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Trazado el gráfico de la función/ x*x*sin(1/x)

Gráfico de la función y = x*x*sin(1/x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
              /1\
f(x) = x*x*sin|-|
              \x/
f(x)=xxsin⁡(1x)f{\left(x \right)} = x x \sin{\left(\frac{1}{x} \right)} 
f = (x*x)*sin(1/x)
Gráfico de la función
02468-8-6-4-2-1010-2020
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:
xxsin⁡(1x)=0x x \sin{\left(\frac{1}{x} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1πx_{1} = \frac{1}{\pi}
Solución numérica
x1=−0.00489707517205832x_{1} = -0.00489707517205832
x2=0.00413389462576352x_{2} = 0.00413389462576352
x3=0.00636619772367581x_{3} = 0.00636619772367581
x4=0.00361715779754308x_{4} = 0.00361715779754308
x5=−0.00624137031732923x_{5} = -0.00624137031732923
x6=0.0530516476972984x_{6} = 0.0530516476972984
x7=−0.00539508281667442x_{7} = -0.00539508281667442
x8=0.00388182788029013x_{8} = 0.00388182788029013
x9=0.00202745150435535x_{9} = 0.00202745150435535
x10=−0.0454728408833987x_{10} = -0.0454728408833987
x11=0.00757880681389978x_{11} = 0.00757880681389978
x12=−0.0117892550438441x_{12} = -0.0117892550438441
x13=0.0040292390656176x_{13} = 0.0040292390656176
x14=0.00521819485547198x_{14} = 0.00521819485547198
x15=0.00691978013443023x_{15} = 0.00691978013443023
x16=0.0795774715459477x_{16} = 0.0795774715459477
x17=0.0117892550438441x_{17} = 0.0117892550438441
x18=−0.00795774715459477x_{18} = -0.00795774715459477
x19=0.00424413181578388x_{19} = 0.00424413181578388
x20=−0.00757880681389978x_{20} = -0.00757880681389978
x21=0.00589462752192205x_{21} = 0.00589462752192205
x22=0.00837657595220502x_{22} = 0.00837657595220502
x23=−0.00723431559508615x_{23} = -0.00723431559508615
x24=−0.00370127774632315x_{24} = -0.00370127774632315
x25=−0.0353677651315323x_{25} = -0.0353677651315323
x26=−0.00418828797610251x_{26} = -0.00418828797610251
x27=−0.0138395602688605x_{27} = -0.0138395602688605
x28=0.0127323954473516x_{28} = 0.0127323954473516
x29=0.0151576136277996x_{29} = 0.0151576136277996
x30=−0.159154943091895x_{30} = -0.159154943091895
x31=−0.01675315190441x_{31} = -0.01675315190441
x32=0.00436040939977795x_{32} = 0.00436040939977795
x33=0.0187241109519877x_{33} = 0.0187241109519877
x34=0.00475089382363867x_{34} = 0.00475089382363867
x35=0.0244853758602916x_{35} = 0.0244853758602916
x36=−0.0212206590789194x_{36} = -0.0212206590789194
x37=−0.0244853758602916x_{37} = -0.0244853758602916
x38=0.00345989006721512x_{38} = 0.00345989006721512
x39=0.000413927030147972x_{39} = 0.000413927030147972
x40=0.00489707517205832x_{40} = 0.00489707517205832
x41=0.0397887357729738x_{41} = 0.0397887357729738
x42=0.00136029865890509x_{42} = 0.00136029865890509
x43=−0.00578745247606892x_{43} = -0.00578745247606892
x44=0.00663145596216231x_{44} = 0.00663145596216231
x45=0.159154943091895x_{45} = 0.159154943091895
x46=−0.00397887357729738x_{46} = -0.00397887357729738
x47=−0.0102680608446384x_{47} = -0.0102680608446384
x48=−0.0187241109519877x_{48} = -0.0187241109519877
x49=0.00909456817667973x_{49} = 0.00909456817667973
x50=−0.00600584690912813x_{50} = -0.00600584690912813
x51=0.0212206590789194x_{51} = 0.0212206590789194
x52=−0.00558438396813668x_{52} = -0.00558438396813668
x53=0.0109762029718549x_{53} = 0.0109762029718549
x54=−0.00461318675628682x_{54} = -0.00461318675628682
x55=−0.00884194128288307x_{55} = -0.00884194128288307
x56=−0.0159154943091895x_{56} = -0.0159154943091895
x57=0.00370127774632315x_{57} = 0.00370127774632315
x58=0.0289372623803446x_{58} = 0.0289372623803446
x59=−0.00475089382363867x_{59} = -0.00475089382363867
x60=0.00338627538493394x_{60} = 0.00338627538493394
x61=0.00530516476972984x_{61} = 0.00530516476972984
x62=−0.0109762029718549x_{62} = -0.0109762029718549
x63=0.00461318675628682x_{63} = 0.00461318675628682
x64=−0.0289372623803446x_{64} = -0.0289372623803446
x65=−0.00312068515866461x_{65} = -0.00312068515866461
x66=−0.00349791083718451x_{66} = -0.00349791083718451
x67=−0.00361715779754308x_{67} = -0.00361715779754308
x68=−0.00408089597671527x_{68} = -0.00408089597671527
x69=−0.00936205547599384x_{69} = -0.00936205547599384
x70=−0.0795774715459477x_{70} = -0.0795774715459477
x71=0.00353677651315323x_{71} = 0.00353677651315323
x72=−0.0127323954473516x_{72} = -0.0127323954473516
x73=−0.00378940340694989x_{73} = -0.00378940340694989
x74=−0.00303152272555991x_{74} = -0.00303152272555991
x75=0.00448323783357452x_{75} = 0.00448323783357452
x76=0.00578745247606892x_{76} = 0.00578745247606892
x77=0.0138395602688605x_{77} = 0.0138395602688605
x78=0.00378940340694989x_{78} = 0.00378940340694989
x79=−0.00837657595220502x_{79} = -0.00837657595220502
x80=−0.00649612012619981x_{80} = -0.00649612012619981
x81=−0.00505253787593319x_{81} = -0.00505253787593319
x82=0.00723431559508615x_{82} = 0.00723431559508615
x83=−0.00448323783357452x_{83} = -0.00448323783357452
x84=0.00558438396813668x_{84} = 0.00558438396813668
x85=0.00964575412678154x_{85} = 0.00964575412678154
x86=0.0102680608446384x_{86} = 0.0102680608446384
x87=0.01675315190441x_{87} = 0.01675315190441
x88=−0.0151576136277996x_{88} = -0.0151576136277996
x89=−0.00691978013443023x_{89} = -0.00691978013443023
x90=0.00539508281667442x_{90} = 0.00539508281667442
x91=0.00795774715459477x_{91} = 0.00795774715459477
x92=−0.00436040939977795x_{92} = -0.00436040939977795
x93=0.00884194128288307x_{93} = 0.00884194128288307
x94=−0.00521819485547198x_{94} = -0.00521819485547198
x95=−0.00335063038088201x_{95} = -0.00335063038088201
x96=0.00505253787593319x_{96} = 0.00505253787593319
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*x)*sin(1/x).
0⋅0sin⁡(10)0 \cdot 0 \sin{\left(\frac{1}{0} \right)}
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
2xsin⁡(1x)−cos⁡(1x)=02 x \sin{\left(\frac{1}{x} \right)} - \cos{\left(\frac{1}{x} \right)} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
2sin⁡(1x)+2cos⁡(1x)−sin⁡(1x)xx−4cos⁡(1x)x=02 \sin{\left(\frac{1}{x} \right)} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x} - \frac{4 \cos{\left(\frac{1}{x} \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−35534.38769619x_{1} = -35534.38769619
x2=−19430.4264107509x_{2} = -19430.4264107509
x3=20409.2087544722x_{3} = 20409.2087544722
x4=15323.9453857711x_{4} = 15323.9453857711
x5=25494.6165959442x_{5} = 25494.6165959442
x6=−14345.2072448485x_{6} = -14345.2072448485
x7=−25363.3880568599x_{7} = -25363.3880568599
x8=24647.0423858719x_{8} = 24647.0423858719
x9=−16887.786230812x_{9} = -16887.786230812
x10=33122.8536469634x_{10} = 33122.8536469634
x11=−23668.2424439769x_{11} = -23668.2424439769
x12=40751.1659842561x_{12} = 40751.1659842561
x13=−11802.729037405x_{13} = -11802.729037405
x14=−20277.9827555881x_{14} = -20277.9827555881
x15=−37229.5676480559x_{15} = -37229.5676480559
x16=11086.4798333592x_{16} = 11086.4798333592
x17=34818.0294026106x_{17} = 34818.0294026106
x18=−38077.1585763586x_{18} = -38077.1585763586
x19=−29601.2835919767x_{19} = -29601.2835919767
x20=28037.3501205005x_{20} = 28037.3501205005
x21=−42315.1212153018x_{21} = -42315.1212153018
x22=−16040.2516648767x_{22} = -16040.2516648767
x23=−21125.5429444287x_{23} = -21125.5429444287
x24=−27906.1207977464x_{24} = -27906.1207977464
x25=17019.0091244462x_{25} = 17019.0091244462
x26=−9260.43473520905x_{26} = -9260.43473520905
x27=9391.63417606635x_{27} = 9391.63417606635
x28=18714.0990935959x_{28} = 18714.0990935959
x29=38208.3896238771x_{29} = 38208.3896238771
x30=−32144.0368179208x_{30} = -32144.0368179208
x31=22104.3335750288x_{31} = 22104.3335750288
x32=−8413.07303917664x_{32} = -8413.07303917664
x33=−41467.5277301779x_{33} = -41467.5277301779
x34=−24515.814163577x_{34} = -24515.814163577
x35=−39772.3421347211x_{35} = -39772.3421347211
x36=−15192.7248763841x_{36} = -15192.7248763841
x37=−7565.76578660753x_{37} = -7565.76578660753
x38=−30448.8668966256x_{38} = -30448.8668966256
x39=12781.4218076587x_{39} = 12781.4218076587
x40=−13497.7004965963x_{40} = -13497.7004965963
x41=−10107.8371566808x_{41} = -10107.8371566808
x42=−6718.53364051853x_{42} = -6718.53364051853
x43=21256.7694965472x_{43} = 21256.7694965472
x44=28884.931060558x_{44} = 28884.931060558
x45=−33839.2106020801x_{45} = -33839.2106020801
x46=−28753.7015215237x_{46} = -28753.7015215237
x47=11933.9413489672x_{47} = 11933.9413489672
x48=32275.2670580593x_{48} = 32275.2670580593
x49=−26210.9639129039x_{49} = -26210.9639129039
x50=35665.6184468128x_{50} = 35665.6184468128
x51=16171.4734604606x_{51} = 16171.4734604606
x52=6849.70308370924x_{52} = 6849.70308370924
x53=10239.0419711546x_{53} = 10239.0419711546
x54=−31296.4513351777x_{54} = -31296.4513351777
x55=−10955.2708382264x_{55} = -10955.2708382264
x56=29732.5133290166x_{56} = 29732.5133290166
x57=−17735.3274585537x_{57} = -17735.3274585537
x58=−40619.9346937544x_{58} = -40619.9346937544
x59=39903.5733493785x_{59} = 39903.5733493785
x60=−21973.1065322912x_{60} = -21973.1065322912
x61=19561.651782658x_{61} = 19561.651782658
x62=−38924.7500842671x_{62} = -38924.7500842671
x63=−22820.6731403276x_{63} = -22820.6731403276
x64=8544.26541208405x_{64} = 8544.26541208405
x65=−32991.6232643634x_{65} = -32991.6232643634
x66=−36381.977339873x_{66} = -36381.977339873
x67=−34686.7987657255x_{67} = -34686.7987657255
x68=39055.9812180799x_{68} = 39055.9812180799
x69=22951.9006201743x_{69} = 22951.9006201743
x70=23799.4703149306x_{70} = 23799.4703149306
x71=13628.917672133x_{71} = 13628.917672133
x72=7696.94860169018x_{72} = 7696.94860169018
x73=26342.1927386122x_{73} = 26342.1927386122
x74=42446.352644067x_{74} = 42446.352644067
x75=27189.7706331009x_{75} = 27189.7706331009
x76=−18582.8744361227x_{76} = -18582.8744361227
x77=31427.6814211469x_{77} = 31427.6814211469
x78=33970.44111659x_{78} = 33970.44111659
x79=37360.7986033292x_{79} = 37360.7986033292
x80=41598.7590919275x_{80} = 41598.7590919275
x81=14476.4262343106x_{81} = 14476.4262343106
x82=17866.5512970788x_{82} = 17866.5512970788
x83=−27058.541547222x_{83} = -27058.541547222
x84=−12650.2068210835x_{84} = -12650.2068210835
x85=30580.0968153986x_{85} = 30580.0968153986
x86=36513.2081963902x_{86} = 36513.2081963902
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

True

True

- 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
[42446.352644067,∞)\left[42446.352644067, \infty\right)
Convexa en los intervalos
(−∞,−33839.2106020801]\left(-\infty, -33839.2106020801\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→−∞(xxsin⁡(1x))=−∞\lim_{x \to -\infty}\left(x x \sin{\left(\frac{1}{x} \right)}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(xxsin⁡(1x))=∞\lim_{x \to \infty}\left(x x \sin{\left(\frac{1}{x} \right)}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x*x)*sin(1/x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(xsin⁡(1x))=1\lim_{x \to -\infty}\left(x \sin{\left(\frac{1}{x} \right)}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xy = x
lim⁡x→∞(xsin⁡(1x))=1\lim_{x \to \infty}\left(x \sin{\left(\frac{1}{x} \right)}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xy = 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:
xxsin⁡(1x)=−x2sin⁡(1x)x x \sin{\left(\frac{1}{x} \right)} = - x^{2} \sin{\left(\frac{1}{x} \right)}
- No
xxsin⁡(1x)=x2sin⁡(1x)x x \sin{\left(\frac{1}{x} \right)} = x^{2} \sin{\left(\frac{1}{x} \right)}
- No
es decir, función
no es
par ni impar
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