Sr Examen

Otras calculadoras

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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         1   cos(x)         
       - - - ------ + sin(x)
         x     x            
f(x) = ---------------------
                 x
f(x)=(−cos⁡(x)x−1x)+sin⁡(x)xf{\left(x \right)} = \frac{\left(- \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}\right) + \sin{\left(x \right)}}{x} 
f = (-cos(x)/x - 1/x + sin(x))/x
Gráfico de la función
02468-8-6-4-2-1010-10001000
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:
(−cos⁡(x)x−1x)+sin⁡(x)x=0\frac{\left(- \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}\right) + \sin{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−97.3893722612836x_{1} = -97.3893722612836
x2=−72.2566310325652x_{2} = -72.2566310325652
x3=−59.6902604182061x_{3} = -59.6902604182061
x4=−75.4247386315323x_{4} = -75.4247386315323
x5=−78.5398163397448x_{5} = -78.5398163397448
x6=97.3893722612836x_{6} = 97.3893722612836
x7=−823.099705078617x_{7} = -823.099705078617
x8=9.42477796076938x_{8} = 9.42477796076938
x9=−56.5840097601383x_{9} = -56.5840097601383
x10=12.7232407841313x_{10} = 12.7232407841313
x11=84.8230016469244x_{11} = 84.8230016469244
x12=−100.550854691761x_{12} = -100.550854691761
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=−18.9549714108416x_{14} = -18.9549714108416
x15=3.14159265358979x_{15} = 3.14159265358979
x16=−25.2120268885508x_{16} = -25.2120268885508
x17=−301.599526030476x_{17} = -301.599526030476
x18=31.4794387120097x_{18} = 31.4794387120097
x19=44.0277152732459x_{19} = 44.0277152732459
x20=28.2743338823081x_{20} = 28.2743338823081
x21=37.7520766759717x_{21} = 37.7520766759717
x22=−28.2743338823081x_{22} = -28.2743338823081
x23=56.5840097601383x_{23} = 56.5840097601383
x24=−40.8407044966673x_{24} = -40.8407044966673
x25=−65.9734457253857x_{25} = -65.9734457253857
x26=6.58462004256417x_{26} = 6.58462004256417
x27=241.902634326414x_{27} = 241.902634326414
x28=−91.106186954104x_{28} = -91.106186954104
x29=−12.7232407841313x_{29} = -12.7232407841313
x30=−50.3052345158712x_{30} = -50.3052345158712
x31=−116.238928182822x_{31} = -116.238928182822
x32=87.9873238688858x_{32} = 87.9873238688858
x33=−37.7520766759717x_{33} = -37.7520766759717
x34=383.279521849928x_{34} = 383.279521849928
x35=40.8407044966673x_{35} = 40.8407044966673
x36=−333.014827001064x_{36} = -333.014827001064
x37=−53.4070751110265x_{37} = -53.4070751110265
x38=−3.14159265358979x_{38} = -3.14159265358979
x39=−84.8230016469244x_{39} = -84.8230016469244
x40=21.9911485751286x_{40} = 21.9911485751286
x41=−94.2689946953541x_{41} = -94.2689946953541
x42=50.3052345158712x_{42} = 50.3052345158712
x43=34.5575191894877x_{43} = 34.5575191894877
x44=−87.9873238688858x_{44} = -87.9873238688858
x45=47.1238898038469x_{45} = 47.1238898038469
x46=−15.707963267949x_{46} = -15.707963267949
x47=285.884931476671x_{47} = 285.884931476671
x48=53.4070751110265x_{48} = 53.4070751110265
x49=−62.8636652691801x_{49} = -62.8636652691801
x50=65.9734457253857x_{50} = 65.9734457253857
x51=25.2120268885508x_{51} = 25.2120268885508
x52=91.106186954104x_{52} = 91.106186954104
x53=81.7058858119467x_{53} = 81.7058858119467
x54=59.6902604182061x_{54} = 59.6902604182061
x55=94.2689946953541x_{55} = 94.2689946953541
x56=−6.58462004256417x_{56} = -6.58462004256417
x57=18.9549714108416x_{57} = 18.9549714108416
x58=100.550854691761x_{58} = 100.550854691761
x59=75.4247386315323x_{59} = 75.4247386315323
x60=−81.7058858119467x_{60} = -81.7058858119467
x61=78.5398163397448x_{61} = 78.5398163397448
x62=15.707963267949x_{62} = 15.707963267949
x63=−69.1439615203373x_{63} = -69.1439615203373
x64=72.2566310325652x_{64} = 72.2566310325652
x65=−47.1238898038469x_{65} = -47.1238898038469
x66=62.8636652691801x_{66} = 62.8636652691801
x67=69.1439615203373x_{67} = 69.1439615203373
x68=−44.0277152732459x_{68} = -44.0277152732459
x69=−9.42477796076938x_{69} = -9.42477796076938
x70=−34.5575191894877x_{70} = -34.5575191894877
x71=−31.4794387120097x_{71} = -31.4794387120097
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 (-1/x - cos(x)/x + sin(x))/x.
(−10−cos⁡(0)0)+sin⁡(0)0\frac{\left(- \frac{1}{0} - \frac{\cos{\left(0 \right)}}{0}\right) + \sin{\left(0 \right)}}{0}
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⁡(x)+sin⁡(x)x+cos⁡(x)x2+1x2x−(−cos⁡(x)x−1x)+sin⁡(x)x2=0\frac{\cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}}{x} - \frac{\left(- \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}\right) + \sin{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−26.7063338692132x_{1} = -26.7063338692132
x2=−98.9599644038221x_{2} = -98.9599644038221
x3=−73.8270605507521x_{3} = -73.8270605507521
x4=−7.88515116887855x_{4} = -7.88515116887855
x5=67.5438038559559x_{5} = 67.5438038559559
x6=80.1103011235726x_{6} = 80.1103011235726
x7=−76.9693574926784x_{7} = -76.9693574926784
x8=98.9599644038221x_{8} = 98.9599644038221
x9=26.7063338692132x_{9} = 26.7063338692132
x10=−14.1470612675535x_{10} = -14.1470612675535
x11=−32.9885573139218x_{11} = -32.9885573139218
x12=7.88515116887855x_{12} = 7.88515116887855
x13=14.1470612675535x_{13} = 14.1470612675535
x14=54.9772101700572x_{14} = 54.9772101700572
x15=−36.1267854631135x_{15} = -36.1267854631135
x16=−45.5540563247426x_{16} = -45.5540563247426
x17=−1.92814221248582x_{17} = -1.92814221248582
x18=−17.2721001030608x_{18} = -17.2721001030608
x19=−42.4103901077775x_{19} = -42.4103901077775
x20=−92.6767504785877x_{20} = -92.6767504785877
x21=32.9885573139218x_{21} = 32.9885573139218
x22=−67.5438038559559x_{22} = -67.5438038559559
x23=−54.9772101700572x_{23} = -54.9772101700572
x24=73.8270605507521x_{24} = 73.8270605507521
x25=−86.3935300869876x_{25} = -86.3935300869876
x26=42.4103901077775x_{26} = 42.4103901077775
x27=−89.5356400464507x_{27} = -89.5356400464507
x28=−83.2524937964679x_{28} = -83.2524937964679
x29=23.5583542044893x_{29} = 23.5583542044893
x30=61.2605241007865x_{30} = 61.2605241007865
x31=1302.19015609242x_{31} = 1302.19015609242
x32=−64.4031313538059x_{32} = -64.4031313538059
x33=−934.623812153386x_{33} = -934.623812153386
x34=86.3935300869876x_{34} = 86.3935300869876
x35=−70.6862348223115x_{35} = -70.6862348223115
x36=−20.4251234219363x_{36} = -20.4251234219363
x37=45.5540563247426x_{37} = 45.5540563247426
x38=51.8370225339475x_{38} = 51.8370225339475
x39=−51.8370225339475x_{39} = -51.8370225339475
x40=−61.2605241007865x_{40} = -61.2605241007865
x41=39.2712033162649x_{41} = 39.2712033162649
x42=−29.8428895570452x_{42} = -29.8428895570452
x43=−58.1200558177705x_{43} = -58.1200558177705
x44=4.62684260763255x_{44} = 4.62684260763255
x45=−300.022076199367x_{45} = -300.022076199367
x46=64.4031313538059x_{46} = 64.4031313538059
x47=108.38511677127x_{47} = 108.38511677127
x48=−80.1103011235726x_{48} = -80.1103011235726
x49=58.1200558177705x_{49} = 58.1200558177705
x50=−95.8187937225186x_{50} = -95.8187937225186
x51=76.9693574926784x_{51} = 76.9693574926784
x52=95.8187937225186x_{52} = 95.8187937225186
x53=−249.756583899011x_{53} = -249.756583899011
x54=−23.5583542044893x_{54} = -23.5583542044893
x55=158.650508459697x_{55} = 158.650508459697
x56=29.8428895570452x_{56} = 29.8428895570452
x57=48.6938433476388x_{57} = 48.6938433476388
x58=1.92814221248582x_{58} = 1.92814221248582
x59=83.2524937964679x_{59} = 83.2524937964679
x60=−48.6938433476388x_{60} = -48.6938433476388
x61=−4.62684260763255x_{61} = -4.62684260763255
x62=92.6767504785877x_{62} = 92.6767504785877
x63=89.5356400464507x_{63} = 89.5356400464507
x64=36.1267854631135x_{64} = 36.1267854631135
x65=10.979252893657x_{65} = 10.979252893657
x66=−39.2712033162649x_{66} = -39.2712033162649
x67=17.2721001030608x_{67} = 17.2721001030608
x68=70.6862348223115x_{68} = 70.6862348223115
x69=−623.606146880461x_{69} = -623.606146880461
x70=−10.979252893657x_{70} = -10.979252893657
x71=20.4251234219363x_{71} = 20.4251234219363
Signos de extremos en los puntos:
(-26.706333869213204, 0.0360459994268291)

(-98.95996440382213, -0.010207188518506)

(-73.82706055075212, -0.0137285722270212)

(-7.885151168878552, 0.111176805721797)

(67.54380385595594, -0.0150243037185957)

(80.11030112357257, -0.0126385600543767)

(-76.96935749267844, 0.0128234426857385)

(98.95996440382213, -0.010207188518506)

(26.706333869213204, 0.0360459994268291)

(-14.147061267553491, 0.0657355160207967)

(-32.9885573139218, 0.0293962653590479)

(7.885151168878552, 0.111176805721797)

(14.147061267553491, 0.0657355160207967)

(54.977210170057226, -0.0185199850140197)

(-36.12678546311355, -0.0284452866980815)

(-45.55405632474259, 0.0214705080029265)

(-1.9281422124858159, 0.310976551731524)

(-17.272100103060776, -0.0612252764721718)

(-42.410390107777516, -0.0241344707854544)

(-92.67675047858768, -0.0109065936002076)

(32.9885573139218, 0.0293962653590479)

(-67.54380385595594, -0.0150243037185957)

(-54.977210170057226, -0.0185199850140197)

(73.82706055075212, -0.0137285722270212)

(-86.39353008698758, -0.0117088837922247)

(42.410390107777516, -0.0241344707854544)

(-89.53564004645072, 0.0110440269065513)

(-83.2524937964679, 0.011867413529818)

(23.558354204489298, -0.0442428581604881)

(61.26052410078646, -0.0165900456152065)

(1302.1901560924223, 0.000767347272002868)

(-64.40313135380586, 0.0152862159223257)

(-934.6238121533861, -0.00107109398854395)

(86.39353008698758, -0.0117088837922247)

(-70.68623482231149, 0.013946966575841)

(-20.42512342193631, 0.0465731774705954)

(45.55405632474259, 0.0214705080029265)

(51.83702253394753, 0.0189193513026445)

(-51.83702253394753, 0.0189193513026445)

(-61.26052410078646, -0.0165900456152065)

(39.27120331626491, 0.0248163567056283)

(-29.842889557045183, -0.0346290604915495)

(-58.12005581777055, 0.0169098984436123)

(4.626842607632548, -0.258060787207482)

(-300.0220761993669, -0.00334419728863768)

(64.40313135380586, 0.0152862159223257)

(108.3851167712697, 0.00914124767807342)

(-80.11030112357257, -0.0126385600543767)

(58.12005581777055, 0.0169098984436123)

(-95.8187937225186, 0.0103274717283076)

(76.96935749267844, 0.0128234426857385)

(95.8187937225186, 0.0103274717283076)

(-249.75658389901088, -0.00401992914018596)

(-23.558354204489298, -0.0442428581604881)

(158.65050845969654, 0.00626343618052408)

(29.842889557045183, -0.0346290604915495)

(48.69384334763882, -0.0209578612824846)

(1.9281422124858159, 0.310976551731524)

(83.2524937964679, 0.011867413529818)

(-48.69384334763882, -0.0209578612824846)

(-4.626842607632548, -0.258060787207482)

(92.67675047858768, -0.0109065936002076)

(89.53564004645072, 0.0110440269065513)

(36.12678546311355, -0.0284452866980815)

(10.979252893656982, -0.0992290820940922)

(-39.27120331626491, 0.0248163567056283)

(17.272100103060776, -0.0612252764721718)

(70.68623482231149, 0.013946966575841)

(-623.606146880461, 0.00160100479603768)

(-10.979252893656982, -0.0992290820940922)

(20.42512342193631, 0.0465731774705954)


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=−98.9599644038221x_{1} = -98.9599644038221
x2=−73.8270605507521x_{2} = -73.8270605507521
x3=67.5438038559559x_{3} = 67.5438038559559
x4=80.1103011235726x_{4} = 80.1103011235726
x5=98.9599644038221x_{5} = 98.9599644038221
x6=54.9772101700572x_{6} = 54.9772101700572
x7=−36.1267854631135x_{7} = -36.1267854631135
x8=−17.2721001030608x_{8} = -17.2721001030608
x9=−42.4103901077775x_{9} = -42.4103901077775
x10=−92.6767504785877x_{10} = -92.6767504785877
x11=−67.5438038559559x_{11} = -67.5438038559559
x12=−54.9772101700572x_{12} = -54.9772101700572
x13=73.8270605507521x_{13} = 73.8270605507521
x14=−86.3935300869876x_{14} = -86.3935300869876
x15=42.4103901077775x_{15} = 42.4103901077775
x16=23.5583542044893x_{16} = 23.5583542044893
x17=61.2605241007865x_{17} = 61.2605241007865
x18=−934.623812153386x_{18} = -934.623812153386
x19=86.3935300869876x_{19} = 86.3935300869876
x20=−61.2605241007865x_{20} = -61.2605241007865
x21=−29.8428895570452x_{21} = -29.8428895570452
x22=4.62684260763255x_{22} = 4.62684260763255
x23=−300.022076199367x_{23} = -300.022076199367
x24=−80.1103011235726x_{24} = -80.1103011235726
x25=−249.756583899011x_{25} = -249.756583899011
x26=−23.5583542044893x_{26} = -23.5583542044893
x27=29.8428895570452x_{27} = 29.8428895570452
x28=48.6938433476388x_{28} = 48.6938433476388
x29=−48.6938433476388x_{29} = -48.6938433476388
x30=−4.62684260763255x_{30} = -4.62684260763255
x31=92.6767504785877x_{31} = 92.6767504785877
x32=36.1267854631135x_{32} = 36.1267854631135
x33=10.979252893657x_{33} = 10.979252893657
x34=17.2721001030608x_{34} = 17.2721001030608
x35=−10.979252893657x_{35} = -10.979252893657
Puntos máximos de la función:
x35=−26.7063338692132x_{35} = -26.7063338692132
x35=−7.88515116887855x_{35} = -7.88515116887855
x35=−76.9693574926784x_{35} = -76.9693574926784
x35=26.7063338692132x_{35} = 26.7063338692132
x35=−14.1470612675535x_{35} = -14.1470612675535
x35=−32.9885573139218x_{35} = -32.9885573139218
x35=7.88515116887855x_{35} = 7.88515116887855
x35=14.1470612675535x_{35} = 14.1470612675535
x35=−45.5540563247426x_{35} = -45.5540563247426
x35=−1.92814221248582x_{35} = -1.92814221248582
x35=32.9885573139218x_{35} = 32.9885573139218
x35=−89.5356400464507x_{35} = -89.5356400464507
x35=−83.2524937964679x_{35} = -83.2524937964679
x35=1302.19015609242x_{35} = 1302.19015609242
x35=−64.4031313538059x_{35} = -64.4031313538059
x35=−70.6862348223115x_{35} = -70.6862348223115
x35=−20.4251234219363x_{35} = -20.4251234219363
x35=45.5540563247426x_{35} = 45.5540563247426
x35=51.8370225339475x_{35} = 51.8370225339475
x35=−51.8370225339475x_{35} = -51.8370225339475
x35=39.2712033162649x_{35} = 39.2712033162649
x35=−58.1200558177705x_{35} = -58.1200558177705
x35=64.4031313538059x_{35} = 64.4031313538059
x35=108.38511677127x_{35} = 108.38511677127
x35=58.1200558177705x_{35} = 58.1200558177705
x35=−95.8187937225186x_{35} = -95.8187937225186
x35=76.9693574926784x_{35} = 76.9693574926784
x35=95.8187937225186x_{35} = 95.8187937225186
x35=158.650508459697x_{35} = 158.650508459697
x35=1.92814221248582x_{35} = 1.92814221248582
x35=83.2524937964679x_{35} = 83.2524937964679
x35=89.5356400464507x_{35} = 89.5356400464507
x35=−39.2712033162649x_{35} = -39.2712033162649
x35=70.6862348223115x_{35} = 70.6862348223115
x35=−623.606146880461x_{35} = -623.606146880461
x35=20.4251234219363x_{35} = 20.4251234219363
Decrece en los intervalos
[98.9599644038221,∞)\left[98.9599644038221, \infty\right)
Crece en los intervalos
(−∞,−934.623812153386]\left(-\infty, -934.623812153386\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⁡(x)+2(cos⁡(x)+sin⁡(x)x+cos⁡(x)x2+1x2)x−cos⁡(x)x+2(−sin⁡(x)+cos⁡(x)x+1x)x2+2sin⁡(x)x2+2cos⁡(x)x3+2x3x=0- \frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}\right)}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(- \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x} + \frac{1}{x}\right)}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{2}} + \frac{2 \cos{\left(x \right)}}{x^{3}} + \frac{2}{x^{3}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−43.959435274256x_{1} = -43.959435274256
x2=−169.640106481498x_{2} = -169.640106481498
x3=40.8162387302113x_{3} = 40.8162387302113
x4=31.3837510681727x_{4} = 31.3837510681727
x5=37.6723867509001x_{5} = 37.6723867509001
x6=2.87544192591899x_{6} = 2.87544192591899
x7=72.2427950087119x_{7} = 72.2427950087119
x8=69.1005374308808x_{8} = 69.1005374308808
x9=−25.0922791572579x_{9} = -25.0922791572579
x10=−59.6735135411868x_{10} = -59.6735135411868
x11=47.1026819038838x_{11} = 47.1026819038838
x12=9.32032144029274x_{12} = 9.32032144029274
x13=12.4813676983046x_{13} = 12.4813676983046
x14=50.2455040576079x_{14} = 50.2455040576079
x15=53.3883597622861x_{15} = 53.3883597622861
x16=21.9458019045592x_{16} = 21.9458019045592
x17=−50.2455040576079x_{17} = -50.2455040576079
x18=−100.521007231031x_{18} = -100.521007231031
x19=−18.7949054335686x_{19} = -18.7949054335686
x20=−4816.06133031462x_{20} = -4816.06133031462
x21=78.527086697811x_{21} = 78.527086697811
x22=−75.3849358830759x_{22} = -75.3849358830759
x23=81.6691467285037x_{23} = 81.6691467285037
x24=−94.2371565349383x_{24} = -94.2371565349383
x25=−12.4813676983046x_{25} = -12.4813676983046
x26=65.9582927584113x_{26} = 65.9582927584113
x27=25.0922791572579x_{27} = 25.0922791572579
x28=6.07880160638625x_{28} = 6.07880160638625
x29=−81.6691467285037x_{29} = -81.6691467285037
x30=28.2390253329945x_{30} = 28.2390253329945
x31=100.521007231031x_{31} = 100.521007231031
x32=−47.1026819038838x_{32} = -47.1026819038838
x33=84.8112145775519x_{33} = 84.8112145775519
x34=−103.662913011277x_{34} = -103.662913011277
x35=−91.0952125149643x_{35} = -91.0952125149643
x36=−34.5286143188912x_{36} = -34.5286143188912
x37=−37.6723867509001x_{37} = -37.6723867509001
x38=−65.9582927584113x_{38} = -65.9582927584113
x39=−40.8162387302113x_{39} = -40.8162387302113
x40=−87.9532104163538x_{40} = -87.9532104163538
x41=−72.2427950087119x_{41} = -72.2427950087119
x42=−53.3883597622861x_{42} = -53.3883597622861
x43=−56.5309248701959x_{43} = -56.5309248701959
x44=−28.2390253329945x_{44} = -28.2390253329945
x45=75.3849358830759x_{45} = 75.3849358830759
x46=−6.07880160638625x_{46} = -6.07880160638625
x47=−78.527086697811x_{47} = -78.527086697811
x48=18.7949054335686x_{48} = 18.7949054335686
x49=91.0952125149643x_{49} = 91.0952125149643
x50=−15.6446495895545x_{50} = -15.6446495895545
x51=−21.9458019045592x_{51} = -21.9458019045592
x52=43.959435274256x_{52} = 43.959435274256
x53=−31.3837510681727x_{53} = -31.3837510681727
x54=97.3791056443682x_{54} = 97.3791056443682
x55=−97.3791056443682x_{55} = -97.3791056443682
x56=59.6735135411868x_{56} = 59.6735135411868
x57=56.5309248701959x_{57} = 56.5309248701959
x58=−84.8112145775519x_{58} = -84.8112145775519
x59=−9.32032144029274x_{59} = -9.32032144029274
x60=−2.87544192591899x_{60} = -2.87544192591899
x61=62.8158945612055x_{61} = 62.8158945612055
x62=257.606715142538x_{62} = 257.606715142538
x63=87.9532104163538x_{63} = 87.9532104163538
x64=−62.8158945612055x_{64} = -62.8158945612055
x65=15.6446495895545x_{65} = 15.6446495895545
x66=34.5286143188912x_{66} = 34.5286143188912
x67=94.2371565349383x_{67} = 94.2371565349383
x68=−69.1005374308808x_{68} = -69.1005374308808
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⁡(x)+2(cos⁡(x)+sin⁡(x)x+cos⁡(x)x2+1x2)x−cos⁡(x)x+2(−sin⁡(x)+cos⁡(x)x+1x)x2+2sin⁡(x)x2+2cos⁡(x)x3+2x3x)=−∞\lim_{x \to 0^-}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}\right)}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(- \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x} + \frac{1}{x}\right)}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{2}} + \frac{2 \cos{\left(x \right)}}{x^{3}} + \frac{2}{x^{3}}}{x}\right) = -\infty
lim⁡x→0+(−sin⁡(x)+2(cos⁡(x)+sin⁡(x)x+cos⁡(x)x2+1x2)x−cos⁡(x)x+2(−sin⁡(x)+cos⁡(x)x+1x)x2+2sin⁡(x)x2+2cos⁡(x)x3+2x3x)=−∞\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}\right)}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(- \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x} + \frac{1}{x}\right)}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{2}} + \frac{2 \cos{\left(x \right)}}{x^{3}} + \frac{2}{x^{3}}}{x}\right) = -\infty
- 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
[97.3791056443682,∞)\left[97.3791056443682, \infty\right)
Convexa en los intervalos
(−∞,−169.640106481498]\left(-\infty, -169.640106481498\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→−∞((−cos⁡(x)x−1x)+sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\left(- \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}\right) + \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((−cos⁡(x)x−1x)+sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\left(- \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}\right) + \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-1/x - cos(x)/x + sin(x))/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−cos⁡(x)x−1x)+sin⁡(x)x2)=0\lim_{x \to -\infty}\left(\frac{\left(- \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}\right) + \sin{\left(x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((−cos⁡(x)x−1x)+sin⁡(x)x2)=0\lim_{x \to \infty}\left(\frac{\left(- \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}\right) + \sin{\left(x \right)}}{x^{2}}\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:
(−cos⁡(x)x−1x)+sin⁡(x)x=−−sin⁡(x)+cos⁡(x)x+1xx\frac{\left(- \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}\right) + \sin{\left(x \right)}}{x} = - \frac{- \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x} + \frac{1}{x}}{x}
- No
(−cos⁡(x)x−1x)+sin⁡(x)x=−sin⁡(x)+cos⁡(x)x+1xx\frac{\left(- \frac{\cos{\left(x \right)}}{x} - \frac{1}{x}\right) + \sin{\left(x \right)}}{x} = \frac{- \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x} + \frac{1}{x}}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор