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)
-
(e^x)*sin(x)
-
4*x*log(x)/(1-log(x^2))
-
(4x^3)/((x^3)-1)
-
Expresiones idénticas
- sin(x)^ dos + dos *x*cos(x)*sin(x)
- seno de (x) al cuadrado más 2 multiplicar por x multiplicar por coseno de (x) multiplicar por seno de (x)
- seno de (x) en el grado dos más dos multiplicar por x multiplicar por coseno de (x) multiplicar por seno de (x)
- sin(x)2+2*x*cos(x)*sin(x)
- sinx2+2*x*cosx*sinx
- sin(x)²+2*x*cos(x)*sin(x)
- sin(x) en el grado 2+2*x*cos(x)*sin(x)
- sin(x)^2+2xcos(x)sin(x)
- sin(x)2+2xcos(x)sin(x)
- sinx2+2xcosxsinx
- sinx^2+2xcosxsinx
-
Expresiones semejantes
- sin(x)^2-2*x*cos(x)*sin(x)
- sinx^2+2*x*cosx*sinx
-
Expresiones con funciones
- Seno sin
- sin(x+1)*sin(x)
- sin(cos(4*x)^3)^2
- sin(x)+(sin(x))^2
- sin(x)-1+cos(x)
- sin(4*a)-cos(4*a)+cos(2*a)
- Seno sin
- sin(x+1)*sin(x)
- sin(cos(4*x)^3)^2
- sin(x)+(sin(x))^2
- sin(x)-1+cos(x)
- sin(4*a)-cos(4*a)+cos(2*a)
Gráfico de la función y = sin(x)^2+2*x*cos(x)*sin(x)
Solución
Ha introducido
[src]
2 f(x) = sin (x) + 2*x*cos(x)*sin(x)
$$f{\left(x \right)} = 2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)}$$
f = ((2*x)*cos(x))*sin(x) + sin(x)^2
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:
$$2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -31.4159265358979$$
$$x_{2} = -73.8341991854591$$
$$x_{3} = 80.1168534696549$$
$$x_{4} = 278.032748190065$$
$$x_{5} = -58.1280655761511$$
$$x_{6} = 45.5640665961997$$
$$x_{7} = -59.6902604182061$$
$$x_{8} = 21.9911485751286$$
$$x_{9} = 43.9822971502571$$
$$x_{10} = 15.707963267949$$
$$x_{11} = 36.1421488970061$$
$$x_{12} = -64.410411962776$$
$$x_{13} = 51.8459224452234$$
$$x_{14} = -53.4070751110265$$
$$x_{15} = 92.682377997352$$
$$x_{16} = -6.28318530717959$$
$$x_{17} = 78.5398163397448$$
$$x_{18} = 29.861872403816$$
$$x_{19} = 58.1280655761511$$
$$x_{20} = 1.83659720315213$$
$$x_{21} = 23.5831433102848$$
$$x_{22} = 86.3995849739529$$
$$x_{23} = 94.2477796076938$$
$$x_{24} = 12.5663706143592$$
$$x_{25} = 81.6814089933346$$
$$x_{26} = -72.2566310325652$$
$$x_{27} = -89.5409746049841$$
$$x_{28} = -1.83659720315213$$
$$x_{29} = 7.91705268466621$$
$$x_{30} = -67.5516436614121$$
$$x_{31} = 48.7049516666752$$
$$x_{32} = 0$$
$$x_{33} = -50.2654824574367$$
$$x_{34} = -81.6814089933346$$
$$x_{35} = -14.1724320747999$$
$$x_{36} = -87.9645943005142$$
$$x_{37} = 6.28318530717959$$
$$x_{38} = -17.3076405374146$$
$$x_{39} = -80.1168534696549$$
$$x_{40} = 20.4448034666183$$
$$x_{41} = -45.5640665961997$$
$$x_{42} = 3.14159265358979$$
$$x_{43} = -37.6991118430775$$
$$x_{44} = 34.5575191894877$$
$$x_{45} = -39.2826357527234$$
$$x_{46} = 65.9734457253857$$
$$x_{47} = -28.2743338823081$$
$$x_{48} = 73.8341991854591$$
$$x_{49} = -51.8459224452234$$
$$x_{50} = -61.2692172687226$$
$$x_{51} = -42.4232862577008$$
$$x_{52} = 72.2566310325652$$
$$x_{53} = -86.3995849739529$$
$$x_{54} = -43.9822971502571$$
$$x_{55} = -83.2582106616487$$
$$x_{56} = -36.1421488970061$$
$$x_{57} = -95.8237937978449$$
$$x_{58} = 70.692907433161$$
$$x_{59} = -9.42477796076938$$
$$x_{60} = 42.4232862577008$$
$$x_{61} = -75.398223686155$$
$$x_{62} = 64.410411962776$$
$$x_{63} = -105.248104538899$$
$$x_{64} = -97.3893722612836$$
$$x_{65} = 26.7222463741877$$
$$x_{66} = 87.9645943005142$$
$$x_{67} = 59.6902604182061$$
$$x_{68} = -29.861872403816$$
$$x_{69} = -20.4448034666183$$
$$x_{70} = 100.530964914873$$
$$x_{71} = 28.2743338823081$$
$$x_{72} = -7.91705268466621$$
$$x_{73} = 25.1327412287183$$
$$x_{74} = -94.2477796076938$$
$$x_{75} = 89.5409746049841$$
$$x_{76} = 67.5516436614121$$
$$x_{77} = -21.9911485751286$$
$$x_{78} = 14.1724320747999$$
$$x_{79} = 56.5486677646163$$
$$x_{80} = -65.9734457253857$$
$$x_{81} = -15.707963267949$$
$$x_{82} = -23.5831433102848$$
$$x_{83} = -84.8230016469244$$
$$x_{84} = -4.81584231784594$$
$$x_{85} = -306.306916073247$$
$$x_{86} = 50.2654824574367$$
$$x_{87} = 37.6991118430775$$
$$x_{88} = 95.8237937978449$$
o sea hay que resolver la ecuación:
$$2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -31.4159265358979$$
$$x_{2} = -73.8341991854591$$
$$x_{3} = 80.1168534696549$$
$$x_{4} = 278.032748190065$$
$$x_{5} = -58.1280655761511$$
$$x_{6} = 45.5640665961997$$
$$x_{7} = -59.6902604182061$$
$$x_{8} = 21.9911485751286$$
$$x_{9} = 43.9822971502571$$
$$x_{10} = 15.707963267949$$
$$x_{11} = 36.1421488970061$$
$$x_{12} = -64.410411962776$$
$$x_{13} = 51.8459224452234$$
$$x_{14} = -53.4070751110265$$
$$x_{15} = 92.682377997352$$
$$x_{16} = -6.28318530717959$$
$$x_{17} = 78.5398163397448$$
$$x_{18} = 29.861872403816$$
$$x_{19} = 58.1280655761511$$
$$x_{20} = 1.83659720315213$$
$$x_{21} = 23.5831433102848$$
$$x_{22} = 86.3995849739529$$
$$x_{23} = 94.2477796076938$$
$$x_{24} = 12.5663706143592$$
$$x_{25} = 81.6814089933346$$
$$x_{26} = -72.2566310325652$$
$$x_{27} = -89.5409746049841$$
$$x_{28} = -1.83659720315213$$
$$x_{29} = 7.91705268466621$$
$$x_{30} = -67.5516436614121$$
$$x_{31} = 48.7049516666752$$
$$x_{32} = 0$$
$$x_{33} = -50.2654824574367$$
$$x_{34} = -81.6814089933346$$
$$x_{35} = -14.1724320747999$$
$$x_{36} = -87.9645943005142$$
$$x_{37} = 6.28318530717959$$
$$x_{38} = -17.3076405374146$$
$$x_{39} = -80.1168534696549$$
$$x_{40} = 20.4448034666183$$
$$x_{41} = -45.5640665961997$$
$$x_{42} = 3.14159265358979$$
$$x_{43} = -37.6991118430775$$
$$x_{44} = 34.5575191894877$$
$$x_{45} = -39.2826357527234$$
$$x_{46} = 65.9734457253857$$
$$x_{47} = -28.2743338823081$$
$$x_{48} = 73.8341991854591$$
$$x_{49} = -51.8459224452234$$
$$x_{50} = -61.2692172687226$$
$$x_{51} = -42.4232862577008$$
$$x_{52} = 72.2566310325652$$
$$x_{53} = -86.3995849739529$$
$$x_{54} = -43.9822971502571$$
$$x_{55} = -83.2582106616487$$
$$x_{56} = -36.1421488970061$$
$$x_{57} = -95.8237937978449$$
$$x_{58} = 70.692907433161$$
$$x_{59} = -9.42477796076938$$
$$x_{60} = 42.4232862577008$$
$$x_{61} = -75.398223686155$$
$$x_{62} = 64.410411962776$$
$$x_{63} = -105.248104538899$$
$$x_{64} = -97.3893722612836$$
$$x_{65} = 26.7222463741877$$
$$x_{66} = 87.9645943005142$$
$$x_{67} = 59.6902604182061$$
$$x_{68} = -29.861872403816$$
$$x_{69} = -20.4448034666183$$
$$x_{70} = 100.530964914873$$
$$x_{71} = 28.2743338823081$$
$$x_{72} = -7.91705268466621$$
$$x_{73} = 25.1327412287183$$
$$x_{74} = -94.2477796076938$$
$$x_{75} = 89.5409746049841$$
$$x_{76} = 67.5516436614121$$
$$x_{77} = -21.9911485751286$$
$$x_{78} = 14.1724320747999$$
$$x_{79} = 56.5486677646163$$
$$x_{80} = -65.9734457253857$$
$$x_{81} = -15.707963267949$$
$$x_{82} = -23.5831433102848$$
$$x_{83} = -84.8230016469244$$
$$x_{84} = -4.81584231784594$$
$$x_{85} = -306.306916073247$$
$$x_{86} = 50.2654824574367$$
$$x_{87} = 37.6991118430775$$
$$x_{88} = 95.8237937978449$$
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
$$2 x \cos^{2}{\left(x \right)} + \left(- 2 x \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) \sin{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -54.2016970313842$$
$$x_{2} = 99.7505790857949$$
$$x_{3} = 32.2168395518658$$
$$x_{4} = 19.6603640661261$$
$$x_{5} = 84.0435524991391$$
$$x_{6} = 5.58635293416499$$
$$x_{7} = 54.2016970313842$$
$$x_{8} = -32.2168395518658$$
$$x_{9} = -79.3315168346756$$
$$x_{10} = 62.0545116429054$$
$$x_{11} = 76.1901839979235$$
$$x_{12} = 77.760847792972$$
$$x_{13} = 98.1798629425939$$
$$x_{14} = -55.7722336752062$$
$$x_{15} = -40.0677825970372$$
$$x_{16} = -21.2292853858495$$
$$x_{17} = 71.4782275499213$$
$$x_{18} = -84.0435524991391$$
$$x_{19} = -68.3369563786298$$
$$x_{20} = -5.58635293416499$$
$$x_{21} = -4.04808180161146$$
$$x_{22} = -35.3570550332742$$
$$x_{23} = 49.4901859325761$$
$$x_{24} = -90.3263240494369$$
$$x_{25} = -93.4677306800165$$
$$x_{26} = -76.1901839979235$$
$$x_{27} = -49.4901859325761$$
$$x_{28} = 82.4728694594266$$
$$x_{29} = 8.69662198229738$$
$$x_{30} = 85.6142396947314$$
$$x_{31} = -85.6142396947314$$
$$x_{32} = -41.6381085824888$$
$$x_{33} = 90.3263240494369$$
$$x_{34} = -82.4728694594266$$
$$x_{35} = 33.7869153354295$$
$$x_{36} = -33.7869153354295$$
$$x_{37} = 25.9374070267134$$
$$x_{38} = -19.6603640661261$$
$$x_{39} = -60.4839244878466$$
$$x_{40} = -57.3427845371101$$
$$x_{41} = 0$$
$$x_{42} = 74.6195257807054$$
$$x_{43} = -13.3890435377793$$
$$x_{44} = -18.0917665453763$$
$$x_{45} = 16.5235843473527$$
$$x_{46} = 11.8231619098018$$
$$x_{47} = 4.04808180161146$$
$$x_{48} = 88.7556256712795$$
$$x_{49} = 96.6091494063022$$
$$x_{50} = 60.4839244878466$$
$$x_{51} = 2.54349254705114$$
$$x_{52} = 27.5071048394191$$
$$x_{53} = 40.0677825970372$$
$$x_{54} = -63.6251091208926$$
$$x_{55} = -71.4782275499213$$
$$x_{56} = 41.6381085824888$$
$$x_{57} = -46.3492776216985$$
$$x_{58} = -24.3678503974527$$
$$x_{59} = 10.2587614549708$$
$$x_{60} = 18.0917665453763$$
$$x_{61} = 47.9197205706165$$
$$x_{62} = 68.3369563786298$$
$$x_{63} = -25.9374070267134$$
$$x_{64} = -98.1798629425939$$
$$x_{65} = 120.170079673253$$
$$x_{66} = -69.9075883539626$$
$$x_{67} = -99.7505790857949$$
$$x_{68} = -58.9133484807877$$
$$x_{69} = -27.5071048394191$$
$$x_{70} = 30.6468374831214$$
$$x_{71} = -10.2587614549708$$
$$x_{72} = 46.3492776216985$$
$$x_{73} = 63.6251091208926$$
$$x_{74} = 24.3678503974527$$
$$x_{75} = -11.8231619098018$$
$$x_{76} = -77.760847792972$$
$$x_{77} = 55.7722336752062$$
$$x_{78} = 52.6311758774383$$
$$x_{79} = -91.8970257752571$$
$$x_{80} = -38.4974949445838$$
$$x_{81} = 91.8970257752571$$
$$x_{82} = 38.4974949445838$$
$$x_{83} = -66.766332133246$$
$$x_{84} = -1.1444648640517$$
$$x_{85} = 66.766332133246$$
$$x_{86} = -47.9197205706165$$
$$x_{87} = -62.0545116429054$$
$$x_{88} = 69.9075883539626$$
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:
$$x_{1} = 99.7505790857949$$
$$x_{2} = 84.0435524991391$$
$$x_{3} = 5.58635293416499$$
$$x_{4} = 62.0545116429054$$
$$x_{5} = 77.760847792972$$
$$x_{6} = -55.7722336752062$$
$$x_{7} = -40.0677825970372$$
$$x_{8} = -21.2292853858495$$
$$x_{9} = 71.4782275499213$$
$$x_{10} = -84.0435524991391$$
$$x_{11} = -68.3369563786298$$
$$x_{12} = -5.58635293416499$$
$$x_{13} = 49.4901859325761$$
$$x_{14} = -90.3263240494369$$
$$x_{15} = -93.4677306800165$$
$$x_{16} = -49.4901859325761$$
$$x_{17} = 8.69662198229738$$
$$x_{18} = 90.3263240494369$$
$$x_{19} = 33.7869153354295$$
$$x_{20} = -33.7869153354295$$
$$x_{21} = 0$$
$$x_{22} = 74.6195257807054$$
$$x_{23} = -18.0917665453763$$
$$x_{24} = 11.8231619098018$$
$$x_{25} = 96.6091494063022$$
$$x_{26} = 2.54349254705114$$
$$x_{27} = 27.5071048394191$$
$$x_{28} = 40.0677825970372$$
$$x_{29} = -71.4782275499213$$
$$x_{30} = -46.3492776216985$$
$$x_{31} = -24.3678503974527$$
$$x_{32} = 18.0917665453763$$
$$x_{33} = 68.3369563786298$$
$$x_{34} = -99.7505790857949$$
$$x_{35} = -58.9133484807877$$
$$x_{36} = -27.5071048394191$$
$$x_{37} = 30.6468374831214$$
$$x_{38} = 46.3492776216985$$
$$x_{39} = 24.3678503974527$$
$$x_{40} = -11.8231619098018$$
$$x_{41} = -77.760847792972$$
$$x_{42} = 55.7722336752062$$
$$x_{43} = 52.6311758774383$$
$$x_{44} = -62.0545116429054$$
Puntos máximos de la función:
$$x_{44} = -54.2016970313842$$
$$x_{44} = 32.2168395518658$$
$$x_{44} = 19.6603640661261$$
$$x_{44} = 54.2016970313842$$
$$x_{44} = -32.2168395518658$$
$$x_{44} = -79.3315168346756$$
$$x_{44} = 76.1901839979235$$
$$x_{44} = 98.1798629425939$$
$$x_{44} = -4.04808180161146$$
$$x_{44} = -35.3570550332742$$
$$x_{44} = -76.1901839979235$$
$$x_{44} = 82.4728694594266$$
$$x_{44} = 85.6142396947314$$
$$x_{44} = -85.6142396947314$$
$$x_{44} = -41.6381085824888$$
$$x_{44} = -82.4728694594266$$
$$x_{44} = 25.9374070267134$$
$$x_{44} = -19.6603640661261$$
$$x_{44} = -60.4839244878466$$
$$x_{44} = -57.3427845371101$$
$$x_{44} = -13.3890435377793$$
$$x_{44} = 16.5235843473527$$
$$x_{44} = 4.04808180161146$$
$$x_{44} = 88.7556256712795$$
$$x_{44} = 60.4839244878466$$
$$x_{44} = -63.6251091208926$$
$$x_{44} = 41.6381085824888$$
$$x_{44} = 10.2587614549708$$
$$x_{44} = 47.9197205706165$$
$$x_{44} = -25.9374070267134$$
$$x_{44} = -98.1798629425939$$
$$x_{44} = 120.170079673253$$
$$x_{44} = -69.9075883539626$$
$$x_{44} = -10.2587614549708$$
$$x_{44} = 63.6251091208926$$
$$x_{44} = -91.8970257752571$$
$$x_{44} = -38.4974949445838$$
$$x_{44} = 91.8970257752571$$
$$x_{44} = 38.4974949445838$$
$$x_{44} = -66.766332133246$$
$$x_{44} = -1.1444648640517$$
$$x_{44} = 66.766332133246$$
$$x_{44} = -47.9197205706165$$
$$x_{44} = 69.9075883539626$$
Decrece en los intervalos
$$\left[99.7505790857949, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7505790857949\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$2 x \cos^{2}{\left(x \right)} + \left(- 2 x \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) \sin{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -54.2016970313842$$
$$x_{2} = 99.7505790857949$$
$$x_{3} = 32.2168395518658$$
$$x_{4} = 19.6603640661261$$
$$x_{5} = 84.0435524991391$$
$$x_{6} = 5.58635293416499$$
$$x_{7} = 54.2016970313842$$
$$x_{8} = -32.2168395518658$$
$$x_{9} = -79.3315168346756$$
$$x_{10} = 62.0545116429054$$
$$x_{11} = 76.1901839979235$$
$$x_{12} = 77.760847792972$$
$$x_{13} = 98.1798629425939$$
$$x_{14} = -55.7722336752062$$
$$x_{15} = -40.0677825970372$$
$$x_{16} = -21.2292853858495$$
$$x_{17} = 71.4782275499213$$
$$x_{18} = -84.0435524991391$$
$$x_{19} = -68.3369563786298$$
$$x_{20} = -5.58635293416499$$
$$x_{21} = -4.04808180161146$$
$$x_{22} = -35.3570550332742$$
$$x_{23} = 49.4901859325761$$
$$x_{24} = -90.3263240494369$$
$$x_{25} = -93.4677306800165$$
$$x_{26} = -76.1901839979235$$
$$x_{27} = -49.4901859325761$$
$$x_{28} = 82.4728694594266$$
$$x_{29} = 8.69662198229738$$
$$x_{30} = 85.6142396947314$$
$$x_{31} = -85.6142396947314$$
$$x_{32} = -41.6381085824888$$
$$x_{33} = 90.3263240494369$$
$$x_{34} = -82.4728694594266$$
$$x_{35} = 33.7869153354295$$
$$x_{36} = -33.7869153354295$$
$$x_{37} = 25.9374070267134$$
$$x_{38} = -19.6603640661261$$
$$x_{39} = -60.4839244878466$$
$$x_{40} = -57.3427845371101$$
$$x_{41} = 0$$
$$x_{42} = 74.6195257807054$$
$$x_{43} = -13.3890435377793$$
$$x_{44} = -18.0917665453763$$
$$x_{45} = 16.5235843473527$$
$$x_{46} = 11.8231619098018$$
$$x_{47} = 4.04808180161146$$
$$x_{48} = 88.7556256712795$$
$$x_{49} = 96.6091494063022$$
$$x_{50} = 60.4839244878466$$
$$x_{51} = 2.54349254705114$$
$$x_{52} = 27.5071048394191$$
$$x_{53} = 40.0677825970372$$
$$x_{54} = -63.6251091208926$$
$$x_{55} = -71.4782275499213$$
$$x_{56} = 41.6381085824888$$
$$x_{57} = -46.3492776216985$$
$$x_{58} = -24.3678503974527$$
$$x_{59} = 10.2587614549708$$
$$x_{60} = 18.0917665453763$$
$$x_{61} = 47.9197205706165$$
$$x_{62} = 68.3369563786298$$
$$x_{63} = -25.9374070267134$$
$$x_{64} = -98.1798629425939$$
$$x_{65} = 120.170079673253$$
$$x_{66} = -69.9075883539626$$
$$x_{67} = -99.7505790857949$$
$$x_{68} = -58.9133484807877$$
$$x_{69} = -27.5071048394191$$
$$x_{70} = 30.6468374831214$$
$$x_{71} = -10.2587614549708$$
$$x_{72} = 46.3492776216985$$
$$x_{73} = 63.6251091208926$$
$$x_{74} = 24.3678503974527$$
$$x_{75} = -11.8231619098018$$
$$x_{76} = -77.760847792972$$
$$x_{77} = 55.7722336752062$$
$$x_{78} = 52.6311758774383$$
$$x_{79} = -91.8970257752571$$
$$x_{80} = -38.4974949445838$$
$$x_{81} = 91.8970257752571$$
$$x_{82} = 38.4974949445838$$
$$x_{83} = -66.766332133246$$
$$x_{84} = -1.1444648640517$$
$$x_{85} = 66.766332133246$$
$$x_{86} = -47.9197205706165$$
$$x_{87} = -62.0545116429054$$
$$x_{88} = 69.9075883539626$$
Signos de extremos en los puntos:
(-54.2016970313842, 54.7016978161202)
(99.75057908579493, -99.2505792117223)
(32.21683955186578, 32.7168432864561)
(19.660364066126064, 20.1603804725165)
(84.04355249913914, -83.5435527096795)
(5.586352934164992, -5.08704763815627)
(54.2016970313842, 54.7016978161202)
(-32.21683955186578, 32.7168432864561)
(-79.33151683467557, 79.8315170850003)
(62.054511642905446, -61.5545121658759)
(76.1901839979235, 76.6901842805014)
(77.76084779297203, -77.2608480587722)
(98.17986294259394, 98.679863074662)
(-55.772233675206174, -55.2722343955109)
(-40.06778259703722, -39.567784539057)
(-21.229285385849522, -20.7292984217544)
(71.47822754992126, -70.9782278921399)
(-84.04355249913914, -83.5435527096795)
(-68.3369563786298, -67.8369567702365)
(-5.586352934164992, -5.08704763815627)
(-4.048081801611461, 4.54985738750427)
(-35.35705503327425, 35.8570578590291)
(49.49018593257614, -48.9901869633787)
(-90.32632404943689, -89.8263242190322)
(-93.46773068001654, -92.9677308330813)
(-76.1901839979235, 76.6901842805014)
(-49.49018593257614, -48.9901869633787)
(82.47286945942662, 82.9728696822254)
(8.696621982297376, -8.1968095460522)
(85.6142396947314, 86.1142398938963)
(-85.6142396947314, 86.1142398938963)
(-41.63810858248877, 42.1381103130485)
(90.32632404943689, -89.8263242190322)
(-82.47286945942662, 82.9728696822254)
(33.7869153354295, -33.2869185734835)
(-33.7869153354295, -33.2869185734835)
(25.937407026713387, 26.4374141796637)
(-19.660364066126064, 20.1603804725165)
(-60.48392448784664, 60.9839250526164)
(-57.3427845371101, 57.8427851998477)
(0, 0)
(74.61952578070536, -74.1195260815031)
(-13.389043537779253, 13.8890953276258)
(-18.09176654537629, -17.5917875900408)
(16.52358434735268, 17.0236119537108)
(11.82316190980181, -11.3232370049634)
(4.048081801611461, 4.54985738750427)
(88.75562567127952, 89.2556258500382)
(96.60914940630224, -96.1091495449167)
(60.48392448784664, 60.9839250526164)
(2.543492547051135, -2.05006361332532)
(27.50710483941906, -27.0071108373536)
(40.06778259703722, -39.567784539057)
(-63.62510912089261, 64.1251096060885)
(-71.47822754992126, -70.9782278921399)
(41.63810858248877, 42.1381103130485)
(-46.34927762169846, -45.8492788765114)
(-24.367850397452695, -23.8678590218251)
(10.258761454970845, 10.7588761423567)
(18.09176654537629, -17.5917875900408)
(47.91972057061652, 48.4197217060929)
(68.3369563786298, -67.8369567702365)
(-25.937407026713387, 26.4374141796637)
(-98.17986294259394, 98.679863074662)
(120.17007967325263, 120.670079745279)
(-69.90758835396257, 70.4075887197664)
(-99.75057908579493, -99.2505792117223)
(-58.91334848078767, -58.413349091932)
(-27.50710483941906, -27.0071108373536)
(30.64683748312145, -30.1468418211343)
(-10.258761454970845, 10.7588761423567)
(46.34927762169846, -45.8492788765114)
(63.62510912089261, 64.1251096060885)
(24.367850397452695, -23.8678590218251)
(-11.82316190980181, -11.3232370049634)
(-77.76084779297203, -77.2608480587722)
(55.772233675206174, -55.2722343955109)
(52.63117587743834, -52.1311767345236)
(-91.89702577525712, 92.3970259363047)
(-38.4974949445838, 38.9974971339563)
(91.89702577525712, 92.3970259363047)
(38.4974949445838, 38.9974971339563)
(-66.766332133246, 67.2663325531403)
(-1.1444648640517021, 1.69081245896345)
(66.766332133246, 67.2663325531403)
(-47.91972057061652, 48.4197217060929)
(-62.054511642905446, -61.5545121658759)
(69.90758835396257, 70.4075887197664)
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:
$$x_{1} = 99.7505790857949$$
$$x_{2} = 84.0435524991391$$
$$x_{3} = 5.58635293416499$$
$$x_{4} = 62.0545116429054$$
$$x_{5} = 77.760847792972$$
$$x_{6} = -55.7722336752062$$
$$x_{7} = -40.0677825970372$$
$$x_{8} = -21.2292853858495$$
$$x_{9} = 71.4782275499213$$
$$x_{10} = -84.0435524991391$$
$$x_{11} = -68.3369563786298$$
$$x_{12} = -5.58635293416499$$
$$x_{13} = 49.4901859325761$$
$$x_{14} = -90.3263240494369$$
$$x_{15} = -93.4677306800165$$
$$x_{16} = -49.4901859325761$$
$$x_{17} = 8.69662198229738$$
$$x_{18} = 90.3263240494369$$
$$x_{19} = 33.7869153354295$$
$$x_{20} = -33.7869153354295$$
$$x_{21} = 0$$
$$x_{22} = 74.6195257807054$$
$$x_{23} = -18.0917665453763$$
$$x_{24} = 11.8231619098018$$
$$x_{25} = 96.6091494063022$$
$$x_{26} = 2.54349254705114$$
$$x_{27} = 27.5071048394191$$
$$x_{28} = 40.0677825970372$$
$$x_{29} = -71.4782275499213$$
$$x_{30} = -46.3492776216985$$
$$x_{31} = -24.3678503974527$$
$$x_{32} = 18.0917665453763$$
$$x_{33} = 68.3369563786298$$
$$x_{34} = -99.7505790857949$$
$$x_{35} = -58.9133484807877$$
$$x_{36} = -27.5071048394191$$
$$x_{37} = 30.6468374831214$$
$$x_{38} = 46.3492776216985$$
$$x_{39} = 24.3678503974527$$
$$x_{40} = -11.8231619098018$$
$$x_{41} = -77.760847792972$$
$$x_{42} = 55.7722336752062$$
$$x_{43} = 52.6311758774383$$
$$x_{44} = -62.0545116429054$$
Puntos máximos de la función:
$$x_{44} = -54.2016970313842$$
$$x_{44} = 32.2168395518658$$
$$x_{44} = 19.6603640661261$$
$$x_{44} = 54.2016970313842$$
$$x_{44} = -32.2168395518658$$
$$x_{44} = -79.3315168346756$$
$$x_{44} = 76.1901839979235$$
$$x_{44} = 98.1798629425939$$
$$x_{44} = -4.04808180161146$$
$$x_{44} = -35.3570550332742$$
$$x_{44} = -76.1901839979235$$
$$x_{44} = 82.4728694594266$$
$$x_{44} = 85.6142396947314$$
$$x_{44} = -85.6142396947314$$
$$x_{44} = -41.6381085824888$$
$$x_{44} = -82.4728694594266$$
$$x_{44} = 25.9374070267134$$
$$x_{44} = -19.6603640661261$$
$$x_{44} = -60.4839244878466$$
$$x_{44} = -57.3427845371101$$
$$x_{44} = -13.3890435377793$$
$$x_{44} = 16.5235843473527$$
$$x_{44} = 4.04808180161146$$
$$x_{44} = 88.7556256712795$$
$$x_{44} = 60.4839244878466$$
$$x_{44} = -63.6251091208926$$
$$x_{44} = 41.6381085824888$$
$$x_{44} = 10.2587614549708$$
$$x_{44} = 47.9197205706165$$
$$x_{44} = -25.9374070267134$$
$$x_{44} = -98.1798629425939$$
$$x_{44} = 120.170079673253$$
$$x_{44} = -69.9075883539626$$
$$x_{44} = -10.2587614549708$$
$$x_{44} = 63.6251091208926$$
$$x_{44} = -91.8970257752571$$
$$x_{44} = -38.4974949445838$$
$$x_{44} = 91.8970257752571$$
$$x_{44} = 38.4974949445838$$
$$x_{44} = -66.766332133246$$
$$x_{44} = -1.1444648640517$$
$$x_{44} = 66.766332133246$$
$$x_{44} = -47.9197205706165$$
$$x_{44} = 69.9075883539626$$
Decrece en los intervalos
$$\left[99.7505790857949, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7505790857949\right]$$
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
$$2 \left(- 2 x \sin{\left(x \right)} \cos{\left(x \right)} - \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)} - \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \sin{\left(x \right)} - \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -36.1490510533536$$
$$x_{2} = 59.702819996638$$
$$x_{3} = 86.4024774084808$$
$$x_{4} = 73.837583391258$$
$$x_{5} = -97.3970720893752$$
$$x_{6} = -23.5936903366458$$
$$x_{7} = -73.837583391258$$
$$x_{8} = -72.2670077223557$$
$$x_{9} = 29.8702177566864$$
$$x_{10} = -102.109105797777$$
$$x_{11} = 92.6850744922969$$
$$x_{12} = -87.973118805972$$
$$x_{13} = -6.39832419514071$$
$$x_{14} = -80.1199725348359$$
$$x_{15} = 80.1199725348359$$
$$x_{16} = -22.0251480595607$$
$$x_{17} = 100.538424195883$$
$$x_{18} = -83.2612121416269$$
$$x_{19} = 45.5695458968334$$
$$x_{20} = 12.6254970626858$$
$$x_{21} = 51.850739347184$$
$$x_{22} = -53.4211108167082$$
$$x_{23} = 36.1490510533536$$
$$x_{24} = 22.0251480595607$$
$$x_{25} = -94.2557360114768$$
$$x_{26} = 64.4142906736553$$
$$x_{27} = -86.4024774084808$$
$$x_{28} = -65.9848100204123$$
$$x_{29} = 9.50305414369817$$
$$x_{30} = 4.86201373808775$$
$$x_{31} = -17.3219495198134$$
$$x_{32} = 7.9472565318421$$
$$x_{33} = 56.5619244608641$$
$$x_{34} = -89.5437656364745$$
$$x_{35} = 87.973118805972$$
$$x_{36} = 28.3008101165524$$
$$x_{37} = -59.702819996638$$
$$x_{38} = 58.1323628207766$$
$$x_{39} = -9.50305414369817$$
$$x_{40} = -43.9993362628856$$
$$x_{41} = -1.90438110959984$$
$$x_{42} = -29.8702177566864$$
$$x_{43} = -95.8264019483769$$
$$x_{44} = 70.6964418518365$$
$$x_{45} = -75.4081682468113$$
$$x_{46} = 67.5553422369627$$
$$x_{47} = 94.2557360114768$$
$$x_{48} = 34.5791949429017$$
$$x_{49} = 50.2803943854432$$
$$x_{50} = 89.5437656364745$$
$$x_{51} = -61.2732945434725$$
$$x_{52} = 1.90438110959984$$
$$x_{53} = -15.755422878338$$
$$x_{54} = 95.8264019483769$$
$$x_{55} = 26.7315648822954$$
$$x_{56} = 15.755422878338$$
$$x_{57} = -28.3008101165524$$
$$x_{58} = 78.5493633153203$$
$$x_{59} = -14.1898261455607$$
$$x_{60} = 48.7100784905706$$
$$x_{61} = -20.4569491126465$$
$$x_{62} = 81.690588945839$$
$$x_{63} = 33.0094280245086$$
$$x_{64} = -67.5553422369627$$
$$x_{65} = 72.2670077223557$$
$$x_{66} = -50.2803943854432$$
$$x_{67} = -3.35197788789037$$
$$x_{68} = 14.1898261455607$$
$$x_{69} = -37.7189852569753$$
$$x_{70} = -51.850739347184$$
$$x_{71} = -42.4291699830311$$
$$x_{72} = -64.4142906736553$$
$$x_{73} = -7.9472565318421$$
$$x_{74} = 3.35197788789037$$
$$x_{75} = 43.9993362628856$$
$$x_{76} = 42.4291699830311$$
$$x_{77} = -39.2889882213125$$
$$x_{78} = 20.4569491126465$$
$$x_{79} = -45.5695458968334$$
$$x_{80} = 37.7189852569753$$
$$x_{81} = 0.596229414668214$$
$$x_{82} = 65.9848100204123$$
$$x_{83} = -58.1323628207766$$
$$x_{84} = -31.4397636015224$$
$$x_{85} = 6.39832419514071$$
$$x_{86} = -81.690588945839$$
$$x_{87} = 23.5936903366458$$
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[95.8264019483769, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -97.3970720893752\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$2 \left(- 2 x \sin{\left(x \right)} \cos{\left(x \right)} - \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)} - \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \sin{\left(x \right)} - \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -36.1490510533536$$
$$x_{2} = 59.702819996638$$
$$x_{3} = 86.4024774084808$$
$$x_{4} = 73.837583391258$$
$$x_{5} = -97.3970720893752$$
$$x_{6} = -23.5936903366458$$
$$x_{7} = -73.837583391258$$
$$x_{8} = -72.2670077223557$$
$$x_{9} = 29.8702177566864$$
$$x_{10} = -102.109105797777$$
$$x_{11} = 92.6850744922969$$
$$x_{12} = -87.973118805972$$
$$x_{13} = -6.39832419514071$$
$$x_{14} = -80.1199725348359$$
$$x_{15} = 80.1199725348359$$
$$x_{16} = -22.0251480595607$$
$$x_{17} = 100.538424195883$$
$$x_{18} = -83.2612121416269$$
$$x_{19} = 45.5695458968334$$
$$x_{20} = 12.6254970626858$$
$$x_{21} = 51.850739347184$$
$$x_{22} = -53.4211108167082$$
$$x_{23} = 36.1490510533536$$
$$x_{24} = 22.0251480595607$$
$$x_{25} = -94.2557360114768$$
$$x_{26} = 64.4142906736553$$
$$x_{27} = -86.4024774084808$$
$$x_{28} = -65.9848100204123$$
$$x_{29} = 9.50305414369817$$
$$x_{30} = 4.86201373808775$$
$$x_{31} = -17.3219495198134$$
$$x_{32} = 7.9472565318421$$
$$x_{33} = 56.5619244608641$$
$$x_{34} = -89.5437656364745$$
$$x_{35} = 87.973118805972$$
$$x_{36} = 28.3008101165524$$
$$x_{37} = -59.702819996638$$
$$x_{38} = 58.1323628207766$$
$$x_{39} = -9.50305414369817$$
$$x_{40} = -43.9993362628856$$
$$x_{41} = -1.90438110959984$$
$$x_{42} = -29.8702177566864$$
$$x_{43} = -95.8264019483769$$
$$x_{44} = 70.6964418518365$$
$$x_{45} = -75.4081682468113$$
$$x_{46} = 67.5553422369627$$
$$x_{47} = 94.2557360114768$$
$$x_{48} = 34.5791949429017$$
$$x_{49} = 50.2803943854432$$
$$x_{50} = 89.5437656364745$$
$$x_{51} = -61.2732945434725$$
$$x_{52} = 1.90438110959984$$
$$x_{53} = -15.755422878338$$
$$x_{54} = 95.8264019483769$$
$$x_{55} = 26.7315648822954$$
$$x_{56} = 15.755422878338$$
$$x_{57} = -28.3008101165524$$
$$x_{58} = 78.5493633153203$$
$$x_{59} = -14.1898261455607$$
$$x_{60} = 48.7100784905706$$
$$x_{61} = -20.4569491126465$$
$$x_{62} = 81.690588945839$$
$$x_{63} = 33.0094280245086$$
$$x_{64} = -67.5553422369627$$
$$x_{65} = 72.2670077223557$$
$$x_{66} = -50.2803943854432$$
$$x_{67} = -3.35197788789037$$
$$x_{68} = 14.1898261455607$$
$$x_{69} = -37.7189852569753$$
$$x_{70} = -51.850739347184$$
$$x_{71} = -42.4291699830311$$
$$x_{72} = -64.4142906736553$$
$$x_{73} = -7.9472565318421$$
$$x_{74} = 3.35197788789037$$
$$x_{75} = 43.9993362628856$$
$$x_{76} = 42.4291699830311$$
$$x_{77} = -39.2889882213125$$
$$x_{78} = 20.4569491126465$$
$$x_{79} = -45.5695458968334$$
$$x_{80} = 37.7189852569753$$
$$x_{81} = 0.596229414668214$$
$$x_{82} = 65.9848100204123$$
$$x_{83} = -58.1323628207766$$
$$x_{84} = -31.4397636015224$$
$$x_{85} = 6.39832419514071$$
$$x_{86} = -81.690588945839$$
$$x_{87} = 23.5936903366458$$
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[95.8264019483769, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -97.3970720893752\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)^2 + ((2*x)*cos(x))*sin(x), dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)}}{x}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)}}{x}\right)$$
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
$$2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)} = 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)}$$
- No
$$2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)} = - 2 x \sin{\left(x \right)} \cos{\left(x \right)} - \sin^{2}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)} = 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)}$$
- No
$$2 x \cos{\left(x \right)} \sin{\left(x \right)} + \sin^{2}{\left(x \right)} = - 2 x \sin{\left(x \right)} \cos{\left(x \right)} - \sin^{2}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar