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sinc(x)
Trazado el gráfico de la función/ sinc(x)

Gráfico de la función y = sinc(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = sinc(x)
f(x)=sinc⁡(x)f{\left(x \right)} = \operatorname{sinc}{\left(x \right)} 
f = sinc(x)
Gráfico de la función
02468-8-6-4-2-10102-1
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:
sinc⁡(x)=0\operatorname{sinc}{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−59.6902604182061x_{1} = -59.6902604182061
x2=−62.8318530717959x_{2} = -62.8318530717959
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=87.9645943005142x_{4} = 87.9645943005142
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=31.4159265358979x_{6} = 31.4159265358979
x7=69.1150383789755x_{7} = 69.1150383789755
x8=2594.95553186517x_{8} = 2594.95553186517
x9=−370.707933123596x_{9} = -370.707933123596
x10=−37.6991118430775x_{10} = -37.6991118430775
x11=−81.6814089933346x_{11} = -81.6814089933346
x12=153.9380400259x_{12} = 153.9380400259
x13=−84.8230016469244x_{13} = -84.8230016469244
x14=−21.9911485751286x_{14} = -21.9911485751286
x15=47.1238898038469x_{15} = 47.1238898038469
x16=−113.097335529233x_{16} = -113.097335529233
x17=−223.053078404875x_{17} = -223.053078404875
x18=−12.5663706143592x_{18} = -12.5663706143592
x19=−15.707963267949x_{19} = -15.707963267949
x20=12.5663706143592x_{20} = 12.5663706143592
x21=−87.9645943005142x_{21} = -87.9645943005142
x22=53.4070751110265x_{22} = 53.4070751110265
x23=72.2566310325652x_{23} = 72.2566310325652
x24=−100.530964914873x_{24} = -100.530964914873
x25=−3.14159265358979x_{25} = -3.14159265358979
x26=34.5575191894877x_{26} = 34.5575191894877
x27=−94.2477796076938x_{27} = -94.2477796076938
x28=6.28318530717959x_{28} = 6.28318530717959
x29=−69.1150383789755x_{29} = -69.1150383789755
x30=97.3893722612836x_{30} = 97.3893722612836
x31=65.9734457253857x_{31} = 65.9734457253857
x32=590.619418874881x_{32} = 590.619418874881
x33=−50.2654824574367x_{33} = -50.2654824574367
x34=15.707963267949x_{34} = 15.707963267949
x35=3.14159265358979x_{35} = 3.14159265358979
x36=−25.1327412287183x_{36} = -25.1327412287183
x37=−18.8495559215388x_{37} = -18.8495559215388
x38=40.8407044966673x_{38} = 40.8407044966673
x39=−53.4070751110265x_{39} = -53.4070751110265
x40=37.6991118430775x_{40} = 37.6991118430775
x41=−43.9822971502571x_{41} = -43.9822971502571
x42=18.8495559215388x_{42} = 18.8495559215388
x43=−78.5398163397448x_{43} = -78.5398163397448
x44=−6.28318530717959x_{44} = -6.28318530717959
x45=−40.8407044966673x_{45} = -40.8407044966673
x46=43.9822971502571x_{46} = 43.9822971502571
x47=56.5486677646163x_{47} = 56.5486677646163
x48=−65.9734457253857x_{48} = -65.9734457253857
x49=25.1327412287183x_{49} = 25.1327412287183
x50=78.5398163397448x_{50} = 78.5398163397448
x51=−28.2743338823081x_{51} = -28.2743338823081
x52=75.398223686155x_{52} = 75.398223686155
x53=59.6902604182061x_{53} = 59.6902604182061
x54=−34.5575191894877x_{54} = -34.5575191894877
x55=81.6814089933346x_{55} = 81.6814089933346
x56=−47.1238898038469x_{56} = -47.1238898038469
x57=100.530964914873x_{57} = 100.530964914873
x58=−9.42477796076938x_{58} = -9.42477796076938
x59=−75.398223686155x_{59} = -75.398223686155
x60=−72.2566310325652x_{60} = -72.2566310325652
x61=−31.4159265358979x_{61} = -31.4159265358979
x62=28.2743338823081x_{62} = 28.2743338823081
x63=−91.106186954104x_{63} = -91.106186954104
x64=21.9911485751286x_{64} = 21.9911485751286
x65=62.8318530717959x_{65} = 62.8318530717959
x66=9.42477796076938x_{66} = 9.42477796076938
x67=50.2654824574367x_{67} = 50.2654824574367
x68=94.2477796076938x_{68} = 94.2477796076938
x69=91.106186954104x_{69} = 91.106186954104
x70=84.8230016469244x_{70} = 84.8230016469244
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 sinc(x).
sinc⁡(0)\operatorname{sinc}{\left(0 \right)}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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⁡(x)x2for x≠00otherwise=0\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=17.2207552719308x_{1} = 17.2207552719308
x2=−80.0981286289451x_{2} = -80.0981286289451
x3=−83.2401924707234x_{3} = -83.2401924707234
x4=98.9500628243319x_{4} = 98.9500628243319
x5=−45.5311340139913x_{5} = -45.5311340139913
x6=−86.3822220347287x_{6} = -86.3822220347287
x7=7.72525183693771x_{7} = 7.72525183693771
x8=−394.267341680887x_{8} = -394.267341680887
x9=−4.49340945790906x_{9} = -4.49340945790906
x10=4.49340945790906x_{10} = 4.49340945790906
x11=108.375719651675x_{11} = 108.375719651675
x12=39.2444323611642x_{12} = 39.2444323611642
x13=−70.6716857116195x_{13} = -70.6716857116195
x14=10.9041216594289x_{14} = 10.9041216594289
x15=−42.3879135681319x_{15} = -42.3879135681319
x16=80.0981286289451x_{16} = 80.0981286289451
x17=89.5242209304172x_{17} = 89.5242209304172
x18=−48.6741442319544x_{18} = -48.6741442319544
x19=3306.52596547105x_{19} = 3306.52596547105
x20=14.0661939128315x_{20} = 14.0661939128315
x21=−36.1006222443756x_{21} = -36.1006222443756
x22=−95.8081387868617x_{22} = -95.8081387868617
x23=64.3871195905574x_{23} = 64.3871195905574
x24=61.2447302603744x_{24} = 61.2447302603744
x25=−54.9596782878889x_{25} = -54.9596782878889
x26=76.9560263103312x_{26} = 76.9560263103312
x27=−76.9560263103312x_{27} = -76.9560263103312
x28=0x_{28} = 0
x29=−98.9500628243319x_{29} = -98.9500628243319
x30=−7.72525183693771x_{30} = -7.72525183693771
x31=−20.3713029592876x_{31} = -20.3713029592876
x32=−39.2444323611642x_{32} = -39.2444323611642
x33=−14.0661939128315x_{33} = -14.0661939128315
x34=−32.9563890398225x_{34} = -32.9563890398225
x35=54.9596782878889x_{35} = 54.9596782878889
x36=73.8138806006806x_{36} = 73.8138806006806
x37=26.6660542588127x_{37} = 26.6660542588127
x38=−26.6660542588127x_{38} = -26.6660542588127
x39=−61.2447302603744x_{39} = -61.2447302603744
x40=−67.5294347771441x_{40} = -67.5294347771441
x41=29.811598790893x_{41} = 29.811598790893
x42=51.8169824872797x_{42} = 51.8169824872797
x43=23.519452498689x_{43} = 23.519452498689
x44=−58.1022547544956x_{44} = -58.1022547544956
x45=67.5294347771441x_{45} = 67.5294347771441
x46=−10.9041216594289x_{46} = -10.9041216594289
x47=−89.5242209304172x_{47} = -89.5242209304172
x48=86.3822220347287x_{48} = 86.3822220347287
x49=−23.519452498689x_{49} = -23.519452498689
x50=−17.2207552719308x_{50} = -17.2207552719308
x51=58.1022547544956x_{51} = 58.1022547544956
x52=−92.6661922776228x_{52} = -92.6661922776228
x53=−29.811598790893x_{53} = -29.811598790893
x54=92.6661922776228x_{54} = 92.6661922776228
x55=−64.3871195905574x_{55} = -64.3871195905574
x56=32.9563890398225x_{56} = 32.9563890398225
x57=20.3713029592876x_{57} = 20.3713029592876
x58=48.6741442319544x_{58} = 48.6741442319544
x59=45.5311340139913x_{59} = 45.5311340139913
x60=36.1006222443756x_{60} = 36.1006222443756
x61=70.6716857116195x_{61} = 70.6716857116195
x62=83.2401924707234x_{62} = 83.2401924707234
x63=95.8081387868617x_{63} = 95.8081387868617
x64=−73.8138806006806x_{64} = -73.8138806006806
x65=42.3879135681319x_{65} = 42.3879135681319
x66=−51.8169824872797x_{66} = -51.8169824872797
Signos de extremos en los puntos:
(17.22075527193077, -0.0579718023461539)

(-80.09812862894512, -0.012483713321779)

(-83.2401924707234, 0.0120125604820527)

(98.95006282433188, -0.010105591736504)

(-45.53113401399128, 0.0219576982284824)

(-86.38222203472871, -0.0115756804584678)

(7.725251836937707, 0.128374553525899)

(-394.26734168088706, -0.00253634191261283)

(-4.493409457909064, -0.217233628211222)

(4.493409457909064, -0.217233628211222)

(108.37571965167469, 0.00922676625078197)

(39.24443236116419, 0.0254730530928808)

(-70.6716857116195, 0.0141485220648664)

(10.904121659428899, -0.0913252028230577)

(-42.38791356813192, -0.0235850682290164)

(80.09812862894512, -0.012483713321779)

(89.52422093041719, 0.0111694646341736)

(-48.674144231954386, -0.0205404540417537)

(3306.525965471054, 0.000302432209730106)

(14.066193912831473, 0.0709134594504622)

(-36.10062224437561, -0.0276897323011492)

(-95.8081387868617, 0.0104369581345658)

(64.38711959055742, 0.0155291838074613)

(61.2447302603744, -0.0163257593209978)

(-54.959678287888934, -0.0181921463218031)

(76.95602631033118, 0.0129933369870427)

(-76.95602631033118, 0.0129933369870427)

(0, 1)

(-98.95006282433188, -0.010105591736504)

(-7.725251836937707, 0.128374553525899)

(-20.37130295928756, 0.0490296240140742)

(-39.24443236116419, 0.0254730530928808)

(-14.066193912831473, 0.0709134594504622)

(-32.956389039822476, 0.0303291711863103)

(54.959678287888934, -0.0181921463218031)

(73.81388060068065, -0.01354634434514)

(26.666054258812675, 0.0374745199939312)

(-26.666054258812675, 0.0374745199939312)

(-61.2447302603744, -0.0163257593209978)

(-67.52943477714412, -0.0148067339465492)

(29.81159879089296, -0.0335251350213988)

(51.81698248727967, 0.019295099487588)

(23.519452498689006, -0.0424796169776126)

(-58.10225475449559, 0.0172084874716279)

(67.52943477714412, -0.0148067339465492)

(-10.904121659428899, -0.0913252028230577)

(-89.52422093041719, 0.0111694646341736)

(86.38222203472871, -0.0115756804584678)

(-23.519452498689006, -0.0424796169776126)

(-17.22075527193077, -0.0579718023461539)

(58.10225475449559, 0.0172084874716279)

(-92.66619227762284, -0.0107907938495342)

(-29.81159879089296, -0.0335251350213988)

(92.66619227762284, -0.0107907938495342)

(-64.38711959055742, 0.0155291838074613)

(32.956389039822476, 0.0303291711863103)

(20.37130295928756, 0.0490296240140742)

(48.674144231954386, -0.0205404540417537)

(45.53113401399128, 0.0219576982284824)

(36.10062224437561, -0.0276897323011492)

(70.6716857116195, 0.0141485220648664)

(83.2401924707234, 0.0120125604820527)

(95.8081387868617, 0.0104369581345658)

(-73.81388060068065, -0.01354634434514)

(42.38791356813192, -0.0235850682290164)

(-51.81698248727967, 0.019295099487588)


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=17.2207552719308x_{1} = 17.2207552719308
x2=−80.0981286289451x_{2} = -80.0981286289451
x3=98.9500628243319x_{3} = 98.9500628243319
x4=−86.3822220347287x_{4} = -86.3822220347287
x5=−394.267341680887x_{5} = -394.267341680887
x6=−4.49340945790906x_{6} = -4.49340945790906
x7=4.49340945790906x_{7} = 4.49340945790906
x8=10.9041216594289x_{8} = 10.9041216594289
x9=−42.3879135681319x_{9} = -42.3879135681319
x10=80.0981286289451x_{10} = 80.0981286289451
x11=−48.6741442319544x_{11} = -48.6741442319544
x12=−36.1006222443756x_{12} = -36.1006222443756
x13=61.2447302603744x_{13} = 61.2447302603744
x14=−54.9596782878889x_{14} = -54.9596782878889
x15=−98.9500628243319x_{15} = -98.9500628243319
x16=54.9596782878889x_{16} = 54.9596782878889
x17=73.8138806006806x_{17} = 73.8138806006806
x18=−61.2447302603744x_{18} = -61.2447302603744
x19=−67.5294347771441x_{19} = -67.5294347771441
x20=29.811598790893x_{20} = 29.811598790893
x21=23.519452498689x_{21} = 23.519452498689
x22=67.5294347771441x_{22} = 67.5294347771441
x23=−10.9041216594289x_{23} = -10.9041216594289
x24=86.3822220347287x_{24} = 86.3822220347287
x25=−23.519452498689x_{25} = -23.519452498689
x26=−17.2207552719308x_{26} = -17.2207552719308
x27=−92.6661922776228x_{27} = -92.6661922776228
x28=−29.811598790893x_{28} = -29.811598790893
x29=92.6661922776228x_{29} = 92.6661922776228
x30=48.6741442319544x_{30} = 48.6741442319544
x31=36.1006222443756x_{31} = 36.1006222443756
x32=−73.8138806006806x_{32} = -73.8138806006806
x33=42.3879135681319x_{33} = 42.3879135681319
Puntos máximos de la función:
x33=−83.2401924707234x_{33} = -83.2401924707234
x33=−45.5311340139913x_{33} = -45.5311340139913
x33=7.72525183693771x_{33} = 7.72525183693771
x33=108.375719651675x_{33} = 108.375719651675
x33=39.2444323611642x_{33} = 39.2444323611642
x33=−70.6716857116195x_{33} = -70.6716857116195
x33=89.5242209304172x_{33} = 89.5242209304172
x33=3306.52596547105x_{33} = 3306.52596547105
x33=14.0661939128315x_{33} = 14.0661939128315
x33=−95.8081387868617x_{33} = -95.8081387868617
x33=64.3871195905574x_{33} = 64.3871195905574
x33=76.9560263103312x_{33} = 76.9560263103312
x33=−76.9560263103312x_{33} = -76.9560263103312
x33=0x_{33} = 0
x33=−7.72525183693771x_{33} = -7.72525183693771
x33=−20.3713029592876x_{33} = -20.3713029592876
x33=−39.2444323611642x_{33} = -39.2444323611642
x33=−14.0661939128315x_{33} = -14.0661939128315
x33=−32.9563890398225x_{33} = -32.9563890398225
x33=26.6660542588127x_{33} = 26.6660542588127
x33=−26.6660542588127x_{33} = -26.6660542588127
x33=51.8169824872797x_{33} = 51.8169824872797
x33=−58.1022547544956x_{33} = -58.1022547544956
x33=−89.5242209304172x_{33} = -89.5242209304172
x33=58.1022547544956x_{33} = 58.1022547544956
x33=−64.3871195905574x_{33} = -64.3871195905574
x33=32.9563890398225x_{33} = 32.9563890398225
x33=20.3713029592876x_{33} = 20.3713029592876
x33=45.5311340139913x_{33} = 45.5311340139913
x33=70.6716857116195x_{33} = 70.6716857116195
x33=83.2401924707234x_{33} = 83.2401924707234
x33=95.8081387868617x_{33} = 95.8081387868617
x33=−51.8169824872797x_{33} = -51.8169824872797
Decrece en los intervalos
[98.9500628243319,∞)\left[98.9500628243319, \infty\right)
Crece en los intervalos
(−∞,−394.267341680887]\left(-\infty, -394.267341680887\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
{−sin⁡(x)+2(xcos⁡(x)−sin⁡(x))x2xfor x≠00otherwise=0\begin{cases} - \frac{\sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−12.404445021902x_{1} = -12.404445021902
x2=−47.0813974121542x_{2} = -47.0813974121542
x3=−18.7426455847748x_{3} = -18.7426455847748
x4=−91.0842274914688x_{4} = -91.0842274914688
x5=9.20584014293667x_{5} = 9.20584014293667
x6=−84.7994143922025x_{6} = -84.7994143922025
x7=21.8996964794928x_{7} = 21.8996964794928
x8=−342.42775856009x_{8} = -342.42775856009
x9=−87.9418500396598x_{9} = -87.9418500396598
x10=−1288.05143523817x_{10} = -1288.05143523817
x11=−59.6567290035279x_{11} = -59.6567290035279
x12=100.511065295271x_{12} = 100.511065295271
x13=−43.9367614714198x_{13} = -43.9367614714198
x14=91.0842274914688x_{14} = 91.0842274914688
x15=−15.5792364103872x_{15} = -15.5792364103872
x16=43.9367614714198x_{16} = 43.9367614714198
x17=−50.2256516491831x_{17} = -50.2256516491831
x18=31.3520917265645x_{18} = 31.3520917265645
x19=53.3695918204908x_{19} = 53.3695918204908
x20=59.6567290035279x_{20} = 59.6567290035279
x21=34.499514921367x_{21} = 34.499514921367
x22=0x_{22} = 0
x23=131.931731514843x_{23} = 131.931731514843
x24=47.0813974121542x_{24} = 47.0813974121542
x25=94.2265525745684x_{25} = 94.2265525745684
x26=−78.5143405319308x_{26} = -78.5143405319308
x27=−40.7916552312719x_{27} = -40.7916552312719
x28=75.3716854092873x_{28} = 75.3716854092873
x29=69.0860849466452x_{29} = 69.0860849466452
x30=−9.20584014293667x_{30} = -9.20584014293667
x31=65.9431119046552x_{31} = 65.9431119046552
x32=−28.2033610039524x_{32} = -28.2033610039524
x33=−81.6569138240367x_{33} = -81.6569138240367
x34=25.052825280993x_{34} = 25.052825280993
x35=−2.0815759778181x_{35} = -2.0815759778181
x36=−62.8000005565198x_{36} = -62.8000005565198
x37=−94.2265525745684x_{37} = -94.2265525745684
x38=−25.052825280993x_{38} = -25.052825280993
x39=−34.499514921367x_{39} = -34.499514921367
x40=−37.6459603230864x_{40} = -37.6459603230864
x41=28.2033610039524x_{41} = 28.2033610039524
x42=81.6569138240367x_{42} = 81.6569138240367
x43=78.5143405319308x_{43} = 78.5143405319308
x44=56.5132704621986x_{44} = 56.5132704621986
x45=15.5792364103872x_{45} = 15.5792364103872
x46=50.2256516491831x_{46} = 50.2256516491831
x47=97.368830362901x_{47} = 97.368830362901
x48=62.8000005565198x_{48} = 62.8000005565198
x49=−1790.70669566846x_{49} = -1790.70669566846
x50=−65.9431119046552x_{50} = -65.9431119046552
x51=87.9418500396598x_{51} = 87.9418500396598
x52=40.7916552312719x_{52} = 40.7916552312719
x53=18.7426455847748x_{53} = 18.7426455847748
x54=84.7994143922025x_{54} = 84.7994143922025
x55=−56.5132704621986x_{55} = -56.5132704621986
x56=37.6459603230864x_{56} = 37.6459603230864
x57=72.2289377620154x_{57} = 72.2289377620154
x58=−100.511065295271x_{58} = -100.511065295271
x59=−53.3695918204908x_{59} = -53.3695918204908
x60=−5.94036999057271x_{60} = -5.94036999057271
x61=−31.3520917265645x_{61} = -31.3520917265645
x62=2.0815759778181x_{62} = 2.0815759778181
x63=5.94036999057271x_{63} = 5.94036999057271
x64=12.404445021902x_{64} = 12.404445021902
x65=−97.368830362901x_{65} = -97.368830362901
x66=−21.8996964794928x_{66} = -21.8996964794928
x67=−69.0860849466452x_{67} = -69.0860849466452
x68=−72.2289377620154x_{68} = -72.2289377620154
x69=−75.3716854092873x_{69} = -75.3716854092873

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
[97.368830362901,∞)\left[97.368830362901, \infty\right)
Convexa en los intervalos
(−∞,−1790.70669566846]\left(-\infty, -1790.70669566846\right]
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sinc(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sinc⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{sinc}{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sinc⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\operatorname{sinc}{\left(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:
sinc⁡(x)=sinc⁡(x)\operatorname{sinc}{\left(x \right)} = \operatorname{sinc}{\left(x \right)}
- Sí
sinc⁡(x)=−sinc⁡(x)\operatorname{sinc}{\left(x \right)} = - \operatorname{sinc}{\left(x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = sinc(x)
    © Sr Examen – Калькулятор