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Trazado el gráfico de la función/ xsin⁡x

Gráfico de la función y = xsin⁡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=31.4159265358979x_{1} = 31.4159265358979
x2=3.14159265358979x_{2} = 3.14159265358979
x3=75.398223686155x_{3} = 75.398223686155
x4=−47.1238898038469x_{4} = -47.1238898038469
x5=−34.5575191894877x_{5} = -34.5575191894877
x6=−69.1150383789755x_{6} = -69.1150383789755
x7=−12.5663706143592x_{7} = -12.5663706143592
x8=−65.9734457253857x_{8} = -65.9734457253857
x9=−50.2654824574367x_{9} = -50.2654824574367
x10=−56.5486677646163x_{10} = -56.5486677646163
x11=59.6902604182061x_{11} = 59.6902604182061
x12=72.2566310325652x_{12} = 72.2566310325652
x13=91.106186954104x_{13} = 91.106186954104
x14=−91.106186954104x_{14} = -91.106186954104
x15=−62.8318530717959x_{15} = -62.8318530717959
x16=−6.28318530717959x_{16} = -6.28318530717959
x17=6.28318530717959x_{17} = 6.28318530717959
x18=62.8318530717959x_{18} = 62.8318530717959
x19=−25.1327412287183x_{19} = -25.1327412287183
x20=94.2477796076938x_{20} = 94.2477796076938
x21=−9.42477796076938x_{21} = -9.42477796076938
x22=−37.6991118430775x_{22} = -37.6991118430775
x23=65.9734457253857x_{23} = 65.9734457253857
x24=−100.530964914873x_{24} = -100.530964914873
x25=−43.9822971502571x_{25} = -43.9822971502571
x26=25.1327412287183x_{26} = 25.1327412287183
x27=21.9911485751286x_{27} = 21.9911485751286
x28=87.9645943005142x_{28} = 87.9645943005142
x29=−40.8407044966673x_{29} = -40.8407044966673
x30=697.433569096934x_{30} = 697.433569096934
x31=−97.3893722612836x_{31} = -97.3893722612836
x32=43.9822971502571x_{32} = 43.9822971502571
x33=−53.4070751110265x_{33} = -53.4070751110265
x34=97.3893722612836x_{34} = 97.3893722612836
x35=100.530964914873x_{35} = 100.530964914873
x36=−94.2477796076938x_{36} = -94.2477796076938
x37=−31.4159265358979x_{37} = -31.4159265358979
x38=18.8495559215388x_{38} = 18.8495559215388
x39=78.5398163397448x_{39} = 78.5398163397448
x40=−18.8495559215388x_{40} = -18.8495559215388
x41=53.4070751110265x_{41} = 53.4070751110265
x42=47.1238898038469x_{42} = 47.1238898038469
x43=12.5663706143592x_{43} = 12.5663706143592
x44=81.6814089933346x_{44} = 81.6814089933346
x45=34.5575191894877x_{45} = 34.5575191894877
x46=−75.398223686155x_{46} = -75.398223686155
x47=−15.707963267949x_{47} = -15.707963267949
x48=50.2654824574367x_{48} = 50.2654824574367
x49=−81.6814089933346x_{49} = -81.6814089933346
x50=−3.14159265358979x_{50} = -3.14159265358979
x51=−59.6902604182061x_{51} = -59.6902604182061
x52=−28.2743338823081x_{52} = -28.2743338823081
x53=−87.9645943005142x_{53} = -87.9645943005142
x54=9.42477796076938x_{54} = 9.42477796076938
x55=−21.9911485751286x_{55} = -21.9911485751286
x56=56.5486677646163x_{56} = 56.5486677646163
x57=15.707963267949x_{57} = 15.707963267949
x58=84.8230016469244x_{58} = 84.8230016469244
x59=−78.5398163397448x_{59} = -78.5398163397448
x60=37.6991118430775x_{60} = 37.6991118430775
x61=−72.2566310325652x_{61} = -72.2566310325652
x62=−84.8230016469244x_{62} = -84.8230016469244
x63=69.1150383789755x_{63} = 69.1150383789755
x64=0x_{64} = 0
x65=28.2743338823081x_{65} = 28.2743338823081
x66=40.8407044966673x_{66} = 40.8407044966673
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=−20.469167402741x_{1} = -20.469167402741
x2=80.1230928148503x_{2} = 80.1230928148503
x3=45.57503179559x_{3} = 45.57503179559
x4=−86.4053708116885x_{4} = -86.4053708116885
x5=−11.085538406497x_{5} = -11.085538406497
x6=95.8290108090195x_{6} = 95.8290108090195
x7=−36.1559664195367x_{7} = -36.1559664195367
x8=−42.4350618814099x_{8} = -42.4350618814099
x9=36.1559664195367x_{9} = 36.1559664195367
x10=7.97866571241324x_{10} = 7.97866571241324
x11=2.02875783811043x_{11} = 2.02875783811043
x12=−80.1230928148503x_{12} = -80.1230928148503
x13=92.687771772017x_{13} = 92.687771772017
x14=86.4053708116885x_{14} = 86.4053708116885
x15=29.8785865061074x_{15} = 29.8785865061074
x16=70.69997803861x_{16} = 70.69997803861
x17=58.1366632448992x_{17} = 58.1366632448992
x18=4.91318043943488x_{18} = 4.91318043943488
x19=73.8409691490209x_{19} = 73.8409691490209
x20=−48.7152107175577x_{20} = -48.7152107175577
x21=54.9960525574964x_{21} = 54.9960525574964
x22=11.085538406497x_{22} = 11.085538406497
x23=−4.91318043943488x_{23} = -4.91318043943488
x24=−76.9820093304187x_{24} = -76.9820093304187
x25=−89.5465575382492x_{25} = -89.5465575382492
x26=−14.2074367251912x_{26} = -14.2074367251912
x27=−33.0170010333572x_{27} = -33.0170010333572
x28=−51.855560729152x_{28} = -51.855560729152
x29=64.4181717218392x_{29} = 64.4181717218392
x30=−67.5590428388084x_{30} = -67.5590428388084
x31=42.4350618814099x_{31} = 42.4350618814099
x32=0x_{32} = 0
x33=−7.97866571241324x_{33} = -7.97866571241324
x34=51.855560729152x_{34} = 51.855560729152
x35=26.7409160147873x_{35} = 26.7409160147873
x36=−26.7409160147873x_{36} = -26.7409160147873
x37=89.5465575382492x_{37} = 89.5465575382492
x38=−83.2642147040886x_{38} = -83.2642147040886
x39=−2.02875783811043x_{39} = -2.02875783811043
x40=83.2642147040886x_{40} = 83.2642147040886
x41=−45.57503179559x_{41} = -45.57503179559
x42=−98.9702722883957x_{42} = -98.9702722883957
x43=67.5590428388084x_{43} = 67.5590428388084
x44=20.469167402741x_{44} = 20.469167402741
x45=−54.9960525574964x_{45} = -54.9960525574964
x46=48.7152107175577x_{46} = 48.7152107175577
x47=39.295350981473x_{47} = 39.295350981473
x48=−17.3363779239834x_{48} = -17.3363779239834
x49=102.111554139654x_{49} = 102.111554139654
x50=−95.8290108090195x_{50} = -95.8290108090195
x51=−64.4181717218392x_{51} = -64.4181717218392
x52=61.2773745335697x_{52} = 61.2773745335697
x53=−29.8785865061074x_{53} = -29.8785865061074
x54=23.6042847729804x_{54} = 23.6042847729804
x55=−39.295350981473x_{55} = -39.295350981473
x56=−58.1366632448992x_{56} = -58.1366632448992
x57=98.9702722883957x_{57} = 98.9702722883957
x58=−23.6042847729804x_{58} = -23.6042847729804
x59=14.2074367251912x_{59} = 14.2074367251912
x60=−73.8409691490209x_{60} = -73.8409691490209
x61=−92.687771772017x_{61} = -92.687771772017
x62=33.0170010333572x_{62} = 33.0170010333572
x63=76.9820093304187x_{63} = 76.9820093304187
x64=−61.2773745335697x_{64} = -61.2773745335697
x65=17.3363779239834x_{65} = 17.3363779239834
x66=−70.69997803861x_{66} = -70.69997803861
Signos de extremos en los puntos:
(-20.46916740274095, 20.4447840582523)

(80.12309281485025, -80.1168531456592)

(45.57503179559002, 45.5640648360268)

(-86.40537081168854, -86.3995847156108)

(-11.085538406497022, -11.04070801593)

(95.82901080901948, 95.8237936084657)

(-36.15596641953672, -36.1421453722421)

(-42.43506188140989, -42.4232840772591)

(36.15596641953672, -36.1421453722421)

(7.978665712413241, 7.91672737158778)

(2.028757838110434, 1.81970574115965)

(-80.12309281485025, -80.1168531456592)

(92.687771772017, -92.6823777880592)

(86.40537081168854, -86.3995847156108)

(29.878586506107393, -29.8618661591868)

(70.69997803861, 70.6929069615931)

(58.13666324489916, 58.1280647280857)

(4.913180439434884, -4.81446988971227)

(73.8409691490209, -73.8341987715416)

(-48.715210717557724, -48.7049502253679)

(54.99605255749639, -54.9869632496976)

(11.085538406497022, -11.04070801593)

(-4.913180439434884, -4.81446988971227)

(-76.98200933041872, 76.9755151282637)

(-89.54655753824919, 89.5409743728852)

(-14.207436725191188, 14.1723741137743)

(-33.017001033357246, 33.0018677308454)

(-51.85556072915197, 51.8459212502015)

(64.41817172183916, 64.4104113393753)

(-67.5590428388084, -67.5516431209725)

(42.43506188140989, -42.4232840772591)

(0, 0)

(-7.978665712413241, 7.91672737158778)

(51.85556072915197, 51.8459212502015)

(26.74091601478731, 26.7222376646974)

(-26.74091601478731, 26.7222376646974)

(89.54655753824919, 89.5409743728852)

(-83.26421470408864, 83.2582103729533)

(-2.028757838110434, 1.81970574115965)

(83.26421470408864, 83.2582103729533)

(-45.57503179559002, 45.5640648360268)

(-98.9702722883957, -98.9652206531187)

(67.5590428388084, -67.5516431209725)

(20.46916740274095, 20.4447840582523)

(-54.99605255749639, -54.9869632496976)

(48.715210717557724, -48.7049502253679)

(39.295350981472986, 39.2826330068918)

(-17.33637792398336, -17.3076086078585)

(102.11155413965392, 102.106657886316)

(-95.82901080901948, 95.8237936084657)

(-64.41817172183916, 64.4104113393753)

(61.277374533569656, -61.2692165444766)

(-29.878586506107393, -29.8618661591868)

(23.604284772980407, -23.5831306496334)

(-39.295350981472986, 39.2826330068918)

(-58.13666324489916, 58.1280647280857)

(98.9702722883957, -98.9652206531187)

(-23.604284772980407, -23.5831306496334)

(14.207436725191188, 14.1723741137743)

(-73.8409691490209, -73.8341987715416)

(-92.687771772017, -92.6823777880592)

(33.017001033357246, 33.0018677308454)

(76.98200933041872, 76.9755151282637)

(-61.277374533569656, -61.2692165444766)

(17.33637792398336, -17.3076086078585)

(-70.69997803861, 70.6929069615931)


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

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
    © Sr Examen – Калькулятор