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

Gráfico de la función y = 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)} = x \sin{\left(x \right)} 
f = x*sin(x)
Gráfico de la función
02468-8-6-4-2-1010-1010
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)=0x \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=−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=18.8495559215388x_{8} = 18.8495559215388
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=−94.2477796076938x_{23} = -94.2477796076938
x24=6.28318530717959x_{24} = 6.28318530717959
x25=−69.1150383789755x_{25} = -69.1150383789755
x26=97.3893722612836x_{26} = 97.3893722612836
x27=0x_{27} = 0
x28=65.9734457253857x_{28} = 65.9734457253857
x29=−50.2654824574367x_{29} = -50.2654824574367
x30=15.707963267949x_{30} = 15.707963267949
x31=3.14159265358979x_{31} = 3.14159265358979
x32=−25.1327412287183x_{32} = -25.1327412287183
x33=−18.8495559215388x_{33} = -18.8495559215388
x34=40.8407044966673x_{34} = 40.8407044966673
x35=−53.4070751110265x_{35} = -53.4070751110265
x36=−43.9822971502571x_{36} = -43.9822971502571
x37=37.6991118430775x_{37} = 37.6991118430775
x38=697.433569096934x_{38} = 697.433569096934
x39=−6.28318530717959x_{39} = -6.28318530717959
x40=−78.5398163397448x_{40} = -78.5398163397448
x41=−40.8407044966673x_{41} = -40.8407044966673
x42=43.9822971502571x_{42} = 43.9822971502571
x43=56.5486677646163x_{43} = 56.5486677646163
x44=−65.9734457253857x_{44} = -65.9734457253857
x45=25.1327412287183x_{45} = 25.1327412287183
x46=78.5398163397448x_{46} = 78.5398163397448
x47=−28.2743338823081x_{47} = -28.2743338823081
x48=75.398223686155x_{48} = 75.398223686155
x49=59.6902604182061x_{49} = 59.6902604182061
x50=−34.5575191894877x_{50} = -34.5575191894877
x51=81.6814089933346x_{51} = 81.6814089933346
x52=−47.1238898038469x_{52} = -47.1238898038469
x53=100.530964914873x_{53} = 100.530964914873
x54=−9.42477796076938x_{54} = -9.42477796076938
x55=−75.398223686155x_{55} = -75.398223686155
x56=−72.2566310325652x_{56} = -72.2566310325652
x57=−31.4159265358979x_{57} = -31.4159265358979
x58=28.2743338823081x_{58} = 28.2743338823081
x59=−91.106186954104x_{59} = -91.106186954104
x60=21.9911485751286x_{60} = 21.9911485751286
x61=62.8318530717959x_{61} = 62.8318530717959
x62=9.42477796076938x_{62} = 9.42477796076938
x63=50.2654824574367x_{63} = 50.2654824574367
x64=94.2477796076938x_{64} = 94.2477796076938
x65=91.106186954104x_{65} = 91.106186954104
x66=84.8230016469244x_{66} = 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 x*sin(x).
0sin⁡(0)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)=0x \cos{\left(x \right)} + \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=29.8785865061074x_{1} = 29.8785865061074
x2=26.7409160147873x_{2} = 26.7409160147873
x3=39.295350981473x_{3} = 39.295350981473
x4=64.4181717218392x_{4} = 64.4181717218392
x5=4.91318043943488x_{5} = 4.91318043943488
x6=−83.2642147040886x_{6} = -83.2642147040886
x7=−54.9960525574964x_{7} = -54.9960525574964
x8=89.5465575382492x_{8} = 89.5465575382492
x9=−2.02875783811043x_{9} = -2.02875783811043
x10=17.3363779239834x_{10} = 17.3363779239834
x11=14.2074367251912x_{11} = 14.2074367251912
x12=−17.3363779239834x_{12} = -17.3363779239834
x13=83.2642147040886x_{13} = 83.2642147040886
x14=−36.1559664195367x_{14} = -36.1559664195367
x15=86.4053708116885x_{15} = 86.4053708116885
x16=−29.8785865061074x_{16} = -29.8785865061074
x17=−64.4181717218392x_{17} = -64.4181717218392
x18=61.2773745335697x_{18} = 61.2773745335697
x19=−42.4350618814099x_{19} = -42.4350618814099
x20=7.97866571241324x_{20} = 7.97866571241324
x21=58.1366632448992x_{21} = 58.1366632448992
x22=−45.57503179559x_{22} = -45.57503179559
x23=92.687771772017x_{23} = 92.687771772017
x24=11.085538406497x_{24} = 11.085538406497
x25=−33.0170010333572x_{25} = -33.0170010333572
x26=102.111554139654x_{26} = 102.111554139654
x27=70.69997803861x_{27} = 70.69997803861
x28=−14.2074367251912x_{28} = -14.2074367251912
x29=42.4350618814099x_{29} = 42.4350618814099
x30=0x_{30} = 0
x31=23.6042847729804x_{31} = 23.6042847729804
x32=76.9820093304187x_{32} = 76.9820093304187
x33=−73.8409691490209x_{33} = -73.8409691490209
x34=−76.9820093304187x_{34} = -76.9820093304187
x35=−7.97866571241324x_{35} = -7.97866571241324
x36=−4.91318043943488x_{36} = -4.91318043943488
x37=−98.9702722883957x_{37} = -98.9702722883957
x38=73.8409691490209x_{38} = 73.8409691490209
x39=48.7152107175577x_{39} = 48.7152107175577
x40=20.469167402741x_{40} = 20.469167402741
x41=−80.1230928148503x_{41} = -80.1230928148503
x42=51.855560729152x_{42} = 51.855560729152
x43=80.1230928148503x_{43} = 80.1230928148503
x44=−23.6042847729804x_{44} = -23.6042847729804
x45=95.8290108090195x_{45} = 95.8290108090195
x46=−86.4053708116885x_{46} = -86.4053708116885
x47=−26.7409160147873x_{47} = -26.7409160147873
x48=67.5590428388084x_{48} = 67.5590428388084
x49=−89.5465575382492x_{49} = -89.5465575382492
x50=2.02875783811043x_{50} = 2.02875783811043
x51=−92.687771772017x_{51} = -92.687771772017
x52=−70.69997803861x_{52} = -70.69997803861
x53=−20.469167402741x_{53} = -20.469167402741
x54=−11.085538406497x_{54} = -11.085538406497
x55=36.1559664195367x_{55} = 36.1559664195367
x56=54.9960525574964x_{56} = 54.9960525574964
x57=33.0170010333572x_{57} = 33.0170010333572
x58=−61.2773745335697x_{58} = -61.2773745335697
x59=45.57503179559x_{59} = 45.57503179559
x60=98.9702722883957x_{60} = 98.9702722883957
x61=−58.1366632448992x_{61} = -58.1366632448992
x62=−67.5590428388084x_{62} = -67.5590428388084
x63=−95.8290108090195x_{63} = -95.8290108090195
x64=−51.855560729152x_{64} = -51.855560729152
x65=−39.295350981473x_{65} = -39.295350981473
x66=−48.7152107175577x_{66} = -48.7152107175577
Signos de extremos en los puntos:
(29.878586506107393, -29.8618661591868)

(26.74091601478731, 26.7222376646974)

(39.295350981472986, 39.2826330068918)

(64.41817172183916, 64.4104113393753)

(4.913180439434884, -4.81446988971227)

(-83.26421470408864, 83.2582103729533)

(-54.99605255749639, -54.9869632496976)

(89.54655753824919, 89.5409743728852)

(-2.028757838110434, 1.81970574115965)

(17.33637792398336, -17.3076086078585)

(14.207436725191188, 14.1723741137743)

(-17.33637792398336, -17.3076086078585)

(83.26421470408864, 83.2582103729533)

(-36.15596641953672, -36.1421453722421)

(86.40537081168854, -86.3995847156108)

(-29.878586506107393, -29.8618661591868)

(-64.41817172183916, 64.4104113393753)

(61.277374533569656, -61.2692165444766)

(-42.43506188140989, -42.4232840772591)

(7.978665712413241, 7.91672737158778)

(58.13666324489916, 58.1280647280857)

(-45.57503179559002, 45.5640648360268)

(92.687771772017, -92.6823777880592)

(11.085538406497022, -11.04070801593)

(-33.017001033357246, 33.0018677308454)

(102.11155413965392, 102.106657886316)

(70.69997803861, 70.6929069615931)

(-14.207436725191188, 14.1723741137743)

(42.43506188140989, -42.4232840772591)

(0, 0)

(23.604284772980407, -23.5831306496334)

(76.98200933041872, 76.9755151282637)

(-73.8409691490209, -73.8341987715416)

(-76.98200933041872, 76.9755151282637)

(-7.978665712413241, 7.91672737158778)

(-4.913180439434884, -4.81446988971227)

(-98.9702722883957, -98.9652206531187)

(73.8409691490209, -73.8341987715416)

(48.715210717557724, -48.7049502253679)

(20.46916740274095, 20.4447840582523)

(-80.12309281485025, -80.1168531456592)

(51.85556072915197, 51.8459212502015)

(80.12309281485025, -80.1168531456592)

(-23.604284772980407, -23.5831306496334)

(95.82901080901948, 95.8237936084657)

(-86.40537081168854, -86.3995847156108)

(-26.74091601478731, 26.7222376646974)

(67.5590428388084, -67.5516431209725)

(-89.54655753824919, 89.5409743728852)

(2.028757838110434, 1.81970574115965)

(-92.687771772017, -92.6823777880592)

(-70.69997803861, 70.6929069615931)

(-20.46916740274095, 20.4447840582523)

(-11.085538406497022, -11.04070801593)

(36.15596641953672, -36.1421453722421)

(54.99605255749639, -54.9869632496976)

(33.017001033357246, 33.0018677308454)

(-61.277374533569656, -61.2692165444766)

(45.57503179559002, 45.5640648360268)

(98.9702722883957, -98.9652206531187)

(-58.13666324489916, 58.1280647280857)

(-67.5590428388084, -67.5516431209725)

(-95.82901080901948, 95.8237936084657)

(-51.85556072915197, 51.8459212502015)

(-39.295350981472986, 39.2826330068918)

(-48.715210717557724, -48.7049502253679)


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=29.8785865061074x_{1} = 29.8785865061074
x2=4.91318043943488x_{2} = 4.91318043943488
x3=−54.9960525574964x_{3} = -54.9960525574964
x4=17.3363779239834x_{4} = 17.3363779239834
x5=−17.3363779239834x_{5} = -17.3363779239834
x6=−36.1559664195367x_{6} = -36.1559664195367
x7=86.4053708116885x_{7} = 86.4053708116885
x8=−29.8785865061074x_{8} = -29.8785865061074
x9=61.2773745335697x_{9} = 61.2773745335697
x10=−42.4350618814099x_{10} = -42.4350618814099
x11=92.687771772017x_{11} = 92.687771772017
x12=11.085538406497x_{12} = 11.085538406497
x13=42.4350618814099x_{13} = 42.4350618814099
x14=0x_{14} = 0
x15=23.6042847729804x_{15} = 23.6042847729804
x16=−73.8409691490209x_{16} = -73.8409691490209
x17=−4.91318043943488x_{17} = -4.91318043943488
x18=−98.9702722883957x_{18} = -98.9702722883957
x19=73.8409691490209x_{19} = 73.8409691490209
x20=48.7152107175577x_{20} = 48.7152107175577
x21=−80.1230928148503x_{21} = -80.1230928148503
x22=80.1230928148503x_{22} = 80.1230928148503
x23=−23.6042847729804x_{23} = -23.6042847729804
x24=−86.4053708116885x_{24} = -86.4053708116885
x25=67.5590428388084x_{25} = 67.5590428388084
x26=−92.687771772017x_{26} = -92.687771772017
x27=−11.085538406497x_{27} = -11.085538406497
x28=36.1559664195367x_{28} = 36.1559664195367
x29=54.9960525574964x_{29} = 54.9960525574964
x30=−61.2773745335697x_{30} = -61.2773745335697
x31=98.9702722883957x_{31} = 98.9702722883957
x32=−67.5590428388084x_{32} = -67.5590428388084
x33=−48.7152107175577x_{33} = -48.7152107175577
Puntos máximos de la función:
x33=26.7409160147873x_{33} = 26.7409160147873
x33=39.295350981473x_{33} = 39.295350981473
x33=64.4181717218392x_{33} = 64.4181717218392
x33=−83.2642147040886x_{33} = -83.2642147040886
x33=89.5465575382492x_{33} = 89.5465575382492
x33=−2.02875783811043x_{33} = -2.02875783811043
x33=14.2074367251912x_{33} = 14.2074367251912
x33=83.2642147040886x_{33} = 83.2642147040886
x33=−64.4181717218392x_{33} = -64.4181717218392
x33=7.97866571241324x_{33} = 7.97866571241324
x33=58.1366632448992x_{33} = 58.1366632448992
x33=−45.57503179559x_{33} = -45.57503179559
x33=−33.0170010333572x_{33} = -33.0170010333572
x33=102.111554139654x_{33} = 102.111554139654
x33=70.69997803861x_{33} = 70.69997803861
x33=−14.2074367251912x_{33} = -14.2074367251912
x33=76.9820093304187x_{33} = 76.9820093304187
x33=−76.9820093304187x_{33} = -76.9820093304187
x33=−7.97866571241324x_{33} = -7.97866571241324
x33=20.469167402741x_{33} = 20.469167402741
x33=51.855560729152x_{33} = 51.855560729152
x33=95.8290108090195x_{33} = 95.8290108090195
x33=−26.7409160147873x_{33} = -26.7409160147873
x33=−89.5465575382492x_{33} = -89.5465575382492
x33=2.02875783811043x_{33} = 2.02875783811043
x33=−70.69997803861x_{33} = -70.69997803861
x33=−20.469167402741x_{33} = -20.469167402741
x33=33.0170010333572x_{33} = 33.0170010333572
x33=45.57503179559x_{33} = 45.57503179559
x33=−58.1366632448992x_{33} = -58.1366632448992
x33=−95.8290108090195x_{33} = -95.8290108090195
x33=−51.855560729152x_{33} = -51.855560729152
x33=−39.295350981473x_{33} = -39.295350981473
Decrece en los intervalos
[98.9702722883957,∞)\left[98.9702722883957, \infty\right)
Crece en los intervalos
(−∞,−98.9702722883957]\left(-\infty, -98.9702722883957\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)+2cos⁡(x)=0- x \sin{\left(x \right)} + 2 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=34.6152330552306x_{1} = 34.6152330552306
x2=−53.4444796697636x_{2} = -53.4444796697636
x3=−100.550852725424x_{3} = -100.550852725424
x4=−97.4099011706723x_{4} = -97.4099011706723
x5=28.3447768697864x_{5} = 28.3447768697864
x6=31.479374920314x_{6} = 31.479374920314
x7=15.8336114149477x_{7} = 15.8336114149477
x8=−28.3447768697864x_{8} = -28.3447768697864
x9=−78.5652673845995x_{9} = -78.5652673845995
x10=56.5839987378634x_{10} = 56.5839987378634
x11=−47.1662676027767x_{11} = -47.1662676027767
x12=12.7222987717666x_{12} = 12.7222987717666
x13=37.7520396346102x_{13} = 37.7520396346102
x14=62.863657228703x_{14} = 62.863657228703
x15=−91.1281305511393x_{15} = -91.1281305511393
x16=−44.0276918992479x_{16} = -44.0276918992479
x17=−81.7058821480364x_{17} = -81.7058821480364
x18=−9.62956034329743x_{18} = -9.62956034329743
x19=40.8895777660408x_{19} = 40.8895777660408
x20=−34.6152330552306x_{20} = -34.6152330552306
x21=−25.2119030642106x_{21} = -25.2119030642106
x22=−56.5839987378634x_{22} = -56.5839987378634
x23=−84.8465692433091x_{23} = -84.8465692433091
x24=−1.0768739863118x_{24} = -1.0768739863118
x25=84.8465692433091x_{25} = 84.8465692433091
x26=6.57833373272234x_{26} = 6.57833373272234
x27=97.4099011706723x_{27} = 97.4099011706723
x28=9.62956034329743x_{28} = 9.62956034329743
x29=50.3052188363296x_{29} = 50.3052188363296
x30=−22.0814757672807x_{30} = -22.0814757672807
x31=53.4444796697636x_{31} = 53.4444796697636
x32=−6.57833373272234x_{32} = -6.57833373272234
x33=81.7058821480364x_{33} = 81.7058821480364
x34=−40.8895777660408x_{34} = -40.8895777660408
x35=91.1281305511393x_{35} = 91.1281305511393
x36=66.0037377708277x_{36} = 66.0037377708277
x37=−128.820822990274x_{37} = -128.820822990274
x38=−66.0037377708277x_{38} = -66.0037377708277
x39=94.2689923093066x_{39} = 94.2689923093066
x40=−37.7520396346102x_{40} = -37.7520396346102
x41=22.0814757672807x_{41} = 22.0814757672807
x42=−12.7222987717666x_{42} = -12.7222987717666
x43=59.7237354324305x_{43} = 59.7237354324305
x44=−75.4247339745236x_{44} = -75.4247339745236
x45=3.6435971674254x_{45} = 3.6435971674254
x46=−50.3052188363296x_{46} = -50.3052188363296
x47=−15.8336114149477x_{47} = -15.8336114149477
x48=69.1439554764926x_{48} = 69.1439554764926
x49=78.5652673845995x_{49} = 78.5652673845995
x50=−3.6435971674254x_{50} = -3.6435971674254
x51=−31.479374920314x_{51} = -31.479374920314
x52=18.954681766529x_{52} = 18.954681766529
x53=−18.954681766529x_{53} = -18.954681766529
x54=75.4247339745236x_{54} = 75.4247339745236
x55=44.0276918992479x_{55} = 44.0276918992479
x56=72.2842925036825x_{56} = 72.2842925036825
x57=−72.2842925036825x_{57} = -72.2842925036825
x58=47.1662676027767x_{58} = 47.1662676027767
x59=25.2119030642106x_{59} = 25.2119030642106
x60=−62.863657228703x_{60} = -62.863657228703
x61=−69.1439554764926x_{61} = -69.1439554764926
x62=−87.9873209346887x_{62} = -87.9873209346887
x63=−59.7237354324305x_{63} = -59.7237354324305
x64=87.9873209346887x_{64} = 87.9873209346887
x65=−94.2689923093066x_{65} = -94.2689923093066
x66=100.550852725424x_{66} = 100.550852725424
x67=1.0768739863118x_{67} = 1.0768739863118

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
[97.4099011706723,∞)\left[97.4099011706723, \infty\right)
Convexa en los intervalos
(−∞,−100.550852725424]\left(-\infty, -100.550852725424\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xsin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(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 izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(xsin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(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 x*sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞sin⁡(x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
lim⁡x→∞sin⁡(x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
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)x \sin{\left(x \right)} = x \sin{\left(x \right)}
- Sí
xsin⁡(x)=−xsin⁡(x)x \sin{\left(x \right)} = - x \sin{\left(x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = x*sin(x)
    © Sr Examen – Калькулятор