Sr Examen
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- 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^4+x^2
-
x-e
-
x*(-8)/(x^2+4)
-
(x+5)^2-9
-
Expresiones idénticas
- abs(cos(dos *x)-sin(x))
- abs( coseno de (2 multiplicar por x) menos seno de (x))
- abs( coseno de (dos multiplicar por x) menos seno de (x))
- abs(cos(2x)-sin(x))
- abscos2x-sinx
-
Expresiones semejantes
- abs(cos(2*x)+sin(x))
- abs(cos(2*x)-sinx)
-
Expresiones con funciones
- abs
- abs((abs(x)+2)/(abs(x)-4))
- abs((x^2-4)/(x^2+1))
- abs(x)*x+3*abs(x)-5x
- abs(x-2)-abs(x+1)+x-2
- abs((1/(x))-1)
- Coseno cos
- cos(2x)/sin(x)
- cos2x+x^3-4
- cos(x+pi/4)+2
- cos(x)*((2*x^3-5)^3)^4
- cos2x-sinx
- Seno sin
- sin(2*x-pi/4)
- sin(5+6*x)^(2)+3^tan(x)
- sin(x)+arcsin(x)
- sin(2*x^3)^2/(x^3)
- sinpiy
Gráfico de la función y = abs(cos(2*x)-sin(x))
Solución
Ha introducido
[src]
f(x) = |cos(2*x) - sin(x)|
$$f{\left(x \right)} = \left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|$$
f = Abs(-sin(x) + cos(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:
$$\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{6}$$
$$x_{3} = \frac{5 \pi}{6}$$
Solución numérica
$$x_{1} = 86.393797888715$$
$$x_{2} = 4.71238877821279$$
$$x_{3} = 84.2994028713261$$
$$x_{4} = -39.2699083757319$$
$$x_{5} = -93.7241808320955$$
$$x_{6} = 92.6769830871924$$
$$x_{7} = 54.9778712411975$$
$$x_{8} = 90.5825881785057$$
$$x_{9} = -64.4026491963026$$
$$x_{10} = 40.317105721069$$
$$x_{11} = 69.6386371545737$$
$$x_{12} = 17.2787597959772$$
$$x_{13} = 48.6946859325274$$
$$x_{14} = 67.5442422659503$$
$$x_{15} = 25.6563400043166$$
$$x_{16} = 80.1106131458253$$
$$x_{17} = -70.6858344924983$$
$$x_{18} = -18.3259571459405$$
$$x_{19} = -5.75958653158129$$
$$x_{20} = -97.9129710368819$$
$$x_{21} = -3.66519142918809$$
$$x_{22} = -26.7035373476123$$
$$x_{23} = -49.7418836818384$$
$$x_{24} = 0.523598775598299$$
$$x_{25} = 61.2610569380464$$
$$x_{26} = -16.2315620435473$$
$$x_{27} = 23.5619451122289$$
$$x_{28} = -64.4026494629427$$
$$x_{29} = 73.8274274783337$$
$$x_{30} = 10.9955740992967$$
$$x_{31} = 34.0339204138894$$
$$x_{32} = -26.7035379915215$$
$$x_{33} = 10.995574056153$$
$$x_{34} = -14.1371670557608$$
$$x_{35} = 27.7507351067098$$
$$x_{36} = 92.6769826185806$$
$$x_{37} = -95.8185758681551$$
$$x_{38} = -32.9867230405965$$
$$x_{39} = -62.3082542961976$$
$$x_{40} = 54.9778721441305$$
$$x_{41} = 54.9778708860144$$
$$x_{42} = -14.1371668400256$$
$$x_{43} = 2.61799387799149$$
$$x_{44} = -41.3643032722656$$
$$x_{45} = 98.9601685995222$$
$$x_{46} = -76.9690201780717$$
$$x_{47} = 78.0162175641465$$
$$x_{48} = -76.9690204511548$$
$$x_{49} = -56.025068989018$$
$$x_{50} = -12.0427718387609$$
$$x_{51} = 88.4881930761125$$
$$x_{52} = -32.98672341235$$
$$x_{53} = 31.9395253114962$$
$$x_{54} = -60.2138591938044$$
$$x_{55} = 46.6002910282486$$
$$x_{56} = 61.2610562112906$$
$$x_{57} = 63.3554518473942$$
$$x_{58} = -51.8362786898924$$
$$x_{59} = 98.9601683847854$$
$$x_{60} = -20.4203520418601$$
$$x_{61} = -7.85398149924071$$
$$x_{62} = -58.1194639999037$$
$$x_{63} = -91.6297857297023$$
$$x_{64} = -47.6474885794452$$
$$x_{65} = -83.2522055292846$$
$$x_{66} = -89.5353907455655$$
$$x_{67} = 36.1283156017834$$
$$x_{68} = 75.9218224617533$$
$$x_{69} = 82.2050077689329$$
$$x_{70} = -1.57079642893127$$
$$x_{71} = -76.9690198122422$$
$$x_{72} = 44.5058959258554$$
$$x_{73} = 71.733032256967$$
$$x_{74} = -100.007366139275$$
$$x_{75} = -45.5530935873709$$
$$x_{76} = 29.8451303193672$$
$$x_{77} = 17.2787598104547$$
$$x_{78} = -85.3466004225227$$
$$x_{79} = 42.4115007297604$$
$$x_{80} = 38.2227106186758$$
$$x_{81} = -95.8185760435073$$
o sea hay que resolver la ecuación:
$$\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{6}$$
$$x_{3} = \frac{5 \pi}{6}$$
Solución numérica
$$x_{1} = 86.393797888715$$
$$x_{2} = 4.71238877821279$$
$$x_{3} = 84.2994028713261$$
$$x_{4} = -39.2699083757319$$
$$x_{5} = -93.7241808320955$$
$$x_{6} = 92.6769830871924$$
$$x_{7} = 54.9778712411975$$
$$x_{8} = 90.5825881785057$$
$$x_{9} = -64.4026491963026$$
$$x_{10} = 40.317105721069$$
$$x_{11} = 69.6386371545737$$
$$x_{12} = 17.2787597959772$$
$$x_{13} = 48.6946859325274$$
$$x_{14} = 67.5442422659503$$
$$x_{15} = 25.6563400043166$$
$$x_{16} = 80.1106131458253$$
$$x_{17} = -70.6858344924983$$
$$x_{18} = -18.3259571459405$$
$$x_{19} = -5.75958653158129$$
$$x_{20} = -97.9129710368819$$
$$x_{21} = -3.66519142918809$$
$$x_{22} = -26.7035373476123$$
$$x_{23} = -49.7418836818384$$
$$x_{24} = 0.523598775598299$$
$$x_{25} = 61.2610569380464$$
$$x_{26} = -16.2315620435473$$
$$x_{27} = 23.5619451122289$$
$$x_{28} = -64.4026494629427$$
$$x_{29} = 73.8274274783337$$
$$x_{30} = 10.9955740992967$$
$$x_{31} = 34.0339204138894$$
$$x_{32} = -26.7035379915215$$
$$x_{33} = 10.995574056153$$
$$x_{34} = -14.1371670557608$$
$$x_{35} = 27.7507351067098$$
$$x_{36} = 92.6769826185806$$
$$x_{37} = -95.8185758681551$$
$$x_{38} = -32.9867230405965$$
$$x_{39} = -62.3082542961976$$
$$x_{40} = 54.9778721441305$$
$$x_{41} = 54.9778708860144$$
$$x_{42} = -14.1371668400256$$
$$x_{43} = 2.61799387799149$$
$$x_{44} = -41.3643032722656$$
$$x_{45} = 98.9601685995222$$
$$x_{46} = -76.9690201780717$$
$$x_{47} = 78.0162175641465$$
$$x_{48} = -76.9690204511548$$
$$x_{49} = -56.025068989018$$
$$x_{50} = -12.0427718387609$$
$$x_{51} = 88.4881930761125$$
$$x_{52} = -32.98672341235$$
$$x_{53} = 31.9395253114962$$
$$x_{54} = -60.2138591938044$$
$$x_{55} = 46.6002910282486$$
$$x_{56} = 61.2610562112906$$
$$x_{57} = 63.3554518473942$$
$$x_{58} = -51.8362786898924$$
$$x_{59} = 98.9601683847854$$
$$x_{60} = -20.4203520418601$$
$$x_{61} = -7.85398149924071$$
$$x_{62} = -58.1194639999037$$
$$x_{63} = -91.6297857297023$$
$$x_{64} = -47.6474885794452$$
$$x_{65} = -83.2522055292846$$
$$x_{66} = -89.5353907455655$$
$$x_{67} = 36.1283156017834$$
$$x_{68} = 75.9218224617533$$
$$x_{69} = 82.2050077689329$$
$$x_{70} = -1.57079642893127$$
$$x_{71} = -76.9690198122422$$
$$x_{72} = 44.5058959258554$$
$$x_{73} = 71.733032256967$$
$$x_{74} = -100.007366139275$$
$$x_{75} = -45.5530935873709$$
$$x_{76} = 29.8451303193672$$
$$x_{77} = 17.2787598104547$$
$$x_{78} = -85.3466004225227$$
$$x_{79} = 42.4115007297604$$
$$x_{80} = 38.2227106186758$$
$$x_{81} = -95.8185760435073$$
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
$$\left(- 2 \sin{\left(2 x \right)} - \cos{\left(x \right)}\right) \operatorname{sign}{\left(- \sin{\left(x \right)} + \cos{\left(2 x \right)} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 1.5707963267949$$
$$x_{2} = 56.2959875094742$$
$$x_{3} = 92.6769832808989$$
$$x_{4} = 14.1371669411541$$
$$x_{5} = -21.7384683199865$$
$$x_{6} = -70.6858347057703$$
$$x_{7} = 53.6597553661686$$
$$x_{8} = -36.1283155162826$$
$$x_{9} = -81.9340892484767$$
$$x_{10} = 7.85398163397448$$
$$x_{11} = -59.437580163064$$
$$x_{12} = 42.4115008234622$$
$$x_{13} = -78.2871360846027$$
$$x_{14} = 28.5270141374502$$
$$x_{15} = 12.3136903592171$$
$$x_{16} = -89.5353906273091$$
$$x_{17} = -0.252680255142079$$
$$x_{18} = 43.729616895115$$
$$x_{19} = 22.2438288302706$$
$$x_{20} = -67.5442420521806$$
$$x_{21} = -51.8362787842316$$
$$x_{22} = -83.2522053201295$$
$$x_{23} = -25.3854214838604$$
$$x_{24} = -14.1371669411541$$
$$x_{25} = -65.7207654702436$$
$$x_{26} = -28.0216536271661$$
$$x_{27} = 64.4026493985908$$
$$x_{28} = 81.4287287381925$$
$$x_{29} = 48.6946861306418$$
$$x_{30} = 67.5442420521806$$
$$x_{31} = 23.5619449019235$$
$$x_{32} = -117.809724509617$$
$$x_{33} = -9.1720977056273$$
$$x_{34} = -94.5004598628359$$
$$x_{35} = 50.0128022022946$$
$$x_{36} = 18.5968756663967$$
$$x_{37} = -80.1106126665397$$
$$x_{38} = 87.7119140453721$$
$$x_{39} = 66.2261259805277$$
$$x_{40} = 62.5791728166538$$
$$x_{41} = -29.845130209103$$
$$x_{42} = -64.4026493985908$$
$$x_{43} = 15.960643523091$$
$$x_{44} = -86.3937979737193$$
$$x_{45} = 20.4203522483337$$
$$x_{46} = -88.2172745556563$$
$$x_{47} = -102.101761241668$$
$$x_{48} = -1.5707963267949$$
$$x_{49} = 86.3937979737193$$
$$x_{50} = -72.0039507774232$$
$$x_{51} = 36.1283155162826$$
$$x_{52} = 100.278284659731$$
$$x_{53} = -44.2349774053992$$
$$x_{54} = 51.8362787842316$$
$$x_{55} = 80.1106126665397$$
$$x_{56} = -15.4552830128069$$
$$x_{57} = 34.8101994446298$$
$$x_{58} = 93.9950993525517$$
$$x_{59} = 89.5353906273091$$
$$x_{60} = 58.1194640914112$$
$$x_{61} = 97.6420525164257$$
$$x_{62} = 73.8274273593601$$
$$x_{63} = -58.1194640914112$$
$$x_{64} = -20.4203522483337$$
$$x_{65} = -50.5181627125788$$
$$x_{66} = -6.53586556232167$$
$$x_{67} = -95.8185759344887$$
$$x_{68} = 29.845130209103$$
$$x_{69} = 26.7035375555132$$
$$x_{70} = -23.5619449019235$$
$$x_{71} = -42.4115008234622$$
$$x_{72} = 37.4464315879354$$
$$x_{73} = -34.3048389343456$$
$$x_{74} = -31.66860679104$$
$$x_{75} = 78.7924965948869$$
$$x_{76} = -97.1366920061415$$
$$x_{77} = -61.261056745001$$
$$x_{78} = 70.6858347057703$$
$$x_{79} = -53.1543948558844$$
$$x_{80} = 59.9429406733481$$
$$x_{81} = -103.419877313321$$
$$x_{82} = 72.5093112877073$$
$$x_{83} = 6.03050505203751$$
$$x_{84} = 45.553093477052$$
$$x_{85} = -37.9517920982196$$
$$x_{86} = -73.8274273593601$$
$$x_{87} = -75.6509039412971$$
$$x_{88} = -45.553093477052$$
$$x_{89} = 9.67745821591146$$
$$x_{90} = -7.85398163397448$$
$$x_{91} = -17.2787595947439$$
$$x_{92} = 95.8185759344887$$
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:
$$x_{1} = 92.6769832808989$$
$$x_{2} = -70.6858347057703$$
$$x_{3} = 42.4115008234622$$
$$x_{4} = -89.5353906273091$$
$$x_{5} = -51.8362787842316$$
$$x_{6} = -83.2522053201295$$
$$x_{7} = -14.1371669411541$$
$$x_{8} = 48.6946861306418$$
$$x_{9} = 67.5442420521806$$
$$x_{10} = 23.5619449019235$$
$$x_{11} = -64.4026493985908$$
$$x_{12} = -102.101761241668$$
$$x_{13} = -1.5707963267949$$
$$x_{14} = 86.3937979737193$$
$$x_{15} = 36.1283155162826$$
$$x_{16} = 80.1106126665397$$
$$x_{17} = 73.8274273593601$$
$$x_{18} = -58.1194640914112$$
$$x_{19} = -20.4203522483337$$
$$x_{20} = -95.8185759344887$$
$$x_{21} = 29.845130209103$$
$$x_{22} = -45.553093477052$$
$$x_{23} = -7.85398163397448$$
Puntos máximos de la función:
$$x_{23} = 1.5707963267949$$
$$x_{23} = 56.2959875094742$$
$$x_{23} = 14.1371669411541$$
$$x_{23} = -21.7384683199865$$
$$x_{23} = 53.6597553661686$$
$$x_{23} = -36.1283155162826$$
$$x_{23} = -81.9340892484767$$
$$x_{23} = 7.85398163397448$$
$$x_{23} = -59.437580163064$$
$$x_{23} = -78.2871360846027$$
$$x_{23} = 28.5270141374502$$
$$x_{23} = 12.3136903592171$$
$$x_{23} = -0.252680255142079$$
$$x_{23} = 43.729616895115$$
$$x_{23} = 22.2438288302706$$
$$x_{23} = -67.5442420521806$$
$$x_{23} = -25.3854214838604$$
$$x_{23} = -65.7207654702436$$
$$x_{23} = -28.0216536271661$$
$$x_{23} = 64.4026493985908$$
$$x_{23} = 81.4287287381925$$
$$x_{23} = -117.809724509617$$
$$x_{23} = -9.1720977056273$$
$$x_{23} = -94.5004598628359$$
$$x_{23} = 50.0128022022946$$
$$x_{23} = 18.5968756663967$$
$$x_{23} = -80.1106126665397$$
$$x_{23} = 87.7119140453721$$
$$x_{23} = 66.2261259805277$$
$$x_{23} = 62.5791728166538$$
$$x_{23} = -29.845130209103$$
$$x_{23} = 15.960643523091$$
$$x_{23} = -86.3937979737193$$
$$x_{23} = 20.4203522483337$$
$$x_{23} = -88.2172745556563$$
$$x_{23} = -72.0039507774232$$
$$x_{23} = 100.278284659731$$
$$x_{23} = -44.2349774053992$$
$$x_{23} = 51.8362787842316$$
$$x_{23} = -15.4552830128069$$
$$x_{23} = 34.8101994446298$$
$$x_{23} = 93.9950993525517$$
$$x_{23} = 89.5353906273091$$
$$x_{23} = 58.1194640914112$$
$$x_{23} = 97.6420525164257$$
$$x_{23} = -50.5181627125788$$
$$x_{23} = -6.53586556232167$$
$$x_{23} = 26.7035375555132$$
$$x_{23} = -23.5619449019235$$
$$x_{23} = -42.4115008234622$$
$$x_{23} = 37.4464315879354$$
$$x_{23} = -34.3048389343456$$
$$x_{23} = -31.66860679104$$
$$x_{23} = 78.7924965948869$$
$$x_{23} = -97.1366920061415$$
$$x_{23} = -61.261056745001$$
$$x_{23} = 70.6858347057703$$
$$x_{23} = -53.1543948558844$$
$$x_{23} = 59.9429406733481$$
$$x_{23} = -103.419877313321$$
$$x_{23} = 72.5093112877073$$
$$x_{23} = 6.03050505203751$$
$$x_{23} = 45.553093477052$$
$$x_{23} = -37.9517920982196$$
$$x_{23} = -73.8274273593601$$
$$x_{23} = -75.6509039412971$$
$$x_{23} = 9.67745821591146$$
$$x_{23} = -17.2787595947439$$
$$x_{23} = 95.8185759344887$$
Decrece en los intervalos
$$\left[92.6769832808989, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -102.101761241668\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\left(- 2 \sin{\left(2 x \right)} - \cos{\left(x \right)}\right) \operatorname{sign}{\left(- \sin{\left(x \right)} + \cos{\left(2 x \right)} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 1.5707963267949$$
$$x_{2} = 56.2959875094742$$
$$x_{3} = 92.6769832808989$$
$$x_{4} = 14.1371669411541$$
$$x_{5} = -21.7384683199865$$
$$x_{6} = -70.6858347057703$$
$$x_{7} = 53.6597553661686$$
$$x_{8} = -36.1283155162826$$
$$x_{9} = -81.9340892484767$$
$$x_{10} = 7.85398163397448$$
$$x_{11} = -59.437580163064$$
$$x_{12} = 42.4115008234622$$
$$x_{13} = -78.2871360846027$$
$$x_{14} = 28.5270141374502$$
$$x_{15} = 12.3136903592171$$
$$x_{16} = -89.5353906273091$$
$$x_{17} = -0.252680255142079$$
$$x_{18} = 43.729616895115$$
$$x_{19} = 22.2438288302706$$
$$x_{20} = -67.5442420521806$$
$$x_{21} = -51.8362787842316$$
$$x_{22} = -83.2522053201295$$
$$x_{23} = -25.3854214838604$$
$$x_{24} = -14.1371669411541$$
$$x_{25} = -65.7207654702436$$
$$x_{26} = -28.0216536271661$$
$$x_{27} = 64.4026493985908$$
$$x_{28} = 81.4287287381925$$
$$x_{29} = 48.6946861306418$$
$$x_{30} = 67.5442420521806$$
$$x_{31} = 23.5619449019235$$
$$x_{32} = -117.809724509617$$
$$x_{33} = -9.1720977056273$$
$$x_{34} = -94.5004598628359$$
$$x_{35} = 50.0128022022946$$
$$x_{36} = 18.5968756663967$$
$$x_{37} = -80.1106126665397$$
$$x_{38} = 87.7119140453721$$
$$x_{39} = 66.2261259805277$$
$$x_{40} = 62.5791728166538$$
$$x_{41} = -29.845130209103$$
$$x_{42} = -64.4026493985908$$
$$x_{43} = 15.960643523091$$
$$x_{44} = -86.3937979737193$$
$$x_{45} = 20.4203522483337$$
$$x_{46} = -88.2172745556563$$
$$x_{47} = -102.101761241668$$
$$x_{48} = -1.5707963267949$$
$$x_{49} = 86.3937979737193$$
$$x_{50} = -72.0039507774232$$
$$x_{51} = 36.1283155162826$$
$$x_{52} = 100.278284659731$$
$$x_{53} = -44.2349774053992$$
$$x_{54} = 51.8362787842316$$
$$x_{55} = 80.1106126665397$$
$$x_{56} = -15.4552830128069$$
$$x_{57} = 34.8101994446298$$
$$x_{58} = 93.9950993525517$$
$$x_{59} = 89.5353906273091$$
$$x_{60} = 58.1194640914112$$
$$x_{61} = 97.6420525164257$$
$$x_{62} = 73.8274273593601$$
$$x_{63} = -58.1194640914112$$
$$x_{64} = -20.4203522483337$$
$$x_{65} = -50.5181627125788$$
$$x_{66} = -6.53586556232167$$
$$x_{67} = -95.8185759344887$$
$$x_{68} = 29.845130209103$$
$$x_{69} = 26.7035375555132$$
$$x_{70} = -23.5619449019235$$
$$x_{71} = -42.4115008234622$$
$$x_{72} = 37.4464315879354$$
$$x_{73} = -34.3048389343456$$
$$x_{74} = -31.66860679104$$
$$x_{75} = 78.7924965948869$$
$$x_{76} = -97.1366920061415$$
$$x_{77} = -61.261056745001$$
$$x_{78} = 70.6858347057703$$
$$x_{79} = -53.1543948558844$$
$$x_{80} = 59.9429406733481$$
$$x_{81} = -103.419877313321$$
$$x_{82} = 72.5093112877073$$
$$x_{83} = 6.03050505203751$$
$$x_{84} = 45.553093477052$$
$$x_{85} = -37.9517920982196$$
$$x_{86} = -73.8274273593601$$
$$x_{87} = -75.6509039412971$$
$$x_{88} = -45.553093477052$$
$$x_{89} = 9.67745821591146$$
$$x_{90} = -7.85398163397448$$
$$x_{91} = -17.2787595947439$$
$$x_{92} = 95.8185759344887$$
Signos de extremos en los puntos:
(1.5707963267948966, 2)
(56.2959875094742, 1.125)
(92.67698328089891, 0)
(14.137166941154069, 2)
(-21.738468319986474, 1.125)
(-70.68583470577035, 0)
(53.65975536616856, 1.125)
(-36.12831551628262, 2)
(-81.93408924847671, 1.125)
(7.853981633974483, 2)
(-59.43758016306399, 1.125)
(42.411500823462205, 0)
(-78.28713608460275, 1.125)
(28.52701413745022, 1.125)
(12.313690359217095, 1.125)
(-89.53539062730911, 0)
(-0.25268025514207865, 1.125)
(43.72961689511503, 1.125)
(22.24382883027063, 1.125)
(-67.54424205218055, 2)
(-51.83627878423159, 0)
(-83.25220532012952, 0)
(-25.385421483860423, 1.125)
(-14.137166941154069, 0)
(-65.72076547024358, 1.125)
(-28.02165362716606, 1.125)
(64.40264939859077, 2)
(81.42872873819255, 1.125)
(48.6946861306418, 0)
(67.54424205218055, 0)
(23.56194490192345, 0)
(-117.80972450961724, 2)
(-9.172097705627301, 1.125)
(-94.50045986283588, 1.125)
(50.012802202294615, 1.125)
(18.59687566639668, 1.125)
(-80.11061266653972, 2)
(87.71191404537213, 1.125)
(66.22612598052774, 1.125)
(62.57917281665379, 1.125)
(-29.845130209103036, 2)
(-64.40264939859077, 0)
(15.960643523091045, 1.125)
(-86.39379797371932, 2)
(20.420352248333657, 2)
(-88.21727455565629, 1.125)
(-102.10176124166829, 0)
(-1.5707963267948966, 0)
(86.39379797371932, 0)
(-72.00395077742317, 1.125)
(36.12831551628262, 0)
(100.2782846597313, 1.125)
(-44.234977405399185, 1.125)
(51.83627878423159, 2)
(80.11061266653972, 0)
(-15.455283012806888, 1.125)
(34.8101994446298, 1.125)
(93.99509935255172, 1.125)
(89.53539062730911, 2)
(58.119464091411174, 2)
(97.64205251642566, 1.125)
(73.82742735936014, 0)
(-58.119464091411174, 0)
(-20.420352248333657, 0)
(-50.51816271257877, 1.125)
(-6.535865562321665, 1.125)
(-95.81857593448869, 0)
(29.845130209103036, 0)
(26.703537555513243, 2)
(-23.56194490192345, 2)
(-42.411500823462205, 2)
(37.44643158793544, 1.125)
(-34.304838934345646, 1.125)
(-31.668606791040013, 1.125)
(78.79249659488691, 1.125)
(-97.13669200614152, 1.125)
(-61.26105674500097, 2)
(70.68583470577035, 2)
(-53.154394855884405, 1.125)
(59.94294067334815, 1.125)
(-103.4198773133211, 1.125)
(72.50931128770732, 1.125)
(6.030505052037507, 1.125)
(45.553093477052, 2)
(-37.9517920982196, 1.125)
(-73.82742735936014, 2)
(-75.65090394129712, 1.125)
(-45.553093477052, 0)
(9.677458215911459, 1.125)
(-7.853981633974483, 0)
(-17.278759594743864, 2)
(95.81857593448869, 2)
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:
$$x_{1} = 92.6769832808989$$
$$x_{2} = -70.6858347057703$$
$$x_{3} = 42.4115008234622$$
$$x_{4} = -89.5353906273091$$
$$x_{5} = -51.8362787842316$$
$$x_{6} = -83.2522053201295$$
$$x_{7} = -14.1371669411541$$
$$x_{8} = 48.6946861306418$$
$$x_{9} = 67.5442420521806$$
$$x_{10} = 23.5619449019235$$
$$x_{11} = -64.4026493985908$$
$$x_{12} = -102.101761241668$$
$$x_{13} = -1.5707963267949$$
$$x_{14} = 86.3937979737193$$
$$x_{15} = 36.1283155162826$$
$$x_{16} = 80.1106126665397$$
$$x_{17} = 73.8274273593601$$
$$x_{18} = -58.1194640914112$$
$$x_{19} = -20.4203522483337$$
$$x_{20} = -95.8185759344887$$
$$x_{21} = 29.845130209103$$
$$x_{22} = -45.553093477052$$
$$x_{23} = -7.85398163397448$$
Puntos máximos de la función:
$$x_{23} = 1.5707963267949$$
$$x_{23} = 56.2959875094742$$
$$x_{23} = 14.1371669411541$$
$$x_{23} = -21.7384683199865$$
$$x_{23} = 53.6597553661686$$
$$x_{23} = -36.1283155162826$$
$$x_{23} = -81.9340892484767$$
$$x_{23} = 7.85398163397448$$
$$x_{23} = -59.437580163064$$
$$x_{23} = -78.2871360846027$$
$$x_{23} = 28.5270141374502$$
$$x_{23} = 12.3136903592171$$
$$x_{23} = -0.252680255142079$$
$$x_{23} = 43.729616895115$$
$$x_{23} = 22.2438288302706$$
$$x_{23} = -67.5442420521806$$
$$x_{23} = -25.3854214838604$$
$$x_{23} = -65.7207654702436$$
$$x_{23} = -28.0216536271661$$
$$x_{23} = 64.4026493985908$$
$$x_{23} = 81.4287287381925$$
$$x_{23} = -117.809724509617$$
$$x_{23} = -9.1720977056273$$
$$x_{23} = -94.5004598628359$$
$$x_{23} = 50.0128022022946$$
$$x_{23} = 18.5968756663967$$
$$x_{23} = -80.1106126665397$$
$$x_{23} = 87.7119140453721$$
$$x_{23} = 66.2261259805277$$
$$x_{23} = 62.5791728166538$$
$$x_{23} = -29.845130209103$$
$$x_{23} = 15.960643523091$$
$$x_{23} = -86.3937979737193$$
$$x_{23} = 20.4203522483337$$
$$x_{23} = -88.2172745556563$$
$$x_{23} = -72.0039507774232$$
$$x_{23} = 100.278284659731$$
$$x_{23} = -44.2349774053992$$
$$x_{23} = 51.8362787842316$$
$$x_{23} = -15.4552830128069$$
$$x_{23} = 34.8101994446298$$
$$x_{23} = 93.9950993525517$$
$$x_{23} = 89.5353906273091$$
$$x_{23} = 58.1194640914112$$
$$x_{23} = 97.6420525164257$$
$$x_{23} = -50.5181627125788$$
$$x_{23} = -6.53586556232167$$
$$x_{23} = 26.7035375555132$$
$$x_{23} = -23.5619449019235$$
$$x_{23} = -42.4115008234622$$
$$x_{23} = 37.4464315879354$$
$$x_{23} = -34.3048389343456$$
$$x_{23} = -31.66860679104$$
$$x_{23} = 78.7924965948869$$
$$x_{23} = -97.1366920061415$$
$$x_{23} = -61.261056745001$$
$$x_{23} = 70.6858347057703$$
$$x_{23} = -53.1543948558844$$
$$x_{23} = 59.9429406733481$$
$$x_{23} = -103.419877313321$$
$$x_{23} = 72.5093112877073$$
$$x_{23} = 6.03050505203751$$
$$x_{23} = 45.553093477052$$
$$x_{23} = -37.9517920982196$$
$$x_{23} = -73.8274273593601$$
$$x_{23} = -75.6509039412971$$
$$x_{23} = 9.67745821591146$$
$$x_{23} = -17.2787595947439$$
$$x_{23} = 95.8185759344887$$
Decrece en los intervalos
$$\left[92.6769832808989, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -102.101761241668\right]$$
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
$$- \left(\sin{\left(x \right)} - 4 \cos{\left(2 x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} - \cos{\left(2 x \right)} \right)} + 2 \left(2 \sin{\left(2 x \right)} + \cos{\left(x \right)}\right)^{2} \delta\left(\sin{\left(x \right)} - \cos{\left(2 x \right)}\right) = 0$$
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
$$- \left(\sin{\left(x \right)} - 4 \cos{\left(2 x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} - \cos{\left(2 x \right)} \right)} + 2 \left(2 \sin{\left(2 x \right)} + \cos{\left(x \right)}\right)^{2} \delta\left(\sin{\left(x \right)} - \cos{\left(2 x \right)}\right) = 0$$
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
$$\lim_{x \to -\infty} \left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = \left|{\left\langle -2, 2\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left|{\left\langle -2, 2\right\rangle}\right|$$
$$\lim_{x \to \infty} \left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = \left|{\left\langle -2, 2\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left|{\left\langle -2, 2\right\rangle}\right|$$
$$\lim_{x \to -\infty} \left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = \left|{\left\langle -2, 2\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left|{\left\langle -2, 2\right\rangle}\right|$$
$$\lim_{x \to \infty} \left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = \left|{\left\langle -2, 2\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left|{\left\langle -2, 2\right\rangle}\right|$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función Abs(cos(2*x) - sin(x)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|}{x}\right) = 0$$
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:
$$\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = \left|{\sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|$$
- No
$$\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = - \left|{\sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = \left|{\sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|$$
- No
$$\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = - \left|{\sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|$$
- No
es decir, función
no es
par ni impar