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Trazado el gráfico de la función/ sin(x*sin(3/x))

Gráfico de la función y = sin(x*sin(3/x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /     /3\\
f(x) = sin|x*sin|-||
          \     \x//
f(x)=sin⁡(xsin⁡(3x))f{\left(x \right)} = \sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)} 
f = sin(x*sin(3/x))
Gráfico de la función
02468-8-6-4-2-10102-2
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:
sin⁡(xsin⁡(3x))=0\sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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*sin(3/x)).
sin⁡(0sin⁡(30))\sin{\left(0 \sin{\left(\frac{3}{0} \right)} \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⁡(3x)−3cos⁡(3x)x)cos⁡(xsin⁡(3x))=0\left(\sin{\left(\frac{3}{x} \right)} - \frac{3 \cos{\left(\frac{3}{x} \right)}}{x}\right) \cos{\left(x \sin{\left(\frac{3}{x} \right)} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=27192.8968063292x_{1} = 27192.8968063292
x2=−27061.6828865696x_{2} = -27061.6828865696
x3=41600.7992951899x_{3} = 41600.7992951899
x4=−33841.720053652x_{4} = -33841.720053652
x5=−39774.4762218498x_{5} = -39774.4762218498
x6=−35536.7770518919x_{6} = -35536.7770518919
x7=21260.7750091757x_{7} = 21260.7750091757
x8=−34689.2466949144x_{8} = -34689.2466949144
x9=−38926.9307606782x_{9} = -38926.9307606782
x10=−27909.1662379044x_{10} = -27909.1662379044
x11=−31299.1654607907x_{11} = -31299.1654607907
x12=28040.3813064297x_{12} = 28040.3813064297
x13=19566.0078246694x_{13} = 19566.0078246694
x14=−25366.740588564x_{14} = -25366.740588564
x15=22108.1842379968x_{15} = 22108.1842379968
x16=−37231.8478930755x_{16} = -37231.8478930755
x17=−28756.6567573918x_{17} = -28756.6567573918
x18=33125.417614296x_{18} = 33125.417614296
x19=−20282.1832785466x_{19} = -20282.1832785466
x20=24650.4930702239x_{20} = 24650.4930702239
x21=33972.9408891771x_{21} = 33972.9408891771
x22=35667.9990274873x_{22} = 35667.9990274873
x23=31430.3842249972x_{23} = 31430.3842249972
x24=34820.4681214307x_{24} = 34820.4681214307
x25=23803.0447297728x_{25} = 23803.0447297728
x26=42448.3520169802x_{26} = 42448.3520169802
x27=−21129.5734293334x_{27} = -21129.5734293334
x28=30582.8748447132x_{28} = 30582.8748447132
x29=28887.8728740778x_{29} = 28887.8728740778
x30=−30451.6568893745x_{30} = -30451.6568893745
x31=−19434.8120251743x_{31} = -19434.8120251743
x32=−38079.3879244013x_{32} = -38079.3879244013
x33=−32994.1974160571x_{33} = -32994.1974160571
x34=39905.7004356266x_{34} = 39905.7004356266
x35=39058.1545847464x_{35} = 39058.1545847464
x36=20413.3821513715x_{36} = 20413.3821513715
x37=36515.5333502675x_{37} = 36515.5333502675
x38=−42317.1267715717x_{38} = -42317.1267715717
x39=−22824.4018926693x_{39} = -22824.4018926693
x40=−36384.3108636297x_{40} = -36384.3108636297
x41=25497.9518533766x_{41} = 25497.9518533766
x42=−32146.6791006688x_{42} = -32146.6791006688
x43=−23671.836715066x_{43} = -23671.836715066
x44=22955.6080017742x_{44} = 22955.6080017742
x45=40753.2487222549x_{45} = 40753.2487222549
x46=−24519.2833456008x_{46} = -24519.2833456008
x47=32277.8986103544x_{47} = 32277.8986103544
x48=−21976.980261313x_{48} = -21976.980261313
x49=−1.62980542332972x_{49} = -1.62980542332972
x50=26345.4200618006x_{50} = 26345.4200618006
x51=38210.6113323835x_{51} = 38210.6113323835
x52=−41469.5743726085x_{52} = -41469.5743726085
x53=−40622.0241429615x_{53} = -40622.0241429615
x54=29735.3709004255x_{54} = 29735.3709004255
x55=−29604.1538247519x_{55} = -29604.1538247519
x56=−26214.20740474x_{56} = -26214.20740474
x57=37363.0708561564x_{57} = 37363.0708561564
Signos de extremos en los puntos:
(27192.896806329158, 0.14112001408454)

(-27061.682886569608, 0.141120014143105)

(41600.79929518987, 0.14112001063406)

(-33841.72005365198, 0.141120011949777)

(-39774.47622184978, 0.141120010875886)

(-35536.77705189194, 0.14112001158754)

(21260.775009175744, 0.141120017915539)

(-34689.24669491438, 0.141120011762023)

(-38926.93076067823, 0.141120010999845)

(-27909.166237904406, 0.14112001377927)

(-31299.16546079066, 0.141120012607432)

(28040.38130642967, 0.141120013725868)

(19566.007824669392, 0.141120019696839)

(-25366.740588563975, 0.141120014983198)

(22108.184237996822, 0.141120017174481)

(-37231.84789307547, 0.141120011273641)

(-28756.65675739183, 0.141120013447123)

(33125.41761429603, 0.141120012119826)

(-20282.183278546625, 0.141120018889532)

(24650.493070223867, 0.141120015391373)

(33972.94088917714, 0.141120011919786)

(35667.99902748726, 0.141120011561631)

(31430.384224997182, 0.14112001256954)

(34820.46812143073, 0.141120011734172)

(23803.044729772777, 0.141120015922706)

(42448.35201698021, 0.14112001053229)

(-21129.573429333373, 0.141120018038314)

(30582.87484471315, 0.141120012822946)

(28887.87287407782, 0.141120013398294)

(-30451.65688937449, 0.141120012864083)

(-19434.812025174313, 0.141120019854481)

(-38079.38792440132, 0.141120011132173)

(-32994.19741605714, 0.141120012152185)

(39905.70043562664, 0.141120010857396)

(39058.15458474644, 0.141120010980124)

(20413.382151371516, 0.141120018750772)

(36515.533350267455, 0.141120011400965)

(-42317.1267715717, 0.141120010547648)

(-22824.401892669306, 0.141120016611433)

(-36384.31086362973, 0.141120011425108)

(25497.951853376628, 0.141120014912127)

(-32146.679100668807, 0.14112001237081)

(-23671.836715066005, 0.141120016010112)

(22955.608001774195, 0.141120016513956)

(40753.2487222549, 0.141120010742245)

(-24519.28334560084, 0.14112001547005)

(32277.898610354423, 0.14112001233583)

(-21976.980261313045, 0.141120017283636)

(-1.6298054233297241, 1)

(26345.42006180063, 0.141120014478376)

(38210.61133238353, 0.141120011111108)

(-41469.57437260855, 0.141120010650377)

(-40622.02414296146, 0.141120010759603)

(29735.370900425496, 0.141120013098326)

(-29604.15382475193, 0.141120013143089)

(-26214.20740473999, 0.141120014542792)

(37363.07085615639, 0.141120011251106)


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=−38926.9307606782x_{1} = -38926.9307606782
x2=−31299.1654607907x_{2} = -31299.1654607907
x3=19566.0078246694x_{3} = 19566.0078246694
x4=23803.0447297728x_{4} = 23803.0447297728
x5=−40622.0241429615x_{5} = -40622.0241429615
x6=−29604.1538247519x_{6} = -29604.1538247519
Puntos máximos de la función:
x6=21260.7750091757x_{6} = 21260.7750091757
x6=−32994.1974160571x_{6} = -32994.1974160571
x6=−22824.4018926693x_{6} = -22824.4018926693
x6=32277.8986103544x_{6} = 32277.8986103544
x6=−1.62980542332972x_{6} = -1.62980542332972
x6=38210.6113323835x_{6} = 38210.6113323835
Decrece en los intervalos
[23803.0447297728,∞)\left[23803.0447297728, \infty\right)
Crece en los intervalos
(−∞,−40622.0241429615]\left(-\infty, -40622.0241429615\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⁡(3x)−3cos⁡(3x)x)2sin⁡(xsin⁡(3x))+9sin⁡(3x)cos⁡(xsin⁡(3x))x3)=0- (\left(\sin{\left(\frac{3}{x} \right)} - \frac{3 \cos{\left(\frac{3}{x} \right)}}{x}\right)^{2} \sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)} + \frac{9 \sin{\left(\frac{3}{x} \right)} \cos{\left(x \sin{\left(\frac{3}{x} \right)} \right)}}{x^{3}}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=8958.73130398003x_{1} = 8958.73130398003
x2=9176.7784709951x_{2} = 9176.7784709951
x3=5252.22574677027x_{3} = 5252.22574677027
x4=3290.6024834855x_{4} = 3290.6024834855
x5=6560.3144793259x_{5} = 6560.3144793259
x6=8086.55514968898x_{6} = 8086.55514968898
x7=−8924.96568848099x_{7} = -8924.96568848099
x8=−5872.49070236622x_{8} = -5872.49070236622
x9=−7398.6753499054x_{9} = -7398.6753499054
x10=−10015.2108203287x_{10} = -10015.2108203287
x11=−8052.79037630445x_{11} = -8052.79037630445
x12=−2385.60811491745x_{12} = -2385.60811491745
x13=8740.68529399448x_{13} = 8740.68529399448
x14=−6090.50755768991x_{14} = -6090.50755768991
x15=2854.88236111151x_{15} = 2854.88236111151
x16=−3256.86154290812x_{16} = -3256.86154290812
x17=5688.23810301977x_{17} = 5688.23810301977
x18=5034.22802339705x_{18} = 5034.22802339705
x19=−3474.75867939155x_{19} = -3474.75867939155
x20=4162.31471795002x_{20} = 4162.31471795002
x21=3508.50316506502x_{21} = 3508.50316506502
x22=7650.47609192686x_{22} = 7650.47609192686
x23=2.15514695190883x_{23} = 2.15514695190883
x24=−9797.15989756068x_{24} = -9797.15989756068
x25=−10669.3683426194x_{25} = -10669.3683426194
x26=−5654.47801622572x_{26} = -5654.47801622572
x27=−8488.87530346657x_{27} = -8488.87530346657
x28=7214.40451531492x_{28} = 7214.40451531492
x29=−7180.64091174915x_{29} = -7180.64091174915
x30=−9579.10985783878x_{30} = -9579.10985783878
x31=9830.92613948657x_{31} = 9830.92613948657
x32=−8270.83210325814x_{32} = -8270.83210325814
x33=−4346.52542424054x_{33} = -4346.52542424054
x34=−2167.9336187525x_{34} = -2167.9336187525
x35=4816.23697990115x_{35} = 4816.23697990115
x36=2637.07724538964x_{36} = 2637.07724538964
x37=−9143.01268188024x_{37} = -9143.01268188024
x38=9612.87595913676x_{38} = 9612.87595913676
x39=3944.36242032907x_{39} = 3944.36242032907
x40=3072.72685601022x_{40} = 3072.72685601022
x41=2419.32299820287x_{41} = 2419.32299820287
x42=6342.29005382536x_{42} = 6342.29005382536
x43=5470.2293355619x_{43} = 5470.2293355619
x44=2201.63614346756x_{44} = 2201.63614346756
x45=6124.26892577724x_{45} = 6124.26892577724
x46=−10887.4222752219x_{46} = -10887.4222752219
x47=7432.43928535128x_{47} = 7432.43928535128
x48=−3038.99028921768x_{48} = -3038.99028921768
x49=10485.0817048096x_{49} = 10485.0817048096
x50=−4782.48060763002x_{50} = -4782.48060763002
x51=10703.1350631848x_{51} = 10703.1350631848
x52=−10233.2625693785x_{52} = -10233.2625693785
x53=7868.51476412975x_{53} = 7868.51476412975
x54=8522.64053051257x_{54} = 8522.64053051257
x55=9394.82671394125x_{55} = 9394.82671394125
x56=−3910.61259408264x_{56} = -3910.61259408264
x57=10921.1890977848x_{57} = 10921.1890977848
x58=−5000.47053475298x_{58} = -5000.47053475298
x59=−2603.35318026324x_{59} = -2603.35318026324
x60=−2.15514695190883x_{60} = -2.15514695190883
x61=4380.27902642189x_{61} = 4380.27902642189
x62=−6744.57904026756x_{62} = -6744.57904026756
x63=−7834.75024669839x_{63} = -7834.75024669839
x64=−3692.67691793631x_{64} = -3692.67691793631
x65=6778.34187999319x_{65} = 6778.34187999319
x66=10048.9771937615x_{66} = 10048.9771937615
x67=3726.42431816781x_{67} = 3726.42431816781
x68=8304.59711249611x_{68} = 8304.59711249611
x69=−5218.46728109055x_{69} = -5218.46728109055
x70=−10451.3150927146x_{70} = -10451.3150927146
x71=4598.25358974522x_{71} = 4598.25358974522
x72=6996.3719743208x_{72} = 6996.3719743208
x73=5906.25146593458x_{73} = 5906.25146593458
x74=−4564.49850115893x_{74} = -4564.49850115893
x75=−6526.55208078714x_{75} = -6526.55208078714
x76=−7616.7118529043x_{76} = -7616.7118529043
x77=−8706.91986538786x_{77} = -8706.91986538786
x78=−5436.47000984016x_{78} = -5436.47000984016
x79=−2821.15128028714x_{79} = -2821.15128028714
x80=10267.02906597x_{80} = 10267.02906597
x81=−6962.60873457935x_{81} = -6962.60873457935
x82=−9361.060763257x_{82} = -9361.060763257
x83=−6308.52814351366x_{83} = -6308.52814351366
x84=−4128.56285025285x_{84} = -4128.56285025285
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
[8958.73130398003,∞)\left[8958.73130398003, \infty\right)
Convexa en los intervalos
(−∞,−4782.48060763002]\left(-\infty, -4782.48060763002\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→−∞sin⁡(xsin⁡(3x))=sin⁡(3)\lim_{x \to -\infty} \sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)} = \sin{\left(3 \right)}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=sin⁡(3)y = \sin{\left(3 \right)}
lim⁡x→∞sin⁡(xsin⁡(3x))=sin⁡(3)\lim_{x \to \infty} \sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)} = \sin{\left(3 \right)}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=sin⁡(3)y = \sin{\left(3 \right)}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x*sin(3/x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(xsin⁡(3x))x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(xsin⁡(3x))x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)}}{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:
sin⁡(xsin⁡(3x))=sin⁡(xsin⁡(3x))\sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)} = \sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)}
- Sí
sin⁡(xsin⁡(3x))=−sin⁡(xsin⁡(3x))\sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)} = - \sin{\left(x \sin{\left(\frac{3}{x} \right)} \right)}
- No
es decir, función
es
par
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