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Trazado el gráfico de la función/ (-sinx-(2cosx)/x+(2sinx)/x^2)/x

Gráfico de la función y = (-sinx-(2cosx)/x+(2sinx)/x^2)/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                 2*cos(x)   2*sin(x)
       -sin(x) - -------- + --------
                    x           2   
                               x    
f(x) = -----------------------------
                     x
f(x)=(−sin⁡(x)−2cos⁡(x)x)+2sin⁡(x)x2xf{\left(x \right)} = \frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} 
f = (-sin(x) - 2*cos(x)/x + (2*sin(x))/x^2)/x
Gráfico de la función
02468-8-6-4-2-10100.5-0.5
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)−2cos⁡(x)x)+2sin⁡(x)x2x=0\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=12.404445021902x_{1} = 12.404445021902
x2=−21.8996964794928x_{2} = -21.8996964794928
x3=50.2256516491831x_{3} = 50.2256516491831
x4=56.5132704621986x_{4} = 56.5132704621986
x5=62.8000005565198x_{5} = 62.8000005565198
x6=−72.2289377620154x_{6} = -72.2289377620154
x7=43.9367614714198x_{7} = 43.9367614714198
x8=−40.7916552312719x_{8} = -40.7916552312719
x9=−62.8000005565198x_{9} = -62.8000005565198
x10=−69.0860849466452x_{10} = -69.0860849466452
x11=−91.0842274914688x_{11} = -91.0842274914688
x12=−100.511065295271x_{12} = -100.511065295271
x13=59.6567290035279x_{13} = 59.6567290035279
x14=−28.2033610039524x_{14} = -28.2033610039524
x15=−53.3695918204908x_{15} = -53.3695918204908
x16=31.3520917265645x_{16} = 31.3520917265645
x17=−18.7426455847748x_{17} = -18.7426455847748
x18=−9.20584014293667x_{18} = -9.20584014293667
x19=−25.052825280993x_{19} = -25.052825280993
x20=−1790.70669566846x_{20} = -1790.70669566846
x21=2.0815759778181x_{21} = 2.0815759778181
x22=91.0842274914688x_{22} = 91.0842274914688
x23=9.20584014293667x_{23} = 9.20584014293667
x24=53.3695918204908x_{24} = 53.3695918204908
x25=40.7916552312719x_{25} = 40.7916552312719
x26=94.2265525745684x_{26} = 94.2265525745684
x27=−94.2265525745684x_{27} = -94.2265525745684
x28=−43.9367614714198x_{28} = -43.9367614714198
x29=−78.5143405319308x_{29} = -78.5143405319308
x30=69.0860849466452x_{30} = 69.0860849466452
x31=81.6569138240367x_{31} = 81.6569138240367
x32=78.5143405319308x_{32} = 78.5143405319308
x33=−65.9431119046552x_{33} = -65.9431119046552
x34=−37.6459603230864x_{34} = -37.6459603230864
x35=97.368830362901x_{35} = 97.368830362901
x36=15.5792364103872x_{36} = 15.5792364103872
x37=−2.0815759778181x_{37} = -2.0815759778181
x38=28.2033610039524x_{38} = 28.2033610039524
x39=75.3716854092873x_{39} = 75.3716854092873
x40=−12.404445021902x_{40} = -12.404445021902
x41=−59.6567290035279x_{41} = -59.6567290035279
x42=37.6459603230864x_{42} = 37.6459603230864
x43=18.7426455847748x_{43} = 18.7426455847748
x44=65.9431119046552x_{44} = 65.9431119046552
x45=−47.0813974121542x_{45} = -47.0813974121542
x46=−81.6569138240367x_{46} = -81.6569138240367
x47=−75.3716854092873x_{47} = -75.3716854092873
x48=−342.42775856009x_{48} = -342.42775856009
x49=25.052825280993x_{49} = 25.052825280993
x50=−87.9418500396598x_{50} = -87.9418500396598
x51=−1288.05143523817x_{51} = -1288.05143523817
x52=84.7994143922025x_{52} = 84.7994143922025
x53=21.8996964794928x_{53} = 21.8996964794928
x54=−34.499514921367x_{54} = -34.499514921367
x55=72.2289377620154x_{55} = 72.2289377620154
x56=100.511065295271x_{56} = 100.511065295271
x57=47.0813974121542x_{57} = 47.0813974121542
x58=34.499514921367x_{58} = 34.499514921367
x59=−50.2256516491831x_{59} = -50.2256516491831
x60=−56.5132704621986x_{60} = -56.5132704621986
x61=131.931731514843x_{61} = 131.931731514843
x62=−15.5792364103872x_{62} = -15.5792364103872
x63=−84.7994143922025x_{63} = -84.7994143922025
x64=−5.94036999057271x_{64} = -5.94036999057271
x65=5.94036999057271x_{65} = 5.94036999057271
x66=87.9418500396598x_{66} = 87.9418500396598
x67=−31.3520917265645x_{67} = -31.3520917265645
x68=−97.368830362901x_{68} = -97.368830362901
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) - 2*cos(x)/x + (2*sin(x))/x^2)/x.
(−sin⁡(0)−2cos⁡(0)0)+2sin⁡(0)020\frac{\left(- \sin{\left(0 \right)} - \frac{2 \cos{\left(0 \right)}}{0}\right) + \frac{2 \sin{\left(0 \right)}}{0^{2}}}{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)+2sin⁡(x)x+4cos⁡(x)x2−4sin⁡(x)x3x−(−sin⁡(x)−2cos⁡(x)x)+2sin⁡(x)x2x2=0\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{4 \cos{\left(x \right)}}{x^{2}} - \frac{4 \sin{\left(x \right)}}{x^{3}}}{x} - \frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−13.9205214266357x_{1} = -13.9205214266357
x2=−3.87023858022217x_{2} = -3.87023858022217
x3=−98.9298409677395x_{3} = -98.9298409677395
x4=36.0450220033785x_{4} = 36.0450220033785
x5=−89.5018675871561x_{5} = -89.5018675871561
x6=−10.7130109882558x_{6} = -10.7130109882558
x7=7.44308705395446x_{7} = 7.44308705395446
x8=−83.2161494163012x_{8} = -83.2161494163012
x9=98.9298409677395x_{9} = 98.9298409677395
x10=26.5905550258527x_{10} = 26.5905550258527
x11=−42.3406072405316x_{11} = -42.3406072405316
x12=51.7783178293904x_{12} = 51.7783178293904
x13=−23.4336891696933x_{13} = -23.4336891696933
x14=−39.1933145376163x_{14} = -39.1933145376163
x15=−17.1027407890472x_{15} = -17.1027407890472
x16=67.499787704256x_{16} = 67.499787704256
x17=−80.0731410729394x_{17} = -80.0731410729394
x18=89.5018675871561x_{18} = 89.5018675871561
x19=−36.0450220033785x_{19} = -36.0450220033785
x20=−92.644597693687x_{20} = -92.644597693687
x21=86.3590546280651x_{21} = 86.3590546280651
x22=−26.5905550258527x_{22} = -26.5905550258527
x23=−61.2120336791686x_{23} = -61.2120336791686
x24=249.744603496217x_{24} = 249.744603496217
x25=−48.6329734602127x_{25} = -48.6329734602127
x26=70.6433593524237x_{26} = 70.6433593524237
x27=−64.356022448055x_{27} = -64.356022448055
x28=−212.043355752614x_{28} = -212.043355752614
x29=73.7867621864701x_{29} = 73.7867621864701
x30=−95.7872531120814x_{30} = -95.7872531120814
x31=45.4871087849823x_{31} = 45.4871087849823
x32=10.7130109882558x_{32} = 10.7130109882558
x33=58.0677849921727x_{33} = 58.0677849921727
x34=23.4336891696933x_{34} = 23.4336891696933
x35=−70.6433593524237x_{35} = -70.6433593524237
x36=−73.7867621864701x_{36} = -73.7867621864701
x37=83.2161494163012x_{37} = 83.2161494163012
x38=3.87023858022217x_{38} = 3.87023858022217
x39=39.1933145376163x_{39} = 39.1933145376163
x40=20.2720010891386x_{40} = 20.2720010891386
x41=17.1027407890472x_{41} = 17.1027407890472
x42=−20.2720010891386x_{42} = -20.2720010891386
x43=61.2120336791686x_{43} = 61.2120336791686
x44=−7.44308705395446x_{44} = -7.44308705395446
x45=92.644597693687x_{45} = 92.644597693687
x46=76.9300169373264x_{46} = 76.9300169373264
x47=48.6329734602127x_{47} = 48.6329734602127
x48=−51.7783178293904x_{48} = -51.7783178293904
x49=−67.499787704256x_{49} = -67.499787704256
x50=−54.9232316104305x_{50} = -54.9232316104305
x51=42.3406072405316x_{51} = 42.3406072405316
x52=95.7872531120814x_{52} = 95.7872531120814
x53=54.9232316104305x_{53} = 54.9232316104305
x54=−86.3590546280651x_{54} = -86.3590546280651
x55=−45.4871087849823x_{55} = -45.4871087849823
x56=29.7441555138788x_{56} = 29.7441555138788
x57=−29.7441555138788x_{57} = -29.7441555138788
x58=1148.2495022125x_{58} = 1148.2495022125
x59=64.356022448055x_{59} = 64.356022448055
x60=80.0731410729394x_{60} = 80.0731410729394
x61=−58.0677849921727x_{61} = -58.0677849921727
x62=13.9205214266357x_{62} = 13.9205214266357
x63=32.8954402073454x_{63} = 32.8954402073454
x64=−76.9300169373264x_{64} = -76.9300169373264
x65=230.894066823744x_{65} = 230.894066823744
x66=−32.8954402073454x_{66} = -32.8954402073454
Signos de extremos en los puntos:
(-13.920521426635718, -0.0716515891912541)

(-3.870238580222165, 0.248692236445789)

(-98.92984096773947, 0.0101076571692611)

(36.04502200337846, 0.027732411296845)

(-89.5018675871561, -0.0111722538908926)

(-10.713010988255775, 0.0929394035698334)

(7.443087053954458, -0.133143202113607)

(-83.21614941630125, -0.012016030684702)

(98.92984096773947, 0.0101076571692611)

(26.590555025852712, -0.0375807705734432)

(-42.34060724053156, 0.0236114046136127)

(51.77831782939038, -0.0193095024183706)

(-23.433689169693317, 0.0426347989465758)

(-39.19331453761631, -0.0255062545726498)

(-17.102740789047186, 0.0583704306894909)

(67.499787704256, 0.0148132358906579)

(-80.07314107293935, 0.012487608370541)

(89.5018675871561, -0.0111722538908926)

(-36.04502200337846, 0.027732411296845)

(-92.64459769368703, 0.0107933087887867)

(86.35905462806514, 0.0115787854311861)

(-26.590555025852712, -0.0375807705734432)

(-61.212033679168634, 0.016334477440491)

(249.7446034962173, 0.00400405842499421)

(-48.63297346021267, 0.0205578354154553)

(70.64335935242372, -0.0141541941579985)

(-64.35602244805503, -0.0155366857700825)

(-212.04335575261447, 0.00471596422453044)

(73.78676218647006, 0.0135513220378694)

(-95.7872531120814, -0.0104392335849293)

(45.48710878498235, -0.02197893994184)

(10.713010988255775, 0.0929394035698334)

(58.06778499217275, -0.0172186996097649)

(23.433689169693317, 0.0426347989465758)

(-70.64335935242372, -0.0141541941579985)

(-73.78676218647006, 0.0135513220378694)

(83.21614941630125, -0.012016030684702)

(3.870238580222165, 0.248692236445789)

(39.19331453761631, -0.0255062545726498)

(20.272001089138612, -0.0492692013251242)

(17.102740789047186, 0.0583704306894909)

(-20.272001089138612, -0.0492692013251242)

(61.212033679168634, 0.016334477440491)

(-7.443087053954458, -0.133143202113607)

(92.64459769368703, 0.0107933087887867)

(76.93001693732643, -0.0129977291788139)

(48.63297346021267, 0.0205578354154553)

(-51.77831782939038, -0.0193095024183706)

(-67.499787704256, 0.0148132358906579)

(-54.92323161043051, 0.0182042144946165)

(42.34060724053156, 0.0236114046136127)

(95.7872531120814, -0.0104392335849293)

(54.92323161043051, 0.0182042144946165)

(-86.35905462806514, 0.0115787854311861)

(-45.48710878498235, -0.02197893994184)

(29.744155513878802, 0.0336010649596176)

(-29.744155513878802, 0.0336010649596176)

(1148.2495022124986, 0.000870890532804869)

(64.35602244805503, -0.0155366857700825)

(80.07314107293935, 0.012487608370541)

(-58.06778499217275, -0.0172186996097649)

(13.920521426635718, -0.0716515891912541)

(32.89544020734542, -0.0303853130040648)

(-76.93001693732643, -0.0129977291788139)

(230.8940668237441, 0.0043309498383777)

(-32.89544020734542, -0.0303853130040648)


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=−13.9205214266357x_{1} = -13.9205214266357
x2=−89.5018675871561x_{2} = -89.5018675871561
x3=7.44308705395446x_{3} = 7.44308705395446
x4=−83.2161494163012x_{4} = -83.2161494163012
x5=26.5905550258527x_{5} = 26.5905550258527
x6=51.7783178293904x_{6} = 51.7783178293904
x7=−39.1933145376163x_{7} = -39.1933145376163
x8=89.5018675871561x_{8} = 89.5018675871561
x9=−26.5905550258527x_{9} = -26.5905550258527
x10=70.6433593524237x_{10} = 70.6433593524237
x11=−64.356022448055x_{11} = -64.356022448055
x12=−95.7872531120814x_{12} = -95.7872531120814
x13=45.4871087849823x_{13} = 45.4871087849823
x14=58.0677849921727x_{14} = 58.0677849921727
x15=−70.6433593524237x_{15} = -70.6433593524237
x16=83.2161494163012x_{16} = 83.2161494163012
x17=39.1933145376163x_{17} = 39.1933145376163
x18=20.2720010891386x_{18} = 20.2720010891386
x19=−20.2720010891386x_{19} = -20.2720010891386
x20=−7.44308705395446x_{20} = -7.44308705395446
x21=76.9300169373264x_{21} = 76.9300169373264
x22=−51.7783178293904x_{22} = -51.7783178293904
x23=95.7872531120814x_{23} = 95.7872531120814
x24=−45.4871087849823x_{24} = -45.4871087849823
x25=64.356022448055x_{25} = 64.356022448055
x26=−58.0677849921727x_{26} = -58.0677849921727
x27=13.9205214266357x_{27} = 13.9205214266357
x28=32.8954402073454x_{28} = 32.8954402073454
x29=−76.9300169373264x_{29} = -76.9300169373264
x30=−32.8954402073454x_{30} = -32.8954402073454
Puntos máximos de la función:
x30=−3.87023858022217x_{30} = -3.87023858022217
x30=−98.9298409677395x_{30} = -98.9298409677395
x30=36.0450220033785x_{30} = 36.0450220033785
x30=−10.7130109882558x_{30} = -10.7130109882558
x30=98.9298409677395x_{30} = 98.9298409677395
x30=−42.3406072405316x_{30} = -42.3406072405316
x30=−23.4336891696933x_{30} = -23.4336891696933
x30=−17.1027407890472x_{30} = -17.1027407890472
x30=67.499787704256x_{30} = 67.499787704256
x30=−80.0731410729394x_{30} = -80.0731410729394
x30=−36.0450220033785x_{30} = -36.0450220033785
x30=−92.644597693687x_{30} = -92.644597693687
x30=86.3590546280651x_{30} = 86.3590546280651
x30=−61.2120336791686x_{30} = -61.2120336791686
x30=249.744603496217x_{30} = 249.744603496217
x30=−48.6329734602127x_{30} = -48.6329734602127
x30=−212.043355752614x_{30} = -212.043355752614
x30=73.7867621864701x_{30} = 73.7867621864701
x30=10.7130109882558x_{30} = 10.7130109882558
x30=23.4336891696933x_{30} = 23.4336891696933
x30=−73.7867621864701x_{30} = -73.7867621864701
x30=3.87023858022217x_{30} = 3.87023858022217
x30=17.1027407890472x_{30} = 17.1027407890472
x30=61.2120336791686x_{30} = 61.2120336791686
x30=92.644597693687x_{30} = 92.644597693687
x30=48.6329734602127x_{30} = 48.6329734602127
x30=−67.499787704256x_{30} = -67.499787704256
x30=−54.9232316104305x_{30} = -54.9232316104305
x30=42.3406072405316x_{30} = 42.3406072405316
x30=54.9232316104305x_{30} = 54.9232316104305
x30=−86.3590546280651x_{30} = -86.3590546280651
x30=29.7441555138788x_{30} = 29.7441555138788
x30=−29.7441555138788x_{30} = -29.7441555138788
x30=1148.2495022125x_{30} = 1148.2495022125
x30=80.0731410729394x_{30} = 80.0731410729394
x30=230.894066823744x_{30} = 230.894066823744
Decrece en los intervalos
[95.7872531120814,∞)\left[95.7872531120814, \infty\right)
Crece en los intervalos
(−∞,−95.7872531120814]\left(-\infty, -95.7872531120814\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)−2sin⁡(x)x−4cos⁡(x)x2+4sin⁡(x)x3)x+2cos⁡(x)x−2(sin⁡(x)+2cos⁡(x)x−2sin⁡(x)x2)x2−6sin⁡(x)x2−12cos⁡(x)x3+12sin⁡(x)x4x=0\frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x} - \frac{4 \cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}}\right)}{x} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}\right)}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{12 \cos{\left(x \right)}}{x^{3}} + \frac{12 \sin{\left(x \right)}}{x^{4}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=81.6324039381741x_{1} = 81.6324039381741
x2=−47.0388279944848x_{2} = -47.0388279944848
x3=84.775813998667x_{3} = 84.775813998667
x4=−15.4482964241208x_{4} = -15.4482964241208
x5=−43.891130947962x_{5} = -43.891130947962
x6=21.8074653570703x_{6} = 21.8074653570703
x7=34.4413140629846x_{7} = 34.4413140629846
x8=28.1320266710925x_{8} = 28.1320266710925
x9=−94.2053159674705x_{9} = -94.2053159674705
x10=37.5926577196659x_{10} = 37.5926577196659
x11=−81.6324039381741x_{11} = -81.6324039381741
x12=72.2012232176578x_{12} = 72.2012232176578
x13=−12.2380147777786x_{13} = -12.2380147777786
x14=59.623159795497x_{14} = 59.623159795497
x15=−65.9127501167638x_{15} = -65.9127501167638
x16=144.485576756624x_{16} = 144.485576756624
x17=−24.9723917634929x_{17} = -24.9723917634929
x18=91.0622574285078x_{18} = 91.0622574285078
x19=15.4482964241208x_{19} = 15.4482964241208
x20=−59.623159795497x_{20} = -59.623159795497
x21=−1.88055044189143x_{21} = -1.88055044189143
x22=78.4888481666001x_{22} = 78.4888481666001
x23=−69.0571071984352x_{23} = -69.0571071984352
x24=−37.5926577196659x_{24} = -37.5926577196659
x25=204.183931989947x_{25} = 204.183931989947
x26=94.2053159674705x_{26} = 94.2053159674705
x27=56.4778286872598x_{27} = 56.4778286872598
x28=−122.489456166536x_{28} = -122.489456166536
x29=40.7424873426775x_{29} = 40.7424873426775
x30=−62.7681156530028x_{30} = -62.7681156530028
x31=−50.1857574315064x_{31} = -50.1857574315064
x32=254.453284801053x_{32} = 254.453284801053
x33=47.0388279944848x_{33} = 47.0388279944848
x34=−28.1320266710925x_{34} = -28.1320266710925
x35=−91.0622574285078x_{35} = -91.0622574285078
x36=100.491157788754x_{36} = 100.491157788754
x37=62.7681156530028x_{37} = 62.7681156530028
x38=−21.8074653570703x_{38} = -21.8074653570703
x39=−75.3451284134656x_{39} = -75.3451284134656
x40=−5.54218959396937x_{40} = -5.54218959396937
x41=−87.9190939999624x_{41} = -87.9190939999624
x42=65.9127501167638x_{42} = 65.9127501167638
x43=53.332055705137x_{43} = 53.332055705137
x44=−40.7424873426775x_{44} = -40.7424873426775
x45=87.9190939999624x_{45} = 87.9190939999624
x46=31.2879944766898x_{46} = 31.2879944766898
x47=5.54218959396937x_{47} = 5.54218959396937
x48=−72.2012232176578x_{48} = -72.2012232176578
x49=24.9723917634929x_{49} = 24.9723917634929
x50=69.0571071984352x_{50} = 69.0571071984352
x51=−31.2879944766898x_{51} = -31.2879944766898
x52=−97.3482797885283x_{52} = -97.3482797885283
x53=18.6344820913471x_{53} = 18.6344820913471
x54=−18.6344820913471x_{54} = -18.6344820913471
x55=97.3482797885283x_{55} = 97.3482797885283
x56=−78.4888481666001x_{56} = -78.4888481666001
x57=−53.332055705137x_{57} = -53.332055705137
x58=−84.775813998667x_{58} = -84.775813998667
x59=−34.4413140629846x_{59} = -34.4413140629846
x60=−56.4778286872598x_{60} = -56.4778286872598
x61=8.97516007482073x_{61} = 8.97516007482073
x62=−8.97516007482073x_{62} = -8.97516007482073
x63=75.3451284134656x_{63} = 75.3451284134656
x64=4156.32611831087x_{64} = 4156.32611831087
x65=43.891130947962x_{65} = 43.891130947962
x66=−100.491157788754x_{66} = -100.491157788754
x67=12.2380147777786x_{67} = 12.2380147777786
x68=50.1857574315064x_{68} = 50.1857574315064
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)−2sin⁡(x)x−4cos⁡(x)x2+4sin⁡(x)x3)x+2cos⁡(x)x−2(sin⁡(x)+2cos⁡(x)x−2sin⁡(x)x2)x2−6sin⁡(x)x2−12cos⁡(x)x3+12sin⁡(x)x4x)=15\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x} - \frac{4 \cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}}\right)}{x} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}\right)}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{12 \cos{\left(x \right)}}{x^{3}} + \frac{12 \sin{\left(x \right)}}{x^{4}}}{x}\right) = \frac{1}{5}
lim⁡x→0+(sin⁡(x)+2(cos⁡(x)−2sin⁡(x)x−4cos⁡(x)x2+4sin⁡(x)x3)x+2cos⁡(x)x−2(sin⁡(x)+2cos⁡(x)x−2sin⁡(x)x2)x2−6sin⁡(x)x2−12cos⁡(x)x3+12sin⁡(x)x4x)=15\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x} - \frac{4 \cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}}\right)}{x} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}\right)}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{12 \cos{\left(x \right)}}{x^{3}} + \frac{12 \sin{\left(x \right)}}{x^{4}}}{x}\right) = \frac{1}{5}
- 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
[144.485576756624,∞)\left[144.485576756624, \infty\right)
Convexa en los intervalos
(−∞,−122.489456166536]\left(-\infty, -122.489456166536\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)−2cos⁡(x)x)+2sin⁡(x)x2x)=0\lim_{x \to -\infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((−sin⁡(x)−2cos⁡(x)x)+2sin⁡(x)x2x)=0\lim_{x \to \infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{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 (-sin(x) - 2*cos(x)/x + (2*sin(x))/x^2)/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−sin⁡(x)−2cos⁡(x)x)+2sin⁡(x)x2x2)=0\lim_{x \to -\infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((−sin⁡(x)−2cos⁡(x)x)+2sin⁡(x)x2x2)=0\lim_{x \to \infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{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:
(−sin⁡(x)−2cos⁡(x)x)+2sin⁡(x)x2x=−sin⁡(x)+2cos⁡(x)x−2sin⁡(x)x2x\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} = - \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}
- No
(−sin⁡(x)−2cos⁡(x)x)+2sin⁡(x)x2x=sin⁡(x)+2cos⁡(x)x−2sin⁡(x)x2x\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} = \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор