Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- 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
- cos(pi*t)/t^ dos
- coseno de ( número pi multiplicar por t) dividir por t al cuadrado
- coseno de ( número pi multiplicar por t) dividir por t en el grado dos
- cos(pi*t)/t2
- cospi*t/t2
- cos(pi*t)/t²
- cos(pi*t)/t en el grado 2
- cos(pit)/t^2
- cos(pit)/t2
- cospit/t2
- cospit/t^2
- cos(pi*t) dividir por t^2
-
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
Gráfico de la función y = cos(pi*t)/t^2
Solución
Ha introducido
[src]
cos(pi*t)
f(t) = ---------
2
t
$$f{\left(t \right)} = \frac{\cos{\left(\pi t \right)}}{t^{2}}$$
f = cos(pi*t)/t^2
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$t_{1} = 0$$
$$t_{1} = 0$$
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:
$$\frac{\cos{\left(\pi t \right)}}{t^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución analítica
$$t_{1} = \frac{1}{2}$$
$$t_{2} = \frac{3}{2}$$
Solución numérica
$$t_{1} = 100.5$$
$$t_{2} = 92.5$$
$$t_{3} = -61.5$$
$$t_{4} = 20.5$$
$$t_{5} = 70.5$$
$$t_{6} = 42.5$$
$$t_{7} = -51.5$$
$$t_{8} = -5.5$$
$$t_{9} = 74.5$$
$$t_{10} = -87.5$$
$$t_{11} = 52.5$$
$$t_{12} = 66.5$$
$$t_{13} = -65.5$$
$$t_{14} = -25.5$$
$$t_{15} = 4.5$$
$$t_{16} = -3.5$$
$$t_{17} = 26.5$$
$$t_{18} = -81.5$$
$$t_{19} = -45.5$$
$$t_{20} = 56.5$$
$$t_{21} = 94.5$$
$$t_{22} = -97.5$$
$$t_{23} = 24.5$$
$$t_{24} = -19.5$$
$$t_{25} = 34.5$$
$$t_{26} = -31.5$$
$$t_{27} = -33.5$$
$$t_{28} = 50.5$$
$$t_{29} = 32.5$$
$$t_{30} = 10.5$$
$$t_{31} = 2.5$$
$$t_{32} = 90.5$$
$$t_{33} = 62.5$$
$$t_{34} = -21.5$$
$$t_{35} = -83.5$$
$$t_{36} = -47.5$$
$$t_{37} = -23.5$$
$$t_{38} = -27.5$$
$$t_{39} = -89.5$$
$$t_{40} = 36.5$$
$$t_{41} = 12.5$$
$$t_{42} = -77.5$$
$$t_{43} = 86.5$$
$$t_{44} = -7.5$$
$$t_{45} = 60.5$$
$$t_{46} = -29.5$$
$$t_{47} = -13.5$$
$$t_{48} = 80.5$$
$$t_{49} = 78.5$$
$$t_{50} = -17.5$$
$$t_{51} = 64.5$$
$$t_{52} = 96.5$$
$$t_{53} = -11.5$$
$$t_{54} = -55.5$$
$$t_{55} = 54.5$$
$$t_{56} = 98.5$$
$$t_{57} = -99.5$$
$$t_{58} = 22.5$$
$$t_{59} = -57.5$$
$$t_{60} = 6.5$$
$$t_{61} = -79.5$$
$$t_{62} = -53.5$$
$$t_{63} = -37.5$$
$$t_{64} = 84.5$$
$$t_{65} = -93.5$$
$$t_{66} = -15.5$$
$$t_{67} = -91.5$$
$$t_{68} = 46.5$$
$$t_{69} = -71.5$$
$$t_{70} = 44.5$$
$$t_{71} = -49.5$$
$$t_{72} = 88.5$$
$$t_{73} = -59.5$$
$$t_{74} = -39.5$$
$$t_{75} = -73.5$$
$$t_{76} = 82.5$$
$$t_{77} = 40.5$$
$$t_{78} = 58.5$$
$$t_{79} = 14.5$$
$$t_{80} = -85.5$$
$$t_{81} = 76.5$$
$$t_{82} = -63.5$$
$$t_{83} = -69.5$$
$$t_{84} = -67.5$$
$$t_{85} = -9.5$$
$$t_{86} = 30.5$$
$$t_{87} = 18.5$$
$$t_{88} = 38.5$$
$$t_{89} = -95.5$$
$$t_{90} = -41.5$$
$$t_{91} = 72.5$$
$$t_{92} = 48.5$$
$$t_{93} = -43.5$$
$$t_{94} = -35.5$$
$$t_{95} = 16.5$$
$$t_{96} = -75.5$$
$$t_{97} = 68.5$$
$$t_{98} = 8.5$$
o sea hay que resolver la ecuación:
$$\frac{\cos{\left(\pi t \right)}}{t^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución analítica
$$t_{1} = \frac{1}{2}$$
$$t_{2} = \frac{3}{2}$$
Solución numérica
$$t_{1} = 100.5$$
$$t_{2} = 92.5$$
$$t_{3} = -61.5$$
$$t_{4} = 20.5$$
$$t_{5} = 70.5$$
$$t_{6} = 42.5$$
$$t_{7} = -51.5$$
$$t_{8} = -5.5$$
$$t_{9} = 74.5$$
$$t_{10} = -87.5$$
$$t_{11} = 52.5$$
$$t_{12} = 66.5$$
$$t_{13} = -65.5$$
$$t_{14} = -25.5$$
$$t_{15} = 4.5$$
$$t_{16} = -3.5$$
$$t_{17} = 26.5$$
$$t_{18} = -81.5$$
$$t_{19} = -45.5$$
$$t_{20} = 56.5$$
$$t_{21} = 94.5$$
$$t_{22} = -97.5$$
$$t_{23} = 24.5$$
$$t_{24} = -19.5$$
$$t_{25} = 34.5$$
$$t_{26} = -31.5$$
$$t_{27} = -33.5$$
$$t_{28} = 50.5$$
$$t_{29} = 32.5$$
$$t_{30} = 10.5$$
$$t_{31} = 2.5$$
$$t_{32} = 90.5$$
$$t_{33} = 62.5$$
$$t_{34} = -21.5$$
$$t_{35} = -83.5$$
$$t_{36} = -47.5$$
$$t_{37} = -23.5$$
$$t_{38} = -27.5$$
$$t_{39} = -89.5$$
$$t_{40} = 36.5$$
$$t_{41} = 12.5$$
$$t_{42} = -77.5$$
$$t_{43} = 86.5$$
$$t_{44} = -7.5$$
$$t_{45} = 60.5$$
$$t_{46} = -29.5$$
$$t_{47} = -13.5$$
$$t_{48} = 80.5$$
$$t_{49} = 78.5$$
$$t_{50} = -17.5$$
$$t_{51} = 64.5$$
$$t_{52} = 96.5$$
$$t_{53} = -11.5$$
$$t_{54} = -55.5$$
$$t_{55} = 54.5$$
$$t_{56} = 98.5$$
$$t_{57} = -99.5$$
$$t_{58} = 22.5$$
$$t_{59} = -57.5$$
$$t_{60} = 6.5$$
$$t_{61} = -79.5$$
$$t_{62} = -53.5$$
$$t_{63} = -37.5$$
$$t_{64} = 84.5$$
$$t_{65} = -93.5$$
$$t_{66} = -15.5$$
$$t_{67} = -91.5$$
$$t_{68} = 46.5$$
$$t_{69} = -71.5$$
$$t_{70} = 44.5$$
$$t_{71} = -49.5$$
$$t_{72} = 88.5$$
$$t_{73} = -59.5$$
$$t_{74} = -39.5$$
$$t_{75} = -73.5$$
$$t_{76} = 82.5$$
$$t_{77} = 40.5$$
$$t_{78} = 58.5$$
$$t_{79} = 14.5$$
$$t_{80} = -85.5$$
$$t_{81} = 76.5$$
$$t_{82} = -63.5$$
$$t_{83} = -69.5$$
$$t_{84} = -67.5$$
$$t_{85} = -9.5$$
$$t_{86} = 30.5$$
$$t_{87} = 18.5$$
$$t_{88} = 38.5$$
$$t_{89} = -95.5$$
$$t_{90} = -41.5$$
$$t_{91} = 72.5$$
$$t_{92} = 48.5$$
$$t_{93} = -43.5$$
$$t_{94} = -35.5$$
$$t_{95} = 16.5$$
$$t_{96} = -75.5$$
$$t_{97} = 68.5$$
$$t_{98} = 8.5$$
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
$$- \frac{\pi \sin{\left(\pi t \right)}}{t^{2}} - \frac{2 \cos{\left(\pi t \right)}}{t^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 13.9855205548016$$
$$t_{2} = -49.9959470431469$$
$$t_{3} = 27.9927621489913$$
$$t_{4} = -39.9949337269324$$
$$t_{5} = -79.9974669436743$$
$$t_{6} = 83.9975875677713$$
$$t_{7} = 73.997261555852$$
$$t_{8} = -65.9969296134898$$
$$t_{9} = 45.9955945905146$$
$$t_{10} = -81.9975287267927$$
$$t_{11} = 1.89693335982618$$
$$t_{12} = 47.9957781602405$$
$$t_{13} = -95.9978891265361$$
$$t_{14} = 91.9977973479946$$
$$t_{15} = -33.9940395820904$$
$$t_{16} = 81.9975287267927$$
$$t_{17} = 9.97972206420178$$
$$t_{18} = 87.9976972257402$$
$$t_{19} = -75.99733362188$$
$$t_{20} = 99.9979735626391$$
$$t_{21} = 85.9976436718834$$
$$t_{22} = -5.96616275923655$$
$$t_{23} = -55.9963813083538$$
$$t_{24} = 51.996102934047$$
$$t_{25} = 77.9974019920879$$
$$t_{26} = -53.9962472766387$$
$$t_{27} = 71.9971854860034$$
$$t_{28} = 67.9970199216538$$
$$t_{29} = -57.9965060959251$$
$$t_{30} = -19.989866170292$$
$$t_{31} = -3.94912445118753$$
$$t_{32} = 75.99733362188$$
$$t_{33} = 17.9887397430726$$
$$t_{34} = 79.9974669436743$$
$$t_{35} = -67.9970199216538$$
$$t_{36} = 35.994370751954$$
$$t_{37} = -87.9976972257402$$
$$t_{38} = -97.9979322061986$$
$$t_{39} = -37.9946670566572$$
$$t_{40} = -29.9932447474161$$
$$t_{41} = 15.9873315091956$$
$$t_{42} = -27.9927621489913$$
$$t_{43} = 19.989866170292$$
$$t_{44} = -91.9977973479946$$
$$t_{45} = -61.9967315172866$$
$$t_{46} = 65.9969296134898$$
$$t_{47} = 21.9907876975926$$
$$t_{48} = -35.994370751954$$
$$t_{49} = 97.9979322061986$$
$$t_{50} = 3.94912445118753$$
$$t_{51} = -89.9977483993647$$
$$t_{52} = 33.9940395820904$$
$$t_{53} = -93.9978442136552$$
$$t_{54} = 23.9915555777301$$
$$t_{55} = -1.89693335982618$$
$$t_{56} = 93.9978442136552$$
$$t_{57} = -45.9955945905146$$
$$t_{58} = 59.9966225638402$$
$$t_{59} = -17.9887397430726$$
$$t_{60} = -51.996102934047$$
$$t_{61} = 29.9932447474161$$
$$t_{62} = 39.9949337269324$$
$$t_{63} = 5.96616275923655$$
$$t_{64} = -15.9873315091956$$
$$t_{65} = -7.97464293590917$$
$$t_{66} = -13.9855205548016$$
$$t_{67} = 57.9965060959251$$
$$t_{68} = -59.9966225638402$$
$$t_{69} = 49.9959470431469$$
$$t_{70} = -77.9974019920879$$
$$t_{71} = -31.9936670082654$$
$$t_{72} = -69.9971050691315$$
$$t_{73} = -11.9831052103186$$
$$t_{74} = -71.9971854860034$$
$$t_{75} = -25.9922052839155$$
$$t_{76} = 7.97464293590917$$
$$t_{77} = -41.9951749969699$$
$$t_{78} = -23.9915555777301$$
$$t_{79} = -47.9957781602405$$
$$t_{80} = -43.9953943309587$$
$$t_{81} = 69.9971050691315$$
$$t_{82} = 11.9831052103186$$
$$t_{83} = 95.9978891265361$$
$$t_{84} = -9.97972206420178$$
$$t_{85} = 61.9967315172866$$
$$t_{86} = 37.9946670566572$$
$$t_{87} = -73.997261555852$$
$$t_{88} = -83.9975875677713$$
$$t_{89} = 41.9951749969699$$
$$t_{90} = 55.9963813083538$$
$$t_{91} = 43.9953943309587$$
$$t_{92} = -63.9968336607947$$
$$t_{93} = -99.9979735626391$$
$$t_{94} = -85.9976436718834$$
$$t_{95} = 63.9968336607947$$
$$t_{96} = -21.9907876975926$$
$$t_{97} = 25.9922052839155$$
$$t_{98} = 53.9962472766387$$
$$t_{99} = 31.9936670082654$$
$$t_{100} = 89.9977483993647$$
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:
La función no tiene puntos mínimos
Puntos máximos de la función:
$$t_{100} = 13.9855205548016$$
$$t_{100} = -49.9959470431469$$
$$t_{100} = 27.9927621489913$$
$$t_{100} = -39.9949337269324$$
$$t_{100} = -79.9974669436743$$
$$t_{100} = 83.9975875677713$$
$$t_{100} = 73.997261555852$$
$$t_{100} = -65.9969296134898$$
$$t_{100} = 45.9955945905146$$
$$t_{100} = -81.9975287267927$$
$$t_{100} = 1.89693335982618$$
$$t_{100} = 47.9957781602405$$
$$t_{100} = -95.9978891265361$$
$$t_{100} = 91.9977973479946$$
$$t_{100} = -33.9940395820904$$
$$t_{100} = 81.9975287267927$$
$$t_{100} = 9.97972206420178$$
$$t_{100} = 87.9976972257402$$
$$t_{100} = -75.99733362188$$
$$t_{100} = 99.9979735626391$$
$$t_{100} = 85.9976436718834$$
$$t_{100} = -5.96616275923655$$
$$t_{100} = -55.9963813083538$$
$$t_{100} = 51.996102934047$$
$$t_{100} = 77.9974019920879$$
$$t_{100} = -53.9962472766387$$
$$t_{100} = 71.9971854860034$$
$$t_{100} = 67.9970199216538$$
$$t_{100} = -57.9965060959251$$
$$t_{100} = -19.989866170292$$
$$t_{100} = -3.94912445118753$$
$$t_{100} = 75.99733362188$$
$$t_{100} = 17.9887397430726$$
$$t_{100} = 79.9974669436743$$
$$t_{100} = -67.9970199216538$$
$$t_{100} = 35.994370751954$$
$$t_{100} = -87.9976972257402$$
$$t_{100} = -97.9979322061986$$
$$t_{100} = -37.9946670566572$$
$$t_{100} = -29.9932447474161$$
$$t_{100} = 15.9873315091956$$
$$t_{100} = -27.9927621489913$$
$$t_{100} = 19.989866170292$$
$$t_{100} = -91.9977973479946$$
$$t_{100} = -61.9967315172866$$
$$t_{100} = 65.9969296134898$$
$$t_{100} = 21.9907876975926$$
$$t_{100} = -35.994370751954$$
$$t_{100} = 97.9979322061986$$
$$t_{100} = 3.94912445118753$$
$$t_{100} = -89.9977483993647$$
$$t_{100} = 33.9940395820904$$
$$t_{100} = -93.9978442136552$$
$$t_{100} = 23.9915555777301$$
$$t_{100} = -1.89693335982618$$
$$t_{100} = 93.9978442136552$$
$$t_{100} = -45.9955945905146$$
$$t_{100} = 59.9966225638402$$
$$t_{100} = -17.9887397430726$$
$$t_{100} = -51.996102934047$$
$$t_{100} = 29.9932447474161$$
$$t_{100} = 39.9949337269324$$
$$t_{100} = 5.96616275923655$$
$$t_{100} = -15.9873315091956$$
$$t_{100} = -7.97464293590917$$
$$t_{100} = -13.9855205548016$$
$$t_{100} = 57.9965060959251$$
$$t_{100} = -59.9966225638402$$
$$t_{100} = 49.9959470431469$$
$$t_{100} = -77.9974019920879$$
$$t_{100} = -31.9936670082654$$
$$t_{100} = -69.9971050691315$$
$$t_{100} = -11.9831052103186$$
$$t_{100} = -71.9971854860034$$
$$t_{100} = -25.9922052839155$$
$$t_{100} = 7.97464293590917$$
$$t_{100} = -41.9951749969699$$
$$t_{100} = -23.9915555777301$$
$$t_{100} = -47.9957781602405$$
$$t_{100} = -43.9953943309587$$
$$t_{100} = 69.9971050691315$$
$$t_{100} = 11.9831052103186$$
$$t_{100} = 95.9978891265361$$
$$t_{100} = -9.97972206420178$$
$$t_{100} = 61.9967315172866$$
$$t_{100} = 37.9946670566572$$
$$t_{100} = -73.997261555852$$
$$t_{100} = -83.9975875677713$$
$$t_{100} = 41.9951749969699$$
$$t_{100} = 55.9963813083538$$
$$t_{100} = 43.9953943309587$$
$$t_{100} = -63.9968336607947$$
$$t_{100} = -99.9979735626391$$
$$t_{100} = -85.9976436718834$$
$$t_{100} = 63.9968336607947$$
$$t_{100} = -21.9907876975926$$
$$t_{100} = 25.9922052839155$$
$$t_{100} = 53.9962472766387$$
$$t_{100} = 31.9936670082654$$
$$t_{100} = 89.9977483993647$$
Decrece en los intervalos
$$\left(-\infty, -99.9979735626391\right]$$
Crece en los intervalos
$$\left[99.9979735626391, \infty\right)$$
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(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
$$- \frac{\pi \sin{\left(\pi t \right)}}{t^{2}} - \frac{2 \cos{\left(\pi t \right)}}{t^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 13.9855205548016$$
$$t_{2} = -49.9959470431469$$
$$t_{3} = 27.9927621489913$$
$$t_{4} = -39.9949337269324$$
$$t_{5} = -79.9974669436743$$
$$t_{6} = 83.9975875677713$$
$$t_{7} = 73.997261555852$$
$$t_{8} = -65.9969296134898$$
$$t_{9} = 45.9955945905146$$
$$t_{10} = -81.9975287267927$$
$$t_{11} = 1.89693335982618$$
$$t_{12} = 47.9957781602405$$
$$t_{13} = -95.9978891265361$$
$$t_{14} = 91.9977973479946$$
$$t_{15} = -33.9940395820904$$
$$t_{16} = 81.9975287267927$$
$$t_{17} = 9.97972206420178$$
$$t_{18} = 87.9976972257402$$
$$t_{19} = -75.99733362188$$
$$t_{20} = 99.9979735626391$$
$$t_{21} = 85.9976436718834$$
$$t_{22} = -5.96616275923655$$
$$t_{23} = -55.9963813083538$$
$$t_{24} = 51.996102934047$$
$$t_{25} = 77.9974019920879$$
$$t_{26} = -53.9962472766387$$
$$t_{27} = 71.9971854860034$$
$$t_{28} = 67.9970199216538$$
$$t_{29} = -57.9965060959251$$
$$t_{30} = -19.989866170292$$
$$t_{31} = -3.94912445118753$$
$$t_{32} = 75.99733362188$$
$$t_{33} = 17.9887397430726$$
$$t_{34} = 79.9974669436743$$
$$t_{35} = -67.9970199216538$$
$$t_{36} = 35.994370751954$$
$$t_{37} = -87.9976972257402$$
$$t_{38} = -97.9979322061986$$
$$t_{39} = -37.9946670566572$$
$$t_{40} = -29.9932447474161$$
$$t_{41} = 15.9873315091956$$
$$t_{42} = -27.9927621489913$$
$$t_{43} = 19.989866170292$$
$$t_{44} = -91.9977973479946$$
$$t_{45} = -61.9967315172866$$
$$t_{46} = 65.9969296134898$$
$$t_{47} = 21.9907876975926$$
$$t_{48} = -35.994370751954$$
$$t_{49} = 97.9979322061986$$
$$t_{50} = 3.94912445118753$$
$$t_{51} = -89.9977483993647$$
$$t_{52} = 33.9940395820904$$
$$t_{53} = -93.9978442136552$$
$$t_{54} = 23.9915555777301$$
$$t_{55} = -1.89693335982618$$
$$t_{56} = 93.9978442136552$$
$$t_{57} = -45.9955945905146$$
$$t_{58} = 59.9966225638402$$
$$t_{59} = -17.9887397430726$$
$$t_{60} = -51.996102934047$$
$$t_{61} = 29.9932447474161$$
$$t_{62} = 39.9949337269324$$
$$t_{63} = 5.96616275923655$$
$$t_{64} = -15.9873315091956$$
$$t_{65} = -7.97464293590917$$
$$t_{66} = -13.9855205548016$$
$$t_{67} = 57.9965060959251$$
$$t_{68} = -59.9966225638402$$
$$t_{69} = 49.9959470431469$$
$$t_{70} = -77.9974019920879$$
$$t_{71} = -31.9936670082654$$
$$t_{72} = -69.9971050691315$$
$$t_{73} = -11.9831052103186$$
$$t_{74} = -71.9971854860034$$
$$t_{75} = -25.9922052839155$$
$$t_{76} = 7.97464293590917$$
$$t_{77} = -41.9951749969699$$
$$t_{78} = -23.9915555777301$$
$$t_{79} = -47.9957781602405$$
$$t_{80} = -43.9953943309587$$
$$t_{81} = 69.9971050691315$$
$$t_{82} = 11.9831052103186$$
$$t_{83} = 95.9978891265361$$
$$t_{84} = -9.97972206420178$$
$$t_{85} = 61.9967315172866$$
$$t_{86} = 37.9946670566572$$
$$t_{87} = -73.997261555852$$
$$t_{88} = -83.9975875677713$$
$$t_{89} = 41.9951749969699$$
$$t_{90} = 55.9963813083538$$
$$t_{91} = 43.9953943309587$$
$$t_{92} = -63.9968336607947$$
$$t_{93} = -99.9979735626391$$
$$t_{94} = -85.9976436718834$$
$$t_{95} = 63.9968336607947$$
$$t_{96} = -21.9907876975926$$
$$t_{97} = 25.9922052839155$$
$$t_{98} = 53.9962472766387$$
$$t_{99} = 31.9936670082654$$
$$t_{100} = 89.9977483993647$$
Signos de extremos en los puntos:
(13.985520554801584, 0.00511261074284191*cos(1.98552055480158*pi))
(-49.99594704314695, 0.000400064855195202*cos(1.99594704314695*pi))
(27.992762148991314, 0.00127616988505787*cos(1.99276214899131*pi))
(-39.994933726932416, 0.000625158351117102*cos(1.99493372693242*pi))
(-79.99746694367427, 0.000156259895221242*cos(1.99746694367427*pi))
(83.99758756777128, 0.000141731496788138*cos(1.99758756777128*pi))
(73.99726155585199, 0.000182628563935817*cos(1.99726155585199*pi))
(-65.99692961348984, 0.000229589772384866*cos(1.99692961348984*pi))
(45.99559459051456, 0.000472680324698444*cos(1.99559459051456*pi))
(-81.9975287267927, 0.000148729963962104*cos(1.99752872679269*pi))
(1.8969333598261786, 0.277904674330611*cos(1.89693335982618*pi))
(47.995778160240505, 0.000434104137674021*cos(1.9957781602405*pi))
(-95.99788912653607, 0.000108511716360785*cos(1.99788912653607*pi))
(91.99779734799456, 0.000118153105559894*cos(1.99779734799456*pi))
(-33.994039582090444, 0.000865355281173681*cos(1.99403958209044*pi))
(81.9975287267927, 0.000148729963962104*cos(1.99752872679269*pi))
(9.979722064201784, 0.0100406795643751*cos(1.97972206420178*pi))
(87.9976972257402, 0.000129138989906118*cos(1.9976972257402*pi))
(-75.99733362188, 0.000173142342717812*cos(1.99733362188*pi))
(99.99797356263906, 0.000100004052997919*cos(1.99797356263906*pi))
(85.99764367188335, 0.000135215630148799*cos(1.99764367188335*pi))
(-5.966162759236554, 0.028093756000948*cos(1.96616275923655*pi))
(-55.996381308353826, 0.00031891876642713*cos(1.99638130835383*pi))
(51.99610293404697, 0.000369877923078159*cos(1.99610293404697*pi))
(77.99740199208787, 0.000164376498835612*cos(1.99740199208787*pi))
(-53.996247276638734, 0.000342983197614567*cos(1.99624727663873*pi))
(71.99718548600342, 0.000192916316652929*cos(1.99718548600342*pi))
(67.99701992165383, 0.000216281932336779*cos(1.99701992165383*pi))
(-57.99650609592515, 0.000297300978102915*cos(1.99650609592515*pi))
(-19.989866170292014, 0.00250253538425064*cos(1.98986617029201*pi))
(-3.9491244511875285, 0.0641207154811537*cos(1.94912445118753*pi))
(75.99733362188, 0.000173142342717812*cos(1.99733362188*pi))
(17.98873974307256, 0.00309028492176371*cos(1.98873974307256*pi))
(79.99746694367427, 0.000156259895221242*cos(1.99746694367427*pi))
(-67.99701992165383, 0.000216281932336779*cos(1.99701992165383*pi))
(35.994370751954015, 0.00077184630352678*cos(1.99437075195402*pi))
(-87.9976972257402, 0.000129138989906118*cos(1.9976972257402*pi))
(-97.99793220619864, 0.000104127676094248*cos(1.99793220619864*pi))
(-37.99466705665724, 0.00069271519413164*cos(1.99466705665724*pi))
(-29.9932447474161, 0.00111161166925489*cos(1.9932447474161*pi))
(15.987331509195563, 0.00391244314096858*cos(1.98733150919556*pi))
(-27.992762148991314, 0.00127616988505787*cos(1.99276214899131*pi))
(19.989866170292014, 0.00250253538425064*cos(1.98986617029201*pi))
(-91.99779734799456, 0.000118153105559894*cos(1.99779734799456*pi))
(-61.99673151728665, 0.000260173112191523*cos(1.99673151728665*pi))
(65.99692961348984, 0.000229589772384866*cos(1.99692961348984*pi))
(21.99078769759256, 0.00206784712471696*cos(1.99078769759256*pi))
(-35.994370751954015, 0.00077184630352678*cos(1.99437075195402*pi))
(97.99793220619864, 0.000104127676094248*cos(1.99793220619864*pi))
(3.9491244511875285, 0.0641207154811537*cos(1.94912445118753*pi))
(-89.99774839936472, 0.000123462967586099*cos(1.99774839936472*pi))
(33.994039582090444, 0.000865355281173681*cos(1.99403958209044*pi))
(-93.99784421365517, 0.000113178572812643*cos(1.99784421365517*pi))
(23.991555577730143, 0.00173733346081139*cos(1.99155557773014*pi))
(-1.8969333598261786, 0.277904674330611*cos(1.89693335982618*pi))
(93.99784421365517, 0.000113178572812643*cos(1.99784421365517*pi))
(-45.99559459051456, 0.000472680324698444*cos(1.99559459051456*pi))
(59.99662256384023, 0.000277809052975537*cos(1.99662256384023*pi))
(-17.98873974307256, 0.00309028492176371*cos(1.98873974307256*pi))
(-51.99610293404697, 0.000369877923078159*cos(1.99610293404697*pi))
(29.9932447474161, 0.00111161166925489*cos(1.9932447474161*pi))
(39.994933726932416, 0.000625158351117102*cos(1.99493372693242*pi))
(5.966162759236554, 0.028093756000948*cos(1.96616275923655*pi))
(-15.987331509195563, 0.00391244314096858*cos(1.98733150919556*pi))
(-7.974642935909171, 0.0157245239628966*cos(1.97464293590917*pi))
(-13.985520554801584, 0.00511261074284191*cos(1.98552055480158*pi))
(57.99650609592515, 0.000297300978102915*cos(1.99650609592515*pi))
(-59.99662256384023, 0.000277809052975537*cos(1.99662256384023*pi))
(49.99594704314695, 0.000400064855195202*cos(1.99594704314695*pi))
(-77.99740199208787, 0.000164376498835612*cos(1.99740199208787*pi))
(-31.9936670082654, 0.000976949149916768*cos(1.9936670082654*pi))
(-69.99710506913149, 0.000204098513763634*cos(1.99710506913149*pi))
(-11.983105210318609, 0.00696403997224107*cos(1.98310521031861*pi))
(-71.99718548600342, 0.000192916316652929*cos(1.99718548600342*pi))
(-25.992205283915453, 0.00148017731255488*cos(1.99220528391545*pi))
(7.974642935909171, 0.0157245239628966*cos(1.97464293590917*pi))
(-41.99517499696988, 0.000567023697079404*cos(1.99517499696988*pi))
(-23.991555577730143, 0.00173733346081139*cos(1.99155557773014*pi))
(-47.995778160240505, 0.000434104137674021*cos(1.9957781602405*pi))
(-43.99539433095872, 0.000516637077204286*cos(1.99539433095872*pi))
(69.99710506913149, 0.000204098513763634*cos(1.99710506913149*pi))
(11.983105210318609, 0.00696403997224107*cos(1.98310521031861*pi))
(95.99788912653607, 0.000108511716360785*cos(1.99788912653607*pi))
(-9.979722064201784, 0.0100406795643751*cos(1.97972206420178*pi))
(61.99673151728665, 0.000260173112191523*cos(1.99673151728665*pi))
(37.99466705665724, 0.00069271519413164*cos(1.99466705665724*pi))
(-73.99726155585199, 0.000182628563935817*cos(1.99726155585199*pi))
(-83.99758756777128, 0.000141731496788138*cos(1.99758756777128*pi))
(41.99517499696988, 0.000567023697079404*cos(1.99517499696988*pi))
(55.996381308353826, 0.00031891876642713*cos(1.99638130835383*pi))
(43.99539433095872, 0.000516637077204286*cos(1.99539433095872*pi))
(-63.99683366079466, 0.000244164784043871*cos(1.99683366079466*pi))
(-99.99797356263906, 0.000100004052997919*cos(1.99797356263906*pi))
(-85.99764367188335, 0.000135215630148799*cos(1.99764367188335*pi))
(63.99683366079466, 0.000244164784043871*cos(1.99683366079466*pi))
(-21.99078769759256, 0.00206784712471696*cos(1.99078769759256*pi))
(25.992205283915453, 0.00148017731255488*cos(1.99220528391545*pi))
(53.996247276638734, 0.000342983197614567*cos(1.99624727663873*pi))
(31.9936670082654, 0.000976949149916768*cos(1.9936670082654*pi))
(89.99774839936472, 0.000123462967586099*cos(1.99774839936472*pi))
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:
La función no tiene puntos mínimos
Puntos máximos de la función:
$$t_{100} = 13.9855205548016$$
$$t_{100} = -49.9959470431469$$
$$t_{100} = 27.9927621489913$$
$$t_{100} = -39.9949337269324$$
$$t_{100} = -79.9974669436743$$
$$t_{100} = 83.9975875677713$$
$$t_{100} = 73.997261555852$$
$$t_{100} = -65.9969296134898$$
$$t_{100} = 45.9955945905146$$
$$t_{100} = -81.9975287267927$$
$$t_{100} = 1.89693335982618$$
$$t_{100} = 47.9957781602405$$
$$t_{100} = -95.9978891265361$$
$$t_{100} = 91.9977973479946$$
$$t_{100} = -33.9940395820904$$
$$t_{100} = 81.9975287267927$$
$$t_{100} = 9.97972206420178$$
$$t_{100} = 87.9976972257402$$
$$t_{100} = -75.99733362188$$
$$t_{100} = 99.9979735626391$$
$$t_{100} = 85.9976436718834$$
$$t_{100} = -5.96616275923655$$
$$t_{100} = -55.9963813083538$$
$$t_{100} = 51.996102934047$$
$$t_{100} = 77.9974019920879$$
$$t_{100} = -53.9962472766387$$
$$t_{100} = 71.9971854860034$$
$$t_{100} = 67.9970199216538$$
$$t_{100} = -57.9965060959251$$
$$t_{100} = -19.989866170292$$
$$t_{100} = -3.94912445118753$$
$$t_{100} = 75.99733362188$$
$$t_{100} = 17.9887397430726$$
$$t_{100} = 79.9974669436743$$
$$t_{100} = -67.9970199216538$$
$$t_{100} = 35.994370751954$$
$$t_{100} = -87.9976972257402$$
$$t_{100} = -97.9979322061986$$
$$t_{100} = -37.9946670566572$$
$$t_{100} = -29.9932447474161$$
$$t_{100} = 15.9873315091956$$
$$t_{100} = -27.9927621489913$$
$$t_{100} = 19.989866170292$$
$$t_{100} = -91.9977973479946$$
$$t_{100} = -61.9967315172866$$
$$t_{100} = 65.9969296134898$$
$$t_{100} = 21.9907876975926$$
$$t_{100} = -35.994370751954$$
$$t_{100} = 97.9979322061986$$
$$t_{100} = 3.94912445118753$$
$$t_{100} = -89.9977483993647$$
$$t_{100} = 33.9940395820904$$
$$t_{100} = -93.9978442136552$$
$$t_{100} = 23.9915555777301$$
$$t_{100} = -1.89693335982618$$
$$t_{100} = 93.9978442136552$$
$$t_{100} = -45.9955945905146$$
$$t_{100} = 59.9966225638402$$
$$t_{100} = -17.9887397430726$$
$$t_{100} = -51.996102934047$$
$$t_{100} = 29.9932447474161$$
$$t_{100} = 39.9949337269324$$
$$t_{100} = 5.96616275923655$$
$$t_{100} = -15.9873315091956$$
$$t_{100} = -7.97464293590917$$
$$t_{100} = -13.9855205548016$$
$$t_{100} = 57.9965060959251$$
$$t_{100} = -59.9966225638402$$
$$t_{100} = 49.9959470431469$$
$$t_{100} = -77.9974019920879$$
$$t_{100} = -31.9936670082654$$
$$t_{100} = -69.9971050691315$$
$$t_{100} = -11.9831052103186$$
$$t_{100} = -71.9971854860034$$
$$t_{100} = -25.9922052839155$$
$$t_{100} = 7.97464293590917$$
$$t_{100} = -41.9951749969699$$
$$t_{100} = -23.9915555777301$$
$$t_{100} = -47.9957781602405$$
$$t_{100} = -43.9953943309587$$
$$t_{100} = 69.9971050691315$$
$$t_{100} = 11.9831052103186$$
$$t_{100} = 95.9978891265361$$
$$t_{100} = -9.97972206420178$$
$$t_{100} = 61.9967315172866$$
$$t_{100} = 37.9946670566572$$
$$t_{100} = -73.997261555852$$
$$t_{100} = -83.9975875677713$$
$$t_{100} = 41.9951749969699$$
$$t_{100} = 55.9963813083538$$
$$t_{100} = 43.9953943309587$$
$$t_{100} = -63.9968336607947$$
$$t_{100} = -99.9979735626391$$
$$t_{100} = -85.9976436718834$$
$$t_{100} = 63.9968336607947$$
$$t_{100} = -21.9907876975926$$
$$t_{100} = 25.9922052839155$$
$$t_{100} = 53.9962472766387$$
$$t_{100} = 31.9936670082654$$
$$t_{100} = 89.9977483993647$$
Decrece en los intervalos
$$\left(-\infty, -99.9979735626391\right]$$
Crece en los intervalos
$$\left[99.9979735626391, \infty\right)$$
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
$$\frac{- \pi^{2} \cos{\left(\pi t \right)} + \frac{4 \pi \sin{\left(\pi t \right)}}{t} + \frac{6 \cos{\left(\pi t \right)}}{t^{2}}}{t^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -27.4852531498735$$
$$t_{2} = -73.494485439293$$
$$t_{3} = 34.488247946803$$
$$t_{4} = -8.4520041886321$$
$$t_{5} = -99.4959265920377$$
$$t_{6} = 80.4949650396326$$
$$t_{7} = -89.4954714116193$$
$$t_{8} = 16.4753945137613$$
$$t_{9} = -65.4938117647479$$
$$t_{10} = -81.4950268271389$$
$$t_{11} = -19.4791902731472$$
$$t_{12} = -33.4878968476656$$
$$t_{13} = 74.494559472942$$
$$t_{14} = -69.4941679940453$$
$$t_{15} = 68.4940828382335$$
$$t_{16} = -31.4871276823421$$
$$t_{17} = 6.43693831017351$$
$$t_{18} = 44.4908903024276$$
$$t_{19} = 96.495799944926$$
$$t_{20} = -51.4921289901534$$
$$t_{21} = 82.4950871164502$$
$$t_{22} = 26.4846959281867$$
$$t_{23} = 72.4944093627926$$
$$t_{24} = 42.4904613912603$$
$$t_{25} = 54.4925623987603$$
$$t_{26} = 36.4888923656265$$
$$t_{27} = 56.4928257561295$$
$$t_{28} = -29.4862540635993$$
$$t_{29} = 62.4935146591758$$
$$t_{30} = 76.49470173224$$
$$t_{31} = 58.4930710984202$$
$$t_{32} = -14.4719862495161$$
$$t_{33} = 30.4867052164678$$
$$t_{34} = 4.40775399379882$$
$$t_{35} = 40.4899900835126$$
$$t_{36} = 88.4954202349846$$
$$t_{37} = 84.4952034133069$$
$$t_{38} = 28.4857711973569$$
$$t_{39} = 8.4520041886321$$
$$t_{40} = 32.4875241142337$$
$$t_{41} = 20.4802077303166$$
$$t_{42} = -71.4943311576956$$
$$t_{43} = -9.45711310059773$$
$$t_{44} = 92.4956183012629$$
$$t_{45} = -21.4811302345078$$
$$t_{46} = 46.4912822926717$$
$$t_{47} = -39.489736516111$$
$$t_{48} = -91.4955704084646$$
$$t_{49} = -83.4951459614086$$
$$t_{50} = -97.4958430267218$$
$$t_{51} = -77.4947701078213$$
$$t_{52} = -41.4902314204746$$
$$t_{53} = 94.495711045555$$
$$t_{54} = 90.4955214570806$$
$$t_{55} = -43.4906807805564$$
$$t_{56} = -2.32394415453121$$
$$t_{57} = 98.4958852336312$$
$$t_{58} = 10.4612348024336$$
$$t_{59} = 100.495967127274$$
$$t_{60} = -75.4946315449401$$
$$t_{61} = -6.43693831017351$$
$$t_{62} = 60.4933002132249$$
$$t_{63} = -45.4910906080887$$
$$t_{64} = 48.4916419334549$$
$$t_{65} = 22.4819704970222$$
$$t_{66} = 18.4780623860658$$
$$t_{67} = -87.4953678883838$$
$$t_{68} = -95.4957559607546$$
$$t_{69} = -61.4934091803175$$
$$t_{70} = 64.493715801469$$
$$t_{71} = -55.492696451127$$
$$t_{72} = -35.488579242766$$
$$t_{73} = -37.4891887712726$$
$$t_{74} = -57.4929505616019$$
$$t_{75} = 38.4894697640766$$
$$t_{76} = -10.4612348024336$$
$$t_{77} = 14.4719862495161$$
$$t_{78} = 24.4834447192119$$
$$t_{79} = 70.4942507333899$$
$$t_{80} = 50.4919730713159$$
$$t_{81} = -15.4738009794179$$
$$t_{82} = 12.4674786415251$$
$$t_{83} = 2.32394415453121$$
$$t_{84} = 66.493904840973$$
$$t_{85} = 78.4948367409432$$
$$t_{86} = -25.4840949108679$$
$$t_{87} = -93.4956651694482$$
$$t_{88} = -59.4931875819289$$
$$t_{89} = -85.4952595210373$$
$$t_{90} = -23.4827390553626$$
$$t_{91} = 52.4922789661076$$
$$t_{92} = -47.4914659011376$$
$$t_{93} = -67.4939951585121$$
$$t_{94} = -79.4949016973791$$
$$t_{95} = 86.4953143312302$$
$$t_{96} = -49.49181084915$$
$$t_{97} = -63.4936168146774$$
$$t_{98} = -53.4924233326508$$
$$t_{99} = -17.4768050298703$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$t_{1} = 0$$
$$\lim_{t \to 0^-}\left(\frac{- \pi^{2} \cos{\left(\pi t \right)} + \frac{4 \pi \sin{\left(\pi t \right)}}{t} + \frac{6 \cos{\left(\pi t \right)}}{t^{2}}}{t^{2}}\right) = \infty$$
$$\lim_{t \to 0^+}\left(\frac{- \pi^{2} \cos{\left(\pi t \right)} + \frac{4 \pi \sin{\left(\pi t \right)}}{t} + \frac{6 \cos{\left(\pi t \right)}}{t^{2}}}{t^{2}}\right) = \infty$$
- 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
$$\left[100.495967127274, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -99.4959265920377\right]$$
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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
$$\frac{- \pi^{2} \cos{\left(\pi t \right)} + \frac{4 \pi \sin{\left(\pi t \right)}}{t} + \frac{6 \cos{\left(\pi t \right)}}{t^{2}}}{t^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -27.4852531498735$$
$$t_{2} = -73.494485439293$$
$$t_{3} = 34.488247946803$$
$$t_{4} = -8.4520041886321$$
$$t_{5} = -99.4959265920377$$
$$t_{6} = 80.4949650396326$$
$$t_{7} = -89.4954714116193$$
$$t_{8} = 16.4753945137613$$
$$t_{9} = -65.4938117647479$$
$$t_{10} = -81.4950268271389$$
$$t_{11} = -19.4791902731472$$
$$t_{12} = -33.4878968476656$$
$$t_{13} = 74.494559472942$$
$$t_{14} = -69.4941679940453$$
$$t_{15} = 68.4940828382335$$
$$t_{16} = -31.4871276823421$$
$$t_{17} = 6.43693831017351$$
$$t_{18} = 44.4908903024276$$
$$t_{19} = 96.495799944926$$
$$t_{20} = -51.4921289901534$$
$$t_{21} = 82.4950871164502$$
$$t_{22} = 26.4846959281867$$
$$t_{23} = 72.4944093627926$$
$$t_{24} = 42.4904613912603$$
$$t_{25} = 54.4925623987603$$
$$t_{26} = 36.4888923656265$$
$$t_{27} = 56.4928257561295$$
$$t_{28} = -29.4862540635993$$
$$t_{29} = 62.4935146591758$$
$$t_{30} = 76.49470173224$$
$$t_{31} = 58.4930710984202$$
$$t_{32} = -14.4719862495161$$
$$t_{33} = 30.4867052164678$$
$$t_{34} = 4.40775399379882$$
$$t_{35} = 40.4899900835126$$
$$t_{36} = 88.4954202349846$$
$$t_{37} = 84.4952034133069$$
$$t_{38} = 28.4857711973569$$
$$t_{39} = 8.4520041886321$$
$$t_{40} = 32.4875241142337$$
$$t_{41} = 20.4802077303166$$
$$t_{42} = -71.4943311576956$$
$$t_{43} = -9.45711310059773$$
$$t_{44} = 92.4956183012629$$
$$t_{45} = -21.4811302345078$$
$$t_{46} = 46.4912822926717$$
$$t_{47} = -39.489736516111$$
$$t_{48} = -91.4955704084646$$
$$t_{49} = -83.4951459614086$$
$$t_{50} = -97.4958430267218$$
$$t_{51} = -77.4947701078213$$
$$t_{52} = -41.4902314204746$$
$$t_{53} = 94.495711045555$$
$$t_{54} = 90.4955214570806$$
$$t_{55} = -43.4906807805564$$
$$t_{56} = -2.32394415453121$$
$$t_{57} = 98.4958852336312$$
$$t_{58} = 10.4612348024336$$
$$t_{59} = 100.495967127274$$
$$t_{60} = -75.4946315449401$$
$$t_{61} = -6.43693831017351$$
$$t_{62} = 60.4933002132249$$
$$t_{63} = -45.4910906080887$$
$$t_{64} = 48.4916419334549$$
$$t_{65} = 22.4819704970222$$
$$t_{66} = 18.4780623860658$$
$$t_{67} = -87.4953678883838$$
$$t_{68} = -95.4957559607546$$
$$t_{69} = -61.4934091803175$$
$$t_{70} = 64.493715801469$$
$$t_{71} = -55.492696451127$$
$$t_{72} = -35.488579242766$$
$$t_{73} = -37.4891887712726$$
$$t_{74} = -57.4929505616019$$
$$t_{75} = 38.4894697640766$$
$$t_{76} = -10.4612348024336$$
$$t_{77} = 14.4719862495161$$
$$t_{78} = 24.4834447192119$$
$$t_{79} = 70.4942507333899$$
$$t_{80} = 50.4919730713159$$
$$t_{81} = -15.4738009794179$$
$$t_{82} = 12.4674786415251$$
$$t_{83} = 2.32394415453121$$
$$t_{84} = 66.493904840973$$
$$t_{85} = 78.4948367409432$$
$$t_{86} = -25.4840949108679$$
$$t_{87} = -93.4956651694482$$
$$t_{88} = -59.4931875819289$$
$$t_{89} = -85.4952595210373$$
$$t_{90} = -23.4827390553626$$
$$t_{91} = 52.4922789661076$$
$$t_{92} = -47.4914659011376$$
$$t_{93} = -67.4939951585121$$
$$t_{94} = -79.4949016973791$$
$$t_{95} = 86.4953143312302$$
$$t_{96} = -49.49181084915$$
$$t_{97} = -63.4936168146774$$
$$t_{98} = -53.4924233326508$$
$$t_{99} = -17.4768050298703$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$t_{1} = 0$$
$$\lim_{t \to 0^-}\left(\frac{- \pi^{2} \cos{\left(\pi t \right)} + \frac{4 \pi \sin{\left(\pi t \right)}}{t} + \frac{6 \cos{\left(\pi t \right)}}{t^{2}}}{t^{2}}\right) = \infty$$
$$\lim_{t \to 0^+}\left(\frac{- \pi^{2} \cos{\left(\pi t \right)} + \frac{4 \pi \sin{\left(\pi t \right)}}{t} + \frac{6 \cos{\left(\pi t \right)}}{t^{2}}}{t^{2}}\right) = \infty$$
- 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
$$\left[100.495967127274, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -99.4959265920377\right]$$
Asíntotas verticales
Hay:
$$t_{1} = 0$$
$$t_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con t->+oo y t->-oo
$$\lim_{t \to -\infty}\left(\frac{\cos{\left(\pi t \right)}}{t^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{t \to \infty}\left(\frac{\cos{\left(\pi t \right)}}{t^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{t \to -\infty}\left(\frac{\cos{\left(\pi t \right)}}{t^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{t \to \infty}\left(\frac{\cos{\left(\pi t \right)}}{t^{2}}\right) = 0$$
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 cos(pi*t)/t^2, dividida por t con t->+oo y t ->-oo
$$\lim_{t \to -\infty}\left(\frac{\cos{\left(\pi t \right)}}{t t^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\cos{\left(\pi t \right)}}{t t^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{t \to -\infty}\left(\frac{\cos{\left(\pi t \right)}}{t t^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\cos{\left(\pi t \right)}}{t t^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-t) и f = -f(-t).
Pues, comprobamos:
$$\frac{\cos{\left(\pi t \right)}}{t^{2}} = \frac{\cos{\left(\pi t \right)}}{t^{2}}$$
- Sí
$$\frac{\cos{\left(\pi t \right)}}{t^{2}} = - \frac{\cos{\left(\pi t \right)}}{t^{2}}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\frac{\cos{\left(\pi t \right)}}{t^{2}} = \frac{\cos{\left(\pi t \right)}}{t^{2}}$$
- Sí
$$\frac{\cos{\left(\pi t \right)}}{t^{2}} = - \frac{\cos{\left(\pi t \right)}}{t^{2}}$$
- No
es decir, función
es
par