Sr Examen
Otras calculadoras
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- 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 =:
-
-exp(x)-exp(-x)-exp(2*x)
-
x^4/(1+x)^3
-
x/2
-
-x+1
- Límite de la función:
-
cos(n)/n
- Suma de la serie:
-
cos(n)/n
-
Expresiones idénticas
- cos(n)/n
- coseno de (n) dividir por n
- cosn/n
- cos(n) dividir por n
-
Expresiones con funciones
- Coseno cos
- cos(x)-2
- cos(x)*x
- cos(x)^(2)
- cos(x)-1
- cosx+x^2
Gráfico de la función y = cos(n)/n
Solución
Ha introducido
[src]
cos(n)
f(n) = ------
n
$$f{\left(n \right)} = \frac{\cos{\left(n \right)}}{n}$$
f = cos(n)/n
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$n_{1} = 0$$
$$n_{1} = 0$$
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje N con f = 0
o sea hay que resolver la ecuación:
$$\frac{\cos{\left(n \right)}}{n} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje N:
Solución analítica
$$n_{1} = \frac{\pi}{2}$$
$$n_{2} = \frac{3 \pi}{2}$$
Solución numérica
$$n_{1} = 17.2787595947439$$
$$n_{2} = -23.5619449019235$$
$$n_{3} = -14.1371669411541$$
$$n_{4} = 51.8362787842316$$
$$n_{5} = -17.2787595947439$$
$$n_{6} = -10.9955742875643$$
$$n_{7} = -36.1283155162826$$
$$n_{8} = -95.8185759344887$$
$$n_{9} = -48.6946861306418$$
$$n_{10} = -26.7035375555132$$
$$n_{11} = 4.71238898038469$$
$$n_{12} = 26.7035375555132$$
$$n_{13} = 89.5353906273091$$
$$n_{14} = 23.5619449019235$$
$$n_{15} = 14.1371669411541$$
$$n_{16} = 42.4115008234622$$
$$n_{17} = 95.8185759344887$$
$$n_{18} = -61.261056745001$$
$$n_{19} = 58.1194640914112$$
$$n_{20} = 36.1283155162826$$
$$n_{21} = 29.845130209103$$
$$n_{22} = -73.8274273593601$$
$$n_{23} = 1173.38485611579$$
$$n_{24} = 61.261056745001$$
$$n_{25} = 48.6946861306418$$
$$n_{26} = -4.71238898038469$$
$$n_{27} = 70.6858347057703$$
$$n_{28} = -7.85398163397448$$
$$n_{29} = -51.8362787842316$$
$$n_{30} = -76.9690200129499$$
$$n_{31} = -89.5353906273091$$
$$n_{32} = -39.2699081698724$$
$$n_{33} = 80.1106126665397$$
$$n_{34} = -42.4115008234622$$
$$n_{35} = 83.2522053201295$$
$$n_{36} = -92.6769832808989$$
$$n_{37} = 347.145988221672$$
$$n_{38} = 32.9867228626928$$
$$n_{39} = 45.553093477052$$
$$n_{40} = 20.4203522483337$$
$$n_{41} = 64.4026493985908$$
$$n_{42} = -32.9867228626928$$
$$n_{43} = 67.5442420521806$$
$$n_{44} = 199.491133502952$$
$$n_{45} = -20.4203522483337$$
$$n_{46} = -422.544211907827$$
$$n_{47} = -80.1106126665397$$
$$n_{48} = 7.85398163397448$$
$$n_{49} = -45.553093477052$$
$$n_{50} = 76.9690200129499$$
$$n_{51} = -1.5707963267949$$
$$n_{52} = 39.2699081698724$$
$$n_{53} = -70.6858347057703$$
$$n_{54} = -67.5442420521806$$
$$n_{55} = -98.9601685880785$$
$$n_{56} = -29.845130209103$$
$$n_{57} = -83.2522053201295$$
$$n_{58} = -86.3937979737193$$
$$n_{59} = 98.9601685880785$$
$$n_{60} = 73.8274273593601$$
$$n_{61} = -58.1194640914112$$
$$n_{62} = 92.6769832808989$$
$$n_{63} = 54.9778714378214$$
$$n_{64} = 86.3937979737193$$
$$n_{65} = 1.5707963267949$$
$$n_{66} = -54.9778714378214$$
$$n_{67} = -64.4026493985908$$
$$n_{68} = 10.9955742875643$$
o sea hay que resolver la ecuación:
$$\frac{\cos{\left(n \right)}}{n} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje N:
Solución analítica
$$n_{1} = \frac{\pi}{2}$$
$$n_{2} = \frac{3 \pi}{2}$$
Solución numérica
$$n_{1} = 17.2787595947439$$
$$n_{2} = -23.5619449019235$$
$$n_{3} = -14.1371669411541$$
$$n_{4} = 51.8362787842316$$
$$n_{5} = -17.2787595947439$$
$$n_{6} = -10.9955742875643$$
$$n_{7} = -36.1283155162826$$
$$n_{8} = -95.8185759344887$$
$$n_{9} = -48.6946861306418$$
$$n_{10} = -26.7035375555132$$
$$n_{11} = 4.71238898038469$$
$$n_{12} = 26.7035375555132$$
$$n_{13} = 89.5353906273091$$
$$n_{14} = 23.5619449019235$$
$$n_{15} = 14.1371669411541$$
$$n_{16} = 42.4115008234622$$
$$n_{17} = 95.8185759344887$$
$$n_{18} = -61.261056745001$$
$$n_{19} = 58.1194640914112$$
$$n_{20} = 36.1283155162826$$
$$n_{21} = 29.845130209103$$
$$n_{22} = -73.8274273593601$$
$$n_{23} = 1173.38485611579$$
$$n_{24} = 61.261056745001$$
$$n_{25} = 48.6946861306418$$
$$n_{26} = -4.71238898038469$$
$$n_{27} = 70.6858347057703$$
$$n_{28} = -7.85398163397448$$
$$n_{29} = -51.8362787842316$$
$$n_{30} = -76.9690200129499$$
$$n_{31} = -89.5353906273091$$
$$n_{32} = -39.2699081698724$$
$$n_{33} = 80.1106126665397$$
$$n_{34} = -42.4115008234622$$
$$n_{35} = 83.2522053201295$$
$$n_{36} = -92.6769832808989$$
$$n_{37} = 347.145988221672$$
$$n_{38} = 32.9867228626928$$
$$n_{39} = 45.553093477052$$
$$n_{40} = 20.4203522483337$$
$$n_{41} = 64.4026493985908$$
$$n_{42} = -32.9867228626928$$
$$n_{43} = 67.5442420521806$$
$$n_{44} = 199.491133502952$$
$$n_{45} = -20.4203522483337$$
$$n_{46} = -422.544211907827$$
$$n_{47} = -80.1106126665397$$
$$n_{48} = 7.85398163397448$$
$$n_{49} = -45.553093477052$$
$$n_{50} = 76.9690200129499$$
$$n_{51} = -1.5707963267949$$
$$n_{52} = 39.2699081698724$$
$$n_{53} = -70.6858347057703$$
$$n_{54} = -67.5442420521806$$
$$n_{55} = -98.9601685880785$$
$$n_{56} = -29.845130209103$$
$$n_{57} = -83.2522053201295$$
$$n_{58} = -86.3937979737193$$
$$n_{59} = 98.9601685880785$$
$$n_{60} = 73.8274273593601$$
$$n_{61} = -58.1194640914112$$
$$n_{62} = 92.6769832808989$$
$$n_{63} = 54.9778714378214$$
$$n_{64} = 86.3937979737193$$
$$n_{65} = 1.5707963267949$$
$$n_{66} = -54.9778714378214$$
$$n_{67} = -64.4026493985908$$
$$n_{68} = 10.9955742875643$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d n} f{\left(n \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d n} f{\left(n \right)} = $$
primera derivada
$$- \frac{\sin{\left(n \right)}}{n} - \frac{\cos{\left(n \right)}}{n^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$n_{1} = 47.1026627703624$$
$$n_{2} = 84.811211299318$$
$$n_{3} = -50.2455828375744$$
$$n_{4} = -100.521017074687$$
$$n_{5} = -65.9582857893902$$
$$n_{6} = 65.9582857893902$$
$$n_{7} = -78.5270825679419$$
$$n_{8} = -21.945612879981$$
$$n_{9} = 28.2389365752603$$
$$n_{10} = 53.3883466217256$$
$$n_{11} = -87.9532251106725$$
$$n_{12} = -25.0929104121121$$
$$n_{13} = -34.5285657554621$$
$$n_{14} = -69.100567727981$$
$$n_{15} = 94.2371684817036$$
$$n_{16} = -135.08108127842$$
$$n_{17} = -62.8159348889734$$
$$n_{18} = 50.2455828375744$$
$$n_{19} = -91.0952098694071$$
$$n_{20} = 91.0952098694071$$
$$n_{21} = 56.5309801938186$$
$$n_{22} = -53.3883466217256$$
$$n_{23} = -169.640108529775$$
$$n_{24} = -109.946647805931$$
$$n_{25} = -6.12125046689807$$
$$n_{26} = 2.79838604578389$$
$$n_{27} = -47.1026627703624$$
$$n_{28} = 62.8159348889734$$
$$n_{29} = 9.31786646179107$$
$$n_{30} = -2.79838604578389$$
$$n_{31} = -81.6691650818489$$
$$n_{32} = 12.4864543952238$$
$$n_{33} = -31.3840740178899$$
$$n_{34} = -94.2371684817036$$
$$n_{35} = 197.91528455229$$
$$n_{36} = 59.6735041304405$$
$$n_{37} = 97.3791034786112$$
$$n_{38} = 75.3849592185347$$
$$n_{39} = -40.8162093266346$$
$$n_{40} = -15.644128370333$$
$$n_{41} = -37.672573565113$$
$$n_{42} = -12.4864543952238$$
$$n_{43} = 15.644128370333$$
$$n_{44} = 69.100567727981$$
$$n_{45} = -84.811211299318$$
$$n_{46} = 31.3840740178899$$
$$n_{47} = 37.672573565113$$
$$n_{48} = -97.3791034786112$$
$$n_{49} = 6.12125046689807$$
$$n_{50} = 72.2427897046973$$
$$n_{51} = -75.3849592185347$$
$$n_{52} = 34.5285657554621$$
$$n_{53} = 43.9595528888955$$
$$n_{54} = 78.5270825679419$$
$$n_{55} = 40.8162093266346$$
$$n_{56} = 100.521017074687$$
$$n_{57} = 21.945612879981$$
$$n_{58} = -9.31786646179107$$
$$n_{59} = 25.0929104121121$$
$$n_{60} = -72.2427897046973$$
$$n_{61} = -18.7964043662102$$
$$n_{62} = 87.9532251106725$$
$$n_{63} = -59.6735041304405$$
$$n_{64} = -28.2389365752603$$
$$n_{65} = 18.7964043662102$$
$$n_{66} = 81.6691650818489$$
$$n_{67} = -56.5309801938186$$
$$n_{68} = -43.9595528888955$$
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:
$$n_{1} = 47.1026627703624$$
$$n_{2} = 84.811211299318$$
$$n_{3} = -50.2455828375744$$
$$n_{4} = -100.521017074687$$
$$n_{5} = 65.9582857893902$$
$$n_{6} = 28.2389365752603$$
$$n_{7} = 53.3883466217256$$
$$n_{8} = -87.9532251106725$$
$$n_{9} = -25.0929104121121$$
$$n_{10} = -69.100567727981$$
$$n_{11} = -62.8159348889734$$
$$n_{12} = 91.0952098694071$$
$$n_{13} = -169.640108529775$$
$$n_{14} = -6.12125046689807$$
$$n_{15} = 2.79838604578389$$
$$n_{16} = 9.31786646179107$$
$$n_{17} = -81.6691650818489$$
$$n_{18} = -31.3840740178899$$
$$n_{19} = -94.2371684817036$$
$$n_{20} = 197.91528455229$$
$$n_{21} = 59.6735041304405$$
$$n_{22} = 97.3791034786112$$
$$n_{23} = -37.672573565113$$
$$n_{24} = -12.4864543952238$$
$$n_{25} = 15.644128370333$$
$$n_{26} = 72.2427897046973$$
$$n_{27} = -75.3849592185347$$
$$n_{28} = 34.5285657554621$$
$$n_{29} = 78.5270825679419$$
$$n_{30} = 40.8162093266346$$
$$n_{31} = 21.945612879981$$
$$n_{32} = -18.7964043662102$$
$$n_{33} = -56.5309801938186$$
$$n_{34} = -43.9595528888955$$
Puntos máximos de la función:
$$n_{34} = -65.9582857893902$$
$$n_{34} = -78.5270825679419$$
$$n_{34} = -21.945612879981$$
$$n_{34} = -34.5285657554621$$
$$n_{34} = 94.2371684817036$$
$$n_{34} = -135.08108127842$$
$$n_{34} = 50.2455828375744$$
$$n_{34} = -91.0952098694071$$
$$n_{34} = 56.5309801938186$$
$$n_{34} = -53.3883466217256$$
$$n_{34} = -109.946647805931$$
$$n_{34} = -47.1026627703624$$
$$n_{34} = 62.8159348889734$$
$$n_{34} = -2.79838604578389$$
$$n_{34} = 12.4864543952238$$
$$n_{34} = 75.3849592185347$$
$$n_{34} = -40.8162093266346$$
$$n_{34} = -15.644128370333$$
$$n_{34} = 69.100567727981$$
$$n_{34} = -84.811211299318$$
$$n_{34} = 31.3840740178899$$
$$n_{34} = 37.672573565113$$
$$n_{34} = -97.3791034786112$$
$$n_{34} = 6.12125046689807$$
$$n_{34} = 43.9595528888955$$
$$n_{34} = 100.521017074687$$
$$n_{34} = -9.31786646179107$$
$$n_{34} = 25.0929104121121$$
$$n_{34} = -72.2427897046973$$
$$n_{34} = 87.9532251106725$$
$$n_{34} = -59.6735041304405$$
$$n_{34} = -28.2389365752603$$
$$n_{34} = 18.7964043662102$$
$$n_{34} = 81.6691650818489$$
Decrece en los intervalos
$$\left[197.91528455229, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -169.640108529775\right]$$
$$\frac{d}{d n} f{\left(n \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d n} f{\left(n \right)} = $$
primera derivada
$$- \frac{\sin{\left(n \right)}}{n} - \frac{\cos{\left(n \right)}}{n^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$n_{1} = 47.1026627703624$$
$$n_{2} = 84.811211299318$$
$$n_{3} = -50.2455828375744$$
$$n_{4} = -100.521017074687$$
$$n_{5} = -65.9582857893902$$
$$n_{6} = 65.9582857893902$$
$$n_{7} = -78.5270825679419$$
$$n_{8} = -21.945612879981$$
$$n_{9} = 28.2389365752603$$
$$n_{10} = 53.3883466217256$$
$$n_{11} = -87.9532251106725$$
$$n_{12} = -25.0929104121121$$
$$n_{13} = -34.5285657554621$$
$$n_{14} = -69.100567727981$$
$$n_{15} = 94.2371684817036$$
$$n_{16} = -135.08108127842$$
$$n_{17} = -62.8159348889734$$
$$n_{18} = 50.2455828375744$$
$$n_{19} = -91.0952098694071$$
$$n_{20} = 91.0952098694071$$
$$n_{21} = 56.5309801938186$$
$$n_{22} = -53.3883466217256$$
$$n_{23} = -169.640108529775$$
$$n_{24} = -109.946647805931$$
$$n_{25} = -6.12125046689807$$
$$n_{26} = 2.79838604578389$$
$$n_{27} = -47.1026627703624$$
$$n_{28} = 62.8159348889734$$
$$n_{29} = 9.31786646179107$$
$$n_{30} = -2.79838604578389$$
$$n_{31} = -81.6691650818489$$
$$n_{32} = 12.4864543952238$$
$$n_{33} = -31.3840740178899$$
$$n_{34} = -94.2371684817036$$
$$n_{35} = 197.91528455229$$
$$n_{36} = 59.6735041304405$$
$$n_{37} = 97.3791034786112$$
$$n_{38} = 75.3849592185347$$
$$n_{39} = -40.8162093266346$$
$$n_{40} = -15.644128370333$$
$$n_{41} = -37.672573565113$$
$$n_{42} = -12.4864543952238$$
$$n_{43} = 15.644128370333$$
$$n_{44} = 69.100567727981$$
$$n_{45} = -84.811211299318$$
$$n_{46} = 31.3840740178899$$
$$n_{47} = 37.672573565113$$
$$n_{48} = -97.3791034786112$$
$$n_{49} = 6.12125046689807$$
$$n_{50} = 72.2427897046973$$
$$n_{51} = -75.3849592185347$$
$$n_{52} = 34.5285657554621$$
$$n_{53} = 43.9595528888955$$
$$n_{54} = 78.5270825679419$$
$$n_{55} = 40.8162093266346$$
$$n_{56} = 100.521017074687$$
$$n_{57} = 21.945612879981$$
$$n_{58} = -9.31786646179107$$
$$n_{59} = 25.0929104121121$$
$$n_{60} = -72.2427897046973$$
$$n_{61} = -18.7964043662102$$
$$n_{62} = 87.9532251106725$$
$$n_{63} = -59.6735041304405$$
$$n_{64} = -28.2389365752603$$
$$n_{65} = 18.7964043662102$$
$$n_{66} = 81.6691650818489$$
$$n_{67} = -56.5309801938186$$
$$n_{68} = -43.9595528888955$$
Signos de extremos en los puntos:
(47.10266277036235, -0.0212254394164143)
(84.81121129931802, -0.0117900744410766)
(-50.24558283757444, -0.0198983065303553)
(-100.52101707468658, -0.00994767611536293)
(-65.95828578939016, 0.0151593553168405)
(65.95828578939016, -0.0151593553168405)
(-78.52708256794193, 0.0127334276777468)
(-21.945612879981045, 0.0455199604051285)
(28.238936575260272, -0.0353899155541688)
(53.38834662172563, -0.0187273944640866)
(-87.95322511067255, -0.0113689449158811)
(-25.092910412112097, -0.0398202855500511)
(-34.52856575546206, 0.0289493889114503)
(-69.10056772798097, -0.0144701459746764)
(94.23716848170359, 0.01061092686295)
(-135.0810812784199, 0.00740275832666827)
(-62.81593488897342, -0.015917510583426)
(50.24558283757444, 0.0198983065303553)
(-91.09520986940714, 0.0109768642483425)
(91.09520986940714, -0.0109768642483425)
(56.53098019381864, 0.0176866485521696)
(-53.38834662172563, 0.0187273944640866)
(-169.6401085297751, -0.00589472993500857)
(-109.94664780593057, 0.00909494432157336)
(-6.1212504668980685, -0.161228034325064)
(2.798386045783887, -0.336508416918395)
(-47.10266277036235, 0.0212254394164143)
(62.81593488897342, 0.015917510583426)
(9.317866461791066, -0.106707947715237)
(-2.798386045783887, 0.336508416918395)
(-81.66916508184887, -0.0122436055670467)
(12.486454395223781, 0.0798311807800032)
(-31.38407401788986, -0.0318471321112693)
(-94.23716848170359, -0.01061092686295)
(197.91528455229027, -0.00505260236866135)
(59.67350413044053, -0.0167555036571887)
(97.3791034786112, -0.0102686022030809)
(75.38495921853475, 0.0132640786518247)
(-40.81620932663458, 0.0244927205346957)
(-15.644128370333028, 0.0637915530395936)
(-37.67257356511297, -0.0265351630103045)
(-12.486454395223781, -0.0798311807800032)
(15.644128370333028, -0.0637915530395936)
(69.10056772798097, 0.0144701459746764)
(-84.81121129931802, 0.0117900744410766)
(31.38407401788986, 0.0318471321112693)
(37.67257356511297, 0.0265351630103045)
(-97.3791034786112, 0.0102686022030809)
(6.1212504668980685, 0.161228034325064)
(72.24278970469729, -0.0138408859131547)
(-75.38495921853475, -0.0132640786518247)
(34.52856575546206, -0.0289493889114503)
(43.959552888895495, 0.0227423004725314)
(78.52708256794193, -0.0127334276777468)
(40.81620932663458, -0.0244927205346957)
(100.52101707468658, 0.00994767611536293)
(21.945612879981045, -0.0455199604051285)
(-9.317866461791066, 0.106707947715237)
(25.092910412112097, 0.0398202855500511)
(-72.24278970469729, 0.0138408859131547)
(-18.796404366210158, -0.0531265325613881)
(87.95322511067255, 0.0113689449158811)
(-59.67350413044053, 0.0167555036571887)
(-28.238936575260272, 0.0353899155541688)
(18.796404366210158, 0.0531265325613881)
(81.66916508184887, 0.0122436055670467)
(-56.53098019381864, -0.0176866485521696)
(-43.959552888895495, -0.0227423004725314)
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:
$$n_{1} = 47.1026627703624$$
$$n_{2} = 84.811211299318$$
$$n_{3} = -50.2455828375744$$
$$n_{4} = -100.521017074687$$
$$n_{5} = 65.9582857893902$$
$$n_{6} = 28.2389365752603$$
$$n_{7} = 53.3883466217256$$
$$n_{8} = -87.9532251106725$$
$$n_{9} = -25.0929104121121$$
$$n_{10} = -69.100567727981$$
$$n_{11} = -62.8159348889734$$
$$n_{12} = 91.0952098694071$$
$$n_{13} = -169.640108529775$$
$$n_{14} = -6.12125046689807$$
$$n_{15} = 2.79838604578389$$
$$n_{16} = 9.31786646179107$$
$$n_{17} = -81.6691650818489$$
$$n_{18} = -31.3840740178899$$
$$n_{19} = -94.2371684817036$$
$$n_{20} = 197.91528455229$$
$$n_{21} = 59.6735041304405$$
$$n_{22} = 97.3791034786112$$
$$n_{23} = -37.672573565113$$
$$n_{24} = -12.4864543952238$$
$$n_{25} = 15.644128370333$$
$$n_{26} = 72.2427897046973$$
$$n_{27} = -75.3849592185347$$
$$n_{28} = 34.5285657554621$$
$$n_{29} = 78.5270825679419$$
$$n_{30} = 40.8162093266346$$
$$n_{31} = 21.945612879981$$
$$n_{32} = -18.7964043662102$$
$$n_{33} = -56.5309801938186$$
$$n_{34} = -43.9595528888955$$
Puntos máximos de la función:
$$n_{34} = -65.9582857893902$$
$$n_{34} = -78.5270825679419$$
$$n_{34} = -21.945612879981$$
$$n_{34} = -34.5285657554621$$
$$n_{34} = 94.2371684817036$$
$$n_{34} = -135.08108127842$$
$$n_{34} = 50.2455828375744$$
$$n_{34} = -91.0952098694071$$
$$n_{34} = 56.5309801938186$$
$$n_{34} = -53.3883466217256$$
$$n_{34} = -109.946647805931$$
$$n_{34} = -47.1026627703624$$
$$n_{34} = 62.8159348889734$$
$$n_{34} = -2.79838604578389$$
$$n_{34} = 12.4864543952238$$
$$n_{34} = 75.3849592185347$$
$$n_{34} = -40.8162093266346$$
$$n_{34} = -15.644128370333$$
$$n_{34} = 69.100567727981$$
$$n_{34} = -84.811211299318$$
$$n_{34} = 31.3840740178899$$
$$n_{34} = 37.672573565113$$
$$n_{34} = -97.3791034786112$$
$$n_{34} = 6.12125046689807$$
$$n_{34} = 43.9595528888955$$
$$n_{34} = 100.521017074687$$
$$n_{34} = -9.31786646179107$$
$$n_{34} = 25.0929104121121$$
$$n_{34} = -72.2427897046973$$
$$n_{34} = 87.9532251106725$$
$$n_{34} = -59.6735041304405$$
$$n_{34} = -28.2389365752603$$
$$n_{34} = 18.7964043662102$$
$$n_{34} = 81.6691650818489$$
Decrece en los intervalos
$$\left[197.91528455229, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -169.640108529775\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0$$
(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:
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = $$
segunda derivada
$$\frac{- \cos{\left(n \right)} + \frac{2 \sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}}}{n} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$n_{1} = -4.2222763997912$$
$$n_{2} = -10.8095072981602$$
$$n_{3} = 42.3642737086586$$
$$n_{4} = -64.3715747870554$$
$$n_{5} = -13.9937625671267$$
$$n_{6} = 76.9430238267933$$
$$n_{7} = -7.5873993379941$$
$$n_{8} = 48.6535676048409$$
$$n_{9} = -80.0856368040887$$
$$n_{10} = 23.4766510546492$$
$$n_{11} = 89.5130456566371$$
$$n_{12} = 4.2222763997912$$
$$n_{13} = 73.8003238908837$$
$$n_{14} = -23.4766510546492$$
$$n_{15} = -86.370639887736$$
$$n_{16} = 58.085025007445$$
$$n_{17} = -20.3217772482235$$
$$n_{18} = -92.655396245836$$
$$n_{19} = 54.9414610202918$$
$$n_{20} = 29.7779159141436$$
$$n_{21} = 230.898398112111$$
$$n_{22} = 17.1619600917303$$
$$n_{23} = 32.9259431758392$$
$$n_{24} = -83.2281726832512$$
$$n_{25} = -73.8003238908837$$
$$n_{26} = 26.6283591640252$$
$$n_{27} = -271.740404503579$$
$$n_{28} = -61.2283863503723$$
$$n_{29} = 70.6575253785884$$
$$n_{30} = -95.7976970894915$$
$$n_{31} = 61.2283863503723$$
$$n_{32} = 83.2281726832512$$
$$n_{33} = 7.5873993379941$$
$$n_{34} = 39.218890250481$$
$$n_{35} = -26.6283591640252$$
$$n_{36} = 13.9937625671267$$
$$n_{37} = -29.7779159141436$$
$$n_{38} = -70.6575253785884$$
$$n_{39} = -42.3642737086586$$
$$n_{40} = -51.7976574095537$$
$$n_{41} = -48.6535676048409$$
$$n_{42} = -54.9414610202918$$
$$n_{43} = 51.7976574095537$$
$$n_{44} = -98.9399529307048$$
$$n_{45} = -67.5146145048817$$
$$n_{46} = 36.0728437679879$$
$$n_{47} = -39.218890250481$$
$$n_{48} = 20.3217772482235$$
$$n_{49} = -45.5091321154553$$
$$n_{50} = -32.9259431758392$$
$$n_{51} = -58.085025007445$$
$$n_{52} = 98.9399529307048$$
$$n_{53} = 45.5091321154553$$
$$n_{54} = 67.5146145048817$$
$$n_{55} = 92.655396245836$$
$$n_{56} = -36.0728437679879$$
$$n_{57} = 80.0856368040887$$
$$n_{58} = 86.370639887736$$
$$n_{59} = 95.7976970894915$$
$$n_{60} = -89.5130456566371$$
$$n_{61} = -17.1619600917303$$
$$n_{62} = 64.3715747870554$$
$$n_{63} = 10.8095072981602$$
$$n_{64} = -76.9430238267933$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$n_{1} = 0$$
$$\lim_{n \to 0^-}\left(\frac{- \cos{\left(n \right)} + \frac{2 \sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}}}{n}\right) = -\infty$$
$$\lim_{n \to 0^+}\left(\frac{- \cos{\left(n \right)} + \frac{2 \sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}}}{n}\right) = \infty$$
- los límites no son iguales, signo
$$n_{1} = 0$$
- 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
$$\left[95.7976970894915, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -271.740404503579\right]$$
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0$$
(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:
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = $$
segunda derivada
$$\frac{- \cos{\left(n \right)} + \frac{2 \sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}}}{n} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$n_{1} = -4.2222763997912$$
$$n_{2} = -10.8095072981602$$
$$n_{3} = 42.3642737086586$$
$$n_{4} = -64.3715747870554$$
$$n_{5} = -13.9937625671267$$
$$n_{6} = 76.9430238267933$$
$$n_{7} = -7.5873993379941$$
$$n_{8} = 48.6535676048409$$
$$n_{9} = -80.0856368040887$$
$$n_{10} = 23.4766510546492$$
$$n_{11} = 89.5130456566371$$
$$n_{12} = 4.2222763997912$$
$$n_{13} = 73.8003238908837$$
$$n_{14} = -23.4766510546492$$
$$n_{15} = -86.370639887736$$
$$n_{16} = 58.085025007445$$
$$n_{17} = -20.3217772482235$$
$$n_{18} = -92.655396245836$$
$$n_{19} = 54.9414610202918$$
$$n_{20} = 29.7779159141436$$
$$n_{21} = 230.898398112111$$
$$n_{22} = 17.1619600917303$$
$$n_{23} = 32.9259431758392$$
$$n_{24} = -83.2281726832512$$
$$n_{25} = -73.8003238908837$$
$$n_{26} = 26.6283591640252$$
$$n_{27} = -271.740404503579$$
$$n_{28} = -61.2283863503723$$
$$n_{29} = 70.6575253785884$$
$$n_{30} = -95.7976970894915$$
$$n_{31} = 61.2283863503723$$
$$n_{32} = 83.2281726832512$$
$$n_{33} = 7.5873993379941$$
$$n_{34} = 39.218890250481$$
$$n_{35} = -26.6283591640252$$
$$n_{36} = 13.9937625671267$$
$$n_{37} = -29.7779159141436$$
$$n_{38} = -70.6575253785884$$
$$n_{39} = -42.3642737086586$$
$$n_{40} = -51.7976574095537$$
$$n_{41} = -48.6535676048409$$
$$n_{42} = -54.9414610202918$$
$$n_{43} = 51.7976574095537$$
$$n_{44} = -98.9399529307048$$
$$n_{45} = -67.5146145048817$$
$$n_{46} = 36.0728437679879$$
$$n_{47} = -39.218890250481$$
$$n_{48} = 20.3217772482235$$
$$n_{49} = -45.5091321154553$$
$$n_{50} = -32.9259431758392$$
$$n_{51} = -58.085025007445$$
$$n_{52} = 98.9399529307048$$
$$n_{53} = 45.5091321154553$$
$$n_{54} = 67.5146145048817$$
$$n_{55} = 92.655396245836$$
$$n_{56} = -36.0728437679879$$
$$n_{57} = 80.0856368040887$$
$$n_{58} = 86.370639887736$$
$$n_{59} = 95.7976970894915$$
$$n_{60} = -89.5130456566371$$
$$n_{61} = -17.1619600917303$$
$$n_{62} = 64.3715747870554$$
$$n_{63} = 10.8095072981602$$
$$n_{64} = -76.9430238267933$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$n_{1} = 0$$
$$\lim_{n \to 0^-}\left(\frac{- \cos{\left(n \right)} + \frac{2 \sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}}}{n}\right) = -\infty$$
$$\lim_{n \to 0^+}\left(\frac{- \cos{\left(n \right)} + \frac{2 \sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}}}{n}\right) = \infty$$
- los límites no son iguales, signo
$$n_{1} = 0$$
- 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
$$\left[95.7976970894915, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -271.740404503579\right]$$
Asíntotas verticales
Hay:
$$n_{1} = 0$$
$$n_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con n->+oo y n->-oo
$$\lim_{n \to -\infty}\left(\frac{\cos{\left(n \right)}}{n}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{n \to \infty}\left(\frac{\cos{\left(n \right)}}{n}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{n \to -\infty}\left(\frac{\cos{\left(n \right)}}{n}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{n \to \infty}\left(\frac{\cos{\left(n \right)}}{n}\right) = 0$$
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(n)/n, dividida por n con n->+oo y n ->-oo
$$\lim_{n \to -\infty}\left(\frac{\cos{\left(n \right)}}{n^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{n \to \infty}\left(\frac{\cos{\left(n \right)}}{n^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{n \to -\infty}\left(\frac{\cos{\left(n \right)}}{n^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{n \to \infty}\left(\frac{\cos{\left(n \right)}}{n^{2}}\right) = 0$$
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(-n) и f = -f(-n).
Pues, comprobamos:
$$\frac{\cos{\left(n \right)}}{n} = - \frac{\cos{\left(n \right)}}{n}$$
- No
$$\frac{\cos{\left(n \right)}}{n} = \frac{\cos{\left(n \right)}}{n}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\cos{\left(n \right)}}{n} = - \frac{\cos{\left(n \right)}}{n}$$
- No
$$\frac{\cos{\left(n \right)}}{n} = \frac{\cos{\left(n \right)}}{n}$$
- No
es decir, función
no es
par ni impar