Sr Examen
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- Puntos de cruce de las funciones
y=(x+1)^3
y=2x
2*x^3-3*x
3-x^2
Gráfico de la función y = 2sintcost^2
Solución
Ha introducido
[src]
2 f(t) = 2*sin (t*cos(t))
$$f{\left(t \right)} = 2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}$$
f = 2*sin(t*cos(t))^2
Gráfico de la función
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje T con f = 0
o sea hay que resolver la ecuación:
$$2 \sin^{2}{\left(t \cos{\left(t \right)} \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = -19.7298536573847$$
$$t_{2} = 8.24489691212938$$
$$t_{3} = 21.9911485835996$$
$$t_{4} = -37.7521014229575$$
$$t_{5} = 30.3883642704472$$
$$t_{6} = -95.7529100122856$$
$$t_{7} = -87.9873258601891$$
$$t_{8} = -7.41657600616137$$
$$t_{9} = 50.2654824368045$$
$$t_{10} = -53.7701140990197$$
$$t_{11} = 72.2566310252921$$
$$t_{12} = 43.9822970917047$$
$$t_{13} = 56.2304338521185$$
$$t_{14} = 24.2660127598138$$
$$t_{15} = -59.7237511124967$$
$$t_{16} = 68.2445855811011$$
$$t_{17} = 82.2544445406972$$
$$t_{18} = 66.0037493498728$$
$$t_{19} = 28.2743338517182$$
$$t_{20} = 20.2646966090641$$
$$t_{21} = -6.5890952493621$$
$$t_{22} = -15.7079632703355$$
$$t_{23} = -64.8009421677305$$
$$t_{24} = 59.6902604291161$$
$$t_{25} = 55.9785194520914$$
$$t_{26} = 76.2492309280351$$
$$t_{27} = -43.9822970841169$$
$$t_{28} = 94.2477796093014$$
$$t_{29} = -17.2787596170566$$
$$t_{30} = 1.57079645531189$$
$$t_{31} = 42.1861902568491$$
$$t_{32} = -39.6769519954494$$
$$t_{33} = -1.57079644811036$$
$$t_{34} = -97.7604616111537$$
$$t_{35} = 57.7871946797054$$
$$t_{36} = -6.28318489514378$$
$$t_{37} = 32.2359651809805$$
$$t_{38} = 183.868704819254$$
$$t_{39} = -50.2654824636369$$
$$t_{40} = -56.5840174314607$$
$$t_{41} = 47.1238898042072$$
$$t_{42} = -66.0037493498799$$
$$t_{43} = 6.28318527830171$$
$$t_{44} = -76.1922897323884$$
$$t_{45} = 83.8946444435889$$
$$t_{46} = -79.7481850773787$$
$$t_{47} = -63.9990786483774$$
$$t_{48} = 0$$
$$t_{49} = -1994.15111674285$$
$$t_{50} = -9.63314741837918$$
$$t_{51} = -93.9990445838144$$
$$t_{52} = 3.69672327529414$$
$$t_{53} = -3.69672296742301$$
$$t_{54} = 11.2778713207023$$
$$t_{55} = -81.7058883814046$$
$$t_{56} = 87.9645936617161$$
$$t_{57} = -71.7526619598438$$
$$t_{58} = 87.9873258391697$$
$$t_{59} = -21.9911485837194$$
$$t_{60} = 86.2475763334824$$
$$t_{61} = 20.5736518008681$$
$$t_{62} = -86.1732747987686$$
$$t_{63} = 15.834439121319$$
$$t_{64} = -86.3574109924251$$
$$t_{65} = 3.69672260904421$$
$$t_{66} = 53.4445015475953$$
$$t_{67} = -24.2660127940325$$
$$t_{68} = -78.565274252698$$
o sea hay que resolver la ecuación:
$$2 \sin^{2}{\left(t \cos{\left(t \right)} \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = -19.7298536573847$$
$$t_{2} = 8.24489691212938$$
$$t_{3} = 21.9911485835996$$
$$t_{4} = -37.7521014229575$$
$$t_{5} = 30.3883642704472$$
$$t_{6} = -95.7529100122856$$
$$t_{7} = -87.9873258601891$$
$$t_{8} = -7.41657600616137$$
$$t_{9} = 50.2654824368045$$
$$t_{10} = -53.7701140990197$$
$$t_{11} = 72.2566310252921$$
$$t_{12} = 43.9822970917047$$
$$t_{13} = 56.2304338521185$$
$$t_{14} = 24.2660127598138$$
$$t_{15} = -59.7237511124967$$
$$t_{16} = 68.2445855811011$$
$$t_{17} = 82.2544445406972$$
$$t_{18} = 66.0037493498728$$
$$t_{19} = 28.2743338517182$$
$$t_{20} = 20.2646966090641$$
$$t_{21} = -6.5890952493621$$
$$t_{22} = -15.7079632703355$$
$$t_{23} = -64.8009421677305$$
$$t_{24} = 59.6902604291161$$
$$t_{25} = 55.9785194520914$$
$$t_{26} = 76.2492309280351$$
$$t_{27} = -43.9822970841169$$
$$t_{28} = 94.2477796093014$$
$$t_{29} = -17.2787596170566$$
$$t_{30} = 1.57079645531189$$
$$t_{31} = 42.1861902568491$$
$$t_{32} = -39.6769519954494$$
$$t_{33} = -1.57079644811036$$
$$t_{34} = -97.7604616111537$$
$$t_{35} = 57.7871946797054$$
$$t_{36} = -6.28318489514378$$
$$t_{37} = 32.2359651809805$$
$$t_{38} = 183.868704819254$$
$$t_{39} = -50.2654824636369$$
$$t_{40} = -56.5840174314607$$
$$t_{41} = 47.1238898042072$$
$$t_{42} = -66.0037493498799$$
$$t_{43} = 6.28318527830171$$
$$t_{44} = -76.1922897323884$$
$$t_{45} = 83.8946444435889$$
$$t_{46} = -79.7481850773787$$
$$t_{47} = -63.9990786483774$$
$$t_{48} = 0$$
$$t_{49} = -1994.15111674285$$
$$t_{50} = -9.63314741837918$$
$$t_{51} = -93.9990445838144$$
$$t_{52} = 3.69672327529414$$
$$t_{53} = -3.69672296742301$$
$$t_{54} = 11.2778713207023$$
$$t_{55} = -81.7058883814046$$
$$t_{56} = 87.9645936617161$$
$$t_{57} = -71.7526619598438$$
$$t_{58} = 87.9873258391697$$
$$t_{59} = -21.9911485837194$$
$$t_{60} = 86.2475763334824$$
$$t_{61} = 20.5736518008681$$
$$t_{62} = -86.1732747987686$$
$$t_{63} = 15.834439121319$$
$$t_{64} = -86.3574109924251$$
$$t_{65} = 3.69672260904421$$
$$t_{66} = 53.4445015475953$$
$$t_{67} = -24.2660127940325$$
$$t_{68} = -78.565274252698$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d t} f{\left(t \right)} = $$
primera derivada
$$4 \left(- t \sin{\left(t \right)} + \cos{\left(t \right)}\right) \sin{\left(t \cos{\left(t \right)} \right)} \cos{\left(t \cos{\left(t \right)} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 9.52933440536196$$
$$t_{2} = 47.1662993169327$$
$$t_{3} = 37.7256128277765$$
$$t_{4} = -68.2746898137795$$
$$t_{5} = 24.1862248303366$$
$$t_{6} = -1.5707963267949$$
$$t_{7} = 28.3449225622678$$
$$t_{8} = -30.5110645458325$$
$$t_{9} = 80.1498192134838$$
$$t_{10} = 34.5575191894877$$
$$t_{11} = -21.9911485751286$$
$$t_{12} = 68.1864393033044$$
$$t_{13} = -72.2843013230124$$
$$t_{14} = -51.3245530867134$$
$$t_{15} = 87.9873258258086$$
$$t_{16} = -75.4114834888481$$
$$t_{17} = -11.7292528690793$$
$$t_{18} = 54.2490766358244$$
$$t_{19} = -53.6691739907764$$
$$t_{20} = -15.7712848748159$$
$$t_{21} = -89.7828454291688$$
$$t_{22} = -37.7521014243949$$
$$t_{23} = 10.1053547868089$$
$$t_{24} = 72.566281542806$$
$$t_{25} = -46.7765007581068$$
$$t_{26} = -31.4159265358979$$
$$t_{27} = -40.5854022598267$$
$$t_{28} = -15.8344387416857$$
$$t_{29} = 100.540910786842$$
$$t_{30} = -59.7237510632525$$
$$t_{31} = -15.707963267949$$
$$t_{32} = -85.7902078153718$$
$$t_{33} = 76.5461072364502$$
$$t_{34} = 63.0720690226406$$
$$t_{35} = 25.1327412287183$$
$$t_{36} = 12.1303742531355$$
$$t_{37} = 0$$
$$t_{38} = 8.24489690812664$$
$$t_{39} = 75.398223686155$$
$$t_{40} = 0.86033358901938$$
$$t_{41} = 67.4976815189735$$
$$t_{42} = -83.9660742234445$$
$$t_{43} = -50.2654824574367$$
$$t_{44} = 20.1847144072068$$
$$t_{45} = 66.0037493532594$$
$$t_{46} = 76.3033501534943$$
$$t_{47} = 59.088081511794$$
$$t_{48} = 97.4099047756117$$
$$t_{49} = 91.3032950363969$$
$$t_{50} = -51.1753017001883$$
$$t_{51} = -39.5928836844695$$
$$t_{52} = 131.946891450771$$
$$t_{53} = 28.2743338823081$$
$$t_{54} = -9.63314724838335$$
$$t_{55} = -96.6873970931048$$
$$t_{56} = 59.7237510632525$$
$$t_{57} = -3.69672292256781$$
$$t_{58} = 4.34224753568081$$
$$t_{59} = 40.8407044966673$$
$$t_{60} = -66.0037493532594$$
$$t_{61} = 2.31611213068295$$
$$t_{62} = -28.309642854452$$
$$t_{63} = 56.3291859523472$$
$$t_{64} = 31.999500727538$$
$$t_{65} = -81.7058882556996$$
$$t_{66} = -135.088484104361$$
$$t_{67} = -87.9873258258086$$
$$t_{68} = 18.9551661778348$$
$$t_{69} = -30.9938864935603$$
$$t_{70} = -91.1171613944647$$
$$t_{71} = 40.8896264144125$$
$$t_{72} = -44.0050179208308$$
$$t_{73} = 94.2477796076938$$
$$t_{74} = 78.2036923334948$$
$$t_{75} = -57.0980924408068$$
$$t_{76} = 72.2566310325652$$
$$t_{77} = -18.90240995686$$
$$t_{78} = 16.2203662438486$$
$$t_{79} = -69.1439655522063$$
$$t_{80} = 48.8237352274238$$
$$t_{81} = 44.0050179208308$$
$$t_{82} = 6.28318530717959$$
$$t_{83} = 64.9078784744433$$
$$t_{84} = -78.100545462825$$
$$t_{85} = 21.9911485751286$$
$$t_{86} = -79.4759164084994$$
$$t_{87} = 50.2654824574367$$
$$t_{88} = -94.2689962866132$$
$$t_{89} = 100.034645727999$$
$$t_{90} = -97.4099047756117$$
$$t_{91} = 53.4070751110265$$
$$t_{92} = -49.9287025648298$$
$$t_{93} = 81.4148630970183$$
$$t_{94} = -37.6991118430775$$
$$t_{95} = -7.85398163397448$$
$$t_{96} = -63.4009836613048$$
$$t_{97} = 14.2476410900056$$
$$t_{98} = -63.9168256049229$$
$$t_{99} = -34.5575191894877$$
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:
$$t_{1} = 47.1662993169327$$
$$t_{2} = -1.5707963267949$$
$$t_{3} = 28.3449225622678$$
$$t_{4} = -30.5110645458325$$
$$t_{5} = 80.1498192134838$$
$$t_{6} = 34.5575191894877$$
$$t_{7} = -21.9911485751286$$
$$t_{8} = 68.1864393033044$$
$$t_{9} = -72.2843013230124$$
$$t_{10} = -51.3245530867134$$
$$t_{11} = 87.9873258258086$$
$$t_{12} = -89.7828454291688$$
$$t_{13} = -37.7521014243949$$
$$t_{14} = 72.566281542806$$
$$t_{15} = -46.7765007581068$$
$$t_{16} = -31.4159265358979$$
$$t_{17} = -15.8344387416857$$
$$t_{18} = -59.7237510632525$$
$$t_{19} = -15.707963267949$$
$$t_{20} = 76.5461072364502$$
$$t_{21} = 25.1327412287183$$
$$t_{22} = 0$$
$$t_{23} = 8.24489690812664$$
$$t_{24} = 75.398223686155$$
$$t_{25} = 67.4976815189735$$
$$t_{26} = -50.2654824574367$$
$$t_{27} = 66.0037493532594$$
$$t_{28} = 76.3033501534943$$
$$t_{29} = 97.4099047756117$$
$$t_{30} = -51.1753017001883$$
$$t_{31} = -39.5928836844695$$
$$t_{32} = 131.946891450771$$
$$t_{33} = 28.2743338823081$$
$$t_{34} = -9.63314724838335$$
$$t_{35} = 59.7237510632525$$
$$t_{36} = -3.69672292256781$$
$$t_{37} = 40.8407044966673$$
$$t_{38} = -66.0037493532594$$
$$t_{39} = -81.7058882556996$$
$$t_{40} = -135.088484104361$$
$$t_{41} = -87.9873258258086$$
$$t_{42} = 18.9551661778348$$
$$t_{43} = -30.9938864935603$$
$$t_{44} = 40.8896264144125$$
$$t_{45} = 94.2477796076938$$
$$t_{46} = 72.2566310325652$$
$$t_{47} = -69.1439655522063$$
$$t_{48} = 48.8237352274238$$
$$t_{49} = 6.28318530717959$$
$$t_{50} = 64.9078784744433$$
$$t_{51} = 21.9911485751286$$
$$t_{52} = -79.4759164084994$$
$$t_{53} = 50.2654824574367$$
$$t_{54} = -94.2689962866132$$
$$t_{55} = 100.034645727999$$
$$t_{56} = -97.4099047756117$$
$$t_{57} = 53.4070751110265$$
$$t_{58} = -49.9287025648298$$
$$t_{59} = 81.4148630970183$$
$$t_{60} = -37.6991118430775$$
$$t_{61} = -7.85398163397448$$
$$t_{62} = -63.4009836613048$$
$$t_{63} = -34.5575191894877$$
Puntos máximos de la función:
$$t_{63} = 9.52933440536196$$
$$t_{63} = 37.7256128277765$$
$$t_{63} = -68.2746898137795$$
$$t_{63} = 24.1862248303366$$
$$t_{63} = -75.4114834888481$$
$$t_{63} = -11.7292528690793$$
$$t_{63} = 54.2490766358244$$
$$t_{63} = -53.6691739907764$$
$$t_{63} = -15.7712848748159$$
$$t_{63} = 10.1053547868089$$
$$t_{63} = -40.5854022598267$$
$$t_{63} = 100.540910786842$$
$$t_{63} = -85.7902078153718$$
$$t_{63} = 63.0720690226406$$
$$t_{63} = 12.1303742531355$$
$$t_{63} = 0.86033358901938$$
$$t_{63} = -83.9660742234445$$
$$t_{63} = 20.1847144072068$$
$$t_{63} = 59.088081511794$$
$$t_{63} = 91.3032950363969$$
$$t_{63} = -96.6873970931048$$
$$t_{63} = 4.34224753568081$$
$$t_{63} = 2.31611213068295$$
$$t_{63} = -28.309642854452$$
$$t_{63} = 56.3291859523472$$
$$t_{63} = 31.999500727538$$
$$t_{63} = -91.1171613944647$$
$$t_{63} = -44.0050179208308$$
$$t_{63} = 78.2036923334948$$
$$t_{63} = -57.0980924408068$$
$$t_{63} = -18.90240995686$$
$$t_{63} = 16.2203662438486$$
$$t_{63} = 44.0050179208308$$
$$t_{63} = -78.100545462825$$
$$t_{63} = 14.2476410900056$$
$$t_{63} = -63.9168256049229$$
Decrece en los intervalos
$$\left[131.946891450771, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -135.088484104361\right]$$
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d t} f{\left(t \right)} = $$
primera derivada
$$4 \left(- t \sin{\left(t \right)} + \cos{\left(t \right)}\right) \sin{\left(t \cos{\left(t \right)} \right)} \cos{\left(t \cos{\left(t \right)} \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 9.52933440536196$$
$$t_{2} = 47.1662993169327$$
$$t_{3} = 37.7256128277765$$
$$t_{4} = -68.2746898137795$$
$$t_{5} = 24.1862248303366$$
$$t_{6} = -1.5707963267949$$
$$t_{7} = 28.3449225622678$$
$$t_{8} = -30.5110645458325$$
$$t_{9} = 80.1498192134838$$
$$t_{10} = 34.5575191894877$$
$$t_{11} = -21.9911485751286$$
$$t_{12} = 68.1864393033044$$
$$t_{13} = -72.2843013230124$$
$$t_{14} = -51.3245530867134$$
$$t_{15} = 87.9873258258086$$
$$t_{16} = -75.4114834888481$$
$$t_{17} = -11.7292528690793$$
$$t_{18} = 54.2490766358244$$
$$t_{19} = -53.6691739907764$$
$$t_{20} = -15.7712848748159$$
$$t_{21} = -89.7828454291688$$
$$t_{22} = -37.7521014243949$$
$$t_{23} = 10.1053547868089$$
$$t_{24} = 72.566281542806$$
$$t_{25} = -46.7765007581068$$
$$t_{26} = -31.4159265358979$$
$$t_{27} = -40.5854022598267$$
$$t_{28} = -15.8344387416857$$
$$t_{29} = 100.540910786842$$
$$t_{30} = -59.7237510632525$$
$$t_{31} = -15.707963267949$$
$$t_{32} = -85.7902078153718$$
$$t_{33} = 76.5461072364502$$
$$t_{34} = 63.0720690226406$$
$$t_{35} = 25.1327412287183$$
$$t_{36} = 12.1303742531355$$
$$t_{37} = 0$$
$$t_{38} = 8.24489690812664$$
$$t_{39} = 75.398223686155$$
$$t_{40} = 0.86033358901938$$
$$t_{41} = 67.4976815189735$$
$$t_{42} = -83.9660742234445$$
$$t_{43} = -50.2654824574367$$
$$t_{44} = 20.1847144072068$$
$$t_{45} = 66.0037493532594$$
$$t_{46} = 76.3033501534943$$
$$t_{47} = 59.088081511794$$
$$t_{48} = 97.4099047756117$$
$$t_{49} = 91.3032950363969$$
$$t_{50} = -51.1753017001883$$
$$t_{51} = -39.5928836844695$$
$$t_{52} = 131.946891450771$$
$$t_{53} = 28.2743338823081$$
$$t_{54} = -9.63314724838335$$
$$t_{55} = -96.6873970931048$$
$$t_{56} = 59.7237510632525$$
$$t_{57} = -3.69672292256781$$
$$t_{58} = 4.34224753568081$$
$$t_{59} = 40.8407044966673$$
$$t_{60} = -66.0037493532594$$
$$t_{61} = 2.31611213068295$$
$$t_{62} = -28.309642854452$$
$$t_{63} = 56.3291859523472$$
$$t_{64} = 31.999500727538$$
$$t_{65} = -81.7058882556996$$
$$t_{66} = -135.088484104361$$
$$t_{67} = -87.9873258258086$$
$$t_{68} = 18.9551661778348$$
$$t_{69} = -30.9938864935603$$
$$t_{70} = -91.1171613944647$$
$$t_{71} = 40.8896264144125$$
$$t_{72} = -44.0050179208308$$
$$t_{73} = 94.2477796076938$$
$$t_{74} = 78.2036923334948$$
$$t_{75} = -57.0980924408068$$
$$t_{76} = 72.2566310325652$$
$$t_{77} = -18.90240995686$$
$$t_{78} = 16.2203662438486$$
$$t_{79} = -69.1439655522063$$
$$t_{80} = 48.8237352274238$$
$$t_{81} = 44.0050179208308$$
$$t_{82} = 6.28318530717959$$
$$t_{83} = 64.9078784744433$$
$$t_{84} = -78.100545462825$$
$$t_{85} = 21.9911485751286$$
$$t_{86} = -79.4759164084994$$
$$t_{87} = 50.2654824574367$$
$$t_{88} = -94.2689962866132$$
$$t_{89} = 100.034645727999$$
$$t_{90} = -97.4099047756117$$
$$t_{91} = 53.4070751110265$$
$$t_{92} = -49.9287025648298$$
$$t_{93} = 81.4148630970183$$
$$t_{94} = -37.6991118430775$$
$$t_{95} = -7.85398163397448$$
$$t_{96} = -63.4009836613048$$
$$t_{97} = 14.2476410900056$$
$$t_{98} = -63.9168256049229$$
$$t_{99} = -34.5575191894877$$
Signos de extremos en los puntos:
(9.529334405361963, 0.0055108542232601)
(47.166299316932744, 5.8097384599231e-29)
(37.7256128277765, 0.000351336059243502)
(-68.2746898137795, 2)
(24.18622483033663, 2)
(-1.5707963267948966, 1.85025446894204e-32)
(28.344922562267822, 1.20102098488764e-29)
(-30.511064545832454, 1.07982704351654e-30)
(80.14981921348375, 1.99547578460227e-25)
(34.55751918948773, 9.72936212698185e-30)
(-21.991148575128552, 1.46976458700862e-30)
(68.18643930330437, 3.37263900646877e-26)
(-72.28430132301244, 1.22976782797384e-28)
(-51.32455308671338, 2.70879499801747e-28)
(87.98732582580864, 2.35162333921379e-29)
(-75.41148348884815, 8.79227763509786e-5)
(-11.729252869079271, 2)
(54.24907663582436, 2)
(-53.669173990776365, 2)
(-15.771284874815882, 0.00201084252147885)
(-89.78284542916876, 5.69808665686428e-25)
(-37.75210142439493, 4.31930817406615e-30)
(10.105354786808867, 2)
(72.56628154280597, 5.49827422663806e-27)
(-46.776500758106835, 4.61251041218787e-27)
(-31.41592653589793, 2.99951956532372e-30)
(-40.585402259826694, 2)
(-15.83443874168566, 7.49879891330929e-31)
(100.54091078684232, 4.94638540860084e-5)
(-59.72375106325252, 6.91380423041194e-29)
(-15.707963267948966, 7.49879891330929e-31)
(-85.79020781537176, 2)
(76.54610723645023, 9.4995845168189e-26)
(63.0720690226406, 2)
(25.132741228718345, 1.91969252180718e-30)
(12.130374253135477, 2)
(0, 0)
(8.244896908126641, 7.21104632442734e-30)
(75.39822368615503, 1.72772326962646e-29)
(0.8603335890193797, 0.566292228465965)
(67.49768151897352, 2.64390428762807e-25)
(-83.9660742234445, 2)
(-50.26548245743669, 7.67877008722871e-30)
(20.184714407206833, 2)
(66.00374935325938, 1.92454699589753e-30)
(76.30335015349426, 2.30309745424817e-25)
(59.08808151179398, 2)
(97.40990477561171, 1.08019092984206e-28)
(91.30329503639689, 2)
(-51.1753017001883, 4.70708922874983e-27)
(-39.59288368446951, 5.95734224048085e-27)
(131.94689145077132, 7.69818798359012e-30)
(28.274333882308138, 2.42961084791221e-30)
(-9.633147248383352, 2.69956760879134e-31)
(-96.6873970931048, 2)
(59.72375106325252, 6.91380423041194e-29)
(-3.6967229225678127, 2.99951956532372e-32)
(4.342247535680812, 2)
(40.840704496667314, 7.68847597031319e-30)
(-66.00374935325938, 1.92454699589753e-30)
(2.316112130682953, 2)
(-28.30964285445201, 0.000623944646835302)
(56.32918595234723, 2)
(31.999500727538045, 2)
(-81.70588825569959, 2.11747992378503e-28)
(-135.0884841043611, 1.55538760012425e-28)
(-87.98732582580864, 2.35162333921379e-29)
(18.95516617783476, 1.07982704351654e-30)
(-30.99388649356027, 4.33361097810928e-29)
(-91.11716139446474, 6.02246101568735e-5)
(40.889626414412525, 7.68847597031319e-30)
(-44.005017920830845, 0.000258216645514578)
(94.2477796076938, 2.35502055154817e-29)
(78.20369233349476, 2)
(-57.098092440806816, 2)
(72.25663103256524, 8.11385462699113e-29)
(-18.902409956860023, 0.00139970121808476)
(16.220366243848606, 2)
(-69.14396555220625, 3.89174485079274e-29)
(48.823735227423796, 4.00248611286014e-26)
(44.005017920830845, 0.000258216645514578)
(6.283185307179586, 1.19980782612949e-31)
(64.9078784744433, 1.58922847564534e-25)
(-78.10054546282495, 2)
(21.991148575128552, 1.46976458700862e-30)
(-79.47591640849937, 1.55626109895089e-28)
(50.26548245743669, 7.67877008722871e-30)
(-94.26899628661319, 2.35502055154817e-29)
(100.03464572799854, 4.2433374167186e-27)
(-97.40990477561171, 1.08019092984206e-28)
(53.40707511102649, 4.32658835265357e-30)
(-49.92870256482977, 3.34837242674972e-27)
(81.41486309701827, 2.56265515794991e-26)
(-37.69911184307752, 4.31930817406615e-30)
(-7.853981633974483, 1.15640904308877e-29)
(-63.4009836613048, 1.05197766907004e-26)
(14.247641090005631, 2)
(-63.9168256049229, 2)
(-34.55751918948773, 9.72936212698185e-30)
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:
$$t_{1} = 47.1662993169327$$
$$t_{2} = -1.5707963267949$$
$$t_{3} = 28.3449225622678$$
$$t_{4} = -30.5110645458325$$
$$t_{5} = 80.1498192134838$$
$$t_{6} = 34.5575191894877$$
$$t_{7} = -21.9911485751286$$
$$t_{8} = 68.1864393033044$$
$$t_{9} = -72.2843013230124$$
$$t_{10} = -51.3245530867134$$
$$t_{11} = 87.9873258258086$$
$$t_{12} = -89.7828454291688$$
$$t_{13} = -37.7521014243949$$
$$t_{14} = 72.566281542806$$
$$t_{15} = -46.7765007581068$$
$$t_{16} = -31.4159265358979$$
$$t_{17} = -15.8344387416857$$
$$t_{18} = -59.7237510632525$$
$$t_{19} = -15.707963267949$$
$$t_{20} = 76.5461072364502$$
$$t_{21} = 25.1327412287183$$
$$t_{22} = 0$$
$$t_{23} = 8.24489690812664$$
$$t_{24} = 75.398223686155$$
$$t_{25} = 67.4976815189735$$
$$t_{26} = -50.2654824574367$$
$$t_{27} = 66.0037493532594$$
$$t_{28} = 76.3033501534943$$
$$t_{29} = 97.4099047756117$$
$$t_{30} = -51.1753017001883$$
$$t_{31} = -39.5928836844695$$
$$t_{32} = 131.946891450771$$
$$t_{33} = 28.2743338823081$$
$$t_{34} = -9.63314724838335$$
$$t_{35} = 59.7237510632525$$
$$t_{36} = -3.69672292256781$$
$$t_{37} = 40.8407044966673$$
$$t_{38} = -66.0037493532594$$
$$t_{39} = -81.7058882556996$$
$$t_{40} = -135.088484104361$$
$$t_{41} = -87.9873258258086$$
$$t_{42} = 18.9551661778348$$
$$t_{43} = -30.9938864935603$$
$$t_{44} = 40.8896264144125$$
$$t_{45} = 94.2477796076938$$
$$t_{46} = 72.2566310325652$$
$$t_{47} = -69.1439655522063$$
$$t_{48} = 48.8237352274238$$
$$t_{49} = 6.28318530717959$$
$$t_{50} = 64.9078784744433$$
$$t_{51} = 21.9911485751286$$
$$t_{52} = -79.4759164084994$$
$$t_{53} = 50.2654824574367$$
$$t_{54} = -94.2689962866132$$
$$t_{55} = 100.034645727999$$
$$t_{56} = -97.4099047756117$$
$$t_{57} = 53.4070751110265$$
$$t_{58} = -49.9287025648298$$
$$t_{59} = 81.4148630970183$$
$$t_{60} = -37.6991118430775$$
$$t_{61} = -7.85398163397448$$
$$t_{62} = -63.4009836613048$$
$$t_{63} = -34.5575191894877$$
Puntos máximos de la función:
$$t_{63} = 9.52933440536196$$
$$t_{63} = 37.7256128277765$$
$$t_{63} = -68.2746898137795$$
$$t_{63} = 24.1862248303366$$
$$t_{63} = -75.4114834888481$$
$$t_{63} = -11.7292528690793$$
$$t_{63} = 54.2490766358244$$
$$t_{63} = -53.6691739907764$$
$$t_{63} = -15.7712848748159$$
$$t_{63} = 10.1053547868089$$
$$t_{63} = -40.5854022598267$$
$$t_{63} = 100.540910786842$$
$$t_{63} = -85.7902078153718$$
$$t_{63} = 63.0720690226406$$
$$t_{63} = 12.1303742531355$$
$$t_{63} = 0.86033358901938$$
$$t_{63} = -83.9660742234445$$
$$t_{63} = 20.1847144072068$$
$$t_{63} = 59.088081511794$$
$$t_{63} = 91.3032950363969$$
$$t_{63} = -96.6873970931048$$
$$t_{63} = 4.34224753568081$$
$$t_{63} = 2.31611213068295$$
$$t_{63} = -28.309642854452$$
$$t_{63} = 56.3291859523472$$
$$t_{63} = 31.999500727538$$
$$t_{63} = -91.1171613944647$$
$$t_{63} = -44.0050179208308$$
$$t_{63} = 78.2036923334948$$
$$t_{63} = -57.0980924408068$$
$$t_{63} = -18.90240995686$$
$$t_{63} = 16.2203662438486$$
$$t_{63} = 44.0050179208308$$
$$t_{63} = -78.100545462825$$
$$t_{63} = 14.2476410900056$$
$$t_{63} = -63.9168256049229$$
Decrece en los intervalos
$$\left[131.946891450771, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -135.088484104361\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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:
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = $$
segunda derivada
$$4 \left(- \left(t \sin{\left(t \right)} - \cos{\left(t \right)}\right)^{2} \sin^{2}{\left(t \cos{\left(t \right)} \right)} + \left(t \sin{\left(t \right)} - \cos{\left(t \right)}\right)^{2} \cos^{2}{\left(t \cos{\left(t \right)} \right)} - \left(t \cos{\left(t \right)} + 2 \sin{\left(t \right)}\right) \sin{\left(t \cos{\left(t \right)} \right)} \cos{\left(t \cos{\left(t \right)} \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -97.7830311590133$$
$$t_{2} = 80.2185167624077$$
$$t_{3} = -94.4849248372462$$
$$t_{4} = -72.0110166705262$$
$$t_{5} = 0.414148600417418$$
$$t_{6} = 16.1155842337616$$
$$t_{7} = 31.4660600301311$$
$$t_{8} = -25.6119389787682$$
$$t_{9} = 4.13936129432959$$
$$t_{10} = 68.8641569033027$$
$$t_{11} = 12.2536883040809$$
$$t_{12} = -13.4417371631342$$
$$t_{13} = 78.3037959425222$$
$$t_{14} = -9.58943734874661$$
$$t_{15} = -45.4666308765017$$
$$t_{16} = 100.162654631$$
$$t_{17} = 28.2892642021859$$
$$t_{18} = -18.8719180936536$$
$$t_{19} = -39.450037546817$$
$$t_{20} = 22.506255410393$$
$$t_{21} = -62.8385781210019$$
$$t_{22} = -72.2784546775147$$
$$t_{23} = -2.00095402802644$$
$$t_{24} = -43.9919019418441$$
$$t_{25} = 25.1953555165641$$
$$t_{26} = -87.9825225189702$$
$$t_{27} = -40.4193507611885$$
$$t_{28} = 22.0103297645576$$
$$t_{29} = -69.2930393823738$$
$$t_{30} = 6.34889349436519$$
$$t_{31} = 66.1560040570041$$
$$t_{32} = -85.2896679607532$$
$$t_{33} = 65.9973464609666$$
$$t_{34} = 84.1098437222526$$
$$t_{35} = -12.0055841162715$$
$$t_{36} = -31.6908651932957$$
$$t_{37} = 37.9474455799602$$
$$t_{38} = -35.7930112666251$$
$$t_{39} = -19.7833627572702$$
$$t_{40} = 60.2505768180223$$
$$t_{41} = 37.7103154345428$$
$$t_{42} = 18.3861090567656$$
$$t_{43} = -56.2726236836019$$
$$t_{44} = -37.7409088447582$$
$$t_{45} = -100.258971024291$$
$$t_{46} = -59.7166750309169$$
$$t_{47} = -65.9973464609666$$
$$t_{48} = 92.0894166836725$$
$$t_{49} = 68.2594676990678$$
$$t_{50} = 2.00095402802644$$
$$t_{51} = -77.9591084572147$$
$$t_{52} = 47.9126824533039$$
$$t_{53} = 90.9746863467404$$
$$t_{54} = 97.4055660614587$$
$$t_{55} = -3.58478420710777$$
$$t_{56} = -91.2595059642504$$
$$t_{57} = -23.9311479752865$$
$$t_{58} = -15.7347663813144$$
$$t_{59} = -75.7959856559523$$
$$t_{60} = -22.0103297645576$$
$$t_{61} = -47.4658677288186$$
$$t_{62} = 87.9825225189702$$
$$t_{63} = 62.0665805687641$$
$$t_{64} = -229.343141411324$$
$$t_{65} = 94.2522635774755$$
$$t_{66} = -57.0108518476812$$
$$t_{67} = 50.2738876235636$$
$$t_{68} = 53.9451697098315$$
$$t_{69} = -49.8106474981215$$
$$t_{70} = -5.87453205244075$$
$$t_{71} = 8.14716168124798$$
$$t_{72} = 72.2624792469954$$
$$t_{73} = -79.9228069810566$$
$$t_{74} = -91.4128413418841$$
$$t_{75} = 10.2398833840334$$
$$t_{76} = 34.2089821268832$$
$$t_{77} = 74.117578238462$$
$$t_{78} = -94.2645130241871$$
$$t_{79} = 40.8792922758249$$
$$t_{80} = 86.1450883352177$$
$$t_{81} = -7.75406280871863$$
$$t_{82} = -28.2892642021859$$
$$t_{83} = -30.0822726898063$$
$$t_{84} = -33.930373820443$$
$$t_{85} = 21.7433169330724$$
$$t_{86} = 45.9570048435807$$
$$t_{87} = 94.2645130241871$$
$$t_{88} = -81.7007157154628$$
$$t_{89} = 57.9972944244986$$
$$t_{90} = -72.9229114257317$$
$$t_{91} = -64.1426117359263$$
$$t_{92} = -51.9423151666049$$
$$t_{93} = 24.8986782897786$$
$$t_{94} = 15.8077847472623$$
$$t_{95} = 43.9919019418441$$
$$t_{96} = 81.8440188512484$$
$$t_{97} = 97.176463821759$$
$$t_{98} = -53.6119622620201$$
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
$$\left[100.162654631, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -97.7830311590133\right]$$
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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:
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = $$
segunda derivada
$$4 \left(- \left(t \sin{\left(t \right)} - \cos{\left(t \right)}\right)^{2} \sin^{2}{\left(t \cos{\left(t \right)} \right)} + \left(t \sin{\left(t \right)} - \cos{\left(t \right)}\right)^{2} \cos^{2}{\left(t \cos{\left(t \right)} \right)} - \left(t \cos{\left(t \right)} + 2 \sin{\left(t \right)}\right) \sin{\left(t \cos{\left(t \right)} \right)} \cos{\left(t \cos{\left(t \right)} \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -97.7830311590133$$
$$t_{2} = 80.2185167624077$$
$$t_{3} = -94.4849248372462$$
$$t_{4} = -72.0110166705262$$
$$t_{5} = 0.414148600417418$$
$$t_{6} = 16.1155842337616$$
$$t_{7} = 31.4660600301311$$
$$t_{8} = -25.6119389787682$$
$$t_{9} = 4.13936129432959$$
$$t_{10} = 68.8641569033027$$
$$t_{11} = 12.2536883040809$$
$$t_{12} = -13.4417371631342$$
$$t_{13} = 78.3037959425222$$
$$t_{14} = -9.58943734874661$$
$$t_{15} = -45.4666308765017$$
$$t_{16} = 100.162654631$$
$$t_{17} = 28.2892642021859$$
$$t_{18} = -18.8719180936536$$
$$t_{19} = -39.450037546817$$
$$t_{20} = 22.506255410393$$
$$t_{21} = -62.8385781210019$$
$$t_{22} = -72.2784546775147$$
$$t_{23} = -2.00095402802644$$
$$t_{24} = -43.9919019418441$$
$$t_{25} = 25.1953555165641$$
$$t_{26} = -87.9825225189702$$
$$t_{27} = -40.4193507611885$$
$$t_{28} = 22.0103297645576$$
$$t_{29} = -69.2930393823738$$
$$t_{30} = 6.34889349436519$$
$$t_{31} = 66.1560040570041$$
$$t_{32} = -85.2896679607532$$
$$t_{33} = 65.9973464609666$$
$$t_{34} = 84.1098437222526$$
$$t_{35} = -12.0055841162715$$
$$t_{36} = -31.6908651932957$$
$$t_{37} = 37.9474455799602$$
$$t_{38} = -35.7930112666251$$
$$t_{39} = -19.7833627572702$$
$$t_{40} = 60.2505768180223$$
$$t_{41} = 37.7103154345428$$
$$t_{42} = 18.3861090567656$$
$$t_{43} = -56.2726236836019$$
$$t_{44} = -37.7409088447582$$
$$t_{45} = -100.258971024291$$
$$t_{46} = -59.7166750309169$$
$$t_{47} = -65.9973464609666$$
$$t_{48} = 92.0894166836725$$
$$t_{49} = 68.2594676990678$$
$$t_{50} = 2.00095402802644$$
$$t_{51} = -77.9591084572147$$
$$t_{52} = 47.9126824533039$$
$$t_{53} = 90.9746863467404$$
$$t_{54} = 97.4055660614587$$
$$t_{55} = -3.58478420710777$$
$$t_{56} = -91.2595059642504$$
$$t_{57} = -23.9311479752865$$
$$t_{58} = -15.7347663813144$$
$$t_{59} = -75.7959856559523$$
$$t_{60} = -22.0103297645576$$
$$t_{61} = -47.4658677288186$$
$$t_{62} = 87.9825225189702$$
$$t_{63} = 62.0665805687641$$
$$t_{64} = -229.343141411324$$
$$t_{65} = 94.2522635774755$$
$$t_{66} = -57.0108518476812$$
$$t_{67} = 50.2738876235636$$
$$t_{68} = 53.9451697098315$$
$$t_{69} = -49.8106474981215$$
$$t_{70} = -5.87453205244075$$
$$t_{71} = 8.14716168124798$$
$$t_{72} = 72.2624792469954$$
$$t_{73} = -79.9228069810566$$
$$t_{74} = -91.4128413418841$$
$$t_{75} = 10.2398833840334$$
$$t_{76} = 34.2089821268832$$
$$t_{77} = 74.117578238462$$
$$t_{78} = -94.2645130241871$$
$$t_{79} = 40.8792922758249$$
$$t_{80} = 86.1450883352177$$
$$t_{81} = -7.75406280871863$$
$$t_{82} = -28.2892642021859$$
$$t_{83} = -30.0822726898063$$
$$t_{84} = -33.930373820443$$
$$t_{85} = 21.7433169330724$$
$$t_{86} = 45.9570048435807$$
$$t_{87} = 94.2645130241871$$
$$t_{88} = -81.7007157154628$$
$$t_{89} = 57.9972944244986$$
$$t_{90} = -72.9229114257317$$
$$t_{91} = -64.1426117359263$$
$$t_{92} = -51.9423151666049$$
$$t_{93} = 24.8986782897786$$
$$t_{94} = 15.8077847472623$$
$$t_{95} = 43.9919019418441$$
$$t_{96} = 81.8440188512484$$
$$t_{97} = 97.176463821759$$
$$t_{98} = -53.6119622620201$$
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
$$\left[100.162654631, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -97.7830311590133\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con t->+oo y t->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{t \to -\infty}\left(2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{t \to \infty}\left(2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{t \to -\infty}\left(2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{t \to \infty}\left(2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 2*sin(t*cos(t))^2, dividida por t con t->+oo y t ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = t \lim_{t \to -\infty}\left(\frac{2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}}{t}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = t \lim_{t \to \infty}\left(\frac{2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}}{t}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = t \lim_{t \to -\infty}\left(\frac{2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}}{t}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = t \lim_{t \to \infty}\left(\frac{2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}}{t}\right)$$
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-t) и f = -f(-t).
Pues, comprobamos:
$$2 \sin^{2}{\left(t \cos{\left(t \right)} \right)} = 2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}$$
- Sí
$$2 \sin^{2}{\left(t \cos{\left(t \right)} \right)} = - 2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$2 \sin^{2}{\left(t \cos{\left(t \right)} \right)} = 2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}$$
- Sí
$$2 \sin^{2}{\left(t \cos{\left(t \right)} \right)} = - 2 \sin^{2}{\left(t \cos{\left(t \right)} \right)}$$
- No
es decir, función
es
par