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Trazado el gráfico de la función/ sin(3*x)^4*atan(2*x)^3

Gráfico de la función y = sin(3*x)^4*atan(2*x)^3

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución analítica
x1=0x_{1} = 0
x2=−2π3x_{2} = - \frac{2 \pi}{3}
x3=−π3x_{3} = - \frac{\pi}{3}
x4=π3x_{4} = \frac{\pi}{3}
x5=2π3x_{5} = \frac{2 \pi}{3}
x6=πx_{6} = \pi
Solución numérica
x1=26.1801496120508x_{1} = 26.1801496120508
x2=−21.9911796977143x_{2} = -21.9911796977143
x3=90.0587669856301x_{3} = 90.0587669856301
x4=−96.342202276036x_{4} = -96.342202276036
x5=−90.0587868567117x_{5} = -90.0587868567117
x6=−64.926143665362x_{6} = -64.926143665362
x7=87.964717895889x_{7} = 87.964717895889
x8=28.2742639284822x_{8} = 28.2742639284822
x9=−68.0676099249042x_{9} = -68.0676099249042
x10=−37.6992569398621x_{10} = -37.6992569398621
x11=−83.7757297758195x_{11} = -83.7757297758195
x12=−31.4160741313719x_{12} = -31.4160741313719
x13=21.9911796911005x_{13} = 21.9911796911005
x14=4.18897101034178x_{14} = 4.18897101034178
x15=−80.6341486197761x_{15} = -80.6341486197761
x16=43.9823593124463x_{16} = 43.9823593124463
x17=−38.7462472898404x_{17} = -38.7462472898404
x18=94.2477860033727x_{18} = 94.2477860033727
x19=50.2654378932694x_{19} = 50.2654378932694
x20=−81.6816027357685x_{20} = -81.6816027357685
x21=−61.7845555812239x_{21} = -61.7845555812239
x22=6.28309012206651x_{22} = 6.28309012206651
x23=−87.9647198294855x_{23} = -87.9647198294855
x24=−15.7080835543402x_{24} = -15.7080835543402
x25=−85.8703547978829x_{25} = -85.8703547978829
x26=−39.7933810638207x_{26} = -39.7933810638207
x27=−59.6904300202992x_{27} = -59.6904300202992
x28=48.1713323440291x_{28} = 48.1713323440291
x29=−65.9735393310389x_{29} = -65.9735393310389
x30=−53.4070285858308x_{30} = -53.4070285858308
x31=−17.8022058539068x_{31} = -17.8022058539068
x32=0x_{32} = 0
x33=−43.9823594210331x_{33} = -43.9823594210331
x34=65.9735387566978x_{34} = 65.9735387566978
x35=72.2566119343772x_{35} = 72.2566119343772
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(3*x)^4*atan(2*x)^3.
sin⁡4(0⋅3)atan⁡3(0⋅2)\sin^{4}{\left(0 \cdot 3 \right)} \operatorname{atan}^{3}{\left(0 \cdot 2 \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
12sin⁡3(3x)cos⁡(3x)atan⁡3(2x)+6sin⁡4(3x)atan⁡2(2x)4x2+1=012 \sin^{3}{\left(3 x \right)} \cos{\left(3 x \right)} \operatorname{atan}^{3}{\left(2 x \right)} + \frac{6 \sin^{4}{\left(3 x \right)} \operatorname{atan}^{2}{\left(2 x \right)}}{4 x^{2} + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=58.1194719868896x_{1} = 58.1194719868896
x2=94.2477801894451x_{2} = 94.2477801894451
x3=−27.7507699394451x_{3} = -27.7507699394451
x4=−78.0162219399394x_{4} = -78.0162219399394
x5=6.28317681033055x_{5} = 6.28317681033055
x6=28.2743275849635x_{6} = 28.2743275849635
x7=−93.7241838620119x_{7} = -93.7241838620119
x8=36.1283360152915x_{8} = 36.1283360152915
x9=43.9823032048267x_{9} = 43.9823032048267
x10=−31.9395515690565x_{10} = -31.9395515690565
x11=16.2316646393078x_{11} = 16.2316646393078
x12=50.2654784216157x_{12} = 50.2654784216157
x13=−49.7418944705192x_{13} = -49.7418944705192
x14=−100.007368799869x_{14} = -100.007368799869
x15=−83.7757974571533x_{15} = -83.7757974571533
x16=100.007368799869x_{16} = 100.007368799869
x17=14.137302547137x_{17} = 14.137302547137
x18=−41.8879092734202x_{18} = -41.8879092734202
x19=−56.0250774875478x_{19} = -56.0250774875478
x20=38.2227289242278x_{20} = 38.2227289242278
x21=−9.9486528927536x_{21} = -9.9486528927536
x22=−65.9734545465241x_{22} = -65.9734545465241
x23=−71.733037434713x_{23} = -71.733037434713
x24=−97.9129738127014x_{24} = -97.9129738127014
x25=80.1106168160882x_{25} = 80.1106168160882
x26=21.9911516357236x_{26} = 21.9911516357236
x27=−5.76042622794097x_{27} = -5.76042622794097
x28=−34.0339435255559x_{28} = -34.0339435255559
x29=48.1710808396552x_{29} = 48.1710808396552
x30=−75.9218270828108x_{30} = -75.9218270828108
x31=60.2138665482131x_{31} = 60.2138665482131
x32=−48.1710843027272x_{32} = -48.1710843027272
x33=−51.8362887162079x_{33} = -51.8362887162079
x34=82.2050117093403x_{34} = 82.2050117093403
x35=−7.85442793232242x_{35} = -7.85442793232242
x36=56.0250774875478x_{36} = 56.0250774875478
x37=−95.818578833185x_{37} = -95.818578833185
x38=−61.7846467566113x_{38} = -61.7846467566113
x39=−63.8790530122287x_{39} = -63.8790530122287
x40=−53.9306830600289x_{40} = -53.9306830600289
x41=72.2566292967368x_{41} = 72.2566292967368
x42=−73.8274322468957x_{42} = -73.8274322468957
x43=65.9734546541161x_{43} = 65.9734546541161
x44=34.0339435255559x_{44} = 34.0339435255559
x45=−43.9823031830836x_{45} = -43.9823031830836
x46=−29.845160301413x_{46} = -29.845160301413
x47=12.0429593731892x_{47} = 12.0429593731892
x48=−85.8701963703783x_{48} = -85.8701963703783
x49=78.0162219399394x_{49} = 78.0162219399394
x50=−21.9911516343345x_{50} = -21.9911516343345
x51=0x_{51} = 0
Signos de extremos en los puntos:
(58.11947198688959, 3.81245334328431)

(94.2477801894451, 3.55947829879546e-23)

(-27.750769939445135, -3.74395320957084)

(-78.0162219399394, -3.82853836791674)

(6.283176810330551, 1.40051218448277e-18)

(28.274327584963466, 4.77227120488344e-19)

(-93.72418386201194, -3.83642962793549)

(36.12833601529148, 3.77424772911856)

(43.982303204826714, 4.12776170044334e-19)

(-31.939551569056547, -3.76106650102085)

(16.23166463930782, 3.65227808246266)

(50.26547842161573, 8.17138455670263e-20)

(-49.74189447051923, -3.80185605166276)

(-100.00736879986853, -3.83889426889535)

(-83.77579745715329, -6.02815288521775e-19)

(100.00736879986853, 3.83889426889535)

(14.137302547137025, 3.61994100445818)

(-41.88790927342023, -8.363571024944e-19)

(-56.02507748754777, -3.81009941798436)

(38.22272892422777, 3.77976426688823)

(-9.948652892753598, -3.51582839238658)

(-65.97345454652411, -1.87344926012261e-18)

(-71.733037434713, -3.82441852772562)

(-97.91297381270144, -3.83810775274243)

(80.11061681608822, 3.82976861597375)

(21.991151635723607, 2.63678727733102e-20)

(-5.760426227940968, -3.26952177038598)

(-34.033943525555884, -3.76805877434123)

(48.17108083965519, 5.54585202683598e-19)

(-75.9218270828108, -3.82724053268093)

(60.2138665482131, 3.81464439955514)

(-48.171084302727216, -2.67131557175863e-20)

(-51.83628871620793, -3.80482449180701)

(82.20501170934025, 3.83093642234329)

(-7.854427932322416, -3.42398322050265)

(56.02507748754777, 3.81009941798436)

(-95.81857883318504, -3.83728696887192)

(-61.784646756611345, -1.82356887284117e-18)

(-63.87905301222868, -1.00779601298045e-20)

(-53.93068306002891, -3.80756376370035)

(72.25662929673679, 2.81267607340039e-21)

(-73.82743224689571, -3.82586938347312)

(65.97345465411614, 1.96653751608659e-18)

(34.033943525555884, 3.76805877434123)

(-43.98230318308357, -4.06878595925984e-19)

(-29.845160301413035, -3.75310367891437)

(12.042959373189229, 3.57667599301974)

(-85.87019637037828, -1.98502365194774e-20)

(78.0162219399394, 3.82853836791674)

(-21.99115163433445, -2.63200335832004e-20)

(0, 0)


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=−27.7507699394451x_{1} = -27.7507699394451
x2=−78.0162219399394x_{2} = -78.0162219399394
x3=−93.7241838620119x_{3} = -93.7241838620119
x4=−31.9395515690565x_{4} = -31.9395515690565
x5=−49.7418944705192x_{5} = -49.7418944705192
x6=−100.007368799869x_{6} = -100.007368799869
x7=−56.0250774875478x_{7} = -56.0250774875478
x8=−9.9486528927536x_{8} = -9.9486528927536
x9=−71.733037434713x_{9} = -71.733037434713
x10=−97.9129738127014x_{10} = -97.9129738127014
x11=−5.76042622794097x_{11} = -5.76042622794097
x12=−34.0339435255559x_{12} = -34.0339435255559
x13=−75.9218270828108x_{13} = -75.9218270828108
x14=−51.8362887162079x_{14} = -51.8362887162079
x15=−7.85442793232242x_{15} = -7.85442793232242
x16=−95.818578833185x_{16} = -95.818578833185
x17=−53.9306830600289x_{17} = -53.9306830600289
x18=−73.8274322468957x_{18} = -73.8274322468957
x19=−29.845160301413x_{19} = -29.845160301413
Puntos máximos de la función:
x19=58.1194719868896x_{19} = 58.1194719868896
x19=36.1283360152915x_{19} = 36.1283360152915
x19=16.2316646393078x_{19} = 16.2316646393078
x19=100.007368799869x_{19} = 100.007368799869
x19=14.137302547137x_{19} = 14.137302547137
x19=38.2227289242278x_{19} = 38.2227289242278
x19=80.1106168160882x_{19} = 80.1106168160882
x19=60.2138665482131x_{19} = 60.2138665482131
x19=82.2050117093403x_{19} = 82.2050117093403
x19=56.0250774875478x_{19} = 56.0250774875478
x19=34.0339435255559x_{19} = 34.0339435255559
x19=12.0429593731892x_{19} = 12.0429593731892
x19=78.0162219399394x_{19} = 78.0162219399394
Decrece en los intervalos
[−5.76042622794097,12.0429593731892]\left[-5.76042622794097, 12.0429593731892\right]
Crece en los intervalos
(−∞,−100.007368799869]\left(-\infty, -100.007368799869\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
12(−3(sin⁡2(3x)−3cos⁡2(3x))atan⁡2(2x)+12sin⁡(3x)cos⁡(3x)atan⁡(2x)4x2+1−2(2xatan⁡(2x)−1)sin⁡2(3x)(4x2+1)2)sin⁡2(3x)atan⁡(2x)=012 \left(- 3 \left(\sin^{2}{\left(3 x \right)} - 3 \cos^{2}{\left(3 x \right)}\right) \operatorname{atan}^{2}{\left(2 x \right)} + \frac{12 \sin{\left(3 x \right)} \cos{\left(3 x \right)} \operatorname{atan}{\left(2 x \right)}}{4 x^{2} + 1} - \frac{2 \left(2 x \operatorname{atan}{\left(2 x \right)} - 1\right) \sin^{2}{\left(3 x \right)}}{\left(4 x^{2} + 1\right)^{2}}\right) \sin^{2}{\left(3 x \right)} \operatorname{atan}{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=40.1425893262862x_{1} = 40.1425893262862
x2=−52.0108216020373x_{2} = -52.0108216020373
x3=28.2743338691272x_{3} = 28.2743338691272
x4=−46.4257704550083x_{4} = -46.4257704550083
x5=89.7099268654689x_{5} = 89.7099268654689
x6=100.181901719611x_{6} = 100.181901719611
x7=−55.8505445935956x_{7} = -55.8505445935956
x8=43.9822971641215x_{8} = 43.9822971641215
x9=−93.8987167805794x_{9} = -93.8987167805794
x10=62.133728260796x_{10} = 62.133728260796
x11=52.7089529888009x_{11} = 52.7089529888009
x12=−69.8131755581685x_{12} = -69.8131755581685
x13=21.9911485838759x_{13} = 21.9911485838759
x14=−75.7472941701292x_{14} = -75.7472941701292
x15=64.2281229206146x_{15} = 64.2281229206146
x16=34.2084763109635x_{16} = 34.2084763109635
x17=76.0963599955179x_{17} = 76.0963599955179
x18=86.2192686239169x_{18} = 86.2192686239169
x19=−27.9253026059486x_{19} = -27.9253026059486
x20=−9.77412583999367x_{20} = -9.77412583999367
x21=−1.75569769451076x_{21} = -1.75569769451076
x22=−30.0196930188992x_{22} = -30.0196930188992
x23=94.2477796093712x_{23} = 94.2477796093712
x24=72.2566310279178x_{24} = 72.2566310279178
x25=−91.8043218180563x_{25} = -91.8043218180563
x26=16.0571330351412x_{26} = 16.0571330351412
x27=13.6135681715903x_{27} = 13.6135681715903
x28=70.1622359258168x_{28} = 70.1622359258168
x29=−53.7561501698762x_{29} = -53.7561501698762
x30=65.9734457408928x_{30} = 65.9734457408928
x31=−74.0019651585066x_{31} = -74.0019651585066
x32=20.2458845618469x_{32} = 20.2458845618469
x33=18.1515054532938x_{33} = 18.1515054532938
x34=−5.93492851760178x_{34} = -5.93492851760178
x35=84.1248737019512x_{35} = 84.1248737019512
x36=10.1231800375708x_{36} = 10.1231800375708
x37=0x_{37} = 0
x38=−71.9075703450956x_{38} = -71.9075703450956
x39=8.72700058844703x_{39} = 8.72700058844703
x40=−99.8328358801346x_{40} = -99.8328358801346
x41=38.0481960979602x_{41} = 38.0481960979602
x42=−21.9911485877595x_{42} = -21.9911485877595
x43=−34.5575191743409x_{43} = -34.5575191743409
x44=6.2831852938859x_{44} = 6.2831852938859
x45=−49.9164273511453x_{45} = -49.9164273511453
x46=−31.7650188139759x_{46} = -31.7650188139759
x47=−11.8684297247241x_{47} = -11.8684297247241
x48=50.265482447645x_{48} = 50.265482447645
x49=−97.7384408933266x_{49} = -97.7384408933266
x50=54.1052159502822x_{50} = 54.1052159502822
x51=−3.84172100507587x_{51} = -3.84172100507587
x52=78.1907548536278x_{52} = 78.1907548536278
x53=82.0304787939956x_{53} = 82.0304787939956
x54=−77.8416890262738x_{54} = -77.8416890262738
x55=−25.8309133898733x_{55} = -25.8309133898733
x56=98.0875067320855x_{56} = 98.0875067320855
x57=−67.3697149862159x_{57} = -67.3697149862159
x58=32.1140843249633x_{58} = 32.1140843249633
x59=87.9645943138781x_{59} = 87.9645943138781
x60=56.1996103815863x_{60} = 56.1996103815863
x61=−47.8220332151124x_{61} = -47.8220332151124
x62=42.236982826926x_{62} = 42.236982826926
x63=3.84172100507587x_{63} = 3.84172100507587
x64=92.502453461464x_{64} = 92.502453461464
x65=−8.02894898136117x_{65} = -8.02894898136117
x66=−95.9931117521786x_{66} = -95.9931117521786
x67=30.7178231002456x_{67} = 30.7178231002456
x68=−15.7079632755833x_{68} = -15.7079632755833
x69=60.0393336481625x_{69} = 60.0393336481625
x70=−33.8594107407881x_{70} = -33.8594107407881
x71=12.2174890640473x_{71} = 12.2174890640473
x72=74.0019651585066x_{72} = 74.0019651585066
x73=−23.3874608157524x_{73} = -23.3874608157524

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
[100.181901719611,∞)\left[100.181901719611, \infty\right)
Convexa en los intervalos
(−∞,−95.9931117521786]\left(-\infty, -95.9931117521786\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡4(3x)atan⁡3(2x))=⟨−18,0⟩π3\lim_{x \to -\infty}\left(\sin^{4}{\left(3 x \right)} \operatorname{atan}^{3}{\left(2 x \right)}\right) = \left\langle - \frac{1}{8}, 0\right\rangle \pi^{3}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−18,0⟩π3y = \left\langle - \frac{1}{8}, 0\right\rangle \pi^{3}
lim⁡x→∞(sin⁡4(3x)atan⁡3(2x))=⟨0,18⟩π3\lim_{x \to \infty}\left(\sin^{4}{\left(3 x \right)} \operatorname{atan}^{3}{\left(2 x \right)}\right) = \left\langle 0, \frac{1}{8}\right\rangle \pi^{3}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨0,18⟩π3y = \left\langle 0, \frac{1}{8}\right\rangle \pi^{3}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(3*x)^4*atan(2*x)^3, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡4(3x)atan⁡3(2x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{4}{\left(3 x \right)} \operatorname{atan}^{3}{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡4(3x)atan⁡3(2x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{4}{\left(3 x \right)} \operatorname{atan}^{3}{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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⁡4(3x)atan⁡3(2x)=−sin⁡4(3x)atan⁡3(2x)\sin^{4}{\left(3 x \right)} \operatorname{atan}^{3}{\left(2 x \right)} = - \sin^{4}{\left(3 x \right)} \operatorname{atan}^{3}{\left(2 x \right)}
- No
sin⁡4(3x)atan⁡3(2x)=sin⁡4(3x)atan⁡3(2x)\sin^{4}{\left(3 x \right)} \operatorname{atan}^{3}{\left(2 x \right)} = \sin^{4}{\left(3 x \right)} \operatorname{atan}^{3}{\left(2 x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор