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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
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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=43.9822971502571x_{1} = 43.9822971502571
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=−43.9822971502571x_{3} = -43.9822971502571
x4=−72.2566310325652x_{4} = -72.2566310325652
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=81.6814089933346x_{6} = 81.6814089933346
x7=−31.4159265358979x_{7} = -31.4159265358979
x8=−78.5398163397448x_{8} = -78.5398163397448
x9=97.3893722612836x_{9} = 97.3893722612836
x10=9.42477796076938x_{10} = 9.42477796076938
x11=−25.1327412287183x_{11} = -25.1327412287183
x12=84.8230016469244x_{12} = 84.8230016469244
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=6.28318530717959x_{15} = 6.28318530717959
x16=119.380520836412x_{16} = 119.380520836412
x17=3.14159265358979x_{17} = 3.14159265358979
x18=131.946891450771x_{18} = 131.946891450771
x19=−50.2654824574367x_{19} = -50.2654824574367
x20=28.2743338823081x_{20} = 28.2743338823081
x21=−75.398223686155x_{21} = -75.398223686155
x22=−28.2743338823081x_{22} = -28.2743338823081
x23=−56.5486677646163x_{23} = -56.5486677646163
x24=−65.9734457253857x_{24} = -65.9734457253857
x25=−40.8407044966673x_{25} = -40.8407044966673
x26=−91.106186954104x_{26} = -91.106186954104
x27=50.2654824574367x_{27} = 50.2654824574367
x28=−69.1150383789755x_{28} = -69.1150383789755
x29=−100.530964914873x_{29} = -100.530964914873
x30=56.5486677646163x_{30} = 56.5486677646163
x31=−62.8318530717959x_{31} = -62.8318530717959
x32=−87.9645943005142x_{32} = -87.9645943005142
x33=40.8407044966673x_{33} = 40.8407044966673
x34=100.530964914873x_{34} = 100.530964914873
x35=18.8495559215388x_{35} = 18.8495559215388
x36=62.8318530717959x_{36} = 62.8318530717959
x37=−53.4070751110265x_{37} = -53.4070751110265
x38=94.2477796076938x_{38} = 94.2477796076938
x39=−3.14159265358979x_{39} = -3.14159265358979
x40=21.9911485751286x_{40} = 21.9911485751286
x41=12.5663706143592x_{41} = 12.5663706143592
x42=−84.8230016469244x_{42} = -84.8230016469244
x43=34.5575191894877x_{43} = 34.5575191894877
x44=47.1238898038469x_{44} = 47.1238898038469
x45=−15.707963267949x_{45} = -15.707963267949
x46=53.4070751110265x_{46} = 53.4070751110265
x47=65.9734457253857x_{47} = 65.9734457253857
x48=87.9645943005142x_{48} = 87.9645943005142
x49=91.106186954104x_{49} = 91.106186954104
x50=59.6902604182061x_{50} = 59.6902604182061
x51=69.1150383789755x_{51} = 69.1150383789755
x52=−6.28318530717959x_{52} = -6.28318530717959
x53=75.398223686155x_{53} = 75.398223686155
x54=−37.6991118430775x_{54} = -37.6991118430775
x55=−12.5663706143592x_{55} = -12.5663706143592
x56=−18.8495559215388x_{56} = -18.8495559215388
x57=31.4159265358979x_{57} = 31.4159265358979
x58=−81.6814089933346x_{58} = -81.6814089933346
x59=78.5398163397448x_{59} = 78.5398163397448
x60=15.707963267949x_{60} = 15.707963267949
x61=72.2566310325652x_{61} = 72.2566310325652
x62=37.6991118430775x_{62} = 37.6991118430775
x63=25.1327412287183x_{63} = 25.1327412287183
x64=−47.1238898038469x_{64} = -47.1238898038469
x65=521.504380495906x_{65} = 521.504380495906
x66=−9.42477796076938x_{66} = -9.42477796076938
x67=−34.5575191894877x_{67} = -34.5575191894877
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=−58.1108600600615x_{1} = -58.1108600600615
x2=−61.2528940466862x_{2} = -61.2528940466862
x3=−14.1017251335659x_{3} = -14.1017251335659
x4=54.9687756155963x_{4} = 54.9687756155963
x5=−246.612995841404x_{5} = -246.612995841404
x6=−36.1144715353049x_{6} = -36.1144715353049
x7=−92.6715879363332x_{7} = -92.6715879363332
x8=−73.8206542907788x_{8} = -73.8206542907788
x9=80.1043708909521x_{9} = 80.1043708909521
x10=26.6848024909251x_{10} = 26.6848024909251
x11=7.78988375114457x_{11} = 7.78988375114457
x12=−45.5421150692309x_{12} = -45.5421150692309
x13=215.19677332017x_{13} = 215.19677332017
x14=−76.9625234358705x_{14} = -76.9625234358705
x15=−98.9551158352145x_{15} = -98.9551158352145
x16=−67.5368388204916x_{16} = -67.5368388204916
x17=−32.9715594404485x_{17} = -32.9715594404485
x18=14.1017251335659x_{18} = 14.1017251335659
x19=−7.78988375114457x_{19} = -7.78988375114457
x20=−20.3958423573092x_{20} = -20.3958423573092
x21=98.9551158352145x_{21} = 98.9551158352145
x22=−17.2497818346079x_{22} = -17.2497818346079
x23=4.60421677720058x_{23} = 4.60421677720058
x24=70.6787605627689x_{24} = 70.6787605627689
x25=64.3948849627586x_{25} = 64.3948849627586
x26=83.2461991121237x_{26} = 83.2461991121237
x27=−1.16556118520721x_{27} = -1.16556118520721
x28=−51.8266315338985x_{28} = -51.8266315338985
x29=92.6715879363332x_{29} = 92.6715879363332
x30=158.64727737108x_{30} = 158.64727737108
x31=−80.1043708909521x_{31} = -80.1043708909521
x32=67.5368388204916x_{32} = 67.5368388204916
x33=61.2528940466862x_{33} = 61.2528940466862
x34=−95.8133575027966x_{34} = -95.8133575027966
x35=−83.2461991121237x_{35} = -83.2461991121237
x36=32.9715594404485x_{36} = 32.9715594404485
x37=39.2571723324086x_{37} = 39.2571723324086
x38=−48.6844162648433x_{38} = -48.6844162648433
x39=−26.6848024909251x_{39} = -26.6848024909251
x40=−10.9499436485412x_{40} = -10.9499436485412
x41=−86.3880101981266x_{41} = -86.3880101981266
x42=95.8133575027966x_{42} = 95.8133575027966
x43=1.16556118520721x_{43} = 1.16556118520721
x44=−64.3948849627586x_{44} = -64.3948849627586
x45=−4.60421677720058x_{45} = -4.60421677720058
x46=45.5421150692309x_{46} = 45.5421150692309
x47=−29.8283692130955x_{47} = -29.8283692130955
x48=89.5298059530594x_{48} = 89.5298059530594
x49=76.9625234358705x_{49} = 76.9625234358705
x50=42.3997088362447x_{50} = 42.3997088362447
x51=73.8206542907788x_{51} = 73.8206542907788
x52=58.1108600600615x_{52} = 58.1108600600615
x53=29.8283692130955x_{53} = 29.8283692130955
x54=36.1144715353049x_{54} = 36.1144715353049
x55=−23.5407082923052x_{55} = -23.5407082923052
x56=−39.2571723324086x_{56} = -39.2571723324086
x57=117.80548025038x_{57} = 117.80548025038
x58=−70.6787605627689x_{58} = -70.6787605627689
x59=20.3958423573092x_{59} = 20.3958423573092
x60=10.9499436485412x_{60} = 10.9499436485412
x61=86.3880101981266x_{61} = 86.3880101981266
x62=−54.9687756155963x_{62} = -54.9687756155963
x63=23.5407082923052x_{63} = 23.5407082923052
x64=−89.5298059530594x_{64} = -89.5298059530594
x65=−42.3997088362447x_{65} = -42.3997088362447
x66=48.6844162648433x_{66} = 48.6844162648433
x67=51.8266315338985x_{67} = 51.8266315338985
x68=17.2497818346079x_{68} = 17.2497818346079
Signos de extremos en los puntos:
(-58.110860060061505, 0.131176268600912*I)

(-61.252894046686194, -0.127768037744087*I)

(-14.101725133565873, 0.266128298234218*I)

(54.96877561559635, -0.134872684738376)

(-246.61299584140428, 0.0636782512070729*I)

(-36.11447153530485, -0.166386370791913*I)

(-92.67158793633321, -0.103877233902111*I)

(-73.82065429077876, -0.116386094038002*I)

(80.1043708909521, -0.111728362291416)

(26.68480249092507, 0.19354937797769)

(7.789883751144573, 0.357554083426262)

(-45.5421150692309, 0.148172370731446*I)

(215.1967733201699, 0.0681680624478802)

(-76.96252343587051, 0.113985913925499*I)

(-98.95511583521451, -0.100525289012326*I)

(-67.53683882049161, -0.121679588990783*I)

(-32.97155944044848, 0.17413269656851*I)

(14.101725133565873, 0.266128298234218)

(-7.789883751144573, 0.357554083426262*I)

(-20.395842357309167, 0.221359780635401*I)

(98.95511583521451, -0.100525289012326)

(-17.249781834607894, -0.240672145897842*I)

(4.604216777200577, -0.463314891176637)

(70.67876056276886, 0.118944583684481)

(64.39488496275855, 0.124612389237314)

(83.24619911212368, 0.109599849994829)

(-1.1655611852072114, 0.851241066782324*I)

(-51.82663153389846, 0.138900336703391*I)

(92.67158793633321, -0.103877233902111)

(158.6472773710796, 0.0793928754394215)

(-80.1043708909521, -0.111728362291416*I)

(67.53683882049161, -0.121679588990783)

(61.252894046686194, -0.127768037744087)

(-95.81335750279658, 0.102160040658152*I)

(-83.24619911212368, 0.109599849994829*I)

(32.97155944044848, 0.17413269656851)

(39.25717233240859, 0.159589851348603)

(-48.68441626484328, -0.143311853691665*I)

(-26.68480249092507, 0.19354937797769*I)

(-10.94994364854116, -0.301885161430297*I)

(-86.38801019812658, -0.107588534144322*I)

(95.81335750279658, 0.102160040658152)

(1.1655611852072114, 0.851241066782324)

(-64.39488496275855, 0.124612389237314*I)

(-4.604216777200577, -0.463314891176637*I)

(45.5421150692309, 0.148172370731446)

(-29.828369213095506, -0.183072974858657*I)

(89.52980595305935, 0.105684039776562)

(76.96252343587051, 0.113985913925499)

(42.39970883624466, -0.15356362930828)

(73.82065429077876, -0.116386094038002)

(58.110860060061505, 0.131176268600912)

(29.828369213095506, -0.183072974858657)

(36.11447153530485, -0.166386370791913)

(-23.54070829230515, -0.206059336815155*I)

(-39.25717233240859, 0.159589851348603*I)

(117.80548025038037, -0.0921326029924126)

(-70.67876056276886, 0.118944583684481*I)

(20.395842357309167, 0.221359780635401)

(10.94994364854116, -0.301885161430297)

(86.38801019812658, -0.107588534144322)

(-54.96877561559635, -0.134872684738376*I)

(23.54070829230515, -0.206059336815155)

(-89.52980595305935, 0.105684039776562*I)

(-42.39970883624466, -0.15356362930828*I)

(48.68441626484328, -0.143311853691665)

(51.82663153389846, 0.138900336703391)

(17.249781834607894, -0.240672145897842)


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=54.9687756155963x_{1} = 54.9687756155963
x2=80.1043708909521x_{2} = 80.1043708909521
x3=98.9551158352145x_{3} = 98.9551158352145
x4=4.60421677720058x_{4} = 4.60421677720058
x5=92.6715879363332x_{5} = 92.6715879363332
x6=67.5368388204916x_{6} = 67.5368388204916
x7=61.2528940466862x_{7} = 61.2528940466862
x8=42.3997088362447x_{8} = 42.3997088362447
x9=73.8206542907788x_{9} = 73.8206542907788
x10=29.8283692130955x_{10} = 29.8283692130955
x11=36.1144715353049x_{11} = 36.1144715353049
x12=117.80548025038x_{12} = 117.80548025038
x13=10.9499436485412x_{13} = 10.9499436485412
x14=86.3880101981266x_{14} = 86.3880101981266
x15=23.5407082923052x_{15} = 23.5407082923052
x16=48.6844162648433x_{16} = 48.6844162648433
x17=17.2497818346079x_{17} = 17.2497818346079
Puntos máximos de la función:
x17=26.6848024909251x_{17} = 26.6848024909251
x17=7.78988375114457x_{17} = 7.78988375114457
x17=215.19677332017x_{17} = 215.19677332017
x17=14.1017251335659x_{17} = 14.1017251335659
x17=70.6787605627689x_{17} = 70.6787605627689
x17=64.3948849627586x_{17} = 64.3948849627586
x17=83.2461991121237x_{17} = 83.2461991121237
x17=158.64727737108x_{17} = 158.64727737108
x17=32.9715594404485x_{17} = 32.9715594404485
x17=39.2571723324086x_{17} = 39.2571723324086
x17=95.8133575027966x_{17} = 95.8133575027966
x17=1.16556118520721x_{17} = 1.16556118520721
x17=45.5421150692309x_{17} = 45.5421150692309
x17=89.5298059530594x_{17} = 89.5298059530594
x17=76.9625234358705x_{17} = 76.9625234358705
x17=58.1108600600615x_{17} = 58.1108600600615
x17=20.3958423573092x_{17} = 20.3958423573092
x17=51.8266315338985x_{17} = 51.8266315338985
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=−72.2427877152145x_{1} = -72.2427877152145
x2=−91.0952088771736x_{2} = -91.0952088771736
x3=−53.3883416918471x_{3} = -53.3883416918471
x4=69.1005654545348x_{4} = 69.1005654545348
x5=−97.3791026663451x_{5} = -97.3791026663451
x6=65.9582831752547x_{6} = 65.9582831752547
x7=37.6725595300203x_{7} = 37.6725595300203
x8=−113.088492608463x_{8} = -113.088492608463
x9=−78.5270810189266x_{9} = -78.5270810189266
x10=−75.384957467622x_{10} = -75.384957467622
x11=43.9595440566684x_{11} = 43.9595440566684
x12=−40.8161982919721x_{12} = -40.8161982919721
x13=25.0928628865337x_{13} = 25.0928628865337
x14=12.4860672578708x_{14} = 12.4860672578708
x15=21.9455418081046x_{15} = 21.9455418081046
x16=9.31693112610028x_{16} = 9.31693112610028
x17=−2.75936321522763x_{17} = -2.75936321522763
x18=−65.9582831752547x_{18} = -65.9582831752547
x19=94.2371675854493x_{19} = 94.2371675854493
x20=50.2455769233645x_{20} = 50.2455769233645
x21=47.1026555912318x_{21} = 47.1026555912318
x22=−9.31693112610028x_{22} = -9.31693112610028
x23=−59.6735006001685x_{23} = -59.6735006001685
x24=91.0952088771736x_{24} = 91.0952088771736
x25=62.8159318625173x_{25} = 62.8159318625173
x26=−37.6725595300203x_{26} = -37.6725595300203
x27=−84.8112100697664x_{27} = -84.8112100697664
x28=56.5309760413753x_{28} = 56.5309760413753
x29=78.5270810189266x_{29} = 78.5270810189266
x30=2.75936321522763x_{30} = 2.75936321522763
x31=6.11791002392407x_{31} = 6.11791002392407
x32=31.3840497369889x_{32} = 31.3840497369889
x33=18.796291187414x_{33} = 18.796291187414
x34=−6.11791002392407x_{34} = -6.11791002392407
x35=−69.1005654545348x_{35} = -69.1005654545348
x36=−56.5309760413753x_{36} = -56.5309760413753
x37=−34.5285475249278x_{37} = -34.5285475249278
x38=84.8112100697664x_{38} = 84.8112100697664
x39=−12.4860672578708x_{39} = -12.4860672578708
x40=28.2389032383054x_{40} = 28.2389032383054
x41=−31.3840497369889x_{41} = -31.3840497369889
x42=40.8161982919721x_{42} = 40.8161982919721
x43=−94.2371675854493x_{43} = -94.2371675854493
x44=−62.8159318625173x_{44} = -62.8159318625173
x45=−50.2455769233645x_{45} = -50.2455769233645
x46=−15.6439318755503x_{46} = -15.6439318755503
x47=−43.9595440566684x_{47} = -43.9595440566684
x48=97.3791026663451x_{48} = 97.3791026663451
x49=100.521016336234x_{49} = 100.521016336234
x50=−81.6691637048431x_{50} = -81.6691637048431
x51=81.6691637048431x_{51} = 81.6691637048431
x52=53.3883416918471x_{52} = 53.3883416918471
x53=−25.0928628865337x_{53} = -25.0928628865337
x54=169.640108376141x_{54} = 169.640108376141
x55=−18.796291187414x_{55} = -18.796291187414
x56=−21.9455418081046x_{56} = -21.9455418081046
x57=−87.95322400825x_{57} = -87.95322400825
x58=75.384957467622x_{58} = 75.384957467622
x59=−100.521016336234x_{59} = -100.521016336234
x60=−147.648081727825x_{60} = -147.648081727825
x61=72.2427877152145x_{61} = 72.2427877152145
x62=59.6735006001685x_{62} = 59.6735006001685
x63=15.6439318755503x_{63} = 15.6439318755503
x64=34.5285475249278x_{64} = 34.5285475249278
x65=−28.2389032383054x_{65} = -28.2389032383054
x66=87.95322400825x_{66} = 87.95322400825
x67=−47.1026555912318x_{67} = -47.1026555912318
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
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⁡(x)x=−sin⁡(x)−x\frac{\sin{\left(x \right)}}{\sqrt{x}} = - \frac{\sin{\left(x \right)}}{\sqrt{- x}}
- No
sin⁡(x)x=sin⁡(x)−x\frac{\sin{\left(x \right)}}{\sqrt{x}} = \frac{\sin{\left(x \right)}}{\sqrt{- x}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор