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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /1\       
f(x) = cos|-|*sin(x)
          \x/
f(x)=sin⁡(x)cos⁡(1x)f{\left(x \right)} = \sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)} 
f = sin(x)*cos(1/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⁡(x)cos⁡(1x)=0\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−2πx_{1} = - \frac{2}{\pi}
x2=2πx_{2} = \frac{2}{\pi}
Solución numérica
x1=12.5663706143592x_{1} = 12.5663706143592
x2=53.4070751110265x_{2} = 53.4070751110265
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=37.6991118430775x_{4} = 37.6991118430775
x5=97.3893722612836x_{5} = 97.3893722612836
x6=78.5398163397448x_{6} = 78.5398163397448
x7=−59.6902604182061x_{7} = -59.6902604182061
x8=−65.9734457253857x_{8} = -65.9734457253857
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=−50.2654824574367x_{10} = -50.2654824574367
x11=−21.9911485751286x_{11} = -21.9911485751286
x12=6.28318530717959x_{12} = 6.28318530717959
x13=−34.5575191894877x_{13} = -34.5575191894877
x14=741.415866247191x_{14} = 741.415866247191
x15=−94.2477796076938x_{15} = -94.2477796076938
x16=−69.1150383789755x_{16} = -69.1150383789755
x17=−15.707963267949x_{17} = -15.707963267949
x18=21.9911485751286x_{18} = 21.9911485751286
x19=62.8318530717959x_{19} = 62.8318530717959
x20=69.1150383789755x_{20} = 69.1150383789755
x21=−119.380520836412x_{21} = -119.380520836412
x22=50.2654824574367x_{22} = 50.2654824574367
x23=81.6814089933346x_{23} = 81.6814089933346
x24=100.530964914873x_{24} = 100.530964914873
x25=−40.8407044966673x_{25} = -40.8407044966673
x26=9.42477796076938x_{26} = 9.42477796076938
x27=−87.9645943005142x_{27} = -87.9645943005142
x28=34.5575191894877x_{28} = 34.5575191894877
x29=65.9734457253857x_{29} = 65.9734457253857
x30=−62.8318530717959x_{30} = -62.8318530717959
x31=−18.8495559215388x_{31} = -18.8495559215388
x32=−28.2743338823081x_{32} = -28.2743338823081
x33=−56.5486677646163x_{33} = -56.5486677646163
x34=549.778714378214x_{34} = 549.778714378214
x35=−53.4070751110265x_{35} = -53.4070751110265
x36=−37.6991118430775x_{36} = -37.6991118430775
x37=−25.1327412287183x_{37} = -25.1327412287183
x38=−100.530964914873x_{38} = -100.530964914873
x39=−9.42477796076938x_{39} = -9.42477796076938
x40=40.8407044966673x_{40} = 40.8407044966673
x41=−91.106186954104x_{41} = -91.106186954104
x42=−75.398223686155x_{42} = -75.398223686155
x43=18.8495559215388x_{43} = 18.8495559215388
x44=87.9645943005142x_{44} = 87.9645943005142
x45=59.6902604182061x_{45} = 59.6902604182061
x46=−6.28318530717959x_{46} = -6.28318530717959
x47=25.1327412287183x_{47} = 25.1327412287183
x48=47.1238898038469x_{48} = 47.1238898038469
x49=91.106186954104x_{49} = 91.106186954104
x50=28.2743338823081x_{50} = 28.2743338823081
x51=−238.761041672824x_{51} = -238.761041672824
x52=−464.955712731289x_{52} = -464.955712731289
x53=−43.9822971502571x_{53} = -43.9822971502571
x54=56.5486677646163x_{54} = 56.5486677646163
x55=−47.1238898038469x_{55} = -47.1238898038469
x56=−141.371669411541x_{56} = -141.371669411541
x57=−3.14159265358979x_{57} = -3.14159265358979
x58=31.4159265358979x_{58} = 31.4159265358979
x59=94.2477796076938x_{59} = 94.2477796076938
x60=−12.5663706143592x_{60} = -12.5663706143592
x61=75.398223686155x_{61} = 75.398223686155
x62=−72.2566310325652x_{62} = -72.2566310325652
x63=−84.8230016469244x_{63} = -84.8230016469244
x64=84.8230016469244x_{64} = 84.8230016469244
x65=72.2566310325652x_{65} = 72.2566310325652
x66=−81.6814089933346x_{66} = -81.6814089933346
x67=43.9822971502571x_{67} = 43.9822971502571
x68=−78.5398163397448x_{68} = -78.5398163397448
x69=15.707963267949x_{69} = 15.707963267949
x70=3.14159265358979x_{70} = 3.14159265358979
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 cos(1/x)*sin(x).
sin⁡(0)cos⁡(10)\sin{\left(0 \right)} \cos{\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
cos⁡(1x)cos⁡(x)+sin⁡(1x)sin⁡(x)x2=0\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
x1=73.8274298446286x_{1} = 73.8274298446286
x2=−1.77032184622602x_{2} = -1.77032184622602
x3=−45.5531040577872x_{3} = -45.5531040577872
x4=−48.6946947926006x_{4} = -48.6946947926006
x5=−89.5353920205745x_{5} = -89.5353920205745
x6=−64.4026531424855x_{6} = -64.4026531424855
x7=−7.85605530705307x_{7} = -7.85605530705307
x8=−73.8274298446286x_{8} = -73.8274298446286
x9=−32.986750731248x_{9} = -32.986750731248
x10=−51.8362859646913x_{10} = -51.8362859646913
x11=−61.2610610949586x_{11} = -61.2610610949586
x12=76.9690222061413x_{12} = 76.9690222061413
x13=−67.5442452975689x_{13} = -67.5442452975689
x14=−26.7035900960034x_{14} = -26.7035900960034
x15=1.77032184622602x_{15} = 1.77032184622602
x16=89.5353920205745x_{16} = 89.5353920205745
x17=58.1194691856341x_{17} = 58.1194691856341
x18=−54.9778774562622x_{18} = -54.9778774562622
x19=10.9963284351197x_{19} = 10.9963284351197
x20=−95.818577071243x_{20} = -95.818577071243
x21=39.2699246862074x_{21} = 39.2699246862074
x22=−76.9690222061413x_{22} = -76.9690222061413
x23=95.818577071243x_{23} = 95.818577071243
x24=−98.9601696199687x_{24} = -98.9601696199687
x25=70.6858375373694x_{25} = 70.6858375373694
x26=83.2522070532694x_{26} = 83.2522070532694
x27=−39.2699246862074x_{27} = -39.2699246862074
x28=23.5620213951938x_{28} = 23.5620213951938
x29=202.632726276733x_{29} = 202.632726276733
x30=−36.1283367275698x_{30} = -36.1283367275698
x31=64.4026531424855x_{31} = 64.4026531424855
x32=1983.91576074208x_{32} = 1983.91576074208
x33=92.6769845372215x_{33} = 92.6769845372215
x34=51.8362859646913x_{34} = 51.8362859646913
x35=61.2610610949586x_{35} = 61.2610610949586
x36=−70.6858375373694x_{36} = -70.6858375373694
x37=−92.6769845372215x_{37} = -92.6769845372215
x38=−86.393799524576x_{38} = -86.393799524576
x39=48.6946947926006x_{39} = 48.6946947926006
x40=−42.4115139342614x_{40} = -42.4115139342614
x41=−58.1194691856341x_{41} = -58.1194691856341
x42=45.5531040577872x_{42} = 45.5531040577872
x43=67.5442452975689x_{43} = 67.5442452975689
x44=−10.9963284351197x_{44} = -10.9963284351197
x45=17.278953653359x_{45} = 17.278953653359
x46=4.72203085083256x_{46} = 4.72203085083256
x47=80.1106146116865x_{47} = 80.1106146116865
x48=−20.4204697787209x_{48} = -20.4204697787209
x49=−83.2522070532694x_{49} = -83.2522070532694
x50=−80.1106146116865x_{50} = -80.1106146116865
x51=−17.278953653359x_{51} = -17.278953653359
x52=20.4204697787209x_{52} = 20.4204697787209
x53=54.9778774562622x_{53} = 54.9778774562622
x54=32.986750731248x_{54} = 32.986750731248
x55=7.85605530705307x_{55} = 7.85605530705307
x56=29.8451678396463x_{56} = 29.8451678396463
x57=−4.72203085083256x_{57} = -4.72203085083256
x58=86.393799524576x_{58} = 86.393799524576
x59=36.1283367275698x_{59} = 36.1283367275698
x60=98.9601696199687x_{60} = 98.9601696199687
x61=−23.5620213951938x_{61} = -23.5620213951938
x62=26.7035900960034x_{62} = 26.7035900960034
x63=14.1375214322216x_{63} = 14.1375214322216
x64=−29.8451678396463x_{64} = -29.8451678396463
x65=42.4115139342614x_{65} = 42.4115139342614
x66=−14.1375214322216x_{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:
x1=73.8274298446286x_{1} = 73.8274298446286
x2=−1.77032184622602x_{2} = -1.77032184622602
x3=−45.5531040577872x_{3} = -45.5531040577872
x4=−89.5353920205745x_{4} = -89.5353920205745
x5=−64.4026531424855x_{5} = -64.4026531424855
x6=−7.85605530705307x_{6} = -7.85605530705307
x7=−32.986750731248x_{7} = -32.986750731248
x8=−51.8362859646913x_{8} = -51.8362859646913
x9=−26.7035900960034x_{9} = -26.7035900960034
x10=10.9963284351197x_{10} = 10.9963284351197
x11=−95.818577071243x_{11} = -95.818577071243
x12=−76.9690222061413x_{12} = -76.9690222061413
x13=−39.2699246862074x_{13} = -39.2699246862074
x14=23.5620213951938x_{14} = 23.5620213951938
x15=1983.91576074208x_{15} = 1983.91576074208
x16=92.6769845372215x_{16} = 92.6769845372215
x17=61.2610610949586x_{17} = 61.2610610949586
x18=−70.6858375373694x_{18} = -70.6858375373694
x19=48.6946947926006x_{19} = 48.6946947926006
x20=−58.1194691856341x_{20} = -58.1194691856341
x21=67.5442452975689x_{21} = 67.5442452975689
x22=17.278953653359x_{22} = 17.278953653359
x23=4.72203085083256x_{23} = 4.72203085083256
x24=80.1106146116865x_{24} = 80.1106146116865
x25=−20.4204697787209x_{25} = -20.4204697787209
x26=−83.2522070532694x_{26} = -83.2522070532694
x27=54.9778774562622x_{27} = 54.9778774562622
x28=29.8451678396463x_{28} = 29.8451678396463
x29=86.393799524576x_{29} = 86.393799524576
x30=36.1283367275698x_{30} = 36.1283367275698
x31=98.9601696199687x_{31} = 98.9601696199687
x32=42.4115139342614x_{32} = 42.4115139342614
x33=−14.1375214322216x_{33} = -14.1375214322216
Puntos máximos de la función:
x33=−48.6946947926006x_{33} = -48.6946947926006
x33=−73.8274298446286x_{33} = -73.8274298446286
x33=−61.2610610949586x_{33} = -61.2610610949586
x33=76.9690222061413x_{33} = 76.9690222061413
x33=−67.5442452975689x_{33} = -67.5442452975689
x33=1.77032184622602x_{33} = 1.77032184622602
x33=89.5353920205745x_{33} = 89.5353920205745
x33=58.1194691856341x_{33} = 58.1194691856341
x33=−54.9778774562622x_{33} = -54.9778774562622
x33=39.2699246862074x_{33} = 39.2699246862074
x33=95.818577071243x_{33} = 95.818577071243
x33=−98.9601696199687x_{33} = -98.9601696199687
x33=70.6858375373694x_{33} = 70.6858375373694
x33=83.2522070532694x_{33} = 83.2522070532694
x33=202.632726276733x_{33} = 202.632726276733
x33=−36.1283367275698x_{33} = -36.1283367275698
x33=64.4026531424855x_{33} = 64.4026531424855
x33=51.8362859646913x_{33} = 51.8362859646913
x33=−92.6769845372215x_{33} = -92.6769845372215
x33=−86.393799524576x_{33} = -86.393799524576
x33=−42.4115139342614x_{33} = -42.4115139342614
x33=45.5531040577872x_{33} = 45.5531040577872
x33=−10.9963284351197x_{33} = -10.9963284351197
x33=−80.1106146116865x_{33} = -80.1106146116865
x33=−17.278953653359x_{33} = -17.278953653359
x33=20.4204697787209x_{33} = 20.4204697787209
x33=32.986750731248x_{33} = 32.986750731248
x33=7.85605530705307x_{33} = 7.85605530705307
x33=−4.72203085083256x_{33} = -4.72203085083256
x33=−23.5620213951938x_{33} = -23.5620213951938
x33=26.7035900960034x_{33} = 26.7035900960034
x33=14.1375214322216x_{33} = 14.1375214322216
x33=−29.8451678396463x_{33} = -29.8451678396463
Decrece en los intervalos
[1983.91576074208,∞)\left[1983.91576074208, \infty\right)
Crece en los intervalos
(−∞,−95.818577071243]\left(-\infty, -95.818577071243\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⁡(x)cos⁡(1x))=⟨−1,1⟩\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=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞(sin⁡(x)cos⁡(1x))=⟨−1,1⟩\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=⟨−1,1⟩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→−∞(sin⁡(x)cos⁡(1x)x)=0\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→∞(sin⁡(x)cos⁡(1x)x)=0\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⁡(x)cos⁡(1x)=−sin⁡(x)cos⁡(1x)\sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)} = - \sin{\left(x \right)} \cos{\left(\frac{1}{x} \right)}
- No
sin⁡(x)cos⁡(1x)=sin⁡(x)cos⁡(1x)\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
    © Sr Examen – Калькулятор