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

Gráfico de la función y = x*sin(1/x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
            /1\
f(x) = x*sin|-|
            \x/
f(x)=xsin⁡(1x)f{\left(x \right)} = x \sin{\left(\frac{1}{x} \right)} 
f = x*sin(1/x)
Gráfico de la función
02468-8-6-4-2-10102-1
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:
xsin⁡(1x)=0x \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=0.318309886183791x_{1} = 0.318309886183791
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(1/x).
0sin⁡(10)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
sin⁡(1x)−cos⁡(1x)x=0\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=42446.0308837659x_{1} = 42446.0308837659
x2=22103.7156394291x_{2} = 22103.7156394291
x3=−37229.2008029351x_{3} = -37229.2008029351
x4=−24515.2570410387x_{4} = -24515.2570410387
x5=−32991.2092909147x_{5} = -32991.2092909147
x6=9390.17882712825x_{6} = 9390.17882712825
x7=24646.4882156886x_{7} = 24646.4882156886
x8=18713.369170442x_{8} = 18713.369170442
x9=−20277.3091678955x_{9} = -20277.3091678955
x10=−16886.9773640064x_{10} = -16886.9773640064
x11=−9258.95883733077x_{11} = -9258.95883733077
x12=−7563.95858775919x_{12} = -7563.95858775919
x13=33122.4413057753x_{13} = 33122.4413057753
x14=−21124.8963893094x_{14} = -21124.8963893094
x15=13627.9152205591x_{15} = 13627.9152205591
x16=−8411.44821783076x_{16} = -8411.44821783076
x17=−21972.4849235058x_{17} = -21972.4849235058
x18=10237.7072403896x_{18} = 10237.7072403896
x19=−33838.8069987955x_{19} = -33838.8069987955
x20=37360.4330405425x_{20} = 37360.4330405425
x21=−29600.822198113x_{21} = -29600.822198113
x22=−12649.1268015143x_{22} = -12649.1268015143
x23=−19429.7234310854x_{23} = -19429.7234310854
x24=34817.6371394969x_{24} = 34817.6371394969
x25=−30448.418348057x_{25} = -30448.418348057
x26=−27905.6313719175x_{26} = -27905.6313719175
x27=−16039.4000376645x_{27} = -16039.4000376645
x28=−23667.6653660369x_{28} = -23667.6653660369
x29=−31296.0149360127x_{29} = -31296.0149360127
x30=−41467.1983793308x_{30} = -41467.1983793308
x31=−11801.5713963528x_{31} = -11801.5713963528
x32=−13496.6883408414x_{32} = -13496.6883408414
x33=−34686.4050257305x_{33} = -34686.4050257305
x34=−40619.5984700161x_{34} = -40619.5984700161
x35=12780.3528304711x_{35} = 12780.3528304711
x36=−17734.5572628217x_{36} = -17734.5572628217
x37=33970.0390649632x_{37} = 33970.0390649632
x38=−27058.0367879985x_{38} = -27058.0367879985
x39=21256.1269144408x_{39} = 21256.1269144408
x40=25494.0808534902x_{40} = 25494.0808534902
x41=16170.6287133314x_{41} = 16170.6287133314
x42=−42314.798461984x_{42} = -42314.798461984
x43=−32143.6119273287x_{43} = -32143.6119273287
x44=17865.7867327802x_{44} = 17865.7867327802
x45=7695.1721243119x_{45} = 7695.1721243119
x46=27189.2682985119x_{46} = 27189.2682985119
x47=−38076.7998978638x_{47} = -38076.7998978638
x48=38208.0321713516x_{48} = 38208.0321713516
x49=40750.8308380171x_{49} = 40750.8308380171
x50=−35534.0033488874x_{50} = -35534.0033488874
x51=39903.2310836223x_{51} = 39903.2310836223
x52=6847.70624844332x_{52} = 6847.70624844332
x53=19560.9534972764x_{53} = 19560.9534972764
x54=17018.2064665824x_{54} = 17018.2064665824
x55=32274.8438867786x_{55} = 32274.8438867786
x56=30579.6501825125x_{56} = 30579.6501825125
x57=11932.7963857739x_{57} = 11932.7963857739
x58=11085.2472504735x_{58} = 11085.2472504735
x59=8542.66546353133x_{59} = 8542.66546353133
x60=26341.6742377005x_{60} = 26341.6742377005
x61=29732.0539618849x_{61} = 29732.0539618849
x62=28036.8629745587x_{62} = 28036.8629745587
x63=−15191.825714434x_{63} = -15191.825714434
x64=22951.3055101033x_{64} = 22951.3055101033
x65=−38924.3992166982x_{65} = -38924.3992166982
x66=−26210.4428283739x_{66} = -26210.4428283739
x67=−36381.6019475619x_{67} = -36381.6019475619
x68=−28753.2265249185x_{68} = -28753.2265249185
x69=−10106.4851636254x_{69} = -10106.4851636254
x70=−25362.8495556371x_{70} = -25362.8495556371
x71=35665.2355068933x_{71} = 35665.2355068933
x72=23798.8964040481x_{72} = 23798.8964040481
x73=36512.8341467568x_{73} = 36512.8341467568
x74=39055.6315237518x_{74} = 39055.6315237518
x75=−10954.0235496446x_{75} = -10954.0235496446
x76=15323.0538900739x_{76} = 15323.0538900739
x77=14475.4825115465x_{77} = 14475.4825115465
x78=31427.2468354985x_{78} = 31427.2468354985
x79=28884.4582116784x_{79} = 28884.4582116784
x80=20408.539477891x_{80} = 20408.539477891
x81=−6716.49789803616x_{81} = -6716.49789803616
x82=−18582.1393820539x_{82} = -18582.1393820539
x83=−22820.0746241929x_{83} = -22820.0746241929
x84=−14344.2549267547x_{84} = -14344.2549267547
x85=−39771.9987451226x_{85} = -39771.9987451226
x86=41598.4307750453x_{86} = 41598.4307750453
Signos de extremos en los puntos:
(42446.030883765874, 0.999999999907493)

(22103.715639429116, 0.999999999658871)

(-37229.200802935076, 0.999999999879751)

(-24515.257041038723, 0.999999999722683)

(-32991.20929091468, 0.999999999846873)

(9390.178827128255, 0.999999998109829)

(24646.48821568855, 0.999999999725629)

(18713.369170442, 0.999999999524068)

(-20277.309167895473, 0.999999999594652)

(-16886.977364006372, 0.999999999415553)

(-9258.958837330767, 0.999999998055874)

(-7563.9585877591935, 0.999999997086933)

(33122.441305775326, 0.999999999848084)

(-21124.896389309444, 0.999999999626527)

(13627.915220559116, 0.999999999102592)

(-8411.448217830764, 0.999999997644369)

(-21972.484923505832, 0.999999999654784)

(10237.707240389647, 0.999999998409831)

(-33838.80699879548, 0.999999999854448)

(37360.433040542535, 0.999999999880594)

(-29600.82219811303, 0.999999999809787)

(-12649.126801514269, 0.999999998958336)

(-19429.72343108544, 0.999999999558516)

(34817.63713949693, 0.999999999862517)

(-30448.418348056977, 0.999999999820229)

(-27905.63137191755, 0.999999999785975)

(-16039.400037664549, 0.999999999352153)

(-23667.665366036923, 0.999999999702465)

(-31296.0149360127, 0.999999999829835)

(-41467.19837933084, 0.999999999903074)

(-11801.571396352814, 0.999999998803345)

(-13496.688340841416, 0.999999999085056)

(-34686.405025730484, 0.999999999861474)

(-40619.59847001615, 0.999999999898987)

(12780.35283047108, 0.999999998979617)

(-17734.557262821672, 0.999999999470083)

(33970.03906496316, 0.99999999985557)

(-27058.03678799853, 0.999999999772356)

(21256.126914440752, 0.999999999631124)

(25494.080853490228, 0.999999999743569)

(16170.628713331358, 0.999999999362625)

(-42314.79846198405, 0.999999999906918)

(-32143.611927328704, 0.999999999838691)

(17865.786732780187, 0.999999999477839)

(7695.172124311905, 0.99999999718543)

(27189.26829851187, 0.999999999774548)

(-38076.799897863806, 0.999999999885045)

(38208.03217135156, 0.999999999885833)

(40750.83083801706, 0.999999999899637)

(-35534.00334888743, 0.999999999868004)

(39903.23108362226, 0.999999999895328)

(6847.706248443317, 0.999999996445664)

(19560.953497276427, 0.999999999564419)

(17018.206466582393, 0.999999999424532)

(32274.8438867786, 0.99999999984)

(30579.650182512534, 0.999999999821769)

(11932.796385773858, 0.999999998829519)

(11085.247250473549, 0.999999998643693)

(8542.665463531333, 0.999999997716179)

(26341.6742377005, 0.999999999759806)

(29732.053961884885, 0.999999999811462)

(28036.86297455866, 0.999999999787974)

(-15191.82571443399, 0.999999999277848)

(22951.305510103328, 0.999999999683602)

(-38924.399216698235, 0.999999999889997)

(-26210.442828373863, 0.999999999757395)

(-36381.60194756191, 0.999999999874083)

(-28753.226524918457, 0.999999999798407)

(-10106.485163625435, 0.999999998368269)

(-25362.84955563705, 0.999999999740909)

(35665.235506893325, 0.999999999868974)

(23798.896404048144, 0.999999999705737)

(36512.83414675676, 0.999999999874986)

(39055.631523751756, 0.999999999890735)

(-10954.023549644575, 0.999999998611003)

(15323.053890073896, 0.999999999290164)

(14475.482511546521, 0.999999999204605)

(31427.246835498532, 0.999999999831253)

(28884.45821167836, 0.999999999800235)

(20408.539477890954, 0.999999999599848)

(-6716.4978980361575, 0.999999996305438)

(-18582.139382053876, 0.999999999517322)

(-22820.074624192912, 0.999999999679952)

(-14344.254926754675, 0.999999999189985)

(-39771.99874512261, 0.999999999894636)

(41598.43077504528, 0.999999999903685)


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=−37229.2008029351x_{1} = -37229.2008029351
x2=−9258.95883733077x_{2} = -9258.95883733077
x3=−7563.95858775919x_{3} = -7563.95858775919
x4=−33838.8069987955x_{4} = -33838.8069987955
x5=34817.6371394969x_{5} = 34817.6371394969
x6=−13496.6883408414x_{6} = -13496.6883408414
x7=−34686.4050257305x_{7} = -34686.4050257305
x8=38208.0321713516x_{8} = 38208.0321713516
x9=11932.7963857739x_{9} = 11932.7963857739
x10=8542.66546353133x_{10} = 8542.66546353133
x11=−28753.2265249185x_{11} = -28753.2265249185
x12=−10954.0235496446x_{12} = -10954.0235496446
x13=−22820.0746241929x_{13} = -22820.0746241929
x14=−14344.2549267547x_{14} = -14344.2549267547
Puntos máximos de la función:
x14=42446.0308837659x_{14} = 42446.0308837659
x14=−21124.8963893094x_{14} = -21124.8963893094
x14=−29600.822198113x_{14} = -29600.822198113
x14=−23667.6653660369x_{14} = -23667.6653660369
x14=40750.8308380171x_{14} = 40750.8308380171
x14=6847.70624844332x_{14} = 6847.70624844332
x14=17018.2064665824x_{14} = 17018.2064665824
x14=28036.8629745587x_{14} = 28036.8629745587
x14=−39771.9987451226x_{14} = -39771.9987451226
x14=41598.4307750453x_{14} = 41598.4307750453
Decrece en los intervalos
[38208.0321713516,∞)\left[38208.0321713516, \infty\right)
Crece en los intervalos
(−∞,−37229.2008029351]\left(-\infty, -37229.2008029351\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
−sin⁡(1x)x3=0- \frac{\sin{\left(\frac{1}{x} \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1πx_{1} = \frac{1}{\pi}
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

lim⁡x→0−(−sin⁡(1x)x3)=⟨−∞,∞⟩\lim_{x \to 0^-}\left(- \frac{\sin{\left(\frac{1}{x} \right)}}{x^{3}}\right) = \left\langle -\infty, \infty\right\rangle
lim⁡x→0+(−sin⁡(1x)x3)=⟨−∞,∞⟩\lim_{x \to 0^+}\left(- \frac{\sin{\left(\frac{1}{x} \right)}}{x^{3}}\right) = \left\langle -\infty, \infty\right\rangle
- los límites son iguales, es decir omitimos el punto correspondiente

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
(−∞,1π]\left(-\infty, \frac{1}{\pi}\right]
Convexa en los intervalos
[1π,∞)\left[\frac{1}{\pi}, \infty\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→−∞(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 horizontal a la izquierda:
y=1y = 1
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 horizontal a la derecha:
y=1y = 1
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x*sin(1/x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞sin⁡(1x)=0\lim_{x \to -\infty} \sin{\left(\frac{1}{x} \right)} = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞sin⁡(1x)=0\lim_{x \to \infty} \sin{\left(\frac{1}{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⁡(1x)=xsin⁡(1x)x \sin{\left(\frac{1}{x} \right)} = x \sin{\left(\frac{1}{x} \right)}
- Sí
xsin⁡(1x)=−xsin⁡(1x)x \sin{\left(\frac{1}{x} \right)} = - x \sin{\left(\frac{1}{x} \right)}
- No
es decir, función
es
par
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