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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
y=(x+1)^3
-
y=2x
-
2*x^3-3*x
-
3-x^2
-
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+cos en el grado 2x/2*x+sinx
- y=sinx+cos^2x/2x+sinx
- y=sinx+cos2x/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+0,5
- Coseno cos
- cos[x]/x^2
- cos(x-pi/3)-1
- cos(x)/(1+cos^2*2x)
- cos(1)/((3*x))
- cos(sqrt(x))+1
- sinx
- sinx+0,5
Gráfico de la función y = y=sinx+cos^2x/2*x+sinx
Solución
Ha introducido
[src]
2
cos (x)
f(x) = sin(x) + -------*x + sin(x)
2
$$f{\left(x \right)} = \left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)}$$
f = x*(cos(x)^2/2) + sin(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:
$$\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 3.78871831657775$$
$$x_{2} = 67.7859501946834$$
$$x_{3} = -92.4697561180394$$
$$x_{4} = -23.9646095965669$$
$$x_{5} = 23.9646095965669$$
$$x_{6} = -3.78871831657775$$
$$x_{7} = 98.7596019103327$$
$$x_{8} = -61.0064157685764$$
$$x_{9} = 0$$
$$x_{10} = -55.2453025786868$$
$$x_{11} = 11.5655493528518$$
$$x_{12} = 42.1057726589812$$
$$x_{13} = 17.7442106228472$$
$$x_{14} = 54.7091495657126$$
$$x_{15} = 30.2049071742994$$
$$x_{16} = -74.0587718328223$$
$$x_{17} = 74.0587718328223$$
$$x_{18} = 124.271833897698$$
$$x_{19} = 29.4810687336138$$
$$x_{20} = 42.7150766041324$$
$$x_{21} = -42.1057726589812$$
$$x_{22} = -98.7596019103327$$
$$x_{23} = -61.5146554458148$$
$$x_{24} = -80.3328196529408$$
$$x_{25} = 61.0064157685764$$
$$x_{26} = 5.50677644245697$$
$$x_{27} = 55.2453025786868$$
$$x_{28} = -67.7859501946834$$
$$x_{29} = 80.3328196529408$$
$$x_{30} = -99.1603323751836$$
$$x_{31} = 10.3966914949526$$
$$x_{32} = 16.8009836799829$$
$$x_{33} = 61.5146554458148$$
$$x_{34} = -17.7442106228472$$
$$x_{35} = -54.7091495657126$$
$$x_{36} = -35.7972276795867$$
$$x_{37} = -10.3966914949526$$
$$x_{38} = -86.1791986560035$$
$$x_{39} = 86.1791986560035$$
$$x_{40} = 92.4697561180394$$
$$x_{41} = 99.1603323751836$$
$$x_{42} = -5.50677644245697$$
$$x_{43} = -48.4092489972788$$
$$x_{44} = -79.8877929071161$$
$$x_{45} = -11.5655493528518$$
$$x_{46} = 48.4092489972788$$
$$x_{47} = -92.8837515060376$$
$$x_{48} = -48.9784834055309$$
$$x_{49} = -30.2049071742994$$
o sea hay que resolver la ecuación:
$$\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 3.78871831657775$$
$$x_{2} = 67.7859501946834$$
$$x_{3} = -92.4697561180394$$
$$x_{4} = -23.9646095965669$$
$$x_{5} = 23.9646095965669$$
$$x_{6} = -3.78871831657775$$
$$x_{7} = 98.7596019103327$$
$$x_{8} = -61.0064157685764$$
$$x_{9} = 0$$
$$x_{10} = -55.2453025786868$$
$$x_{11} = 11.5655493528518$$
$$x_{12} = 42.1057726589812$$
$$x_{13} = 17.7442106228472$$
$$x_{14} = 54.7091495657126$$
$$x_{15} = 30.2049071742994$$
$$x_{16} = -74.0587718328223$$
$$x_{17} = 74.0587718328223$$
$$x_{18} = 124.271833897698$$
$$x_{19} = 29.4810687336138$$
$$x_{20} = 42.7150766041324$$
$$x_{21} = -42.1057726589812$$
$$x_{22} = -98.7596019103327$$
$$x_{23} = -61.5146554458148$$
$$x_{24} = -80.3328196529408$$
$$x_{25} = 61.0064157685764$$
$$x_{26} = 5.50677644245697$$
$$x_{27} = 55.2453025786868$$
$$x_{28} = -67.7859501946834$$
$$x_{29} = 80.3328196529408$$
$$x_{30} = -99.1603323751836$$
$$x_{31} = 10.3966914949526$$
$$x_{32} = 16.8009836799829$$
$$x_{33} = 61.5146554458148$$
$$x_{34} = -17.7442106228472$$
$$x_{35} = -54.7091495657126$$
$$x_{36} = -35.7972276795867$$
$$x_{37} = -10.3966914949526$$
$$x_{38} = -86.1791986560035$$
$$x_{39} = 86.1791986560035$$
$$x_{40} = 92.4697561180394$$
$$x_{41} = 99.1603323751836$$
$$x_{42} = -5.50677644245697$$
$$x_{43} = -48.4092489972788$$
$$x_{44} = -79.8877929071161$$
$$x_{45} = -11.5655493528518$$
$$x_{46} = 48.4092489972788$$
$$x_{47} = -92.8837515060376$$
$$x_{48} = -48.9784834055309$$
$$x_{49} = -30.2049071742994$$
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
$$- x \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{2} + 2 \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -44.0390771234017$$
$$x_{2} = 23.5619449019235$$
$$x_{3} = 2.3860860459443$$
$$x_{4} = -86.3937979737193$$
$$x_{5} = 65.9506975390986$$
$$x_{6} = -53.3789667537027$$
$$x_{7} = 44.0390771234017$$
$$x_{8} = -95.8185759344887$$
$$x_{9} = 56.5928487077727$$
$$x_{10} = 21.9226189316584$$
$$x_{11} = -83.2522053201295$$
$$x_{12} = 95.8185759344887$$
$$x_{13} = 303.163691071415$$
$$x_{14} = -97.373966514501$$
$$x_{15} = -94.274299213244$$
$$x_{16} = 37.7653294920198$$
$$x_{17} = 9.26136941301364$$
$$x_{18} = 45.553093477052$$
$$x_{19} = 20.4203522483337$$
$$x_{20} = -80.1106126665397$$
$$x_{21} = 72.2358627380718$$
$$x_{22} = -14.1371669411541$$
$$x_{23} = 81.7120061622563$$
$$x_{24} = -29.845130209103$$
$$x_{25} = 70.6858347057703$$
$$x_{26} = -102.101761241668$$
$$x_{27} = -15.6115829776934$$
$$x_{28} = 48.6946861306418$$
$$x_{29} = -81.7120061622563$$
$$x_{30} = -31.4953367711689$$
$$x_{31} = -20.4203522483337$$
$$x_{32} = -61.261056745001$$
$$x_{33} = 26.7035375555132$$
$$x_{34} = 94.274299213244$$
$$x_{35} = -9.26136941301364$$
$$x_{36} = -87.993007178388$$
$$x_{37} = -51.8362787842316$$
$$x_{38} = 15.6115829776934$$
$$x_{39} = -42.4115008234622$$
$$x_{40} = -17.2787595947439$$
$$x_{41} = -28.2211320612647$$
$$x_{42} = 51.8362787842316$$
$$x_{43} = 73.8274273593601$$
$$x_{44} = -65.9506975390986$$
$$x_{45} = 34.5140312012932$$
$$x_{46} = 59.665114800435$$
$$x_{47} = 29.845130209103$$
$$x_{48} = -58.1194640914112$$
$$x_{49} = 32.9867228626928$$
$$x_{50} = 42.4115008234622$$
$$x_{51} = 12.7627599249538$$
$$x_{52} = -23.5619449019235$$
$$x_{53} = -64.4026493985908$$
$$x_{54} = -73.8274273593601$$
$$x_{55} = -89.5353906273091$$
$$x_{56} = -72.2358627380718$$
$$x_{57} = 64.4026493985908$$
$$x_{58} = 80.1106126665397$$
$$x_{59} = 78.5207107750708$$
$$x_{60} = -50.3151774380626$$
$$x_{61} = -39.2699081698724$$
$$x_{62} = 28.2211320612647$$
$$x_{63} = -37.7653294920198$$
$$x_{64} = 36.1283155162826$$
$$x_{65} = -45.553093477052$$
$$x_{66} = -75.4313688242127$$
$$x_{67} = 14.1371669411541$$
$$x_{68} = 1.95517406323303$$
$$x_{69} = 50.3151774380626$$
$$x_{70} = 87.993007178388$$
$$x_{71} = 144.530559790087$$
$$x_{72} = 92.6769832808989$$
$$x_{73} = 67.5442420521806$$
$$x_{74} = -21.9226189316584$$
$$x_{75} = 58.1194640914112$$
$$x_{76} = 6.66212078172724$$
$$x_{77} = -7.85398163397448$$
$$x_{78} = 7.85398163397448$$
$$x_{79} = 100.555827750706$$
$$x_{80} = -67.5442420521806$$
$$x_{81} = 89.5353906273091$$
$$x_{82} = 86.3937979737193$$
$$x_{83} = -36.1283155162826$$
$$x_{84} = -59.665114800435$$
$$x_{85} = 4.71238898038469$$
$$x_{86} = 1.5707963267949$$
$$x_{87} = -1.95517406323303$$
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:
$$x_{1} = -44.0390771234017$$
$$x_{2} = 23.5619449019235$$
$$x_{3} = -53.3789667537027$$
$$x_{4} = 95.8185759344887$$
$$x_{5} = 303.163691071415$$
$$x_{6} = -97.373966514501$$
$$x_{7} = -94.274299213244$$
$$x_{8} = 45.553093477052$$
$$x_{9} = 20.4203522483337$$
$$x_{10} = 70.6858347057703$$
$$x_{11} = -15.6115829776934$$
$$x_{12} = 48.6946861306418$$
$$x_{13} = -81.7120061622563$$
$$x_{14} = -31.4953367711689$$
$$x_{15} = 26.7035375555132$$
$$x_{16} = -9.26136941301364$$
$$x_{17} = -87.993007178388$$
$$x_{18} = -28.2211320612647$$
$$x_{19} = 51.8362787842316$$
$$x_{20} = 73.8274273593601$$
$$x_{21} = -65.9506975390986$$
$$x_{22} = 29.845130209103$$
$$x_{23} = 32.9867228626928$$
$$x_{24} = 42.4115008234622$$
$$x_{25} = -72.2358627380718$$
$$x_{26} = 64.4026493985908$$
$$x_{27} = 80.1106126665397$$
$$x_{28} = -50.3151774380626$$
$$x_{29} = -37.7653294920198$$
$$x_{30} = 36.1283155162826$$
$$x_{31} = -75.4313688242127$$
$$x_{32} = 14.1371669411541$$
$$x_{33} = 1.95517406323303$$
$$x_{34} = 92.6769832808989$$
$$x_{35} = 67.5442420521806$$
$$x_{36} = -21.9226189316584$$
$$x_{37} = 58.1194640914112$$
$$x_{38} = 7.85398163397448$$
$$x_{39} = 89.5353906273091$$
$$x_{40} = 86.3937979737193$$
$$x_{41} = -59.665114800435$$
$$x_{42} = 4.71238898038469$$
Puntos máximos de la función:
$$x_{42} = 2.3860860459443$$
$$x_{42} = -86.3937979737193$$
$$x_{42} = 65.9506975390986$$
$$x_{42} = 44.0390771234017$$
$$x_{42} = -95.8185759344887$$
$$x_{42} = 56.5928487077727$$
$$x_{42} = 21.9226189316584$$
$$x_{42} = -83.2522053201295$$
$$x_{42} = 37.7653294920198$$
$$x_{42} = 9.26136941301364$$
$$x_{42} = -80.1106126665397$$
$$x_{42} = 72.2358627380718$$
$$x_{42} = -14.1371669411541$$
$$x_{42} = 81.7120061622563$$
$$x_{42} = -29.845130209103$$
$$x_{42} = -102.101761241668$$
$$x_{42} = -20.4203522483337$$
$$x_{42} = -61.261056745001$$
$$x_{42} = 94.274299213244$$
$$x_{42} = -51.8362787842316$$
$$x_{42} = 15.6115829776934$$
$$x_{42} = -42.4115008234622$$
$$x_{42} = -17.2787595947439$$
$$x_{42} = 34.5140312012932$$
$$x_{42} = 59.665114800435$$
$$x_{42} = -58.1194640914112$$
$$x_{42} = 12.7627599249538$$
$$x_{42} = -23.5619449019235$$
$$x_{42} = -64.4026493985908$$
$$x_{42} = -73.8274273593601$$
$$x_{42} = -89.5353906273091$$
$$x_{42} = 78.5207107750708$$
$$x_{42} = -39.2699081698724$$
$$x_{42} = 28.2211320612647$$
$$x_{42} = -45.553093477052$$
$$x_{42} = 50.3151774380626$$
$$x_{42} = 87.993007178388$$
$$x_{42} = 144.530559790087$$
$$x_{42} = 6.66212078172724$$
$$x_{42} = -7.85398163397448$$
$$x_{42} = 100.555827750706$$
$$x_{42} = -67.5442420521806$$
$$x_{42} = -36.1283155162826$$
$$x_{42} = 1.5707963267949$$
$$x_{42} = -1.95517406323303$$
Decrece en los intervalos
$$\left[303.163691071415, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.373966514501\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- x \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{2} + 2 \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -44.0390771234017$$
$$x_{2} = 23.5619449019235$$
$$x_{3} = 2.3860860459443$$
$$x_{4} = -86.3937979737193$$
$$x_{5} = 65.9506975390986$$
$$x_{6} = -53.3789667537027$$
$$x_{7} = 44.0390771234017$$
$$x_{8} = -95.8185759344887$$
$$x_{9} = 56.5928487077727$$
$$x_{10} = 21.9226189316584$$
$$x_{11} = -83.2522053201295$$
$$x_{12} = 95.8185759344887$$
$$x_{13} = 303.163691071415$$
$$x_{14} = -97.373966514501$$
$$x_{15} = -94.274299213244$$
$$x_{16} = 37.7653294920198$$
$$x_{17} = 9.26136941301364$$
$$x_{18} = 45.553093477052$$
$$x_{19} = 20.4203522483337$$
$$x_{20} = -80.1106126665397$$
$$x_{21} = 72.2358627380718$$
$$x_{22} = -14.1371669411541$$
$$x_{23} = 81.7120061622563$$
$$x_{24} = -29.845130209103$$
$$x_{25} = 70.6858347057703$$
$$x_{26} = -102.101761241668$$
$$x_{27} = -15.6115829776934$$
$$x_{28} = 48.6946861306418$$
$$x_{29} = -81.7120061622563$$
$$x_{30} = -31.4953367711689$$
$$x_{31} = -20.4203522483337$$
$$x_{32} = -61.261056745001$$
$$x_{33} = 26.7035375555132$$
$$x_{34} = 94.274299213244$$
$$x_{35} = -9.26136941301364$$
$$x_{36} = -87.993007178388$$
$$x_{37} = -51.8362787842316$$
$$x_{38} = 15.6115829776934$$
$$x_{39} = -42.4115008234622$$
$$x_{40} = -17.2787595947439$$
$$x_{41} = -28.2211320612647$$
$$x_{42} = 51.8362787842316$$
$$x_{43} = 73.8274273593601$$
$$x_{44} = -65.9506975390986$$
$$x_{45} = 34.5140312012932$$
$$x_{46} = 59.665114800435$$
$$x_{47} = 29.845130209103$$
$$x_{48} = -58.1194640914112$$
$$x_{49} = 32.9867228626928$$
$$x_{50} = 42.4115008234622$$
$$x_{51} = 12.7627599249538$$
$$x_{52} = -23.5619449019235$$
$$x_{53} = -64.4026493985908$$
$$x_{54} = -73.8274273593601$$
$$x_{55} = -89.5353906273091$$
$$x_{56} = -72.2358627380718$$
$$x_{57} = 64.4026493985908$$
$$x_{58} = 80.1106126665397$$
$$x_{59} = 78.5207107750708$$
$$x_{60} = -50.3151774380626$$
$$x_{61} = -39.2699081698724$$
$$x_{62} = 28.2211320612647$$
$$x_{63} = -37.7653294920198$$
$$x_{64} = 36.1283155162826$$
$$x_{65} = -45.553093477052$$
$$x_{66} = -75.4313688242127$$
$$x_{67} = 14.1371669411541$$
$$x_{68} = 1.95517406323303$$
$$x_{69} = 50.3151774380626$$
$$x_{70} = 87.993007178388$$
$$x_{71} = 144.530559790087$$
$$x_{72} = 92.6769832808989$$
$$x_{73} = 67.5442420521806$$
$$x_{74} = -21.9226189316584$$
$$x_{75} = 58.1194640914112$$
$$x_{76} = 6.66212078172724$$
$$x_{77} = -7.85398163397448$$
$$x_{78} = 7.85398163397448$$
$$x_{79} = 100.555827750706$$
$$x_{80} = -67.5442420521806$$
$$x_{81} = 89.5353906273091$$
$$x_{82} = 86.3937979737193$$
$$x_{83} = -36.1283155162826$$
$$x_{84} = -59.665114800435$$
$$x_{85} = 4.71238898038469$$
$$x_{86} = 1.5707963267949$$
$$x_{87} = -1.95517406323303$$
Signos de extremos en los puntos:
(-44.03907712340174, -22.0621235268108)
(23.56194490192345, -2)
(2.386086045944296, 2.00347724185052)
(-86.39379797371932, 2)
(65.95069753909861, 33.0037800787516)
(-53.37896675370266, -26.7246114216225)
(44.03907712340174, 22.0621235268108)
(-95.81857593448869, -2)
(56.59284870777266, 28.3295600570462)
(21.92261893165838, 11.0468643042567)
(-83.25220532012952, -2)
(95.81857593448869, 2)
(303.16369107141503, 2)
(-97.37396651450105, -48.7062392219801)
(-94.27429921324396, -47.1570393104567)
(37.76532949201978, 18.9323279509052)
(9.261369413013638, 4.8334957711568)
(45.553093477052, 2)
(20.420352248333657, 2)
(-80.11061266653972, 2)
(72.23586273807183, 36.1438887513725)
(-14.137166941154069, -2)
(81.71200616225626, 40.878950957149)
(-29.845130209103036, 2)
(70.68583470577035, 2)
(-102.10176124166829, -2)
(-15.611582977693413, -7.92596876727007)
(48.6946861306418, -2)
(-81.71200616225626, -40.878950957149)
(-31.4953367711689, -15.8072259831523)
(-20.420352248333657, -2)
(-61.26105674500097, 2)
(26.703537555513243, 2)
(94.27429921324396, 47.1570393104567)
(-9.261369413013638, -4.8334957711568)
(-87.99300717838803, -44.0178132471365)
(-51.83627878423159, -2)
(15.611582977693413, 7.92596876727007)
(-42.411500823462205, 2)
(-17.278759594743864, 2)
(-28.221132061264726, -14.1770181299482)
(51.83627878423159, 2)
(73.82742735936014, -2)
(-65.95069753909861, -33.0037800787516)
(34.51403120129323, 17.3113481774677)
(59.66511480043501, 29.8639841229746)
(29.845130209103036, -2)
(-58.119464091411174, -2)
(32.98672286269283, 2)
(42.411500823462205, -2)
(12.762759924953802, 6.52866467541177)
(-23.56194490192345, 2)
(-64.40264939859077, -2)
(-73.82742735936014, 2)
(-89.53539062730911, -2)
(-72.23586273807183, -36.1438887513725)
(64.40264939859077, 2)
(80.11061266653972, -2)
(78.52071077507081, 39.2842350188297)
(-50.31517743806264, -25.1948599469259)
(-39.269908169872416, -2)
(28.221132061264726, 14.1770181299482)
(-37.76532949201978, -18.9323279509052)
(36.12831551628262, -2)
(-45.553093477052, -2)
(-75.43136882421273, -37.7405432646497)
(14.137166941154069, 2)
(1.9551740632330274, 1.99152413666373)
(50.31517743806264, 25.1948599469259)
(87.99300717838803, 44.0178132471365)
(144.53055979008684, 72.278253188722)
(92.6769832808989, -2)
(67.54424205218055, -2)
(-21.92261893165838, -11.0468643042567)
(58.119464091411174, 2)
(6.662120781727237, 3.61506997914071)
(-7.853981633974483, -2)
(7.853981633974483, 2)
(100.55582775070604, 50.29656100199)
(-67.54424205218055, 2)
(89.53539062730911, 2)
(86.39379797371932, -2)
(-36.12831551628262, 2)
(-59.66511480043501, -29.8639841229746)
(4.71238898038469, -2)
(1.5707963267948966, 2)
(-1.9551740632330274, -1.99152413666373)
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:
$$x_{1} = -44.0390771234017$$
$$x_{2} = 23.5619449019235$$
$$x_{3} = -53.3789667537027$$
$$x_{4} = 95.8185759344887$$
$$x_{5} = 303.163691071415$$
$$x_{6} = -97.373966514501$$
$$x_{7} = -94.274299213244$$
$$x_{8} = 45.553093477052$$
$$x_{9} = 20.4203522483337$$
$$x_{10} = 70.6858347057703$$
$$x_{11} = -15.6115829776934$$
$$x_{12} = 48.6946861306418$$
$$x_{13} = -81.7120061622563$$
$$x_{14} = -31.4953367711689$$
$$x_{15} = 26.7035375555132$$
$$x_{16} = -9.26136941301364$$
$$x_{17} = -87.993007178388$$
$$x_{18} = -28.2211320612647$$
$$x_{19} = 51.8362787842316$$
$$x_{20} = 73.8274273593601$$
$$x_{21} = -65.9506975390986$$
$$x_{22} = 29.845130209103$$
$$x_{23} = 32.9867228626928$$
$$x_{24} = 42.4115008234622$$
$$x_{25} = -72.2358627380718$$
$$x_{26} = 64.4026493985908$$
$$x_{27} = 80.1106126665397$$
$$x_{28} = -50.3151774380626$$
$$x_{29} = -37.7653294920198$$
$$x_{30} = 36.1283155162826$$
$$x_{31} = -75.4313688242127$$
$$x_{32} = 14.1371669411541$$
$$x_{33} = 1.95517406323303$$
$$x_{34} = 92.6769832808989$$
$$x_{35} = 67.5442420521806$$
$$x_{36} = -21.9226189316584$$
$$x_{37} = 58.1194640914112$$
$$x_{38} = 7.85398163397448$$
$$x_{39} = 89.5353906273091$$
$$x_{40} = 86.3937979737193$$
$$x_{41} = -59.665114800435$$
$$x_{42} = 4.71238898038469$$
Puntos máximos de la función:
$$x_{42} = 2.3860860459443$$
$$x_{42} = -86.3937979737193$$
$$x_{42} = 65.9506975390986$$
$$x_{42} = 44.0390771234017$$
$$x_{42} = -95.8185759344887$$
$$x_{42} = 56.5928487077727$$
$$x_{42} = 21.9226189316584$$
$$x_{42} = -83.2522053201295$$
$$x_{42} = 37.7653294920198$$
$$x_{42} = 9.26136941301364$$
$$x_{42} = -80.1106126665397$$
$$x_{42} = 72.2358627380718$$
$$x_{42} = -14.1371669411541$$
$$x_{42} = 81.7120061622563$$
$$x_{42} = -29.845130209103$$
$$x_{42} = -102.101761241668$$
$$x_{42} = -20.4203522483337$$
$$x_{42} = -61.261056745001$$
$$x_{42} = 94.274299213244$$
$$x_{42} = -51.8362787842316$$
$$x_{42} = 15.6115829776934$$
$$x_{42} = -42.4115008234622$$
$$x_{42} = -17.2787595947439$$
$$x_{42} = 34.5140312012932$$
$$x_{42} = 59.665114800435$$
$$x_{42} = -58.1194640914112$$
$$x_{42} = 12.7627599249538$$
$$x_{42} = -23.5619449019235$$
$$x_{42} = -64.4026493985908$$
$$x_{42} = -73.8274273593601$$
$$x_{42} = -89.5353906273091$$
$$x_{42} = 78.5207107750708$$
$$x_{42} = -39.2699081698724$$
$$x_{42} = 28.2211320612647$$
$$x_{42} = -45.553093477052$$
$$x_{42} = 50.3151774380626$$
$$x_{42} = 87.993007178388$$
$$x_{42} = 144.530559790087$$
$$x_{42} = 6.66212078172724$$
$$x_{42} = -7.85398163397448$$
$$x_{42} = 100.555827750706$$
$$x_{42} = -67.5442420521806$$
$$x_{42} = -36.1283155162826$$
$$x_{42} = 1.5707963267949$$
$$x_{42} = -1.95517406323303$$
Decrece en los intervalos
$$\left[303.163691071415, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.373966514501\right]$$
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
$$x \sin^{2}{\left(x \right)} - x \cos^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -5.68333365815917$$
$$x_{2} = -85.6060002756458$$
$$x_{3} = -35.3371707368851$$
$$x_{4} = -49.5041184575701$$
$$x_{5} = -77.7517299620525$$
$$x_{6} = 30.6689740642012$$
$$x_{7} = 3.88150931827571$$
$$x_{8} = 98.1726758587925$$
$$x_{9} = 99.7575811912676$$
$$x_{10} = -27.4811986158069$$
$$x_{11} = 77.7517299620525$$
$$x_{12} = 10.1911466105742$$
$$x_{13} = 91.8893484152259$$
$$x_{14} = -84.0351179851916$$
$$x_{15} = -33.7658552816908$$
$$x_{16} = 47.9050272821379$$
$$x_{17} = 46.3339521422813$$
$$x_{18} = -13.4461636797092$$
$$x_{19} = 24.3953781648349$$
$$x_{20} = 96.6018143285984$$
$$x_{21} = -11.8764761105503$$
$$x_{22} = -69.9178751833658$$
$$x_{23} = 25.9658809624812$$
$$x_{24} = 55.7846329514975$$
$$x_{25} = -18.1281211978358$$
$$x_{26} = -3.88150931827571$$
$$x_{27} = 52.6176870231887$$
$$x_{28} = 88.7636996098794$$
$$x_{29} = -19.6984034202597$$
$$x_{30} = -47.9050272821379$$
$$x_{31} = -90.3184775775828$$
$$x_{32} = -41.6212093511959$$
$$x_{33} = -38.5164216985664$$
$$x_{34} = 90.3184775775828$$
$$x_{35} = 93.4751971840106$$
$$x_{36} = 68.3471183238555$$
$$x_{37} = -76.1996097673336$$
$$x_{38} = -40.0500416376345$$
$$x_{39} = 38.5164216985664$$
$$x_{40} = 76.1996097673336$$
$$x_{41} = -21.1956380532273$$
$$x_{42} = 69.9178751833658$$
$$x_{43} = -68.3471183238555$$
$$x_{44} = 5.68333365815917$$
$$x_{45} = 27.4811986158069$$
$$x_{46} = 84.0351179851916$$
$$x_{47} = -25.9658809624812$$
$$x_{48} = 8.61311439171713$$
$$x_{49} = -71.4683059330294$$
$$x_{50} = -46.3339521422813$$
$$x_{51} = 60.4722731965474$$
$$x_{52} = -1.73907577125217$$
$$x_{53} = -99.7575811912676$$
$$x_{54} = -63.6364300110522$$
$$x_{55} = 74.6288466175257$$
$$x_{56} = -55.7846329514975$$
$$x_{57} = 54.1887003463867$$
$$x_{58} = 85.6060002756458$$
$$x_{59} = -96.6018143285984$$
$$x_{60} = -10.1911466105742$$
$$x_{61} = -91.8893484152259$$
$$x_{62} = 63.6364300110522$$
$$x_{63} = 41.6212093511959$$
$$x_{64} = 33.7658552816908$$
$$x_{65} = 32.2395812279394$$
$$x_{66} = -98.1726758587925$$
$$x_{67} = -32.2395812279394$$
$$x_{68} = 82.4815671238687$$
$$x_{69} = 19.6984034202597$$
$$x_{70} = 0$$
$$x_{71} = 40.0500416376345$$
$$x_{72} = 11.8764761105503$$
$$x_{73} = -60.4722731965474$$
$$x_{74} = 18.1281211978358$$
$$x_{75} = 49.5041184575701$$
$$x_{76} = -2.20509043068849$$
$$x_{77} = -57.3553703951451$$
$$x_{78} = 16.4813056239467$$
$$x_{79} = -82.4815671238687$$
$$x_{80} = -79.3226265646267$$
$$x_{81} = -62.0656814141177$$
$$x_{82} = -93.4751971840106$$
$$x_{83} = -54.1887003463867$$
$$x_{84} = 66.7557738425105$$
$$x_{85} = 62.0656814141177$$
$$x_{86} = 2.20509043068849$$
$$x_{87} = -44.7950656622447$$
$$x_{88} = -24.3953781648349$$
$$x_{89} = -16.4813056239467$$
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
$$\left[98.1726758587925, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -98.1726758587925\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$x \sin^{2}{\left(x \right)} - x \cos^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -5.68333365815917$$
$$x_{2} = -85.6060002756458$$
$$x_{3} = -35.3371707368851$$
$$x_{4} = -49.5041184575701$$
$$x_{5} = -77.7517299620525$$
$$x_{6} = 30.6689740642012$$
$$x_{7} = 3.88150931827571$$
$$x_{8} = 98.1726758587925$$
$$x_{9} = 99.7575811912676$$
$$x_{10} = -27.4811986158069$$
$$x_{11} = 77.7517299620525$$
$$x_{12} = 10.1911466105742$$
$$x_{13} = 91.8893484152259$$
$$x_{14} = -84.0351179851916$$
$$x_{15} = -33.7658552816908$$
$$x_{16} = 47.9050272821379$$
$$x_{17} = 46.3339521422813$$
$$x_{18} = -13.4461636797092$$
$$x_{19} = 24.3953781648349$$
$$x_{20} = 96.6018143285984$$
$$x_{21} = -11.8764761105503$$
$$x_{22} = -69.9178751833658$$
$$x_{23} = 25.9658809624812$$
$$x_{24} = 55.7846329514975$$
$$x_{25} = -18.1281211978358$$
$$x_{26} = -3.88150931827571$$
$$x_{27} = 52.6176870231887$$
$$x_{28} = 88.7636996098794$$
$$x_{29} = -19.6984034202597$$
$$x_{30} = -47.9050272821379$$
$$x_{31} = -90.3184775775828$$
$$x_{32} = -41.6212093511959$$
$$x_{33} = -38.5164216985664$$
$$x_{34} = 90.3184775775828$$
$$x_{35} = 93.4751971840106$$
$$x_{36} = 68.3471183238555$$
$$x_{37} = -76.1996097673336$$
$$x_{38} = -40.0500416376345$$
$$x_{39} = 38.5164216985664$$
$$x_{40} = 76.1996097673336$$
$$x_{41} = -21.1956380532273$$
$$x_{42} = 69.9178751833658$$
$$x_{43} = -68.3471183238555$$
$$x_{44} = 5.68333365815917$$
$$x_{45} = 27.4811986158069$$
$$x_{46} = 84.0351179851916$$
$$x_{47} = -25.9658809624812$$
$$x_{48} = 8.61311439171713$$
$$x_{49} = -71.4683059330294$$
$$x_{50} = -46.3339521422813$$
$$x_{51} = 60.4722731965474$$
$$x_{52} = -1.73907577125217$$
$$x_{53} = -99.7575811912676$$
$$x_{54} = -63.6364300110522$$
$$x_{55} = 74.6288466175257$$
$$x_{56} = -55.7846329514975$$
$$x_{57} = 54.1887003463867$$
$$x_{58} = 85.6060002756458$$
$$x_{59} = -96.6018143285984$$
$$x_{60} = -10.1911466105742$$
$$x_{61} = -91.8893484152259$$
$$x_{62} = 63.6364300110522$$
$$x_{63} = 41.6212093511959$$
$$x_{64} = 33.7658552816908$$
$$x_{65} = 32.2395812279394$$
$$x_{66} = -98.1726758587925$$
$$x_{67} = -32.2395812279394$$
$$x_{68} = 82.4815671238687$$
$$x_{69} = 19.6984034202597$$
$$x_{70} = 0$$
$$x_{71} = 40.0500416376345$$
$$x_{72} = 11.8764761105503$$
$$x_{73} = -60.4722731965474$$
$$x_{74} = 18.1281211978358$$
$$x_{75} = 49.5041184575701$$
$$x_{76} = -2.20509043068849$$
$$x_{77} = -57.3553703951451$$
$$x_{78} = 16.4813056239467$$
$$x_{79} = -82.4815671238687$$
$$x_{80} = -79.3226265646267$$
$$x_{81} = -62.0656814141177$$
$$x_{82} = -93.4751971840106$$
$$x_{83} = -54.1887003463867$$
$$x_{84} = 66.7557738425105$$
$$x_{85} = 62.0656814141177$$
$$x_{86} = 2.20509043068849$$
$$x_{87} = -44.7950656622447$$
$$x_{88} = -24.3953781648349$$
$$x_{89} = -16.4813056239467$$
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
$$\left[98.1726758587925, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -98.1726758587925\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)}\right) = \left\langle -\infty, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)}\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle 0, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)}\right) = \left\langle -\infty, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)}\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle 0, \infty\right\rangle$$
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
$$\lim_{x \to -\infty}\left(\frac{\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)}}{x}\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle 0, \infty\right\rangle x$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)}}{x}\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle 0, \infty\right\rangle x$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)}}{x}\right)$$
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:
$$\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)} = - \frac{x \cos^{2}{\left(x \right)}}{2} - 2 \sin{\left(x \right)}$$
- No
$$\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)} = \frac{x \cos^{2}{\left(x \right)}}{2} + 2 \sin{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)} = - \frac{x \cos^{2}{\left(x \right)}}{2} - 2 \sin{\left(x \right)}$$
- No
$$\left(x \frac{\cos^{2}{\left(x \right)}}{2} + \sin{\left(x \right)}\right) + \sin{\left(x \right)} = \frac{x \cos^{2}{\left(x \right)}}{2} + 2 \sin{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Gráfico