 20.4379300963978
x16=58.1253545010966x_{16} = 58.1253545010966
x16=−95.8221117869534x_{16} = -95.8221117869534
x16=−45.5606634099038x_{16} = -45.5606634099038
x16=51.8429042612973x_{16} = 51.8429042612973
x16=−89.5391788958977x_{16} = -89.5391788958977
x16=64.4079514776374x_{16} = 64.4079514776374
x16=7.9047266518689x_{16} = 7.9047266518689
x16=−14.1633536976303x_{16} = -14.1633536976303
x16=−1.88455017054346x_{16} = -1.88455017054346
x16=1.88455017054346x_{16} = 1.88455017054346
x16=−58.1253545010966x_{16} = -58.1253545010966
Decrece en los intervalos
[92.680640992997,∞)\left[92.680640992997, \infty\right)
Crece en los intervalos
(−∞,−80.1148552083624]\left(-\infty, -80.1148552083624\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
(2xsin⁡2(x)3(x2+1)2+2(1−1x2+1)sin⁡(x)cos⁡(x)−(x−atan⁡(x))(sin⁡2(x)−2cos⁡2(x)))sin⁡(x)=0\left(\frac{2 x \sin^{2}{\left(x \right)}}{3 \left(x^{2} + 1\right)^{2}} + 2 \left(1 - \frac{1}{x^{2} + 1}\right) \sin{\left(x \right)} \cos{\left(x \right)} - \left(x - \operatorname{atan}{\left(x \right)}\right) \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right)\right) \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−54.3686863927772x_{1} = -54.3686863927772
x2=−50.2654824574367x_{2} = -50.2654824574367
x3=−5.40936978846495x_{3} = -5.40936978846495
x4=−2.4141722102196x_{4} = -2.4141722102196
x5=85.782269852414x_{5} = 85.782269852414
x6=−19.8229153148452x_{6} = -19.8229153148452
x7=−55.599529764795x_{7} = -55.599529764795
x8=76.35798871949x_{8} = 76.35798871949
x9=−39.8941003995144x_{9} = -39.8941003995144
x10=72.2566310325652x_{10} = 72.2566310325652
x11=−83.8717397472183x_{11} = -83.8717397472183
x12=−43.9822971502571x_{12} = -43.9822971502571
x13=0.000406608608637562x_{13} = 0.000406608608637562
x14=−63.7925138805732x_{14} = -63.7925138805732
x15=−37.6991118430775x_{15} = -37.6991118430775
x16=−12.5663706143592x_{16} = -12.5663706143592
x17=−94.2477796076938x_{17} = -94.2477796076938
x18=−68.1647337177157x_{18} = -68.1647337177157
x19=−21.9911485751286x_{19} = -21.9911485751286
x20=50.2654824574367x_{20} = 50.2654824574367
x21=−15.707963267949x_{21} = -15.707963267949
x22=21.9911485751286x_{22} = 21.9911485751286
x23=−87.9645943005142x_{23} = -87.9645943005142
x24=−72.2566310325652x_{24} = -72.2566310325652
x25=−24.1921794603074x_{25} = -24.1921794603074
x26=4.20190394940552x_{26} = 4.20190394940552
x27=−4.20190394940552x_{27} = -4.20190394940552
x28=79.4994024460026x_{28} = 79.4994024460026
x29=84.8230016469244x_{29} = 84.8230016469244
x30=96.4375728840975x_{30} = 96.4375728840975
x31=−92.0651813066377x_{31} = -92.0651813066377
x32=68.1647337177157x_{32} = 68.1647337177157
x33=54.3686863927772x_{33} = 54.3686863927772
x34=−13.548906564136x_{34} = -13.548906564136
x35=26.1015373766289x_{35} = 26.1015373766289
x36=−33.6126232938398x_{36} = -33.6126232938398
x37=36.7532857268625x_{37} = 36.7532857268625
x38=30.4721625114502x_{38} = 30.4721625114502
x39=−77.5888898474853x_{39} = -77.5888898474853
x40=−82.6408300049229x_{40} = -82.6408300049229
x41=83.8717397472183x_{41} = 83.8717397472183
x42=−57.5099267847335x_{42} = -57.5099267847335
x43=−46.1760581313926x_{43} = -46.1760581313926
x44=−65.9734457253857x_{44} = -65.9734457253857
x45=−70.0752103496049x_{45} = -70.0752103496049
x46=15.707963267949x_{46} = 15.707963267949
x47=−17.9146341508486x_{47} = -17.9146341508486
x48=28.2743338823081x_{48} = 28.2743338823081
x49=94.2477796076938x_{49} = 94.2477796076938
x50=6.28318530717959x_{50} = 6.28318530717959
x51=32.3819978743262x_{51} = 32.3819978743262
x52=−51.2274903560907x_{52} = -51.2274903560907
x53=41.8042715244945x_{53} = 41.8042715244945
x54=52.4583188066494x_{54} = 52.4583188066494
x55=−79.4994024460026x_{55} = -79.4994024460026
x56=19.8229153148452x_{56} = 19.8229153148452
x57=−75.398223686155x_{57} = -75.398223686155
x58=70.0752103496049x_{58} = 70.0752103496049
x59=−99.5790527132113x_{59} = -99.5790527132113
x60=0x_{60} = 0
x61=61.8820709651112x_{61} = 61.8820709651112
x62=−41.8042715244945x_{62} = -41.8042715244945
x63=−34.5575191894877x_{63} = -34.5575191894877
x64=98.348128249113x_{64} = 98.348128249113
x65=−348.716784548467x_{65} = -348.716784548467
x66=22.9618974346408x_{66} = 22.9618974346408
x67=−28.2743338823081x_{67} = -28.2743338823081
x68=39.8941003995144x_{68} = 39.8941003995144
x69=10.4165024727153x_{69} = 10.4165024727153
x70=46.1760581313926x_{70} = 46.1760581313926
x71=−85.782269852414x_{71} = -85.782269852414
x72=−61.8820709651112x_{72} = -61.8820709651112
x73=8.51661058959281x_{73} = 8.51661058959281
x74=63.7925138805732x_{74} = 63.7925138805732
x75=57.5099267847335x_{75} = 57.5099267847335
x76=−26.1015373766289x_{76} = -26.1015373766289
x77=90.1546372456078x_{77} = 90.1546372456078
x78=−8.51661058959281x_{78} = -8.51661058959281
x79=12.5663706143592x_{79} = 12.5663706143592
x80=87.9645943005142x_{80} = 87.9645943005142
x81=−100.530964914873x_{81} = -100.530964914873
x82=−11.6439446877948x_{82} = -11.6439446877948
x83=43.9822971502571x_{83} = 43.9822971502571
x84=−81.6814089933346x_{84} = -81.6814089933346
x85=−35.522602636666x_{85} = -35.522602636666
x86=24.1921794603074x_{86} = 24.1921794603074
x87=48.086347606213x_{87} = 48.086347606213
x88=−59.6902604182061x_{88} = -59.6902604182061
x89=−48.086347606213x_{89} = -48.086347606213
x90=65.9734457253857x_{90} = 65.9734457253857
x91=74.4474865946004x_{91} = 74.4474865946004
x92=17.9146341508486x_{92} = 17.9146341508486
x93=−90.1546372456078x_{93} = -90.1546372456078
x94=2.4141722102196x_{94} = 2.4141722102196
x95=92.0651813066377x_{95} = 92.0651813066377
x96=−47.1238898038469x_{96} = -47.1238898038469

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
[98.348128249113,∞)\left[98.348128249113, \infty\right)
Convexa en los intervalos
(−∞,−348.716784548467]\left(-\infty, -348.716784548467\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((x3+(−1)atan⁡(x)3)sin⁡3(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞((x3+(−1)atan⁡(x)3)sin⁡3(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)^3*((-atan(x))/3 + x/3), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x3+(−1)atan⁡(x)3)sin⁡3(x)x)=⟨−13,13⟩\lim_{x \to -\infty}\left(\frac{\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}}{x}\right) = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−13,13⟩xy = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle x
lim⁡x→∞((x3+(−1)atan⁡(x)3)sin⁡3(x)x)=⟨−13,13⟩\lim_{x \to \infty}\left(\frac{\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}}{x}\right) = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−13,13⟩xy = \left\langle - \frac{1}{3}, \frac{1}{3}\right\rangle x
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:
(x3+(−1)atan⁡(x)3)sin⁡3(x)=−(−x3+atan⁡(x)3)sin⁡3(x)\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} = - \left(- \frac{x}{3} + \frac{\operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}
- No
(x3+(−1)atan⁡(x)3)sin⁡3(x)=(−x3+atan⁡(x)3)sin⁡3(x)\left(\frac{x}{3} + \frac{\left(-1\right) \operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)} = \left(- \frac{x}{3} + \frac{\operatorname{atan}{\left(x \right)}}{3}\right) \sin^{3}{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор