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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
y=(x+1)^3
-
y=3x^2-x^3
-
3-x^2
-
1/((x-1)^2)
-
Expresiones idénticas
- cos(x)/ cinco *atan(siete *x)
- coseno de (x) dividir por 5 multiplicar por arco tangente de gente de (7 multiplicar por x)
- coseno de (x) dividir por cinco multiplicar por arco tangente de gente de (siete multiplicar por x)
- cos(x)/5atan(7x)
- cosx/5atan7x
- cos(x) dividir por 5*atan(7*x)
-
Expresiones semejantes
- cos(x)/5*arctan(7*x)
- cosx/5*atan(7*x)
-
Expresiones con funciones
- Coseno cos
- cos(x^3)+x^3
- cos(x)*sin(x^2)
- cos[x]/x^2
- cos(x-pi/3)-1
- cos(x)/(1+cos^2*2x)
- Arcotangente arctan
- atan(36/(x-158))-atan(102/(x-113))-157*2/50/180
- atan(9*x)/((9*x))
- atan(x/10^6)
- atan(x^2+3*x)
- atan(-36/(x-158))-atan(-30/(x+40))-157*2/50/180
Gráfico de la función y = cos(x)/5*atan(7*x)
Solución
Ha introducido
[src]
cos(x)
f(x) = ------*atan(7*x)
5
$$f{\left(x \right)} = \frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)}$$
f = (cos(x)/5)*atan(7*x)
Gráfico de la función
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:
$$\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
Solución numérica
$$x_{1} = 48.6946861306418$$
$$x_{2} = 54.9778714378214$$
$$x_{3} = -98.9601685880785$$
$$x_{4} = 67.5442420521806$$
$$x_{5} = 76.9690200129499$$
$$x_{6} = 36.1283155162826$$
$$x_{7} = 58.1194640914112$$
$$x_{8} = 14.1371669411541$$
$$x_{9} = -29.845130209103$$
$$x_{10} = 61.261056745001$$
$$x_{11} = -36.1283155162826$$
$$x_{12} = -4.71238898038469$$
$$x_{13} = -39.2699081698724$$
$$x_{14} = 1.5707963267949$$
$$x_{15} = -14.1371669411541$$
$$x_{16} = -64.4026493985908$$
$$x_{17} = -67.5442420521806$$
$$x_{18} = 92.6769832808989$$
$$x_{19} = -51.8362787842316$$
$$x_{20} = -86.3937979737193$$
$$x_{21} = 42.4115008234622$$
$$x_{22} = -17.2787595947439$$
$$x_{23} = -45.553093477052$$
$$x_{24} = 9658.82661346182$$
$$x_{25} = -89.5353906273091$$
$$x_{26} = -1.5707963267949$$
$$x_{27} = 39.2699081698724$$
$$x_{28} = 23.5619449019235$$
$$x_{29} = 7.85398163397448$$
$$x_{30} = -58.1194640914112$$
$$x_{31} = -61.261056745001$$
$$x_{32} = -73.8274273593601$$
$$x_{33} = 73.8274273593601$$
$$x_{34} = 29.845130209103$$
$$x_{35} = 4.71238898038469$$
$$x_{36} = 0$$
$$x_{37} = 86.3937979737193$$
$$x_{38} = -453.960138443725$$
$$x_{39} = 64.4026493985908$$
$$x_{40} = 89.5353906273091$$
$$x_{41} = -20.4203522483337$$
$$x_{42} = -26.7035375555132$$
$$x_{43} = 98.9601685880785$$
$$x_{44} = 51.8362787842316$$
$$x_{45} = 83.2522053201295$$
$$x_{46} = -48.6946861306418$$
$$x_{47} = -54.9778714378214$$
$$x_{48} = 70.6858347057703$$
$$x_{49} = -95.8185759344887$$
$$x_{50} = 26.7035375555132$$
$$x_{51} = 80.1106126665397$$
$$x_{52} = 5945.46409691868$$
$$x_{53} = -23.5619449019235$$
$$x_{54} = -7.85398163397448$$
$$x_{55} = -83.2522053201295$$
$$x_{56} = -76.9690200129499$$
$$x_{57} = -42.4115008234622$$
$$x_{58} = -32.9867228626928$$
$$x_{59} = 17.2787595947439$$
$$x_{60} = 32.9867228626928$$
$$x_{61} = 20.4203522483337$$
$$x_{62} = -70.6858347057703$$
$$x_{63} = -10.9955742875643$$
$$x_{64} = -92.6769832808989$$
$$x_{65} = 45.553093477052$$
$$x_{66} = 10.9955742875643$$
$$x_{67} = -80.1106126665397$$
$$x_{68} = 95.8185759344887$$
o sea hay que resolver la ecuación:
$$\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
Solución numérica
$$x_{1} = 48.6946861306418$$
$$x_{2} = 54.9778714378214$$
$$x_{3} = -98.9601685880785$$
$$x_{4} = 67.5442420521806$$
$$x_{5} = 76.9690200129499$$
$$x_{6} = 36.1283155162826$$
$$x_{7} = 58.1194640914112$$
$$x_{8} = 14.1371669411541$$
$$x_{9} = -29.845130209103$$
$$x_{10} = 61.261056745001$$
$$x_{11} = -36.1283155162826$$
$$x_{12} = -4.71238898038469$$
$$x_{13} = -39.2699081698724$$
$$x_{14} = 1.5707963267949$$
$$x_{15} = -14.1371669411541$$
$$x_{16} = -64.4026493985908$$
$$x_{17} = -67.5442420521806$$
$$x_{18} = 92.6769832808989$$
$$x_{19} = -51.8362787842316$$
$$x_{20} = -86.3937979737193$$
$$x_{21} = 42.4115008234622$$
$$x_{22} = -17.2787595947439$$
$$x_{23} = -45.553093477052$$
$$x_{24} = 9658.82661346182$$
$$x_{25} = -89.5353906273091$$
$$x_{26} = -1.5707963267949$$
$$x_{27} = 39.2699081698724$$
$$x_{28} = 23.5619449019235$$
$$x_{29} = 7.85398163397448$$
$$x_{30} = -58.1194640914112$$
$$x_{31} = -61.261056745001$$
$$x_{32} = -73.8274273593601$$
$$x_{33} = 73.8274273593601$$
$$x_{34} = 29.845130209103$$
$$x_{35} = 4.71238898038469$$
$$x_{36} = 0$$
$$x_{37} = 86.3937979737193$$
$$x_{38} = -453.960138443725$$
$$x_{39} = 64.4026493985908$$
$$x_{40} = 89.5353906273091$$
$$x_{41} = -20.4203522483337$$
$$x_{42} = -26.7035375555132$$
$$x_{43} = 98.9601685880785$$
$$x_{44} = 51.8362787842316$$
$$x_{45} = 83.2522053201295$$
$$x_{46} = -48.6946861306418$$
$$x_{47} = -54.9778714378214$$
$$x_{48} = 70.6858347057703$$
$$x_{49} = -95.8185759344887$$
$$x_{50} = 26.7035375555132$$
$$x_{51} = 80.1106126665397$$
$$x_{52} = 5945.46409691868$$
$$x_{53} = -23.5619449019235$$
$$x_{54} = -7.85398163397448$$
$$x_{55} = -83.2522053201295$$
$$x_{56} = -76.9690200129499$$
$$x_{57} = -42.4115008234622$$
$$x_{58} = -32.9867228626928$$
$$x_{59} = 17.2787595947439$$
$$x_{60} = 32.9867228626928$$
$$x_{61} = 20.4203522483337$$
$$x_{62} = -70.6858347057703$$
$$x_{63} = -10.9955742875643$$
$$x_{64} = -92.6769832808989$$
$$x_{65} = 45.553093477052$$
$$x_{66} = 10.9955742875643$$
$$x_{67} = -80.1106126665397$$
$$x_{68} = 95.8185759344887$$
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
$$- \frac{\sin{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5} + \frac{7 \cos{\left(x \right)}}{5 \left(49 x^{2} + 1\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 40.8407591425359$$
$$x_{2} = 100.530973921756$$
$$x_{3} = -28.2744480075961$$
$$x_{4} = 69.1150574426549$$
$$x_{5} = 3.15100482407892$$
$$x_{6} = 62.8318761418628$$
$$x_{7} = 84.823014300677$$
$$x_{8} = -34.5575955435353$$
$$x_{9} = -59.69028598254$$
$$x_{10} = -50.2655185173536$$
$$x_{11} = -72.256648473579$$
$$x_{12} = 50.2655185173536$$
$$x_{13} = -47.1239308369192$$
$$x_{14} = 97.3893818589064$$
$$x_{15} = -65.9734666491843$$
$$x_{16} = 87.964606066119$$
$$x_{17} = 37.6991759878229$$
$$x_{18} = -18.8498131052308$$
$$x_{19} = 25.1328857253434$$
$$x_{20} = 34.5575955435353$$
$$x_{21} = 21.9913374004364$$
$$x_{22} = -78.5398311003412$$
$$x_{23} = -84.823014300677$$
$$x_{24} = 43.9823442609731$$
$$x_{25} = 12.5669506041308$$
$$x_{26} = -53.4071070500068$$
$$x_{27} = 18.8498131052308$$
$$x_{28} = -125.663711906957$$
$$x_{29} = -6.28551985758014$$
$$x_{30} = -43.9823442609731$$
$$x_{31} = -91.1061979218887$$
$$x_{32} = -9.42581132885092$$
$$x_{33} = 91.1061979218887$$
$$x_{34} = -75.3982397031955$$
$$x_{35} = 72.256648473579$$
$$x_{36} = -37.6991759878229$$
$$x_{37} = -94.2477898561398$$
$$x_{38} = -97.3893818589064$$
$$x_{39} = 56.548696250726$$
$$x_{40} = 65.9734666491843$$
$$x_{41} = 31.4160189482124$$
$$x_{42} = -81.6814226397312$$
$$x_{43} = 15.7083339551082$$
$$x_{44} = 9.42581132885092$$
$$x_{45} = 47.1239308369192$$
$$x_{46} = 28.2744480075961$$
$$x_{47} = -21.9913374004364$$
$$x_{48} = 78.5398311003412$$
$$x_{49} = -69.1150574426549$$
$$x_{50} = -62.8318761418628$$
$$x_{51} = 81.6814226397312$$
$$x_{52} = -25.1328857253434$$
$$x_{53} = -40.8407591425359$$
$$x_{54} = 53.4071070500068$$
$$x_{55} = -15.7083339551082$$
$$x_{56} = 0.456935759159445$$
$$x_{57} = 75.3982397031955$$
$$x_{58} = -1448.27421334826$$
$$x_{59} = 94.2477898561398$$
$$x_{60} = -3.15100482407892$$
$$x_{61} = -12.5669506041308$$
$$x_{62} = -56.548696250726$$
$$x_{63} = -100.530973921756$$
$$x_{64} = -31.4160189482124$$
$$x_{65} = 6.28551985758014$$
$$x_{66} = -87.964606066119$$
$$x_{67} = 59.69028598254$$
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:
$$x_{1} = 40.8407591425359$$
$$x_{2} = 3.15100482407892$$
$$x_{3} = 84.823014300677$$
$$x_{4} = -50.2655185173536$$
$$x_{5} = 97.3893818589064$$
$$x_{6} = -18.8498131052308$$
$$x_{7} = 34.5575955435353$$
$$x_{8} = 21.9913374004364$$
$$x_{9} = -125.663711906957$$
$$x_{10} = -6.28551985758014$$
$$x_{11} = -43.9823442609731$$
$$x_{12} = 91.1061979218887$$
$$x_{13} = -75.3982397031955$$
$$x_{14} = 72.256648473579$$
$$x_{15} = -37.6991759878229$$
$$x_{16} = -94.2477898561398$$
$$x_{17} = 65.9734666491843$$
$$x_{18} = -81.6814226397312$$
$$x_{19} = 15.7083339551082$$
$$x_{20} = 9.42581132885092$$
$$x_{21} = 47.1239308369192$$
$$x_{22} = 28.2744480075961$$
$$x_{23} = 78.5398311003412$$
$$x_{24} = -69.1150574426549$$
$$x_{25} = -62.8318761418628$$
$$x_{26} = -25.1328857253434$$
$$x_{27} = 53.4071070500068$$
$$x_{28} = -12.5669506041308$$
$$x_{29} = -56.548696250726$$
$$x_{30} = -100.530973921756$$
$$x_{31} = -31.4160189482124$$
$$x_{32} = -87.964606066119$$
$$x_{33} = 59.69028598254$$
Puntos máximos de la función:
$$x_{33} = 100.530973921756$$
$$x_{33} = -28.2744480075961$$
$$x_{33} = 69.1150574426549$$
$$x_{33} = 62.8318761418628$$
$$x_{33} = -34.5575955435353$$
$$x_{33} = -59.69028598254$$
$$x_{33} = -72.256648473579$$
$$x_{33} = 50.2655185173536$$
$$x_{33} = -47.1239308369192$$
$$x_{33} = -65.9734666491843$$
$$x_{33} = 87.964606066119$$
$$x_{33} = 37.6991759878229$$
$$x_{33} = 25.1328857253434$$
$$x_{33} = -78.5398311003412$$
$$x_{33} = -84.823014300677$$
$$x_{33} = 43.9823442609731$$
$$x_{33} = 12.5669506041308$$
$$x_{33} = -53.4071070500068$$
$$x_{33} = 18.8498131052308$$
$$x_{33} = -91.1061979218887$$
$$x_{33} = -9.42581132885092$$
$$x_{33} = -97.3893818589064$$
$$x_{33} = 56.548696250726$$
$$x_{33} = 31.4160189482124$$
$$x_{33} = -21.9913374004364$$
$$x_{33} = 81.6814226397312$$
$$x_{33} = -40.8407591425359$$
$$x_{33} = -15.7083339551082$$
$$x_{33} = 0.456935759159445$$
$$x_{33} = 75.3982397031955$$
$$x_{33} = -1448.27421334826$$
$$x_{33} = 94.2477898561398$$
$$x_{33} = -3.15100482407892$$
$$x_{33} = 6.28551985758014$$
Decrece en los intervalos
$$\left[97.3893818589064, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -125.663711906957\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{\sin{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5} + \frac{7 \cos{\left(x \right)}}{5 \left(49 x^{2} + 1\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 40.8407591425359$$
$$x_{2} = 100.530973921756$$
$$x_{3} = -28.2744480075961$$
$$x_{4} = 69.1150574426549$$
$$x_{5} = 3.15100482407892$$
$$x_{6} = 62.8318761418628$$
$$x_{7} = 84.823014300677$$
$$x_{8} = -34.5575955435353$$
$$x_{9} = -59.69028598254$$
$$x_{10} = -50.2655185173536$$
$$x_{11} = -72.256648473579$$
$$x_{12} = 50.2655185173536$$
$$x_{13} = -47.1239308369192$$
$$x_{14} = 97.3893818589064$$
$$x_{15} = -65.9734666491843$$
$$x_{16} = 87.964606066119$$
$$x_{17} = 37.6991759878229$$
$$x_{18} = -18.8498131052308$$
$$x_{19} = 25.1328857253434$$
$$x_{20} = 34.5575955435353$$
$$x_{21} = 21.9913374004364$$
$$x_{22} = -78.5398311003412$$
$$x_{23} = -84.823014300677$$
$$x_{24} = 43.9823442609731$$
$$x_{25} = 12.5669506041308$$
$$x_{26} = -53.4071070500068$$
$$x_{27} = 18.8498131052308$$
$$x_{28} = -125.663711906957$$
$$x_{29} = -6.28551985758014$$
$$x_{30} = -43.9823442609731$$
$$x_{31} = -91.1061979218887$$
$$x_{32} = -9.42581132885092$$
$$x_{33} = 91.1061979218887$$
$$x_{34} = -75.3982397031955$$
$$x_{35} = 72.256648473579$$
$$x_{36} = -37.6991759878229$$
$$x_{37} = -94.2477898561398$$
$$x_{38} = -97.3893818589064$$
$$x_{39} = 56.548696250726$$
$$x_{40} = 65.9734666491843$$
$$x_{41} = 31.4160189482124$$
$$x_{42} = -81.6814226397312$$
$$x_{43} = 15.7083339551082$$
$$x_{44} = 9.42581132885092$$
$$x_{45} = 47.1239308369192$$
$$x_{46} = 28.2744480075961$$
$$x_{47} = -21.9913374004364$$
$$x_{48} = 78.5398311003412$$
$$x_{49} = -69.1150574426549$$
$$x_{50} = -62.8318761418628$$
$$x_{51} = 81.6814226397312$$
$$x_{52} = -25.1328857253434$$
$$x_{53} = -40.8407591425359$$
$$x_{54} = 53.4071070500068$$
$$x_{55} = -15.7083339551082$$
$$x_{56} = 0.456935759159445$$
$$x_{57} = 75.3982397031955$$
$$x_{58} = -1448.27421334826$$
$$x_{59} = 94.2477898561398$$
$$x_{60} = -3.15100482407892$$
$$x_{61} = -12.5669506041308$$
$$x_{62} = -56.548696250726$$
$$x_{63} = -100.530973921756$$
$$x_{64} = -31.4160189482124$$
$$x_{65} = 6.28551985758014$$
$$x_{66} = -87.964606066119$$
$$x_{67} = 59.69028598254$$
Signos de extremos en los puntos:
(40.840759142535944, -0.313459686512763)
(100.53097392175573, 0.313875060307489)
(-28.274448007596067, 0.313148768421786)
(69.1150574426549, 0.313745876542117)
(3.1510048240789175, -0.305084553805719)
(62.83187614186276, 0.313704537817189)
(84.82301430067703, -0.313822429844181)
(-34.557595543535314, 0.313332492056769)
(-59.69028598254, 0.313680604892405)
(-50.265518517353634, -0.313590856582214)
(-72.25664847357905, 0.313763849914225)
(50.265518517353634, 0.313590856582214)
(-47.12393083691916, 0.313552962935163)
(97.38938185890642, -0.313865892416862)
(-65.97346664918427, 0.313726191429446)
(87.96460606611903, 0.313834459659947)
(37.69917598782292, 0.313401388949929)
(-18.849813105230847, -0.312643543356069)
(25.13288572534344, 0.313022459847738)
(34.557595543535314, -0.313332492056769)
(21.991337400436436, -0.312860065186453)
(-78.53983110034122, 0.313795483067281)
(-84.82301430067703, 0.313822429844181)
(43.98234426097308, 0.313509655978696)
(12.566950604130808, 0.311885773715038)
(-53.40710705000676, 0.313624292196214)
(18.849813105230847, 0.312643543356069)
(-125.66371190695742, -0.313931901257722)
(-6.285519857580139, -0.309613608941595)
(-43.98234426097308, -0.313509655978696)
(-91.10619792188866, 0.313845659835682)
(-9.42581132885092, 0.311128140924491)
(91.10619792188866, -0.313845659835682)
(-75.39823970319554, -0.313780325511985)
(72.25664847357905, -0.313763849914225)
(-37.69917598782292, -0.313401388949929)
(-94.24778985613978, -0.313856113335073)
(-97.38938185890642, 0.313865892416862)
(56.54869625072598, 0.31365401277349)
(65.97346664918427, -0.313726191429446)
(31.416018948212397, 0.313249816147344)
(-81.68142263973122, -0.313809474661132)
(15.708333955108206, -0.312340423329518)
(9.42581132885092, -0.311128140924491)
(47.12393083691916, -0.313552962935163)
(28.274448007596067, -0.313148768421786)
(-21.991337400436436, 0.312860065186453)
(78.53983110034122, -0.313795483067281)
(-69.1150574426549, -0.313745876542117)
(-62.83187614186276, -0.313704537817189)
(81.68142263973122, 0.313809474661132)
(-25.13288572534344, -0.313022459847738)
(-40.840759142535944, 0.313459686512763)
(53.40710705000676, -0.313624292196214)
(-15.708333955108206, 0.312340423329518)
(0.4569357591594449, 0.227543784365642)
(75.39823970319554, 0.313780325511985)
(-1448.2742133482566, 0.314139537445428)
(94.24778985613978, 0.313856113335073)
(-3.1510048240789175, 0.305084553805719)
(-12.566950604130808, -0.311885773715038)
(-56.54869625072598, -0.31365401277349)
(-100.53097392175573, -0.313875060307489)
(-31.416018948212397, -0.313249816147344)
(6.285519857580139, 0.309613608941595)
(-87.96460606611903, -0.313834459659947)
(59.69028598254, -0.313680604892405)
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:
$$x_{1} = 40.8407591425359$$
$$x_{2} = 3.15100482407892$$
$$x_{3} = 84.823014300677$$
$$x_{4} = -50.2655185173536$$
$$x_{5} = 97.3893818589064$$
$$x_{6} = -18.8498131052308$$
$$x_{7} = 34.5575955435353$$
$$x_{8} = 21.9913374004364$$
$$x_{9} = -125.663711906957$$
$$x_{10} = -6.28551985758014$$
$$x_{11} = -43.9823442609731$$
$$x_{12} = 91.1061979218887$$
$$x_{13} = -75.3982397031955$$
$$x_{14} = 72.256648473579$$
$$x_{15} = -37.6991759878229$$
$$x_{16} = -94.2477898561398$$
$$x_{17} = 65.9734666491843$$
$$x_{18} = -81.6814226397312$$
$$x_{19} = 15.7083339551082$$
$$x_{20} = 9.42581132885092$$
$$x_{21} = 47.1239308369192$$
$$x_{22} = 28.2744480075961$$
$$x_{23} = 78.5398311003412$$
$$x_{24} = -69.1150574426549$$
$$x_{25} = -62.8318761418628$$
$$x_{26} = -25.1328857253434$$
$$x_{27} = 53.4071070500068$$
$$x_{28} = -12.5669506041308$$
$$x_{29} = -56.548696250726$$
$$x_{30} = -100.530973921756$$
$$x_{31} = -31.4160189482124$$
$$x_{32} = -87.964606066119$$
$$x_{33} = 59.69028598254$$
Puntos máximos de la función:
$$x_{33} = 100.530973921756$$
$$x_{33} = -28.2744480075961$$
$$x_{33} = 69.1150574426549$$
$$x_{33} = 62.8318761418628$$
$$x_{33} = -34.5575955435353$$
$$x_{33} = -59.69028598254$$
$$x_{33} = -72.256648473579$$
$$x_{33} = 50.2655185173536$$
$$x_{33} = -47.1239308369192$$
$$x_{33} = -65.9734666491843$$
$$x_{33} = 87.964606066119$$
$$x_{33} = 37.6991759878229$$
$$x_{33} = 25.1328857253434$$
$$x_{33} = -78.5398311003412$$
$$x_{33} = -84.823014300677$$
$$x_{33} = 43.9823442609731$$
$$x_{33} = 12.5669506041308$$
$$x_{33} = -53.4071070500068$$
$$x_{33} = 18.8498131052308$$
$$x_{33} = -91.1061979218887$$
$$x_{33} = -9.42581132885092$$
$$x_{33} = -97.3893818589064$$
$$x_{33} = 56.548696250726$$
$$x_{33} = 31.4160189482124$$
$$x_{33} = -21.9913374004364$$
$$x_{33} = 81.6814226397312$$
$$x_{33} = -40.8407591425359$$
$$x_{33} = -15.7083339551082$$
$$x_{33} = 0.456935759159445$$
$$x_{33} = 75.3982397031955$$
$$x_{33} = -1448.27421334826$$
$$x_{33} = 94.2477898561398$$
$$x_{33} = -3.15100482407892$$
$$x_{33} = 6.28551985758014$$
Decrece en los intervalos
$$\left[97.3893818589064, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -125.663711906957\right]$$
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
$$- \frac{\frac{686 x \cos{\left(x \right)}}{\left(49 x^{2} + 1\right)^{2}} + \cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)} + \frac{14 \sin{\left(x \right)}}{49 x^{2} + 1}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 108.384962045496$$
$$x_{2} = -98.9601871785062$$
$$x_{3} = 73.8274607719173$$
$$x_{4} = -70.6858711563115$$
$$x_{5} = 51.8363465955313$$
$$x_{6} = -7.85696048829761$$
$$x_{7} = 26.7037934913803$$
$$x_{8} = 95.8185957645393$$
$$x_{9} = 64.4026933138351$$
$$x_{10} = -23.5622737801745$$
$$x_{11} = -1.63889828017201$$
$$x_{12} = 7.85696048829761$$
$$x_{13} = 48.6947629826139$$
$$x_{14} = -36.1284552172578$$
$$x_{15} = 89.5354133396834$$
$$x_{16} = -73.8274607719173$$
$$x_{17} = 54.9779317147184$$
$$x_{18} = 17.2793719491254$$
$$x_{19} = -20.420790349936$$
$$x_{20} = 1.63889828017201$$
$$x_{21} = 23.5622737801745$$
$$x_{22} = 76.9690507520825$$
$$x_{23} = -10.9970903927802$$
$$x_{24} = 4.72068894233548$$
$$x_{25} = 42.4116021606933$$
$$x_{26} = -14.1380826577008$$
$$x_{27} = -48.6947629826139$$
$$x_{28} = -61.2611052833421$$
$$x_{29} = -54.9779317147184$$
$$x_{30} = -89.5354133396834$$
$$x_{31} = 70.6858711563115$$
$$x_{32} = -17.2793719491254$$
$$x_{33} = 67.544281974692$$
$$x_{34} = -64.4026933138351$$
$$x_{35} = 86.3938223688308$$
$$x_{36} = 10.9970903927802$$
$$x_{37} = -95.8185957645393$$
$$x_{38} = -86.3938223688308$$
$$x_{39} = 98.9601871785062$$
$$x_{40} = 29.8453350287588$$
$$x_{41} = 83.2522315921566$$
$$x_{42} = 0$$
$$x_{43} = -39.2700263894989$$
$$x_{44} = -92.6770044788244$$
$$x_{45} = -32.9868904796577$$
$$x_{46} = -83.2522315921566$$
$$x_{47} = -76.9690507520825$$
$$x_{48} = 14.1380826577008$$
$$x_{49} = -67.544281974692$$
$$x_{50} = -42.4116021606933$$
$$x_{51} = 61.2611052833421$$
$$x_{52} = -26.7037934913803$$
$$x_{53} = 92.6770044788244$$
$$x_{54} = 80.1106410407278$$
$$x_{55} = 20.420790349936$$
$$x_{56} = -4.72068894233548$$
$$x_{57} = -300.022100439155$$
$$x_{58} = 45.5531813059564$$
$$x_{59} = 32.9868904796577$$
$$x_{60} = -80.1106410407278$$
$$x_{61} = -51.8363465955313$$
$$x_{62} = -58.1195180232416$$
$$x_{63} = -29.8453350287588$$
$$x_{64} = 36.1284552172578$$
$$x_{65} = 58.1195180232416$$
$$x_{66} = -45.5531813059564$$
$$x_{67} = 39.2700263894989$$
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
$$\left[108.384962045496, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.8185957645393\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\frac{686 x \cos{\left(x \right)}}{\left(49 x^{2} + 1\right)^{2}} + \cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)} + \frac{14 \sin{\left(x \right)}}{49 x^{2} + 1}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 108.384962045496$$
$$x_{2} = -98.9601871785062$$
$$x_{3} = 73.8274607719173$$
$$x_{4} = -70.6858711563115$$
$$x_{5} = 51.8363465955313$$
$$x_{6} = -7.85696048829761$$
$$x_{7} = 26.7037934913803$$
$$x_{8} = 95.8185957645393$$
$$x_{9} = 64.4026933138351$$
$$x_{10} = -23.5622737801745$$
$$x_{11} = -1.63889828017201$$
$$x_{12} = 7.85696048829761$$
$$x_{13} = 48.6947629826139$$
$$x_{14} = -36.1284552172578$$
$$x_{15} = 89.5354133396834$$
$$x_{16} = -73.8274607719173$$
$$x_{17} = 54.9779317147184$$
$$x_{18} = 17.2793719491254$$
$$x_{19} = -20.420790349936$$
$$x_{20} = 1.63889828017201$$
$$x_{21} = 23.5622737801745$$
$$x_{22} = 76.9690507520825$$
$$x_{23} = -10.9970903927802$$
$$x_{24} = 4.72068894233548$$
$$x_{25} = 42.4116021606933$$
$$x_{26} = -14.1380826577008$$
$$x_{27} = -48.6947629826139$$
$$x_{28} = -61.2611052833421$$
$$x_{29} = -54.9779317147184$$
$$x_{30} = -89.5354133396834$$
$$x_{31} = 70.6858711563115$$
$$x_{32} = -17.2793719491254$$
$$x_{33} = 67.544281974692$$
$$x_{34} = -64.4026933138351$$
$$x_{35} = 86.3938223688308$$
$$x_{36} = 10.9970903927802$$
$$x_{37} = -95.8185957645393$$
$$x_{38} = -86.3938223688308$$
$$x_{39} = 98.9601871785062$$
$$x_{40} = 29.8453350287588$$
$$x_{41} = 83.2522315921566$$
$$x_{42} = 0$$
$$x_{43} = -39.2700263894989$$
$$x_{44} = -92.6770044788244$$
$$x_{45} = -32.9868904796577$$
$$x_{46} = -83.2522315921566$$
$$x_{47} = -76.9690507520825$$
$$x_{48} = 14.1380826577008$$
$$x_{49} = -67.544281974692$$
$$x_{50} = -42.4116021606933$$
$$x_{51} = 61.2611052833421$$
$$x_{52} = -26.7037934913803$$
$$x_{53} = 92.6770044788244$$
$$x_{54} = 80.1106410407278$$
$$x_{55} = 20.420790349936$$
$$x_{56} = -4.72068894233548$$
$$x_{57} = -300.022100439155$$
$$x_{58} = 45.5531813059564$$
$$x_{59} = 32.9868904796577$$
$$x_{60} = -80.1106410407278$$
$$x_{61} = -51.8363465955313$$
$$x_{62} = -58.1195180232416$$
$$x_{63} = -29.8453350287588$$
$$x_{64} = 36.1284552172578$$
$$x_{65} = 58.1195180232416$$
$$x_{66} = -45.5531813059564$$
$$x_{67} = 39.2700263894989$$
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
$$\left[108.384962045496, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.8185957645393\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)}\right) = \left\langle - \frac{1}{10}, \frac{1}{10}\right\rangle \pi$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{1}{10}, \frac{1}{10}\right\rangle \pi$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)}\right) = \left\langle - \frac{1}{10}, \frac{1}{10}\right\rangle \pi$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{1}{10}, \frac{1}{10}\right\rangle \pi$$
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)}\right) = \left\langle - \frac{1}{10}, \frac{1}{10}\right\rangle \pi$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{1}{10}, \frac{1}{10}\right\rangle \pi$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)}\right) = \left\langle - \frac{1}{10}, \frac{1}{10}\right\rangle \pi$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{1}{10}, \frac{1}{10}\right\rangle \pi$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (cos(x)/5)*atan(7*x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5 x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5 x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5 x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5 x}\right) = 0$$
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:
$$\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)} = - \frac{\cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5}$$
- No
$$\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)} = \frac{\cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)} = - \frac{\cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5}$$
- No
$$\frac{\cos{\left(x \right)}}{5} \operatorname{atan}{\left(7 x \right)} = \frac{\cos{\left(x \right)} \operatorname{atan}{\left(7 x \right)}}{5}$$
- No
es decir, función
no es
par ni impar