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-1)/(x+2)
-
-1-x
-
-exp(-x)+(-cos(x*sqrt(3)/2)-sin(x*sqrt(3)/2))*exp(x/2)
-
3/(x^2+1)
- Límite de la función:
-
(-tan(x)+sin(x))/x^2
-
Expresiones idénticas
- (-tan(x)+sin(x))/x^ dos
- ( menos tangente de (x) más seno de (x)) dividir por x al cuadrado
- ( menos tangente de (x) más seno de (x)) dividir por x en el grado dos
- (-tan(x)+sin(x))/x2
- -tanx+sinx/x2
- (-tan(x)+sin(x))/x²
- (-tan(x)+sin(x))/x en el grado 2
- -tanx+sinx/x^2
- (-tan(x)+sin(x)) dividir por x^2
-
Expresiones semejantes
- (tan(x)+sin(x))/x^2
- (-tan(x)-sin(x))/x^2
- (-tan(x)+sinx)/x^2
-
Expresiones con funciones
- Tangente tan
- tan(4)^2/sin(8*x)
- tan(x)^(2)
- tan(2*x+pi/6)-2
- tan(pi/3-2*x)
- tan(2)^(6)*x*cos(7*x)^2
- Seno sin
- sin(x)+x
- sin(x)*sin(3*x)
- sin(x)/sin(x+pi/4)
- sinx+cos^2x
- sin(x)*cos(x)^(2)
Gráfico de la función y = (-tan(x)+sin(x))/x^2
Solución
Ha introducido
[src]
-tan(x) + sin(x)
f(x) = ----------------
2
x
$$f{\left(x \right)} = \frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}}$$
f = (sin(x) - tan(x))/x^2
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \pi$$
$$x_{2} = \pi$$
$$x_{3} = 2 \pi$$
Solución numérica
$$x_{1} = 18.8499705557576$$
$$x_{2} = -69.115190002488$$
$$x_{3} = -15.707963267949$$
$$x_{4} = -12.5662132152219$$
$$x_{5} = 84.8230016469244$$
$$x_{6} = 21.9911485751286$$
$$x_{7} = -69.1146296243476$$
$$x_{8} = 81.6814952966863$$
$$x_{9} = -21.9911485751286$$
$$x_{10} = 69.1152917430574$$
$$x_{11} = 28.2743338823081$$
$$x_{12} = -56.5485217299594$$
$$x_{13} = 72.2566310325652$$
$$x_{14} = -12.5667507896127$$
$$x_{15} = 43.9823032529319$$
$$x_{16} = -40.8407044966673$$
$$x_{17} = -25.1328857738208$$
$$x_{18} = -62.8316150966479$$
$$x_{19} = -97.3893722612836$$
$$x_{20} = -43.9823032328965$$
$$x_{21} = 59.6902604182061$$
$$x_{22} = -18.8497845786293$$
$$x_{23} = -91.106186954104$$
$$x_{24} = 94.2477801894711$$
$$x_{25} = -62.8320963183562$$
$$x_{26} = -53.4070751110265$$
$$x_{27} = 62.831712710125$$
$$x_{28} = 12.566293169695$$
$$x_{29} = 100.53089752613$$
$$x_{30} = 25.132981592904$$
$$x_{31} = 53.4070751110265$$
$$x_{32} = -28.2743338823081$$
$$x_{33} = -94.2477097292694$$
$$x_{34} = -72.2566310325652$$
$$x_{35} = 75.3983812155628$$
$$x_{36} = -37.6991249673766$$
$$x_{37} = 50.2654784069467$$
$$x_{38} = -6.28310226447675$$
$$x_{39} = 9.42477796076938$$
$$x_{40} = 65.9734457253857$$
$$x_{41} = 25.1325002165247$$
$$x_{42} = 31.4155161069179$$
$$x_{43} = -84.8230016469244$$
$$x_{44} = -75.3962870884282$$
$$x_{45} = 87.9646063264249$$
$$x_{46} = 31.4160773323774$$
$$x_{47} = 3.14159265358979$$
$$x_{48} = 40.8407044966673$$
$$x_{49} = -78.5398163397448$$
$$x_{50} = 47.1238898038469$$
$$x_{51} = -18.8493020291405$$
$$x_{52} = 91.106186954104$$
$$x_{53} = 69.1148097354674$$
$$x_{54} = 81.6801148690188$$
$$x_{55} = -56.5490909250485$$
$$x_{56} = 78.5398163397448$$
$$x_{57} = 15.707963267949$$
$$x_{58} = 62.8323048592662$$
$$x_{59} = -3.14159265358979$$
$$x_{60} = 56.5485961317107$$
$$x_{61} = -81.6814265554779$$
$$x_{62} = -25.1323022679877$$
$$x_{63} = -65.9734457253857$$
$$x_{64} = -34.5575191894877$$
$$x_{65} = 6.2831766239039$$
$$x_{66} = -59.6902604182061$$
$$x_{67} = 18.8494072742531$$
$$x_{68} = 37.6991934748829$$
$$x_{69} = -75.3983065960259$$
$$x_{70} = -31.4160048810679$$
$$x_{71} = 34.5575191894877$$
$$x_{72} = -87.9646059894675$$
$$x_{73} = -47.1238898038469$$
$$x_{74} = -50.2654080159534$$
$$x_{75} = -9.42477796076938$$
$$x_{76} = 97.3893722612836$$
$$x_{77} = -100.530824375464$$
$$x_{78} = 75.3978374769382$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \pi$$
$$x_{2} = \pi$$
$$x_{3} = 2 \pi$$
Solución numérica
$$x_{1} = 18.8499705557576$$
$$x_{2} = -69.115190002488$$
$$x_{3} = -15.707963267949$$
$$x_{4} = -12.5662132152219$$
$$x_{5} = 84.8230016469244$$
$$x_{6} = 21.9911485751286$$
$$x_{7} = -69.1146296243476$$
$$x_{8} = 81.6814952966863$$
$$x_{9} = -21.9911485751286$$
$$x_{10} = 69.1152917430574$$
$$x_{11} = 28.2743338823081$$
$$x_{12} = -56.5485217299594$$
$$x_{13} = 72.2566310325652$$
$$x_{14} = -12.5667507896127$$
$$x_{15} = 43.9823032529319$$
$$x_{16} = -40.8407044966673$$
$$x_{17} = -25.1328857738208$$
$$x_{18} = -62.8316150966479$$
$$x_{19} = -97.3893722612836$$
$$x_{20} = -43.9823032328965$$
$$x_{21} = 59.6902604182061$$
$$x_{22} = -18.8497845786293$$
$$x_{23} = -91.106186954104$$
$$x_{24} = 94.2477801894711$$
$$x_{25} = -62.8320963183562$$
$$x_{26} = -53.4070751110265$$
$$x_{27} = 62.831712710125$$
$$x_{28} = 12.566293169695$$
$$x_{29} = 100.53089752613$$
$$x_{30} = 25.132981592904$$
$$x_{31} = 53.4070751110265$$
$$x_{32} = -28.2743338823081$$
$$x_{33} = -94.2477097292694$$
$$x_{34} = -72.2566310325652$$
$$x_{35} = 75.3983812155628$$
$$x_{36} = -37.6991249673766$$
$$x_{37} = 50.2654784069467$$
$$x_{38} = -6.28310226447675$$
$$x_{39} = 9.42477796076938$$
$$x_{40} = 65.9734457253857$$
$$x_{41} = 25.1325002165247$$
$$x_{42} = 31.4155161069179$$
$$x_{43} = -84.8230016469244$$
$$x_{44} = -75.3962870884282$$
$$x_{45} = 87.9646063264249$$
$$x_{46} = 31.4160773323774$$
$$x_{47} = 3.14159265358979$$
$$x_{48} = 40.8407044966673$$
$$x_{49} = -78.5398163397448$$
$$x_{50} = 47.1238898038469$$
$$x_{51} = -18.8493020291405$$
$$x_{52} = 91.106186954104$$
$$x_{53} = 69.1148097354674$$
$$x_{54} = 81.6801148690188$$
$$x_{55} = -56.5490909250485$$
$$x_{56} = 78.5398163397448$$
$$x_{57} = 15.707963267949$$
$$x_{58} = 62.8323048592662$$
$$x_{59} = -3.14159265358979$$
$$x_{60} = 56.5485961317107$$
$$x_{61} = -81.6814265554779$$
$$x_{62} = -25.1323022679877$$
$$x_{63} = -65.9734457253857$$
$$x_{64} = -34.5575191894877$$
$$x_{65} = 6.2831766239039$$
$$x_{66} = -59.6902604182061$$
$$x_{67} = 18.8494072742531$$
$$x_{68} = 37.6991934748829$$
$$x_{69} = -75.3983065960259$$
$$x_{70} = -31.4160048810679$$
$$x_{71} = 34.5575191894877$$
$$x_{72} = -87.9646059894675$$
$$x_{73} = -47.1238898038469$$
$$x_{74} = -50.2654080159534$$
$$x_{75} = -9.42477796076938$$
$$x_{76} = 97.3893722612836$$
$$x_{77} = -100.530824375464$$
$$x_{78} = 75.3978374769382$$
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
$$\frac{\cos{\left(x \right)} - \tan^{2}{\left(x \right)} - 1}{x^{2}} - \frac{2 \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -1156.10608070793$$
$$x_{2} = 18.8495554664323$$
$$x_{3} = 62.8318545472756$$
$$x_{4} = 11447.9637185929$$
$$x_{5} = -87.9645943580839$$
$$x_{6} = 56.5486904064043$$
$$x_{7} = 81.6814092168155$$
$$x_{8} = 6.28318571765258$$
$$x_{9} = 125.663722285741$$
$$x_{10} = 87.9645943361183$$
$$x_{11} = -43.9822971743842$$
$$x_{12} = -56.5486704437837$$
$$x_{13} = 18.8495589320187$$
$$x_{14} = -62.8318521213994$$
$$x_{15} = 446.106360287555$$
$$x_{16} = -62.8318540471604$$
$$x_{17} = 94.2477796093519$$
$$x_{18} = -18.849556783468$$
$$x_{19} = -56.5486708391066$$
$$x_{20} = -69.1150272493433$$
$$x_{21} = -37.6991118773074$$
$$x_{22} = 37.6991109715204$$
$$x_{23} = 100.530964744173$$
$$x_{24} = -87.9646453390757$$
$$x_{25} = 75.3982211826328$$
$$x_{26} = -31.4159193329553$$
$$x_{27} = 81.6814033208689$$
$$x_{28} = 62.8318566731154$$
$$x_{29} = -69.1150355143013$$
$$x_{30} = 12.5663722955771$$
$$x_{31} = -113.097343315674$$
$$x_{32} = -797.964534586798$$
$$x_{33} = -31.4159153392304$$
$$x_{34} = 12.566370255665$$
$$x_{35} = -18.8495548402421$$
$$x_{36} = 69.1150315456483$$
$$x_{37} = 69.1150321122122$$
$$x_{38} = -12.5663730816243$$
$$x_{39} = 31.4159236906738$$
$$x_{40} = -75.398213104482$$
$$x_{41} = -75.3982240366897$$
$$x_{42} = 62.8318518591974$$
$$x_{43} = -150.796496172887$$
$$x_{44} = 12.5663699149855$$
$$x_{45} = -81.6814093267247$$
$$x_{46} = -383.274319745291$$
$$x_{47} = -31.4159267321981$$
$$x_{48} = 94.2477798839718$$
$$x_{49} = 25.1327402547553$$
$$x_{50} = 6.28318528363928$$
$$x_{51} = 43.982294196519$$
$$x_{52} = -56.5486673282356$$
$$x_{53} = -75.3982238973503$$
$$x_{54} = 31.4159265700968$$
$$x_{55} = 69.1150394396063$$
$$x_{56} = 56.5486675815965$$
$$x_{57} = -119.380522005839$$
$$x_{58} = -31.4159249350464$$
$$x_{59} = 43.9822874055792$$
$$x_{60} = -94.247800478884$$
$$x_{61} = -25.1327416418926$$
$$x_{62} = -43.9822967106817$$
$$x_{63} = 37.6991120515325$$
$$x_{64} = -50.2654822633581$$
$$x_{65} = -12.5663701097364$$
$$x_{66} = 31.415926978842$$
$$x_{67} = 94.247788860923$$
$$x_{68} = -31.4159325737557$$
$$x_{69} = -56.5486676993217$$
$$x_{70} = 69.1150375000423$$
$$x_{71} = 12.5663704080972$$
$$x_{72} = -94.2477794270741$$
$$x_{73} = 62.8318526612428$$
$$x_{74} = 100.530963894357$$
$$x_{75} = 43.9822971694917$$
$$x_{76} = 75.3982241632366$$
$$x_{77} = 81.6814063718606$$
$$x_{78} = -94.2477804566038$$
$$x_{79} = 50.2654824463089$$
$$x_{80} = 56.5486678154258$$
$$x_{81} = -25.1327379006296$$
$$x_{82} = 25.1327409532572$$
$$x_{83} = 87.9646020448506$$
$$x_{84} = -100.530964503023$$
$$x_{85} = -69.1150388279099$$
$$x_{86} = -81.6814090385276$$
$$x_{87} = -37.6991120138589$$
$$x_{88} = 25.1327421838749$$
$$x_{89} = -6.2831850697993$$
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
La función no tiene puntos máximos
Decrece en todo el eje numérico
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\cos{\left(x \right)} - \tan^{2}{\left(x \right)} - 1}{x^{2}} - \frac{2 \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -1156.10608070793$$
$$x_{2} = 18.8495554664323$$
$$x_{3} = 62.8318545472756$$
$$x_{4} = 11447.9637185929$$
$$x_{5} = -87.9645943580839$$
$$x_{6} = 56.5486904064043$$
$$x_{7} = 81.6814092168155$$
$$x_{8} = 6.28318571765258$$
$$x_{9} = 125.663722285741$$
$$x_{10} = 87.9645943361183$$
$$x_{11} = -43.9822971743842$$
$$x_{12} = -56.5486704437837$$
$$x_{13} = 18.8495589320187$$
$$x_{14} = -62.8318521213994$$
$$x_{15} = 446.106360287555$$
$$x_{16} = -62.8318540471604$$
$$x_{17} = 94.2477796093519$$
$$x_{18} = -18.849556783468$$
$$x_{19} = -56.5486708391066$$
$$x_{20} = -69.1150272493433$$
$$x_{21} = -37.6991118773074$$
$$x_{22} = 37.6991109715204$$
$$x_{23} = 100.530964744173$$
$$x_{24} = -87.9646453390757$$
$$x_{25} = 75.3982211826328$$
$$x_{26} = -31.4159193329553$$
$$x_{27} = 81.6814033208689$$
$$x_{28} = 62.8318566731154$$
$$x_{29} = -69.1150355143013$$
$$x_{30} = 12.5663722955771$$
$$x_{31} = -113.097343315674$$
$$x_{32} = -797.964534586798$$
$$x_{33} = -31.4159153392304$$
$$x_{34} = 12.566370255665$$
$$x_{35} = -18.8495548402421$$
$$x_{36} = 69.1150315456483$$
$$x_{37} = 69.1150321122122$$
$$x_{38} = -12.5663730816243$$
$$x_{39} = 31.4159236906738$$
$$x_{40} = -75.398213104482$$
$$x_{41} = -75.3982240366897$$
$$x_{42} = 62.8318518591974$$
$$x_{43} = -150.796496172887$$
$$x_{44} = 12.5663699149855$$
$$x_{45} = -81.6814093267247$$
$$x_{46} = -383.274319745291$$
$$x_{47} = -31.4159267321981$$
$$x_{48} = 94.2477798839718$$
$$x_{49} = 25.1327402547553$$
$$x_{50} = 6.28318528363928$$
$$x_{51} = 43.982294196519$$
$$x_{52} = -56.5486673282356$$
$$x_{53} = -75.3982238973503$$
$$x_{54} = 31.4159265700968$$
$$x_{55} = 69.1150394396063$$
$$x_{56} = 56.5486675815965$$
$$x_{57} = -119.380522005839$$
$$x_{58} = -31.4159249350464$$
$$x_{59} = 43.9822874055792$$
$$x_{60} = -94.247800478884$$
$$x_{61} = -25.1327416418926$$
$$x_{62} = -43.9822967106817$$
$$x_{63} = 37.6991120515325$$
$$x_{64} = -50.2654822633581$$
$$x_{65} = -12.5663701097364$$
$$x_{66} = 31.415926978842$$
$$x_{67} = 94.247788860923$$
$$x_{68} = -31.4159325737557$$
$$x_{69} = -56.5486676993217$$
$$x_{70} = 69.1150375000423$$
$$x_{71} = 12.5663704080972$$
$$x_{72} = -94.2477794270741$$
$$x_{73} = 62.8318526612428$$
$$x_{74} = 100.530963894357$$
$$x_{75} = 43.9822971694917$$
$$x_{76} = 75.3982241632366$$
$$x_{77} = 81.6814063718606$$
$$x_{78} = -94.2477804566038$$
$$x_{79} = 50.2654824463089$$
$$x_{80} = 56.5486678154258$$
$$x_{81} = -25.1327379006296$$
$$x_{82} = 25.1327409532572$$
$$x_{83} = 87.9646020448506$$
$$x_{84} = -100.530964503023$$
$$x_{85} = -69.1150388279099$$
$$x_{86} = -81.6814090385276$$
$$x_{87} = -37.6991120138589$$
$$x_{88} = 25.1327421838749$$
$$x_{89} = -6.2831850697993$$
Signos de extremos en los puntos:
(-1156.106080707933, -1.4791990664389e-21)
(18.849555466432278, 1.32607504514291e-22)
(62.831854547275576, -4.06798065885684e-22)
(11447.963718592906, -2.68157684858385e-21)
(-87.96459435808394, 1.19729883771663e-26)
(56.5486904064043, -1.81491833854508e-18)
(81.6814092168155, -8.37117337725645e-25)
(6.2831857176525805, -8.75656364687435e-22)
(125.66372228574147, -1.33178707198341e-19)
(87.96459433611828, -2.56564036781697e-27)
(-43.98229717438416, 3.42085382277207e-27)
(-56.54867044378374, 3.00696164976335e-21)
(18.84955893201873, -3.8394775346208e-20)
(-62.83185212139935, -1.0872623196502e-22)
(446.1063602875554, -2.11662688328545e-17)
(-62.83185404716038, 1.17523019059098e-22)
(94.24777960935188, 0)
(-18.84955678346803, 9.01134915986591e-22)
(-56.54867083910665, 4.54395022019461e-21)
(-69.11502724934327, -1.44300449926285e-19)
(-37.699111877307445, 1.3968486432945e-26)
(37.69911097152045, 2.32957115598771e-22)
(100.53096474417282, 2.46194574663805e-25)
(-87.96464533907574, 8.59110372107709e-18)
(75.39822118263282, 1.38008655372991e-21)
(-31.415919332955337, -1.89321684187199e-19)
(81.68140332086892, 1.36785151426515e-20)
(62.831856673115375, -5.91552147446529e-21)
(-69.11503551430128, -2.46065553116755e-21)
(12.566372295577121, -1.50457321081269e-20)
(-113.0973433156742, 1.84536635182418e-20)
(-797.9645345867976, 1.49320513219997e-25)
(-31.41591533923042, -7.11109306159166e-19)
(12.566370255665037, 1.46166250644098e-22)
(-18.84955484024213, -1.77902663943669e-21)
(69.11503154564825, 3.33980343388759e-20)
(69.11503211221218, 2.57605497910039e-20)
(-12.566373081624262, 4.75563080864335e-20)
(31.415923690673846, 1.1668842119061e-20)
(-75.39821310448201, -1.04210153131107e-19)
(-75.39822403668971, 3.79011595711375e-24)
(62.8318518591974, 2.25820148195894e-22)
(-150.79649617288732, 2.55542052520409e-18)
(12.56636991498552, 1.08350767820219e-21)
(-81.68140932672466, 2.77716651578661e-24)
(-383.2743197452912, 1.39607239040796e-20)
(-31.41592673219809, 3.83518759406612e-24)
(94.24777988397176, -1.18601761660029e-24)
(25.132740254755262, 7.31166511843277e-22)
(6.283185283639279, 1.67621838755743e-25)
(43.98229419651897, 6.66086853101854e-21)
(-56.548667328235624, -1.29793108768428e-23)
(-75.39822389735026, 8.28796858550101e-25)
(31.41592657009683, -2.01146204561751e-26)
(69.11503943960635, -1.24876879800169e-22)
(56.54866758159649, 9.60204112386715e-25)
(-119.38052200583878, 5.61054888419897e-23)
(-31.415924935046437, -2.07840371883782e-21)
(43.982287405579214, 2.39175696670774e-19)
(-94.24780047888402, 5.11763207243069e-19)
(-25.132741641892622, 5.58180700521486e-23)
(-43.98229671068173, -2.19481985897024e-23)
(37.699112051532516, -3.18481487727445e-24)
(-50.26548226335814, -1.4457383595994e-24)
(-12.566370109736413, -4.06985854135691e-22)
(31.415926978842, -4.40376079061269e-23)
(94.24778886092298, -4.45971146345717e-20)
(-31.415932573755715, 1.11511551181653e-19)
(-56.54866769932167, -4.55269189322245e-26)
(69.11503750004229, 7.10605784716989e-23)
(12.566370408097232, 2.78252259383914e-23)
(-94.24777942707406, -3.3077376060062e-25)
(62.83185266124285, 8.76997465259479e-24)
(100.5309638943567, 5.25703998531663e-23)
(43.98229716949169, -1.71042691176656e-27)
(75.39822416323665, -9.55444461657348e-24)
(81.68140637186058, 1.3500521538547e-21)
(-94.24778045660378, 3.44362296755962e-23)
(50.265482446308916, 6.54772803023241e-28)
(56.54866781542581, -2.06940539751252e-26)
(-25.132737900629582, -2.91789467123813e-20)
(25.132740953257205, 1.65945622762347e-23)
(87.96460204485058, -3.0012869445813e-20)
(-100.53096450302264, -3.45720042675861e-24)
(-69.11503882790991, 9.47548226118709e-24)
(-81.68140903852758, 6.94291633846001e-27)
(-37.699112013858915, 1.75071695357976e-24)
(25.132742183874935, -6.89931430707941e-22)
(-6.283185069799302, -1.69633312367307e-22)
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
La función no tiene puntos máximos
Decrece en todo el eje numérico
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
$$\frac{- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} + \frac{4 \left(- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{x^{2}}}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -43.9822971502571$$
$$x_{2} = -31.4159265358979$$
$$x_{3} = -53.5485033332354$$
$$x_{4} = 34.7628137456958$$
$$x_{5} = -97.4699892938459$$
$$x_{6} = -22.2791470609148$$
$$x_{7} = -28.5145580814312$$
$$x_{8} = 59.8181276206052$$
$$x_{9} = 6.28318530717959$$
$$x_{10} = 84.9150381857849$$
$$x_{11} = 3.81019252919313$$
$$x_{12} = 47.2819227750125$$
$$x_{13} = 16.064895766025$$
$$x_{14} = 94.2477796076938$$
$$x_{15} = 50.2654824574367$$
$$x_{16} = 56.5486677646163$$
$$x_{17} = 43.9822971502571$$
$$x_{18} = -66.090050226592$$
$$x_{19} = -50.2654824574367$$
$$x_{20} = 37.6991118430775$$
$$x_{21} = -56.5486677646163$$
$$x_{22} = -62.8318530717959$$
$$x_{23} = 22.2791470609148$$
$$x_{24} = 53.5485033332354$$
$$x_{25} = -81.6814089933346$$
$$x_{26} = -6.28318530717959$$
$$x_{27} = -91.1921433712405$$
$$x_{28} = 91.1921433712405$$
$$x_{29} = -78.6388353763319$$
$$x_{30} = 97.4699892938459$$
$$x_{31} = 66.090050226592$$
$$x_{32} = -25.1327412287183$$
$$x_{33} = -75.398223686155$$
$$x_{34} = -72.363745968728$$
$$x_{35} = -69.1150383789755$$
$$x_{36} = 78.6388353763319$$
$$x_{37} = -47.2819227750125$$
$$x_{38} = 62.8318530717959$$
$$x_{39} = -18.8495559215388$$
$$x_{40} = 9.89036297373071$$
$$x_{41} = 25.1327412287183$$
$$x_{42} = 100.530964914873$$
$$x_{43} = -87.9645943005142$$
$$x_{44} = -34.7628137456958$$
$$x_{45} = 41.0194757257576$$
$$x_{46} = 81.6814089933346$$
$$x_{47} = 75.398223686155$$
$$x_{48} = -9.89036297373071$$
$$x_{49} = -41.0194757257576$$
$$x_{50} = 12.5663706143592$$
$$x_{51} = 87.9645943005142$$
$$x_{52} = 69.1150383789755$$
$$x_{53} = -16.064895766025$$
$$x_{54} = -37.6991118430775$$
$$x_{55} = 31.4159265358979$$
$$x_{56} = -12.5663706143592$$
$$x_{57} = -84.9150381857849$$
$$x_{58} = -94.2477796076938$$
$$x_{59} = 72.363745968728$$
$$x_{60} = 28.5145580814312$$
$$x_{61} = -100.530964914873$$
$$x_{62} = -3.81019252919313$$
$$x_{63} = 18.8495559215388$$
$$x_{64} = -59.8181276206052$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} + \frac{4 \left(- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{x^{2}}}{x^{2}}\right) = 0$$
$$\lim_{x \to 0^+}\left(\frac{- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} + \frac{4 \left(- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{x^{2}}}{x^{2}}\right) = 0$$
- 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
$$\left(-\infty, -100.530964914873\right]$$
Convexa en los intervalos
$$\left[100.530964914873, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} + \frac{4 \left(- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{x^{2}}}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -43.9822971502571$$
$$x_{2} = -31.4159265358979$$
$$x_{3} = -53.5485033332354$$
$$x_{4} = 34.7628137456958$$
$$x_{5} = -97.4699892938459$$
$$x_{6} = -22.2791470609148$$
$$x_{7} = -28.5145580814312$$
$$x_{8} = 59.8181276206052$$
$$x_{9} = 6.28318530717959$$
$$x_{10} = 84.9150381857849$$
$$x_{11} = 3.81019252919313$$
$$x_{12} = 47.2819227750125$$
$$x_{13} = 16.064895766025$$
$$x_{14} = 94.2477796076938$$
$$x_{15} = 50.2654824574367$$
$$x_{16} = 56.5486677646163$$
$$x_{17} = 43.9822971502571$$
$$x_{18} = -66.090050226592$$
$$x_{19} = -50.2654824574367$$
$$x_{20} = 37.6991118430775$$
$$x_{21} = -56.5486677646163$$
$$x_{22} = -62.8318530717959$$
$$x_{23} = 22.2791470609148$$
$$x_{24} = 53.5485033332354$$
$$x_{25} = -81.6814089933346$$
$$x_{26} = -6.28318530717959$$
$$x_{27} = -91.1921433712405$$
$$x_{28} = 91.1921433712405$$
$$x_{29} = -78.6388353763319$$
$$x_{30} = 97.4699892938459$$
$$x_{31} = 66.090050226592$$
$$x_{32} = -25.1327412287183$$
$$x_{33} = -75.398223686155$$
$$x_{34} = -72.363745968728$$
$$x_{35} = -69.1150383789755$$
$$x_{36} = 78.6388353763319$$
$$x_{37} = -47.2819227750125$$
$$x_{38} = 62.8318530717959$$
$$x_{39} = -18.8495559215388$$
$$x_{40} = 9.89036297373071$$
$$x_{41} = 25.1327412287183$$
$$x_{42} = 100.530964914873$$
$$x_{43} = -87.9645943005142$$
$$x_{44} = -34.7628137456958$$
$$x_{45} = 41.0194757257576$$
$$x_{46} = 81.6814089933346$$
$$x_{47} = 75.398223686155$$
$$x_{48} = -9.89036297373071$$
$$x_{49} = -41.0194757257576$$
$$x_{50} = 12.5663706143592$$
$$x_{51} = 87.9645943005142$$
$$x_{52} = 69.1150383789755$$
$$x_{53} = -16.064895766025$$
$$x_{54} = -37.6991118430775$$
$$x_{55} = 31.4159265358979$$
$$x_{56} = -12.5663706143592$$
$$x_{57} = -84.9150381857849$$
$$x_{58} = -94.2477796076938$$
$$x_{59} = 72.363745968728$$
$$x_{60} = 28.5145580814312$$
$$x_{61} = -100.530964914873$$
$$x_{62} = -3.81019252919313$$
$$x_{63} = 18.8495559215388$$
$$x_{64} = -59.8181276206052$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} + \frac{4 \left(- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{x^{2}}}{x^{2}}\right) = 0$$
$$\lim_{x \to 0^+}\left(\frac{- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)} + \frac{4 \left(- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{x^{2}}}{x^{2}}\right) = 0$$
- 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
$$\left(-\infty, -100.530964914873\right]$$
Convexa en los intervalos
$$\left[100.530964914873, \infty\right)$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-tan(x) + sin(x))/x^2, dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x x^{2}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x x^{2}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x x^{2}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x x^{2}}\right)$$
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:
$$\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}} = \frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{x^{2}}$$
- No
$$\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}} = - \frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{x^{2}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}} = \frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{x^{2}}$$
- No
$$\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{2}} = - \frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{x^{2}}$$
- No
es decir, función
no es
par ni impar