Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ arctg(x/sinx)

Gráfico de la función y = arctg(x/sinx)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           /  x   \
f(x) = atan|------|
           \sin(x)/
f(x)=atan⁡(xsin⁡(x))f{\left(x \right)} = \operatorname{atan}{\left(\frac{x}{\sin{\left(x \right)}} \right)} 
f = atan(x/sin(x))
Gráfico de la función
02468-8-6-4-2-10105-5
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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:
atan⁡(xsin⁡(x))=0\operatorname{atan}{\left(\frac{x}{\sin{\left(x \right)}} \right)} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 atan(x/sin(x)).
atan⁡(0sin⁡(0))\operatorname{atan}{\left(\frac{0}{\sin{\left(0 \right)}} \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
−xcos⁡(x)sin⁡2(x)+1sin⁡(x)x2sin⁡2(x)+1=0\frac{- \frac{x \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}}{\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=20.3713029592876x_{1} = 20.3713029592876
x2=−4.49340945790906x_{2} = -4.49340945790906
x3=−7.72525183693771x_{3} = -7.72525183693771
x4=−42.3879135681319x_{4} = -42.3879135681319
x5=−67.5294347771441x_{5} = -67.5294347771441
x6=−73.8138806006806x_{6} = -73.8138806006806
x7=6.31689619156083⋅10−18x_{7} = 6.31689619156083 \cdot 10^{-18}
x8=111.517572246131x_{8} = 111.517572246131
x9=70.6716857116195x_{9} = 70.6716857116195
x10=−92.6661922776228x_{10} = -92.6661922776228
x11=−95.8081387868617x_{11} = -95.8081387868617
x12=10.9041216594289x_{12} = 10.9041216594289
x13=26.6660542588127x_{13} = 26.6660542588127
x14=−10.9041216594289x_{14} = -10.9041216594289
x15=1.19664026863252⋅10−16x_{15} = 1.19664026863252 \cdot 10^{-16}
x16=80.0981286289451x_{16} = 80.0981286289451
x17=−36.1006222443756x_{17} = -36.1006222443756
x18=83.2401924707234x_{18} = 83.2401924707234
x19=48.6741442319544x_{19} = 48.6741442319544
x20=−14.0661939128315x_{20} = -14.0661939128315
x21=14.0661939128315x_{21} = 14.0661939128315
x22=−29.811598790893x_{22} = -29.811598790893
x23=23.519452498689x_{23} = 23.519452498689
x24=76.9560263103312x_{24} = 76.9560263103312
x25=67.5294347771441x_{25} = 67.5294347771441
x26=−17.2207552719308x_{26} = -17.2207552719308
x27=39.2444323611642x_{27} = 39.2444323611642
x28=−23.519452498689x_{28} = -23.519452498689
x29=17.2207552719308x_{29} = 17.2207552719308
x30=29.811598790893x_{30} = 29.811598790893
x31=−48.6741442319544x_{31} = -48.6741442319544
x32=−86.3822220347287x_{32} = -86.3822220347287
x33=89.5242209304172x_{33} = 89.5242209304172
x34=61.2447302603744x_{34} = 61.2447302603744
x35=−39.2444323611642x_{35} = -39.2444323611642
x36=−76.9560263103312x_{36} = -76.9560263103312
x37=−58.1022547544956x_{37} = -58.1022547544956
x38=−54.9596782878889x_{38} = -54.9596782878889
x39=54.9596782878889x_{39} = 54.9596782878889
x40=95.8081387868617x_{40} = 95.8081387868617
x41=92.6661922776228x_{41} = 92.6661922776228
x42=−70.6716857116195x_{42} = -70.6716857116195
x43=86.3822220347287x_{43} = 86.3822220347287
x44=−26.6660542588127x_{44} = -26.6660542588127
x45=−20.3713029592876x_{45} = -20.3713029592876
x46=−89.5242209304172x_{46} = -89.5242209304172
x47=−83.2401924707234x_{47} = -83.2401924707234
x48=45.5311340139913x_{48} = 45.5311340139913
x49=36.1006222443756x_{49} = 36.1006222443756
x50=−7.43532772224002⋅10−17x_{50} = -7.43532772224002 \cdot 10^{-17}
x51=−32.9563890398225x_{51} = -32.9563890398225
x52=4.49340945790906x_{52} = 4.49340945790906
x53=98.9500628243319x_{53} = 98.9500628243319
x54=7.72525183693771x_{54} = 7.72525183693771
x55=−98.9500628243319x_{55} = -98.9500628243319
x56=−444.533110935535x_{56} = -444.533110935535
x57=−61.2447302603744x_{57} = -61.2447302603744
x58=42.3879135681319x_{58} = 42.3879135681319
x59=51.8169824872797x_{59} = 51.8169824872797
x60=−114.659410595023x_{60} = -114.659410595023
x61=−51.8169824872797x_{61} = -51.8169824872797
x62=−108.375719651675x_{62} = -108.375719651675
x63=−80.0981286289451x_{63} = -80.0981286289451
x64=58.1022547544956x_{64} = 58.1022547544956
x65=32.9563890398225x_{65} = 32.9563890398225
x66=64.3871195905574x_{66} = 64.3871195905574
x67=347.143107573282x_{67} = 347.143107573282
x68=−45.5311340139913x_{68} = -45.5311340139913
x69=−64.3871195905574x_{69} = -64.3871195905574
x70=73.8138806006806x_{70} = 73.8138806006806
Signos de extremos en los puntos:
(20.37130295928756, 1.52180593371551)

(-4.493409457909064, -1.35688620894552)

(-7.725251836937707, 1.44312008655917)

(-42.38791356813192, -1.54721563021445)

(-67.52943477714412, -1.55599067477904)

(-73.81388060068065, -1.55725081095882)

(6.316896191560833e-18, 0.785398163397448)

(111.51757224613101, -1.56182973096642)

(70.6716857116195, 1.55664874870356)

(-92.66619227762284, -1.56000595174721)

(-95.8081387868617, 1.56035974760184)

(10.904121659428899, -1.47972375394027)

(26.666054258812675, 1.53333933435263)

(-10.904121659428899, -1.47972375394027)

(1.1966402686325174e-16, 0.785398163397448)

(80.09812862894512, -1.55831326191267)

(-36.10062224437561, -1.54311366800893)

(83.2401924707234, 1.55878434407342)

(48.674144231954386, -1.5502587607648)

(-14.066193912831473, 1.50000137792287)

(14.066193912831473, 1.50000137792287)

(-29.81159879089296, -1.53728374333092)

(23.519452498689006, -1.52834223393023)

(76.95602631033118, 1.55780372094166)

(67.52943477714412, -1.55599067477904)

(-17.22075527193077, -1.51288933633225)

(39.24443236116419, 1.54532878117926)

(-23.519452498689006, -1.52834223393023)

(17.22075527193077, -1.51288933633225)

(29.81159879089296, -1.53728374333092)

(-48.674144231954386, -1.5502587607648)

(-86.38222203472871, -1.55922116332795)

(89.52422093041719, 1.55962732661537)

(61.2447302603744, -1.55447201767914)

(-39.24443236116419, 1.54532878117926)

(-76.95602631033118, 1.55780372094166)

(-58.10225475449559, 1.55358953768302)

(-54.959678287888934, -1.552606186997)

(54.959678287888934, -1.552606186997)

(95.8081387868617, 1.56035974760184)

(92.66619227762284, -1.56000595174721)

(-70.6716857116195, 1.55664874870356)

(86.38222203472871, -1.55922116332795)

(-26.666054258812675, 1.53333933435263)

(-20.37130295928756, 1.52180593371551)

(-89.52422093041719, 1.55962732661537)

(-83.2401924707234, 1.55878434407342)

(45.53113401399128, 1.54884215644453)

(36.10062224437561, -1.54311366800893)

(-7.435327722240017e-17, 0.785398163397448)

(-32.956389039822476, 1.54047644999598)

(4.493409457909064, -1.35688620894552)

(98.95006282433188, -1.56069107904171)

(7.725251836937707, 1.44312008655917)

(-98.95006282433188, -1.56069107904171)

(-444.5331109355349, -1.56854678506592)

(-61.2447302603744, -1.55447201767914)

(42.38791356813192, -1.54721563021445)

(51.81698248727967, 1.55150362129997)

(-114.65941059502308, 1.56207539744288)

(-51.81698248727967, 1.55150362129997)

(-108.37571965167469, 1.5615698223655)

(-80.09812862894512, -1.55831326191267)

(58.10225475449559, 1.55358953768302)

(32.956389039822476, 1.54047644999598)

(64.38711959055742, 1.55526839112313)

(347.14310757328207, 1.56791569035676)

(-45.53113401399128, 1.54884215644453)

(-64.38711959055742, 1.55526839112313)

(73.81388060068065, -1.55725081095882)


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=20.3713029592876x_{1} = 20.3713029592876
x2=−7.72525183693771x_{2} = -7.72525183693771
x3=6.31689619156083⋅10−18x_{3} = 6.31689619156083 \cdot 10^{-18}
x4=70.6716857116195x_{4} = 70.6716857116195
x5=−95.8081387868617x_{5} = -95.8081387868617
x6=26.6660542588127x_{6} = 26.6660542588127
x7=1.19664026863252⋅10−16x_{7} = 1.19664026863252 \cdot 10^{-16}
x8=83.2401924707234x_{8} = 83.2401924707234
x9=−14.0661939128315x_{9} = -14.0661939128315
x10=14.0661939128315x_{10} = 14.0661939128315
x11=76.9560263103312x_{11} = 76.9560263103312
x12=39.2444323611642x_{12} = 39.2444323611642
x13=89.5242209304172x_{13} = 89.5242209304172
x14=−39.2444323611642x_{14} = -39.2444323611642
x15=−76.9560263103312x_{15} = -76.9560263103312
x16=−58.1022547544956x_{16} = -58.1022547544956
x17=95.8081387868617x_{17} = 95.8081387868617
x18=−70.6716857116195x_{18} = -70.6716857116195
x19=−26.6660542588127x_{19} = -26.6660542588127
x20=−20.3713029592876x_{20} = -20.3713029592876
x21=−89.5242209304172x_{21} = -89.5242209304172
x22=−83.2401924707234x_{22} = -83.2401924707234
x23=45.5311340139913x_{23} = 45.5311340139913
x24=−7.43532772224002⋅10−17x_{24} = -7.43532772224002 \cdot 10^{-17}
x25=−32.9563890398225x_{25} = -32.9563890398225
x26=7.72525183693771x_{26} = 7.72525183693771
x27=51.8169824872797x_{27} = 51.8169824872797
x28=−114.659410595023x_{28} = -114.659410595023
x29=−51.8169824872797x_{29} = -51.8169824872797
x30=−108.375719651675x_{30} = -108.375719651675
x31=58.1022547544956x_{31} = 58.1022547544956
x32=32.9563890398225x_{32} = 32.9563890398225
x33=64.3871195905574x_{33} = 64.3871195905574
x34=347.143107573282x_{34} = 347.143107573282
x35=−45.5311340139913x_{35} = -45.5311340139913
x36=−64.3871195905574x_{36} = -64.3871195905574
Puntos máximos de la función:
x36=−4.49340945790906x_{36} = -4.49340945790906
x36=−42.3879135681319x_{36} = -42.3879135681319
x36=−67.5294347771441x_{36} = -67.5294347771441
x36=−73.8138806006806x_{36} = -73.8138806006806
x36=111.517572246131x_{36} = 111.517572246131
x36=−92.6661922776228x_{36} = -92.6661922776228
x36=10.9041216594289x_{36} = 10.9041216594289
x36=−10.9041216594289x_{36} = -10.9041216594289
x36=80.0981286289451x_{36} = 80.0981286289451
x36=−36.1006222443756x_{36} = -36.1006222443756
x36=48.6741442319544x_{36} = 48.6741442319544
x36=−29.811598790893x_{36} = -29.811598790893
x36=23.519452498689x_{36} = 23.519452498689
x36=67.5294347771441x_{36} = 67.5294347771441
x36=−17.2207552719308x_{36} = -17.2207552719308
x36=−23.519452498689x_{36} = -23.519452498689
x36=17.2207552719308x_{36} = 17.2207552719308
x36=29.811598790893x_{36} = 29.811598790893
x36=−48.6741442319544x_{36} = -48.6741442319544
x36=−86.3822220347287x_{36} = -86.3822220347287
x36=61.2447302603744x_{36} = 61.2447302603744
x36=−54.9596782878889x_{36} = -54.9596782878889
x36=54.9596782878889x_{36} = 54.9596782878889
x36=92.6661922776228x_{36} = 92.6661922776228
x36=86.3822220347287x_{36} = 86.3822220347287
x36=36.1006222443756x_{36} = 36.1006222443756
x36=4.49340945790906x_{36} = 4.49340945790906
x36=98.9500628243319x_{36} = 98.9500628243319
x36=−98.9500628243319x_{36} = -98.9500628243319
x36=−444.533110935535x_{36} = -444.533110935535
x36=−61.2447302603744x_{36} = -61.2447302603744
x36=42.3879135681319x_{36} = 42.3879135681319
x36=−80.0981286289451x_{36} = -80.0981286289451
x36=73.8138806006806x_{36} = 73.8138806006806
Decrece en los intervalos
[347.143107573282,∞)\left[347.143107573282, \infty\right)
Crece en los intervalos
(−∞,−114.659410595023]\left(-\infty, -114.659410595023\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
x+2xcos⁡2(x)sin⁡2(x)−2x(xcos⁡(x)sin⁡(x)−1)2(x2sin⁡2(x)+1)sin⁡2(x)−2cos⁡(x)sin⁡(x)(x2sin⁡2(x)+1)sin⁡(x)=0\frac{x + \frac{2 x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 x \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} - 1\right)^{2}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−9.21096438740149x_{1} = -9.21096438740149
x2=−34.4996123350132x_{2} = -34.4996123350132
x3=−72.2289483771681x_{3} = -72.2289483771681
x4=−5.95939190757933x_{4} = -5.95939190757933
x5=21.9000773156394x_{5} = 21.9000773156394
x6=50.2256832197934x_{6} = 50.2256832197934
x7=−75.3716947511882x_{7} = -75.3716947511882
x8=72.2289483771681x_{8} = 72.2289483771681
x9=28.2035393053095x_{9} = 28.2035393053095
x10=157.066899940015x_{10} = 157.066899940015
x11=100.511069234565x_{11} = 100.511069234565
x12=−65.9431258539286x_{12} = -65.9431258539286
x13=−94.2265573558031x_{13} = -94.2265573558031
x14=40.7917141624847x_{14} = 40.7917141624847
x15=34.4996123350132x_{15} = 34.4996123350132
x16=−62.8000167068325x_{16} = -62.8000167068325
x17=15.5802941824244x_{17} = 15.5802941824244
x18=84.7994209518635x_{18} = 84.7994209518635
x19=−84.7994209518635x_{19} = -84.7994209518635
x20=91.0842327848165x_{20} = 91.0842327848165
x21=−18.7432530945386x_{21} = -18.7432530945386
x22=−56.5132926241755x_{22} = -56.5132926241755
x23=81.6569211705466x_{23} = 81.6569211705466
x24=106.795425016936x_{24} = 106.795425016936
x25=94.2265573558031x_{25} = 94.2265573558031
x26=−78.5143487963623x_{26} = -78.5143487963623
x27=65.9431258539286x_{27} = 65.9431258539286
x28=−12.4065403639626x_{28} = -12.4065403639626
x29=59.6567478435559x_{29} = 59.6567478435559
x30=−28.2035393053095x_{30} = -28.2035393053095
x31=−50.2256832197934x_{31} = -50.2256832197934
x32=12.4065403639626x_{32} = 12.4065403639626
x33=69.0860970774096x_{33} = 69.0860970774096
x34=−15.5802941824244x_{34} = -15.5802941824244
x35=62.8000167068325x_{35} = 62.8000167068325
x36=2.45871417599962x_{36} = 2.45871417599962
x37=53.3696181339615x_{37} = 53.3696181339615
x38=−53.3696181339615x_{38} = -53.3696181339615
x39=37.6460352959305x_{39} = 37.6460352959305
x40=−21.9000773156394x_{40} = -21.9000773156394
x41=−87.9418559209576x_{41} = -87.9418559209576
x42=5.95939190757933x_{42} = 5.95939190757933
x43=−25.053079662454x_{43} = -25.053079662454
x44=25.053079662454x_{44} = 25.053079662454
x45=47.0814357397523x_{45} = 47.0814357397523
x46=−31.3522215217643x_{46} = -31.3522215217643
x47=31.3522215217643x_{47} = 31.3522215217643
x48=−100.511069234565x_{48} = -100.511069234565
x49=75.3716947511882x_{49} = 75.3716947511882
x50=−40.7917141624847x_{50} = -40.7917141624847
x51=−47.0814357397523x_{51} = -47.0814357397523
x52=−37.6460352959305x_{52} = -37.6460352959305
x53=18.7432530945386x_{53} = 18.7432530945386
x54=−69.0860970774096x_{54} = -69.0860970774096
x55=97.3688346960149x_{55} = 97.3688346960149
x56=9.21096438740149x_{56} = 9.21096438740149
x57=56.5132926241755x_{57} = 56.5132926241755
x58=−91.0842327848165x_{58} = -91.0842327848165
x59=78.5143487963623x_{59} = 78.5143487963623
x60=−59.6567478435559x_{60} = -59.6567478435559
x61=−43.9368086315937x_{61} = -43.9368086315937
x62=43.9368086315937x_{62} = 43.9368086315937
x63=−2.45871417599962x_{63} = -2.45871417599962
x64=−81.6569211705466x_{64} = -81.6569211705466
x65=87.9418559209576x_{65} = 87.9418559209576
x66=−97.3688346960149x_{66} = -97.3688346960149
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979

lim⁡x→0−(x+2xcos⁡2(x)sin⁡2(x)−2x(xcos⁡(x)sin⁡(x)−1)2(x2sin⁡2(x)+1)sin⁡2(x)−2cos⁡(x)sin⁡(x)(x2sin⁡2(x)+1)sin⁡(x))=16\lim_{x \to 0^-}\left(\frac{x + \frac{2 x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 x \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} - 1\right)^{2}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}\right) = \frac{1}{6}
lim⁡x→0+(x+2xcos⁡2(x)sin⁡2(x)−2x(xcos⁡(x)sin⁡(x)−1)2(x2sin⁡2(x)+1)sin⁡2(x)−2cos⁡(x)sin⁡(x)(x2sin⁡2(x)+1)sin⁡(x))=16\lim_{x \to 0^+}\left(\frac{x + \frac{2 x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 x \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} - 1\right)^{2}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}\right) = \frac{1}{6}
- los límites son iguales, es decir omitimos el punto correspondiente
lim⁡x→3.14159265358979−(x+2xcos⁡2(x)sin⁡2(x)−2x(xcos⁡(x)sin⁡(x)−1)2(x2sin⁡2(x)+1)sin⁡2(x)−2cos⁡(x)sin⁡(x)(x2sin⁡2(x)+1)sin⁡(x))=0.894109817441799\lim_{x \to 3.14159265358979^-}\left(\frac{x + \frac{2 x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 x \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} - 1\right)^{2}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}\right) = 0.894109817441799
lim⁡x→3.14159265358979+(x+2xcos⁡2(x)sin⁡2(x)−2x(xcos⁡(x)sin⁡(x)−1)2(x2sin⁡2(x)+1)sin⁡2(x)−2cos⁡(x)sin⁡(x)(x2sin⁡2(x)+1)sin⁡(x))=0.894109817441799\lim_{x \to 3.14159265358979^+}\left(\frac{x + \frac{2 x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 x \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} - 1\right)^{2}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\left(\frac{x^{2}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}\right) = 0.894109817441799
- los límites son iguales, es decir omitimos el punto correspondiente

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
[157.066899940015,∞)\left[157.066899940015, \infty\right)
Convexa en los intervalos
(−∞,−97.3688346960149]\left(-\infty, -97.3688346960149\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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→−∞atan⁡(xsin⁡(x))y = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x}{\sin{\left(x \right)}} \right)}
True

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

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