Sr Examen
Otras calculadoras
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- Integrales paso a paso
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- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3+3*x^2+2
-
x*(-1-log(x))
-
e^3*x+10*e^2*x
-
-cos(2*x)-sin(2*x)
-
Expresiones idénticas
- abs(cos(x)*sin(x))^ tres
- abs( coseno de (x) multiplicar por seno de (x)) al cubo
- abs( coseno de (x) multiplicar por seno de (x)) en el grado tres
- abs(cos(x)*sin(x))3
- abscosx*sinx3
- abs(cos(x)*sin(x))³
- abs(cos(x)*sin(x)) en el grado 3
- abs(cos(x)sin(x))^3
- abs(cos(x)sin(x))3
- abscosxsinx3
- abscosxsinx^3
-
Expresiones semejantes
- abs(cosx*sinx)^3
-
Expresiones con funciones
- abs
- abs(|2*x-5|-1)
- abs(log(1-x))
- abs(log[2,abs(x)]-1)
- absolute(3-(1/2)*e^absolute(x+2))
- abs(arccos(3abs(x-1)+1)-(pi/3))
- Coseno cos
- cos(2*x)-sqrt(3)/2
- cos(2*x*sqrt(2))+sin(2*x*sqrt(2))
- cos(2/x)-2*sin(1/x)+1/x
- cos(4*x)+sin(4*x)
- cos(3*x-1)
- Seno sin
- sin(x)/2+(1+x)*exp(-x)
- sin(x)^1
- sin((x^2+1)/(x-1))
- sin(asin(x))
- sin(40*x+sin(40*x)^(2)*2)^(50)
Gráfico de la función y = abs(cos(x)*sin(x))^3
Solución
Ha introducido
[src]
3 f(x) = |cos(x)*sin(x)|
$$f{\left(x \right)} = \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3}$$
f = Abs(sin(x)*cos(x))^3
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:
$$\left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = 0$$
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
$$x_{1} = 95.8186236594161$$
$$x_{2} = 45.5531529759555$$
$$x_{3} = 34.5574601966861$$
$$x_{4} = -50.2654228305023$$
$$x_{5} = 58.1195414662104$$
$$x_{6} = 72.2566292958971$$
$$x_{7} = 51.8363232535927$$
$$x_{8} = 7.85402257865455$$
$$x_{9} = 48.6946268124239$$
$$x_{10} = -12.5664346805496$$
$$x_{11} = -75.3982859319017$$
$$x_{12} = -43.9823033627095$$
$$x_{13} = 9.42468091566607$$
$$x_{14} = 73.8274734896477$$
$$x_{15} = 80.11070769314$$
$$x_{16} = -45.5531374165576$$
$$x_{17} = 87.9646062618766$$
$$x_{18} = 12.566310817148$$
$$x_{19} = 31.4158351320415$$
$$x_{20} = -36.1282783755472$$
$$x_{21} = 6.28317669991181$$
$$x_{22} = 14.1372124791535$$
$$x_{23} = 78.5397591585436$$
$$x_{24} = -28.2742733484704$$
$$x_{25} = -86.3937330346576$$
$$x_{26} = 84.8230905147235$$
$$x_{27} = 65.9734547917135$$
$$x_{28} = 4.71233320272705$$
$$x_{29} = 81.6814729895525$$
$$x_{30} = -80.1105796228002$$
$$x_{31} = 0$$
$$x_{32} = -61.2611031899243$$
$$x_{33} = 1523.67246905558$$
$$x_{34} = -23.5619872943506$$
$$x_{35} = -83.2522470339395$$
$$x_{36} = -14.1371278627626$$
$$x_{37} = -78.5398182904347$$
$$x_{38} = 15.7080254064876$$
$$x_{39} = -10.9956645364959$$
$$x_{40} = -100.530953779175$$
$$x_{41} = 94.2477801895426$$
$$x_{42} = -64.402584893839$$
$$x_{43} = -29.8450972978891$$
$$x_{44} = 59.6903238713162$$
$$x_{45} = -59.6902756491872$$
$$x_{46} = -20.4202890998685$$
$$x_{47} = 64.4026136371408$$
$$x_{48} = 67.5442997419149$$
$$x_{49} = 20.4203130843561$$
$$x_{50} = 36.1282758392829$$
$$x_{51} = -42.4114369087883$$
$$x_{52} = -97.3894349001807$$
$$x_{53} = -89.5354375223026$$
$$x_{54} = -95.8185559065024$$
$$x_{55} = 42.4114633393296$$
$$x_{56} = 89.5354461079601$$
$$x_{57} = -51.836250913374$$
$$x_{58} = -72.2565723804793$$
$$x_{59} = -94.2477219973583$$
$$x_{60} = -15.7079741153866$$
$$x_{61} = 23.562005879767$$
$$x_{62} = -9.42483849004633$$
$$x_{63} = 1.57085850721285$$
$$x_{64} = 43.9823032463116$$
$$x_{65} = 50.265478410943$$
$$x_{66} = 28.2743275435142$$
$$x_{67} = -53.4071368698507$$
$$x_{68} = -58.1194289615388$$
$$x_{69} = -67.5442874928606$$
$$x_{70} = -87.9646072874727$$
$$x_{71} = -45.5530734085886$$
$$x_{72} = -39.2699584208524$$
$$x_{73} = -34.5575569038925$$
$$x_{74} = -37.6991248991848$$
$$x_{75} = -7.85394271974308$$
$$x_{76} = -1.57083712711347$$
$$x_{77} = 37.6991746761598$$
$$x_{78} = -65.9734551999195$$
$$x_{79} = 92.6769217869229$$
$$x_{80} = 86.3937639771802$$
$$x_{81} = 56.5486096448912$$
$$x_{82} = -20.4203887877688$$
$$x_{83} = 100.530908734808$$
$$x_{84} = -6.28312393564209$$
$$x_{85} = 37.6991277169061$$
$$x_{86} = -31.4159877206731$$
$$x_{87} = 21.9911516409413$$
$$x_{88} = -21.9911516551834$$
$$x_{89} = -73.8274037159877$$
$$x_{90} = 62.8319614833021$$
$$x_{91} = 26.7034798052225$$
$$x_{92} = 70.6857741572608$$
$$x_{93} = -81.6814263637279$$
$$x_{94} = 29.8451729503133$$
$$x_{95} = -56.5486855373134$$
$$x_{96} = -17.2788129093397$$
o sea hay que resolver la ecuación:
$$\left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = 0$$
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
$$x_{1} = 95.8186236594161$$
$$x_{2} = 45.5531529759555$$
$$x_{3} = 34.5574601966861$$
$$x_{4} = -50.2654228305023$$
$$x_{5} = 58.1195414662104$$
$$x_{6} = 72.2566292958971$$
$$x_{7} = 51.8363232535927$$
$$x_{8} = 7.85402257865455$$
$$x_{9} = 48.6946268124239$$
$$x_{10} = -12.5664346805496$$
$$x_{11} = -75.3982859319017$$
$$x_{12} = -43.9823033627095$$
$$x_{13} = 9.42468091566607$$
$$x_{14} = 73.8274734896477$$
$$x_{15} = 80.11070769314$$
$$x_{16} = -45.5531374165576$$
$$x_{17} = 87.9646062618766$$
$$x_{18} = 12.566310817148$$
$$x_{19} = 31.4158351320415$$
$$x_{20} = -36.1282783755472$$
$$x_{21} = 6.28317669991181$$
$$x_{22} = 14.1372124791535$$
$$x_{23} = 78.5397591585436$$
$$x_{24} = -28.2742733484704$$
$$x_{25} = -86.3937330346576$$
$$x_{26} = 84.8230905147235$$
$$x_{27} = 65.9734547917135$$
$$x_{28} = 4.71233320272705$$
$$x_{29} = 81.6814729895525$$
$$x_{30} = -80.1105796228002$$
$$x_{31} = 0$$
$$x_{32} = -61.2611031899243$$
$$x_{33} = 1523.67246905558$$
$$x_{34} = -23.5619872943506$$
$$x_{35} = -83.2522470339395$$
$$x_{36} = -14.1371278627626$$
$$x_{37} = -78.5398182904347$$
$$x_{38} = 15.7080254064876$$
$$x_{39} = -10.9956645364959$$
$$x_{40} = -100.530953779175$$
$$x_{41} = 94.2477801895426$$
$$x_{42} = -64.402584893839$$
$$x_{43} = -29.8450972978891$$
$$x_{44} = 59.6903238713162$$
$$x_{45} = -59.6902756491872$$
$$x_{46} = -20.4202890998685$$
$$x_{47} = 64.4026136371408$$
$$x_{48} = 67.5442997419149$$
$$x_{49} = 20.4203130843561$$
$$x_{50} = 36.1282758392829$$
$$x_{51} = -42.4114369087883$$
$$x_{52} = -97.3894349001807$$
$$x_{53} = -89.5354375223026$$
$$x_{54} = -95.8185559065024$$
$$x_{55} = 42.4114633393296$$
$$x_{56} = 89.5354461079601$$
$$x_{57} = -51.836250913374$$
$$x_{58} = -72.2565723804793$$
$$x_{59} = -94.2477219973583$$
$$x_{60} = -15.7079741153866$$
$$x_{61} = 23.562005879767$$
$$x_{62} = -9.42483849004633$$
$$x_{63} = 1.57085850721285$$
$$x_{64} = 43.9823032463116$$
$$x_{65} = 50.265478410943$$
$$x_{66} = 28.2743275435142$$
$$x_{67} = -53.4071368698507$$
$$x_{68} = -58.1194289615388$$
$$x_{69} = -67.5442874928606$$
$$x_{70} = -87.9646072874727$$
$$x_{71} = -45.5530734085886$$
$$x_{72} = -39.2699584208524$$
$$x_{73} = -34.5575569038925$$
$$x_{74} = -37.6991248991848$$
$$x_{75} = -7.85394271974308$$
$$x_{76} = -1.57083712711347$$
$$x_{77} = 37.6991746761598$$
$$x_{78} = -65.9734551999195$$
$$x_{79} = 92.6769217869229$$
$$x_{80} = 86.3937639771802$$
$$x_{81} = 56.5486096448912$$
$$x_{82} = -20.4203887877688$$
$$x_{83} = 100.530908734808$$
$$x_{84} = -6.28312393564209$$
$$x_{85} = 37.6991277169061$$
$$x_{86} = -31.4159877206731$$
$$x_{87} = 21.9911516409413$$
$$x_{88} = -21.9911516551834$$
$$x_{89} = -73.8274037159877$$
$$x_{90} = 62.8319614833021$$
$$x_{91} = 26.7034798052225$$
$$x_{92} = 70.6857741572608$$
$$x_{93} = -81.6814263637279$$
$$x_{94} = 29.8451729503133$$
$$x_{95} = -56.5486855373134$$
$$x_{96} = -17.2788129093397$$
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
$$3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} \cos{\left(x \right)} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 43.9822971689083$$
$$x_{2} = -94.2477796324164$$
$$x_{3} = 32.2013246992954$$
$$x_{4} = 10.2101761241668$$
$$x_{5} = -41.6261026600648$$
$$x_{6} = -36.1283154463325$$
$$x_{7} = 94.2477796093316$$
$$x_{8} = 36.1283155197863$$
$$x_{9} = -72.2566310809142$$
$$x_{10} = 64.4026493330202$$
$$x_{11} = 79.3252145031423$$
$$x_{12} = -80.1106125993813$$
$$x_{13} = -65.9734457500487$$
$$x_{14} = 100.53096496827$$
$$x_{15} = -19.6349540849362$$
$$x_{16} = -3.92699081698724$$
$$x_{17} = -69.9004365423729$$
$$x_{18} = 21.9911485850651$$
$$x_{19} = -77.7544181763474$$
$$x_{20} = -81.6814090348242$$
$$x_{21} = 7.8539817047048$$
$$x_{22} = -29.8451301842373$$
$$x_{23} = -91.8915851175014$$
$$x_{24} = -45.5530935337225$$
$$x_{25} = -59.6902604553434$$
$$x_{26} = 91.8915851175014$$
$$x_{27} = -30.6305283725005$$
$$x_{28} = 50.2654824464602$$
$$x_{29} = -37.69911187557$$
$$x_{30} = -84.037603483527$$
$$x_{31} = -55.7632696012188$$
$$x_{32} = -73.8274273285119$$
$$x_{33} = 84.037603483527$$
$$x_{34} = 29.8451302793894$$
$$x_{35} = 6.28318528484945$$
$$x_{36} = 87.9645943337222$$
$$x_{37} = 15.7079633386939$$
$$x_{38} = 3.92699081698724$$
$$x_{39} = 58.1194640856367$$
$$x_{40} = -58.1194640226157$$
$$x_{41} = -15.7079632955062$$
$$x_{42} = 90.3207887907066$$
$$x_{43} = -6.28318531248459$$
$$x_{44} = -43.9822971683559$$
$$x_{45} = 98.174770424681$$
$$x_{46} = -23.5619449626207$$
$$x_{47} = -18.0641577581413$$
$$x_{48} = 76.1836218495525$$
$$x_{49} = 42.411500757839$$
$$x_{50} = -14.1371668705497$$
$$x_{51} = 24.3473430653209$$
$$x_{52} = -37.6991124071589$$
$$x_{53} = -51.8362787555325$$
$$x_{54} = 80.1106126524571$$
$$x_{55} = -40.0553063332699$$
$$x_{56} = -21.9911485849892$$
$$x_{57} = -67.5442421030672$$
$$x_{58} = -133.517687526032$$
$$x_{59} = -89.5353906700869$$
$$x_{60} = 46.3384916404494$$
$$x_{61} = 37.6991119490999$$
$$x_{62} = -50.2654825209008$$
$$x_{63} = 40.0553063332699$$
$$x_{64} = -33.7721210260903$$
$$x_{65} = -25.9181393921158$$
$$x_{66} = 72.256631027738$$
$$x_{67} = 73.827427426781$$
$$x_{68} = -1.57079639015459$$
$$x_{69} = 0$$
$$x_{70} = -7.85398161466544$$
$$x_{71} = 56.5486677534171$$
$$x_{72} = -84.8230018018197$$
$$x_{73} = 14.1371669532624$$
$$x_{74} = 25.9181393921158$$
$$x_{75} = -95.8185759030611$$
$$x_{76} = 51.8362788534439$$
$$x_{77} = -99.7455667514759$$
$$x_{78} = 13.3517687777566$$
$$x_{79} = 62.0464549083984$$
$$x_{80} = 54.1924732744239$$
$$x_{81} = 68.329640215578$$
$$x_{82} = 28.2743338655019$$
$$x_{83} = 65.9734457517426$$
$$x_{84} = 18.0641577581413$$
$$x_{85} = -29.8451302277112$$
$$x_{86} = -28.2743339383473$$
$$x_{87} = 59.6902604133739$$
$$x_{88} = 2.35619449019234$$
$$x_{89} = -11.7809724509617$$
$$x_{90} = -62.0464549083984$$
$$x_{91} = 69.9004365423729$$
$$x_{92} = -87.9645943300798$$
$$x_{93} = -63.6172512351933$$
$$x_{94} = 86.3937979088331$$
$$x_{95} = 95.8185759992737$$
$$x_{96} = 20.4203521833985$$
$$x_{97} = 47.9092879672443$$
$$x_{98} = 34.5575192485904$$
$$x_{99} = 78.5398163564575$$
$$x_{100} = -47.9092879672443$$
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:
$$x_{1} = 43.9822971689083$$
$$x_{2} = -94.2477796324164$$
$$x_{3} = -36.1283154463325$$
$$x_{4} = 94.2477796093316$$
$$x_{5} = 36.1283155197863$$
$$x_{6} = -72.2566310809142$$
$$x_{7} = 64.4026493330202$$
$$x_{8} = -80.1106125993813$$
$$x_{9} = -65.9734457500487$$
$$x_{10} = 100.53096496827$$
$$x_{11} = 21.9911485850651$$
$$x_{12} = -81.6814090348242$$
$$x_{13} = 7.8539817047048$$
$$x_{14} = -29.8451301842373$$
$$x_{15} = -45.5530935337225$$
$$x_{16} = -59.6902604553434$$
$$x_{17} = 50.2654824464602$$
$$x_{18} = -37.69911187557$$
$$x_{19} = -73.8274273285119$$
$$x_{20} = 29.8451302793894$$
$$x_{21} = 6.28318528484945$$
$$x_{22} = 87.9645943337222$$
$$x_{23} = 15.7079633386939$$
$$x_{24} = 58.1194640856367$$
$$x_{25} = -58.1194640226157$$
$$x_{26} = -15.7079632955062$$
$$x_{27} = -6.28318531248459$$
$$x_{28} = -43.9822971683559$$
$$x_{29} = -23.5619449626207$$
$$x_{30} = 42.411500757839$$
$$x_{31} = -14.1371668705497$$
$$x_{32} = -51.8362787555325$$
$$x_{33} = 80.1106126524571$$
$$x_{34} = -21.9911485849892$$
$$x_{35} = -67.5442421030672$$
$$x_{36} = -133.517687526032$$
$$x_{37} = -89.5353906700869$$
$$x_{38} = 37.6991119490999$$
$$x_{39} = -50.2654825209008$$
$$x_{40} = 72.256631027738$$
$$x_{41} = 73.827427426781$$
$$x_{42} = -1.57079639015459$$
$$x_{43} = 0$$
$$x_{44} = -7.85398161466544$$
$$x_{45} = 56.5486677534171$$
$$x_{46} = -84.8230018018197$$
$$x_{47} = 14.1371669532624$$
$$x_{48} = -95.8185759030611$$
$$x_{49} = 51.8362788534439$$
$$x_{50} = 28.2743338655019$$
$$x_{51} = 65.9734457517426$$
$$x_{52} = -29.8451302277112$$
$$x_{53} = -28.2743339383473$$
$$x_{54} = 59.6902604133739$$
$$x_{55} = -87.9645943300798$$
$$x_{56} = 86.3937979088331$$
$$x_{57} = 95.8185759992737$$
$$x_{58} = 20.4203521833985$$
$$x_{59} = 34.5575192485904$$
$$x_{60} = 78.5398163564575$$
Puntos máximos de la función:
$$x_{60} = 32.2013246992954$$
$$x_{60} = 10.2101761241668$$
$$x_{60} = -41.6261026600648$$
$$x_{60} = 79.3252145031423$$
$$x_{60} = -19.6349540849362$$
$$x_{60} = -3.92699081698724$$
$$x_{60} = -69.9004365423729$$
$$x_{60} = -77.7544181763474$$
$$x_{60} = -91.8915851175014$$
$$x_{60} = 91.8915851175014$$
$$x_{60} = -30.6305283725005$$
$$x_{60} = -84.037603483527$$
$$x_{60} = -55.7632696012188$$
$$x_{60} = 84.037603483527$$
$$x_{60} = 3.92699081698724$$
$$x_{60} = 90.3207887907066$$
$$x_{60} = 98.174770424681$$
$$x_{60} = -18.0641577581413$$
$$x_{60} = 76.1836218495525$$
$$x_{60} = 24.3473430653209$$
$$x_{60} = -40.0553063332699$$
$$x_{60} = 46.3384916404494$$
$$x_{60} = 40.0553063332699$$
$$x_{60} = -33.7721210260903$$
$$x_{60} = -25.9181393921158$$
$$x_{60} = 25.9181393921158$$
$$x_{60} = -99.7455667514759$$
$$x_{60} = 13.3517687777566$$
$$x_{60} = 62.0464549083984$$
$$x_{60} = 54.1924732744239$$
$$x_{60} = 68.329640215578$$
$$x_{60} = 18.0641577581413$$
$$x_{60} = 2.35619449019234$$
$$x_{60} = -11.7809724509617$$
$$x_{60} = -62.0464549083984$$
$$x_{60} = 69.9004365423729$$
$$x_{60} = -63.6172512351933$$
$$x_{60} = 47.9092879672443$$
$$x_{60} = -47.9092879672443$$
Decrece en los intervalos
$$\left[100.53096496827, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -133.517687526032\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} \cos{\left(x \right)} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 43.9822971689083$$
$$x_{2} = -94.2477796324164$$
$$x_{3} = 32.2013246992954$$
$$x_{4} = 10.2101761241668$$
$$x_{5} = -41.6261026600648$$
$$x_{6} = -36.1283154463325$$
$$x_{7} = 94.2477796093316$$
$$x_{8} = 36.1283155197863$$
$$x_{9} = -72.2566310809142$$
$$x_{10} = 64.4026493330202$$
$$x_{11} = 79.3252145031423$$
$$x_{12} = -80.1106125993813$$
$$x_{13} = -65.9734457500487$$
$$x_{14} = 100.53096496827$$
$$x_{15} = -19.6349540849362$$
$$x_{16} = -3.92699081698724$$
$$x_{17} = -69.9004365423729$$
$$x_{18} = 21.9911485850651$$
$$x_{19} = -77.7544181763474$$
$$x_{20} = -81.6814090348242$$
$$x_{21} = 7.8539817047048$$
$$x_{22} = -29.8451301842373$$
$$x_{23} = -91.8915851175014$$
$$x_{24} = -45.5530935337225$$
$$x_{25} = -59.6902604553434$$
$$x_{26} = 91.8915851175014$$
$$x_{27} = -30.6305283725005$$
$$x_{28} = 50.2654824464602$$
$$x_{29} = -37.69911187557$$
$$x_{30} = -84.037603483527$$
$$x_{31} = -55.7632696012188$$
$$x_{32} = -73.8274273285119$$
$$x_{33} = 84.037603483527$$
$$x_{34} = 29.8451302793894$$
$$x_{35} = 6.28318528484945$$
$$x_{36} = 87.9645943337222$$
$$x_{37} = 15.7079633386939$$
$$x_{38} = 3.92699081698724$$
$$x_{39} = 58.1194640856367$$
$$x_{40} = -58.1194640226157$$
$$x_{41} = -15.7079632955062$$
$$x_{42} = 90.3207887907066$$
$$x_{43} = -6.28318531248459$$
$$x_{44} = -43.9822971683559$$
$$x_{45} = 98.174770424681$$
$$x_{46} = -23.5619449626207$$
$$x_{47} = -18.0641577581413$$
$$x_{48} = 76.1836218495525$$
$$x_{49} = 42.411500757839$$
$$x_{50} = -14.1371668705497$$
$$x_{51} = 24.3473430653209$$
$$x_{52} = -37.6991124071589$$
$$x_{53} = -51.8362787555325$$
$$x_{54} = 80.1106126524571$$
$$x_{55} = -40.0553063332699$$
$$x_{56} = -21.9911485849892$$
$$x_{57} = -67.5442421030672$$
$$x_{58} = -133.517687526032$$
$$x_{59} = -89.5353906700869$$
$$x_{60} = 46.3384916404494$$
$$x_{61} = 37.6991119490999$$
$$x_{62} = -50.2654825209008$$
$$x_{63} = 40.0553063332699$$
$$x_{64} = -33.7721210260903$$
$$x_{65} = -25.9181393921158$$
$$x_{66} = 72.256631027738$$
$$x_{67} = 73.827427426781$$
$$x_{68} = -1.57079639015459$$
$$x_{69} = 0$$
$$x_{70} = -7.85398161466544$$
$$x_{71} = 56.5486677534171$$
$$x_{72} = -84.8230018018197$$
$$x_{73} = 14.1371669532624$$
$$x_{74} = 25.9181393921158$$
$$x_{75} = -95.8185759030611$$
$$x_{76} = 51.8362788534439$$
$$x_{77} = -99.7455667514759$$
$$x_{78} = 13.3517687777566$$
$$x_{79} = 62.0464549083984$$
$$x_{80} = 54.1924732744239$$
$$x_{81} = 68.329640215578$$
$$x_{82} = 28.2743338655019$$
$$x_{83} = 65.9734457517426$$
$$x_{84} = 18.0641577581413$$
$$x_{85} = -29.8451302277112$$
$$x_{86} = -28.2743339383473$$
$$x_{87} = 59.6902604133739$$
$$x_{88} = 2.35619449019234$$
$$x_{89} = -11.7809724509617$$
$$x_{90} = -62.0464549083984$$
$$x_{91} = 69.9004365423729$$
$$x_{92} = -87.9645943300798$$
$$x_{93} = -63.6172512351933$$
$$x_{94} = 86.3937979088331$$
$$x_{95} = 95.8185759992737$$
$$x_{96} = 20.4203521833985$$
$$x_{97} = 47.9092879672443$$
$$x_{98} = 34.5575192485904$$
$$x_{99} = 78.5398163564575$$
$$x_{100} = -47.9092879672443$$
Signos de extremos en los puntos:
(43.98229716890828, 6.48811751416143e-24)
(-94.24777963241641, 1.51106491072184e-23)
(32.201324699295384, 0.125)
(10.210176124166829, 0.125)
(-41.62610266006476, 0.125)
(-36.12831544633253, 3.4226689515248e-22)
(94.24777960933163, 4.39348087029501e-27)
(36.128315519786256, 4.30086627024243e-26)
(-72.25663108091422, 1.13021684441833e-22)
(64.40264933302024, 2.81920071935721e-22)
(79.32521450314228, 0.125)
(-80.11061259938134, 3.02901042262663e-22)
(-65.97344575004867, 1.50016233681479e-23)
(100.53096496827025, 1.522465342001e-22)
(-19.634954084936208, 0.125)
(-3.9269908169872414, 0.125)
(-69.9004365423729, 0.125)
(21.99114858506508, 9.81078764488066e-25)
(-77.75441817634739, 0.125)
(-81.68140903482424, 7.1419744348026e-23)
(7.853981704704805, 3.53848126740524e-22)
(-29.8451301842373, 1.53746070030667e-23)
(-91.89158511750145, 0.125)
(-45.5530935337225, 1.8199990461979e-22)
(-59.69026045534336, 5.1218952261819e-23)
(91.89158511750145, 0.125)
(-30.630528372500486, 0.125)
(50.26548244646023, 1.32247290487336e-24)
(-37.69911187557003, 3.43044088063799e-23)
(-84.03760348352696, 0.125)
(-55.76326960121883, 0.125)
(-73.82742732851193, 2.93555298760904e-23)
(84.03760348352696, 0.125)
(29.845130279389437, 3.47227348186348e-22)
(6.283185284849447, 1.11345924036983e-23)
(87.96459433372223, 3.66208987524238e-23)
(15.70796333869387, 3.5406703384702e-22)
(3.9269908169872414, 0.125)
(58.11946408563672, 1.9254517670331e-25)
(-58.119464022615695, 3.25596491118918e-22)
(-15.70796329550622, 2.09270398257639e-23)
(90.32078879070656, 0.125)
(-6.2831853124845924, 1.49299251428044e-25)
(-43.98229716835591, 5.92856939408264e-24)
(98.17477042468104, 0.125)
(-23.561944962620714, 2.23618312697362e-22)
(-18.06415775814131, 0.125)
(76.18362184955248, 0.125)
(42.41150075783897, 2.82600507338299e-22)
(-14.137166870549688, 3.51961337875311e-22)
(24.3473430653209, 0.125)
(-37.69911240715892, 1.794838364877e-19)
(-51.83627875553255, 2.36375348300805e-23)
(80.11061265245715, 2.79284296658744e-24)
(-40.05530633326986, 0.125)
(-21.991148584989208, 9.58776431871893e-25)
(-67.54424210306725, 1.31768829189417e-22)
(-133.51768752603243, 1.59143517847928e-20)
(-89.53539067008688, 7.82806832480277e-23)
(46.33849164044945, 0.125)
(37.699111949099915, 1.19177110082757e-21)
(-50.26548252090079, 2.55613820489159e-22)
(40.05530633326986, 0.125)
(-33.772121026090275, 0.125)
(-25.918139392115794, 0.125)
(72.256631027738, 1.12486161483226e-25)
(73.827427426781, 3.06466382975157e-22)
(-1.5707963901545945, 2.54354422397756e-22)
(0, 0)
(-7.853981614665438, 7.19916962931526e-24)
(56.54866775341709, 1.40462170038278e-24)
(-84.82300180181973, 3.71633463879623e-21)
(14.137166953262355, 1.7752026792127e-24)
(25.918139392115794, 0.125)
(-95.81857590306113, 3.10407435051095e-23)
(51.83627885344392, 3.31551100186431e-22)
(-99.74556675147593, 0.125)
(13.351768777756622, 0.125)
(62.04645490839842, 0.125)
(54.19247327442393, 0.125)
(68.329640215578, 0.125)
(28.274333865501934, 4.74688828769133e-24)
(65.9734457517426, 1.8309866387567e-23)
(18.06415775814131, 0.125)
(-29.845130227711206, 6.44333903921372e-24)
(-28.274333938347255, 1.75984254372543e-22)
(59.69026041337386, 1.12833193434337e-25)
(2.356194490192345, 0.125)
(-11.780972450961725, 0.125)
(-62.04645490839842, 0.125)
(69.9004365423729, 0.125)
(-87.96459433007985, 2.58441188123515e-23)
(-63.617251235193315, 0.125)
(86.39379790883312, 2.73185010774034e-22)
(95.81857599927372, 2.7190919575737e-22)
(20.420352183398542, 2.73803391707347e-22)
(47.909287967244346, 0.125)
(34.55751924859039, 2.06452973686466e-22)
(78.53981635645745, 4.66802919974114e-24)
(-47.909287967244346, 0.125)
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:
$$x_{1} = 43.9822971689083$$
$$x_{2} = -94.2477796324164$$
$$x_{3} = -36.1283154463325$$
$$x_{4} = 94.2477796093316$$
$$x_{5} = 36.1283155197863$$
$$x_{6} = -72.2566310809142$$
$$x_{7} = 64.4026493330202$$
$$x_{8} = -80.1106125993813$$
$$x_{9} = -65.9734457500487$$
$$x_{10} = 100.53096496827$$
$$x_{11} = 21.9911485850651$$
$$x_{12} = -81.6814090348242$$
$$x_{13} = 7.8539817047048$$
$$x_{14} = -29.8451301842373$$
$$x_{15} = -45.5530935337225$$
$$x_{16} = -59.6902604553434$$
$$x_{17} = 50.2654824464602$$
$$x_{18} = -37.69911187557$$
$$x_{19} = -73.8274273285119$$
$$x_{20} = 29.8451302793894$$
$$x_{21} = 6.28318528484945$$
$$x_{22} = 87.9645943337222$$
$$x_{23} = 15.7079633386939$$
$$x_{24} = 58.1194640856367$$
$$x_{25} = -58.1194640226157$$
$$x_{26} = -15.7079632955062$$
$$x_{27} = -6.28318531248459$$
$$x_{28} = -43.9822971683559$$
$$x_{29} = -23.5619449626207$$
$$x_{30} = 42.411500757839$$
$$x_{31} = -14.1371668705497$$
$$x_{32} = -51.8362787555325$$
$$x_{33} = 80.1106126524571$$
$$x_{34} = -21.9911485849892$$
$$x_{35} = -67.5442421030672$$
$$x_{36} = -133.517687526032$$
$$x_{37} = -89.5353906700869$$
$$x_{38} = 37.6991119490999$$
$$x_{39} = -50.2654825209008$$
$$x_{40} = 72.256631027738$$
$$x_{41} = 73.827427426781$$
$$x_{42} = -1.57079639015459$$
$$x_{43} = 0$$
$$x_{44} = -7.85398161466544$$
$$x_{45} = 56.5486677534171$$
$$x_{46} = -84.8230018018197$$
$$x_{47} = 14.1371669532624$$
$$x_{48} = -95.8185759030611$$
$$x_{49} = 51.8362788534439$$
$$x_{50} = 28.2743338655019$$
$$x_{51} = 65.9734457517426$$
$$x_{52} = -29.8451302277112$$
$$x_{53} = -28.2743339383473$$
$$x_{54} = 59.6902604133739$$
$$x_{55} = -87.9645943300798$$
$$x_{56} = 86.3937979088331$$
$$x_{57} = 95.8185759992737$$
$$x_{58} = 20.4203521833985$$
$$x_{59} = 34.5575192485904$$
$$x_{60} = 78.5398163564575$$
Puntos máximos de la función:
$$x_{60} = 32.2013246992954$$
$$x_{60} = 10.2101761241668$$
$$x_{60} = -41.6261026600648$$
$$x_{60} = 79.3252145031423$$
$$x_{60} = -19.6349540849362$$
$$x_{60} = -3.92699081698724$$
$$x_{60} = -69.9004365423729$$
$$x_{60} = -77.7544181763474$$
$$x_{60} = -91.8915851175014$$
$$x_{60} = 91.8915851175014$$
$$x_{60} = -30.6305283725005$$
$$x_{60} = -84.037603483527$$
$$x_{60} = -55.7632696012188$$
$$x_{60} = 84.037603483527$$
$$x_{60} = 3.92699081698724$$
$$x_{60} = 90.3207887907066$$
$$x_{60} = 98.174770424681$$
$$x_{60} = -18.0641577581413$$
$$x_{60} = 76.1836218495525$$
$$x_{60} = 24.3473430653209$$
$$x_{60} = -40.0553063332699$$
$$x_{60} = 46.3384916404494$$
$$x_{60} = 40.0553063332699$$
$$x_{60} = -33.7721210260903$$
$$x_{60} = -25.9181393921158$$
$$x_{60} = 25.9181393921158$$
$$x_{60} = -99.7455667514759$$
$$x_{60} = 13.3517687777566$$
$$x_{60} = 62.0464549083984$$
$$x_{60} = 54.1924732744239$$
$$x_{60} = 68.329640215578$$
$$x_{60} = 18.0641577581413$$
$$x_{60} = 2.35619449019234$$
$$x_{60} = -11.7809724509617$$
$$x_{60} = -62.0464549083984$$
$$x_{60} = 69.9004365423729$$
$$x_{60} = -63.6172512351933$$
$$x_{60} = 47.9092879672443$$
$$x_{60} = -47.9092879672443$$
Decrece en los intervalos
$$\left[100.53096496827, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -133.517687526032\right]$$
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
$$6 \left(\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)} \delta\left(\sin{\left(x \right)} \cos{\left(x \right)}\right) + \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)^{2} \operatorname{sign}{\left(\sin{\left(x \right)} \cos{\left(x \right)} \right)} - 2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \cos{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} = 0$$
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
$$6 \left(\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)} \delta\left(\sin{\left(x \right)} \cos{\left(x \right)}\right) + \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)^{2} \operatorname{sign}{\left(\sin{\left(x \right)} \cos{\left(x \right)} \right)} - 2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \cos{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} = 0$$
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
$$\lim_{x \to -\infty} \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = \left\langle 0, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle 0, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty} \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = \left\langle 0, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle 0, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to -\infty} \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = \left\langle 0, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle 0, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty} \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = \left\langle 0, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle 0, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función Abs(cos(x)*sin(x))^3, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|}{x}\right) = 0$$
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:
$$\left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3}$$
- Sí
$$\left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = - \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3}$$
- Sí
$$\left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3} = - \left|{\sin{\left(x \right)} \cos{\left(x \right)}}\right|^{3}$$
- No
es decir, función
es
par