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Trazado el gráfico de la función/ x^(-2)(cosx-1)

Gráfico de la función y = x^(-2)(cosx-1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x) - 1
f(x) = ----------
            2    
           x
f(x)=cos⁡(x)−1x2f{\left(x \right)} = \frac{\cos{\left(x \right)} - 1}{x^{2}} 
f = (cos(x) - 1)/x^2
Gráfico de la función
02468-8-6-4-2-1010-1.00.5
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:
cos⁡(x)−1x2=0\frac{\cos{\left(x \right)} - 1}{x^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=2πx_{1} = 2 \pi
Solución numérica
x1=−87.964593868177x_{1} = -87.964593868177
x2=−43.982296668693x_{2} = -43.982296668693
x3=37.699113477718x_{3} = 37.699113477718
x4=62.8318535339473x_{4} = 62.8318535339473
x5=−62.8318534649472x_{5} = -62.8318534649472
x6=6.28318517164063x_{6} = 6.28318517164063
x7=−37.699111545658x_{7} = -37.699111545658
x8=100.530964985289x_{8} = 100.530964985289
x9=−62.8318526597593x_{9} = -62.8318526597593
x10=37.6991120269657x_{10} = 37.6991120269657
x11=−113.097337167454x_{11} = -113.097337167454
x12=−56.5486682178103x_{12} = -56.5486682178103
x13=43.9822965913968x_{13} = 43.9822965913968
x14=25.132741604817x_{14} = 25.132741604817
x15=56.5486677662762x_{15} = 56.5486677662762
x16=−6.28318370279736x_{16} = -6.28318370279736
x17=−6.28318551307897x_{17} = -6.28318551307897
x18=−50.26548292433x_{18} = -50.26548292433
x19=6.28318325977485x_{19} = 6.28318325977485
x20=−94.2477799320641x_{20} = -94.2477799320641
x21=−6.2831851024088x_{21} = -6.2831851024088
x22=−100.530964622049x_{22} = -100.530964622049
x23=81.6814091875845x_{23} = 81.6814091875845
x24=106.814150266336x_{24} = 106.814150266336
x25=43.9822973193343x_{25} = 43.9822973193343
x26=−56.5486689506328x_{26} = -56.5486689506328
x27=56.5486682432518x_{27} = 56.5486682432518
x28=12.5663709771202x_{28} = 12.5663709771202
x29=31.4159260193027x_{29} = 31.4159260193027
x30=−94.2477800936746x_{30} = -94.2477800936746
x31=−25.1327415130761x_{31} = -25.1327415130761
x32=−12.5663702736785x_{32} = -12.5663702736785
x33=37.6991108186645x_{33} = 37.6991108186645
x34=−87.9645947288122x_{34} = -87.9645947288122
x35=−31.4159259505911x_{35} = -31.4159259505911
x36=75.3982239968282x_{36} = 75.3982239968282
x37=−94.2477794440236x_{37} = -94.2477794440236
x38=615.75223387666x_{38} = 615.75223387666
x39=999.026539452294x_{39} = 999.026539452294
x40=−37.6991119218115x_{40} = -37.6991119218115
x41=−31.4159267112335x_{41} = -31.4159267112335
x42=−6.28318343659183x_{42} = -6.28318343659183
x43=25.1327407984158x_{43} = 25.1327407984158
x44=94.2477792039061x_{44} = 94.2477792039061
x45=−87.9645943584489x_{45} = -87.9645943584489
x46=−25.1327406912539x_{46} = -25.1327406912539
x47=43.9822971694279x_{47} = 43.9822971694279
x48=87.9645945259691x_{48} = 87.9645945259691
x49=87.9645937762976x_{49} = 87.9645937762976
x50=−18.849556280551x_{50} = -18.849556280551
x51=75.3982238735113x_{51} = 75.3982238735113
x52=69.1150387810133x_{52} = 69.1150387810133
x53=−18.8495554685467x_{53} = -18.8495554685467
x54=31.4159268317665x_{54} = 31.4159268317665
x55=37.6991112773793x_{55} = 37.6991112773793
x56=6.28318528383988x_{56} = 6.28318528383988
x57=−12.5663701928159x_{57} = -12.5663701928159
x58=12.5663704316993x_{58} = 12.5663704316993
x59=−75.3982231443749x_{59} = -75.3982231443749
x60=−50.2654822837128x_{60} = -50.2654822837128
x61=50.2654824463285x_{61} = 50.2654824463285
x62=−100.530966107056x_{62} = -100.530966107056
x63=−62.8318531787238x_{63} = -62.8318531787238
x64=−56.5486674611179x_{64} = -56.5486674611179
x65=94.2477800532258x_{65} = 94.2477800532258
x66=18.8495562402486x_{66} = 18.8495562402486
x67=6.2831839162719x_{67} = 6.2831839162719
x68=−43.9822971744808x_{68} = -43.9822971744808
x69=81.6814084607256x_{69} = 81.6814084607256
x70=−50.2654823437827x_{70} = -50.2654823437827
x71=18.8495563328137x_{71} = 18.8495563328137
x72=−12.5663709886702x_{72} = -12.5663709886702
x73=−75.3982238720728x_{73} = -75.3982238720728
x74=−43.9822975465957x_{74} = -43.9822975465957
x75=50.2654820137177x_{75} = 50.2654820137177
x76=−81.6814084568261x_{76} = -81.6814084568261
x77=18.8495556046711x_{77} = 18.8495556046711
x78=−62.831863538801x_{78} = -62.831863538801
x79=−37.6991112623766x_{79} = -37.6991112623766
x80=−238.761043156455x_{80} = -238.761043156455
x81=62.8318527786103x_{81} = 62.8318527786103
x82=87.9645943358427x_{82} = 87.9645943358427
x83=50.265482878775x_{83} = 50.265482878775
x84=−69.1150378843794x_{84} = -69.1150378843794
x85=−81.6814091610451x_{85} = -81.6814091610451
x86=62.8318537989466x_{86} = 62.8318537989466
x87=100.530964758741x_{87} = 100.530964758741
x88=−81.6814090381197x_{88} = -81.6814090381197
x89=−37.6991118771277x_{89} = -37.6991118771277
x90=−69.1150386804025x_{90} = -69.1150386804025
x91=75.3982232009371x_{91} = 75.3982232009371
x92=56.5486675990625x_{92} = 56.5486675990625
x93=81.6814082753175x_{93} = 81.6814082753175
x94=−75.3982227106821x_{94} = -75.3982227106821
x95=−31.4159251397495x_{95} = -31.4159251397495
x96=−94.2477795742125x_{96} = -94.2477795742125
x97=94.2477796093522x_{97} = 94.2477796093522
x98=69.1150379770714x_{98} = 69.1150379770714
x99=12.5663699205299x_{99} = 12.5663699205299
x100=−43.9823106482567x_{100} = -43.9823106482567
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(x) - 1)/x^2.
−1+cos⁡(0)0\frac{-1 + \cos{\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
−sin⁡(x)x2−2(cos⁡(x)−1)x3=0- \frac{\sin{\left(x \right)}}{x^{2}} - \frac{2 \left(\cos{\left(x \right)} - 1\right)}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=12.5663706143592x_{1} = 12.5663706143592
x2=47.038904997378x_{2} = 47.038904997378
x3=15.4505036738754x_{3} = 15.4505036738754
x4=−719.419157645404x_{4} = -719.419157645404
x5=37.6991118430775x_{5} = 37.6991118430775
x6=−40.7426059185751x_{6} = -40.7426059185751
x7=−31.4159265358979x_{7} = -31.4159265358979
x8=34.4415105438615x_{8} = 34.4415105438615
x9=−50.2654824574367x_{9} = -50.2654824574367
x10=6.28318530717959x_{10} = 6.28318530717959
x11=97.3482884639088x_{11} = 97.3482884639088
x12=−94.2477796076938x_{12} = -94.2477796076938
x13=−69.1150383789755x_{13} = -69.1150383789755
x14=−34.4415105438615x_{14} = -34.4415105438615
x15=69.1150383789755x_{15} = 69.1150383789755
x16=62.8318530717959x_{16} = 62.8318530717959
x17=8.98681891581813x_{17} = 8.98681891581813
x18=50.2654824574367x_{18} = 50.2654824574367
x19=81.6814089933346x_{19} = 81.6814089933346
x20=100.530964914873x_{20} = 100.530964914873
x21=−87.9645943005142x_{21} = -87.9645943005142
x22=−91.0622680279826x_{22} = -91.0622680279826
x23=84.7758271362638x_{23} = 84.7758271362638
x24=−62.8318530717959x_{24} = -62.8318530717959
x25=−8.98681891581813x_{25} = -8.98681891581813
x26=21.8082433188578x_{26} = 21.8082433188578
x27=−18.8495559215388x_{27} = -18.8495559215388
x28=−21.8082433188578x_{28} = -21.8082433188578
x29=91.0622680279826x_{29} = 91.0622680279826
x30=28.1323878256629x_{30} = 28.1323878256629
x31=−15.4505036738754x_{31} = -15.4505036738754
x32=125.663706143592x_{32} = 125.663706143592
x33=−59.6231975817859x_{33} = -59.6231975817859
x34=−56.5486677646163x_{34} = -56.5486677646163
x35=59.6231975817859x_{35} = 59.6231975817859
x36=−37.6991118430775x_{36} = -37.6991118430775
x37=−25.1327412287183x_{37} = -25.1327412287183
x38=−100.530964914873x_{38} = -100.530964914873
x39=−197.900125648664x_{39} = -197.900125648664
x40=−75.398223686155x_{40} = -75.398223686155
x41=−84.7758271362638x_{41} = -84.7758271362638
x42=−28.1323878256629x_{42} = -28.1323878256629
x43=18.8495559215388x_{43} = 18.8495559215388
x44=169.646003293849x_{44} = 169.646003293849
x45=−6.28318530717959x_{45} = -6.28318530717959
x46=65.912778079645x_{46} = 65.912778079645
x47=25.1327412287183x_{47} = 25.1327412287183
x48=−103.633964974559x_{48} = -103.633964974559
x49=−97.3482884639088x_{49} = -97.3482884639088
x50=78.4888647223284x_{50} = 78.4888647223284
x51=56.5486677646163x_{51} = 56.5486677646163
x52=−43.9822971502571x_{52} = -43.9822971502571
x53=−78.4888647223284x_{53} = -78.4888647223284
x54=172.764444069457x_{54} = 172.764444069457
x55=−72.2012444887512x_{55} = -72.2012444887512
x56=−53.3321085176254x_{56} = -53.3321085176254
x57=72.2012444887512x_{57} = 72.2012444887512
x58=31.4159265358979x_{58} = 31.4159265358979
x59=94.2477796076938x_{59} = 94.2477796076938
x60=40.7426059185751x_{60} = 40.7426059185751
x61=−12.5663706143592x_{61} = -12.5663706143592
x62=75.398223686155x_{62} = 75.398223686155
x63=245.044226980004x_{63} = 245.044226980004
x64=53.3321085176254x_{64} = 53.3321085176254
x65=−47.038904997378x_{65} = -47.038904997378
x66=−81.6814089933346x_{66} = -81.6814089933346
x67=43.9822971502571x_{67} = 43.9822971502571
x68=−65.912778079645x_{68} = -65.912778079645
x69=87.9645943005142x_{69} = 87.9645943005142
Signos de extremos en los puntos:
(12.566370614359172, 0)

(47.03890499737801, -0.000902258929282338)

(15.450503673875414, -0.00824001299648697)

(-719.4191576454039, -3.86422710095088e-6)

(37.69911184307752, 0)

(-40.74260591857512, -0.00120195201548074)

(-31.41592653589793, 0)

(34.44151054386154, -0.00168036493363077)

(-50.26548245743669, 0)

(6.283185307179586, 0)

(97.34828846390877, -0.000210955126120699)

(-94.2477796076938, 0)

(-69.11503837897546, 0)

(-34.44151054386154, -0.00168036493363077)

(69.11503837897546, 0)

(62.83185307179586, 0)

(8.986818915818128, -0.0235952246129056)

(50.26548245743669, 0)

(81.68140899333463, 0)

(100.53096491487338, 0)

(-87.96459430051421, 0)

(-91.06226802798255, -0.00024107025574655)

(84.77582713626384, -0.000278127721683679)

(-62.83185307179586, 0)

(-8.986818915818128, -0.0235952246129056)

(21.808243318857798, -0.00417014633533631)

(-18.84955592153876, 0)

(-21.808243318857798, -0.00417014633533631)

(91.06226802798255, -0.00024107025574655)

(28.132387825662946, -0.00251435936561617)

(-15.450503673875414, -0.00824001299648697)

(125.66370614359172, 0)

(-59.62319758178592, -0.000561967339101509)

(-56.548667764616276, 0)

(59.62319758178592, -0.000561967339101509)

(-37.69911184307752, 0)

(-25.132741228718345, 0)

(-100.53096491487338, 0)

(-197.90012564866376, -5.10614921724493e-5)

(-75.39822368615503, 0)

(-84.77582713626384, -0.000278127721683679)

(-28.132387825662946, -0.00251435936561617)

(18.84955592153876, 0)

(169.64600329384882, 0)

(-6.283185307179586, 0)

(65.91277807964495, -0.000459929312424257)

(25.132741228718345, 0)

(-103.63396497455933, -0.000186150432117959)

(-97.34828846390877, -0.000210955126120699)

(78.48886472232839, -0.000324438216936361)

(56.548667764616276, 0)

(-43.982297150257104, 0)

(-78.48886472232839, -0.000324438216936361)

(172.76444406945743, -6.69981890382762e-5)

(-72.20124448875121, -0.000383360637454652)

(-53.33210851762535, -0.000702169824387774)

(72.20124448875121, -0.000383360637454652)

(31.41592653589793, 0)

(94.2477796076938, 0)

(40.74260591857512, -0.00120195201548074)

(-12.566370614359172, 0)

(75.39822368615503, 0)

(245.04422698000388, 0)

(53.33210851762535, -0.000702169824387774)

(-47.03890499737801, -0.000902258929282338)

(-81.68140899333463, 0)

(43.982297150257104, 0)

(-65.91277807964495, -0.000459929312424257)

(87.96459430051421, 0)


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=47.038904997378x_{1} = 47.038904997378
x2=15.4505036738754x_{2} = 15.4505036738754
x3=−719.419157645404x_{3} = -719.419157645404
x4=−40.7426059185751x_{4} = -40.7426059185751
x5=34.4415105438615x_{5} = 34.4415105438615
x6=97.3482884639088x_{6} = 97.3482884639088
x7=−34.4415105438615x_{7} = -34.4415105438615
x8=8.98681891581813x_{8} = 8.98681891581813
x9=−91.0622680279826x_{9} = -91.0622680279826
x10=84.7758271362638x_{10} = 84.7758271362638
x11=−8.98681891581813x_{11} = -8.98681891581813
x12=21.8082433188578x_{12} = 21.8082433188578
x13=−21.8082433188578x_{13} = -21.8082433188578
x14=91.0622680279826x_{14} = 91.0622680279826
x15=28.1323878256629x_{15} = 28.1323878256629
x16=−15.4505036738754x_{16} = -15.4505036738754
x17=−59.6231975817859x_{17} = -59.6231975817859
x18=59.6231975817859x_{18} = 59.6231975817859
x19=−197.900125648664x_{19} = -197.900125648664
x20=−84.7758271362638x_{20} = -84.7758271362638
x21=−28.1323878256629x_{21} = -28.1323878256629
x22=65.912778079645x_{22} = 65.912778079645
x23=−103.633964974559x_{23} = -103.633964974559
x24=−97.3482884639088x_{24} = -97.3482884639088
x25=78.4888647223284x_{25} = 78.4888647223284
x26=−78.4888647223284x_{26} = -78.4888647223284
x27=172.764444069457x_{27} = 172.764444069457
x28=−72.2012444887512x_{28} = -72.2012444887512
x29=−53.3321085176254x_{29} = -53.3321085176254
x30=72.2012444887512x_{30} = 72.2012444887512
x31=40.7426059185751x_{31} = 40.7426059185751
x32=53.3321085176254x_{32} = 53.3321085176254
x33=−47.038904997378x_{33} = -47.038904997378
x34=−65.912778079645x_{34} = -65.912778079645
Puntos máximos de la función:
x34=12.5663706143592x_{34} = 12.5663706143592
x34=37.6991118430775x_{34} = 37.6991118430775
x34=−31.4159265358979x_{34} = -31.4159265358979
x34=−50.2654824574367x_{34} = -50.2654824574367
x34=6.28318530717959x_{34} = 6.28318530717959
x34=−94.2477796076938x_{34} = -94.2477796076938
x34=−69.1150383789755x_{34} = -69.1150383789755
x34=69.1150383789755x_{34} = 69.1150383789755
x34=62.8318530717959x_{34} = 62.8318530717959
x34=50.2654824574367x_{34} = 50.2654824574367
x34=81.6814089933346x_{34} = 81.6814089933346
x34=100.530964914873x_{34} = 100.530964914873
x34=−87.9645943005142x_{34} = -87.9645943005142
x34=−62.8318530717959x_{34} = -62.8318530717959
x34=−18.8495559215388x_{34} = -18.8495559215388
x34=125.663706143592x_{34} = 125.663706143592
x34=−56.5486677646163x_{34} = -56.5486677646163
x34=−37.6991118430775x_{34} = -37.6991118430775
x34=−25.1327412287183x_{34} = -25.1327412287183
x34=−100.530964914873x_{34} = -100.530964914873
x34=−75.398223686155x_{34} = -75.398223686155
x34=18.8495559215388x_{34} = 18.8495559215388
x34=169.646003293849x_{34} = 169.646003293849
x34=−6.28318530717959x_{34} = -6.28318530717959
x34=25.1327412287183x_{34} = 25.1327412287183
x34=56.5486677646163x_{34} = 56.5486677646163
x34=−43.9822971502571x_{34} = -43.9822971502571
x34=31.4159265358979x_{34} = 31.4159265358979
x34=94.2477796076938x_{34} = 94.2477796076938
x34=−12.5663706143592x_{34} = -12.5663706143592
x34=75.398223686155x_{34} = 75.398223686155
x34=245.044226980004x_{34} = 245.044226980004
x34=−81.6814089933346x_{34} = -81.6814089933346
x34=43.9822971502571x_{34} = 43.9822971502571
x34=87.9645943005142x_{34} = 87.9645943005142
Decrece en los intervalos
[172.764444069457,∞)\left[172.764444069457, \infty\right)
Crece en los intervalos
(−∞,−719.419157645404]\left(-\infty, -719.419157645404\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
−cos⁡(x)+4sin⁡(x)x+6(cos⁡(x)−1)x2x2=0\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=92.6330997569468x_{1} = 92.6330997569468
x2=2.60616342185561x_{2} = 2.60616342185561
x3=−142.91418226402x_{3} = -142.91418226402
x4=−64.3419201670335x_{4} = -64.3419201670335
x5=−36.0125674161738x_{5} = -36.0125674161738
x6=70.6303965638466x_{6} = 70.6303965638466
x7=26.5612831467777x_{7} = 26.5612831467777
x8=80.0597089609147x_{8} = 80.0597089609147
x9=17.0227015753487x_{9} = 17.0227015753487
x10=32.8705018721625x_{10} = 32.8705018721625
x11=58.0523264799188x_{11} = 58.0523264799188
x12=−61.194077945117x_{12} = -61.194077945117
x13=−20.2369613974299x_{13} = -20.2369613974299
x14=−10.5620648126053x_{14} = -10.5620648126053
x15=−76.9180245407825x_{15} = -76.9180245407825
x16=98.9191157001159x_{16} = 98.9191157001159
x17=−45.4679907819814x_{17} = -45.4679907819814
x18=67.4836428454212x_{18} = 67.4836428454212
x19=−29.7035811594339x_{19} = -29.7035811594339
x20=42.3135862158589x_{20} = 42.3135862158589
x21=−13.8788496402477x_{21} = -13.8788496402477
x22=45.4679907819814x_{22} = 45.4679907819814
x23=−237.173273178795x_{23} = -237.173273178795
x24=51.7612194989417x_{24} = 51.7612194989417
x25=−86.3466644132886x_{25} = -86.3466644132886
x26=−70.6303965638466x_{26} = -70.6303965638466
x27=48.6098380914692x_{27} = 48.6098380914692
x28=−26.5612831467777x_{28} = -26.5612831467777
x29=89.4914388962793x_{29} = 89.4914388962793
x30=29.7035811594339x_{30} = 29.7035811594339
x31=−58.0523264799188x_{31} = -58.0523264799188
x32=13.8788496402477x_{32} = 13.8788496402477
x33=−92.6330997569468x_{33} = -92.6330997569468
x34=−190.045473675927x_{34} = -190.045473675927
x35=−39.1716546993425x_{35} = -39.1716546993425
x36=73.7720976264654x_{36} = 73.7720976264654
x37=−67.4836428454212x_{37} = -67.4836428454212
x38=−83.2049930785316x_{38} = -83.2049930785316
x39=−54.9030104821856x_{39} = -54.9030104821856
x40=−48.6098380914692x_{40} = -48.6098380914692
x41=−51.7612194989417x_{41} = -51.7612194989417
x42=36.0125674161738x_{42} = 36.0125674161738
x43=76.9180245407825x_{43} = 76.9180245407825
x44=−89.4914388962793x_{44} = -89.4914388962793
x45=1277.05428515371x_{45} = 1277.05428515371
x46=−17.0227015753487x_{46} = -17.0227015753487
x47=83.2049930785316x_{47} = 83.2049930785316
x48=86.3466644132886x_{48} = 86.3466644132886
x49=20.2369613974299x_{49} = 20.2369613974299
x50=−23.3797164409097x_{50} = -23.3797164409097
x51=23.3797164409097x_{51} = 23.3797164409097
x52=61.194077945117x_{52} = 61.194077945117
x53=−42.3135862158589x_{53} = -42.3135862158589
x54=−73.7720976264654x_{54} = -73.7720976264654
x55=−98.9191157001159x_{55} = -98.9191157001159
x56=−80.0597089609147x_{56} = -80.0597089609147
x57=887.495425187411x_{57} = 887.495425187411
x58=−95.7774633534361x_{58} = -95.7774633534361
x59=39.1716546993425x_{59} = 39.1716546993425
x60=54.9030104821856x_{60} = 54.9030104821856
x61=95.7774633534361x_{61} = 95.7774633534361
x62=7.4144569281165x_{62} = 7.4144569281165
x63=10.5620648126053x_{63} = 10.5620648126053
x64=64.3419201670335x_{64} = 64.3419201670335
x65=−2.60616342185561x_{65} = -2.60616342185561
x66=−32.8705018721625x_{66} = -32.8705018721625
x67=−7.4144569281165x_{67} = -7.4144569281165
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−(−cos⁡(x)+4sin⁡(x)x+6(cos⁡(x)−1)x2x2)=112\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{2}}\right) = \frac{1}{12}
lim⁡x→0+(−cos⁡(x)+4sin⁡(x)x+6(cos⁡(x)−1)x2x2)=112\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{2}}\right) = \frac{1}{12}
- 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
[1277.05428515371,∞)\left[1277.05428515371, \infty\right)
Convexa en los intervalos
(−∞,−237.173273178795]\left(-\infty, -237.173273178795\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)−1x2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} - 1}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(x)−1x2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} - 1}{x^{2}}\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 (cos(x) - 1)/x^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)−1x3)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} - 1}{x^{3}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x)−1x3)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} - 1}{x^{3}}\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)−1x2=cos⁡(x)−1x2\frac{\cos{\left(x \right)} - 1}{x^{2}} = \frac{\cos{\left(x \right)} - 1}{x^{2}}
- Sí
cos⁡(x)−1x2=−cos⁡(x)−1x2\frac{\cos{\left(x \right)} - 1}{x^{2}} = - \frac{\cos{\left(x \right)} - 1}{x^{2}}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор