Sr Examen
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- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x+5
-
lnx/x
-
e^5*x
-
5x-3
- Límite de la función:
-
cos(1/x)*sin(x)
-
Expresiones idénticas
- cos(uno /x)*sin(x)
- coseno de (1 dividir por x) multiplicar por seno de (x)
- coseno de (uno dividir por x) multiplicar por seno de (x)
- cos(1/x)sin(x)
- cos1/xsinx
- cos(1 dividir por x)*sin(x)
-
Expresiones semejantes
- cos(1/x)*sinx
-
Expresiones con funciones
- Coseno cos
- cos(pi/x)
- cos(2*x)+3
- cos(2x)
- cos(x)^(x^(-2))
- cos(x-pi/4)
- Seno sin
- sin(x)^2
- sin(9*x)
- sin(x)/x
- sin(x)+cos(x)
- sinx+x
Gráfico de la función y = cos(1/x)*sin(x)
Solución
Ha introducido
[src]
/1\
f(x) = cos|-|*sin(x)
\x/
$$f{\left(x \right)} = \sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}$$
f = sin(x)*cos(1/x)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \frac{2}{\pi}$$
$$x_{2} = \frac{2}{\pi}$$
Solución numérica
$$x_{1} = 12.5663706143592$$
$$x_{2} = 53.4070751110265$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 37.6991118430775$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 78.5398163397448$$
$$x_{7} = -59.6902604182061$$
$$x_{8} = -65.9734457253857$$
$$x_{9} = -31.4159265358979$$
$$x_{10} = -50.2654824574367$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 6.28318530717959$$
$$x_{13} = -34.5575191894877$$
$$x_{14} = 741.415866247191$$
$$x_{15} = -94.2477796076938$$
$$x_{16} = -69.1150383789755$$
$$x_{17} = -15.707963267949$$
$$x_{18} = 21.9911485751286$$
$$x_{19} = 62.8318530717959$$
$$x_{20} = 69.1150383789755$$
$$x_{21} = -119.380520836412$$
$$x_{22} = 50.2654824574367$$
$$x_{23} = 81.6814089933346$$
$$x_{24} = 100.530964914873$$
$$x_{25} = -40.8407044966673$$
$$x_{26} = 9.42477796076938$$
$$x_{27} = -87.9645943005142$$
$$x_{28} = 34.5575191894877$$
$$x_{29} = 65.9734457253857$$
$$x_{30} = -62.8318530717959$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = -28.2743338823081$$
$$x_{33} = -56.5486677646163$$
$$x_{34} = 549.778714378214$$
$$x_{35} = -53.4070751110265$$
$$x_{36} = -37.6991118430775$$
$$x_{37} = -25.1327412287183$$
$$x_{38} = -100.530964914873$$
$$x_{39} = -9.42477796076938$$
$$x_{40} = 40.8407044966673$$
$$x_{41} = -91.106186954104$$
$$x_{42} = -75.398223686155$$
$$x_{43} = 18.8495559215388$$
$$x_{44} = 87.9645943005142$$
$$x_{45} = 59.6902604182061$$
$$x_{46} = -6.28318530717959$$
$$x_{47} = 25.1327412287183$$
$$x_{48} = 47.1238898038469$$
$$x_{49} = 91.106186954104$$
$$x_{50} = 28.2743338823081$$
$$x_{51} = -238.761041672824$$
$$x_{52} = -464.955712731289$$
$$x_{53} = -43.9822971502571$$
$$x_{54} = 56.5486677646163$$
$$x_{55} = -47.1238898038469$$
$$x_{56} = -141.371669411541$$
$$x_{57} = -3.14159265358979$$
$$x_{58} = 31.4159265358979$$
$$x_{59} = 94.2477796076938$$
$$x_{60} = -12.5663706143592$$
$$x_{61} = 75.398223686155$$
$$x_{62} = -72.2566310325652$$
$$x_{63} = -84.8230016469244$$
$$x_{64} = 84.8230016469244$$
$$x_{65} = 72.2566310325652$$
$$x_{66} = -81.6814089933346$$
$$x_{67} = 43.9822971502571$$
$$x_{68} = -78.5398163397448$$
$$x_{69} = 15.707963267949$$
$$x_{70} = 3.14159265358979$$
o sea hay que resolver la ecuación:
$$\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \frac{2}{\pi}$$
$$x_{2} = \frac{2}{\pi}$$
Solución numérica
$$x_{1} = 12.5663706143592$$
$$x_{2} = 53.4070751110265$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 37.6991118430775$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 78.5398163397448$$
$$x_{7} = -59.6902604182061$$
$$x_{8} = -65.9734457253857$$
$$x_{9} = -31.4159265358979$$
$$x_{10} = -50.2654824574367$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 6.28318530717959$$
$$x_{13} = -34.5575191894877$$
$$x_{14} = 741.415866247191$$
$$x_{15} = -94.2477796076938$$
$$x_{16} = -69.1150383789755$$
$$x_{17} = -15.707963267949$$
$$x_{18} = 21.9911485751286$$
$$x_{19} = 62.8318530717959$$
$$x_{20} = 69.1150383789755$$
$$x_{21} = -119.380520836412$$
$$x_{22} = 50.2654824574367$$
$$x_{23} = 81.6814089933346$$
$$x_{24} = 100.530964914873$$
$$x_{25} = -40.8407044966673$$
$$x_{26} = 9.42477796076938$$
$$x_{27} = -87.9645943005142$$
$$x_{28} = 34.5575191894877$$
$$x_{29} = 65.9734457253857$$
$$x_{30} = -62.8318530717959$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = -28.2743338823081$$
$$x_{33} = -56.5486677646163$$
$$x_{34} = 549.778714378214$$
$$x_{35} = -53.4070751110265$$
$$x_{36} = -37.6991118430775$$
$$x_{37} = -25.1327412287183$$
$$x_{38} = -100.530964914873$$
$$x_{39} = -9.42477796076938$$
$$x_{40} = 40.8407044966673$$
$$x_{41} = -91.106186954104$$
$$x_{42} = -75.398223686155$$
$$x_{43} = 18.8495559215388$$
$$x_{44} = 87.9645943005142$$
$$x_{45} = 59.6902604182061$$
$$x_{46} = -6.28318530717959$$
$$x_{47} = 25.1327412287183$$
$$x_{48} = 47.1238898038469$$
$$x_{49} = 91.106186954104$$
$$x_{50} = 28.2743338823081$$
$$x_{51} = -238.761041672824$$
$$x_{52} = -464.955712731289$$
$$x_{53} = -43.9822971502571$$
$$x_{54} = 56.5486677646163$$
$$x_{55} = -47.1238898038469$$
$$x_{56} = -141.371669411541$$
$$x_{57} = -3.14159265358979$$
$$x_{58} = 31.4159265358979$$
$$x_{59} = 94.2477796076938$$
$$x_{60} = -12.5663706143592$$
$$x_{61} = 75.398223686155$$
$$x_{62} = -72.2566310325652$$
$$x_{63} = -84.8230016469244$$
$$x_{64} = 84.8230016469244$$
$$x_{65} = 72.2566310325652$$
$$x_{66} = -81.6814089933346$$
$$x_{67} = 43.9822971502571$$
$$x_{68} = -78.5398163397448$$
$$x_{69} = 15.707963267949$$
$$x_{70} = 3.14159265358979$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\cos{\left(\frac{1}{x} \right)} \cos{\left(x \right)} + \frac{\sin{\left(\frac{1}{x} \right)} \sin{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 73.8274298446286$$
$$x_{2} = -1.77032184622602$$
$$x_{3} = -45.5531040577872$$
$$x_{4} = -48.6946947926006$$
$$x_{5} = -89.5353920205745$$
$$x_{6} = -64.4026531424855$$
$$x_{7} = -7.85605530705307$$
$$x_{8} = -73.8274298446286$$
$$x_{9} = -32.986750731248$$
$$x_{10} = -51.8362859646913$$
$$x_{11} = -61.2610610949586$$
$$x_{12} = 76.9690222061413$$
$$x_{13} = -67.5442452975689$$
$$x_{14} = -26.7035900960034$$
$$x_{15} = 1.77032184622602$$
$$x_{16} = 89.5353920205745$$
$$x_{17} = 58.1194691856341$$
$$x_{18} = -54.9778774562622$$
$$x_{19} = 10.9963284351197$$
$$x_{20} = -95.818577071243$$
$$x_{21} = 39.2699246862074$$
$$x_{22} = -76.9690222061413$$
$$x_{23} = 95.818577071243$$
$$x_{24} = -98.9601696199687$$
$$x_{25} = 70.6858375373694$$
$$x_{26} = 83.2522070532694$$
$$x_{27} = -39.2699246862074$$
$$x_{28} = 23.5620213951938$$
$$x_{29} = 202.632726276733$$
$$x_{30} = -36.1283367275698$$
$$x_{31} = 64.4026531424855$$
$$x_{32} = 1983.91576074208$$
$$x_{33} = 92.6769845372215$$
$$x_{34} = 51.8362859646913$$
$$x_{35} = 61.2610610949586$$
$$x_{36} = -70.6858375373694$$
$$x_{37} = -92.6769845372215$$
$$x_{38} = -86.393799524576$$
$$x_{39} = 48.6946947926006$$
$$x_{40} = -42.4115139342614$$
$$x_{41} = -58.1194691856341$$
$$x_{42} = 45.5531040577872$$
$$x_{43} = 67.5442452975689$$
$$x_{44} = -10.9963284351197$$
$$x_{45} = 17.278953653359$$
$$x_{46} = 4.72203085083256$$
$$x_{47} = 80.1106146116865$$
$$x_{48} = -20.4204697787209$$
$$x_{49} = -83.2522070532694$$
$$x_{50} = -80.1106146116865$$
$$x_{51} = -17.278953653359$$
$$x_{52} = 20.4204697787209$$
$$x_{53} = 54.9778774562622$$
$$x_{54} = 32.986750731248$$
$$x_{55} = 7.85605530705307$$
$$x_{56} = 29.8451678396463$$
$$x_{57} = -4.72203085083256$$
$$x_{58} = 86.393799524576$$
$$x_{59} = 36.1283367275698$$
$$x_{60} = 98.9601696199687$$
$$x_{61} = -23.5620213951938$$
$$x_{62} = 26.7035900960034$$
$$x_{63} = 14.1375214322216$$
$$x_{64} = -29.8451678396463$$
$$x_{65} = 42.4115139342614$$
$$x_{66} = -14.1375214322216$$
Signos de extremos en los puntos:
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:
$$x_{1} = 73.8274298446286$$
$$x_{2} = -1.77032184622602$$
$$x_{3} = -45.5531040577872$$
$$x_{4} = -89.5353920205745$$
$$x_{5} = -64.4026531424855$$
$$x_{6} = -7.85605530705307$$
$$x_{7} = -32.986750731248$$
$$x_{8} = -51.8362859646913$$
$$x_{9} = -26.7035900960034$$
$$x_{10} = 10.9963284351197$$
$$x_{11} = -95.818577071243$$
$$x_{12} = -76.9690222061413$$
$$x_{13} = -39.2699246862074$$
$$x_{14} = 23.5620213951938$$
$$x_{15} = 1983.91576074208$$
$$x_{16} = 92.6769845372215$$
$$x_{17} = 61.2610610949586$$
$$x_{18} = -70.6858375373694$$
$$x_{19} = 48.6946947926006$$
$$x_{20} = -58.1194691856341$$
$$x_{21} = 67.5442452975689$$
$$x_{22} = 17.278953653359$$
$$x_{23} = 4.72203085083256$$
$$x_{24} = 80.1106146116865$$
$$x_{25} = -20.4204697787209$$
$$x_{26} = -83.2522070532694$$
$$x_{27} = 54.9778774562622$$
$$x_{28} = 29.8451678396463$$
$$x_{29} = 86.393799524576$$
$$x_{30} = 36.1283367275698$$
$$x_{31} = 98.9601696199687$$
$$x_{32} = 42.4115139342614$$
$$x_{33} = -14.1375214322216$$
Puntos máximos de la función:
$$x_{33} = -48.6946947926006$$
$$x_{33} = -73.8274298446286$$
$$x_{33} = -61.2610610949586$$
$$x_{33} = 76.9690222061413$$
$$x_{33} = -67.5442452975689$$
$$x_{33} = 1.77032184622602$$
$$x_{33} = 89.5353920205745$$
$$x_{33} = 58.1194691856341$$
$$x_{33} = -54.9778774562622$$
$$x_{33} = 39.2699246862074$$
$$x_{33} = 95.818577071243$$
$$x_{33} = -98.9601696199687$$
$$x_{33} = 70.6858375373694$$
$$x_{33} = 83.2522070532694$$
$$x_{33} = 202.632726276733$$
$$x_{33} = -36.1283367275698$$
$$x_{33} = 64.4026531424855$$
$$x_{33} = 51.8362859646913$$
$$x_{33} = -92.6769845372215$$
$$x_{33} = -86.393799524576$$
$$x_{33} = -42.4115139342614$$
$$x_{33} = 45.5531040577872$$
$$x_{33} = -10.9963284351197$$
$$x_{33} = -80.1106146116865$$
$$x_{33} = -17.278953653359$$
$$x_{33} = 20.4204697787209$$
$$x_{33} = 32.986750731248$$
$$x_{33} = 7.85605530705307$$
$$x_{33} = -4.72203085083256$$
$$x_{33} = -23.5620213951938$$
$$x_{33} = 26.7035900960034$$
$$x_{33} = 14.1375214322216$$
$$x_{33} = -29.8451678396463$$
Decrece en los intervalos
$$\left[1983.91576074208, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.818577071243\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\cos{\left(\frac{1}{x} \right)} \cos{\left(x \right)} + \frac{\sin{\left(\frac{1}{x} \right)} \sin{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 73.8274298446286$$
$$x_{2} = -1.77032184622602$$
$$x_{3} = -45.5531040577872$$
$$x_{4} = -48.6946947926006$$
$$x_{5} = -89.5353920205745$$
$$x_{6} = -64.4026531424855$$
$$x_{7} = -7.85605530705307$$
$$x_{8} = -73.8274298446286$$
$$x_{9} = -32.986750731248$$
$$x_{10} = -51.8362859646913$$
$$x_{11} = -61.2610610949586$$
$$x_{12} = 76.9690222061413$$
$$x_{13} = -67.5442452975689$$
$$x_{14} = -26.7035900960034$$
$$x_{15} = 1.77032184622602$$
$$x_{16} = 89.5353920205745$$
$$x_{17} = 58.1194691856341$$
$$x_{18} = -54.9778774562622$$
$$x_{19} = 10.9963284351197$$
$$x_{20} = -95.818577071243$$
$$x_{21} = 39.2699246862074$$
$$x_{22} = -76.9690222061413$$
$$x_{23} = 95.818577071243$$
$$x_{24} = -98.9601696199687$$
$$x_{25} = 70.6858375373694$$
$$x_{26} = 83.2522070532694$$
$$x_{27} = -39.2699246862074$$
$$x_{28} = 23.5620213951938$$
$$x_{29} = 202.632726276733$$
$$x_{30} = -36.1283367275698$$
$$x_{31} = 64.4026531424855$$
$$x_{32} = 1983.91576074208$$
$$x_{33} = 92.6769845372215$$
$$x_{34} = 51.8362859646913$$
$$x_{35} = 61.2610610949586$$
$$x_{36} = -70.6858375373694$$
$$x_{37} = -92.6769845372215$$
$$x_{38} = -86.393799524576$$
$$x_{39} = 48.6946947926006$$
$$x_{40} = -42.4115139342614$$
$$x_{41} = -58.1194691856341$$
$$x_{42} = 45.5531040577872$$
$$x_{43} = 67.5442452975689$$
$$x_{44} = -10.9963284351197$$
$$x_{45} = 17.278953653359$$
$$x_{46} = 4.72203085083256$$
$$x_{47} = 80.1106146116865$$
$$x_{48} = -20.4204697787209$$
$$x_{49} = -83.2522070532694$$
$$x_{50} = -80.1106146116865$$
$$x_{51} = -17.278953653359$$
$$x_{52} = 20.4204697787209$$
$$x_{53} = 54.9778774562622$$
$$x_{54} = 32.986750731248$$
$$x_{55} = 7.85605530705307$$
$$x_{56} = 29.8451678396463$$
$$x_{57} = -4.72203085083256$$
$$x_{58} = 86.393799524576$$
$$x_{59} = 36.1283367275698$$
$$x_{60} = 98.9601696199687$$
$$x_{61} = -23.5620213951938$$
$$x_{62} = 26.7035900960034$$
$$x_{63} = 14.1375214322216$$
$$x_{64} = -29.8451678396463$$
$$x_{65} = 42.4115139342614$$
$$x_{66} = -14.1375214322216$$
Signos de extremos en los puntos:
(73.82742984462863, -0.999908266517767)
(-1.7703218462260202, -0.827901301432779)
(-45.55310405778716, -0.999759055668925)
(-48.69469479260059, 0.99978914130114)
(-89.53539202057446, -0.999937629960711)
(-64.40265314248548, -0.999879453727449)
(-7.856055307053065, -0.991907384132263)
(-73.82742984462863, 0.999908266517767)
(-32.98675073124796, -0.999540529076331)
(-51.83628596469129, -0.999813924651387)
(-61.26106109495865, 0.99986677327201)
(76.9690222061413, 0.999915602036213)
(-67.5442452975689, 0.999890406351476)
(-26.703590096003428, -0.999298898656899)
(1.7703218462260202, 0.827901301432779)
(89.53539202057446, 0.999937629960711)
(58.11946918563406, 0.999851981482606)
(-54.97787745626216, 0.999834582238175)
(10.996328435119693, -0.995867574424807)
(-95.81857707124303, -0.999945541379844)
(39.269924686207396, 0.999675789868926)
(-76.9690222061413, -0.999915602036213)
(95.81857707124303, 0.999945541379844)
(-98.9601696199687, 0.999948944157013)
(70.68583753736935, 0.999899931368176)
(83.25220705326942, 0.999927860474679)
(-39.269924686207396, -0.999675789868926)
(23.56202139519381, -0.99909950536129)
(202.63272627673345, 0.99998782272965)
(-36.128336727569845, 0.999616957824081)
(64.40265314248548, 0.999879453727449)
(1983.9157607420825, -0.999999872964957)
(92.6769845372215, -0.999941786728398)
(51.83628596469129, 0.999813924651387)
(61.26106109495865, -0.99986677327201)
(-70.68583753736935, -0.999899931368176)
(-92.6769845372215, 0.999941786728398)
(-86.39379952457602, 0.999933011536799)
(48.69469479260059, -0.99978914130114)
(-42.41151393426139, 0.999722039894878)
(-58.11946918563406, -0.999851981482606)
(45.55310405778716, 0.999759055668925)
(67.5442452975689, -0.999890406351476)
(-10.996328435119693, 0.995867574424807)
(17.278953653359007, -0.998325755891758)
(4.722030850832559, -0.977614273674147)
(80.11061461168646, -0.999922091606656)
(-20.42046977872091, -0.998801179244391)
(-83.25220705326942, -0.999927860474679)
(-80.11061461168646, 0.999922091606656)
(-17.278953653359007, 0.998325755891758)
(20.42046977872091, 0.998801179244391)
(54.97787745626216, -0.999834582238175)
(32.98675073124796, 0.999540529076331)
(7.856055307053065, 0.991907384132263)
(29.845167839646347, -0.999438717024592)
(-4.722030850832559, 0.977614273674147)
(86.39379952457602, -0.999933011536799)
(36.128336727569845, -0.999616957824081)
(98.9601696199687, -0.999948944157013)
(-23.56202139519381, 0.99909950536129)
(26.703590096003428, 0.999298898656899)
(14.13752143222161, 0.997499348016665)
(-29.845167839646347, 0.999438717024592)
(42.41151393426139, -0.999722039894878)
(-14.13752143222161, -0.997499348016665)
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:
$$x_{1} = 73.8274298446286$$
$$x_{2} = -1.77032184622602$$
$$x_{3} = -45.5531040577872$$
$$x_{4} = -89.5353920205745$$
$$x_{5} = -64.4026531424855$$
$$x_{6} = -7.85605530705307$$
$$x_{7} = -32.986750731248$$
$$x_{8} = -51.8362859646913$$
$$x_{9} = -26.7035900960034$$
$$x_{10} = 10.9963284351197$$
$$x_{11} = -95.818577071243$$
$$x_{12} = -76.9690222061413$$
$$x_{13} = -39.2699246862074$$
$$x_{14} = 23.5620213951938$$
$$x_{15} = 1983.91576074208$$
$$x_{16} = 92.6769845372215$$
$$x_{17} = 61.2610610949586$$
$$x_{18} = -70.6858375373694$$
$$x_{19} = 48.6946947926006$$
$$x_{20} = -58.1194691856341$$
$$x_{21} = 67.5442452975689$$
$$x_{22} = 17.278953653359$$
$$x_{23} = 4.72203085083256$$
$$x_{24} = 80.1106146116865$$
$$x_{25} = -20.4204697787209$$
$$x_{26} = -83.2522070532694$$
$$x_{27} = 54.9778774562622$$
$$x_{28} = 29.8451678396463$$
$$x_{29} = 86.393799524576$$
$$x_{30} = 36.1283367275698$$
$$x_{31} = 98.9601696199687$$
$$x_{32} = 42.4115139342614$$
$$x_{33} = -14.1375214322216$$
Puntos máximos de la función:
$$x_{33} = -48.6946947926006$$
$$x_{33} = -73.8274298446286$$
$$x_{33} = -61.2610610949586$$
$$x_{33} = 76.9690222061413$$
$$x_{33} = -67.5442452975689$$
$$x_{33} = 1.77032184622602$$
$$x_{33} = 89.5353920205745$$
$$x_{33} = 58.1194691856341$$
$$x_{33} = -54.9778774562622$$
$$x_{33} = 39.2699246862074$$
$$x_{33} = 95.818577071243$$
$$x_{33} = -98.9601696199687$$
$$x_{33} = 70.6858375373694$$
$$x_{33} = 83.2522070532694$$
$$x_{33} = 202.632726276733$$
$$x_{33} = -36.1283367275698$$
$$x_{33} = 64.4026531424855$$
$$x_{33} = 51.8362859646913$$
$$x_{33} = -92.6769845372215$$
$$x_{33} = -86.393799524576$$
$$x_{33} = -42.4115139342614$$
$$x_{33} = 45.5531040577872$$
$$x_{33} = -10.9963284351197$$
$$x_{33} = -80.1106146116865$$
$$x_{33} = -17.278953653359$$
$$x_{33} = 20.4204697787209$$
$$x_{33} = 32.986750731248$$
$$x_{33} = 7.85605530705307$$
$$x_{33} = -4.72203085083256$$
$$x_{33} = -23.5620213951938$$
$$x_{33} = 26.7035900960034$$
$$x_{33} = 14.1375214322216$$
$$x_{33} = -29.8451678396463$$
Decrece en los intervalos
$$\left[1983.91576074208, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.818577071243\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}\right) = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}\right) = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}\right) = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}\right) = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 1\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(1/x)*sin(x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}}{x}\right) = 0$$
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:
$$\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)} = - \sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}$$
- No
$$\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)} = \sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}$$
- Sí
es decir, función
es
impar
Pues, comprobamos:
$$\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)} = - \sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}$$
- No
$$\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)} = \sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}$$
- Sí
es decir, función
es
impar