Sr Examen

Otras calculadoras

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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         ___       
f(x) = \/ x *sin(x)
f(x)=xsin⁡(x)f{\left(x \right)} = \sqrt{x} \sin{\left(x \right)} 
f = sqrt(x)*sin(x)
Gráfico de la función
02468-8-6-4-2-10105-5
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:
xsin⁡(x)=0\sqrt{x} \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=−43.9822971502571x_{4} = -43.9822971502571
x5=81.6814089933346x_{5} = 81.6814089933346
x6=−100.530964914873x_{6} = -100.530964914873
x7=−47.1238898038469x_{7} = -47.1238898038469
x8=28.2743338823081x_{8} = 28.2743338823081
x9=65.9734457253857x_{9} = 65.9734457253857
x10=−31.4159265358979x_{10} = -31.4159265358979
x11=−9.42477796076938x_{11} = -9.42477796076938
x12=40.8407044966673x_{12} = 40.8407044966673
x13=56.5486677646163x_{13} = 56.5486677646163
x14=−56.5486677646163x_{14} = -56.5486677646163
x15=12.5663706143592x_{15} = 12.5663706143592
x16=43.9822971502571x_{16} = 43.9822971502571
x17=100.530964914873x_{17} = 100.530964914873
x18=−3.14159265358979x_{18} = -3.14159265358979
x19=−15.707963267949x_{19} = -15.707963267949
x20=59.6902604182061x_{20} = 59.6902604182061
x21=6.28318530717959x_{21} = 6.28318530717959
x22=9.42477796076938x_{22} = 9.42477796076938
x23=−53.4070751110265x_{23} = -53.4070751110265
x24=−6.28318530717959x_{24} = -6.28318530717959
x25=−87.9645943005142x_{25} = -87.9645943005142
x26=69.1150383789755x_{26} = 69.1150383789755
x27=21.9911485751286x_{27} = 21.9911485751286
x28=87.9645943005142x_{28} = 87.9645943005142
x29=18.8495559215388x_{29} = 18.8495559215388
x30=−84.8230016469244x_{30} = -84.8230016469244
x31=−72.2566310325652x_{31} = -72.2566310325652
x32=25.1327412287183x_{32} = 25.1327412287183
x33=37.6991118430775x_{33} = 37.6991118430775
x34=−25.1327412287183x_{34} = -25.1327412287183
x35=0x_{35} = 0
x36=50.2654824574367x_{36} = 50.2654824574367
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=−21.9911485751286x_{38} = -21.9911485751286
x39=−62.8318530717959x_{39} = -62.8318530717959
x40=75.398223686155x_{40} = 75.398223686155
x41=84.8230016469244x_{41} = 84.8230016469244
x42=53.4070751110265x_{42} = 53.4070751110265
x43=34.5575191894877x_{43} = 34.5575191894877
x44=−28.2743338823081x_{44} = -28.2743338823081
x45=15.707963267949x_{45} = 15.707963267949
x46=−91.106186954104x_{46} = -91.106186954104
x47=47.1238898038469x_{47} = 47.1238898038469
x48=97.3893722612836x_{48} = 97.3893722612836
x49=−69.1150383789755x_{49} = -69.1150383789755
x50=94.2477796076938x_{50} = 94.2477796076938
x51=−18.8495559215388x_{51} = -18.8495559215388
x52=−50.2654824574367x_{52} = -50.2654824574367
x53=−37.6991118430775x_{53} = -37.6991118430775
x54=−81.6814089933346x_{54} = -81.6814089933346
x55=62.8318530717959x_{55} = 62.8318530717959
x56=78.5398163397448x_{56} = 78.5398163397448
x57=31.4159265358979x_{57} = 31.4159265358979
x58=−78.5398163397448x_{58} = -78.5398163397448
x59=−40.8407044966673x_{59} = -40.8407044966673
x60=−97.3893722612836x_{60} = -97.3893722612836
x61=−223.053078404875x_{61} = -223.053078404875
x62=141.371669411541x_{62} = 141.371669411541
x63=−75.398223686155x_{63} = -75.398223686155
x64=91.106186954104x_{64} = 91.106186954104
x65=−12.5663706143592x_{65} = -12.5663706143592
x66=−94.2477796076938x_{66} = -94.2477796076938
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 sqrt(x)*sin(x).
0sin⁡(0)\sqrt{0} \sin{\left(0 \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
xcos⁡(x)+sin⁡(x)2x=0\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=89.5409746049841x_{1} = 89.5409746049841
x2=−4.81584231784594x_{2} = -4.81584231784594
x3=36.1421488970061x_{3} = 36.1421488970061
x4=83.2582106616487x_{4} = 83.2582106616487
x5=−23.5831433102848x_{5} = -23.5831433102848
x6=−11.0408298179713x_{6} = -11.0408298179713
x7=61.2692172687226x_{7} = 61.2692172687226
x8=−64.410411962776x_{8} = -64.410411962776
x9=−48.7049516666752x_{9} = -48.7049516666752
x10=11.0408298179713x_{10} = 11.0408298179713
x11=−1.83659720315213x_{11} = -1.83659720315213
x12=−36.1421488970061x_{12} = -36.1421488970061
x13=23.5831433102848x_{13} = 23.5831433102848
x14=−73.8341991854591x_{14} = -73.8341991854591
x15=20.4448034666183x_{15} = 20.4448034666183
x16=86.3995849739529x_{16} = 86.3995849739529
x17=70.692907433161x_{17} = 70.692907433161
x18=−39.2826357527234x_{18} = -39.2826357527234
x19=48.7049516666752x_{19} = 48.7049516666752
x20=39.2826357527234x_{20} = 39.2826357527234
x21=−17.3076405374146x_{21} = -17.3076405374146
x22=54.9869642514883x_{22} = 54.9869642514883
x23=4.81584231784594x_{23} = 4.81584231784594
x24=−92.682377997352x_{24} = -92.682377997352
x25=45.5640665961997x_{25} = 45.5640665961997
x26=−33.0018723591446x_{26} = -33.0018723591446
x27=51.8459224452234x_{27} = 51.8459224452234
x28=−58.1280655761511x_{28} = -58.1280655761511
x29=−67.5516436614121x_{29} = -67.5516436614121
x30=98.9652208250325x_{30} = 98.9652208250325
x31=26.7222463741877x_{31} = 26.7222463741877
x32=−54.9869642514883x_{32} = -54.9869642514883
x33=−45.5640665961997x_{33} = -45.5640665961997
x34=76.9755154935569x_{34} = 76.9755154935569
x35=73.8341991854591x_{35} = 73.8341991854591
x36=7.91705268466621x_{36} = 7.91705268466621
x37=42.4232862577008x_{37} = 42.4232862577008
x38=−86.3995849739529x_{38} = -86.3995849739529
x39=80.1168534696549x_{39} = 80.1168534696549
x40=92.682377997352x_{40} = 92.682377997352
x41=95.8237937978449x_{41} = 95.8237937978449
x42=29.861872403816x_{42} = 29.861872403816
x43=−76.9755154935569x_{43} = -76.9755154935569
x44=67.5516436614121x_{44} = 67.5516436614121
x45=−83.2582106616487x_{45} = -83.2582106616487
x46=−42.4232862577008x_{46} = -42.4232862577008
x47=−61.2692172687226x_{47} = -61.2692172687226
x48=33.0018723591446x_{48} = 33.0018723591446
x49=14.1724320747999x_{49} = 14.1724320747999
x50=−98.9652208250325x_{50} = -98.9652208250325
x51=−70.692907433161x_{51} = -70.692907433161
x52=1.83659720315213x_{52} = 1.83659720315213
x53=−26.7222463741877x_{53} = -26.7222463741877
x54=−29.861872403816x_{54} = -29.861872403816
x55=−7.91705268466621x_{55} = -7.91705268466621
x56=−80.1168534696549x_{56} = -80.1168534696549
x57=−89.5409746049841x_{57} = -89.5409746049841
x58=58.1280655761511x_{58} = 58.1280655761511
x59=−14.1724320747999x_{59} = -14.1724320747999
x60=−20.4448034666183x_{60} = -20.4448034666183
x61=17.3076405374146x_{61} = 17.3076405374146
x62=64.410411962776x_{62} = 64.410411962776
x63=−95.8237937978449x_{63} = -95.8237937978449
x64=−51.8459224452234x_{64} = -51.8459224452234
Signos de extremos en los puntos:
(89.54097460498406, 9.46246176606193)

(-4.815842317845935, 2.18276978467772*I)

(36.142148897006074, -6.01125886058877)

(83.25821066164869, 9.12442919108264)

(-23.583143310284843, 4.85515677204621*I)

(-11.040829817971295, 3.31937237072132*I)

(61.269217268722585, -7.82720494097395)

(-64.41041196277601, -8.02536795646149*I)

(-48.70495166667517, 6.97852557917854*I)

(11.040829817971295, -3.31937237072132)

(-1.8365972031521258, -1.30761941299144*I)

(-36.142148897006074, 6.01125886058877*I)

(23.583143310284843, -4.85515677204621)

(-73.83419918545908, 8.59248586707723*I)

(20.4448034666183, 4.52024144595309)

(86.3995849739529, -9.29498206229774)

(70.692907433161, 8.40769713937167)

(-39.282635752723394, -6.26707847792961*I)

(48.70495166667517, -6.97852557917854)

(39.282635752723394, 6.26707847792961)

(-17.307640537414635, 4.15851032158028*I)

(54.98696425148828, -7.4150130205716)

(4.815842317845935, -2.18276978467772)

(-92.68237799735202, 9.6270286533*I)

(45.56406659619972, 6.74970965872142)

(-33.00187235914463, -5.74406639671223*I)

(51.84592244522343, 7.20007645193272)

(-58.12806557615112, -7.6238943490782*I)

(-67.5516436614121, 8.21875556224649*I)

(98.96522082503246, -9.94799953505937)

(26.72224637418772, 5.16845181340769)

(-54.98696425148828, 7.4150130205716*I)

(-45.56406659619972, -6.74970965872142*I)

(76.97551549355693, 8.77338405887965)

(73.83419918545908, -8.59248586707723)

(7.917052684666207, 2.808131180007)

(42.423286257700816, -6.51286373926386)

(-86.3995849739529, 9.29498206229774*I)

(80.11685346965491, -8.95062752823053)

(92.68237799735202, -9.6270286533)

(95.82379379784489, 9.78882959875799)

(29.861872403816044, -5.46383591176171)

(-76.97551549355693, -8.77338405887965*I)

(67.5516436614121, -8.21875556224649)

(-83.25821066164869, -9.12442919108264*I)

(-42.423286257700816, 6.51286373926386*I)

(-61.269217268722585, 7.82720494097395*I)

(33.00187235914463, 5.74406639671223)

(14.172432074799941, 3.76228841574689)

(-98.96522082503246, 9.94799953505937*I)

(-70.692907433161, -8.40769713937167*I)

(1.8365972031521258, 1.30761941299144)

(-26.72224637418772, -5.16845181340769*I)

(-29.861872403816044, 5.46383591176171*I)

(-7.917052684666207, -2.808131180007*I)

(-80.11685346965491, 8.95062752823053*I)

(-89.54097460498406, -9.46246176606193*I)

(58.12806557615112, 7.6238943490782)

(-14.172432074799941, -3.76228841574689*I)

(-20.4448034666183, -4.52024144595309*I)

(17.307640537414635, -4.15851032158028)

(64.41041196277601, 8.02536795646149)

(-95.82379379784489, -9.78882959875799*I)

(-51.84592244522343, -7.20007645193272*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=36.1421488970061x_{1} = 36.1421488970061
x2=61.2692172687226x_{2} = 61.2692172687226
x3=11.0408298179713x_{3} = 11.0408298179713
x4=23.5831433102848x_{4} = 23.5831433102848
x5=86.3995849739529x_{5} = 86.3995849739529
x6=48.7049516666752x_{6} = 48.7049516666752
x7=54.9869642514883x_{7} = 54.9869642514883
x8=4.81584231784594x_{8} = 4.81584231784594
x9=98.9652208250325x_{9} = 98.9652208250325
x10=73.8341991854591x_{10} = 73.8341991854591
x11=42.4232862577008x_{11} = 42.4232862577008
x12=80.1168534696549x_{12} = 80.1168534696549
x13=92.682377997352x_{13} = 92.682377997352
x14=29.861872403816x_{14} = 29.861872403816
x15=67.5516436614121x_{15} = 67.5516436614121
x16=17.3076405374146x_{16} = 17.3076405374146
Puntos máximos de la función:
x16=89.5409746049841x_{16} = 89.5409746049841
x16=83.2582106616487x_{16} = 83.2582106616487
x16=20.4448034666183x_{16} = 20.4448034666183
x16=70.692907433161x_{16} = 70.692907433161
x16=39.2826357527234x_{16} = 39.2826357527234
x16=45.5640665961997x_{16} = 45.5640665961997
x16=51.8459224452234x_{16} = 51.8459224452234
x16=26.7222463741877x_{16} = 26.7222463741877
x16=76.9755154935569x_{16} = 76.9755154935569
x16=7.91705268466621x_{16} = 7.91705268466621
x16=95.8237937978449x_{16} = 95.8237937978449
x16=33.0018723591446x_{16} = 33.0018723591446
x16=14.1724320747999x_{16} = 14.1724320747999
x16=1.83659720315213x_{16} = 1.83659720315213
x16=58.1280655761511x_{16} = 58.1280655761511
x16=64.410411962776x_{16} = 64.410411962776
Decrece en los intervalos
[98.9652208250325,∞)\left[98.9652208250325, \infty\right)
Crece en los intervalos
(−∞,4.81584231784594]\left(-\infty, 4.81584231784594\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
−xsin⁡(x)+cos⁡(x)x−sin⁡(x)4x32=0- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sqrt{x}} - \frac{\sin{\left(x \right)}}{4 x^{\frac{3}{2}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=50.2853643733782x_{1} = 50.2853643733782
x2=−72.2704663982901x_{2} = -72.2704663982901
x3=−75.4114829061337x_{3} = -75.4114829061337
x4=91.1171610640786x_{4} = 91.1171610640786
x5=53.4257888392775x_{5} = 53.4257888392775
x6=−91.1171610640786x_{6} = -91.1171610640786
x7=−69.1295022175061x_{7} = -69.1295022175061
x8=−50.2853643733782x_{8} = -50.2853643733782
x9=−47.1450953533935x_{9} = -47.1450953533935
x10=15.7712217163826x_{10} = 15.7712217163826
x11=69.1295022175061x_{11} = 69.1295022175061
x12=−53.4257888392775x_{12} = -53.4257888392775
x13=−100.54091054091x_{13} = -100.54091054091
x14=56.5663428995631x_{14} = 56.5663428995631
x15=−18.9023731724419x_{15} = -18.9023731724419
x16=−59.7070061315463x_{16} = -59.7070061315463
x17=84.834788308704x_{17} = 84.834788308704
x18=97.3996386085752x_{18} = 97.3996386085752
x19=−84.834788308704x_{19} = -84.834788308704
x20=−3.42038548945687x_{20} = -3.42038548945687
x21=65.9885978289116x_{21} = 65.9885978289116
x22=−6.43640901362357x_{22} = -6.43640901362357
x23=−25.1724307086655x_{23} = -25.1724307086655
x24=0.746349736778129x_{24} = 0.746349736778129
x25=−65.9885978289116x_{25} = -65.9885978289116
x26=−40.8651666720526x_{26} = -40.8651666720526
x27=25.1724307086655x_{27} = 25.1724307086655
x28=−9.52905247096223x_{28} = -9.52905247096223
x29=47.1450953533935x_{29} = 47.1450953533935
x30=44.0050149904158x_{30} = 44.0050149904158
x31=−62.8477621879326x_{31} = -62.8477621879326
x32=81.6936487772184x_{32} = 81.6936487772184
x33=−81.6936487772184x_{33} = -81.6936487772184
x34=−12.64516529855x_{34} = -12.64516529855
x35=−97.3996386085752x_{35} = -97.3996386085752
x36=100.54091054091x_{36} = 100.54091054091
x37=−31.4477066173312x_{37} = -31.4477066173312
x38=−116.247530144815x_{38} = -116.247530144815
x39=31.4477066173312x_{39} = 31.4477066173312
x40=12.64516529855x_{40} = 12.64516529855
x41=75.4114829061337x_{41} = 75.4114829061337
x42=3.42038548945687x_{42} = 3.42038548945687
x43=−22.0364734735106x_{43} = -22.0364734735106
x44=34.5864181840427x_{44} = 34.5864181840427
x45=6.43640901362357x_{45} = 6.43640901362357
x46=−78.5525454686572x_{46} = -78.5525454686572
x47=−44.0050149904158x_{47} = -44.0050149904158
x48=22.0364734735106x_{48} = 22.0364734735106
x49=18.9023731724419x_{49} = 18.9023731724419
x50=37.7256081789305x_{50} = 37.7256081789305
x51=28.3096318664276x_{51} = 28.3096318664276
x52=72.2704663982901x_{52} = 72.2704663982901
x53=−0.746349736778129x_{53} = -0.746349736778129
x54=59.7070061315463x_{54} = 59.7070061315463
x55=−94.2583880465909x_{55} = -94.2583880465909
x56=−56.5663428995631x_{56} = -56.5663428995631
x57=62.8477621879326x_{57} = 62.8477621879326
x58=9.52905247096223x_{58} = 9.52905247096223
x59=78.5525454686572x_{59} = 78.5525454686572
x60=40.8651666720526x_{60} = 40.8651666720526
x61=−28.3096318664276x_{61} = -28.3096318664276
x62=−34.5864181840427x_{62} = -34.5864181840427
x63=94.2583880465909x_{63} = 94.2583880465909
x64=−15.7712217163826x_{64} = -15.7712217163826
x65=87.9759601854462x_{65} = 87.9759601854462
x66=−87.9759601854462x_{66} = -87.9759601854462
x67=−37.7256081789305x_{67} = -37.7256081789305

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
[97.3996386085752,∞)\left[97.3996386085752, \infty\right)
Convexa en los intervalos
(−∞,3.42038548945687]\left(-\infty, 3.42038548945687\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xsin⁡(x))=⟨−∞,∞⟩i\lim_{x \to -\infty}\left(\sqrt{x} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle i
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩iy = \left\langle -\infty, \infty\right\rangle i
lim⁡x→∞(xsin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\sqrt{x} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sqrt(x)*sin(x), dividida por x 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,
la inclinada coincide con la asíntota horizontal a la derecha
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,
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:
xsin⁡(x)=−−xsin⁡(x)\sqrt{x} \sin{\left(x \right)} = - \sqrt{- x} \sin{\left(x \right)}
- No
xsin⁡(x)=−xsin⁡(x)\sqrt{x} \sin{\left(x \right)} = \sqrt{- x} \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sqrt(x)*sin(x)
    © Sr Examen – Калькулятор