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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x/(x^3+2)
-
x*e^(-x)^2
-
2*x^3-15*x^2+36*x-32
-
x*(x-4)
- Límite de la función:
-
(1-cos(x))/x^3
- Integral de d{x}:
- (1-cos(x))/x^3
-
Expresiones idénticas
- (uno -cos(x))/x^ tres
- (1 menos coseno de (x)) dividir por x al cubo
- (uno menos coseno de (x)) dividir por x en el grado tres
- (1-cos(x))/x3
- 1-cosx/x3
- (1-cos(x))/x³
- (1-cos(x))/x en el grado 3
- 1-cosx/x^3
- (1-cos(x)) dividir por x^3
-
Expresiones semejantes
- (1+cos(x))/x^3
- (1-cosx)/x^3
-
Expresiones con funciones
- Coseno cos
- cos(x),x
- cos(x)-sqrt(3)*sin(x)
- cos(3x)+3cos(x)
- cos(x)*e^x
- cos(x)+sin(2*x)/2
Gráfico de la función y = (1-cos(x))/x^3
Solución
Ha introducido
[src]
1 - cos(x)
f(x) = ----------
3
x
$$f{\left(x \right)} = \frac{1 - \cos{\left(x \right)}}{x^{3}}$$
f = (1 - cos(x))/x^3
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{1 - \cos{\left(x \right)}}{x^{3}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 2 \pi$$
Solución numérica
$$x_{1} = 87.9645944825602$$
$$x_{2} = -25.132741504951$$
$$x_{3} = -43.9822971744519$$
$$x_{4} = 37.6991104084362$$
$$x_{5} = 62.8318527754349$$
$$x_{6} = 56.5486676457973$$
$$x_{7} = -100.530966103549$$
$$x_{8} = 131.946891706466$$
$$x_{9} = 94.2477791750714$$
$$x_{10} = 25.1327415885691$$
$$x_{11} = -18.8495562601109$$
$$x_{12} = 50.2654828447704$$
$$x_{13} = -94.2477795072385$$
$$x_{14} = 37.699107022324$$
$$x_{15} = -6.28317650290564$$
$$x_{16} = 188.495557225622$$
$$x_{17} = 62.8318541837957$$
$$x_{18} = 100.530964911463$$
$$x_{19} = 69.1150379711617$$
$$x_{20} = 87.9645943358115$$
$$x_{21} = -12.5663709397984$$
$$x_{22} = -18.8495567199802$$
$$x_{23} = -50.2654822302291$$
$$x_{24} = -31.4159267089767$$
$$x_{25} = -6.28318520513552$$
$$x_{26} = 100.530964758203$$
$$x_{27} = -43.9822975047697$$
$$x_{28} = -75.398223871027$$
$$x_{29} = 56.5486675979903$$
$$x_{30} = -56.5486682055439$$
$$x_{31} = 81.6814084479926$$
$$x_{32} = 18.8495556776517$$
$$x_{33} = -113.097335767285$$
$$x_{34} = -12.5663702538165$$
$$x_{35} = -37.6991117821226$$
$$x_{36} = -50.2654822823858$$
$$x_{37} = -56.5486674573404$$
$$x_{38} = 69.1150387745763$$
$$x_{39} = 18.8495555928333$$
$$x_{40} = 43.9822972235274$$
$$x_{41} = 87.9645937585325$$
$$x_{42} = -37.6991112196296$$
$$x_{43} = -25.1327398426202$$
$$x_{44} = 50.2654819602861$$
$$x_{45} = -87.964594358366$$
$$x_{46} = -69.1150378741625$$
$$x_{47} = -50.2654829031067$$
$$x_{48} = 12.566369611541$$
$$x_{49} = 6.28317806579271$$
$$x_{50} = -12.5663697046189$$
$$x_{51} = -100.53096462006$$
$$x_{52} = -81.6814090380658$$
$$x_{53} = 94.2477800348352$$
$$x_{54} = -31.4159259143681$$
$$x_{55} = 12.5663704260691$$
$$x_{56} = -81.6814091069125$$
$$x_{57} = 31.4159268247988$$
$$x_{58} = 18.8495562989474$$
$$x_{59} = -100.530963503519$$
$$x_{60} = 25.1327407806025$$
$$x_{61} = 12.5663708959515$$
$$x_{62} = -6.28318508907611$$
$$x_{63} = 94.2477796093521$$
$$x_{64} = -75.3982231303996$$
$$x_{65} = -94.2477794433926$$
$$x_{66} = 81.6814080870153$$
$$x_{67} = -25.1327406602488$$
$$x_{68} = -94.2477800819314$$
$$x_{69} = -18.8495554428749$$
$$x_{70} = 6.28318528365757$$
$$x_{71} = 81.68140918653$$
$$x_{72} = 31.415925995796$$
$$x_{73} = -62.8318526529414$$
$$x_{74} = 6.28318323792045$$
$$x_{75} = 75.3982231918613$$
$$x_{76} = -56.5486687262309$$
$$x_{77} = -62.8318534581452$$
$$x_{78} = -81.6814084380207$$
$$x_{79} = 37.6991112480929$$
$$x_{80} = 6.28318254900838$$
$$x_{81} = 37.699112024902$$
$$x_{82} = 62.8318542396042$$
$$x_{83} = -87.9645938395231$$
$$x_{84} = 56.5486682245241$$
$$x_{85} = 50.2654824463231$$
$$x_{86} = -43.9822966117866$$
$$x_{87} = -113.09733805159$$
$$x_{88} = -37.6991118770599$$
$$x_{89} = -87.9645947079771$$
$$x_{90} = 31.4159281957973$$
$$x_{91} = 43.9822971694095$$
$$x_{92} = -37.6991119922907$$
$$x_{93} = -37.6991134136969$$
$$x_{94} = -6.28318272233945$$
$$x_{95} = -75.3982224893229$$
$$x_{96} = 62.8318535226653$$
$$x_{97} = 43.982296554597$$
$$x_{98} = 75.3982239936905$$
$$x_{99} = -69.1150386771791$$
$$x_{100} = 75.3982254008243$$
o sea hay que resolver la ecuación:
$$\frac{1 - \cos{\left(x \right)}}{x^{3}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 2 \pi$$
Solución numérica
$$x_{1} = 87.9645944825602$$
$$x_{2} = -25.132741504951$$
$$x_{3} = -43.9822971744519$$
$$x_{4} = 37.6991104084362$$
$$x_{5} = 62.8318527754349$$
$$x_{6} = 56.5486676457973$$
$$x_{7} = -100.530966103549$$
$$x_{8} = 131.946891706466$$
$$x_{9} = 94.2477791750714$$
$$x_{10} = 25.1327415885691$$
$$x_{11} = -18.8495562601109$$
$$x_{12} = 50.2654828447704$$
$$x_{13} = -94.2477795072385$$
$$x_{14} = 37.699107022324$$
$$x_{15} = -6.28317650290564$$
$$x_{16} = 188.495557225622$$
$$x_{17} = 62.8318541837957$$
$$x_{18} = 100.530964911463$$
$$x_{19} = 69.1150379711617$$
$$x_{20} = 87.9645943358115$$
$$x_{21} = -12.5663709397984$$
$$x_{22} = -18.8495567199802$$
$$x_{23} = -50.2654822302291$$
$$x_{24} = -31.4159267089767$$
$$x_{25} = -6.28318520513552$$
$$x_{26} = 100.530964758203$$
$$x_{27} = -43.9822975047697$$
$$x_{28} = -75.398223871027$$
$$x_{29} = 56.5486675979903$$
$$x_{30} = -56.5486682055439$$
$$x_{31} = 81.6814084479926$$
$$x_{32} = 18.8495556776517$$
$$x_{33} = -113.097335767285$$
$$x_{34} = -12.5663702538165$$
$$x_{35} = -37.6991117821226$$
$$x_{36} = -50.2654822823858$$
$$x_{37} = -56.5486674573404$$
$$x_{38} = 69.1150387745763$$
$$x_{39} = 18.8495555928333$$
$$x_{40} = 43.9822972235274$$
$$x_{41} = 87.9645937585325$$
$$x_{42} = -37.6991112196296$$
$$x_{43} = -25.1327398426202$$
$$x_{44} = 50.2654819602861$$
$$x_{45} = -87.964594358366$$
$$x_{46} = -69.1150378741625$$
$$x_{47} = -50.2654829031067$$
$$x_{48} = 12.566369611541$$
$$x_{49} = 6.28317806579271$$
$$x_{50} = -12.5663697046189$$
$$x_{51} = -100.53096462006$$
$$x_{52} = -81.6814090380658$$
$$x_{53} = 94.2477800348352$$
$$x_{54} = -31.4159259143681$$
$$x_{55} = 12.5663704260691$$
$$x_{56} = -81.6814091069125$$
$$x_{57} = 31.4159268247988$$
$$x_{58} = 18.8495562989474$$
$$x_{59} = -100.530963503519$$
$$x_{60} = 25.1327407806025$$
$$x_{61} = 12.5663708959515$$
$$x_{62} = -6.28318508907611$$
$$x_{63} = 94.2477796093521$$
$$x_{64} = -75.3982231303996$$
$$x_{65} = -94.2477794433926$$
$$x_{66} = 81.6814080870153$$
$$x_{67} = -25.1327406602488$$
$$x_{68} = -94.2477800819314$$
$$x_{69} = -18.8495554428749$$
$$x_{70} = 6.28318528365757$$
$$x_{71} = 81.68140918653$$
$$x_{72} = 31.415925995796$$
$$x_{73} = -62.8318526529414$$
$$x_{74} = 6.28318323792045$$
$$x_{75} = 75.3982231918613$$
$$x_{76} = -56.5486687262309$$
$$x_{77} = -62.8318534581452$$
$$x_{78} = -81.6814084380207$$
$$x_{79} = 37.6991112480929$$
$$x_{80} = 6.28318254900838$$
$$x_{81} = 37.699112024902$$
$$x_{82} = 62.8318542396042$$
$$x_{83} = -87.9645938395231$$
$$x_{84} = 56.5486682245241$$
$$x_{85} = 50.2654824463231$$
$$x_{86} = -43.9822966117866$$
$$x_{87} = -113.09733805159$$
$$x_{88} = -37.6991118770599$$
$$x_{89} = -87.9645947079771$$
$$x_{90} = 31.4159281957973$$
$$x_{91} = 43.9822971694095$$
$$x_{92} = -37.6991119922907$$
$$x_{93} = -37.6991134136969$$
$$x_{94} = -6.28318272233945$$
$$x_{95} = -75.3982224893229$$
$$x_{96} = 62.8318535226653$$
$$x_{97} = 43.982296554597$$
$$x_{98} = 75.3982239936905$$
$$x_{99} = -69.1150386771791$$
$$x_{100} = 75.3982254008243$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{\sin{\left(x \right)}}{x^{3}} - \frac{3 \left(1 - \cos{\left(x \right)}\right)}{x^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 69.1150383789755$$
$$x_{2} = 34.3834574736429$$
$$x_{3} = -358.141562509236$$
$$x_{4} = 53.2946120595863$$
$$x_{5} = 12.5663706143592$$
$$x_{6} = -31.4159265358979$$
$$x_{7} = -34.3834574736429$$
$$x_{8} = -100.530964914873$$
$$x_{9} = -69.1150383789755$$
$$x_{10} = 172.752867742679$$
$$x_{11} = -25.1327412287183$$
$$x_{12} = 28.0613255359845$$
$$x_{13} = -103.614666881545$$
$$x_{14} = -81.6814089933346$$
$$x_{15} = 97.3277443984361$$
$$x_{16} = -46.9963934217376$$
$$x_{17} = 18.8495559215388$$
$$x_{18} = 94.2477796076938$$
$$x_{19} = 21.7165998839246$$
$$x_{20} = 8.76525105319464$$
$$x_{21} = -91.0403059179293$$
$$x_{22} = -62.8318530717959$$
$$x_{23} = -97.3277443984361$$
$$x_{24} = -59.5896567409223$$
$$x_{25} = -84.7522366006475$$
$$x_{26} = 78.4633847807352$$
$$x_{27} = 40.6935271463195$$
$$x_{28} = -56.5486677646163$$
$$x_{29} = -40.6935271463195$$
$$x_{30} = 72.1735459082524$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = 6.28318530717959$$
$$x_{33} = 56.5486677646163$$
$$x_{34} = 87.9645943005142$$
$$x_{35} = 31.4159265358979$$
$$x_{36} = 25.1327412287183$$
$$x_{37} = 43.9822971502571$$
$$x_{38} = -12.5663706143592$$
$$x_{39} = -15.3212429040887$$
$$x_{40} = -50.2654824574367$$
$$x_{41} = -8.76525105319464$$
$$x_{42} = 65.8824372805376$$
$$x_{43} = 100.530964914873$$
$$x_{44} = 59.5896567409223$$
$$x_{45} = 81.6814089933346$$
$$x_{46} = 191.605840141864$$
$$x_{47} = -75.398223686155$$
$$x_{48} = -65.8824372805376$$
$$x_{49} = 103.614666881545$$
$$x_{50} = 91.0403059179293$$
$$x_{51} = -87.9645943005142$$
$$x_{52} = 37.6991118430775$$
$$x_{53} = -78.4633847807352$$
$$x_{54} = -6.28318530717959$$
$$x_{55} = -53.2946120595863$$
$$x_{56} = 84.7522366006475$$
$$x_{57} = 50.2654824574367$$
$$x_{58} = -37.6991118430775$$
$$x_{59} = 15.3212429040887$$
$$x_{60} = -28.0613255359845$$
$$x_{61} = 46.9963934217376$$
$$x_{62} = 62.8318530717959$$
$$x_{63} = -72.1735459082524$$
$$x_{64} = -94.2477796076938$$
$$x_{65} = 75.398223686155$$
$$x_{66} = -43.9822971502571$$
$$x_{67} = -21.7165998839246$$
Signos de extremos en los puntos:
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:
$$x_{1} = 69.1150383789755$$
$$x_{2} = 12.5663706143592$$
$$x_{3} = -34.3834574736429$$
$$x_{4} = -103.614666881545$$
$$x_{5} = -46.9963934217376$$
$$x_{6} = 18.8495559215388$$
$$x_{7} = 94.2477796076938$$
$$x_{8} = -91.0403059179293$$
$$x_{9} = -97.3277443984361$$
$$x_{10} = -59.5896567409223$$
$$x_{11} = -84.7522366006475$$
$$x_{12} = -40.6935271463195$$
$$x_{13} = 6.28318530717959$$
$$x_{14} = 56.5486677646163$$
$$x_{15} = 87.9645943005142$$
$$x_{16} = 31.4159265358979$$
$$x_{17} = 25.1327412287183$$
$$x_{18} = 43.9822971502571$$
$$x_{19} = -15.3212429040887$$
$$x_{20} = -8.76525105319464$$
$$x_{21} = 100.530964914873$$
$$x_{22} = 81.6814089933346$$
$$x_{23} = -65.8824372805376$$
$$x_{24} = 37.6991118430775$$
$$x_{25} = -78.4633847807352$$
$$x_{26} = -53.2946120595863$$
$$x_{27} = 50.2654824574367$$
$$x_{28} = -28.0613255359845$$
$$x_{29} = 62.8318530717959$$
$$x_{30} = -72.1735459082524$$
$$x_{31} = 75.398223686155$$
$$x_{32} = -21.7165998839246$$
Puntos máximos de la función:
$$x_{32} = 34.3834574736429$$
$$x_{32} = -358.141562509236$$
$$x_{32} = 53.2946120595863$$
$$x_{32} = -31.4159265358979$$
$$x_{32} = -100.530964914873$$
$$x_{32} = -69.1150383789755$$
$$x_{32} = 172.752867742679$$
$$x_{32} = -25.1327412287183$$
$$x_{32} = 28.0613255359845$$
$$x_{32} = -81.6814089933346$$
$$x_{32} = 97.3277443984361$$
$$x_{32} = 21.7165998839246$$
$$x_{32} = 8.76525105319464$$
$$x_{32} = -62.8318530717959$$
$$x_{32} = 78.4633847807352$$
$$x_{32} = 40.6935271463195$$
$$x_{32} = -56.5486677646163$$
$$x_{32} = 72.1735459082524$$
$$x_{32} = -18.8495559215388$$
$$x_{32} = -12.5663706143592$$
$$x_{32} = -50.2654824574367$$
$$x_{32} = 65.8824372805376$$
$$x_{32} = 59.5896567409223$$
$$x_{32} = 191.605840141864$$
$$x_{32} = -75.398223686155$$
$$x_{32} = 103.614666881545$$
$$x_{32} = 91.0403059179293$$
$$x_{32} = -87.9645943005142$$
$$x_{32} = -6.28318530717959$$
$$x_{32} = 84.7522366006475$$
$$x_{32} = -37.6991118430775$$
$$x_{32} = 15.3212429040887$$
$$x_{32} = 46.9963934217376$$
$$x_{32} = -94.2477796076938$$
$$x_{32} = -43.9822971502571$$
Decrece en los intervalos
$$\left[100.530964914873, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -103.614666881545\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{\sin{\left(x \right)}}{x^{3}} - \frac{3 \left(1 - \cos{\left(x \right)}\right)}{x^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 69.1150383789755$$
$$x_{2} = 34.3834574736429$$
$$x_{3} = -358.141562509236$$
$$x_{4} = 53.2946120595863$$
$$x_{5} = 12.5663706143592$$
$$x_{6} = -31.4159265358979$$
$$x_{7} = -34.3834574736429$$
$$x_{8} = -100.530964914873$$
$$x_{9} = -69.1150383789755$$
$$x_{10} = 172.752867742679$$
$$x_{11} = -25.1327412287183$$
$$x_{12} = 28.0613255359845$$
$$x_{13} = -103.614666881545$$
$$x_{14} = -81.6814089933346$$
$$x_{15} = 97.3277443984361$$
$$x_{16} = -46.9963934217376$$
$$x_{17} = 18.8495559215388$$
$$x_{18} = 94.2477796076938$$
$$x_{19} = 21.7165998839246$$
$$x_{20} = 8.76525105319464$$
$$x_{21} = -91.0403059179293$$
$$x_{22} = -62.8318530717959$$
$$x_{23} = -97.3277443984361$$
$$x_{24} = -59.5896567409223$$
$$x_{25} = -84.7522366006475$$
$$x_{26} = 78.4633847807352$$
$$x_{27} = 40.6935271463195$$
$$x_{28} = -56.5486677646163$$
$$x_{29} = -40.6935271463195$$
$$x_{30} = 72.1735459082524$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = 6.28318530717959$$
$$x_{33} = 56.5486677646163$$
$$x_{34} = 87.9645943005142$$
$$x_{35} = 31.4159265358979$$
$$x_{36} = 25.1327412287183$$
$$x_{37} = 43.9822971502571$$
$$x_{38} = -12.5663706143592$$
$$x_{39} = -15.3212429040887$$
$$x_{40} = -50.2654824574367$$
$$x_{41} = -8.76525105319464$$
$$x_{42} = 65.8824372805376$$
$$x_{43} = 100.530964914873$$
$$x_{44} = 59.5896567409223$$
$$x_{45} = 81.6814089933346$$
$$x_{46} = 191.605840141864$$
$$x_{47} = -75.398223686155$$
$$x_{48} = -65.8824372805376$$
$$x_{49} = 103.614666881545$$
$$x_{50} = 91.0403059179293$$
$$x_{51} = -87.9645943005142$$
$$x_{52} = 37.6991118430775$$
$$x_{53} = -78.4633847807352$$
$$x_{54} = -6.28318530717959$$
$$x_{55} = -53.2946120595863$$
$$x_{56} = 84.7522366006475$$
$$x_{57} = 50.2654824574367$$
$$x_{58} = -37.6991118430775$$
$$x_{59} = 15.3212429040887$$
$$x_{60} = -28.0613255359845$$
$$x_{61} = 46.9963934217376$$
$$x_{62} = 62.8318530717959$$
$$x_{63} = -72.1735459082524$$
$$x_{64} = -94.2477796076938$$
$$x_{65} = 75.398223686155$$
$$x_{66} = -43.9822971502571$$
$$x_{67} = -21.7165998839246$$
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:
$$x_{1} = 69.1150383789755$$
$$x_{2} = 12.5663706143592$$
$$x_{3} = -34.3834574736429$$
$$x_{4} = -103.614666881545$$
$$x_{5} = -46.9963934217376$$
$$x_{6} = 18.8495559215388$$
$$x_{7} = 94.2477796076938$$
$$x_{8} = -91.0403059179293$$
$$x_{9} = -97.3277443984361$$
$$x_{10} = -59.5896567409223$$
$$x_{11} = -84.7522366006475$$
$$x_{12} = -40.6935271463195$$
$$x_{13} = 6.28318530717959$$
$$x_{14} = 56.5486677646163$$
$$x_{15} = 87.9645943005142$$
$$x_{16} = 31.4159265358979$$
$$x_{17} = 25.1327412287183$$
$$x_{18} = 43.9822971502571$$
$$x_{19} = -15.3212429040887$$
$$x_{20} = -8.76525105319464$$
$$x_{21} = 100.530964914873$$
$$x_{22} = 81.6814089933346$$
$$x_{23} = -65.8824372805376$$
$$x_{24} = 37.6991118430775$$
$$x_{25} = -78.4633847807352$$
$$x_{26} = -53.2946120595863$$
$$x_{27} = 50.2654824574367$$
$$x_{28} = -28.0613255359845$$
$$x_{29} = 62.8318530717959$$
$$x_{30} = -72.1735459082524$$
$$x_{31} = 75.398223686155$$
$$x_{32} = -21.7165998839246$$
Puntos máximos de la función:
$$x_{32} = 34.3834574736429$$
$$x_{32} = -358.141562509236$$
$$x_{32} = 53.2946120595863$$
$$x_{32} = -31.4159265358979$$
$$x_{32} = -100.530964914873$$
$$x_{32} = -69.1150383789755$$
$$x_{32} = 172.752867742679$$
$$x_{32} = -25.1327412287183$$
$$x_{32} = 28.0613255359845$$
$$x_{32} = -81.6814089933346$$
$$x_{32} = 97.3277443984361$$
$$x_{32} = 21.7165998839246$$
$$x_{32} = 8.76525105319464$$
$$x_{32} = -62.8318530717959$$
$$x_{32} = 78.4633847807352$$
$$x_{32} = 40.6935271463195$$
$$x_{32} = -56.5486677646163$$
$$x_{32} = 72.1735459082524$$
$$x_{32} = -18.8495559215388$$
$$x_{32} = -12.5663706143592$$
$$x_{32} = -50.2654824574367$$
$$x_{32} = 65.8824372805376$$
$$x_{32} = 59.5896567409223$$
$$x_{32} = 191.605840141864$$
$$x_{32} = -75.398223686155$$
$$x_{32} = 103.614666881545$$
$$x_{32} = 91.0403059179293$$
$$x_{32} = -87.9645943005142$$
$$x_{32} = -6.28318530717959$$
$$x_{32} = 84.7522366006475$$
$$x_{32} = -37.6991118430775$$
$$x_{32} = 15.3212429040887$$
$$x_{32} = 46.9963934217376$$
$$x_{32} = -94.2477796076938$$
$$x_{32} = -43.9822971502571$$
Decrece en los intervalos
$$\left[100.530964914873, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -103.614666881545\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -13.7629054364474$$
$$x_{2} = -180.607988647138$$
$$x_{3} = -58.0196094603257$$
$$x_{4} = 26.4940342762617$$
$$x_{5} = -95.757225336625$$
$$x_{6} = 95.757225336625$$
$$x_{7} = 13.7629054364474$$
$$x_{8} = 142.899890829779$$
$$x_{9} = 64.3122518347769$$
$$x_{10} = -39.1243616720552$$
$$x_{11} = -80.0337727043067$$
$$x_{12} = -92.6107978934392$$
$$x_{13} = 54.8645329044238$$
$$x_{14} = -35.9521892251906$$
$$x_{15} = -7.23596086240901$$
$$x_{16} = -70.6032573008857$$
$$x_{17} = -29.6290578288832$$
$$x_{18} = -54.8645329044238$$
$$x_{19} = -26.4940342762617$$
$$x_{20} = 98.8982740744881$$
$$x_{21} = 67.4526569322989$$
$$x_{22} = 168.039076105639$$
$$x_{23} = -10.3082396070295$$
$$x_{24} = 23.2823711761139$$
$$x_{25} = 80.0337727043067$$
$$x_{26} = 61.1597502405898$$
$$x_{27} = -51.7247558065069$$
$$x_{28} = 16.8822581120733$$
$$x_{29} = 32.8149658360757$$
$$x_{30} = 51.7247558065069$$
$$x_{31} = 76.893017111242$$
$$x_{32} = -23.2823711761139$$
$$x_{33} = -73.7438605372815$$
$$x_{34} = 7.23596086240901$$
$$x_{35} = 35.9521892251906$$
$$x_{36} = -67.4526569322989$$
$$x_{37} = -45.426811974001$$
$$x_{38} = 70.6032573008857$$
$$x_{39} = 92.6107978934392$$
$$x_{40} = -42.2628375864088$$
$$x_{41} = -48.5660690324093$$
$$x_{42} = 29.6290578288832$$
$$x_{43} = -64.3122518347769$$
$$x_{44} = -76.893017111242$$
$$x_{45} = -32.8149658360757$$
$$x_{46} = -164.897669333511$$
$$x_{47} = -83.1818070038329$$
$$x_{48} = 42.2628375864088$$
$$x_{49} = 73.7438605372815$$
$$x_{50} = 89.4698269104847$$
$$x_{51} = -61.1597502405898$$
$$x_{52} = 58.0196094603257$$
$$x_{53} = 45.426811974001$$
$$x_{54} = 39.1243616720552$$
$$x_{55} = 48.5660690324093$$
$$x_{56} = 10.3082396070295$$
$$x_{57} = -16.8822581120733$$
$$x_{58} = -20.1517770766287$$
$$x_{59} = 20.1517770766287$$
$$x_{60} = -89.4698269104847$$
$$x_{61} = 83.1818070038329$$
$$x_{62} = -86.3226822703887$$
$$x_{63} = 86.3226822703887$$
$$x_{64} = -98.8982740744881$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{3}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{3}}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = 0$$
- 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
$$\left[168.039076105639, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -180.607988647138\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -13.7629054364474$$
$$x_{2} = -180.607988647138$$
$$x_{3} = -58.0196094603257$$
$$x_{4} = 26.4940342762617$$
$$x_{5} = -95.757225336625$$
$$x_{6} = 95.757225336625$$
$$x_{7} = 13.7629054364474$$
$$x_{8} = 142.899890829779$$
$$x_{9} = 64.3122518347769$$
$$x_{10} = -39.1243616720552$$
$$x_{11} = -80.0337727043067$$
$$x_{12} = -92.6107978934392$$
$$x_{13} = 54.8645329044238$$
$$x_{14} = -35.9521892251906$$
$$x_{15} = -7.23596086240901$$
$$x_{16} = -70.6032573008857$$
$$x_{17} = -29.6290578288832$$
$$x_{18} = -54.8645329044238$$
$$x_{19} = -26.4940342762617$$
$$x_{20} = 98.8982740744881$$
$$x_{21} = 67.4526569322989$$
$$x_{22} = 168.039076105639$$
$$x_{23} = -10.3082396070295$$
$$x_{24} = 23.2823711761139$$
$$x_{25} = 80.0337727043067$$
$$x_{26} = 61.1597502405898$$
$$x_{27} = -51.7247558065069$$
$$x_{28} = 16.8822581120733$$
$$x_{29} = 32.8149658360757$$
$$x_{30} = 51.7247558065069$$
$$x_{31} = 76.893017111242$$
$$x_{32} = -23.2823711761139$$
$$x_{33} = -73.7438605372815$$
$$x_{34} = 7.23596086240901$$
$$x_{35} = 35.9521892251906$$
$$x_{36} = -67.4526569322989$$
$$x_{37} = -45.426811974001$$
$$x_{38} = 70.6032573008857$$
$$x_{39} = 92.6107978934392$$
$$x_{40} = -42.2628375864088$$
$$x_{41} = -48.5660690324093$$
$$x_{42} = 29.6290578288832$$
$$x_{43} = -64.3122518347769$$
$$x_{44} = -76.893017111242$$
$$x_{45} = -32.8149658360757$$
$$x_{46} = -164.897669333511$$
$$x_{47} = -83.1818070038329$$
$$x_{48} = 42.2628375864088$$
$$x_{49} = 73.7438605372815$$
$$x_{50} = 89.4698269104847$$
$$x_{51} = -61.1597502405898$$
$$x_{52} = 58.0196094603257$$
$$x_{53} = 45.426811974001$$
$$x_{54} = 39.1243616720552$$
$$x_{55} = 48.5660690324093$$
$$x_{56} = 10.3082396070295$$
$$x_{57} = -16.8822581120733$$
$$x_{58} = -20.1517770766287$$
$$x_{59} = 20.1517770766287$$
$$x_{60} = -89.4698269104847$$
$$x_{61} = 83.1818070038329$$
$$x_{62} = -86.3226822703887$$
$$x_{63} = 86.3226822703887$$
$$x_{64} = -98.8982740744881$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{3}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{x} - \frac{12 \left(\cos{\left(x \right)} - 1\right)}{x^{2}}}{x^{3}}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = 0$$
- 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
$$\left[168.039076105639, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -180.607988647138\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x^{3}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{x^{3}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x^{3}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{x^{3}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x x^{3}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{x x^{3}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x x^{3}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{x x^{3}}\right) = 0$$
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:
$$\frac{1 - \cos{\left(x \right)}}{x^{3}} = - \frac{1 - \cos{\left(x \right)}}{x^{3}}$$
- No
$$\frac{1 - \cos{\left(x \right)}}{x^{3}} = \frac{1 - \cos{\left(x \right)}}{x^{3}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{1 - \cos{\left(x \right)}}{x^{3}} = - \frac{1 - \cos{\left(x \right)}}{x^{3}}$$
- No
$$\frac{1 - \cos{\left(x \right)}}{x^{3}} = \frac{1 - \cos{\left(x \right)}}{x^{3}}$$
- No
es decir, función
no es
par ni impar
Gráfico