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 =:
-
y=x^3-6x^2+8x
-
y=x^3(x+2)^2
-
y=x^2+2x
-
y=x^2+2x-8
-
Expresiones idénticas
- |sin(x)|*cos(x)
- módulo de seno de (x)| multiplicar por coseno de (x)
- |sin(x)|cos(x)
- |sinx|cosx
-
Expresiones semejantes
- |sinx|*cosx
-
Expresiones con funciones
- Módulo |
- |2x+1|/x-1
- |3cos(2x)-2|
- |3x+1|-|3x-1|
- |x-491/200|
- |(|x+4|)^2-2|
Gráfico de la función y = |sin(x)|*cos(x)
Solución
Ha introducido
[src]
f(x) = |sin(x)|*cos(x)
$$f{\left(x \right)} = \cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|$$
f = cos(x)*Abs(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:
$$\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{2}$$
$$x_{3} = \pi$$
$$x_{4} = \frac{3 \pi}{2}$$
Solución numérica
$$x_{1} = 48.6946861306418$$
$$x_{2} = 81.6814089933346$$
$$x_{3} = -14.1371669411541$$
$$x_{4} = -86.3937979737193$$
$$x_{5} = -1.5707963267949$$
$$x_{6} = 23.5619449019235$$
$$x_{7} = 59.6902604182061$$
$$x_{8} = 73.8274273593601$$
$$x_{9} = 4.71238898038469$$
$$x_{10} = 34.5575191894877$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = -20.4203522483337$$
$$x_{13} = 15.707963267949$$
$$x_{14} = -95.8185759344887$$
$$x_{15} = 26.7035375555132$$
$$x_{16} = -81.6814089933346$$
$$x_{17} = 20.4203522483337$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = 67.5442420521806$$
$$x_{20} = -59.6902604182061$$
$$x_{21} = 36.1283155162826$$
$$x_{22} = -43.9822971502571$$
$$x_{23} = 58.1194640914112$$
$$x_{24} = -29.845130209103$$
$$x_{25} = 12.5663706143592$$
$$x_{26} = 43.9822971502571$$
$$x_{27} = 7.85398163397448$$
$$x_{28} = -15.707963267949$$
$$x_{29} = 0$$
$$x_{30} = 89.5353906273091$$
$$x_{31} = -65.9734457253857$$
$$x_{32} = -28.2743338823081$$
$$x_{33} = 51.8362787842316$$
$$x_{34} = -69.1150383789755$$
$$x_{35} = 70.6858347057703$$
$$x_{36} = -50.2654824574367$$
$$x_{37} = -18.8495559215388$$
$$x_{38} = 80.1106126665397$$
$$x_{39} = 45.553093477052$$
$$x_{40} = 14.1371669411541$$
$$x_{41} = 28.2743338823081$$
$$x_{42} = 65.9734457253857$$
$$x_{43} = -67.5442420521806$$
$$x_{44} = 42.4115008234622$$
$$x_{45} = -45.553093477052$$
$$x_{46} = -102.101761241668$$
$$x_{47} = -58.1194640914112$$
$$x_{48} = -87.9645943005142$$
$$x_{49} = -6.28318530717959$$
$$x_{50} = -83.2522053201295$$
$$x_{51} = 94.2477796076938$$
$$x_{52} = -17.2787595947439$$
$$x_{53} = 95.8185759344887$$
$$x_{54} = -39.2699081698724$$
$$x_{55} = 72.2566310325652$$
$$x_{56} = -36.1283155162826$$
$$x_{57} = 1.5707963267949$$
$$x_{58} = -9.42477796076938$$
$$x_{59} = -64.4026493985908$$
$$x_{60} = 56.5486677646163$$
$$x_{61} = 92.6769832808989$$
$$x_{62} = -51.8362787842316$$
$$x_{63} = 100.530964914873$$
$$x_{64} = -89.5353906273091$$
$$x_{65} = 6.28318530717959$$
$$x_{66} = -61.261056745001$$
$$x_{67} = -73.8274273593601$$
$$x_{68} = 21.9911485751286$$
$$x_{69} = 29.845130209103$$
$$x_{70} = 87.9645943005142$$
$$x_{71} = -72.2566310325652$$
$$x_{72} = 37.6991118430775$$
$$x_{73} = 50.2654824574367$$
$$x_{74} = 86.3937979737193$$
$$x_{75} = 64.4026493985908$$
$$x_{76} = -37.6991118430775$$
$$x_{77} = -23.5619449019235$$
$$x_{78} = -76.9690200129499$$
$$x_{79} = 78.5398163397448$$
$$x_{80} = -42.4115008234622$$
$$x_{81} = -80.1106126665397$$
$$x_{82} = -7.85398163397448$$
o sea hay que resolver la ecuación:
$$\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{2}$$
$$x_{3} = \pi$$
$$x_{4} = \frac{3 \pi}{2}$$
Solución numérica
$$x_{1} = 48.6946861306418$$
$$x_{2} = 81.6814089933346$$
$$x_{3} = -14.1371669411541$$
$$x_{4} = -86.3937979737193$$
$$x_{5} = -1.5707963267949$$
$$x_{6} = 23.5619449019235$$
$$x_{7} = 59.6902604182061$$
$$x_{8} = 73.8274273593601$$
$$x_{9} = 4.71238898038469$$
$$x_{10} = 34.5575191894877$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = -20.4203522483337$$
$$x_{13} = 15.707963267949$$
$$x_{14} = -95.8185759344887$$
$$x_{15} = 26.7035375555132$$
$$x_{16} = -81.6814089933346$$
$$x_{17} = 20.4203522483337$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = 67.5442420521806$$
$$x_{20} = -59.6902604182061$$
$$x_{21} = 36.1283155162826$$
$$x_{22} = -43.9822971502571$$
$$x_{23} = 58.1194640914112$$
$$x_{24} = -29.845130209103$$
$$x_{25} = 12.5663706143592$$
$$x_{26} = 43.9822971502571$$
$$x_{27} = 7.85398163397448$$
$$x_{28} = -15.707963267949$$
$$x_{29} = 0$$
$$x_{30} = 89.5353906273091$$
$$x_{31} = -65.9734457253857$$
$$x_{32} = -28.2743338823081$$
$$x_{33} = 51.8362787842316$$
$$x_{34} = -69.1150383789755$$
$$x_{35} = 70.6858347057703$$
$$x_{36} = -50.2654824574367$$
$$x_{37} = -18.8495559215388$$
$$x_{38} = 80.1106126665397$$
$$x_{39} = 45.553093477052$$
$$x_{40} = 14.1371669411541$$
$$x_{41} = 28.2743338823081$$
$$x_{42} = 65.9734457253857$$
$$x_{43} = -67.5442420521806$$
$$x_{44} = 42.4115008234622$$
$$x_{45} = -45.553093477052$$
$$x_{46} = -102.101761241668$$
$$x_{47} = -58.1194640914112$$
$$x_{48} = -87.9645943005142$$
$$x_{49} = -6.28318530717959$$
$$x_{50} = -83.2522053201295$$
$$x_{51} = 94.2477796076938$$
$$x_{52} = -17.2787595947439$$
$$x_{53} = 95.8185759344887$$
$$x_{54} = -39.2699081698724$$
$$x_{55} = 72.2566310325652$$
$$x_{56} = -36.1283155162826$$
$$x_{57} = 1.5707963267949$$
$$x_{58} = -9.42477796076938$$
$$x_{59} = -64.4026493985908$$
$$x_{60} = 56.5486677646163$$
$$x_{61} = 92.6769832808989$$
$$x_{62} = -51.8362787842316$$
$$x_{63} = 100.530964914873$$
$$x_{64} = -89.5353906273091$$
$$x_{65} = 6.28318530717959$$
$$x_{66} = -61.261056745001$$
$$x_{67} = -73.8274273593601$$
$$x_{68} = 21.9911485751286$$
$$x_{69} = 29.845130209103$$
$$x_{70} = 87.9645943005142$$
$$x_{71} = -72.2566310325652$$
$$x_{72} = 37.6991118430775$$
$$x_{73} = 50.2654824574367$$
$$x_{74} = 86.3937979737193$$
$$x_{75} = 64.4026493985908$$
$$x_{76} = -37.6991118430775$$
$$x_{77} = -23.5619449019235$$
$$x_{78} = -76.9690200129499$$
$$x_{79} = 78.5398163397448$$
$$x_{80} = -42.4115008234622$$
$$x_{81} = -80.1106126665397$$
$$x_{82} = -7.85398163397448$$
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
$$- \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + \cos^{2}{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 32.2013246992954$$
$$x_{2} = 66.7588438887831$$
$$x_{3} = -27.4889357189107$$
$$x_{4} = 10.2101761241668$$
$$x_{5} = -32.2013246992954$$
$$x_{6} = -41.6261026600648$$
$$x_{7} = 55.7632696012188$$
$$x_{8} = -38.484510006475$$
$$x_{9} = 44.7676953136546$$
$$x_{10} = 30.6305283725005$$
$$x_{11} = -21.2057504117311$$
$$x_{12} = 110.74114103904$$
$$x_{13} = 27.4889357189107$$
$$x_{14} = -19.6349540849362$$
$$x_{15} = -3.92699081698724$$
$$x_{16} = 49.4800842940392$$
$$x_{17} = 82.4668071567321$$
$$x_{18} = -69.9004365423729$$
$$x_{19} = 52.621676947629$$
$$x_{20} = -164.148216150067$$
$$x_{21} = -109.170344712245$$
$$x_{22} = -77.7544181763474$$
$$x_{23} = 88.7499924639117$$
$$x_{24} = 16.4933614313464$$
$$x_{25} = 41.6261026600648$$
$$x_{26} = 96.6039740978861$$
$$x_{27} = 0.785398163397448$$
$$x_{28} = -91.8915851175014$$
$$x_{29} = -46.3384916404494$$
$$x_{30} = -5.49778714378214$$
$$x_{31} = -84.037603483527$$
$$x_{32} = 91.8915851175014$$
$$x_{33} = -55.7632696012188$$
$$x_{34} = 77.7544181763474$$
$$x_{35} = 84.037603483527$$
$$x_{36} = -71.4712328691678$$
$$x_{37} = 8.63937979737193$$
$$x_{38} = -76.1836218495525$$
$$x_{39} = -65.1880475619882$$
$$x_{40} = 60.4756585816035$$
$$x_{41} = 3.92699081698724$$
$$x_{42} = 38.484510006475$$
$$x_{43} = 90.3207887907066$$
$$x_{44} = -90.3207887907066$$
$$x_{45} = -79.3252145031423$$
$$x_{46} = -16.4933614313464$$
$$x_{47} = 98.174770424681$$
$$x_{48} = -18.0641577581413$$
$$x_{49} = 76.1836218495525$$
$$x_{50} = 1601.4268551674$$
$$x_{51} = -13.3517687777566$$
$$x_{52} = 24.3473430653209$$
$$x_{53} = -40.0553063332699$$
$$x_{54} = -82.4668071567321$$
$$x_{55} = -98.174770424681$$
$$x_{56} = -68.329640215578$$
$$x_{57} = 46.3384916404494$$
$$x_{58} = 40.0553063332699$$
$$x_{59} = -33.7721210260903$$
$$x_{60} = -25.9181393921158$$
$$x_{61} = -60.4756585816035$$
$$x_{62} = 21.2057504117311$$
$$x_{63} = -49.4800842940392$$
$$x_{64} = -87.1791961371168$$
$$x_{65} = -57.3340659280137$$
$$x_{66} = 0$$
$$x_{67} = 25.9181393921158$$
$$x_{68} = 85.6083998103219$$
$$x_{69} = -10.2101761241668$$
$$x_{70} = -35.3429173528852$$
$$x_{71} = -99.7455667514759$$
$$x_{72} = 74.6128255227576$$
$$x_{73} = 62.0464549083984$$
$$x_{74} = 54.1924732744239$$
$$x_{75} = -24.3473430653209$$
$$x_{76} = 68.329640215578$$
$$x_{77} = 63.6172512351933$$
$$x_{78} = 19.6349540849362$$
$$x_{79} = -85.6083998103219$$
$$x_{80} = 18.0641577581413$$
$$x_{81} = -93.4623814442964$$
$$x_{82} = -54.1924732744239$$
$$x_{83} = -43.1968989868597$$
$$x_{84} = 2.35619449019234$$
$$x_{85} = 99.7455667514759$$
$$x_{86} = -11.7809724509617$$
$$x_{87} = 11.7809724509617$$
$$x_{88} = -62.0464549083984$$
$$x_{89} = 69.9004365423729$$
$$x_{90} = -63.6172512351933$$
$$x_{91} = -2.35619449019234$$
$$x_{92} = 33.7721210260903$$
$$x_{93} = 47.9092879672443$$
$$x_{94} = 5.49778714378214$$
$$x_{95} = -47.9092879672443$$
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:
$$x_{1} = 66.7588438887831$$
$$x_{2} = -27.4889357189107$$
$$x_{3} = 10.2101761241668$$
$$x_{4} = -41.6261026600648$$
$$x_{5} = -21.2057504117311$$
$$x_{6} = 110.74114103904$$
$$x_{7} = 27.4889357189107$$
$$x_{8} = -3.92699081698724$$
$$x_{9} = 52.621676947629$$
$$x_{10} = -109.170344712245$$
$$x_{11} = -77.7544181763474$$
$$x_{12} = 16.4933614313464$$
$$x_{13} = 41.6261026600648$$
$$x_{14} = 96.6039740978861$$
$$x_{15} = -91.8915851175014$$
$$x_{16} = -46.3384916404494$$
$$x_{17} = -84.037603483527$$
$$x_{18} = 91.8915851175014$$
$$x_{19} = 77.7544181763474$$
$$x_{20} = 84.037603483527$$
$$x_{21} = -71.4712328691678$$
$$x_{22} = 8.63937979737193$$
$$x_{23} = -65.1880475619882$$
$$x_{24} = 60.4756585816035$$
$$x_{25} = 3.92699081698724$$
$$x_{26} = 90.3207887907066$$
$$x_{27} = -90.3207887907066$$
$$x_{28} = -79.3252145031423$$
$$x_{29} = -16.4933614313464$$
$$x_{30} = 98.174770424681$$
$$x_{31} = -40.0553063332699$$
$$x_{32} = -98.174770424681$$
$$x_{33} = 46.3384916404494$$
$$x_{34} = 40.0553063332699$$
$$x_{35} = -33.7721210260903$$
$$x_{36} = -60.4756585816035$$
$$x_{37} = 21.2057504117311$$
$$x_{38} = 0$$
$$x_{39} = 85.6083998103219$$
$$x_{40} = -10.2101761241668$$
$$x_{41} = -35.3429173528852$$
$$x_{42} = 54.1924732744239$$
$$x_{43} = -85.6083998103219$$
$$x_{44} = -54.1924732744239$$
$$x_{45} = 2.35619449019234$$
$$x_{46} = -2.35619449019234$$
$$x_{47} = 33.7721210260903$$
$$x_{48} = 47.9092879672443$$
$$x_{49} = -47.9092879672443$$
Puntos máximos de la función:
$$x_{49} = 32.2013246992954$$
$$x_{49} = -32.2013246992954$$
$$x_{49} = 55.7632696012188$$
$$x_{49} = -38.484510006475$$
$$x_{49} = 44.7676953136546$$
$$x_{49} = 30.6305283725005$$
$$x_{49} = -19.6349540849362$$
$$x_{49} = 49.4800842940392$$
$$x_{49} = 82.4668071567321$$
$$x_{49} = -69.9004365423729$$
$$x_{49} = -164.148216150067$$
$$x_{49} = 88.7499924639117$$
$$x_{49} = 0.785398163397448$$
$$x_{49} = -5.49778714378214$$
$$x_{49} = -55.7632696012188$$
$$x_{49} = -76.1836218495525$$
$$x_{49} = 38.484510006475$$
$$x_{49} = -18.0641577581413$$
$$x_{49} = 76.1836218495525$$
$$x_{49} = 1601.4268551674$$
$$x_{49} = -13.3517687777566$$
$$x_{49} = 24.3473430653209$$
$$x_{49} = -82.4668071567321$$
$$x_{49} = -68.329640215578$$
$$x_{49} = -25.9181393921158$$
$$x_{49} = -49.4800842940392$$
$$x_{49} = -87.1791961371168$$
$$x_{49} = -57.3340659280137$$
$$x_{49} = 25.9181393921158$$
$$x_{49} = -99.7455667514759$$
$$x_{49} = 74.6128255227576$$
$$x_{49} = 62.0464549083984$$
$$x_{49} = -24.3473430653209$$
$$x_{49} = 68.329640215578$$
$$x_{49} = 63.6172512351933$$
$$x_{49} = 19.6349540849362$$
$$x_{49} = 18.0641577581413$$
$$x_{49} = -93.4623814442964$$
$$x_{49} = -43.1968989868597$$
$$x_{49} = 99.7455667514759$$
$$x_{49} = -11.7809724509617$$
$$x_{49} = 11.7809724509617$$
$$x_{49} = -62.0464549083984$$
$$x_{49} = 69.9004365423729$$
$$x_{49} = -63.6172512351933$$
$$x_{49} = 5.49778714378214$$
Decrece en los intervalos
$$\left[110.74114103904, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -109.170344712245\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + \cos^{2}{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 32.2013246992954$$
$$x_{2} = 66.7588438887831$$
$$x_{3} = -27.4889357189107$$
$$x_{4} = 10.2101761241668$$
$$x_{5} = -32.2013246992954$$
$$x_{6} = -41.6261026600648$$
$$x_{7} = 55.7632696012188$$
$$x_{8} = -38.484510006475$$
$$x_{9} = 44.7676953136546$$
$$x_{10} = 30.6305283725005$$
$$x_{11} = -21.2057504117311$$
$$x_{12} = 110.74114103904$$
$$x_{13} = 27.4889357189107$$
$$x_{14} = -19.6349540849362$$
$$x_{15} = -3.92699081698724$$
$$x_{16} = 49.4800842940392$$
$$x_{17} = 82.4668071567321$$
$$x_{18} = -69.9004365423729$$
$$x_{19} = 52.621676947629$$
$$x_{20} = -164.148216150067$$
$$x_{21} = -109.170344712245$$
$$x_{22} = -77.7544181763474$$
$$x_{23} = 88.7499924639117$$
$$x_{24} = 16.4933614313464$$
$$x_{25} = 41.6261026600648$$
$$x_{26} = 96.6039740978861$$
$$x_{27} = 0.785398163397448$$
$$x_{28} = -91.8915851175014$$
$$x_{29} = -46.3384916404494$$
$$x_{30} = -5.49778714378214$$
$$x_{31} = -84.037603483527$$
$$x_{32} = 91.8915851175014$$
$$x_{33} = -55.7632696012188$$
$$x_{34} = 77.7544181763474$$
$$x_{35} = 84.037603483527$$
$$x_{36} = -71.4712328691678$$
$$x_{37} = 8.63937979737193$$
$$x_{38} = -76.1836218495525$$
$$x_{39} = -65.1880475619882$$
$$x_{40} = 60.4756585816035$$
$$x_{41} = 3.92699081698724$$
$$x_{42} = 38.484510006475$$
$$x_{43} = 90.3207887907066$$
$$x_{44} = -90.3207887907066$$
$$x_{45} = -79.3252145031423$$
$$x_{46} = -16.4933614313464$$
$$x_{47} = 98.174770424681$$
$$x_{48} = -18.0641577581413$$
$$x_{49} = 76.1836218495525$$
$$x_{50} = 1601.4268551674$$
$$x_{51} = -13.3517687777566$$
$$x_{52} = 24.3473430653209$$
$$x_{53} = -40.0553063332699$$
$$x_{54} = -82.4668071567321$$
$$x_{55} = -98.174770424681$$
$$x_{56} = -68.329640215578$$
$$x_{57} = 46.3384916404494$$
$$x_{58} = 40.0553063332699$$
$$x_{59} = -33.7721210260903$$
$$x_{60} = -25.9181393921158$$
$$x_{61} = -60.4756585816035$$
$$x_{62} = 21.2057504117311$$
$$x_{63} = -49.4800842940392$$
$$x_{64} = -87.1791961371168$$
$$x_{65} = -57.3340659280137$$
$$x_{66} = 0$$
$$x_{67} = 25.9181393921158$$
$$x_{68} = 85.6083998103219$$
$$x_{69} = -10.2101761241668$$
$$x_{70} = -35.3429173528852$$
$$x_{71} = -99.7455667514759$$
$$x_{72} = 74.6128255227576$$
$$x_{73} = 62.0464549083984$$
$$x_{74} = 54.1924732744239$$
$$x_{75} = -24.3473430653209$$
$$x_{76} = 68.329640215578$$
$$x_{77} = 63.6172512351933$$
$$x_{78} = 19.6349540849362$$
$$x_{79} = -85.6083998103219$$
$$x_{80} = 18.0641577581413$$
$$x_{81} = -93.4623814442964$$
$$x_{82} = -54.1924732744239$$
$$x_{83} = -43.1968989868597$$
$$x_{84} = 2.35619449019234$$
$$x_{85} = 99.7455667514759$$
$$x_{86} = -11.7809724509617$$
$$x_{87} = 11.7809724509617$$
$$x_{88} = -62.0464549083984$$
$$x_{89} = 69.9004365423729$$
$$x_{90} = -63.6172512351933$$
$$x_{91} = -2.35619449019234$$
$$x_{92} = 33.7721210260903$$
$$x_{93} = 47.9092879672443$$
$$x_{94} = 5.49778714378214$$
$$x_{95} = -47.9092879672443$$
Signos de extremos en los puntos:
(32.201324699295384, 0.5)
(66.7588438887831, -0.5)
(-27.488935718910692, -0.5)
(10.210176124166829, -0.5)
(-32.201324699295384, 0.5)
(-41.62610266006476, -0.5)
(55.76326960121883, 0.5)
(-38.48451000647497, 0.5)
(44.767695313654556, 0.5)
(30.630528372500486, 0.5)
(-21.205750411731103, -0.5)
(110.74114103904022, -0.5)
(27.488935718910692, -0.5)
(-19.634954084936208, 0.5)
(-3.9269908169872414, -0.5)
(49.480084294039244, 0.5)
(82.46680715673207, 0.5)
(-69.9004365423729, 0.5)
(52.621676947629034, -0.5)
(-164.1482161500667, 0.5)
(-109.17034471224531, -0.5)
(-77.75441817634739, -0.5)
(88.74999246391165, 0.5)
(16.493361431346415, -0.5)
(41.62610266006476, -0.5)
(96.60397409788614, -0.5)
(0.7853981633974483, 0.5)
(-91.89158511750145, -0.5)
(-46.33849164044945, -0.5)
(-5.497787143782138, 0.5)
(-84.03760348352696, -0.5)
(91.89158511750145, -0.5)
(-55.76326960121883, 0.5)
(77.75441817634739, -0.5)
(84.03760348352696, -0.5)
(-71.47123286916779, -0.5)
(8.639379797371932, -0.5)
(-76.18362184955248, 0.5)
(-65.18804756198821, -0.5)
(60.47565858160352, -0.5)
(3.9269908169872414, -0.5)
(38.48451000647497, 0.5)
(90.32078879070656, -0.5)
(-90.32078879070656, -0.5)
(-79.32521450314228, -0.5)
(-16.493361431346415, -0.5)
(98.17477042468104, -0.5)
(-18.06415775814131, 0.5)
(76.18362184955248, 0.5)
(1601.4268551673972, 0.5)
(-13.351768777756622, 0.5)
(24.3473430653209, 0.5)
(-40.05530633326986, -0.5)
(-82.46680715673207, 0.5)
(-98.17477042468104, -0.5)
(-68.329640215578, 0.5)
(46.33849164044945, -0.5)
(40.05530633326986, -0.5)
(-33.772121026090275, -0.5)
(-25.918139392115794, 0.5)
(-60.47565858160352, -0.5)
(21.205750411731103, -0.5)
(-49.480084294039244, 0.5)
(-87.17919613711676, 0.5)
(-57.33406592801373, 0.5)
(0, 0)
(25.918139392115794, 0.5)
(85.60839981032187, -0.5)
(-10.210176124166829, -0.5)
(-35.34291735288517, -0.5)
(-99.74556675147593, 0.5)
(74.61282552275759, 0.5)
(62.04645490839842, 0.5)
(54.19247327442393, -0.5)
(-24.3473430653209, 0.5)
(68.329640215578, 0.5)
(63.617251235193315, 0.5)
(19.634954084936208, 0.5)
(-85.60839981032187, -0.5)
(18.06415775814131, 0.5)
(-93.46238144429635, 0.5)
(-54.19247327442393, -0.5)
(-43.19689898685966, 0.5)
(2.356194490192345, -0.5)
(99.74556675147593, 0.5)
(-11.780972450961725, 0.5)
(11.780972450961725, 0.5)
(-62.04645490839842, 0.5)
(69.9004365423729, 0.5)
(-63.617251235193315, 0.5)
(-2.356194490192345, -0.5)
(33.772121026090275, -0.5)
(47.909287967244346, -0.5)
(5.497787143782138, 0.5)
(-47.909287967244346, -0.5)
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:
$$x_{1} = 66.7588438887831$$
$$x_{2} = -27.4889357189107$$
$$x_{3} = 10.2101761241668$$
$$x_{4} = -41.6261026600648$$
$$x_{5} = -21.2057504117311$$
$$x_{6} = 110.74114103904$$
$$x_{7} = 27.4889357189107$$
$$x_{8} = -3.92699081698724$$
$$x_{9} = 52.621676947629$$
$$x_{10} = -109.170344712245$$
$$x_{11} = -77.7544181763474$$
$$x_{12} = 16.4933614313464$$
$$x_{13} = 41.6261026600648$$
$$x_{14} = 96.6039740978861$$
$$x_{15} = -91.8915851175014$$
$$x_{16} = -46.3384916404494$$
$$x_{17} = -84.037603483527$$
$$x_{18} = 91.8915851175014$$
$$x_{19} = 77.7544181763474$$
$$x_{20} = 84.037603483527$$
$$x_{21} = -71.4712328691678$$
$$x_{22} = 8.63937979737193$$
$$x_{23} = -65.1880475619882$$
$$x_{24} = 60.4756585816035$$
$$x_{25} = 3.92699081698724$$
$$x_{26} = 90.3207887907066$$
$$x_{27} = -90.3207887907066$$
$$x_{28} = -79.3252145031423$$
$$x_{29} = -16.4933614313464$$
$$x_{30} = 98.174770424681$$
$$x_{31} = -40.0553063332699$$
$$x_{32} = -98.174770424681$$
$$x_{33} = 46.3384916404494$$
$$x_{34} = 40.0553063332699$$
$$x_{35} = -33.7721210260903$$
$$x_{36} = -60.4756585816035$$
$$x_{37} = 21.2057504117311$$
$$x_{38} = 0$$
$$x_{39} = 85.6083998103219$$
$$x_{40} = -10.2101761241668$$
$$x_{41} = -35.3429173528852$$
$$x_{42} = 54.1924732744239$$
$$x_{43} = -85.6083998103219$$
$$x_{44} = -54.1924732744239$$
$$x_{45} = 2.35619449019234$$
$$x_{46} = -2.35619449019234$$
$$x_{47} = 33.7721210260903$$
$$x_{48} = 47.9092879672443$$
$$x_{49} = -47.9092879672443$$
Puntos máximos de la función:
$$x_{49} = 32.2013246992954$$
$$x_{49} = -32.2013246992954$$
$$x_{49} = 55.7632696012188$$
$$x_{49} = -38.484510006475$$
$$x_{49} = 44.7676953136546$$
$$x_{49} = 30.6305283725005$$
$$x_{49} = -19.6349540849362$$
$$x_{49} = 49.4800842940392$$
$$x_{49} = 82.4668071567321$$
$$x_{49} = -69.9004365423729$$
$$x_{49} = -164.148216150067$$
$$x_{49} = 88.7499924639117$$
$$x_{49} = 0.785398163397448$$
$$x_{49} = -5.49778714378214$$
$$x_{49} = -55.7632696012188$$
$$x_{49} = -76.1836218495525$$
$$x_{49} = 38.484510006475$$
$$x_{49} = -18.0641577581413$$
$$x_{49} = 76.1836218495525$$
$$x_{49} = 1601.4268551674$$
$$x_{49} = -13.3517687777566$$
$$x_{49} = 24.3473430653209$$
$$x_{49} = -82.4668071567321$$
$$x_{49} = -68.329640215578$$
$$x_{49} = -25.9181393921158$$
$$x_{49} = -49.4800842940392$$
$$x_{49} = -87.1791961371168$$
$$x_{49} = -57.3340659280137$$
$$x_{49} = 25.9181393921158$$
$$x_{49} = -99.7455667514759$$
$$x_{49} = 74.6128255227576$$
$$x_{49} = 62.0464549083984$$
$$x_{49} = -24.3473430653209$$
$$x_{49} = 68.329640215578$$
$$x_{49} = 63.6172512351933$$
$$x_{49} = 19.6349540849362$$
$$x_{49} = 18.0641577581413$$
$$x_{49} = -93.4623814442964$$
$$x_{49} = -43.1968989868597$$
$$x_{49} = 99.7455667514759$$
$$x_{49} = -11.7809724509617$$
$$x_{49} = 11.7809724509617$$
$$x_{49} = -62.0464549083984$$
$$x_{49} = 69.9004365423729$$
$$x_{49} = -63.6172512351933$$
$$x_{49} = 5.49778714378214$$
Decrece en los intervalos
$$\left[110.74114103904, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -109.170344712245\right]$$
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
$$- \left(3 \sin{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} - 2 \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\right) + \left|{\sin{\left(x \right)}}\right|\right) \cos{\left(x \right)} = 0$$
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
$$- \left(3 \sin{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} - 2 \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\right) + \left|{\sin{\left(x \right)}}\right|\right) \cos{\left(x \right)} = 0$$
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
$$\lim_{x \to -\infty}\left(\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty}\left(\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to -\infty}\left(\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty}\left(\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función Abs(sin(x))*cos(x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|}{x}\right) = 0$$
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:
$$\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = \cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|$$
- Sí
$$\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = - \cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = \cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|$$
- Sí
$$\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = - \cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|$$
- No
es decir, función
es
par