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sin(x)/sqrt(x)
Trazado el gráfico de la función/ sin(x)/sqrt(x)

Gráfico de la función y = sin(x)/sqrt(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(x)
f(x) = ------
         ___ 
       \/ x
f(x)=sin⁡(x)xf{\left(x \right)} = \frac{\sin{\left(x \right)}}{\sqrt{x}} 
f = sin(x)/sqrt(x)
Gráfico de la función
02468-8-6-4-2-10101-1
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:
sin⁡(x)x=0\frac{\sin{\left(x \right)}}{\sqrt{x}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=πx_{1} = \pi
Solución numérica
x1=521.504380495906x_{1} = 521.504380495906
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=−62.8318530717959x_{3} = -62.8318530717959
x4=−97.3893722612836x_{4} = -97.3893722612836
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=87.9645943005142x_{6} = 87.9645943005142
x7=69.1150383789755x_{7} = 69.1150383789755
x8=31.4159265358979x_{8} = 31.4159265358979
x9=−37.6991118430775x_{9} = -37.6991118430775
x10=−81.6814089933346x_{10} = -81.6814089933346
x11=−84.8230016469244x_{11} = -84.8230016469244
x12=−21.9911485751286x_{12} = -21.9911485751286
x13=47.1238898038469x_{13} = 47.1238898038469
x14=−15.707963267949x_{14} = -15.707963267949
x15=−12.5663706143592x_{15} = -12.5663706143592
x16=12.5663706143592x_{16} = 12.5663706143592
x17=−87.9645943005142x_{17} = -87.9645943005142
x18=53.4070751110265x_{18} = 53.4070751110265
x19=72.2566310325652x_{19} = 72.2566310325652
x20=−100.530964914873x_{20} = -100.530964914873
x21=−3.14159265358979x_{21} = -3.14159265358979
x22=34.5575191894877x_{22} = 34.5575191894877
x23=131.946891450771x_{23} = 131.946891450771
x24=−94.2477796076938x_{24} = -94.2477796076938
x25=6.28318530717959x_{25} = 6.28318530717959
x26=−69.1150383789755x_{26} = -69.1150383789755
x27=97.3893722612836x_{27} = 97.3893722612836
x28=65.9734457253857x_{28} = 65.9734457253857
x29=15.707963267949x_{29} = 15.707963267949
x30=−50.2654824574367x_{30} = -50.2654824574367
x31=−25.1327412287183x_{31} = -25.1327412287183
x32=3.14159265358979x_{32} = 3.14159265358979
x33=−18.8495559215388x_{33} = -18.8495559215388
x34=40.8407044966673x_{34} = 40.8407044966673
x35=119.380520836412x_{35} = 119.380520836412
x36=−53.4070751110265x_{36} = -53.4070751110265
x37=37.6991118430775x_{37} = 37.6991118430775
x38=−43.9822971502571x_{38} = -43.9822971502571
x39=18.8495559215388x_{39} = 18.8495559215388
x40=−78.5398163397448x_{40} = -78.5398163397448
x41=−6.28318530717959x_{41} = -6.28318530717959
x42=−40.8407044966673x_{42} = -40.8407044966673
x43=43.9822971502571x_{43} = 43.9822971502571
x44=56.5486677646163x_{44} = 56.5486677646163
x45=−65.9734457253857x_{45} = -65.9734457253857
x46=25.1327412287183x_{46} = 25.1327412287183
x47=78.5398163397448x_{47} = 78.5398163397448
x48=−28.2743338823081x_{48} = -28.2743338823081
x49=75.398223686155x_{49} = 75.398223686155
x50=59.6902604182061x_{50} = 59.6902604182061
x51=−34.5575191894877x_{51} = -34.5575191894877
x52=81.6814089933346x_{52} = 81.6814089933346
x53=−47.1238898038469x_{53} = -47.1238898038469
x54=100.530964914873x_{54} = 100.530964914873
x55=−9.42477796076938x_{55} = -9.42477796076938
x56=−75.398223686155x_{56} = -75.398223686155
x57=−72.2566310325652x_{57} = -72.2566310325652
x58=−31.4159265358979x_{58} = -31.4159265358979
x59=28.2743338823081x_{59} = 28.2743338823081
x60=−91.106186954104x_{60} = -91.106186954104
x61=21.9911485751286x_{61} = 21.9911485751286
x62=62.8318530717959x_{62} = 62.8318530717959
x63=9.42477796076938x_{63} = 9.42477796076938
x64=50.2654824574367x_{64} = 50.2654824574367
x65=94.2477796076938x_{65} = 94.2477796076938
x66=91.106186954104x_{66} = 91.106186954104
x67=84.8230016469244x_{67} = 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 sin(x)/sqrt(x).
sin⁡(0)0\frac{\sin{\left(0 \right)}}{\sqrt{0}}
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
cos⁡(x)x−sin⁡(x)2x32=0\frac{\cos{\left(x \right)}}{\sqrt{x}} - \frac{\sin{\left(x \right)}}{2 x^{\frac{3}{2}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=39.2571723324086x_{1} = 39.2571723324086
x2=92.6715879363332x_{2} = 92.6715879363332
x3=67.5368388204916x_{3} = 67.5368388204916
x4=−36.1144715353049x_{4} = -36.1144715353049
x5=58.1108600600615x_{5} = 58.1108600600615
x6=17.2497818346079x_{6} = 17.2497818346079
x7=−54.9687756155963x_{7} = -54.9687756155963
x8=−73.8206542907788x_{8} = -73.8206542907788
x9=−80.1043708909521x_{9} = -80.1043708909521
x10=−61.2528940466862x_{10} = -61.2528940466862
x11=14.1017251335659x_{11} = 14.1017251335659
x12=10.9499436485412x_{12} = 10.9499436485412
x13=95.8133575027966x_{13} = 95.8133575027966
x14=86.3880101981266x_{14} = 86.3880101981266
x15=51.8266315338985x_{15} = 51.8266315338985
x16=215.19677332017x_{16} = 215.19677332017
x17=61.2528940466862x_{17} = 61.2528940466862
x18=−7.78988375114457x_{18} = -7.78988375114457
x19=−95.8133575027966x_{19} = -95.8133575027966
x20=−10.9499436485412x_{20} = -10.9499436485412
x21=−23.5407082923052x_{21} = -23.5407082923052
x22=42.3997088362447x_{22} = 42.3997088362447
x23=−98.9551158352145x_{23} = -98.9551158352145
x24=−83.2461991121237x_{24} = -83.2461991121237
x25=−20.3958423573092x_{25} = -20.3958423573092
x26=−76.9625234358705x_{26} = -76.9625234358705
x27=−92.6715879363332x_{27} = -92.6715879363332
x28=64.3948849627586x_{28} = 64.3948849627586
x29=−51.8266315338985x_{29} = -51.8266315338985
x30=36.1144715353049x_{30} = 36.1144715353049
x31=158.64727737108x_{31} = 158.64727737108
x32=83.2461991121237x_{32} = 83.2461991121237
x33=98.9551158352145x_{33} = 98.9551158352145
x34=70.6787605627689x_{34} = 70.6787605627689
x35=20.3958423573092x_{35} = 20.3958423573092
x36=−29.8283692130955x_{36} = -29.8283692130955
x37=−45.5421150692309x_{37} = -45.5421150692309
x38=73.8206542907788x_{38} = 73.8206542907788
x39=48.6844162648433x_{39} = 48.6844162648433
x40=−39.2571723324086x_{40} = -39.2571723324086
x41=32.9715594404485x_{41} = 32.9715594404485
x42=−67.5368388204916x_{42} = -67.5368388204916
x43=−70.6787605627689x_{43} = -70.6787605627689
x44=−246.612995841404x_{44} = -246.612995841404
x45=−4.60421677720058x_{45} = -4.60421677720058
x46=−26.6848024909251x_{46} = -26.6848024909251
x47=1.16556118520721x_{47} = 1.16556118520721
x48=89.5298059530594x_{48} = 89.5298059530594
x49=−1.16556118520721x_{49} = -1.16556118520721
x50=−32.9715594404485x_{50} = -32.9715594404485
x51=−14.1017251335659x_{51} = -14.1017251335659
x52=23.5407082923052x_{52} = 23.5407082923052
x53=−17.2497818346079x_{53} = -17.2497818346079
x54=54.9687756155963x_{54} = 54.9687756155963
x55=7.78988375114457x_{55} = 7.78988375114457
x56=−89.5298059530594x_{56} = -89.5298059530594
x57=80.1043708909521x_{57} = 80.1043708909521
x58=−86.3880101981266x_{58} = -86.3880101981266
x59=−64.3948849627586x_{59} = -64.3948849627586
x60=45.5421150692309x_{60} = 45.5421150692309
x61=76.9625234358705x_{61} = 76.9625234358705
x62=4.60421677720058x_{62} = 4.60421677720058
x63=29.8283692130955x_{63} = 29.8283692130955
x64=−48.6844162648433x_{64} = -48.6844162648433
x65=−42.3997088362447x_{65} = -42.3997088362447
x66=26.6848024909251x_{66} = 26.6848024909251
x67=117.80548025038x_{67} = 117.80548025038
x68=−58.1108600600615x_{68} = -58.1108600600615
Signos de extremos en los puntos:
(39.25717233240859, 0.159589851348603)

(92.67158793633321, -0.103877233902111)

(67.53683882049161, -0.121679588990783)

(-36.11447153530485, -0.166386370791913*I)

(58.110860060061505, 0.131176268600912)

(17.249781834607894, -0.240672145897842)

(-54.96877561559635, -0.134872684738376*I)

(-73.82065429077876, -0.116386094038002*I)

(-80.1043708909521, -0.111728362291416*I)

(-61.252894046686194, -0.127768037744087*I)

(14.101725133565873, 0.266128298234218)

(10.94994364854116, -0.301885161430297)

(95.81335750279658, 0.102160040658152)

(86.38801019812658, -0.107588534144322)

(51.82663153389846, 0.138900336703391)

(215.1967733201699, 0.0681680624478802)

(61.252894046686194, -0.127768037744087)

(-7.789883751144573, 0.357554083426262*I)

(-95.81335750279658, 0.102160040658152*I)

(-10.94994364854116, -0.301885161430297*I)

(-23.54070829230515, -0.206059336815155*I)

(42.39970883624466, -0.15356362930828)

(-98.95511583521451, -0.100525289012326*I)

(-83.24619911212368, 0.109599849994829*I)

(-20.395842357309167, 0.221359780635401*I)

(-76.96252343587051, 0.113985913925499*I)

(-92.67158793633321, -0.103877233902111*I)

(64.39488496275855, 0.124612389237314)

(-51.82663153389846, 0.138900336703391*I)

(36.11447153530485, -0.166386370791913)

(158.6472773710796, 0.0793928754394215)

(83.24619911212368, 0.109599849994829)

(98.95511583521451, -0.100525289012326)

(70.67876056276886, 0.118944583684481)

(20.395842357309167, 0.221359780635401)

(-29.828369213095506, -0.183072974858657*I)

(-45.5421150692309, 0.148172370731446*I)

(73.82065429077876, -0.116386094038002)

(48.68441626484328, -0.143311853691665)

(-39.25717233240859, 0.159589851348603*I)

(32.97155944044848, 0.17413269656851)

(-67.53683882049161, -0.121679588990783*I)

(-70.67876056276886, 0.118944583684481*I)

(-246.61299584140428, 0.0636782512070729*I)

(-4.604216777200577, -0.463314891176637*I)

(-26.68480249092507, 0.19354937797769*I)

(1.1655611852072114, 0.851241066782324)

(89.52980595305935, 0.105684039776562)

(-1.1655611852072114, 0.851241066782324*I)

(-32.97155944044848, 0.17413269656851*I)

(-14.101725133565873, 0.266128298234218*I)

(23.54070829230515, -0.206059336815155)

(-17.249781834607894, -0.240672145897842*I)

(54.96877561559635, -0.134872684738376)

(7.789883751144573, 0.357554083426262)

(-89.52980595305935, 0.105684039776562*I)

(80.1043708909521, -0.111728362291416)

(-86.38801019812658, -0.107588534144322*I)

(-64.39488496275855, 0.124612389237314*I)

(45.5421150692309, 0.148172370731446)

(76.96252343587051, 0.113985913925499)

(4.604216777200577, -0.463314891176637)

(29.828369213095506, -0.183072974858657)

(-48.68441626484328, -0.143311853691665*I)

(-42.39970883624466, -0.15356362930828*I)

(26.68480249092507, 0.19354937797769)

(117.80548025038037, -0.0921326029924126)

(-58.110860060061505, 0.131176268600912*I)


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=92.6715879363332x_{1} = 92.6715879363332
x2=67.5368388204916x_{2} = 67.5368388204916
x3=17.2497818346079x_{3} = 17.2497818346079
x4=10.9499436485412x_{4} = 10.9499436485412
x5=86.3880101981266x_{5} = 86.3880101981266
x6=61.2528940466862x_{6} = 61.2528940466862
x7=42.3997088362447x_{7} = 42.3997088362447
x8=36.1144715353049x_{8} = 36.1144715353049
x9=98.9551158352145x_{9} = 98.9551158352145
x10=73.8206542907788x_{10} = 73.8206542907788
x11=48.6844162648433x_{11} = 48.6844162648433
x12=23.5407082923052x_{12} = 23.5407082923052
x13=54.9687756155963x_{13} = 54.9687756155963
x14=80.1043708909521x_{14} = 80.1043708909521
x15=4.60421677720058x_{15} = 4.60421677720058
x16=29.8283692130955x_{16} = 29.8283692130955
x17=117.80548025038x_{17} = 117.80548025038
Puntos máximos de la función:
x17=39.2571723324086x_{17} = 39.2571723324086
x17=58.1108600600615x_{17} = 58.1108600600615
x17=14.1017251335659x_{17} = 14.1017251335659
x17=95.8133575027966x_{17} = 95.8133575027966
x17=51.8266315338985x_{17} = 51.8266315338985
x17=215.19677332017x_{17} = 215.19677332017
x17=64.3948849627586x_{17} = 64.3948849627586
x17=158.64727737108x_{17} = 158.64727737108
x17=83.2461991121237x_{17} = 83.2461991121237
x17=70.6787605627689x_{17} = 70.6787605627689
x17=20.3958423573092x_{17} = 20.3958423573092
x17=32.9715594404485x_{17} = 32.9715594404485
x17=1.16556118520721x_{17} = 1.16556118520721
x17=89.5298059530594x_{17} = 89.5298059530594
x17=7.78988375114457x_{17} = 7.78988375114457
x17=45.5421150692309x_{17} = 45.5421150692309
x17=76.9625234358705x_{17} = 76.9625234358705
x17=26.6848024909251x_{17} = 26.6848024909251
Decrece en los intervalos
[117.80548025038,∞)\left[117.80548025038, \infty\right)
Crece en los intervalos
(−∞,4.60421677720058]\left(-\infty, 4.60421677720058\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)−cos⁡(x)x+3sin⁡(x)4x2x=0\frac{- \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{4 x^{2}}}{\sqrt{x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=47.1026555912318x_{1} = 47.1026555912318
x2=94.2371675854493x_{2} = 94.2371675854493
x3=100.521016336234x_{3} = 100.521016336234
x4=50.2455769233645x_{4} = 50.2455769233645
x5=−113.088492608463x_{5} = -113.088492608463
x6=−18.796291187414x_{6} = -18.796291187414
x7=−9.31693112610028x_{7} = -9.31693112610028
x8=−91.0952088771736x_{8} = -91.0952088771736
x9=18.796291187414x_{9} = 18.796291187414
x10=87.95322400825x_{10} = 87.95322400825
x11=2.75936321522763x_{11} = 2.75936321522763
x12=−15.6439318755503x_{12} = -15.6439318755503
x13=91.0952088771736x_{13} = 91.0952088771736
x14=72.2427877152145x_{14} = 72.2427877152145
x15=−59.6735006001685x_{15} = -59.6735006001685
x16=62.8159318625173x_{16} = 62.8159318625173
x17=169.640108376141x_{17} = 169.640108376141
x18=−97.3791026663451x_{18} = -97.3791026663451
x19=56.5309760413753x_{19} = 56.5309760413753
x20=15.6439318755503x_{20} = 15.6439318755503
x21=−12.4860672578708x_{21} = -12.4860672578708
x22=−69.1005654545348x_{22} = -69.1005654545348
x23=6.11791002392407x_{23} = 6.11791002392407
x24=−56.5309760413753x_{24} = -56.5309760413753
x25=−34.5285475249278x_{25} = -34.5285475249278
x26=34.5285475249278x_{26} = 34.5285475249278
x27=25.0928628865337x_{27} = 25.0928628865337
x28=97.3791026663451x_{28} = 97.3791026663451
x29=43.9595440566684x_{29} = 43.9595440566684
x30=−21.9455418081046x_{30} = -21.9455418081046
x31=69.1005654545348x_{31} = 69.1005654545348
x32=−87.95322400825x_{32} = -87.95322400825
x33=−31.3840497369889x_{33} = -31.3840497369889
x34=75.384957467622x_{34} = 75.384957467622
x35=−28.2389032383054x_{35} = -28.2389032383054
x36=−37.6725595300203x_{36} = -37.6725595300203
x37=−43.9595440566684x_{37} = -43.9595440566684
x38=53.3883416918471x_{38} = 53.3883416918471
x39=−94.2371675854493x_{39} = -94.2371675854493
x40=−147.648081727825x_{40} = -147.648081727825
x41=40.8161982919721x_{41} = 40.8161982919721
x42=−25.0928628865337x_{42} = -25.0928628865337
x43=21.9455418081046x_{43} = 21.9455418081046
x44=−81.6691637048431x_{44} = -81.6691637048431
x45=28.2389032383054x_{45} = 28.2389032383054
x46=12.4860672578708x_{46} = 12.4860672578708
x47=84.8112100697664x_{47} = 84.8112100697664
x48=−6.11791002392407x_{48} = -6.11791002392407
x49=−62.8159318625173x_{49} = -62.8159318625173
x50=−72.2427877152145x_{50} = -72.2427877152145
x51=78.5270810189266x_{51} = 78.5270810189266
x52=−75.384957467622x_{52} = -75.384957467622
x53=−40.8161982919721x_{53} = -40.8161982919721
x54=−50.2455769233645x_{54} = -50.2455769233645
x55=−100.521016336234x_{55} = -100.521016336234
x56=9.31693112610028x_{56} = 9.31693112610028
x57=31.3840497369889x_{57} = 31.3840497369889
x58=−78.5270810189266x_{58} = -78.5270810189266
x59=−47.1026555912318x_{59} = -47.1026555912318
x60=65.9582831752547x_{60} = 65.9582831752547
x61=−53.3883416918471x_{61} = -53.3883416918471
x62=−2.75936321522763x_{62} = -2.75936321522763
x63=37.6725595300203x_{63} = 37.6725595300203
x64=59.6735006001685x_{64} = 59.6735006001685
x65=−65.9582831752547x_{65} = -65.9582831752547
x66=−84.8112100697664x_{66} = -84.8112100697664
x67=81.6691637048431x_{67} = 81.6691637048431
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

lim⁡x→0−(−sin⁡(x)−cos⁡(x)x+3sin⁡(x)4x2x)=−∞i\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{4 x^{2}}}{\sqrt{x}}\right) = - \infty i
lim⁡x→0+(−sin⁡(x)−cos⁡(x)x+3sin⁡(x)4x2x)=−∞\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{4 x^{2}}}{\sqrt{x}}\right) = -\infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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
[97.3791026663451,∞)\left[97.3791026663451, \infty\right)
Convexa en los intervalos
(−∞,2.75936321522763]\left(-\infty, 2.75936321522763\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{x}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{x}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)/sqrt(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x)xx)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{x} x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x)xx)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{x} x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Gráfico
Gráfico de la función y = sin(x)/sqrt(x)
    © Sr Examen – Калькулятор