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Trazado el gráfico de la función/ sin(2)^(2)*x*cos(x)/2

Gráfico de la función y = sin(2)^(2)*x*cos(x)/2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2            
       sin (2)*x*cos(x)
f(x) = ----------------
              2
$$f{\left(x \right)} = \frac{x \sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2}$$ 
f = ((x*sin(2)^2)*cos(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:
$$\frac{x \sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
Solución numérica
$$x_{1} = 48.6946861306418$$
$$x_{2} = 92.6769832808989$$
$$x_{3} = 86.3937979737193$$
$$x_{4} = -7.85398163397448$$
$$x_{5} = -86.3937979737193$$
$$x_{6} = 1.5707963267949$$
$$x_{7} = -64.4026493985908$$
$$x_{8} = -58.1194640914112$$
$$x_{9} = -83.2522053201295$$
$$x_{10} = -54.9778714378214$$
$$x_{11} = 54.9778714378214$$
$$x_{12} = 89.5353906273091$$
$$x_{13} = -20.4203522483337$$
$$x_{14} = 32.9867228626928$$
$$x_{15} = -17.2787595947439$$
$$x_{16} = 23.5619449019235$$
$$x_{17} = -45.553093477052$$
$$x_{18} = 64.4026493985908$$
$$x_{19} = 45.553093477052$$
$$x_{20} = 83.2522053201295$$
$$x_{21} = -29.845130209103$$
$$x_{22} = -51.8362787842316$$
$$x_{23} = 80.1106126665397$$
$$x_{24} = -39.2699081698724$$
$$x_{25} = -92.6769832808989$$
$$x_{26} = 4.71238898038469$$
$$x_{27} = 70.6858347057703$$
$$x_{28} = 36.1283155162826$$
$$x_{29} = -70.6858347057703$$
$$x_{30} = -48.6946861306418$$
$$x_{31} = 42.4115008234622$$
$$x_{32} = -42.4115008234622$$
$$x_{33} = -67.5442420521806$$
$$x_{34} = 10.9955742875643$$
$$x_{35} = 98.9601685880785$$
$$x_{36} = -23.5619449019235$$
$$x_{37} = 20.4203522483337$$
$$x_{38} = -61.261056745001$$
$$x_{39} = -10.9955742875643$$
$$x_{40} = 17.2787595947439$$
$$x_{41} = -95.8185759344887$$
$$x_{42} = -36.1283155162826$$
$$x_{43} = 61.261056745001$$
$$x_{44} = 114.668131856027$$
$$x_{45} = 73.8274273593601$$
$$x_{46} = 14.1371669411541$$
$$x_{47} = -26.7035375555132$$
$$x_{48} = 51.8362787842316$$
$$x_{49} = -89.5353906273091$$
$$x_{50} = 39.2699081698724$$
$$x_{51} = -32.9867228626928$$
$$x_{52} = -14.1371669411541$$
$$x_{53} = -4.71238898038469$$
$$x_{54} = -76.9690200129499$$
$$x_{55} = 95.8185759344887$$
$$x_{56} = -114.668131856027$$
$$x_{57} = 76.9690200129499$$
$$x_{58} = 58.1194640914112$$
$$x_{59} = -80.1106126665397$$
$$x_{60} = -73.8274273593601$$
$$x_{61} = 7.85398163397448$$
$$x_{62} = -1.5707963267949$$
$$x_{63} = 29.845130209103$$
$$x_{64} = 0$$
$$x_{65} = 67.5442420521806$$
$$x_{66} = 26.7035375555132$$
$$x_{67} = -98.9601685880785$$
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en ((sin(2)^2*x)*cos(x))/2.
$$\frac{0 \sin^{2}{\left(2 \right)} \cos{\left(0 \right)}}{2}$$
Resultado:
$$f{\left(0 \right)} = 0$$
Punto:
(0, 0)
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
$$- \frac{x \sin^{2}{\left(2 \right)} \sin{\left(x \right)}}{2} + \frac{\sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 59.7070073053355$$
$$x_{2} = 84.8347887180423$$
$$x_{3} = -78.5525459842429$$
$$x_{4} = 6.43729817917195$$
$$x_{5} = -47.145097736761$$
$$x_{6} = -69.1295029738953$$
$$x_{7} = -9.52933440536196$$
$$x_{8} = 40.8651703304881$$
$$x_{9} = 22.0364967279386$$
$$x_{10} = 18.90240995686$$
$$x_{11} = -40.8651703304881$$
$$x_{12} = 0.86033358901938$$
$$x_{13} = 28.309642854452$$
$$x_{14} = 25.1724463266467$$
$$x_{15} = 65.9885986984904$$
$$x_{16} = -147.661626855354$$
$$x_{17} = -100.540910786842$$
$$x_{18} = -0.86033358901938$$
$$x_{19} = 44.0050179208308$$
$$x_{20} = 12.6452872238566$$
$$x_{21} = 3.42561845948173$$
$$x_{22} = 97.3996388790738$$
$$x_{23} = -15.7712848748159$$
$$x_{24} = 72.270467060309$$
$$x_{25} = -84.8347887180423$$
$$x_{26} = 9.52933440536196$$
$$x_{27} = -44.0050179208308$$
$$x_{28} = -75.4114834888481$$
$$x_{29} = -65.9885986984904$$
$$x_{30} = -56.5663442798215$$
$$x_{31} = 53.4257904773947$$
$$x_{32} = -53.4257904773947$$
$$x_{33} = 87.9759605524932$$
$$x_{34} = 69.1295029738953$$
$$x_{35} = 47.145097736761$$
$$x_{36} = -6.43729817917195$$
$$x_{37} = -97.3996388790738$$
$$x_{38} = -18.90240995686$$
$$x_{39} = 78.5525459842429$$
$$x_{40} = -25.1724463266467$$
$$x_{41} = 15.7712848748159$$
$$x_{42} = -91.1171613944647$$
$$x_{43} = -28.309642854452$$
$$x_{44} = -81.6936492356017$$
$$x_{45} = -3.42561845948173$$
$$x_{46} = -12.6452872238566$$
$$x_{47} = -59.7070073053355$$
$$x_{48} = 91.1171613944647$$
$$x_{49} = 31.4477146375462$$
$$x_{50} = -72.270467060309$$
$$x_{51} = 81.6936492356017$$
$$x_{52} = -116.247530303932$$
$$x_{53} = -94.2583883450399$$
$$x_{54} = -50.2853663377737$$
$$x_{55} = -31.4477146375462$$
$$x_{56} = -37.7256128277765$$
$$x_{57} = -87.9759605524932$$
$$x_{58} = 34.5864242152889$$
$$x_{59} = 94.2583883450399$$
$$x_{60} = 62.8477631944545$$
$$x_{61} = -34.5864242152889$$
$$x_{62} = -62.8477631944545$$
$$x_{63} = 37.7256128277765$$
$$x_{64} = -22.0364967279386$$
$$x_{65} = 50.2853663377737$$
$$x_{66} = 75.4114834888481$$
$$x_{67} = 56.5663442798215$$
$$x_{68} = 100.540910786842$$
Signos de extremos en los puntos:
                                         2    
(59.70700730533546, -29.8493174201329*sin (2))

                                         2    
(84.83478871804229, -42.4144477618284*sin (2))

                                         2    
(-78.55254598424293, 39.2730907958671*sin (2))

                                        2    
(6.437298179171947, 3.18050197241693*sin (2))

                                        2    
(-47.14509773676103, 23.567247878771*sin (2))

                                          2    
(-69.12950297389526, -34.5611356534609*sin (2))

                                        2    
(-9.529334405361963, 4.7386471297399*sin (2))

                                         2    
(40.86517033048807, -20.4264702322587*sin (2))

                                          2    
(22.036496727938566, -11.0069210395792*sin (2))

                                         2    
(18.902409956860023, 9.43800684898451*sin (2))

                                         2    
(-40.86517033048807, 20.4264702322587*sin (2))

                                          2    
(0.8603335890193797, 0.280548169095523*sin (2))

                                         2    
(28.30964285445201, -14.1459987695472*sin (2))

                                         2    
(25.172446326646664, 12.5763034089358*sin (2))

                                         2    
(65.98859869849039, -32.9905114683958*sin (2))

                                         2    
(-147.66162685535437, 73.829120425871*sin (2))

                                          2    
(-100.54091078684232, -50.267969027913*sin (2))

                                            2    
(-0.8603335890193797, -0.280548169095523*sin (2))

                                         2    
(44.005017920830845, 21.9968299895532*sin (2))

                                         2    
(12.645287223856643, 6.30296564894634*sin (2))

                                          2    
(3.4256184594817283, -1.64418569779545*sin (2))

                                         2    
(97.39963887907376, -48.6972528978117*sin (2))

                                          2    
(-15.771284874815882, 7.86983848106687*sin (2))

                                         2    
(72.27046706030896, -36.1317747991247*sin (2))

                                         2    
(-84.83478871804229, 42.4144477618284*sin (2))

                                        2    
(9.529334405361963, -4.7386471297399*sin (2))

                                           2    
(-44.005017920830845, -21.9968299895532*sin (2))

                                         2    
(-75.41148348884815, -37.702427036601*sin (2))

                                         2    
(-65.98859869849039, 32.9905114683958*sin (2))

                                          2    
(-56.56634427982152, -28.2787535864381*sin (2))

                                         2    
(53.42579047739466, -26.7082170799481*sin (2))

                                         2    
(-53.42579047739466, 26.7082170799481*sin (2))

                                        2    
(87.97596055249322, 43.9851388662124*sin (2))

                                        2    
(69.12950297389526, 34.5611356534609*sin (2))

                                        2    
(47.14509773676103, -23.567247878771*sin (2))

                                          2    
(-6.437298179171947, -3.18050197241693*sin (2))

                                         2    
(-97.39963887907376, 48.6972528978117*sin (2))

                                           2    
(-18.902409956860023, -9.43800684898451*sin (2))

                                         2    
(78.55254598424293, -39.2730907958671*sin (2))

                                           2    
(-25.172446326646664, -12.5763034089358*sin (2))

                                          2    
(15.771284874815882, -7.86983848106687*sin (2))

                                         2    
(-91.11716139446474, 45.5558372248235*sin (2))

                                         2    
(-28.30964285445201, 14.1459987695472*sin (2))

                                          2    
(-81.69364923560168, -40.8437647482623*sin (2))

                                          2    
(-3.4256184594817283, 1.64418569779545*sin (2))

                                           2    
(-12.645287223856643, -6.30296564894634*sin (2))

                                         2    
(-59.70700730533546, 29.8493174201329*sin (2))

                                         2    
(91.11716139446474, -45.5558372248235*sin (2))

                                         2    
(31.447714637546234, 15.7159136392673*sin (2))

                                         2    
(-72.27046706030896, 36.1317747991247*sin (2))

                                        2    
(81.69364923560168, 40.8437647482623*sin (2))

                                         2    
(-116.2475303039321, 58.1216146879934*sin (2))

                                          2    
(-94.25838834503986, -47.1265421125543*sin (2))

                                          2    
(-50.28536633777365, -25.1377130176986*sin (2))

                                           2    
(-31.447714637546234, -15.7159136392673*sin (2))

                                        2    
(-37.7256128277765, -18.856183106405*sin (2))

                                          2    
(-87.97596055249322, -43.9851388662124*sin (2))

                                         2    
(34.58642421528892, -17.2859883667942*sin (2))

                                        2    
(94.25838834503986, 47.1265421125543*sin (2))

                                        2    
(62.84776319445445, 31.4199044860773*sin (2))

                                         2    
(-34.58642421528892, 17.2859883667942*sin (2))

                                          2    
(-62.84776319445445, -31.4199044860773*sin (2))

                                      2    
(37.7256128277765, 18.856183106405*sin (2))

                                          2    
(-22.036496727938566, 11.0069210395792*sin (2))

                                        2    
(50.28536633777365, 25.1377130176986*sin (2))

                                       2    
(75.41148348884815, 37.702427036601*sin (2))

                                        2    
(56.56634427982152, 28.2787535864381*sin (2))

                                        2    
(100.54091078684232, 50.267969027913*sin (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:
$$x_{1} = 59.7070073053355$$
$$x_{2} = 84.8347887180423$$
$$x_{3} = -69.1295029738953$$
$$x_{4} = 40.8651703304881$$
$$x_{5} = 22.0364967279386$$
$$x_{6} = 28.309642854452$$
$$x_{7} = 65.9885986984904$$
$$x_{8} = -100.540910786842$$
$$x_{9} = -0.86033358901938$$
$$x_{10} = 3.42561845948173$$
$$x_{11} = 97.3996388790738$$
$$x_{12} = 72.270467060309$$
$$x_{13} = 9.52933440536196$$
$$x_{14} = -44.0050179208308$$
$$x_{15} = -75.4114834888481$$
$$x_{16} = -56.5663442798215$$
$$x_{17} = 53.4257904773947$$
$$x_{18} = 47.145097736761$$
$$x_{19} = -6.43729817917195$$
$$x_{20} = -18.90240995686$$
$$x_{21} = 78.5525459842429$$
$$x_{22} = -25.1724463266467$$
$$x_{23} = 15.7712848748159$$
$$x_{24} = -81.6936492356017$$
$$x_{25} = -12.6452872238566$$
$$x_{26} = 91.1171613944647$$
$$x_{27} = -94.2583883450399$$
$$x_{28} = -50.2853663377737$$
$$x_{29} = -31.4477146375462$$
$$x_{30} = -37.7256128277765$$
$$x_{31} = -87.9759605524932$$
$$x_{32} = 34.5864242152889$$
$$x_{33} = -62.8477631944545$$
Puntos máximos de la función:
$$x_{33} = -78.5525459842429$$
$$x_{33} = 6.43729817917195$$
$$x_{33} = -47.145097736761$$
$$x_{33} = -9.52933440536196$$
$$x_{33} = 18.90240995686$$
$$x_{33} = -40.8651703304881$$
$$x_{33} = 0.86033358901938$$
$$x_{33} = 25.1724463266467$$
$$x_{33} = -147.661626855354$$
$$x_{33} = 44.0050179208308$$
$$x_{33} = 12.6452872238566$$
$$x_{33} = -15.7712848748159$$
$$x_{33} = -84.8347887180423$$
$$x_{33} = -65.9885986984904$$
$$x_{33} = -53.4257904773947$$
$$x_{33} = 87.9759605524932$$
$$x_{33} = 69.1295029738953$$
$$x_{33} = -97.3996388790738$$
$$x_{33} = -91.1171613944647$$
$$x_{33} = -28.309642854452$$
$$x_{33} = -3.42561845948173$$
$$x_{33} = -59.7070073053355$$
$$x_{33} = 31.4477146375462$$
$$x_{33} = -72.270467060309$$
$$x_{33} = 81.6936492356017$$
$$x_{33} = -116.247530303932$$
$$x_{33} = 94.2583883450399$$
$$x_{33} = 62.8477631944545$$
$$x_{33} = -34.5864242152889$$
$$x_{33} = 37.7256128277765$$
$$x_{33} = -22.0364967279386$$
$$x_{33} = 50.2853663377737$$
$$x_{33} = 75.4114834888481$$
$$x_{33} = 56.5663442798215$$
$$x_{33} = 100.540910786842$$
Decrece en los intervalos
$$\left[97.3996388790738, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.540910786842\right]$$
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
$$- \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \sin^{2}{\left(2 \right)}}{2} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 33.0471686947054$$
$$x_{2} = -76.9949898891676$$
$$x_{3} = -98.9803718651523$$
$$x_{4} = 5.08698509410227$$
$$x_{5} = -64.4336791037316$$
$$x_{6} = 92.6985552433969$$
$$x_{7} = 42.458570771699$$
$$x_{8} = 45.5969279840735$$
$$x_{9} = 51.8748140534268$$
$$x_{10} = -95.839441141233$$
$$x_{11} = 64.4336791037316$$
$$x_{12} = 48.7357007949054$$
$$x_{13} = -48.7357007949054$$
$$x_{14} = -29.9118938695518$$
$$x_{15} = -17.3932439645948$$
$$x_{16} = -80.1355651940744$$
$$x_{17} = -5.08698509410227$$
$$x_{18} = -33.0471686947054$$
$$x_{19} = 58.153842078645$$
$$x_{20} = -8.09616360322292$$
$$x_{21} = 70.7141100665485$$
$$x_{22} = 61.2936749662429$$
$$x_{23} = 29.9118938695518$$
$$x_{24} = 86.4169374541167$$
$$x_{25} = 23.6463238196036$$
$$x_{26} = -23.6463238196036$$
$$x_{27} = -58.153842078645$$
$$x_{28} = 20.5175229099417$$
$$x_{29} = 36.1835330907526$$
$$x_{30} = -89.5577188827244$$
$$x_{31} = 14.2763529183365$$
$$x_{32} = -39.3207281322521$$
$$x_{33} = 11.17270586833$$
$$x_{34} = 17.3932439645948$$
$$x_{35} = 98.9803718651523$$
$$x_{36} = 80.1355651940744$$
$$x_{37} = -2.2889297281034$$
$$x_{38} = -67.573830670859$$
$$x_{39} = 2.2889297281034$$
$$x_{40} = -36.1835330907526$$
$$x_{41} = -14.2763529183365$$
$$x_{42} = -73.8545010149048$$
$$x_{43} = -42.458570771699$$
$$x_{44} = 55.0142096788381$$
$$x_{45} = 89.5577188827244$$
$$x_{46} = -92.6985552433969$$
$$x_{47} = 26.7780870755585$$
$$x_{48} = -26.7780870755585$$
$$x_{49} = -61.2936749662429$$
$$x_{50} = 8.09616360322292$$
$$x_{51} = -51.8748140534268$$
$$x_{52} = -83.2762171649775$$
$$x_{53} = 95.839441141233$$
$$x_{54} = -45.5969279840735$$
$$x_{55} = 73.8545010149048$$
$$x_{56} = -70.7141100665485$$
$$x_{57} = -20.5175229099417$$
$$x_{58} = 67.573830670859$$
$$x_{59} = 39.3207281322521$$
$$x_{60} = 76.9949898891676$$
$$x_{61} = 83.2762171649775$$
$$x_{62} = -55.0142096788381$$
$$x_{63} = 0$$
$$x_{64} = -86.4169374541167$$
$$x_{65} = -11.17270586833$$

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
$$\left[95.839441141233, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.839441141233\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{x \sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2}\right) = \left\langle -\infty, \infty\right\rangle \sin^{2}{\left(2 \right)}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle \sin^{2}{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(\frac{x \sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2}\right) = \left\langle -\infty, \infty\right\rangle \sin^{2}{\left(2 \right)}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle \sin^{2}{\left(2 \right)}$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((sin(2)^2*x)*cos(x))/2, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \sin^{2}{\left(2 \right)}$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle x \sin^{2}{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \sin^{2}{\left(2 \right)}$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle x \sin^{2}{\left(2 \right)}$$
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:
$$\frac{x \sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2} = - \frac{x \sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2}$$
- No
$$\frac{x \sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2} = \frac{x \sin^{2}{\left(2 \right)} \cos{\left(x \right)}}{2}$$
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор