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(1+sin(x)^2/cos(x)^2+2*sin(x)/cos(x))/x^2

Gráfico de la función y = (1+sin(x)^2/cos(x)^2+2*sin(x)/cos(x))/x^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src]
              2              
           sin (x)   2*sin(x)
       1 + ------- + --------
              2       cos(x) 
           cos (x)           
f(x) = ----------------------
                  2          
                 x           
$$f{\left(x \right)} = \frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{x^{2}}$$
f = (sin(x)^2/cos(x)^2 + 1 + (2*sin(x))/cos(x))/x^2
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = 4.71238898038469$$
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:
$$\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{x^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
$$x_{1} = - \frac{\pi}{4}$$
$$x_{2} = \frac{3 \pi}{4}$$
Solución numérica
$$x_{1} = -76.1836217656357$$
$$x_{2} = 36.913713940677$$
$$x_{3} = -47.9092876749821$$
$$x_{4} = -32.2013246101807$$
$$x_{5} = 24.3473437194866$$
$$x_{6} = -22.7765465295636$$
$$x_{7} = -85.6083991237338$$
$$x_{8} = 87.1791963332033$$
$$x_{9} = 33.7721211162059$$
$$x_{10} = -25.9181372926128$$
$$x_{11} = 30.6305287539402$$
$$x_{12} = 80.896011106149$$
$$x_{13} = -76.1836175363196$$
$$x_{14} = -3.92699148930622$$
$$x_{15} = 74.6128259361381$$
$$x_{16} = -91.8915842141087$$
$$x_{17} = 62.0464547630837$$
$$x_{18} = -41.6261027701397$$
$$x_{19} = 49.4800844307755$$
$$x_{20} = -73.0420289245632$$
$$x_{21} = 52.6216762089118$$
$$x_{22} = 84.0376033955208$$
$$x_{23} = 90.3207887333398$$
$$x_{24} = -47.909286073736$$
$$x_{25} = -54.1924700173873$$
$$x_{26} = 11.7809725354481$$
$$x_{27} = 18.0641594503677$$
$$x_{28} = 55.7632696938929$$
$$x_{29} = -95.0331775046949$$
$$x_{30} = 96.6039745209417$$
$$x_{31} = 84.0376057158837$$
$$x_{32} = -41.6261019164888$$
$$x_{33} = 52.6216773488977$$
$$x_{34} = 14.9225653404353$$
$$x_{35} = -91.8915834294438$$
$$x_{36} = -32.2013226266388$$
$$x_{37} = 65.1880477544721$$
$$x_{38} = -13.351768277298$$
$$x_{39} = -57.334066921177$$
$$x_{40} = 46.338491566334$$
$$x_{41} = 27.4889358506937$$
$$x_{42} = -7.06858308172745$$
$$x_{43} = -10.2101750589792$$
$$x_{44} = 18.0641574833979$$
$$x_{45} = -85.6083999414882$$
$$x_{46} = 93.4623815860925$$
$$x_{47} = 71.4712330087456$$
$$x_{48} = -19.6349532575436$$
$$x_{49} = -66.7588436961489$$
$$x_{50} = 90.3207895546285$$
$$x_{51} = 74.6128248314062$$
$$x_{52} = 30.6305275786058$$
$$x_{53} = -29.05973175402$$
$$x_{54} = -10.2101760280894$$
$$x_{55} = 80.8960080702851$$
$$x_{56} = 46.3384923479958$$
$$x_{57} = -82.4668070215542$$
$$x_{58} = 2.35619438891923$$
$$x_{59} = 58.9048625249913$$
$$x_{60} = 5.49778724526066$$
$$x_{61} = -98.1747703430083$$
$$x_{62} = -25.9181388181681$$
$$x_{63} = -19.634954184651$$
$$x_{64} = 2.35619473376461$$
$$x_{65} = -51.0508803425116$$
$$x_{66} = 68.3296409542489$$
$$x_{67} = -88.7499922747363$$
$$x_{68} = -60.4756584439888$$
$$x_{69} = 43.1968991742637$$
$$x_{70} = 21.2057505881215$$
$$x_{71} = -98.1747651988069$$
$$x_{72} = -101.316362685381$$
$$x_{73} = 14.9225655842303$$
$$x_{74} = 40.055308278761$$
$$x_{75} = -54.1924731881503$$
$$x_{76} = -35.3429182768592$$
$$x_{77} = 8.63938010203419$$
$$x_{78} = 68.3296401500172$$
$$x_{79} = 62.0464570060655$$
$$x_{80} = 36.913713375867$$
$$x_{81} = 40.0553060662772$$
$$x_{82} = -13.3517696183926$$
$$x_{83} = 99.7455668480587$$
$$x_{84} = -69.9004347650231$$
$$x_{85} = 8.63937890238161$$
$$x_{86} = 58.9048607247935$$
$$x_{87} = 77.7544182710727$$
$$x_{88} = -79.3252155672327$$
$$x_{89} = -63.617251355641$$
$$x_{90} = -16.4933612804894$$
$$x_{91} = -38.4845098652527$$
$$x_{92} = -79.3252141018114$$
$$x_{93} = -44.7676951159933$$
$$x_{94} = -69.9004363165473$$
$$x_{95} = 24.3473429820664$$
$$x_{96} = -57.3340655156019$$
$$x_{97} = 96.6039734495431$$
$$x_{98} = -35.3429169217907$$
$$x_{99} = -63.617250525522$$
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (1 + sin(x)^2/cos(x)^2 + (2*sin(x))/cos(x))/x^2.
$$\frac{\frac{2 \sin{\left(0 \right)}}{\cos{\left(0 \right)}} + \left(\frac{\sin^{2}{\left(0 \right)}}{\cos^{2}{\left(0 \right)}} + 1\right)}{0^{2}}$$
Resultado:
$$f{\left(0 \right)} = \tilde{\infty}$$
signof no cruza Y
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
$$\frac{\frac{2 \sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2}{x^{2}} - \frac{2 \left(\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 21.2057504117311$$
$$x_{2} = 68.329640215578$$
$$x_{3} = 84.037603483527$$
$$x_{4} = 62.0464549083984$$
$$x_{5} = -63.6172512351933$$
$$x_{6} = -73.0420291959627$$
$$x_{7} = -35.3429173528852$$
$$x_{8} = -98.174770424681$$
$$x_{9} = -38.484510006475$$
$$x_{10} = -91.8915851175014$$
$$x_{11} = 8.63937979737193$$
$$x_{12} = 74.6128255227576$$
$$x_{13} = -51.0508806208341$$
$$x_{14} = -10.2101761241668$$
$$x_{15} = 58.9048622548086$$
$$x_{16} = 80.8960108299372$$
$$x_{17} = -54.1924732744239$$
$$x_{18} = -85.6083998103219$$
$$x_{19} = 18.0641577581413$$
$$x_{20} = -13.3517687777566$$
$$x_{21} = -47.9092879672443$$
$$x_{22} = 55.7632696012188$$
$$x_{23} = 46.3384916404494$$
$$x_{24} = 40.0553063332699$$
$$x_{25} = -79.3252145031423$$
$$x_{26} = 33.7721210260903$$
$$x_{27} = -101.316363078271$$
$$x_{28} = -16.4933614313464$$
$$x_{29} = -66.7588438887831$$
$$x_{30} = -57.3340659280137$$
$$x_{31} = 65.1880475619882$$
$$x_{32} = 96.6039740978861$$
$$x_{33} = -41.6261026600648$$
$$x_{34} = -29.0597320457056$$
$$x_{35} = -44.7676953136546$$
$$x_{36} = -22.776546738526$$
$$x_{37} = -69.9004365423729$$
$$x_{38} = 52.621676947629$$
$$x_{39} = -76.1836218495525$$
$$x_{40} = -25.9181393921158$$
$$x_{41} = 36.9137136796801$$
$$x_{42} = 30.6305283725005$$
$$x_{43} = 14.9225651045515$$
$$x_{44} = -19.6349540849362$$
$$x_{45} = 99.7455667514759$$
$$x_{46} = 2.35619449019234$$
$$x_{47} = -258.39599575776$$
$$x_{48} = 87.1791961371168$$
$$x_{49} = -82.4668071567321$$
$$x_{50} = -60.4756585816035$$
$$x_{51} = -3.92699081698724$$
$$x_{52} = -88.7499924639117$$
$$x_{53} = 90.3207887907066$$
$$x_{54} = -32.2013246992954$$
$$x_{55} = 43.1968989868597$$
$$x_{56} = -95.0331777710912$$
$$x_{57} = 77.7544181763474$$
$$x_{58} = -7.06858347057703$$
$$x_{59} = 24.3473430653209$$
Signos de extremos en los puntos:
(21.205750411731103, 9.875587244949e-19)

(68.329640215578, 0)

(84.03760348352696, 6.28815014548678e-20)

(62.04645490839842, 0)

(-63.617251235193315, 0)

(-73.0420291959627, 0)

(-35.34291735288517, 0)

(-98.17477042468104, 4.6075539850034e-20)

(-38.48451000647497, 0)

(-91.89158511750145, 0)

(8.639379797371932, 0)

(74.61282552275759, 0)

(-51.05088062083414, 0)

(-10.210176124166829, 0)

(58.90486225480862, 0)

(80.89601082993718, 0)

(-54.19247327442393, -7.56070479055642e-20)

(-85.60839981032187, 0)

(18.06415775814131, 0)

(-13.351768777756622, 2.49110833964284e-18)

(-47.909287967244346, 9.6738821574413e-20)

(55.76326960121883, 0)

(46.33849164044945, 0)

(40.05530633326986, 0)

(-79.32521450314228, 3.52872419447496e-20)

(33.772121026090275, 0)

(-101.31636307827083, 0)

(-16.493361431346415, 0)

(-66.7588438887831, 0)

(-57.33406592801373, 0)

(65.18804756198821, 5.22521636055147e-20)

(96.60397409788614, 0)

(-41.62610266006476, 1.28147082619577e-19)

(-29.059732045705587, 0)

(-44.767695313654556, 0)

(-22.776546738526, 0)

(-69.9004365423729, 9.08888158258783e-20)

(52.621676947629034, 0)

(-76.18362184955248, 3.82575358782433e-20)

(-25.918139392115794, 0)

(36.91371367968007, -1.62953895463282e-19)

(30.630528372500486, 2.36663481313867e-19)

(14.922565104551518, 0)

(-19.634954084936208, 0)

(99.74556675147593, 0)

(2.356194490192345, 0)

(-258.3959957577605, 3.3255897033323e-21)

(87.17919613711676, 0)

(-82.46680715673207, 0)

(-60.47565858160352, 6.07126252451326e-20)

(-3.9269908169872414, 0)

(-88.74999246391165, 0)

(90.32078879070656, 0)

(-32.201324699295384, 0)

(43.19689898685966, 1.1899674548046e-19)

(-95.03317777109125, 0)

(77.75441817634739, 0)

(-7.0685834705770345, 0)

(24.3473430653209, 0)


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:
$$x_{1} = 21.2057504117311$$
$$x_{2} = 68.329640215578$$
$$x_{3} = 84.037603483527$$
$$x_{4} = 62.0464549083984$$
$$x_{5} = -63.6172512351933$$
$$x_{6} = -73.0420291959627$$
$$x_{7} = -35.3429173528852$$
$$x_{8} = -98.174770424681$$
$$x_{9} = -38.484510006475$$
$$x_{10} = -91.8915851175014$$
$$x_{11} = 8.63937979737193$$
$$x_{12} = 74.6128255227576$$
$$x_{13} = -51.0508806208341$$
$$x_{14} = -10.2101761241668$$
$$x_{15} = 58.9048622548086$$
$$x_{16} = 80.8960108299372$$
$$x_{17} = -54.1924732744239$$
$$x_{18} = -85.6083998103219$$
$$x_{19} = 18.0641577581413$$
$$x_{20} = -13.3517687777566$$
$$x_{21} = -47.9092879672443$$
$$x_{22} = 55.7632696012188$$
$$x_{23} = 46.3384916404494$$
$$x_{24} = 40.0553063332699$$
$$x_{25} = -79.3252145031423$$
$$x_{26} = 33.7721210260903$$
$$x_{27} = -101.316363078271$$
$$x_{28} = -16.4933614313464$$
$$x_{29} = -66.7588438887831$$
$$x_{30} = -57.3340659280137$$
$$x_{31} = 65.1880475619882$$
$$x_{32} = 96.6039740978861$$
$$x_{33} = -41.6261026600648$$
$$x_{34} = -29.0597320457056$$
$$x_{35} = -44.7676953136546$$
$$x_{36} = -22.776546738526$$
$$x_{37} = -69.9004365423729$$
$$x_{38} = 52.621676947629$$
$$x_{39} = -76.1836218495525$$
$$x_{40} = -25.9181393921158$$
$$x_{41} = 36.9137136796801$$
$$x_{42} = 30.6305283725005$$
$$x_{43} = 14.9225651045515$$
$$x_{44} = -19.6349540849362$$
$$x_{45} = 99.7455667514759$$
$$x_{46} = 2.35619449019234$$
$$x_{47} = -258.39599575776$$
$$x_{48} = 87.1791961371168$$
$$x_{49} = -82.4668071567321$$
$$x_{50} = -60.4756585816035$$
$$x_{51} = -3.92699081698724$$
$$x_{52} = -88.7499924639117$$
$$x_{53} = 90.3207887907066$$
$$x_{54} = -32.2013246992954$$
$$x_{55} = 43.1968989868597$$
$$x_{56} = -95.0331777710912$$
$$x_{57} = 77.7544181763474$$
$$x_{58} = -7.06858347057703$$
$$x_{59} = 24.3473430653209$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[99.7455667514759, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -258.39599575776\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = 4.71238898038469$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{x^{2}}\right)$$
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{x^{2}}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (1 + sin(x)^2/cos(x)^2 + (2*sin(x))/cos(x))/x^2, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{x x^{2}}\right)$$
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{x x^{2}}\right)$$
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:
$$\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{x^{2}} = \frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1}{x^{2}}$$
- No
$$\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{x^{2}} = - \frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1}{x^{2}}$$
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (1+sin(x)^2/cos(x)^2+2*sin(x)/cos(x))/x^2