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- 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
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- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
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- Gráfico de la función y =:
-
7-x-2*x^2
-
y=x^3
-
(3*x-4)^40/(x^2-2)^36
-
y=x^2-x
-
Expresiones idénticas
- y=(sinx+cos^ dos x)/(2*x+sinx)
- y es igual a ( seno de x más coseno de al cuadrado x) dividir por (2 multiplicar por x más seno de x)
- y es igual a ( seno de x más coseno de en el grado dos x) dividir por (2 multiplicar por x más seno de x)
- y=(sinx+cos2x)/(2*x+sinx)
- y=sinx+cos2x/2*x+sinx
- y=(sinx+cos²x)/(2*x+sinx)
- y=(sinx+cos en el grado 2x)/(2*x+sinx)
- y=(sinx+cos^2x)/(2x+sinx)
- y=(sinx+cos2x)/(2x+sinx)
- y=sinx+cos2x/2x+sinx
- y=sinx+cos^2x/2x+sinx
- y=(sinx+cos^2x) dividir por (2*x+sinx)
-
Expresiones semejantes
- y=(sinx+cos^2x)/(2*x-sinx)
- y=(sinx-cos^2x)/(2*x+sinx)
-
Expresiones con funciones
- sinx
- sinx*sinx+cosx
- sinx+sin2x
- sinx:2
- sinx^2-cosx
- sinx/(x^2-4)
- Coseno cos
- cos(log(x))+sin(log(x))
- cos(3)/x
- cos((x+pi)/6)-1
- cos(x)*sqrt(x^2+1)
- cos(3*x^2+5*x-4)
- sinx
- sinx*sinx+cosx
- sinx+sin2x
- sinx:2
- sinx^2-cosx
- sinx/(x^2-4)
Gráfico de la función y = y=(sinx+cos^2x)/(2*x+sinx)
Solución
Ha introducido
[src]
2
sin(x) + cos (x)
f(x) = ----------------
2*x + sin(x)
$$f{\left(x \right)} = \frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}}$$
f = (sin(x) + cos(x)^2)/(2*x + sin(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:
$$\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)}$$
$$x_{2} = - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
Solución numérica
$$x_{1} = -27.6080944498156$$
$$x_{2} = 10.0910173932619$$
$$x_{3} = 81.0151695608421$$
$$x_{4} = -84.1567622144319$$
$$x_{5} = 93.5815401752013$$
$$x_{6} = 85.4892410794169$$
$$x_{7} = 68.4487989464829$$
$$x_{8} = 22.6573880076211$$
$$x_{9} = -59.0240209857136$$
$$x_{10} = -77.8735769072523$$
$$x_{11} = 62.1656136393034$$
$$x_{12} = -76.0644631186476$$
$$x_{13} = -103.006318135971$$
$$x_{14} = 98.0556116937761$$
$$x_{15} = 28.9405733148007$$
$$x_{16} = -63.4980925042884$$
$$x_{17} = 37.032872410585$$
$$x_{18} = 87.2983548680217$$
$$x_{19} = 47.7901292363394$$
$$x_{20} = 160.887464765572$$
$$x_{21} = -6.9494247396721$$
$$x_{22} = -15.0417238354565$$
$$x_{23} = -33.8912797569952$$
$$x_{24} = 11.9001311818667$$
$$x_{25} = -44.6485365827496$$
$$x_{26} = -88.6308337330067$$
$$x_{27} = 5.61694587468707$$
$$x_{28} = -19.5157953540313$$
$$x_{29} = 43.3160577177646$$
$$x_{30} = -13.2326100468517$$
$$x_{31} = -21.324909142636$$
$$x_{32} = 66.6396851578782$$
$$x_{33} = 60.3564998506986$$
$$x_{34} = -32.0821659683904$$
$$x_{35} = -40.1744650641748$$
$$x_{36} = -94.9140190401863$$
$$x_{37} = -38.36535127557$$
$$x_{38} = 49.5992430249442$$
$$x_{39} = 3.80783208608231$$
$$x_{40} = 35.2237586219802$$
$$x_{41} = 72.9228704650578$$
$$x_{42} = -46.4576503713544$$
$$x_{43} = -65.3072062928931$$
$$x_{44} = 30.7496871034054$$
$$x_{45} = 99.8647254823809$$
$$x_{46} = -82.3476484258271$$
$$x_{47} = 514.554955756234$$
$$x_{48} = -2.47535322109728$$
$$x_{49} = -25.7989806612109$$
$$x_{50} = -71.5903916000727$$
$$x_{51} = 16.3742027004415$$
$$x_{52} = 55.8824283321238$$
$$x_{53} = -57.2149071971088$$
$$x_{54} = -69.781277811468$$
$$x_{55} = -8.75853852827686$$
$$x_{56} = 91.7724263865965$$
$$x_{57} = -226.860910490958$$
$$x_{58} = 74.7319842536625$$
$$x_{59} = 18.1833164890462$$
$$x_{60} = 41.5069439291598$$
$$x_{61} = 24.4665017962258$$
$$x_{62} = 54.073314543519$$
$$x_{63} = -90.4399475216115$$
$$x_{64} = -50.9317218899292$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} + \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} \right)}$$
$$x_{2} = - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
Solución numérica
$$x_{1} = -27.6080944498156$$
$$x_{2} = 10.0910173932619$$
$$x_{3} = 81.0151695608421$$
$$x_{4} = -84.1567622144319$$
$$x_{5} = 93.5815401752013$$
$$x_{6} = 85.4892410794169$$
$$x_{7} = 68.4487989464829$$
$$x_{8} = 22.6573880076211$$
$$x_{9} = -59.0240209857136$$
$$x_{10} = -77.8735769072523$$
$$x_{11} = 62.1656136393034$$
$$x_{12} = -76.0644631186476$$
$$x_{13} = -103.006318135971$$
$$x_{14} = 98.0556116937761$$
$$x_{15} = 28.9405733148007$$
$$x_{16} = -63.4980925042884$$
$$x_{17} = 37.032872410585$$
$$x_{18} = 87.2983548680217$$
$$x_{19} = 47.7901292363394$$
$$x_{20} = 160.887464765572$$
$$x_{21} = -6.9494247396721$$
$$x_{22} = -15.0417238354565$$
$$x_{23} = -33.8912797569952$$
$$x_{24} = 11.9001311818667$$
$$x_{25} = -44.6485365827496$$
$$x_{26} = -88.6308337330067$$
$$x_{27} = 5.61694587468707$$
$$x_{28} = -19.5157953540313$$
$$x_{29} = 43.3160577177646$$
$$x_{30} = -13.2326100468517$$
$$x_{31} = -21.324909142636$$
$$x_{32} = 66.6396851578782$$
$$x_{33} = 60.3564998506986$$
$$x_{34} = -32.0821659683904$$
$$x_{35} = -40.1744650641748$$
$$x_{36} = -94.9140190401863$$
$$x_{37} = -38.36535127557$$
$$x_{38} = 49.5992430249442$$
$$x_{39} = 3.80783208608231$$
$$x_{40} = 35.2237586219802$$
$$x_{41} = 72.9228704650578$$
$$x_{42} = -46.4576503713544$$
$$x_{43} = -65.3072062928931$$
$$x_{44} = 30.7496871034054$$
$$x_{45} = 99.8647254823809$$
$$x_{46} = -82.3476484258271$$
$$x_{47} = 514.554955756234$$
$$x_{48} = -2.47535322109728$$
$$x_{49} = -25.7989806612109$$
$$x_{50} = -71.5903916000727$$
$$x_{51} = 16.3742027004415$$
$$x_{52} = 55.8824283321238$$
$$x_{53} = -57.2149071971088$$
$$x_{54} = -69.781277811468$$
$$x_{55} = -8.75853852827686$$
$$x_{56} = 91.7724263865965$$
$$x_{57} = -226.860910490958$$
$$x_{58} = 74.7319842536625$$
$$x_{59} = 18.1833164890462$$
$$x_{60} = 41.5069439291598$$
$$x_{61} = 24.4665017962258$$
$$x_{62} = 54.073314543519$$
$$x_{63} = -90.4399475216115$$
$$x_{64} = -50.9317218899292$$
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
$$\frac{- 2 \sin{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}}{2 x + \sin{\left(x \right)}} + \frac{\left(\sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \left(- \cos{\left(x \right)} - 2\right)}{\left(2 x + \sin{\left(x \right)}\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -41.3529774647475$$
$$x_{2} = 2.01479446037585$$
$$x_{3} = 65.4425860714719$$
$$x_{4} = 94.7589162310949$$
$$x_{5} = -9.90232850558352$$
$$x_{6} = 42.4035780651284$$
$$x_{7} = 51.8552096349014$$
$$x_{8} = 58.1363821829774$$
$$x_{9} = 31.9032825418075$$
$$x_{10} = 8.84534205100899$$
$$x_{11} = 40.3052742161767$$
$$x_{12} = -11.9316926800394$$
$$x_{13} = 36.1190016096917$$
$$x_{14} = 48.6877929199837$$
$$x_{15} = 23.5475888754376$$
$$x_{16} = -5.46404383047132$$
$$x_{17} = 78.0101320100576$$
$$x_{18} = -1.38328650289807$$
$$x_{19} = -14.1141070265638$$
$$x_{20} = 64.4179416566111$$
$$x_{21} = -53.9219704221859$$
$$x_{22} = 7.9678434923823$$
$$x_{23} = -20.4042825379789$$
$$x_{24} = -83.2482172625626$$
$$x_{25} = -80.1232545556264$$
$$x_{26} = -26.6912049585744$$
$$x_{27} = -3.54385575865849$$
$$x_{28} = -49.7173223129056$$
$$x_{29} = 44.4796697569725$$
$$x_{30} = -28.7817213408721$$
$$x_{31} = -37.142376006547$$
$$x_{32} = -99.9952926545408$$
$$x_{33} = 92.6733734818711$$
$$x_{34} = -45.5458283974807$$
$$x_{35} = 13.0047539475155$$
$$x_{36} = -66.4899772181092$$
$$x_{37} = -81.1428931093307$$
$$x_{38} = 82.1906640745$$
$$x_{39} = 59.1586246267012$$
$$x_{40} = -64.3975000055593$$
$$x_{41} = 73.8228916609809$$
$$x_{42} = -36.1567983317979$$
$$x_{43} = 76.9818470135987$$
$$x_{44} = 84.2937729640729$$
$$x_{45} = -39.2614901567144$$
$$x_{46} = 38.192280316289$$
$$x_{47} = 29.8338324102276$$
$$x_{48} = -43.430487276986$$
$$x_{49} = 6.64985262818388$$
$$x_{50} = 102.111461001086$$
$$x_{51} = 21.4451270784094$$
$$x_{52} = -23.6063277093999$$
$$x_{53} = 75.9063074576005$$
$$x_{54} = -29.8798267244363$$
$$x_{55} = 88.4748562048021$$
$$x_{56} = -16.2030378728615$$
$$x_{57} = 95.8289052600865$$
$$x_{58} = 14.2034149195081$$
$$x_{59} = -149.232397721914$$
$$x_{60} = 86.3899246117355$$
$$x_{61} = -162.83183633842$$
$$x_{62} = -47.6376456519054$$
$$x_{63} = -60.2060587086157$$
$$x_{64} = -97.9081631868469$$
$$x_{65} = 20.4671027647044$$
$$x_{66} = 71.72641079795$$
$$x_{67} = 34.0198786602713$$
$$x_{68} = -67.5592719381135$$
$$x_{69} = 67.53928229591$$
$$x_{70} = 101.042868557548$$
$$x_{71} = -73.841160194753$$
$$x_{72} = 246.619061877752$$
$$x_{73} = 50.7660377399756$$
$$x_{74} = -51.8298888465831$$
$$x_{75} = -93.7112881414842$$
$$x_{76} = -56.0033205465944$$
$$x_{77} = -86.4055095065621$$
$$x_{78} = -87.4271643315042$$
$$x_{79} = -91.6246494719537$$
$$x_{80} = 80.1064342216035$$
$$x_{81} = -62.2887403282097$$
$$x_{82} = -72.7737697226529$$
$$x_{83} = -18.2561916922943$$
$$x_{84} = -95.8151091146332$$
$$x_{85} = -89.5316814113144$$
$$x_{86} = -58.1137611132666$$
$$x_{87} = 27.733466786878$$
$$x_{88} = -7.81313193138257$$
$$x_{89} = 4.63537240878903$$
$$x_{90} = -22.49407589545$$
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:
$$x_{1} = -41.3529774647475$$
$$x_{2} = 2.01479446037585$$
$$x_{3} = -9.90232850558352$$
$$x_{4} = 42.4035780651284$$
$$x_{5} = 51.8552096349014$$
$$x_{6} = 58.1363821829774$$
$$x_{7} = -11.9316926800394$$
$$x_{8} = 36.1190016096917$$
$$x_{9} = 48.6877929199837$$
$$x_{10} = 23.5475888754376$$
$$x_{11} = -5.46404383047132$$
$$x_{12} = 64.4179416566111$$
$$x_{13} = -53.9219704221859$$
$$x_{14} = 7.9678434923823$$
$$x_{15} = -3.54385575865849$$
$$x_{16} = -49.7173223129056$$
$$x_{17} = -28.7817213408721$$
$$x_{18} = -37.142376006547$$
$$x_{19} = -99.9952926545408$$
$$x_{20} = 92.6733734818711$$
$$x_{21} = -66.4899772181092$$
$$x_{22} = -81.1428931093307$$
$$x_{23} = 73.8228916609809$$
$$x_{24} = 76.9818470135987$$
$$x_{25} = 29.8338324102276$$
$$x_{26} = -43.430487276986$$
$$x_{27} = 102.111461001086$$
$$x_{28} = -16.2030378728615$$
$$x_{29} = 95.8289052600865$$
$$x_{30} = 14.2034149195081$$
$$x_{31} = 86.3899246117355$$
$$x_{32} = -162.83183633842$$
$$x_{33} = -47.6376456519054$$
$$x_{34} = -60.2060587086157$$
$$x_{35} = -97.9081631868469$$
$$x_{36} = 20.4671027647044$$
$$x_{37} = 67.53928229591$$
$$x_{38} = 246.619061877752$$
$$x_{39} = -93.7112881414842$$
$$x_{40} = -56.0033205465944$$
$$x_{41} = -87.4271643315042$$
$$x_{42} = -91.6246494719537$$
$$x_{43} = 80.1064342216035$$
$$x_{44} = -62.2887403282097$$
$$x_{45} = -72.7737697226529$$
$$x_{46} = -18.2561916922943$$
$$x_{47} = 4.63537240878903$$
$$x_{48} = -22.49407589545$$
Puntos máximos de la función:
$$x_{48} = 65.4425860714719$$
$$x_{48} = 94.7589162310949$$
$$x_{48} = 31.9032825418075$$
$$x_{48} = 8.84534205100899$$
$$x_{48} = 40.3052742161767$$
$$x_{48} = 78.0101320100576$$
$$x_{48} = -1.38328650289807$$
$$x_{48} = -14.1141070265638$$
$$x_{48} = -20.4042825379789$$
$$x_{48} = -83.2482172625626$$
$$x_{48} = -80.1232545556264$$
$$x_{48} = -26.6912049585744$$
$$x_{48} = 44.4796697569725$$
$$x_{48} = -45.5458283974807$$
$$x_{48} = 13.0047539475155$$
$$x_{48} = 82.1906640745$$
$$x_{48} = 59.1586246267012$$
$$x_{48} = -64.3975000055593$$
$$x_{48} = -36.1567983317979$$
$$x_{48} = 84.2937729640729$$
$$x_{48} = -39.2614901567144$$
$$x_{48} = 38.192280316289$$
$$x_{48} = 6.64985262818388$$
$$x_{48} = 21.4451270784094$$
$$x_{48} = -23.6063277093999$$
$$x_{48} = 75.9063074576005$$
$$x_{48} = -29.8798267244363$$
$$x_{48} = 88.4748562048021$$
$$x_{48} = -149.232397721914$$
$$x_{48} = 71.72641079795$$
$$x_{48} = 34.0198786602713$$
$$x_{48} = -67.5592719381135$$
$$x_{48} = 101.042868557548$$
$$x_{48} = -73.841160194753$$
$$x_{48} = 50.7660377399756$$
$$x_{48} = -51.8298888465831$$
$$x_{48} = -86.4055095065621$$
$$x_{48} = -95.8151091146332$$
$$x_{48} = -89.5316814113144$$
$$x_{48} = -58.1137611132666$$
$$x_{48} = 27.733466786878$$
$$x_{48} = -7.81313193138257$$
Decrece en los intervalos
$$\left[246.619061877752, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -162.83183633842\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{- 2 \sin{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}}{2 x + \sin{\left(x \right)}} + \frac{\left(\sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \left(- \cos{\left(x \right)} - 2\right)}{\left(2 x + \sin{\left(x \right)}\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -41.3529774647475$$
$$x_{2} = 2.01479446037585$$
$$x_{3} = 65.4425860714719$$
$$x_{4} = 94.7589162310949$$
$$x_{5} = -9.90232850558352$$
$$x_{6} = 42.4035780651284$$
$$x_{7} = 51.8552096349014$$
$$x_{8} = 58.1363821829774$$
$$x_{9} = 31.9032825418075$$
$$x_{10} = 8.84534205100899$$
$$x_{11} = 40.3052742161767$$
$$x_{12} = -11.9316926800394$$
$$x_{13} = 36.1190016096917$$
$$x_{14} = 48.6877929199837$$
$$x_{15} = 23.5475888754376$$
$$x_{16} = -5.46404383047132$$
$$x_{17} = 78.0101320100576$$
$$x_{18} = -1.38328650289807$$
$$x_{19} = -14.1141070265638$$
$$x_{20} = 64.4179416566111$$
$$x_{21} = -53.9219704221859$$
$$x_{22} = 7.9678434923823$$
$$x_{23} = -20.4042825379789$$
$$x_{24} = -83.2482172625626$$
$$x_{25} = -80.1232545556264$$
$$x_{26} = -26.6912049585744$$
$$x_{27} = -3.54385575865849$$
$$x_{28} = -49.7173223129056$$
$$x_{29} = 44.4796697569725$$
$$x_{30} = -28.7817213408721$$
$$x_{31} = -37.142376006547$$
$$x_{32} = -99.9952926545408$$
$$x_{33} = 92.6733734818711$$
$$x_{34} = -45.5458283974807$$
$$x_{35} = 13.0047539475155$$
$$x_{36} = -66.4899772181092$$
$$x_{37} = -81.1428931093307$$
$$x_{38} = 82.1906640745$$
$$x_{39} = 59.1586246267012$$
$$x_{40} = -64.3975000055593$$
$$x_{41} = 73.8228916609809$$
$$x_{42} = -36.1567983317979$$
$$x_{43} = 76.9818470135987$$
$$x_{44} = 84.2937729640729$$
$$x_{45} = -39.2614901567144$$
$$x_{46} = 38.192280316289$$
$$x_{47} = 29.8338324102276$$
$$x_{48} = -43.430487276986$$
$$x_{49} = 6.64985262818388$$
$$x_{50} = 102.111461001086$$
$$x_{51} = 21.4451270784094$$
$$x_{52} = -23.6063277093999$$
$$x_{53} = 75.9063074576005$$
$$x_{54} = -29.8798267244363$$
$$x_{55} = 88.4748562048021$$
$$x_{56} = -16.2030378728615$$
$$x_{57} = 95.8289052600865$$
$$x_{58} = 14.2034149195081$$
$$x_{59} = -149.232397721914$$
$$x_{60} = 86.3899246117355$$
$$x_{61} = -162.83183633842$$
$$x_{62} = -47.6376456519054$$
$$x_{63} = -60.2060587086157$$
$$x_{64} = -97.9081631868469$$
$$x_{65} = 20.4671027647044$$
$$x_{66} = 71.72641079795$$
$$x_{67} = 34.0198786602713$$
$$x_{68} = -67.5592719381135$$
$$x_{69} = 67.53928229591$$
$$x_{70} = 101.042868557548$$
$$x_{71} = -73.841160194753$$
$$x_{72} = 246.619061877752$$
$$x_{73} = 50.7660377399756$$
$$x_{74} = -51.8298888465831$$
$$x_{75} = -93.7112881414842$$
$$x_{76} = -56.0033205465944$$
$$x_{77} = -86.4055095065621$$
$$x_{78} = -87.4271643315042$$
$$x_{79} = -91.6246494719537$$
$$x_{80} = 80.1064342216035$$
$$x_{81} = -62.2887403282097$$
$$x_{82} = -72.7737697226529$$
$$x_{83} = -18.2561916922943$$
$$x_{84} = -95.8151091146332$$
$$x_{85} = -89.5316814113144$$
$$x_{86} = -58.1137611132666$$
$$x_{87} = 27.733466786878$$
$$x_{88} = -7.81313193138257$$
$$x_{89} = 4.63537240878903$$
$$x_{90} = -22.49407589545$$
Signos de extremos en los puntos:
(-41.35297746474752, -0.0152027133614035)
(2.0147944603758536, 0.220482274888344)
(65.44258607147187, 0.00951325717485948)
(94.75891623109493, 0.00657808753630484)
(-9.902328505583515, -0.0645316571557262)
(42.40357806512839, -0.0119310273495031)
(51.85520963490141, 0.00955187525373224)
(58.136382182977364, 0.00852835961442124)
(31.903282541807478, 0.0194320863770111)
(8.845342051008988, 0.0684133486818883)
(40.30527421617674, 0.0154078405932773)
(-11.931692680039395, -0.0533451348390638)
(36.119001609691715, -0.0140356161819472)
(48.68779291998369, -0.0103753297176253)
(23.547588875437643, -0.0216874885107848)
(-5.464043830471319, -0.117365912127011)
(78.01013201005762, 0.00798574112803653)
(-1.3832865028980688, 0.252790196972477)
(-14.114107026563767, 0.0341865416009588)
(64.41794165661113, 0.00770293840013173)
(-53.92197042218591, -0.0116434603759953)
(7.967843492382297, 0.0594495094375276)
(-20.404282537978926, 0.0239093534269942)
(-83.24821726256265, 0.00597013419122284)
(-80.12325455562645, -0.00628007109848922)
(-26.691204958574446, 0.0183841295155401)
(-3.54385575865849, -0.184914779153016)
(-49.71732231290559, -0.0126327922775973)
(44.47966975697249, 0.0139705490691659)
(-28.781721340872146, -0.0218965446993551)
(-37.142376006547046, -0.0169367477684101)
(-99.99529265454082, -0.0062657429254339)
(92.67337348187107, -0.00542445383614381)
(-45.545828397480655, 0.0108578903922337)
(13.004753947515454, 0.0470718357843687)
(-66.48997721810916, -0.00943466905134992)
(-81.14289310933074, -0.00772585750200205)
(82.19066407449996, 0.00758084007393295)
(59.158624626701204, 0.0105193373987631)
(-64.39750000555927, 0.00770415137676712)
(73.82289166098089, -0.00681894186471197)
(-36.15679833179786, -0.01402817695801)
(76.9818470135987, 0.00645365867963294)
(84.29377296407291, 0.00739226721528381)
(-39.261490156714416, 0.0125736503967048)
(38.19228031628896, 0.0162545706074322)
(29.833832410227643, -0.0170418826311983)
(-43.430487276986035, -0.0144713926406672)
(6.649852628183878, 0.0900542156831534)
(102.11146100108562, 0.00487298036478682)
(21.44512707840941, 0.0287869368499406)
(-23.606327709399945, -0.0216599207857995)
(75.90630745760052, 0.00820633675619302)
(-29.87982672443633, -0.0170285424415272)
(88.47485620480208, 0.007043951926617)
(-16.203037872861493, -0.0391275155457851)
(95.82890526008649, 0.00519082833121826)
(14.203414919508088, 0.0340826791629927)
(-149.23239772191351, -0.00336181859844665)
(86.38992461173554, -0.00582127331312244)
(-162.83183633841955, -0.0038441681091659)
(-47.63764565190537, -0.0131871311770171)
(-60.20605870861569, -0.0104233301405802)
(-97.90816318684689, -0.00639964899703906)
(20.467102764704443, 0.0238735295997772)
(71.72641079794995, 0.00868282771096448)
(34.01987866027134, 0.0182322285559569)
(-67.55927193811354, -0.00745692625876442)
(67.53928229591003, -0.0074580378509206)
(101.0428685575477, 0.006170026813837)
(-73.84116019475299, -0.00681809259255276)
(246.6190618777524, 0.00202333272846652)
(50.76603773997558, 0.0122495064026728)
(-51.82988884658312, 0.00955418567442668)
(-93.71128814148425, -0.00668699669612042)
(-56.00332054659435, -0.0112088348328392)
(-86.40550950656214, -0.00582074526462848)
(-87.42716433150423, -0.00716898290594765)
(-91.62464947195366, -0.00683969620672395)
(80.10643422160345, -0.00628073450318515)
(-62.28874032820971, -0.0100734388621746)
(-72.77376972265293, -0.00861731463254976)
(-18.25619169229427, -0.0346700666526081)
(-95.81510911463316, 0.00519120007329383)
(-89.53168141131435, 0.00555348620160029)
(-58.11376111326663, 0.00853000502082749)
(27.733466786878015, 0.0223247253197552)
(-7.813131931382567, 0.0599983290002394)
(4.635372408789032, -0.119790995838967)
(-22.49407589544997, -0.0280787141914041)
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:
$$x_{1} = -41.3529774647475$$
$$x_{2} = 2.01479446037585$$
$$x_{3} = -9.90232850558352$$
$$x_{4} = 42.4035780651284$$
$$x_{5} = 51.8552096349014$$
$$x_{6} = 58.1363821829774$$
$$x_{7} = -11.9316926800394$$
$$x_{8} = 36.1190016096917$$
$$x_{9} = 48.6877929199837$$
$$x_{10} = 23.5475888754376$$
$$x_{11} = -5.46404383047132$$
$$x_{12} = 64.4179416566111$$
$$x_{13} = -53.9219704221859$$
$$x_{14} = 7.9678434923823$$
$$x_{15} = -3.54385575865849$$
$$x_{16} = -49.7173223129056$$
$$x_{17} = -28.7817213408721$$
$$x_{18} = -37.142376006547$$
$$x_{19} = -99.9952926545408$$
$$x_{20} = 92.6733734818711$$
$$x_{21} = -66.4899772181092$$
$$x_{22} = -81.1428931093307$$
$$x_{23} = 73.8228916609809$$
$$x_{24} = 76.9818470135987$$
$$x_{25} = 29.8338324102276$$
$$x_{26} = -43.430487276986$$
$$x_{27} = 102.111461001086$$
$$x_{28} = -16.2030378728615$$
$$x_{29} = 95.8289052600865$$
$$x_{30} = 14.2034149195081$$
$$x_{31} = 86.3899246117355$$
$$x_{32} = -162.83183633842$$
$$x_{33} = -47.6376456519054$$
$$x_{34} = -60.2060587086157$$
$$x_{35} = -97.9081631868469$$
$$x_{36} = 20.4671027647044$$
$$x_{37} = 67.53928229591$$
$$x_{38} = 246.619061877752$$
$$x_{39} = -93.7112881414842$$
$$x_{40} = -56.0033205465944$$
$$x_{41} = -87.4271643315042$$
$$x_{42} = -91.6246494719537$$
$$x_{43} = 80.1064342216035$$
$$x_{44} = -62.2887403282097$$
$$x_{45} = -72.7737697226529$$
$$x_{46} = -18.2561916922943$$
$$x_{47} = 4.63537240878903$$
$$x_{48} = -22.49407589545$$
Puntos máximos de la función:
$$x_{48} = 65.4425860714719$$
$$x_{48} = 94.7589162310949$$
$$x_{48} = 31.9032825418075$$
$$x_{48} = 8.84534205100899$$
$$x_{48} = 40.3052742161767$$
$$x_{48} = 78.0101320100576$$
$$x_{48} = -1.38328650289807$$
$$x_{48} = -14.1141070265638$$
$$x_{48} = -20.4042825379789$$
$$x_{48} = -83.2482172625626$$
$$x_{48} = -80.1232545556264$$
$$x_{48} = -26.6912049585744$$
$$x_{48} = 44.4796697569725$$
$$x_{48} = -45.5458283974807$$
$$x_{48} = 13.0047539475155$$
$$x_{48} = 82.1906640745$$
$$x_{48} = 59.1586246267012$$
$$x_{48} = -64.3975000055593$$
$$x_{48} = -36.1567983317979$$
$$x_{48} = 84.2937729640729$$
$$x_{48} = -39.2614901567144$$
$$x_{48} = 38.192280316289$$
$$x_{48} = 6.64985262818388$$
$$x_{48} = 21.4451270784094$$
$$x_{48} = -23.6063277093999$$
$$x_{48} = 75.9063074576005$$
$$x_{48} = -29.8798267244363$$
$$x_{48} = 88.4748562048021$$
$$x_{48} = -149.232397721914$$
$$x_{48} = 71.72641079795$$
$$x_{48} = 34.0198786602713$$
$$x_{48} = -67.5592719381135$$
$$x_{48} = 101.042868557548$$
$$x_{48} = -73.841160194753$$
$$x_{48} = 50.7660377399756$$
$$x_{48} = -51.8298888465831$$
$$x_{48} = -86.4055095065621$$
$$x_{48} = -95.8151091146332$$
$$x_{48} = -89.5316814113144$$
$$x_{48} = -58.1137611132666$$
$$x_{48} = 27.733466786878$$
$$x_{48} = -7.81313193138257$$
Decrece en los intervalos
$$\left[246.619061877752, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -162.83183633842\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (sin(x) + cos(x)^2)/(2*x + sin(x)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{x \left(2 x + \sin{\left(x \right)}\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{x \left(2 x + \sin{\left(x \right)}\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{x \left(2 x + \sin{\left(x \right)}\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{x \left(2 x + \sin{\left(x \right)}\right)}\right) = 0$$
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:
$$\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}} = \frac{- \sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{- 2 x - \sin{\left(x \right)}}$$
- No
$$\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}} = - \frac{- \sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{- 2 x - \sin{\left(x \right)}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}} = \frac{- \sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{- 2 x - \sin{\left(x \right)}}$$
- No
$$\frac{\sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 x + \sin{\left(x \right)}} = - \frac{- \sin{\left(x \right)} + \cos^{2}{\left(x \right)}}{- 2 x - \sin{\left(x \right)}}$$
- No
es decir, función
no es
par ni impar
Gráfico