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 =:
-
-sqrt(-1-x^2)
-
7-x-2*x^2
-
y=x^3+x
-
y=(x^3)/(x^2-4)
-
Expresiones idénticas
- (uno +x)*cos(dos *x)+(uno + dos *x^ dos)*sin(dos *x)
- (1 más x) multiplicar por coseno de (2 multiplicar por x) más (1 más 2 multiplicar por x al cuadrado ) multiplicar por seno de (2 multiplicar por x)
- (uno más x) multiplicar por coseno de (dos multiplicar por x) más (uno más dos multiplicar por x en el grado dos) multiplicar por seno de (dos multiplicar por x)
- (1+x)*cos(2*x)+(1+2*x2)*sin(2*x)
- 1+x*cos2*x+1+2*x2*sin2*x
- (1+x)*cos(2*x)+(1+2*x²)*sin(2*x)
- (1+x)*cos(2*x)+(1+2*x en el grado 2)*sin(2*x)
- (1+x)cos(2x)+(1+2x^2)sin(2x)
- (1+x)cos(2x)+(1+2x2)sin(2x)
- 1+xcos2x+1+2x2sin2x
- 1+xcos2x+1+2x^2sin2x
-
Expresiones semejantes
- (1-x)*cos(2*x)+(1+2*x^2)*sin(2*x)
- (1+x)*cos(2*x)+(1-2*x^2)*sin(2*x)
- (1+x)*cos(2*x)-(1+2*x^2)*sin(2*x)
-
Expresiones con funciones
- Coseno cos
- cos(log(x))+sin(log(x))
- cos(3)/x
- cos((x+pi)/6)-1
- cos(x)*sqrt(x^2+1)
- cos(x^3)+x^3
- Seno sin
- sin(x)^2/(x^2*cos(x)^2)
- sin(3*x)+x^4
- sin(pi*x)/sin(pi*x)
- sin(4*x)*cos(x)+2*sin(3*x)-cos(4*x)*sin(x)
- sin((pi*x^2)/(1+x^2))
Gráfico de la función y = (1+x)*cos(2*x)+(1+2*x^2)*sin(2*x)
Solución
Ha introducido
[src]
/ 2\ f(x) = (1 + x)*cos(2*x) + \1 + 2*x /*sin(2*x)
$$f{\left(x \right)} = \left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}$$
f = (x + 1)*cos(2*x) + (2*x^2 + 1)*sin(2*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:
$$\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 72.2531234800138$$
$$x_{2} = 7.81826876924305$$
$$x_{3} = -21.9803051356946$$
$$x_{4} = -0.278931041060777$$
$$x_{5} = 78.5365928844032$$
$$x_{6} = 86.3908708820984$$
$$x_{7} = -64.3988281569855$$
$$x_{8} = -102.099336763716$$
$$x_{9} = -14.1207633382095$$
$$x_{10} = -73.8240871584911$$
$$x_{11} = -20.4087188664462$$
$$x_{12} = -43.9767432605192$$
$$x_{13} = -6.25005875094726$$
$$x_{14} = 50.260410575023$$
$$x_{15} = 29.8364760980318$$
$$x_{16} = 87.9617200613318$$
$$x_{17} = -10.9749679908989$$
$$x_{18} = 59.6860023549882$$
$$x_{19} = 64.398707608287$$
$$x_{20} = 37.6923060370727$$
$$x_{21} = 45.5474857799052$$
$$x_{22} = 67.540486247005$$
$$x_{23} = -53.4024822783076$$
$$x_{24} = -95.8159941613389$$
$$x_{25} = -72.2532192467265$$
$$x_{26} = 20.4075198021992$$
$$x_{27} = -62.8279378598643$$
$$x_{28} = 34.550077551014$$
$$x_{29} = 95.8159397022241$$
$$x_{30} = 6.23739713436489$$
$$x_{31} = 1.3342573809468$$
$$x_{32} = -75.3949521287807$$
$$x_{33} = 94.2450989800716$$
$$x_{34} = -67.5405958426583$$
$$x_{35} = 80.1074531887516$$
$$x_{36} = 100.528453464286$$
$$x_{37} = -58.1152370419602$$
$$x_{38} = 12.5449349818577$$
$$x_{39} = 56.5441690748166$$
$$x_{40} = -7.82637353588674$$
$$x_{41} = -86.3909378713102$$
$$x_{42} = 42.4054683014633$$
$$x_{43} = -59.6861426890327$$
$$x_{44} = 14.1182616079552$$
$$x_{45} = -50.2606084679462$$
$$x_{46} = 51.8313634647791$$
$$x_{47} = -94.2451552696178$$
$$x_{48} = 26.6938292810617$$
$$x_{49} = 89.5325673660738$$
$$x_{50} = 21.9792712479986$$
$$x_{51} = 23.5508906211554$$
$$x_{52} = -36.1215890359586$$
$$x_{53} = -87.9617846794332$$
$$x_{54} = -97.3868316945517$$
$$x_{55} = -81.678385945106$$
$$x_{56} = 4.64886631898059$$
$$x_{57} = 48.6894473887194$$
$$x_{58} = -1.5244984437257$$
$$x_{59} = 73.8239954236046$$
$$x_{60} = -9.40116539740112$$
$$x_{61} = 9.39553675396715$$
$$x_{62} = -23.551791252621$$
$$x_{63} = -29.8370374339198$$
$$x_{64} = 43.9764847881422$$
$$x_{65} = -25.1231950750204$$
$$x_{66} = -65.9697140504635$$
$$x_{67} = -15.6930824323661$$
$$x_{68} = 58.1150890197838$$
$$x_{69} = 28.2651829057899$$
$$x_{70} = -83.2492386112819$$
$$x_{71} = 36.1212059692001$$
$$x_{72} = -80.1075310981577$$
$$x_{73} = -39.2637054615986$$
$$x_{74} = -37.6926578466619$$
$$x_{75} = 92.6742567372551$$
$$x_{76} = -45.5477267339441$$
$$x_{77} = 81.6783110034029$$
$$x_{78} = 70.6822481300311$$
$$x_{79} = -31.4082247777527$$
$$x_{80} = -28.265808344767$$
$$x_{81} = -17.2651444495442$$
$$x_{82} = 15.6910560232405$$
$$x_{83} = -42.4057462744947$$
$$x_{84} = -89.532629736762$$
$$x_{85} = -4.67136240857354$$
$$x_{86} = -61.2570428262825$$
$$x_{87} = -51.8315495459132$$
$$x_{88} = 65.9695991738407$$
o sea hay que resolver la ecuación:
$$\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 72.2531234800138$$
$$x_{2} = 7.81826876924305$$
$$x_{3} = -21.9803051356946$$
$$x_{4} = -0.278931041060777$$
$$x_{5} = 78.5365928844032$$
$$x_{6} = 86.3908708820984$$
$$x_{7} = -64.3988281569855$$
$$x_{8} = -102.099336763716$$
$$x_{9} = -14.1207633382095$$
$$x_{10} = -73.8240871584911$$
$$x_{11} = -20.4087188664462$$
$$x_{12} = -43.9767432605192$$
$$x_{13} = -6.25005875094726$$
$$x_{14} = 50.260410575023$$
$$x_{15} = 29.8364760980318$$
$$x_{16} = 87.9617200613318$$
$$x_{17} = -10.9749679908989$$
$$x_{18} = 59.6860023549882$$
$$x_{19} = 64.398707608287$$
$$x_{20} = 37.6923060370727$$
$$x_{21} = 45.5474857799052$$
$$x_{22} = 67.540486247005$$
$$x_{23} = -53.4024822783076$$
$$x_{24} = -95.8159941613389$$
$$x_{25} = -72.2532192467265$$
$$x_{26} = 20.4075198021992$$
$$x_{27} = -62.8279378598643$$
$$x_{28} = 34.550077551014$$
$$x_{29} = 95.8159397022241$$
$$x_{30} = 6.23739713436489$$
$$x_{31} = 1.3342573809468$$
$$x_{32} = -75.3949521287807$$
$$x_{33} = 94.2450989800716$$
$$x_{34} = -67.5405958426583$$
$$x_{35} = 80.1074531887516$$
$$x_{36} = 100.528453464286$$
$$x_{37} = -58.1152370419602$$
$$x_{38} = 12.5449349818577$$
$$x_{39} = 56.5441690748166$$
$$x_{40} = -7.82637353588674$$
$$x_{41} = -86.3909378713102$$
$$x_{42} = 42.4054683014633$$
$$x_{43} = -59.6861426890327$$
$$x_{44} = 14.1182616079552$$
$$x_{45} = -50.2606084679462$$
$$x_{46} = 51.8313634647791$$
$$x_{47} = -94.2451552696178$$
$$x_{48} = 26.6938292810617$$
$$x_{49} = 89.5325673660738$$
$$x_{50} = 21.9792712479986$$
$$x_{51} = 23.5508906211554$$
$$x_{52} = -36.1215890359586$$
$$x_{53} = -87.9617846794332$$
$$x_{54} = -97.3868316945517$$
$$x_{55} = -81.678385945106$$
$$x_{56} = 4.64886631898059$$
$$x_{57} = 48.6894473887194$$
$$x_{58} = -1.5244984437257$$
$$x_{59} = 73.8239954236046$$
$$x_{60} = -9.40116539740112$$
$$x_{61} = 9.39553675396715$$
$$x_{62} = -23.551791252621$$
$$x_{63} = -29.8370374339198$$
$$x_{64} = 43.9764847881422$$
$$x_{65} = -25.1231950750204$$
$$x_{66} = -65.9697140504635$$
$$x_{67} = -15.6930824323661$$
$$x_{68} = 58.1150890197838$$
$$x_{69} = 28.2651829057899$$
$$x_{70} = -83.2492386112819$$
$$x_{71} = 36.1212059692001$$
$$x_{72} = -80.1075310981577$$
$$x_{73} = -39.2637054615986$$
$$x_{74} = -37.6926578466619$$
$$x_{75} = 92.6742567372551$$
$$x_{76} = -45.5477267339441$$
$$x_{77} = 81.6783110034029$$
$$x_{78} = 70.6822481300311$$
$$x_{79} = -31.4082247777527$$
$$x_{80} = -28.265808344767$$
$$x_{81} = -17.2651444495442$$
$$x_{82} = 15.6910560232405$$
$$x_{83} = -42.4057462744947$$
$$x_{84} = -89.532629736762$$
$$x_{85} = -4.67136240857354$$
$$x_{86} = -61.2570428262825$$
$$x_{87} = -51.8315495459132$$
$$x_{88} = 65.9695991738407$$
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
$$4 x \sin{\left(2 x \right)} - 2 \left(x + 1\right) \sin{\left(2 x \right)} + 2 \left(2 x^{2} + 1\right) \cos{\left(2 x \right)} + \cos{\left(2 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -46.3440003353635$$
$$x_{2} = 73.0454043442496$$
$$x_{3} = 71.4746811135369$$
$$x_{4} = 55.7676709251147$$
$$x_{5} = 46.3437676695875$$
$$x_{6} = 62.0504181043217$$
$$x_{7} = -35.3501846363215$$
$$x_{8} = -4.00104956279273$$
$$x_{9} = -99.7480979789309$$
$$x_{10} = 99.7480477323609$$
$$x_{11} = 49.4850325574119$$
$$x_{12} = -82.4698749476771$$
$$x_{13} = -91.8943349662814$$
$$x_{14} = -49.4852366366831$$
$$x_{15} = -85.6113537567511$$
$$x_{16} = 91.8942757653517$$
$$x_{17} = -40.0616991407757$$
$$x_{18} = 76.1868597211761$$
$$x_{19} = -54.1971698352737$$
$$x_{20} = 54.1969996847081$$
$$x_{21} = 90.3235256964552$$
$$x_{22} = 18.0771894448225$$
$$x_{23} = 8.66462883323801$$
$$x_{24} = 85.6112855490263$$
$$x_{25} = -18.0787135522592$$
$$x_{26} = 69.9039611175754$$
$$x_{27} = -98.1773425510901$$
$$x_{28} = -41.6322488892228$$
$$x_{29} = 52.6263357564021$$
$$x_{30} = -5.54950466269413$$
$$x_{31} = 38.4908327786634$$
$$x_{32} = -69.9040634126507$$
$$x_{33} = -84.0406132763715$$
$$x_{34} = 63.6211181901024$$
$$x_{35} = -13.3717684541969$$
$$x_{36} = -62.0505479234573$$
$$x_{37} = 47.9143948649121$$
$$x_{38} = 10.2320504954601$$
$$x_{39} = 74.6161306084371$$
$$x_{40} = 60.4797229283736$$
$$x_{41} = -79.3284052625226$$
$$x_{42} = -60.4798595756766$$
$$x_{43} = 66.7625317281172$$
$$x_{44} = 96.6065348991784$$
$$x_{45} = 77.7575915122693$$
$$x_{46} = -38.4911699769947$$
$$x_{47} = -57.3385009130025$$
$$x_{48} = -47.9146125354541$$
$$x_{49} = -197.136213576985$$
$$x_{50} = -19.6482963154439$$
$$x_{51} = 2.40973954080437$$
$$x_{52} = -27.4983473448304$$
$$x_{53} = -24.3580128816892$$
$$x_{54} = 16.5075457300265$$
$$x_{55} = 82.4698014455491$$
$$x_{56} = -77.7576741910705$$
$$x_{57} = -21.2180651155129$$
$$x_{58} = 0.733973382411932$$
$$x_{59} = 41.6319606186974$$
$$x_{60} = 88.7527772454802$$
$$x_{61} = -55.7678316300603$$
$$x_{62} = -68.3333516264169$$
$$x_{63} = 5.53385248103157$$
$$x_{64} = -71.47477896306$$
$$x_{65} = 40.0613878425751$$
$$x_{66} = 25.9273984152002$$
$$x_{67} = -10.2367679865919$$
$$x_{68} = 32.2088395595391$$
$$x_{69} = 30.6384147857828$$
$$x_{70} = 98.1772906840609$$
$$x_{71} = -11.803807466993$$
$$x_{72} = 24.3571718975006$$
$$x_{73} = -63.6212416802983$$
$$x_{74} = 3.97182867347168$$
$$x_{75} = -90.3235869740081$$
$$x_{76} = -2.47971406845198$$
$$x_{77} = -32.2093209423531$$
$$x_{78} = -33.7797354116201$$
$$x_{79} = 68.3332445758418$$
$$x_{80} = 33.7792977002729$$
$$x_{81} = -93.4650845996383$$
$$x_{82} = 84.0405424956326$$
$$x_{83} = 19.6470052563392$$
$$x_{84} = -76.1869458435232$$
$$x_{85} = -25.9281408073249$$
$$x_{86} = -8.67117538237253$$
$$x_{87} = 27.4976871832371$$
$$x_{88} = 11.8002496199353$$
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:
$$x_{1} = 71.4746811135369$$
$$x_{2} = 55.7676709251147$$
$$x_{3} = 46.3437676695875$$
$$x_{4} = 62.0504181043217$$
$$x_{5} = -35.3501846363215$$
$$x_{6} = -4.00104956279273$$
$$x_{7} = 99.7480477323609$$
$$x_{8} = 49.4850325574119$$
$$x_{9} = -82.4698749476771$$
$$x_{10} = -91.8943349662814$$
$$x_{11} = -85.6113537567511$$
$$x_{12} = -54.1971698352737$$
$$x_{13} = 90.3235256964552$$
$$x_{14} = 18.0771894448225$$
$$x_{15} = 8.66462883323801$$
$$x_{16} = -98.1773425510901$$
$$x_{17} = -41.6322488892228$$
$$x_{18} = 52.6263357564021$$
$$x_{19} = -69.9040634126507$$
$$x_{20} = -13.3717684541969$$
$$x_{21} = 74.6161306084371$$
$$x_{22} = -79.3284052625226$$
$$x_{23} = -60.4798595756766$$
$$x_{24} = 96.6065348991784$$
$$x_{25} = 77.7575915122693$$
$$x_{26} = -38.4911699769947$$
$$x_{27} = -57.3385009130025$$
$$x_{28} = -47.9146125354541$$
$$x_{29} = -19.6482963154439$$
$$x_{30} = 2.40973954080437$$
$$x_{31} = 5.53385248103157$$
$$x_{32} = 40.0613878425751$$
$$x_{33} = -10.2367679865919$$
$$x_{34} = 30.6384147857828$$
$$x_{35} = 24.3571718975006$$
$$x_{36} = -63.6212416802983$$
$$x_{37} = -32.2093209423531$$
$$x_{38} = 68.3332445758418$$
$$x_{39} = 33.7792977002729$$
$$x_{40} = 84.0405424956326$$
$$x_{41} = -76.1869458435232$$
$$x_{42} = -25.9281408073249$$
$$x_{43} = 27.4976871832371$$
$$x_{44} = 11.8002496199353$$
Puntos máximos de la función:
$$x_{44} = -46.3440003353635$$
$$x_{44} = 73.0454043442496$$
$$x_{44} = -99.7480979789309$$
$$x_{44} = -49.4852366366831$$
$$x_{44} = 91.8942757653517$$
$$x_{44} = -40.0616991407757$$
$$x_{44} = 76.1868597211761$$
$$x_{44} = 54.1969996847081$$
$$x_{44} = 85.6112855490263$$
$$x_{44} = -18.0787135522592$$
$$x_{44} = 69.9039611175754$$
$$x_{44} = -5.54950466269413$$
$$x_{44} = 38.4908327786634$$
$$x_{44} = -84.0406132763715$$
$$x_{44} = 63.6211181901024$$
$$x_{44} = -62.0505479234573$$
$$x_{44} = 47.9143948649121$$
$$x_{44} = 10.2320504954601$$
$$x_{44} = 60.4797229283736$$
$$x_{44} = 66.7625317281172$$
$$x_{44} = -197.136213576985$$
$$x_{44} = -27.4983473448304$$
$$x_{44} = -24.3580128816892$$
$$x_{44} = 16.5075457300265$$
$$x_{44} = 82.4698014455491$$
$$x_{44} = -77.7576741910705$$
$$x_{44} = -21.2180651155129$$
$$x_{44} = 0.733973382411932$$
$$x_{44} = 41.6319606186974$$
$$x_{44} = 88.7527772454802$$
$$x_{44} = -55.7678316300603$$
$$x_{44} = -68.3333516264169$$
$$x_{44} = -71.47477896306$$
$$x_{44} = 25.9273984152002$$
$$x_{44} = 32.2088395595391$$
$$x_{44} = 98.1772906840609$$
$$x_{44} = -11.803807466993$$
$$x_{44} = 3.97182867347168$$
$$x_{44} = -90.3235869740081$$
$$x_{44} = -2.47971406845198$$
$$x_{44} = -33.7797354116201$$
$$x_{44} = -93.4650845996383$$
$$x_{44} = 19.6470052563392$$
$$x_{44} = -8.67117538237253$$
Decrece en los intervalos
$$\left[99.7480477323609, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.1773425510901\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$4 x \sin{\left(2 x \right)} - 2 \left(x + 1\right) \sin{\left(2 x \right)} + 2 \left(2 x^{2} + 1\right) \cos{\left(2 x \right)} + \cos{\left(2 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -46.3440003353635$$
$$x_{2} = 73.0454043442496$$
$$x_{3} = 71.4746811135369$$
$$x_{4} = 55.7676709251147$$
$$x_{5} = 46.3437676695875$$
$$x_{6} = 62.0504181043217$$
$$x_{7} = -35.3501846363215$$
$$x_{8} = -4.00104956279273$$
$$x_{9} = -99.7480979789309$$
$$x_{10} = 99.7480477323609$$
$$x_{11} = 49.4850325574119$$
$$x_{12} = -82.4698749476771$$
$$x_{13} = -91.8943349662814$$
$$x_{14} = -49.4852366366831$$
$$x_{15} = -85.6113537567511$$
$$x_{16} = 91.8942757653517$$
$$x_{17} = -40.0616991407757$$
$$x_{18} = 76.1868597211761$$
$$x_{19} = -54.1971698352737$$
$$x_{20} = 54.1969996847081$$
$$x_{21} = 90.3235256964552$$
$$x_{22} = 18.0771894448225$$
$$x_{23} = 8.66462883323801$$
$$x_{24} = 85.6112855490263$$
$$x_{25} = -18.0787135522592$$
$$x_{26} = 69.9039611175754$$
$$x_{27} = -98.1773425510901$$
$$x_{28} = -41.6322488892228$$
$$x_{29} = 52.6263357564021$$
$$x_{30} = -5.54950466269413$$
$$x_{31} = 38.4908327786634$$
$$x_{32} = -69.9040634126507$$
$$x_{33} = -84.0406132763715$$
$$x_{34} = 63.6211181901024$$
$$x_{35} = -13.3717684541969$$
$$x_{36} = -62.0505479234573$$
$$x_{37} = 47.9143948649121$$
$$x_{38} = 10.2320504954601$$
$$x_{39} = 74.6161306084371$$
$$x_{40} = 60.4797229283736$$
$$x_{41} = -79.3284052625226$$
$$x_{42} = -60.4798595756766$$
$$x_{43} = 66.7625317281172$$
$$x_{44} = 96.6065348991784$$
$$x_{45} = 77.7575915122693$$
$$x_{46} = -38.4911699769947$$
$$x_{47} = -57.3385009130025$$
$$x_{48} = -47.9146125354541$$
$$x_{49} = -197.136213576985$$
$$x_{50} = -19.6482963154439$$
$$x_{51} = 2.40973954080437$$
$$x_{52} = -27.4983473448304$$
$$x_{53} = -24.3580128816892$$
$$x_{54} = 16.5075457300265$$
$$x_{55} = 82.4698014455491$$
$$x_{56} = -77.7576741910705$$
$$x_{57} = -21.2180651155129$$
$$x_{58} = 0.733973382411932$$
$$x_{59} = 41.6319606186974$$
$$x_{60} = 88.7527772454802$$
$$x_{61} = -55.7678316300603$$
$$x_{62} = -68.3333516264169$$
$$x_{63} = 5.53385248103157$$
$$x_{64} = -71.47477896306$$
$$x_{65} = 40.0613878425751$$
$$x_{66} = 25.9273984152002$$
$$x_{67} = -10.2367679865919$$
$$x_{68} = 32.2088395595391$$
$$x_{69} = 30.6384147857828$$
$$x_{70} = 98.1772906840609$$
$$x_{71} = -11.803807466993$$
$$x_{72} = 24.3571718975006$$
$$x_{73} = -63.6212416802983$$
$$x_{74} = 3.97182867347168$$
$$x_{75} = -90.3235869740081$$
$$x_{76} = -2.47971406845198$$
$$x_{77} = -32.2093209423531$$
$$x_{78} = -33.7797354116201$$
$$x_{79} = 68.3332445758418$$
$$x_{80} = 33.7792977002729$$
$$x_{81} = -93.4650845996383$$
$$x_{82} = 84.0405424956326$$
$$x_{83} = 19.6470052563392$$
$$x_{84} = -76.1869458435232$$
$$x_{85} = -25.9281408073249$$
$$x_{86} = -8.67117538237253$$
$$x_{87} = 27.4976871832371$$
$$x_{88} = 11.8002496199353$$
Signos de extremos en los puntos:
(-46.34400033536347, 4295.77241162879)
(73.04540434424959, 10671.5192190717)
(71.47468111353693, -10217.5172664874)
(55.767670925114686, -6220.32551835937)
(46.34376766958748, -4295.75084259667)
(62.05041810432167, -7700.76708402805)
(-35.35018463632153, -2499.50776913704)
(-4.001049562792727, -32.2123981849912)
(-99.74809797893093, 19899.6111879493)
(99.7480477323609, -19899.6011635745)
(49.48503255741195, -4897.79739366201)
(-82.46987494767706, -13602.8046311193)
(-91.89433496628143, -16889.3822743653)
(-49.48523663668308, 4897.8175945321)
(-85.61135375675114, -14658.8520793698)
(91.89427576535168, 16889.3713934257)
(-40.0616991407757, 3210.11762106072)
(76.18685972117615, 11609.1319188942)
(-54.19716983527374, -5874.90755088552)
(54.196999684708075, 5874.88910520378)
(90.3235256964552, -16316.9342436795)
(18.07718944482248, -653.850085049453)
(8.664628833238012, -150.471066416477)
(85.6112855490263, 14658.8404000658)
(-18.078713552259234, 653.905253247471)
(69.9039611175754, 9773.38491156615)
(-98.1773425510901, -19277.8261909414)
(-41.63224888922282, -3466.72686420296)
(52.62633575640211, -5539.32228101307)
(-5.549504662694135, 61.7897223119221)
(38.49083277866341, 2963.35205599715)
(-69.90406341265069, -9773.39921433547)
(-84.0406132763715, 14125.8935509627)
(63.62111819010238, 8095.55145854756)
(-13.37176845419685, -357.826813224645)
(-62.050547923457316, 7700.78319627622)
(47.91439486491214, 4591.83932709046)
(10.23205049546007, 209.69717748404)
(74.61613060843712, -11135.390769744)
(60.479722928373576, 7315.85230361728)
(-79.32840526252257, -12586.2356181124)
(-60.47985957567662, -7315.86883411131)
(66.76253172811717, 8914.72899281355)
(96.60653489917844, -18665.9004506519)
(77.75759151226926, -12092.7426668807)
(-38.49116997699469, -2963.37802075856)
(-57.33850091300247, -6575.64895741656)
(-47.9146125354541, -4591.86018964566)
(-197.13621357698503, 77725.6208960592)
(-19.648296315443943, -772.338300928009)
(2.4097395408043725, -12.1769778715047)
(-27.498347344830403, 1512.55137017073)
(-24.35801288168923, 1186.85676933945)
(16.507545730026536, 545.28184543525)
(82.4698014455491, 13602.7925070339)
(-77.75767419107052, 12092.7555254935)
(-21.218065115512903, 900.641277777628)
(0.7339733824119316, 2.24448011058458)
(41.631960618697356, 3466.70285640032)
(88.75277724548016, 15754.3666949933)
(-55.76783163006025, 6220.34344482565)
(-68.33335162641693, 9339.13678525602)
(5.533852481031567, -61.6143036102879)
(-71.47477896306, 10217.5312550501)
(40.061387842575144, -3210.0926730697)
(25.927398415200248, 1344.73067819968)
(-10.236767986591886, -209.794075954155)
(32.20883955953906, 2075.0851392377)
(30.63841478578284, -1877.69226026885)
(98.1772906840609, 19277.8160062138)
(-11.803807466992989, 278.874902986725)
(24.357171897500567, -1186.81577486029)
(-63.62124168029834, -8095.56717318632)
(3.9718286734716837, 31.9747961447626)
(-90.32358697400808, 16316.9453138012)
(-2.479714068451985, 12.5324063062493)
(-32.20932094235307, -2075.11616021353)
(-33.77973541162014, 2282.37713042123)
(68.33324457584182, -9339.12215384402)
(33.77929770027289, -2282.34754937004)
(-93.46508459963829, 17471.6888425143)
(84.04054249563256, -14125.8816534229)
(19.647005256339227, 772.287519427001)
(-76.18694584352323, -11609.1450425313)
(-25.928140807324862, -1344.76919673357)
(-8.671175382372533, 150.5851027317)
(27.497687183237108, -1512.51504595851)
(11.80024961993525, -278.790701557359)
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:
$$x_{1} = 71.4746811135369$$
$$x_{2} = 55.7676709251147$$
$$x_{3} = 46.3437676695875$$
$$x_{4} = 62.0504181043217$$
$$x_{5} = -35.3501846363215$$
$$x_{6} = -4.00104956279273$$
$$x_{7} = 99.7480477323609$$
$$x_{8} = 49.4850325574119$$
$$x_{9} = -82.4698749476771$$
$$x_{10} = -91.8943349662814$$
$$x_{11} = -85.6113537567511$$
$$x_{12} = -54.1971698352737$$
$$x_{13} = 90.3235256964552$$
$$x_{14} = 18.0771894448225$$
$$x_{15} = 8.66462883323801$$
$$x_{16} = -98.1773425510901$$
$$x_{17} = -41.6322488892228$$
$$x_{18} = 52.6263357564021$$
$$x_{19} = -69.9040634126507$$
$$x_{20} = -13.3717684541969$$
$$x_{21} = 74.6161306084371$$
$$x_{22} = -79.3284052625226$$
$$x_{23} = -60.4798595756766$$
$$x_{24} = 96.6065348991784$$
$$x_{25} = 77.7575915122693$$
$$x_{26} = -38.4911699769947$$
$$x_{27} = -57.3385009130025$$
$$x_{28} = -47.9146125354541$$
$$x_{29} = -19.6482963154439$$
$$x_{30} = 2.40973954080437$$
$$x_{31} = 5.53385248103157$$
$$x_{32} = 40.0613878425751$$
$$x_{33} = -10.2367679865919$$
$$x_{34} = 30.6384147857828$$
$$x_{35} = 24.3571718975006$$
$$x_{36} = -63.6212416802983$$
$$x_{37} = -32.2093209423531$$
$$x_{38} = 68.3332445758418$$
$$x_{39} = 33.7792977002729$$
$$x_{40} = 84.0405424956326$$
$$x_{41} = -76.1869458435232$$
$$x_{42} = -25.9281408073249$$
$$x_{43} = 27.4976871832371$$
$$x_{44} = 11.8002496199353$$
Puntos máximos de la función:
$$x_{44} = -46.3440003353635$$
$$x_{44} = 73.0454043442496$$
$$x_{44} = -99.7480979789309$$
$$x_{44} = -49.4852366366831$$
$$x_{44} = 91.8942757653517$$
$$x_{44} = -40.0616991407757$$
$$x_{44} = 76.1868597211761$$
$$x_{44} = 54.1969996847081$$
$$x_{44} = 85.6112855490263$$
$$x_{44} = -18.0787135522592$$
$$x_{44} = 69.9039611175754$$
$$x_{44} = -5.54950466269413$$
$$x_{44} = 38.4908327786634$$
$$x_{44} = -84.0406132763715$$
$$x_{44} = 63.6211181901024$$
$$x_{44} = -62.0505479234573$$
$$x_{44} = 47.9143948649121$$
$$x_{44} = 10.2320504954601$$
$$x_{44} = 60.4797229283736$$
$$x_{44} = 66.7625317281172$$
$$x_{44} = -197.136213576985$$
$$x_{44} = -27.4983473448304$$
$$x_{44} = -24.3580128816892$$
$$x_{44} = 16.5075457300265$$
$$x_{44} = 82.4698014455491$$
$$x_{44} = -77.7576741910705$$
$$x_{44} = -21.2180651155129$$
$$x_{44} = 0.733973382411932$$
$$x_{44} = 41.6319606186974$$
$$x_{44} = 88.7527772454802$$
$$x_{44} = -55.7678316300603$$
$$x_{44} = -68.3333516264169$$
$$x_{44} = -71.47477896306$$
$$x_{44} = 25.9273984152002$$
$$x_{44} = 32.2088395595391$$
$$x_{44} = 98.1772906840609$$
$$x_{44} = -11.803807466993$$
$$x_{44} = 3.97182867347168$$
$$x_{44} = -90.3235869740081$$
$$x_{44} = -2.47971406845198$$
$$x_{44} = -33.7797354116201$$
$$x_{44} = -93.4650845996383$$
$$x_{44} = 19.6470052563392$$
$$x_{44} = -8.67117538237253$$
Decrece en los intervalos
$$\left[99.7480477323609, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.1773425510901\right]$$
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
$$4 \left(4 x \cos{\left(2 x \right)} - \left(x + 1\right) \cos{\left(2 x \right)} - \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}\right) = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$4 \left(4 x \cos{\left(2 x \right)} - \left(x + 1\right) \cos{\left(2 x \right)} - \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}\right) = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (1 + x)*cos(2*x) + (1 + 2*x^2)*sin(2*x), dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}}{x}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}}{x}\right)$$
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:
$$\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)} = \left(1 - x\right) \cos{\left(2 x \right)} - \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}$$
- No
$$\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)} = - \left(1 - x\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)} = \left(1 - x\right) \cos{\left(2 x \right)} - \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}$$
- No
$$\left(x + 1\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)} = - \left(1 - x\right) \cos{\left(2 x \right)} + \left(2 x^{2} + 1\right) \sin{\left(2 x \right)}$$
- No
es decir, función
no es
par ni impar