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Trazado el gráfico de la función/ sinx(x)

Gráfico de la función y = sinx(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = sin(x)*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=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=25.1327412287183x_{3} = 25.1327412287183
x4=3.14159265358979x_{4} = 3.14159265358979
x5=−43.9822971502571x_{5} = -43.9822971502571
x6=81.6814089933346x_{6} = 81.6814089933346
x7=−100.530964914873x_{7} = -100.530964914873
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=−47.1238898038469x_{24} = -47.1238898038469
x25=69.1150383789755x_{25} = 69.1150383789755
x26=21.9911485751286x_{26} = 21.9911485751286
x27=87.9645943005142x_{27} = 87.9645943005142
x28=18.8495559215388x_{28} = 18.8495559215388
x29=−87.9645943005142x_{29} = -87.9645943005142
x30=−72.2566310325652x_{30} = -72.2566310325652
x31=−84.8230016469244x_{31} = -84.8230016469244
x32=37.6991118430775x_{32} = 37.6991118430775
x33=697.433569096934x_{33} = 697.433569096934
x34=0x_{34} = 0
x35=50.2654824574367x_{35} = 50.2654824574367
x36=−6.28318530717959x_{36} = -6.28318530717959
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=−25.1327412287183x_{47} = -25.1327412287183
x48=47.1238898038469x_{48} = 47.1238898038469
x49=97.3893722612836x_{49} = 97.3893722612836
x50=−69.1150383789755x_{50} = -69.1150383789755
x51=94.2477796076938x_{51} = 94.2477796076938
x52=−18.8495559215388x_{52} = -18.8495559215388
x53=−50.2654824574367x_{53} = -50.2654824574367
x54=−37.6991118430775x_{54} = -37.6991118430775
x55=−81.6814089933346x_{55} = -81.6814089933346
x56=62.8318530717959x_{56} = 62.8318530717959
x57=78.5398163397448x_{57} = 78.5398163397448
x58=31.4159265358979x_{58} = 31.4159265358979
x59=−78.5398163397448x_{59} = -78.5398163397448
x60=−40.8407044966673x_{60} = -40.8407044966673
x61=−97.3893722612836x_{61} = -97.3893722612836
x62=−75.398223686155x_{62} = -75.398223686155
x63=91.106186954104x_{63} = 91.106186954104
x64=−12.5663706143592x_{64} = -12.5663706143592
x65=−94.2477796076938x_{65} = -94.2477796076938
x66=−34.5575191894877x_{66} = -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)*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=−67.5590428388084x_{1} = -67.5590428388084
x2=−17.3363779239834x_{2} = -17.3363779239834
x3=−7.97866571241324x_{3} = -7.97866571241324
x4=26.7409160147873x_{4} = 26.7409160147873
x5=2.02875783811043x_{5} = 2.02875783811043
x6=−92.687771772017x_{6} = -92.687771772017
x7=−14.2074367251912x_{7} = -14.2074367251912
x8=64.4181717218392x_{8} = 64.4181717218392
x9=−89.5465575382492x_{9} = -89.5465575382492
x10=−39.295350981473x_{10} = -39.295350981473
x11=33.0170010333572x_{11} = 33.0170010333572
x12=54.9960525574964x_{12} = 54.9960525574964
x13=76.9820093304187x_{13} = 76.9820093304187
x14=−33.0170010333572x_{14} = -33.0170010333572
x15=58.1366632448992x_{15} = 58.1366632448992
x16=17.3363779239834x_{16} = 17.3363779239834
x17=−2.02875783811043x_{17} = -2.02875783811043
x18=−61.2773745335697x_{18} = -61.2773745335697
x19=−86.4053708116885x_{19} = -86.4053708116885
x20=−80.1230928148503x_{20} = -80.1230928148503
x21=−70.69997803861x_{21} = -70.69997803861
x22=95.8290108090195x_{22} = 95.8290108090195
x23=29.8785865061074x_{23} = 29.8785865061074
x24=23.6042847729804x_{24} = 23.6042847729804
x25=86.4053708116885x_{25} = 86.4053708116885
x26=4.91318043943488x_{26} = 4.91318043943488
x27=−26.7409160147873x_{27} = -26.7409160147873
x28=−98.9702722883957x_{28} = -98.9702722883957
x29=−29.8785865061074x_{29} = -29.8785865061074
x30=11.085538406497x_{30} = 11.085538406497
x31=67.5590428388084x_{31} = 67.5590428388084
x32=14.2074367251912x_{32} = 14.2074367251912
x33=−51.855560729152x_{33} = -51.855560729152
x34=51.855560729152x_{34} = 51.855560729152
x35=70.69997803861x_{35} = 70.69997803861
x36=0x_{36} = 0
x37=−95.8290108090195x_{37} = -95.8290108090195
x38=−45.57503179559x_{38} = -45.57503179559
x39=−42.4350618814099x_{39} = -42.4350618814099
x40=20.469167402741x_{40} = 20.469167402741
x41=−23.6042847729804x_{41} = -23.6042847729804
x42=36.1559664195367x_{42} = 36.1559664195367
x43=−83.2642147040886x_{43} = -83.2642147040886
x44=−76.9820093304187x_{44} = -76.9820093304187
x45=−36.1559664195367x_{45} = -36.1559664195367
x46=42.4350618814099x_{46} = 42.4350618814099
x47=61.2773745335697x_{47} = 61.2773745335697
x48=−58.1366632448992x_{48} = -58.1366632448992
x49=39.295350981473x_{49} = 39.295350981473
x50=−64.4181717218392x_{50} = -64.4181717218392
x51=−48.7152107175577x_{51} = -48.7152107175577
x52=−73.8409691490209x_{52} = -73.8409691490209
x53=7.97866571241324x_{53} = 7.97866571241324
x54=80.1230928148503x_{54} = 80.1230928148503
x55=−4.91318043943488x_{55} = -4.91318043943488
x56=45.57503179559x_{56} = 45.57503179559
x57=98.9702722883957x_{57} = 98.9702722883957
x58=73.8409691490209x_{58} = 73.8409691490209
x59=−20.469167402741x_{59} = -20.469167402741
x60=−54.9960525574964x_{60} = -54.9960525574964
x61=92.687771772017x_{61} = 92.687771772017
x62=89.5465575382492x_{62} = 89.5465575382492
x63=−11.085538406497x_{63} = -11.085538406497
x64=102.111554139654x_{64} = 102.111554139654
x65=83.2642147040886x_{65} = 83.2642147040886
x66=48.7152107175577x_{66} = 48.7152107175577
Signos de extremos en los puntos:
(-67.5590428388084, -67.5516431209725)

(-17.33637792398336, -17.3076086078585)

(-7.978665712413241, 7.91672737158778)

(26.74091601478731, 26.7222376646974)

(2.028757838110434, 1.81970574115965)

(-92.687771772017, -92.6823777880592)

(-14.207436725191188, 14.1723741137743)

(64.41817172183916, 64.4104113393753)

(-89.54655753824919, 89.5409743728852)

(-39.295350981472986, 39.2826330068918)

(33.017001033357246, 33.0018677308454)

(54.99605255749639, -54.9869632496976)

(76.98200933041872, 76.9755151282637)

(-33.017001033357246, 33.0018677308454)

(58.13666324489916, 58.1280647280857)

(17.33637792398336, -17.3076086078585)

(-2.028757838110434, 1.81970574115965)

(-61.277374533569656, -61.2692165444766)

(-86.40537081168854, -86.3995847156108)

(-80.12309281485025, -80.1168531456592)

(-70.69997803861, 70.6929069615931)

(95.82901080901948, 95.8237936084657)

(29.878586506107393, -29.8618661591868)

(23.604284772980407, -23.5831306496334)

(86.40537081168854, -86.3995847156108)

(4.913180439434884, -4.81446988971227)

(-26.74091601478731, 26.7222376646974)

(-98.9702722883957, -98.9652206531187)

(-29.878586506107393, -29.8618661591868)

(11.085538406497022, -11.04070801593)

(67.5590428388084, -67.5516431209725)

(14.207436725191188, 14.1723741137743)

(-51.85556072915197, 51.8459212502015)

(51.85556072915197, 51.8459212502015)

(70.69997803861, 70.6929069615931)

(0, 0)

(-95.82901080901948, 95.8237936084657)

(-45.57503179559002, 45.5640648360268)

(-42.43506188140989, -42.4232840772591)

(20.46916740274095, 20.4447840582523)

(-23.604284772980407, -23.5831306496334)

(36.15596641953672, -36.1421453722421)

(-83.26421470408864, 83.2582103729533)

(-76.98200933041872, 76.9755151282637)

(-36.15596641953672, -36.1421453722421)

(42.43506188140989, -42.4232840772591)

(61.277374533569656, -61.2692165444766)

(-58.13666324489916, 58.1280647280857)

(39.295350981472986, 39.2826330068918)

(-64.41817172183916, 64.4104113393753)

(-48.715210717557724, -48.7049502253679)

(-73.8409691490209, -73.8341987715416)

(7.978665712413241, 7.91672737158778)

(80.12309281485025, -80.1168531456592)

(-4.913180439434884, -4.81446988971227)

(45.57503179559002, 45.5640648360268)

(98.9702722883957, -98.9652206531187)

(73.8409691490209, -73.8341987715416)

(-20.46916740274095, 20.4447840582523)

(-54.99605255749639, -54.9869632496976)

(92.687771772017, -92.6823777880592)

(89.54655753824919, 89.5409743728852)

(-11.085538406497022, -11.04070801593)

(102.11155413965392, 102.106657886316)

(83.26421470408864, 83.2582103729533)

(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=−67.5590428388084x_{1} = -67.5590428388084
x2=−17.3363779239834x_{2} = -17.3363779239834
x3=−92.687771772017x_{3} = -92.687771772017
x4=54.9960525574964x_{4} = 54.9960525574964
x5=17.3363779239834x_{5} = 17.3363779239834
x6=−61.2773745335697x_{6} = -61.2773745335697
x7=−86.4053708116885x_{7} = -86.4053708116885
x8=−80.1230928148503x_{8} = -80.1230928148503
x9=29.8785865061074x_{9} = 29.8785865061074
x10=23.6042847729804x_{10} = 23.6042847729804
x11=86.4053708116885x_{11} = 86.4053708116885
x12=4.91318043943488x_{12} = 4.91318043943488
x13=−98.9702722883957x_{13} = -98.9702722883957
x14=−29.8785865061074x_{14} = -29.8785865061074
x15=11.085538406497x_{15} = 11.085538406497
x16=67.5590428388084x_{16} = 67.5590428388084
x17=0x_{17} = 0
x18=−42.4350618814099x_{18} = -42.4350618814099
x19=−23.6042847729804x_{19} = -23.6042847729804
x20=36.1559664195367x_{20} = 36.1559664195367
x21=−36.1559664195367x_{21} = -36.1559664195367
x22=42.4350618814099x_{22} = 42.4350618814099
x23=61.2773745335697x_{23} = 61.2773745335697
x24=−48.7152107175577x_{24} = -48.7152107175577
x25=−73.8409691490209x_{25} = -73.8409691490209
x26=80.1230928148503x_{26} = 80.1230928148503
x27=−4.91318043943488x_{27} = -4.91318043943488
x28=98.9702722883957x_{28} = 98.9702722883957
x29=73.8409691490209x_{29} = 73.8409691490209
x30=−54.9960525574964x_{30} = -54.9960525574964
x31=92.687771772017x_{31} = 92.687771772017
x32=−11.085538406497x_{32} = -11.085538406497
x33=48.7152107175577x_{33} = 48.7152107175577
Puntos máximos de la función:
x33=−7.97866571241324x_{33} = -7.97866571241324
x33=26.7409160147873x_{33} = 26.7409160147873
x33=2.02875783811043x_{33} = 2.02875783811043
x33=−14.2074367251912x_{33} = -14.2074367251912
x33=64.4181717218392x_{33} = 64.4181717218392
x33=−89.5465575382492x_{33} = -89.5465575382492
x33=−39.295350981473x_{33} = -39.295350981473
x33=33.0170010333572x_{33} = 33.0170010333572
x33=76.9820093304187x_{33} = 76.9820093304187
x33=−33.0170010333572x_{33} = -33.0170010333572
x33=58.1366632448992x_{33} = 58.1366632448992
x33=−2.02875783811043x_{33} = -2.02875783811043
x33=−70.69997803861x_{33} = -70.69997803861
x33=95.8290108090195x_{33} = 95.8290108090195
x33=−26.7409160147873x_{33} = -26.7409160147873
x33=14.2074367251912x_{33} = 14.2074367251912
x33=−51.855560729152x_{33} = -51.855560729152
x33=51.855560729152x_{33} = 51.855560729152
x33=70.69997803861x_{33} = 70.69997803861
x33=−95.8290108090195x_{33} = -95.8290108090195
x33=−45.57503179559x_{33} = -45.57503179559
x33=20.469167402741x_{33} = 20.469167402741
x33=−83.2642147040886x_{33} = -83.2642147040886
x33=−76.9820093304187x_{33} = -76.9820093304187
x33=−58.1366632448992x_{33} = -58.1366632448992
x33=39.295350981473x_{33} = 39.295350981473
x33=−64.4181717218392x_{33} = -64.4181717218392
x33=7.97866571241324x_{33} = 7.97866571241324
x33=45.57503179559x_{33} = 45.57503179559
x33=−20.469167402741x_{33} = -20.469167402741
x33=89.5465575382492x_{33} = 89.5465575382492
x33=102.111554139654x_{33} = 102.111554139654
x33=83.2642147040886x_{33} = 83.2642147040886
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=6.57833373272234x_{1} = 6.57833373272234
x2=−87.9873209346887x_{2} = -87.9873209346887
x3=78.5652673845995x_{3} = 78.5652673845995
x4=−91.1281305511393x_{4} = -91.1281305511393
x5=−9.62956034329743x_{5} = -9.62956034329743
x6=−72.2842925036825x_{6} = -72.2842925036825
x7=9.62956034329743x_{7} = 9.62956034329743
x8=62.863657228703x_{8} = 62.863657228703
x9=72.2842925036825x_{9} = 72.2842925036825
x10=−78.5652673845995x_{10} = -78.5652673845995
x11=100.550852725424x_{11} = 100.550852725424
x12=−37.7520396346102x_{12} = -37.7520396346102
x13=−53.4444796697636x_{13} = -53.4444796697636
x14=40.8895777660408x_{14} = 40.8895777660408
x15=31.479374920314x_{15} = 31.479374920314
x16=69.1439554764926x_{16} = 69.1439554764926
x17=1.0768739863118x_{17} = 1.0768739863118
x18=−12.7222987717666x_{18} = -12.7222987717666
x19=84.8465692433091x_{19} = 84.8465692433091
x20=94.2689923093066x_{20} = 94.2689923093066
x21=−15.8336114149477x_{21} = -15.8336114149477
x22=56.5839987378634x_{22} = 56.5839987378634
x23=−75.4247339745236x_{23} = -75.4247339745236
x24=25.2119030642106x_{24} = 25.2119030642106
x25=−59.7237354324305x_{25} = -59.7237354324305
x26=91.1281305511393x_{26} = 91.1281305511393
x27=50.3052188363296x_{27} = 50.3052188363296
x28=59.7237354324305x_{28} = 59.7237354324305
x29=22.0814757672807x_{29} = 22.0814757672807
x30=−94.2689923093066x_{30} = -94.2689923093066
x31=−34.6152330552306x_{31} = -34.6152330552306
x32=15.8336114149477x_{32} = 15.8336114149477
x33=−28.3447768697864x_{33} = -28.3447768697864
x34=75.4247339745236x_{34} = 75.4247339745236
x35=−97.4099011706723x_{35} = -97.4099011706723
x36=81.7058821480364x_{36} = 81.7058821480364
x37=34.6152330552306x_{37} = 34.6152330552306
x38=12.7222987717666x_{38} = 12.7222987717666
x39=−31.479374920314x_{39} = -31.479374920314
x40=−100.550852725424x_{40} = -100.550852725424
x41=44.0276918992479x_{41} = 44.0276918992479
x42=18.954681766529x_{42} = 18.954681766529
x43=−66.0037377708277x_{43} = -66.0037377708277
x44=28.3447768697864x_{44} = 28.3447768697864
x45=−69.1439554764926x_{45} = -69.1439554764926
x46=−81.7058821480364x_{46} = -81.7058821480364
x47=66.0037377708277x_{47} = 66.0037377708277
x48=−1.0768739863118x_{48} = -1.0768739863118
x49=−40.8895777660408x_{49} = -40.8895777660408
x50=−44.0276918992479x_{50} = -44.0276918992479
x51=−56.5839987378634x_{51} = -56.5839987378634
x52=−62.863657228703x_{52} = -62.863657228703
x53=−25.2119030642106x_{53} = -25.2119030642106
x54=47.1662676027767x_{54} = 47.1662676027767
x55=−3.6435971674254x_{55} = -3.6435971674254
x56=87.9873209346887x_{56} = 87.9873209346887
x57=−18.954681766529x_{57} = -18.954681766529
x58=−84.8465692433091x_{58} = -84.8465692433091
x59=37.7520396346102x_{59} = 37.7520396346102
x60=−50.3052188363296x_{60} = -50.3052188363296
x61=−47.1662676027767x_{61} = -47.1662676027767
x62=97.4099011706723x_{62} = 97.4099011706723
x63=−22.0814757672807x_{63} = -22.0814757672807
x64=53.4444796697636x_{64} = 53.4444796697636
x65=−128.820822990274x_{65} = -128.820822990274
x66=−6.57833373272234x_{66} = -6.57833373272234
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 sin(x)*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)}
- No
xsin⁡(x)=−xsin⁡(x)x \sin{\left(x \right)} = - x \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор