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-8/x^4
-
y=x²-2x-8
-
-x^4+x^2
-
x*e^(-x^1)
-
Expresiones idénticas
- -(cero . uno *e^(cero . uno *x)*(sin(x)+cos(x)))/(dos *(uno -e^(cero . uno *x)*sin(x)))
- menos (0.1 multiplicar por e en el grado (0.1 multiplicar por x) multiplicar por ( seno de (x) más coseno de (x))) dividir por (2 multiplicar por (1 menos e en el grado (0.1 multiplicar por x) multiplicar por seno de (x)))
- menos (cero . uno multiplicar por e en el grado (cero . uno multiplicar por x) multiplicar por ( seno de (x) más coseno de (x))) dividir por (dos multiplicar por (uno menos e en el grado (cero . uno multiplicar por x) multiplicar por seno de (x)))
- -(0.1*e(0.1*x)*(sin(x)+cos(x)))/(2*(1-e(0.1*x)*sin(x)))
- -0.1*e0.1*x*sinx+cosx/2*1-e0.1*x*sinx
- -(0.1e^(0.1x)(sin(x)+cos(x)))/(2(1-e^(0.1x)sin(x)))
- -(0.1e(0.1x)(sin(x)+cos(x)))/(2(1-e(0.1x)sin(x)))
- -0.1e0.1xsinx+cosx/21-e0.1xsinx
- -0.1e^0.1xsinx+cosx/21-e^0.1xsinx
- -(0.1*e^(0.1*x)*(sin(x)+cos(x))) dividir por (2*(1-e^(0.1*x)*sin(x)))
-
Expresiones semejantes
- -(0.1*e^(0.1*x)*(sin(x)-cos(x)))/(2*(1-e^(0.1*x)*sin(x)))
- (0.1*e^(0.1*x)*(sin(x)+cos(x)))/(2*(1-e^(0.1*x)*sin(x)))
- -(0.1*e^(0.1*x)*(sin(x)+cos(x)))/(2*(1+e^(0.1*x)*sin(x)))
-
Expresiones con funciones
- Seno sin
- sin(5+6*x)^(2)+3^tan(x)
- sin(x)+arcsin(x)
- sin(2*x^3)^2/(x^3)
- sin(x)*sin(1.5*pi+x)-6*x
- sin(x)+sin(10/3*x)+log(x)-(0,84*x+3)
- Coseno cos
- cos(3*z)
- cos(x)*((2*x^3-5)^3)^4
- cos2x-sinx
- cos(x-1)-2
- cos(x)-sin(2)*5*x+e/x
- Seno sin
- sin(5+6*x)^(2)+3^tan(x)
- sin(x)+arcsin(x)
- sin(2*x^3)^2/(x^3)
- sin(x)*sin(1.5*pi+x)-6*x
- sin(x)+sin(10/3*x)+log(x)-(0,84*x+3)
Gráfico de la función y = -(0.1*e^(0.1*x)*(sin(x)+cos(x)))/(2*(1-e^(0.1*x)*sin(x)))
Solución
Ha introducido
[src]
x
--
10
E
- ---*(sin(x) + cos(x))
10
f(x) = ------------------------
/ x \
| -- |
| 10 |
2*\1 - E *sin(x)/
$$f{\left(x \right)} = \frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)}$$
f = (-E^(x/10)/10*(sin(x) + cos(x)))/((2*(-E^(x/10)*sin(x) + 1)))
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:
$$\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} = 0$$
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
$$x_{1} = 93.4623814442964$$
$$x_{2} = 21.2057504117311$$
$$x_{3} = 43.1968989868597$$
$$x_{4} = 90.3207887907066$$
$$x_{5} = -10.2101761241668$$
$$x_{6} = -19.6349540849362$$
$$x_{7} = 71.4712328691678$$
$$x_{8} = -44.7676953136546$$
$$x_{9} = -104.457955731861$$
$$x_{10} = 11.7809724509617$$
$$x_{11} = -41.6261026600648$$
$$x_{12} = 18.0641577581413$$
$$x_{13} = 8.63937979737193$$
$$x_{14} = -66.7588438887831$$
$$x_{15} = -60.4756585816035$$
$$x_{16} = 27.4889357189107$$
$$x_{17} = 74.6128255227576$$
$$x_{18} = -29.0597320457056$$
$$x_{19} = 40.0553063332699$$
$$x_{20} = -85.6083998103219$$
$$x_{21} = 33.7721210260903$$
$$x_{22} = -47.9092879672443$$
$$x_{23} = -57.3340659280137$$
$$x_{24} = 84.037603483527$$
$$x_{25} = -54.1924732744239$$
$$x_{26} = -76.1836218495525$$
$$x_{27} = -32.2013246992954$$
$$x_{28} = -73.0420291959627$$
$$x_{29} = 30.6305283725005$$
$$x_{30} = -82.4668071567321$$
$$x_{31} = -38.484510006475$$
$$x_{32} = 96.6039740978861$$
$$x_{33} = -22.776546738526$$
$$x_{34} = -79.3252145031423$$
$$x_{35} = 2.35619449019234$$
$$x_{36} = 68.329640215578$$
$$x_{37} = 65.1880475619882$$
$$x_{38} = -13.3517687777566$$
$$x_{39} = -88.7499924639117$$
$$x_{40} = 80.8960108299372$$
$$x_{41} = -51.0508806208341$$
$$x_{42} = 36.9137136796801$$
$$x_{43} = -69.9004365423729$$
$$x_{44} = -7.06858347057703$$
$$x_{45} = -16.4933614313464$$
$$x_{46} = -3.92699081698724$$
$$x_{47} = 62.0464549083984$$
$$x_{48} = 49.4800842940392$$
$$x_{49} = -98.174770424681$$
$$x_{50} = 46.3384916404494$$
$$x_{51} = -25.9181393921158$$
$$x_{52} = 52.621676947629$$
$$x_{53} = 24.3473430653209$$
$$x_{54} = -0.785398163397448$$
$$x_{55} = 5.49778714378214$$
$$x_{56} = -63.6172512351933$$
$$x_{57} = 99.7455667514759$$
$$x_{58} = 58.9048622548086$$
$$x_{59} = 14.9225651045515$$
$$x_{60} = -91.8915851175014$$
$$x_{61} = 55.7632696012188$$
$$x_{62} = -95.0331777710912$$
$$x_{63} = -35.3429173528852$$
$$x_{64} = 77.7544181763474$$
$$x_{65} = 87.1791961371168$$
o sea hay que resolver la ecuación:
$$\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} = 0$$
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
$$x_{1} = 93.4623814442964$$
$$x_{2} = 21.2057504117311$$
$$x_{3} = 43.1968989868597$$
$$x_{4} = 90.3207887907066$$
$$x_{5} = -10.2101761241668$$
$$x_{6} = -19.6349540849362$$
$$x_{7} = 71.4712328691678$$
$$x_{8} = -44.7676953136546$$
$$x_{9} = -104.457955731861$$
$$x_{10} = 11.7809724509617$$
$$x_{11} = -41.6261026600648$$
$$x_{12} = 18.0641577581413$$
$$x_{13} = 8.63937979737193$$
$$x_{14} = -66.7588438887831$$
$$x_{15} = -60.4756585816035$$
$$x_{16} = 27.4889357189107$$
$$x_{17} = 74.6128255227576$$
$$x_{18} = -29.0597320457056$$
$$x_{19} = 40.0553063332699$$
$$x_{20} = -85.6083998103219$$
$$x_{21} = 33.7721210260903$$
$$x_{22} = -47.9092879672443$$
$$x_{23} = -57.3340659280137$$
$$x_{24} = 84.037603483527$$
$$x_{25} = -54.1924732744239$$
$$x_{26} = -76.1836218495525$$
$$x_{27} = -32.2013246992954$$
$$x_{28} = -73.0420291959627$$
$$x_{29} = 30.6305283725005$$
$$x_{30} = -82.4668071567321$$
$$x_{31} = -38.484510006475$$
$$x_{32} = 96.6039740978861$$
$$x_{33} = -22.776546738526$$
$$x_{34} = -79.3252145031423$$
$$x_{35} = 2.35619449019234$$
$$x_{36} = 68.329640215578$$
$$x_{37} = 65.1880475619882$$
$$x_{38} = -13.3517687777566$$
$$x_{39} = -88.7499924639117$$
$$x_{40} = 80.8960108299372$$
$$x_{41} = -51.0508806208341$$
$$x_{42} = 36.9137136796801$$
$$x_{43} = -69.9004365423729$$
$$x_{44} = -7.06858347057703$$
$$x_{45} = -16.4933614313464$$
$$x_{46} = -3.92699081698724$$
$$x_{47} = 62.0464549083984$$
$$x_{48} = 49.4800842940392$$
$$x_{49} = -98.174770424681$$
$$x_{50} = 46.3384916404494$$
$$x_{51} = -25.9181393921158$$
$$x_{52} = 52.621676947629$$
$$x_{53} = 24.3473430653209$$
$$x_{54} = -0.785398163397448$$
$$x_{55} = 5.49778714378214$$
$$x_{56} = -63.6172512351933$$
$$x_{57} = 99.7455667514759$$
$$x_{58} = 58.9048622548086$$
$$x_{59} = 14.9225651045515$$
$$x_{60} = -91.8915851175014$$
$$x_{61} = 55.7632696012188$$
$$x_{62} = -95.0331777710912$$
$$x_{63} = -35.3429173528852$$
$$x_{64} = 77.7544181763474$$
$$x_{65} = 87.1791961371168$$
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
$$\frac{1}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} \left(- \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{10} - \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{100}\right) - \frac{\left(\frac{e^{\frac{x}{10}} \sin{\left(x \right)}}{5} + 2 e^{\frac{x}{10}} \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{40 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -1924.82277051613$$
$$x_{2} = -124.778636646949$$
$$x_{3} = -2330.50523719648$$
$$x_{4} = -83.9380940296237$$
$$x_{5} = -2.81619635397987$$
$$x_{6} = -30.4975236705277$$
$$x_{7} = -36.7962906872222$$
$$x_{8} = -71.3721235521918$$
$$x_{9} = 1.90312690149253$$
$$x_{10} = -582.0968365869$$
$$x_{11} = -74.5127482905657$$
$$x_{12} = -486.062167657715$$
$$x_{13} = -61.9453505282283$$
$$x_{14} = -11.4556169574971$$
$$x_{15} = -153.052973368617$$
$$x_{16} = -822.325556649958$$
$$x_{17} = -49.375369247982$$
$$x_{18} = -898.224266525647$$
$$x_{19} = -27.4345573572623$$
$$x_{20} = -39.9685654523554$$
$$x_{21} = -52.5256909712287$$
$$x_{22} = -80.7961242092005$$
$$x_{23} = -55.6609093999258$$
$$x_{24} = -77.6550479304676$$
$$x_{25} = -435.796312033093$$
$$x_{26} = -2733.12508743085$$
$$x_{27} = -567.743203483864$$
$$x_{28} = -14.9808472121154$$
$$x_{29} = -39.968565452453$$
$$x_{30} = -90.2212050693967$$
$$x_{31} = 2.82358414056662$$
$$x_{32} = -4047.08933720441$$
$$x_{33} = -3830.03441994575$$
$$x_{34} = -21.1907151682264$$
$$x_{35} = -43.0877668109453$$
$$x_{36} = -24.1849742273843$$
$$x_{37} = -21.1907151682264$$
$$x_{38} = -33.6966592867304$$
$$x_{39} = -234.734382203391$$
$$x_{40} = -93.3626507569302$$
$$x_{41} = -5821.90700008095$$
$$x_{42} = -102.787514924946$$
$$x_{43} = -87.0794112023455$$
$$x_{44} = -68.2292056709602$$
$$x_{45} = -12204.9931764696$$
$$x_{46} = -65.0894273082213$$
$$x_{47} = -99.6458650042135$$
$$x_{48} = -131.061823204732$$
$$x_{49} = -8.83480155558271$$
$$x_{50} = -476.63701652976$$
$$x_{51} = -1074.39547911081$$
$$x_{52} = -46.2457238206076$$
$$x_{53} = -564.547138764174$$
$$x_{54} = -442.080574029362$$
$$x_{55} = -17.8461051872707$$
$$x_{56} = -96.5043507553482$$
$$x_{57} = -58.8071586004732$$
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:
$$x_{1} = -124.778636646949$$
$$x_{2} = -30.4975236705277$$
$$x_{3} = -36.7962906872222$$
$$x_{4} = 1.90312690149253$$
$$x_{5} = -74.5127482905657$$
$$x_{6} = -61.9453505282283$$
$$x_{7} = -11.4556169574971$$
$$x_{8} = -49.375369247982$$
$$x_{9} = -80.7961242092005$$
$$x_{10} = -55.6609093999258$$
$$x_{11} = -43.0877668109453$$
$$x_{12} = -24.1849742273843$$
$$x_{13} = -93.3626507569302$$
$$x_{14} = -87.0794112023455$$
$$x_{15} = -68.2292056709602$$
$$x_{16} = -99.6458650042135$$
$$x_{17} = -131.061823204732$$
$$x_{18} = -476.63701652976$$
$$x_{19} = -17.8461051872707$$
Puntos máximos de la función:
$$x_{19} = -83.9380940296237$$
$$x_{19} = -2.81619635397987$$
$$x_{19} = -71.3721235521918$$
$$x_{19} = -153.052973368617$$
$$x_{19} = -27.4345573572623$$
$$x_{19} = -39.9685654523554$$
$$x_{19} = -52.5256909712287$$
$$x_{19} = -77.6550479304676$$
$$x_{19} = -435.796312033093$$
$$x_{19} = -567.743203483864$$
$$x_{19} = -14.9808472121154$$
$$x_{19} = -39.968565452453$$
$$x_{19} = -90.2212050693967$$
$$x_{19} = 2.82358414056662$$
$$x_{19} = -21.1907151682264$$
$$x_{19} = -21.1907151682264$$
$$x_{19} = -33.6966592867304$$
$$x_{19} = -234.734382203391$$
$$x_{19} = -102.787514924946$$
$$x_{19} = -65.0894273082213$$
$$x_{19} = -8.83480155558271$$
$$x_{19} = -46.2457238206076$$
$$x_{19} = -96.5043507553482$$
$$x_{19} = -58.8071586004732$$
Decrece en los intervalos
$$\left[1.90312690149253, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -476.63701652976\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{1}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} \left(- \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{10} - \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{100}\right) - \frac{\left(\frac{e^{\frac{x}{10}} \sin{\left(x \right)}}{5} + 2 e^{\frac{x}{10}} \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{40 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -1924.82277051613$$
$$x_{2} = -124.778636646949$$
$$x_{3} = -2330.50523719648$$
$$x_{4} = -83.9380940296237$$
$$x_{5} = -2.81619635397987$$
$$x_{6} = -30.4975236705277$$
$$x_{7} = -36.7962906872222$$
$$x_{8} = -71.3721235521918$$
$$x_{9} = 1.90312690149253$$
$$x_{10} = -582.0968365869$$
$$x_{11} = -74.5127482905657$$
$$x_{12} = -486.062167657715$$
$$x_{13} = -61.9453505282283$$
$$x_{14} = -11.4556169574971$$
$$x_{15} = -153.052973368617$$
$$x_{16} = -822.325556649958$$
$$x_{17} = -49.375369247982$$
$$x_{18} = -898.224266525647$$
$$x_{19} = -27.4345573572623$$
$$x_{20} = -39.9685654523554$$
$$x_{21} = -52.5256909712287$$
$$x_{22} = -80.7961242092005$$
$$x_{23} = -55.6609093999258$$
$$x_{24} = -77.6550479304676$$
$$x_{25} = -435.796312033093$$
$$x_{26} = -2733.12508743085$$
$$x_{27} = -567.743203483864$$
$$x_{28} = -14.9808472121154$$
$$x_{29} = -39.968565452453$$
$$x_{30} = -90.2212050693967$$
$$x_{31} = 2.82358414056662$$
$$x_{32} = -4047.08933720441$$
$$x_{33} = -3830.03441994575$$
$$x_{34} = -21.1907151682264$$
$$x_{35} = -43.0877668109453$$
$$x_{36} = -24.1849742273843$$
$$x_{37} = -21.1907151682264$$
$$x_{38} = -33.6966592867304$$
$$x_{39} = -234.734382203391$$
$$x_{40} = -93.3626507569302$$
$$x_{41} = -5821.90700008095$$
$$x_{42} = -102.787514924946$$
$$x_{43} = -87.0794112023455$$
$$x_{44} = -68.2292056709602$$
$$x_{45} = -12204.9931764696$$
$$x_{46} = -65.0894273082213$$
$$x_{47} = -99.6458650042135$$
$$x_{48} = -131.061823204732$$
$$x_{49} = -8.83480155558271$$
$$x_{50} = -476.63701652976$$
$$x_{51} = -1074.39547911081$$
$$x_{52} = -46.2457238206076$$
$$x_{53} = -564.547138764174$$
$$x_{54} = -442.080574029362$$
$$x_{55} = -17.8461051872707$$
$$x_{56} = -96.5043507553482$$
$$x_{57} = -58.8071586004732$$
Signos de extremos en los puntos:
(-1924.8227705161255, 1.7691438401015e-85)
(-124.77863664694901, -2.68076129189499e-7)
(-2330.505237196484, -4.22135311569299e-103)
(-83.93809402962373, 1.59173249107843e-5)
(-2.8161963539798656, 0.0385179368983429)
(-30.497523670527663, -0.00344988864624691)
(-36.79629068722217, -0.00180771879571231)
(-71.37212355219184, 5.5902289213802e-5)
(1.9031269014925303, 0.261012330751754)
(-582.0968365869004, -4.32301071762198e-28)
(-74.51274829056572, -4.08746729062026e-5)
(-486.06216765771495, 5.46926244404683e-23)
(-61.94535052822834, -0.000143779294596033)
(-11.455616957497138, -0.0298024142051202)
(-153.05297336861673, 1.58605436357488e-8)
(-822.3255566499581, -1.36867144921588e-37)
(-49.37536924798202, -0.000507203209118152)
(-898.2242665256471, -6.0251928423631e-41)
(-27.434557357262268, 0.00433563189151737)
(-39.968565452355406, 0.00127634792817982)
(-52.52569097122874, 0.000366918027567138)
(-80.79612420920053, -2.18015769450153e-5)
(-55.660909399925835, -0.000269881932289126)
(-77.65504793046756, 2.98317600251334e-5)
(-435.79631203309265, 8.33548665036684e-21)
(-2733.1250874308507, -1.060741158431e-120)
(-567.743203483864, 1.55078029557284e-26)
(-14.980847212115394, 0.0137393797338713)
(-39.96856545245299, 0.00127634792817982)
(-90.22120506939667, 8.49239687877201e-6)
(2.8235841405666235, 0.0721894678915088)
(-4047.089337204406, -8.2215084778855e-179)
(-3830.034419945751, 1.12522718794921e-168)
(-21.190715168226397, 0.00782017337166575)
(-43.08776681094528, -0.00095542252796214)
(-24.184974227384313, -0.00669881624888215)
(-21.1907151682264, 0.00782017337166574)
(-33.69665928673042, 0.00236392962632975)
(-234.73438220339085, 4.4971733134714e-12)
(-93.36265075693017, -6.203872366296e-6)
(-5821.907000080949, 2.51175383826673e-255)
(-102.78751492494612, 2.41717872746826e-6)
(-87.0794112023455, -1.16295803774142e-5)
(-68.22920567096024, -7.66479316941045e-5)
(-12204.993176469627, 4.78805184243152e-532)
(-65.08942730822133, 0.000104730111883607)
(-99.6458650042135, -3.30958665751948e-6)
(-131.0618232047325, -1.43015225686589e-7)
(-8.834801555582711, 0.0233108239607265)
(-476.63701652976, -1.40359463750799e-22)
(-1074.3954791108067, -1.12431696078796e-48)
(-46.245723820607644, 0.000685364558211034)
(-564.547138764174, -2.11999310241646e-26)
(-442.08057402936174, 4.44688025828588e-21)
(-17.84610518727068, -0.0134996580421051)
(-96.50435075534824, 4.5307900389751e-6)
(-58.8071586004732, 0.000196114702958014)
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:
$$x_{1} = -124.778636646949$$
$$x_{2} = -30.4975236705277$$
$$x_{3} = -36.7962906872222$$
$$x_{4} = 1.90312690149253$$
$$x_{5} = -74.5127482905657$$
$$x_{6} = -61.9453505282283$$
$$x_{7} = -11.4556169574971$$
$$x_{8} = -49.375369247982$$
$$x_{9} = -80.7961242092005$$
$$x_{10} = -55.6609093999258$$
$$x_{11} = -43.0877668109453$$
$$x_{12} = -24.1849742273843$$
$$x_{13} = -93.3626507569302$$
$$x_{14} = -87.0794112023455$$
$$x_{15} = -68.2292056709602$$
$$x_{16} = -99.6458650042135$$
$$x_{17} = -131.061823204732$$
$$x_{18} = -476.63701652976$$
$$x_{19} = -17.8461051872707$$
Puntos máximos de la función:
$$x_{19} = -83.9380940296237$$
$$x_{19} = -2.81619635397987$$
$$x_{19} = -71.3721235521918$$
$$x_{19} = -153.052973368617$$
$$x_{19} = -27.4345573572623$$
$$x_{19} = -39.9685654523554$$
$$x_{19} = -52.5256909712287$$
$$x_{19} = -77.6550479304676$$
$$x_{19} = -435.796312033093$$
$$x_{19} = -567.743203483864$$
$$x_{19} = -14.9808472121154$$
$$x_{19} = -39.968565452453$$
$$x_{19} = -90.2212050693967$$
$$x_{19} = 2.82358414056662$$
$$x_{19} = -21.1907151682264$$
$$x_{19} = -21.1907151682264$$
$$x_{19} = -33.6966592867304$$
$$x_{19} = -234.734382203391$$
$$x_{19} = -102.787514924946$$
$$x_{19} = -65.0894273082213$$
$$x_{19} = -8.83480155558271$$
$$x_{19} = -46.2457238206076$$
$$x_{19} = -96.5043507553482$$
$$x_{19} = -58.8071586004732$$
Decrece en los intervalos
$$\left[1.90312690149253, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -476.63701652976\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)}\right) = \left\langle -2, 2\right\rangle \lim_{x \to 0^+}\left(\frac{e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -2, 2\right\rangle \lim_{x \to 0^+}\left(\frac{e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)}\right) = \left\langle -2, 2\right\rangle \lim_{x \to 0^+}\left(\frac{e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -2, 2\right\rangle \lim_{x \to 0^+}\left(\frac{e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-E^(x/10)/10*(sin(x) + cos(x)))/((2*(1 - E^(x/10)*sin(x)))), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(- \frac{\frac{1}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{10 x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(- \frac{\frac{1}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{10 x}\right) = \left\langle -2, 2\right\rangle \lim_{x \to 0^+}\left(\frac{x e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -2, 2\right\rangle x \lim_{x \to 0^+}\left(\frac{x e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)$$
$$\lim_{x \to -\infty}\left(- \frac{\frac{1}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{10 x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(- \frac{\frac{1}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{10 x}\right) = \left\langle -2, 2\right\rangle \lim_{x \to 0^+}\left(\frac{x e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -2, 2\right\rangle x \lim_{x \to 0^+}\left(\frac{x e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)$$
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:
$$\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} = - \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- \frac{x}{10}}}{10 \left(2 + 2 e^{- \frac{x}{10}} \sin{\left(x \right)}\right)}$$
- No
$$\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} = \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- \frac{x}{10}}}{10 \left(2 + 2 e^{- \frac{x}{10}} \sin{\left(x \right)}\right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} = - \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- \frac{x}{10}}}{10 \left(2 + 2 e^{- \frac{x}{10}} \sin{\left(x \right)}\right)}$$
- No
$$\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} = \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- \frac{x}{10}}}{10 \left(2 + 2 e^{- \frac{x}{10}} \sin{\left(x \right)}\right)}$$
- No
es decir, función
no es
par ni impar