Sr Examen
Otras calculadoras
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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^4-2x^2+10
-
x^3+3*x^2-9*x
-
x^3-6*x+cos(x)+sin(x)
-
-x^3+6x^2
-
Expresiones idénticas
- ((dos *(uno -cos(t)))* dos)*(t-sin(t))
- ((2 multiplicar por (1 menos coseno de (t))) multiplicar por 2) multiplicar por (t menos seno de (t))
- ((dos multiplicar por (uno menos coseno de (t))) multiplicar por dos) multiplicar por (t menos seno de (t))
- ((2(1-cos(t)))2)(t-sin(t))
- 21-cost2t-sint
-
Expresiones semejantes
- ((2*(1-cos(t)))*2)*(t+sin(t))
- ((2*(1+cos(t)))*2)*(t-sin(t))
-
Expresiones con funciones
- Coseno cos
- cos(sqrt(x^2))
- cos(x):|cos(x)|
- cos(2*x-pi/3)*(-3)
- cos(2π*20.4*t+6.6*sin2πt)
- cos(x-pi)-2
- Seno sin
- sin(-x-(pi/3))+1
- sin(x+2)/(x^2-6*x-16)
- sin(x-pi/4)*(-2)
- sin(pi*x)/asin(x)
- sin(x)*e^((1/2)*cot(x)^(2))
Gráfico de la función y = ((2*(1-cos(t)))*2)*(t-sin(t))
Solución
Ha introducido
[src]
f(t) = 2*(1 - cos(t))*2*(t - sin(t))
$$f{\left(t \right)} = 2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right)$$
f = (2*(2*(1 - cos(t))))*(t - sin(t))
Gráfico de la función
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje T con f = 0
o sea hay que resolver la ecuación:
$$2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right) = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = 87.9645943359053$$
$$t_{2} = 43.9822971694648$$
$$t_{3} = 94.2477796093523$$
$$t_{4} = 50.2654824463392$$
$$t_{5} = 12.5663704429179$$
$$t_{6} = -81.6814090382281$$
$$t_{7} = 6.2831852842096$$
$$t_{8} = -37.6991118772636$$
$$t_{9} = 0.00293944616148423$$
$$t_{10} = 87.9645942831369$$
$$t_{11} = 0$$
$$t_{12} = -0.00394279252748134$$
$$t_{13} = -31.4159265504176$$
$$t_{14} = -6.28318512814589$$
$$t_{15} = -43.9822971745393$$
$$t_{16} = -87.9645943586162$$
$$t_{17} = -75.3982236369254$$
$$t_{18} = 37.6991118863561$$
o sea hay que resolver la ecuación:
$$2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right) = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = 87.9645943359053$$
$$t_{2} = 43.9822971694648$$
$$t_{3} = 94.2477796093523$$
$$t_{4} = 50.2654824463392$$
$$t_{5} = 12.5663704429179$$
$$t_{6} = -81.6814090382281$$
$$t_{7} = 6.2831852842096$$
$$t_{8} = -37.6991118772636$$
$$t_{9} = 0.00293944616148423$$
$$t_{10} = 87.9645942831369$$
$$t_{11} = 0$$
$$t_{12} = -0.00394279252748134$$
$$t_{13} = -31.4159265504176$$
$$t_{14} = -6.28318512814589$$
$$t_{15} = -43.9822971745393$$
$$t_{16} = -87.9645943586162$$
$$t_{17} = -75.3982236369254$$
$$t_{18} = 37.6991118863561$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d t} f{\left(t \right)} = $$
primera derivada
$$4 \left(1 - \cos{\left(t \right)}\right)^{2} + 4 \left(t - \sin{\left(t \right)}\right) \sin{\left(t \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 69.1150383789755$$
$$t_{2} = 59.7570242163071$$
$$t_{3} = -78.5906363373572$$
$$t_{4} = 91.150021477956$$
$$t_{5} = 34.672005585407$$
$$t_{6} = -97.4303869376198$$
$$t_{7} = 12.5663706143592$$
$$t_{8} = -31.4159265358979$$
$$t_{9} = -69.1150383789755$$
$$t_{10} = -100.530964914873$$
$$t_{11} = 97.4303869376198$$
$$t_{12} = -66.0338916411917$$
$$t_{13} = -47.208269164396$$
$$t_{14} = -25.1327412287183$$
$$t_{15} = -3.87436681728653$$
$$t_{16} = 84.8700716191843$$
$$t_{17} = -81.6814089933346$$
$$t_{18} = 66.0338916411917$$
$$t_{19} = -34.672005585407$$
$$t_{20} = 72.3118486603673$$
$$t_{21} = 53.4816248697994$$
$$t_{22} = 18.8495559215388$$
$$t_{23} = 78.5906363373572$$
$$t_{24} = 94.2477796076938$$
$$t_{25} = -28.4135252130045$$
$$t_{26} = 0$$
$$t_{27} = 22.1682978517915$$
$$t_{28} = -84.8700716191843$$
$$t_{29} = -62.8318530717959$$
$$t_{30} = -40.9378760535004$$
$$t_{31} = 28.4135252130045$$
$$t_{32} = -56.5486677646163$$
$$t_{33} = 47.208269164396$$
$$t_{34} = -18.8495559215388$$
$$t_{35} = 6.28318530717959$$
$$t_{36} = 56.5486677646163$$
$$t_{37} = 87.9645943005142$$
$$t_{38} = 31.4159265358979$$
$$t_{39} = 25.1327412287183$$
$$t_{40} = 43.9822971502571$$
$$t_{41} = -22.1682978517915$$
$$t_{42} = -12.5663706143592$$
$$t_{43} = 100.530964914873$$
$$t_{44} = -50.2654824574367$$
$$t_{45} = -72.3118486603673$$
$$t_{46} = 40.9378760535004$$
$$t_{47} = 3.87436681728653$$
$$t_{48} = 81.6814089933346$$
$$t_{49} = -91.150021477956$$
$$t_{50} = -53.4816248697994$$
$$t_{51} = -75.398223686155$$
$$t_{52} = -87.9645943005142$$
$$t_{53} = 15.9502279739122$$
$$t_{54} = 9.80006416821608$$
$$t_{55} = 37.6991118430775$$
$$t_{56} = -59.7570242163071$$
$$t_{57} = -6.28318530717959$$
$$t_{58} = 50.2654824574367$$
$$t_{59} = -37.6991118430775$$
$$t_{60} = -43.9822971502571$$
$$t_{61} = 62.8318530717959$$
$$t_{62} = -9.80006416821608$$
$$t_{63} = -94.2477796076938$$
$$t_{64} = 75.398223686155$$
$$t_{65} = -15.9502279739122$$
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:
$$t_{1} = 69.1150383789755$$
$$t_{2} = -78.5906363373572$$
$$t_{3} = -97.4303869376198$$
$$t_{4} = 12.5663706143592$$
$$t_{5} = -66.0338916411917$$
$$t_{6} = -47.208269164396$$
$$t_{7} = -3.87436681728653$$
$$t_{8} = -34.672005585407$$
$$t_{9} = 18.8495559215388$$
$$t_{10} = 94.2477796076938$$
$$t_{11} = -28.4135252130045$$
$$t_{12} = -84.8700716191843$$
$$t_{13} = -40.9378760535004$$
$$t_{14} = 6.28318530717959$$
$$t_{15} = 56.5486677646163$$
$$t_{16} = 87.9645943005142$$
$$t_{17} = 31.4159265358979$$
$$t_{18} = 25.1327412287183$$
$$t_{19} = 43.9822971502571$$
$$t_{20} = -22.1682978517915$$
$$t_{21} = 100.530964914873$$
$$t_{22} = -72.3118486603673$$
$$t_{23} = 81.6814089933346$$
$$t_{24} = -91.150021477956$$
$$t_{25} = -53.4816248697994$$
$$t_{26} = 37.6991118430775$$
$$t_{27} = -59.7570242163071$$
$$t_{28} = 50.2654824574367$$
$$t_{29} = 62.8318530717959$$
$$t_{30} = -9.80006416821608$$
$$t_{31} = 75.398223686155$$
$$t_{32} = -15.9502279739122$$
Puntos máximos de la función:
$$t_{32} = 59.7570242163071$$
$$t_{32} = 91.150021477956$$
$$t_{32} = 34.672005585407$$
$$t_{32} = -31.4159265358979$$
$$t_{32} = -69.1150383789755$$
$$t_{32} = -100.530964914873$$
$$t_{32} = 97.4303869376198$$
$$t_{32} = -25.1327412287183$$
$$t_{32} = 84.8700716191843$$
$$t_{32} = -81.6814089933346$$
$$t_{32} = 66.0338916411917$$
$$t_{32} = 72.3118486603673$$
$$t_{32} = 53.4816248697994$$
$$t_{32} = 78.5906363373572$$
$$t_{32} = 22.1682978517915$$
$$t_{32} = -62.8318530717959$$
$$t_{32} = 28.4135252130045$$
$$t_{32} = -56.5486677646163$$
$$t_{32} = 47.208269164396$$
$$t_{32} = -18.8495559215388$$
$$t_{32} = -12.5663706143592$$
$$t_{32} = -50.2654824574367$$
$$t_{32} = 40.9378760535004$$
$$t_{32} = 3.87436681728653$$
$$t_{32} = -75.398223686155$$
$$t_{32} = -87.9645943005142$$
$$t_{32} = 15.9502279739122$$
$$t_{32} = 9.80006416821608$$
$$t_{32} = -6.28318530717959$$
$$t_{32} = -37.6991118430775$$
$$t_{32} = -43.9822971502571$$
$$t_{32} = -94.2477796076938$$
Decrece en los intervalos
$$\left[100.530964914873, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.4303869376198\right]$$
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d t} f{\left(t \right)} = $$
primera derivada
$$4 \left(1 - \cos{\left(t \right)}\right)^{2} + 4 \left(t - \sin{\left(t \right)}\right) \sin{\left(t \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 69.1150383789755$$
$$t_{2} = 59.7570242163071$$
$$t_{3} = -78.5906363373572$$
$$t_{4} = 91.150021477956$$
$$t_{5} = 34.672005585407$$
$$t_{6} = -97.4303869376198$$
$$t_{7} = 12.5663706143592$$
$$t_{8} = -31.4159265358979$$
$$t_{9} = -69.1150383789755$$
$$t_{10} = -100.530964914873$$
$$t_{11} = 97.4303869376198$$
$$t_{12} = -66.0338916411917$$
$$t_{13} = -47.208269164396$$
$$t_{14} = -25.1327412287183$$
$$t_{15} = -3.87436681728653$$
$$t_{16} = 84.8700716191843$$
$$t_{17} = -81.6814089933346$$
$$t_{18} = 66.0338916411917$$
$$t_{19} = -34.672005585407$$
$$t_{20} = 72.3118486603673$$
$$t_{21} = 53.4816248697994$$
$$t_{22} = 18.8495559215388$$
$$t_{23} = 78.5906363373572$$
$$t_{24} = 94.2477796076938$$
$$t_{25} = -28.4135252130045$$
$$t_{26} = 0$$
$$t_{27} = 22.1682978517915$$
$$t_{28} = -84.8700716191843$$
$$t_{29} = -62.8318530717959$$
$$t_{30} = -40.9378760535004$$
$$t_{31} = 28.4135252130045$$
$$t_{32} = -56.5486677646163$$
$$t_{33} = 47.208269164396$$
$$t_{34} = -18.8495559215388$$
$$t_{35} = 6.28318530717959$$
$$t_{36} = 56.5486677646163$$
$$t_{37} = 87.9645943005142$$
$$t_{38} = 31.4159265358979$$
$$t_{39} = 25.1327412287183$$
$$t_{40} = 43.9822971502571$$
$$t_{41} = -22.1682978517915$$
$$t_{42} = -12.5663706143592$$
$$t_{43} = 100.530964914873$$
$$t_{44} = -50.2654824574367$$
$$t_{45} = -72.3118486603673$$
$$t_{46} = 40.9378760535004$$
$$t_{47} = 3.87436681728653$$
$$t_{48} = 81.6814089933346$$
$$t_{49} = -91.150021477956$$
$$t_{50} = -53.4816248697994$$
$$t_{51} = -75.398223686155$$
$$t_{52} = -87.9645943005142$$
$$t_{53} = 15.9502279739122$$
$$t_{54} = 9.80006416821608$$
$$t_{55} = 37.6991118430775$$
$$t_{56} = -59.7570242163071$$
$$t_{57} = -6.28318530717959$$
$$t_{58} = 50.2654824574367$$
$$t_{59} = -37.6991118430775$$
$$t_{60} = -43.9822971502571$$
$$t_{61} = 62.8318530717959$$
$$t_{62} = -9.80006416821608$$
$$t_{63} = -94.2477796076938$$
$$t_{64} = 75.398223686155$$
$$t_{65} = -15.9502279739122$$
Signos de extremos en los puntos:
(69.11503837897546, 0)
(59.757024216307144, 478.056788254046)
(-78.5906363373572, -628.725353032197)
(91.15002147795605, 729.200340195794)
(34.672005585407014, 277.379036039187)
(-97.43038693761983, -779.443233433015)
(12.566370614359172, 0)
(-31.41592653589793, 0)
(-69.11503837897546, 0)
(-100.53096491487338, 0)
(97.43038693761983, 779.443233433015)
(-66.03389164119173, -528.271574429759)
(-47.20826916439604, -377.667352719234)
(-25.132741228718345, 0)
(-3.874366817286529, -31.6817386662676)
(84.87007161918429, 678.960781412765)
(-81.68140899333463, 0)
(66.03389164119173, 528.271574429759)
(-34.672005585407014, -277.379036039187)
(72.31184866036726, 578.495125741951)
(53.4816248697994, 427.8538264531)
(18.84955592153876, 0)
(78.5906363373572, 628.725353032197)
(94.2477796076938, 0)
(-28.413525213004526, -227.313569079866)
(0, 0)
(22.16829785179154, 177.357414407187)
(-84.87007161918429, -678.960781412765)
(-62.83185307179586, 0)
(-40.937876053500375, -327.504839148526)
(28.413525213004526, 227.313569079866)
(-56.548667764616276, 0)
(47.20826916439604, 377.667352719234)
(-18.84955592153876, 0)
(6.283185307179586, 0)
(56.548667764616276, 0)
(87.96459430051421, 0)
(31.41592653589793, 0)
(25.132741228718345, 0)
(43.982297150257104, 0)
(-22.16829785179154, -177.357414407187)
(-12.566370614359172, 0)
(100.53096491487338, 0)
(-50.26548245743669, 0)
(-72.31184866036726, -578.495125741951)
(40.937876053500375, 327.504839148526)
(3.874366817286529, 31.6817386662676)
(81.68140899333463, 0)
(-91.15002147795605, -729.200340195794)
(-53.4816248697994, -427.8538264531)
(-75.39822368615503, 0)
(-87.96459430051421, 0)
(15.950227973912167, 127.629847050074)
(9.800064168216084, 78.5025537190268)
(37.69911184307752, 0)
(-59.757024216307144, -478.056788254046)
(-6.283185307179586, 0)
(50.26548245743669, 0)
(-37.69911184307752, 0)
(-43.982297150257104, 0)
(62.83185307179586, 0)
(-9.800064168216084, -78.5025537190268)
(-94.2477796076938, 0)
(75.39822368615503, 0)
(-15.950227973912167, -127.629847050074)
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:
$$t_{1} = 69.1150383789755$$
$$t_{2} = -78.5906363373572$$
$$t_{3} = -97.4303869376198$$
$$t_{4} = 12.5663706143592$$
$$t_{5} = -66.0338916411917$$
$$t_{6} = -47.208269164396$$
$$t_{7} = -3.87436681728653$$
$$t_{8} = -34.672005585407$$
$$t_{9} = 18.8495559215388$$
$$t_{10} = 94.2477796076938$$
$$t_{11} = -28.4135252130045$$
$$t_{12} = -84.8700716191843$$
$$t_{13} = -40.9378760535004$$
$$t_{14} = 6.28318530717959$$
$$t_{15} = 56.5486677646163$$
$$t_{16} = 87.9645943005142$$
$$t_{17} = 31.4159265358979$$
$$t_{18} = 25.1327412287183$$
$$t_{19} = 43.9822971502571$$
$$t_{20} = -22.1682978517915$$
$$t_{21} = 100.530964914873$$
$$t_{22} = -72.3118486603673$$
$$t_{23} = 81.6814089933346$$
$$t_{24} = -91.150021477956$$
$$t_{25} = -53.4816248697994$$
$$t_{26} = 37.6991118430775$$
$$t_{27} = -59.7570242163071$$
$$t_{28} = 50.2654824574367$$
$$t_{29} = 62.8318530717959$$
$$t_{30} = -9.80006416821608$$
$$t_{31} = 75.398223686155$$
$$t_{32} = -15.9502279739122$$
Puntos máximos de la función:
$$t_{32} = 59.7570242163071$$
$$t_{32} = 91.150021477956$$
$$t_{32} = 34.672005585407$$
$$t_{32} = -31.4159265358979$$
$$t_{32} = -69.1150383789755$$
$$t_{32} = -100.530964914873$$
$$t_{32} = 97.4303869376198$$
$$t_{32} = -25.1327412287183$$
$$t_{32} = 84.8700716191843$$
$$t_{32} = -81.6814089933346$$
$$t_{32} = 66.0338916411917$$
$$t_{32} = 72.3118486603673$$
$$t_{32} = 53.4816248697994$$
$$t_{32} = 78.5906363373572$$
$$t_{32} = 22.1682978517915$$
$$t_{32} = -62.8318530717959$$
$$t_{32} = 28.4135252130045$$
$$t_{32} = -56.5486677646163$$
$$t_{32} = 47.208269164396$$
$$t_{32} = -18.8495559215388$$
$$t_{32} = -12.5663706143592$$
$$t_{32} = -50.2654824574367$$
$$t_{32} = 40.9378760535004$$
$$t_{32} = 3.87436681728653$$
$$t_{32} = -75.398223686155$$
$$t_{32} = -87.9645943005142$$
$$t_{32} = 15.9502279739122$$
$$t_{32} = 9.80006416821608$$
$$t_{32} = -6.28318530717959$$
$$t_{32} = -37.6991118430775$$
$$t_{32} = -43.9822971502571$$
$$t_{32} = -94.2477796076938$$
Decrece en los intervalos
$$\left[100.530964914873, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.4303869376198\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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:
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = $$
segunda derivada
$$4 \left(\left(t - \sin{\left(t \right)}\right) \cos{\left(t \right)} - 3 \left(\cos{\left(t \right)} - 1\right) \sin{\left(t \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -20.5984961227107$$
$$t_{2} = -23.6700340653395$$
$$t_{3} = -55.0286549509357$$
$$t_{4} = -45.6250232943739$$
$$t_{5} = 58.1747776805578$$
$$t_{6} = 77.0100852297287$$
$$t_{7} = 92.7079951723293$$
$$t_{8} = 64.4522306554298$$
$$t_{9} = 51.8988189202009$$
$$t_{10} = -64.4522306554298$$
$$t_{11} = 55.0286549509357$$
$$t_{12} = 42.4759723749004$$
$$t_{13} = 39.3545260587032$$
$$t_{14} = 5.03999883727742$$
$$t_{15} = -73.8659375362638$$
$$t_{16} = -51.8988189202009$$
$$t_{17} = 2.75187606274332$$
$$t_{18} = 29.9333497797328$$
$$t_{19} = -86.4269625619413$$
$$t_{20} = -58.1747776805578$$
$$t_{21} = 17.4181715623283$$
$$t_{22} = 89.570434107269$$
$$t_{23} = -36.2028195898461$$
$$t_{24} = 23.6700340653395$$
$$t_{25} = -48.7515022934728$$
$$t_{26} = 14.4138179153909$$
$$t_{27} = -89.570434107269$$
$$t_{28} = 8.39688009979201$$
$$t_{29} = -33.0894159738636$$
$$t_{30} = 20.5984961227107$$
$$t_{31} = 0$$
$$t_{32} = 70.7307585212674$$
$$t_{33} = -14.4138179153909$$
$$t_{34} = -26.8339805376221$$
$$t_{35} = -70.7307585212674$$
$$t_{36} = -11.1915066200626$$
$$t_{37} = -2.75187606274332$$
$$t_{38} = -83.2900217007128$$
$$t_{39} = -61.3069643399117$$
$$t_{40} = -95.8512250837506$$
$$t_{41} = -92.7079951723293$$
$$t_{42} = -5.03999883727742$$
$$t_{43} = 95.8512250837506$$
$$t_{44} = -102.132322109174$$
$$t_{45} = 73.8659375362638$$
$$t_{46} = 26.8339805376221$$
$$t_{47} = -67.5861271560931$$
$$t_{48} = -77.0100852297287$$
$$t_{49} = 67.5861271560931$$
$$t_{50} = 36.2028195898461$$
$$t_{51} = 83.2900217007128$$
$$t_{52} = 80.1462508595454$$
$$t_{53} = -42.4759723749004$$
$$t_{54} = 48.7515022934728$$
$$t_{55} = -80.1462508595454$$
$$t_{56} = -8.39688009979201$$
$$t_{57} = -98.9892900751667$$
$$t_{58} = -111.552495900049$$
$$t_{59} = 33.0894159738636$$
$$t_{60} = 11.1915066200626$$
$$t_{61} = 86.4269625619413$$
$$t_{62} = 61.3069643399117$$
$$t_{63} = -39.3545260587032$$
$$t_{64} = -17.4181715623283$$
$$t_{65} = 45.6250232943739$$
$$t_{66} = -29.9333497797328$$
$$t_{67} = 98.9892900751667$$
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
$$\left[98.9892900751667, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -111.552495900049\right]$$
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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:
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = $$
segunda derivada
$$4 \left(\left(t - \sin{\left(t \right)}\right) \cos{\left(t \right)} - 3 \left(\cos{\left(t \right)} - 1\right) \sin{\left(t \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -20.5984961227107$$
$$t_{2} = -23.6700340653395$$
$$t_{3} = -55.0286549509357$$
$$t_{4} = -45.6250232943739$$
$$t_{5} = 58.1747776805578$$
$$t_{6} = 77.0100852297287$$
$$t_{7} = 92.7079951723293$$
$$t_{8} = 64.4522306554298$$
$$t_{9} = 51.8988189202009$$
$$t_{10} = -64.4522306554298$$
$$t_{11} = 55.0286549509357$$
$$t_{12} = 42.4759723749004$$
$$t_{13} = 39.3545260587032$$
$$t_{14} = 5.03999883727742$$
$$t_{15} = -73.8659375362638$$
$$t_{16} = -51.8988189202009$$
$$t_{17} = 2.75187606274332$$
$$t_{18} = 29.9333497797328$$
$$t_{19} = -86.4269625619413$$
$$t_{20} = -58.1747776805578$$
$$t_{21} = 17.4181715623283$$
$$t_{22} = 89.570434107269$$
$$t_{23} = -36.2028195898461$$
$$t_{24} = 23.6700340653395$$
$$t_{25} = -48.7515022934728$$
$$t_{26} = 14.4138179153909$$
$$t_{27} = -89.570434107269$$
$$t_{28} = 8.39688009979201$$
$$t_{29} = -33.0894159738636$$
$$t_{30} = 20.5984961227107$$
$$t_{31} = 0$$
$$t_{32} = 70.7307585212674$$
$$t_{33} = -14.4138179153909$$
$$t_{34} = -26.8339805376221$$
$$t_{35} = -70.7307585212674$$
$$t_{36} = -11.1915066200626$$
$$t_{37} = -2.75187606274332$$
$$t_{38} = -83.2900217007128$$
$$t_{39} = -61.3069643399117$$
$$t_{40} = -95.8512250837506$$
$$t_{41} = -92.7079951723293$$
$$t_{42} = -5.03999883727742$$
$$t_{43} = 95.8512250837506$$
$$t_{44} = -102.132322109174$$
$$t_{45} = 73.8659375362638$$
$$t_{46} = 26.8339805376221$$
$$t_{47} = -67.5861271560931$$
$$t_{48} = -77.0100852297287$$
$$t_{49} = 67.5861271560931$$
$$t_{50} = 36.2028195898461$$
$$t_{51} = 83.2900217007128$$
$$t_{52} = 80.1462508595454$$
$$t_{53} = -42.4759723749004$$
$$t_{54} = 48.7515022934728$$
$$t_{55} = -80.1462508595454$$
$$t_{56} = -8.39688009979201$$
$$t_{57} = -98.9892900751667$$
$$t_{58} = -111.552495900049$$
$$t_{59} = 33.0894159738636$$
$$t_{60} = 11.1915066200626$$
$$t_{61} = 86.4269625619413$$
$$t_{62} = 61.3069643399117$$
$$t_{63} = -39.3545260587032$$
$$t_{64} = -17.4181715623283$$
$$t_{65} = 45.6250232943739$$
$$t_{66} = -29.9333497797328$$
$$t_{67} = 98.9892900751667$$
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
$$\left[98.9892900751667, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -111.552495900049\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con t->+oo y t->-oo
$$\lim_{t \to -\infty}\left(2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right)\right) = \left\langle -\infty, 0\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, 0\right\rangle$$
$$\lim_{t \to \infty}\left(2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right)\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle 0, \infty\right\rangle$$
$$\lim_{t \to -\infty}\left(2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right)\right) = \left\langle -\infty, 0\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, 0\right\rangle$$
$$\lim_{t \to \infty}\left(2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right)\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle 0, \infty\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((2*(1 - cos(t)))*2)*(t - sin(t)), dividida por t con t->+oo y t ->-oo
$$\lim_{t \to -\infty}\left(\frac{4 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right)}{t}\right) = \left\langle 0, 8\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle 0, 8\right\rangle t$$
$$\lim_{t \to \infty}\left(\frac{4 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right)}{t}\right) = \left\langle 0, 8\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle 0, 8\right\rangle t$$
$$\lim_{t \to -\infty}\left(\frac{4 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right)}{t}\right) = \left\langle 0, 8\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle 0, 8\right\rangle t$$
$$\lim_{t \to \infty}\left(\frac{4 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right)}{t}\right) = \left\langle 0, 8\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle 0, 8\right\rangle t$$
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-t) и f = -f(-t).
Pues, comprobamos:
$$2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right) = 4 \left(1 - \cos{\left(t \right)}\right) \left(- t + \sin{\left(t \right)}\right)$$
- No
$$2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right) = - 4 \left(1 - \cos{\left(t \right)}\right) \left(- t + \sin{\left(t \right)}\right)$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right) = 4 \left(1 - \cos{\left(t \right)}\right) \left(- t + \sin{\left(t \right)}\right)$$
- No
$$2 \cdot 2 \left(1 - \cos{\left(t \right)}\right) \left(t - \sin{\left(t \right)}\right) = - 4 \left(1 - \cos{\left(t \right)}\right) \left(- t + \sin{\left(t \right)}\right)$$
- No
es decir, función
no es
par ni impar