Sr Examen

Otras calculadoras

(1-cos(x))/x^3
Trazado el gráfico de la función/ (1-cos(x))/x^3

Gráfico de la función y = (1-cos(x))/x^3

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       1 - cos(x)
f(x) = ----------
            3    
           x
f(x)=1−cos⁡(x)x3f{\left(x \right)} = \frac{1 - \cos{\left(x \right)}}{x^{3}} 
f = (1 - cos(x))/x^3
Gráfico de la función
02468-8-6-4-2-1010-2020
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:
1−cos⁡(x)x3=0\frac{1 - \cos{\left(x \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=2πx_{1} = 2 \pi
Solución numérica
x1=87.9645944825602x_{1} = 87.9645944825602
x2=−25.132741504951x_{2} = -25.132741504951
x3=−43.9822971744519x_{3} = -43.9822971744519
x4=37.6991104084362x_{4} = 37.6991104084362
x5=62.8318527754349x_{5} = 62.8318527754349
x6=56.5486676457973x_{6} = 56.5486676457973
x7=−100.530966103549x_{7} = -100.530966103549
x8=131.946891706466x_{8} = 131.946891706466
x9=94.2477791750714x_{9} = 94.2477791750714
x10=25.1327415885691x_{10} = 25.1327415885691
x11=−18.8495562601109x_{11} = -18.8495562601109
x12=50.2654828447704x_{12} = 50.2654828447704
x13=−94.2477795072385x_{13} = -94.2477795072385
x14=37.699107022324x_{14} = 37.699107022324
x15=−6.28317650290564x_{15} = -6.28317650290564
x16=188.495557225622x_{16} = 188.495557225622
x17=62.8318541837957x_{17} = 62.8318541837957
x18=100.530964911463x_{18} = 100.530964911463
x19=69.1150379711617x_{19} = 69.1150379711617
x20=87.9645943358115x_{20} = 87.9645943358115
x21=−12.5663709397984x_{21} = -12.5663709397984
x22=−18.8495567199802x_{22} = -18.8495567199802
x23=−50.2654822302291x_{23} = -50.2654822302291
x24=−31.4159267089767x_{24} = -31.4159267089767
x25=−6.28318520513552x_{25} = -6.28318520513552
x26=100.530964758203x_{26} = 100.530964758203
x27=−43.9822975047697x_{27} = -43.9822975047697
x28=−75.398223871027x_{28} = -75.398223871027
x29=56.5486675979903x_{29} = 56.5486675979903
x30=−56.5486682055439x_{30} = -56.5486682055439
x31=81.6814084479926x_{31} = 81.6814084479926
x32=18.8495556776517x_{32} = 18.8495556776517
x33=−113.097335767285x_{33} = -113.097335767285
x34=−12.5663702538165x_{34} = -12.5663702538165
x35=−37.6991117821226x_{35} = -37.6991117821226
x36=−50.2654822823858x_{36} = -50.2654822823858
x37=−56.5486674573404x_{37} = -56.5486674573404
x38=69.1150387745763x_{38} = 69.1150387745763
x39=18.8495555928333x_{39} = 18.8495555928333
x40=43.9822972235274x_{40} = 43.9822972235274
x41=87.9645937585325x_{41} = 87.9645937585325
x42=−37.6991112196296x_{42} = -37.6991112196296
x43=−25.1327398426202x_{43} = -25.1327398426202
x44=50.2654819602861x_{44} = 50.2654819602861
x45=−87.964594358366x_{45} = -87.964594358366
x46=−69.1150378741625x_{46} = -69.1150378741625
x47=−50.2654829031067x_{47} = -50.2654829031067
x48=12.566369611541x_{48} = 12.566369611541
x49=6.28317806579271x_{49} = 6.28317806579271
x50=−12.5663697046189x_{50} = -12.5663697046189
x51=−100.53096462006x_{51} = -100.53096462006
x52=−81.6814090380658x_{52} = -81.6814090380658
x53=94.2477800348352x_{53} = 94.2477800348352
x54=−31.4159259143681x_{54} = -31.4159259143681
x55=12.5663704260691x_{55} = 12.5663704260691
x56=−81.6814091069125x_{56} = -81.6814091069125
x57=31.4159268247988x_{57} = 31.4159268247988
x58=18.8495562989474x_{58} = 18.8495562989474
x59=−100.530963503519x_{59} = -100.530963503519
x60=25.1327407806025x_{60} = 25.1327407806025
x61=12.5663708959515x_{61} = 12.5663708959515
x62=−6.28318508907611x_{62} = -6.28318508907611
x63=94.2477796093521x_{63} = 94.2477796093521
x64=−75.3982231303996x_{64} = -75.3982231303996
x65=−94.2477794433926x_{65} = -94.2477794433926
x66=81.6814080870153x_{66} = 81.6814080870153
x67=−25.1327406602488x_{67} = -25.1327406602488
x68=−94.2477800819314x_{68} = -94.2477800819314
x69=−18.8495554428749x_{69} = -18.8495554428749
x70=6.28318528365757x_{70} = 6.28318528365757
x71=81.68140918653x_{71} = 81.68140918653
x72=31.415925995796x_{72} = 31.415925995796
x73=−62.8318526529414x_{73} = -62.8318526529414
x74=6.28318323792045x_{74} = 6.28318323792045
x75=75.3982231918613x_{75} = 75.3982231918613
x76=−56.5486687262309x_{76} = -56.5486687262309
x77=−62.8318534581452x_{77} = -62.8318534581452
x78=−81.6814084380207x_{78} = -81.6814084380207
x79=37.6991112480929x_{79} = 37.6991112480929
x80=6.28318254900838x_{80} = 6.28318254900838
x81=37.699112024902x_{81} = 37.699112024902
x82=62.8318542396042x_{82} = 62.8318542396042
x83=−87.9645938395231x_{83} = -87.9645938395231
x84=56.5486682245241x_{84} = 56.5486682245241
x85=50.2654824463231x_{85} = 50.2654824463231
x86=−43.9822966117866x_{86} = -43.9822966117866
x87=−113.09733805159x_{87} = -113.09733805159
x88=−37.6991118770599x_{88} = -37.6991118770599
x89=−87.9645947079771x_{89} = -87.9645947079771
x90=31.4159281957973x_{90} = 31.4159281957973
x91=43.9822971694095x_{91} = 43.9822971694095
x92=−37.6991119922907x_{92} = -37.6991119922907
x93=−37.6991134136969x_{93} = -37.6991134136969
x94=−6.28318272233945x_{94} = -6.28318272233945
x95=−75.3982224893229x_{95} = -75.3982224893229
x96=62.8318535226653x_{96} = 62.8318535226653
x97=43.982296554597x_{97} = 43.982296554597
x98=75.3982239936905x_{98} = 75.3982239936905
x99=−69.1150386771791x_{99} = -69.1150386771791
x100=75.3982254008243x_{100} = 75.3982254008243
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 - cos(x))/x^3.
1−cos⁡(0)03\frac{1 - \cos{\left(0 \right)}}{0^{3}}
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)x3−3(1−cos⁡(x))x4=0\frac{\sin{\left(x \right)}}{x^{3}} - \frac{3 \left(1 - \cos{\left(x \right)}\right)}{x^{4}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=69.1150383789755x_{1} = 69.1150383789755
x2=34.3834574736429x_{2} = 34.3834574736429
x3=−358.141562509236x_{3} = -358.141562509236
x4=53.2946120595863x_{4} = 53.2946120595863
x5=12.5663706143592x_{5} = 12.5663706143592
x6=−31.4159265358979x_{6} = -31.4159265358979
x7=−34.3834574736429x_{7} = -34.3834574736429
x8=−100.530964914873x_{8} = -100.530964914873
x9=−69.1150383789755x_{9} = -69.1150383789755
x10=172.752867742679x_{10} = 172.752867742679
x11=−25.1327412287183x_{11} = -25.1327412287183
x12=28.0613255359845x_{12} = 28.0613255359845
x13=−103.614666881545x_{13} = -103.614666881545
x14=−81.6814089933346x_{14} = -81.6814089933346
x15=97.3277443984361x_{15} = 97.3277443984361
x16=−46.9963934217376x_{16} = -46.9963934217376
x17=18.8495559215388x_{17} = 18.8495559215388
x18=94.2477796076938x_{18} = 94.2477796076938
x19=21.7165998839246x_{19} = 21.7165998839246
x20=8.76525105319464x_{20} = 8.76525105319464
x21=−91.0403059179293x_{21} = -91.0403059179293
x22=−62.8318530717959x_{22} = -62.8318530717959
x23=−97.3277443984361x_{23} = -97.3277443984361
x24=−59.5896567409223x_{24} = -59.5896567409223
x25=−84.7522366006475x_{25} = -84.7522366006475
x26=78.4633847807352x_{26} = 78.4633847807352
x27=40.6935271463195x_{27} = 40.6935271463195
x28=−56.5486677646163x_{28} = -56.5486677646163
x29=−40.6935271463195x_{29} = -40.6935271463195
x30=72.1735459082524x_{30} = 72.1735459082524
x31=−18.8495559215388x_{31} = -18.8495559215388
x32=6.28318530717959x_{32} = 6.28318530717959
x33=56.5486677646163x_{33} = 56.5486677646163
x34=87.9645943005142x_{34} = 87.9645943005142
x35=31.4159265358979x_{35} = 31.4159265358979
x36=25.1327412287183x_{36} = 25.1327412287183
x37=43.9822971502571x_{37} = 43.9822971502571
x38=−12.5663706143592x_{38} = -12.5663706143592
x39=−15.3212429040887x_{39} = -15.3212429040887
x40=−50.2654824574367x_{40} = -50.2654824574367
x41=−8.76525105319464x_{41} = -8.76525105319464
x42=65.8824372805376x_{42} = 65.8824372805376
x43=100.530964914873x_{43} = 100.530964914873
x44=59.5896567409223x_{44} = 59.5896567409223
x45=81.6814089933346x_{45} = 81.6814089933346
x46=191.605840141864x_{46} = 191.605840141864
x47=−75.398223686155x_{47} = -75.398223686155
x48=−65.8824372805376x_{48} = -65.8824372805376
x49=103.614666881545x_{49} = 103.614666881545
x50=91.0403059179293x_{50} = 91.0403059179293
x51=−87.9645943005142x_{51} = -87.9645943005142
x52=37.6991118430775x_{52} = 37.6991118430775
x53=−78.4633847807352x_{53} = -78.4633847807352
x54=−6.28318530717959x_{54} = -6.28318530717959
x55=−53.2946120595863x_{55} = -53.2946120595863
x56=84.7522366006475x_{56} = 84.7522366006475
x57=50.2654824574367x_{57} = 50.2654824574367
x58=−37.6991118430775x_{58} = -37.6991118430775
x59=15.3212429040887x_{59} = 15.3212429040887
x60=−28.0613255359845x_{60} = -28.0613255359845
x61=46.9963934217376x_{61} = 46.9963934217376
x62=62.8318530717959x_{62} = 62.8318530717959
x63=−72.1735459082524x_{63} = -72.1735459082524
x64=−94.2477796076938x_{64} = -94.2477796076938
x65=75.398223686155x_{65} = 75.398223686155
x66=−43.9822971502571x_{66} = -43.9822971502571
x67=−21.7165998839246x_{67} = -21.7165998839246
Signos de extremos en los puntos:
(69.11503837897546, 0)

(34.38345747364294, 4.88301086492743e-5)

(-358.14156250923645, 0)

(53.294612059586285, 1.3170617075317e-5)

(12.566370614359172, 0)

(-31.41592653589793, 0)

(-34.38345747364294, -4.88301086492743e-5)

(-100.53096491487338, 0)

(-69.11503837897546, 0)

(172.75286774267903, 3.87813789774886e-7)

(-25.132741228718345, 0)

(28.06132553598445, 8.9489039255268e-5)

(-103.61466688154489, -1.79639721612586e-6)

(-81.68140899333463, 0)

(97.3277443984361, 2.16724291945259e-6)

(-46.99639342173757, -1.91897937474958e-5)

(18.84955592153876, 0)

(94.2477796076938, 0)

(21.716599883924637, 0.000191621714203828)

(8.76525105319464, 0.00265844950588122)

(-91.04030591792932, -2.64763151549218e-6)

(-62.83185307179586, 0)

(-97.3277443984361, -2.16724291945259e-6)

(-59.58965674092231, -9.42796575308398e-6)

(-84.75223660064755, -3.28119978897628e-6)

(78.46338478073515, 4.13422835053337e-6)

(40.69352714631952, 2.95188908411357e-5)

(-56.548667764616276, 0)

(-40.69352714631952, -2.95188908411357e-5)

(72.17354590825245, 5.31063132429645e-6)

(-18.84955592153876, 0)

(6.283185307179586, 0)

(56.548667764616276, 0)

(87.96459430051421, 0)

(31.41592653589793, 0)

(25.132741228718345, 0)

(43.982297150257104, 0)

(-12.566370614359172, 0)

(-15.32124290408871, -0.000535560240964847)

(-50.26548245743669, 0)

(-8.76525105319464, -0.00265844950588122)

(65.88243728053762, 6.97945398544749e-6)

(100.53096491487338, 0)

(59.58965674092231, 9.42796575308398e-6)

(81.68140899333463, 0)

(191.60584014186395, 2.84247933754335e-7)

(-75.39822368615503, 0)

(-65.88243728053762, -6.97945398544749e-6)

(103.61466688154489, 1.79639721612586e-6)

(91.04030591792932, 2.64763151549218e-6)

(-87.96459430051421, 0)

(37.69911184307752, 0)

(-78.46338478073515, -4.13422835053337e-6)

(-6.283185307179586, 0)

(-53.294612059586285, -1.3170617075317e-5)

(84.75223660064755, 3.28119978897628e-6)

(50.26548245743669, 0)

(-37.69911184307752, 0)

(15.32124290408871, 0.000535560240964847)

(-28.06132553598445, -8.9489039255268e-5)

(46.99639342173757, 1.91897937474958e-5)

(62.83185307179586, 0)

(-72.17354590825245, -5.31063132429645e-6)

(-94.2477796076938, 0)

(75.39822368615503, 0)

(-43.982297150257104, 0)

(-21.716599883924637, -0.000191621714203828)


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=69.1150383789755x_{1} = 69.1150383789755
x2=12.5663706143592x_{2} = 12.5663706143592
x3=−34.3834574736429x_{3} = -34.3834574736429
x4=−103.614666881545x_{4} = -103.614666881545
x5=−46.9963934217376x_{5} = -46.9963934217376
x6=18.8495559215388x_{6} = 18.8495559215388
x7=94.2477796076938x_{7} = 94.2477796076938
x8=−91.0403059179293x_{8} = -91.0403059179293
x9=−97.3277443984361x_{9} = -97.3277443984361
x10=−59.5896567409223x_{10} = -59.5896567409223
x11=−84.7522366006475x_{11} = -84.7522366006475
x12=−40.6935271463195x_{12} = -40.6935271463195
x13=6.28318530717959x_{13} = 6.28318530717959
x14=56.5486677646163x_{14} = 56.5486677646163
x15=87.9645943005142x_{15} = 87.9645943005142
x16=31.4159265358979x_{16} = 31.4159265358979
x17=25.1327412287183x_{17} = 25.1327412287183
x18=43.9822971502571x_{18} = 43.9822971502571
x19=−15.3212429040887x_{19} = -15.3212429040887
x20=−8.76525105319464x_{20} = -8.76525105319464
x21=100.530964914873x_{21} = 100.530964914873
x22=81.6814089933346x_{22} = 81.6814089933346
x23=−65.8824372805376x_{23} = -65.8824372805376
x24=37.6991118430775x_{24} = 37.6991118430775
x25=−78.4633847807352x_{25} = -78.4633847807352
x26=−53.2946120595863x_{26} = -53.2946120595863
x27=50.2654824574367x_{27} = 50.2654824574367
x28=−28.0613255359845x_{28} = -28.0613255359845
x29=62.8318530717959x_{29} = 62.8318530717959
x30=−72.1735459082524x_{30} = -72.1735459082524
x31=75.398223686155x_{31} = 75.398223686155
x32=−21.7165998839246x_{32} = -21.7165998839246
Puntos máximos de la función:
x32=34.3834574736429x_{32} = 34.3834574736429
x32=−358.141562509236x_{32} = -358.141562509236
x32=53.2946120595863x_{32} = 53.2946120595863
x32=−31.4159265358979x_{32} = -31.4159265358979
x32=−100.530964914873x_{32} = -100.530964914873
x32=−69.1150383789755x_{32} = -69.1150383789755
x32=172.752867742679x_{32} = 172.752867742679
x32=−25.1327412287183x_{32} = -25.1327412287183
x32=28.0613255359845x_{32} = 28.0613255359845
x32=−81.6814089933346x_{32} = -81.6814089933346
x32=97.3277443984361x_{32} = 97.3277443984361
x32=21.7165998839246x_{32} = 21.7165998839246
x32=8.76525105319464x_{32} = 8.76525105319464
x32=−62.8318530717959x_{32} = -62.8318530717959
x32=78.4633847807352x_{32} = 78.4633847807352
x32=40.6935271463195x_{32} = 40.6935271463195
x32=−56.5486677646163x_{32} = -56.5486677646163
x32=72.1735459082524x_{32} = 72.1735459082524
x32=−18.8495559215388x_{32} = -18.8495559215388
x32=−12.5663706143592x_{32} = -12.5663706143592
x32=−50.2654824574367x_{32} = -50.2654824574367
x32=65.8824372805376x_{32} = 65.8824372805376
x32=59.5896567409223x_{32} = 59.5896567409223
x32=191.605840141864x_{32} = 191.605840141864
x32=−75.398223686155x_{32} = -75.398223686155
x32=103.614666881545x_{32} = 103.614666881545
x32=91.0403059179293x_{32} = 91.0403059179293
x32=−87.9645943005142x_{32} = -87.9645943005142
x32=−6.28318530717959x_{32} = -6.28318530717959
x32=84.7522366006475x_{32} = 84.7522366006475
x32=−37.6991118430775x_{32} = -37.6991118430775
x32=15.3212429040887x_{32} = 15.3212429040887
x32=46.9963934217376x_{32} = 46.9963934217376
x32=−94.2477796076938x_{32} = -94.2477796076938
x32=−43.9822971502571x_{32} = -43.9822971502571
Decrece en los intervalos
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Crece en los intervalos
(−∞,−103.614666881545]\left(-\infty, -103.614666881545\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)−6sin⁡(x)x−12(cos⁡(x)−1)x2x3=0\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−13.7629054364474x_{1} = -13.7629054364474
x2=−180.607988647138x_{2} = -180.607988647138
x3=−58.0196094603257x_{3} = -58.0196094603257
x4=26.4940342762617x_{4} = 26.4940342762617
x5=−95.757225336625x_{5} = -95.757225336625
x6=95.757225336625x_{6} = 95.757225336625
x7=13.7629054364474x_{7} = 13.7629054364474
x8=142.899890829779x_{8} = 142.899890829779
x9=64.3122518347769x_{9} = 64.3122518347769
x10=−39.1243616720552x_{10} = -39.1243616720552
x11=−80.0337727043067x_{11} = -80.0337727043067
x12=−92.6107978934392x_{12} = -92.6107978934392
x13=54.8645329044238x_{13} = 54.8645329044238
x14=−35.9521892251906x_{14} = -35.9521892251906
x15=−7.23596086240901x_{15} = -7.23596086240901
x16=−70.6032573008857x_{16} = -70.6032573008857
x17=−29.6290578288832x_{17} = -29.6290578288832
x18=−54.8645329044238x_{18} = -54.8645329044238
x19=−26.4940342762617x_{19} = -26.4940342762617
x20=98.8982740744881x_{20} = 98.8982740744881
x21=67.4526569322989x_{21} = 67.4526569322989
x22=168.039076105639x_{22} = 168.039076105639
x23=−10.3082396070295x_{23} = -10.3082396070295
x24=23.2823711761139x_{24} = 23.2823711761139
x25=80.0337727043067x_{25} = 80.0337727043067
x26=61.1597502405898x_{26} = 61.1597502405898
x27=−51.7247558065069x_{27} = -51.7247558065069
x28=16.8822581120733x_{28} = 16.8822581120733
x29=32.8149658360757x_{29} = 32.8149658360757
x30=51.7247558065069x_{30} = 51.7247558065069
x31=76.893017111242x_{31} = 76.893017111242
x32=−23.2823711761139x_{32} = -23.2823711761139
x33=−73.7438605372815x_{33} = -73.7438605372815
x34=7.23596086240901x_{34} = 7.23596086240901
x35=35.9521892251906x_{35} = 35.9521892251906
x36=−67.4526569322989x_{36} = -67.4526569322989
x37=−45.426811974001x_{37} = -45.426811974001
x38=70.6032573008857x_{38} = 70.6032573008857
x39=92.6107978934392x_{39} = 92.6107978934392
x40=−42.2628375864088x_{40} = -42.2628375864088
x41=−48.5660690324093x_{41} = -48.5660690324093
x42=29.6290578288832x_{42} = 29.6290578288832
x43=−64.3122518347769x_{43} = -64.3122518347769
x44=−76.893017111242x_{44} = -76.893017111242
x45=−32.8149658360757x_{45} = -32.8149658360757
x46=−164.897669333511x_{46} = -164.897669333511
x47=−83.1818070038329x_{47} = -83.1818070038329
x48=42.2628375864088x_{48} = 42.2628375864088
x49=73.7438605372815x_{49} = 73.7438605372815
x50=89.4698269104847x_{50} = 89.4698269104847
x51=−61.1597502405898x_{51} = -61.1597502405898
x52=58.0196094603257x_{52} = 58.0196094603257
x53=45.426811974001x_{53} = 45.426811974001
x54=39.1243616720552x_{54} = 39.1243616720552
x55=48.5660690324093x_{55} = 48.5660690324093
x56=10.3082396070295x_{56} = 10.3082396070295
x57=−16.8822581120733x_{57} = -16.8822581120733
x58=−20.1517770766287x_{58} = -20.1517770766287
x59=20.1517770766287x_{59} = 20.1517770766287
x60=−89.4698269104847x_{60} = -89.4698269104847
x61=83.1818070038329x_{61} = 83.1818070038329
x62=−86.3226822703887x_{62} = -86.3226822703887
x63=86.3226822703887x_{63} = 86.3226822703887
x64=−98.8982740744881x_{64} = -98.8982740744881
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)−6sin⁡(x)x−12(cos⁡(x)−1)x2x3)=−∞\lim_{x \to 0^-}\left(\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{3}}\right) = -\infty
lim⁡x→0+(cos⁡(x)−6sin⁡(x)x−12(cos⁡(x)−1)x2x3)=∞\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{3}}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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
[168.039076105639,∞)\left[168.039076105639, \infty\right)
Convexa en los intervalos
(−∞,−180.607988647138]\left(-\infty, -180.607988647138\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→−∞(1−cos⁡(x)x3)=0\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x^{3}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(1−cos⁡(x)x3)=0\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{x^{3}}\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 - cos(x))/x^3, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(1−cos⁡(x)xx3)=0\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x x^{3}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(1−cos⁡(x)xx3)=0\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{x 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:
1−cos⁡(x)x3=−1−cos⁡(x)x3\frac{1 - \cos{\left(x \right)}}{x^{3}} = - \frac{1 - \cos{\left(x \right)}}{x^{3}}
- No
1−cos⁡(x)x3=1−cos⁡(x)x3\frac{1 - \cos{\left(x \right)}}{x^{3}} = \frac{1 - \cos{\left(x \right)}}{x^{3}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (1-cos(x))/x^3
    © Sr Examen – Калькулятор