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-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3-12
-
x^3-12*x+7
-
x^3/3-2*x^2
-
x^2*e^((-1)/x)
-
Expresiones idénticas
- cos(tres , ocho *t)+cos(tres *t)
- coseno de (3,8 multiplicar por t) más coseno de (3 multiplicar por t)
- coseno de (tres , ocho multiplicar por t) más coseno de (tres multiplicar por t)
- cos(3,8t)+cos(3t)
- cos3,8t+cos3t
-
Expresiones semejantes
- cos(3,8*t)-cos(3*t)
-
Expresiones con funciones
- Coseno cos
- cos(2-sqrt(x))/(1+sqrt(x))
- cos(x^π)
- cos(3x/2)-4tg(7x/5)+2cos(3x)
- cos(x-1)*sin(x+1)
- cos(3*x^5)
- Coseno cos
- cos(2-sqrt(x))/(1+sqrt(x))
- cos(x^π)
- cos(3x/2)-4tg(7x/5)+2cos(3x)
- cos(x-1)*sin(x+1)
- cos(3*x^5)
Gráfico de la función y = cos(3,8*t)+cos(3*t)
Solución
Ha introducido
[src]
/19*t\
f(t) = cos|----| + cos(3*t)
\ 5 /
$$f{\left(t \right)} = \cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}$$
f = cos(3*t) + cos(19*t/5)
Gráfico de la función
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
$$- 3 \sin{\left(3 t \right)} - \frac{19 \sin{\left(\frac{19 t}{5} \right)}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 46.2130785585998$$
$$t_{2} = -11.6625933562164$$
$$t_{3} = 80.3571482096613$$
$$t_{4} = 42.4028265792191$$
$$t_{5} = 51.8449530284747$$
$$t_{6} = 4.04536991173257$$
$$t_{7} = 8.30960459256762$$
$$t_{8} = -43.6089593406824$$
$$t_{9} = -66.3467835349603$$
$$t_{10} = 17.5252951378654$$
$$t_{11} = -13.8906313980325$$
$$t_{12} = -72.0508422820368$$
$$t_{13} = -64.6491849417123$$
$$t_{14} = -70.2302117471772$$
$$t_{15} = -5.58784584818021$$
$$t_{16} = -129.709076055324$$
$$t_{17} = 98.2931495194264$$
$$t_{18} = 70.2302117471772$$
$$t_{19} = -39.7255311284656$$
$$t_{20} = 61.9210418265488$$
$$t_{21} = 10.1201174197688$$
$$t_{22} = -46.2130785585998$$
$$t_{23} = 99.835625455874$$
$$t_{24} = 94.2477796076938$$
$$t_{25} = 31.4159265358979$$
$$t_{26} = 7.39835867538134$$
$$t_{27} = 36.1369897605258$$
$$t_{28} = -29.5985946659815$$
$$t_{29} = 72.0508422820368$$
$$t_{30} = 37.904900593606$$
$$t_{31} = 58.110789847168$$
$$t_{32} = 1.81733186991646$$
$$t_{33} = 43.0785198921143$$
$$t_{34} = -91.5399833739277$$
$$t_{35} = 0$$
$$t_{36} = 18.415759501715$$
$$t_{37} = 34.123722769664$$
$$t_{38} = 60.1240568380298$$
$$t_{39} = -51.8449530284747$$
$$t_{40} = -40.6349157461388$$
$$t_{41} = -23.1063219433303$$
$$t_{42} = -73.818753115117$$
$$t_{43} = 49.831686037613$$
$$t_{44} = -82.0547468029093$$
$$t_{45} = 13.8906313980325$$
$$t_{46} = -1.81733186991646$$
$$t_{47} = 84.127662187925$$
$$t_{48} = 92.4304477377773$$
$$t_{49} = 27.9009960727335$$
$$t_{50} = 12.1930328047845$$
$$t_{51} = 85.9381750151262$$
$$t_{52} = 82.0547468029093$$
$$t_{53} = -84.127662187925$$
$$t_{54} = -4.04536991173257$$
$$t_{55} = -75.8320201059788$$
$$t_{56} = -87.7588055499857$$
$$t_{57} = 66.3467835349603$$
$$t_{58} = 96.0651114776103$$
$$t_{59} = -61.9210418265488$$
$$t_{60} = -97.7627100708583$$
$$t_{61} = 68.4196989199761$$
$$t_{62} = 44.4160935700808$$
$$t_{63} = -99.835625455874$$
$$t_{64} = -25.8280806877177$$
$$t_{65} = 48.034701049094$$
$$t_{66} = -36.1369897605258$$
$$t_{67} = -48.034701049094$$
$$t_{68} = 25.8280806877177$$
$$t_{69} = 56.3428790140878$$
$$t_{70} = -22.196937325657$$
$$t_{71} = -77.6290050944977$$
$$t_{72} = 24.0175678605166$$
$$t_{73} = 90.2024096959612$$
$$t_{74} = -10.1201174197688$$
$$t_{75} = -85.9381750151262$$
$$t_{76} = -96.0651114776103$$
$$t_{77} = -27.9009960727335$$
$$t_{78} = -94.2477796076938$$
$$t_{79} = -37.904900593606$$
$$t_{80} = -19.7533331796815$$
$$t_{81} = -49.831686037613$$
$$t_{82} = 86.8494209323125$$
$$t_{83} = -24.0175678605166$$
$$t_{84} = -53.612863861555$$
$$t_{85} = -15.707963267949$$
$$t_{86} = 32.326737781145$$
$$t_{87} = 75.8320201059788$$
$$t_{88} = 22.196937325657$$
$$t_{89} = -67.5529162964237$$
$$t_{90} = -63.742664317043$$
$$t_{91} = -34.123722769664$$
$$t_{92} = -60.1240568380298$$
$$t_{93} = -58.110789847168$$
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:
$$t_{1} = 80.3571482096613$$
$$t_{2} = 42.4028265792191$$
$$t_{3} = 51.8449530284747$$
$$t_{4} = 4.04536991173257$$
$$t_{5} = -43.6089593406824$$
$$t_{6} = 17.5252951378654$$
$$t_{7} = -13.8906313980325$$
$$t_{8} = -72.0508422820368$$
$$t_{9} = -70.2302117471772$$
$$t_{10} = -5.58784584818021$$
$$t_{11} = -129.709076055324$$
$$t_{12} = 98.2931495194264$$
$$t_{13} = 70.2302117471772$$
$$t_{14} = 61.9210418265488$$
$$t_{15} = 99.835625455874$$
$$t_{16} = 7.39835867538134$$
$$t_{17} = 72.0508422820368$$
$$t_{18} = -91.5399833739277$$
$$t_{19} = 34.123722769664$$
$$t_{20} = 60.1240568380298$$
$$t_{21} = -51.8449530284747$$
$$t_{22} = -40.6349157461388$$
$$t_{23} = -73.818753115117$$
$$t_{24} = -82.0547468029093$$
$$t_{25} = 13.8906313980325$$
$$t_{26} = 12.1930328047845$$
$$t_{27} = 82.0547468029093$$
$$t_{28} = -4.04536991173257$$
$$t_{29} = -61.9210418265488$$
$$t_{30} = 68.4196989199761$$
$$t_{31} = -99.835625455874$$
$$t_{32} = -25.8280806877177$$
$$t_{33} = 25.8280806877177$$
$$t_{34} = -22.196937325657$$
$$t_{35} = 24.0175678605166$$
$$t_{36} = 90.2024096959612$$
$$t_{37} = 86.8494209323125$$
$$t_{38} = -24.0175678605166$$
$$t_{39} = -53.612863861555$$
$$t_{40} = -15.707963267949$$
$$t_{41} = 32.326737781145$$
$$t_{42} = 22.196937325657$$
$$t_{43} = -63.742664317043$$
$$t_{44} = -34.123722769664$$
$$t_{45} = -60.1240568380298$$
Puntos máximos de la función:
$$t_{45} = 46.2130785585998$$
$$t_{45} = -11.6625933562164$$
$$t_{45} = 8.30960459256762$$
$$t_{45} = -66.3467835349603$$
$$t_{45} = -64.6491849417123$$
$$t_{45} = -39.7255311284656$$
$$t_{45} = 10.1201174197688$$
$$t_{45} = -46.2130785585998$$
$$t_{45} = 94.2477796076938$$
$$t_{45} = 31.4159265358979$$
$$t_{45} = 36.1369897605258$$
$$t_{45} = -29.5985946659815$$
$$t_{45} = 37.904900593606$$
$$t_{45} = 58.110789847168$$
$$t_{45} = 1.81733186991646$$
$$t_{45} = 43.0785198921143$$
$$t_{45} = 0$$
$$t_{45} = 18.415759501715$$
$$t_{45} = -23.1063219433303$$
$$t_{45} = 49.831686037613$$
$$t_{45} = -1.81733186991646$$
$$t_{45} = 84.127662187925$$
$$t_{45} = 92.4304477377773$$
$$t_{45} = 27.9009960727335$$
$$t_{45} = 85.9381750151262$$
$$t_{45} = -84.127662187925$$
$$t_{45} = -75.8320201059788$$
$$t_{45} = -87.7588055499857$$
$$t_{45} = 66.3467835349603$$
$$t_{45} = 96.0651114776103$$
$$t_{45} = -97.7627100708583$$
$$t_{45} = 44.4160935700808$$
$$t_{45} = 48.034701049094$$
$$t_{45} = -36.1369897605258$$
$$t_{45} = -48.034701049094$$
$$t_{45} = 56.3428790140878$$
$$t_{45} = -77.6290050944977$$
$$t_{45} = -10.1201174197688$$
$$t_{45} = -85.9381750151262$$
$$t_{45} = -96.0651114776103$$
$$t_{45} = -27.9009960727335$$
$$t_{45} = -94.2477796076938$$
$$t_{45} = -37.904900593606$$
$$t_{45} = -19.7533331796815$$
$$t_{45} = -49.831686037613$$
$$t_{45} = 75.8320201059788$$
$$t_{45} = -67.5529162964237$$
$$t_{45} = -58.110789847168$$
Decrece en los intervalos
$$\left[99.835625455874, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -129.709076055324\right]$$
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(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
$$- 3 \sin{\left(3 t \right)} - \frac{19 \sin{\left(\frac{19 t}{5} \right)}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 46.2130785585998$$
$$t_{2} = -11.6625933562164$$
$$t_{3} = 80.3571482096613$$
$$t_{4} = 42.4028265792191$$
$$t_{5} = 51.8449530284747$$
$$t_{6} = 4.04536991173257$$
$$t_{7} = 8.30960459256762$$
$$t_{8} = -43.6089593406824$$
$$t_{9} = -66.3467835349603$$
$$t_{10} = 17.5252951378654$$
$$t_{11} = -13.8906313980325$$
$$t_{12} = -72.0508422820368$$
$$t_{13} = -64.6491849417123$$
$$t_{14} = -70.2302117471772$$
$$t_{15} = -5.58784584818021$$
$$t_{16} = -129.709076055324$$
$$t_{17} = 98.2931495194264$$
$$t_{18} = 70.2302117471772$$
$$t_{19} = -39.7255311284656$$
$$t_{20} = 61.9210418265488$$
$$t_{21} = 10.1201174197688$$
$$t_{22} = -46.2130785585998$$
$$t_{23} = 99.835625455874$$
$$t_{24} = 94.2477796076938$$
$$t_{25} = 31.4159265358979$$
$$t_{26} = 7.39835867538134$$
$$t_{27} = 36.1369897605258$$
$$t_{28} = -29.5985946659815$$
$$t_{29} = 72.0508422820368$$
$$t_{30} = 37.904900593606$$
$$t_{31} = 58.110789847168$$
$$t_{32} = 1.81733186991646$$
$$t_{33} = 43.0785198921143$$
$$t_{34} = -91.5399833739277$$
$$t_{35} = 0$$
$$t_{36} = 18.415759501715$$
$$t_{37} = 34.123722769664$$
$$t_{38} = 60.1240568380298$$
$$t_{39} = -51.8449530284747$$
$$t_{40} = -40.6349157461388$$
$$t_{41} = -23.1063219433303$$
$$t_{42} = -73.818753115117$$
$$t_{43} = 49.831686037613$$
$$t_{44} = -82.0547468029093$$
$$t_{45} = 13.8906313980325$$
$$t_{46} = -1.81733186991646$$
$$t_{47} = 84.127662187925$$
$$t_{48} = 92.4304477377773$$
$$t_{49} = 27.9009960727335$$
$$t_{50} = 12.1930328047845$$
$$t_{51} = 85.9381750151262$$
$$t_{52} = 82.0547468029093$$
$$t_{53} = -84.127662187925$$
$$t_{54} = -4.04536991173257$$
$$t_{55} = -75.8320201059788$$
$$t_{56} = -87.7588055499857$$
$$t_{57} = 66.3467835349603$$
$$t_{58} = 96.0651114776103$$
$$t_{59} = -61.9210418265488$$
$$t_{60} = -97.7627100708583$$
$$t_{61} = 68.4196989199761$$
$$t_{62} = 44.4160935700808$$
$$t_{63} = -99.835625455874$$
$$t_{64} = -25.8280806877177$$
$$t_{65} = 48.034701049094$$
$$t_{66} = -36.1369897605258$$
$$t_{67} = -48.034701049094$$
$$t_{68} = 25.8280806877177$$
$$t_{69} = 56.3428790140878$$
$$t_{70} = -22.196937325657$$
$$t_{71} = -77.6290050944977$$
$$t_{72} = 24.0175678605166$$
$$t_{73} = 90.2024096959612$$
$$t_{74} = -10.1201174197688$$
$$t_{75} = -85.9381750151262$$
$$t_{76} = -96.0651114776103$$
$$t_{77} = -27.9009960727335$$
$$t_{78} = -94.2477796076938$$
$$t_{79} = -37.904900593606$$
$$t_{80} = -19.7533331796815$$
$$t_{81} = -49.831686037613$$
$$t_{82} = 86.8494209323125$$
$$t_{83} = -24.0175678605166$$
$$t_{84} = -53.612863861555$$
$$t_{85} = -15.707963267949$$
$$t_{86} = 32.326737781145$$
$$t_{87} = 75.8320201059788$$
$$t_{88} = 22.196937325657$$
$$t_{89} = -67.5529162964237$$
$$t_{90} = -63.742664317043$$
$$t_{91} = -34.123722769664$$
$$t_{92} = -60.1240568380298$$
$$t_{93} = -58.110789847168$$
Signos de extremos en los puntos:
(46.2130785585998, 1.86685146053908)
(-11.662593356216396, 0.0353698741883327)
(80.35714820966129, -1.48631817948336)
(42.40282657921908, -0.58810826364559)
(51.84495302847472, -0.588108263645574)
(4.045369911732569, -0.0353698741883335)
(8.309604592567624, 1.96641492229032)
(-43.60895934068243, -0.267915527280037)
(-66.34678353496032, 0.267915527280091)
(17.525295137865424, -1.48631817948333)
(-13.890631398032507, -1.48631817948334)
(-72.05084228203678, -1.70486332547012)
(-64.64918494171232, 1.48631817948335)
(-70.23021174717721, -1.96641492229031)
(-5.587845848180208, -1.21939794095159)
(-129.7090760553243, -0.0353698741883438)
(98.29314951942636, -0.0353698741883313)
(70.23021174717721, -1.96641492229031)
(-39.725531128465555, 1.96641492229032)
(61.921041826548766, -1.86685146053908)
(10.120117419768759, 1.21939794095159)
(-46.2130785585998, 1.86685146053908)
(99.83562545587401, -1.21939794095161)
(94.2477796076938, 2)
(31.41592653589793, 2)
(7.398358675381342, -1.96641492229032)
(36.13698976052575, 0.588108263645588)
(-29.598594665981473, 1.48631817948333)
(72.05084228203678, -1.70486332547012)
(37.90490059360599, 1.70486332547013)
(58.11078984716804, 0.588108263645612)
(1.8173318699164585, 1.48631817948333)
(43.07851989211433, 0.0353698741883345)
(-91.53998337392774, -0.914915940591087)
(0, 2)
(18.41575950171503, 0.914915940591107)
(34.12372276966399, -0.914915940591114)
(60.1240568380298, -0.914915940591091)
(-51.84495302847472, -0.588108263645574)
(-40.634915746138844, -1.70486332547014)
(-23.106321943330308, 1.96641492229031)
(-73.81875311511702, -0.588108263645603)
(49.83168603761296, 0.914915940591105)
(-82.05474680290929, -0.267915527280083)
(13.890631398032507, -1.48631817948334)
(-1.8173318699164585, 1.48631817948333)
(84.12766218792504, 1.21939794095158)
(92.43044773777734, 1.48631817948333)
(27.900996072733466, 0.267915527280059)
(12.193032804784503, -0.267915527280059)
(85.93817501512618, 1.96641492229031)
(82.05474680290929, -0.267915527280083)
(-84.12766218792504, 1.21939794095158)
(-4.045369911732569, -0.0353698741883335)
(-75.83202010597877, 0.914915940591078)
(-87.75880554998574, 1.70486332547017)
(66.34678353496032, 0.267915527280091)
(96.06511147761026, 1.48631817948337)
(-61.921041826548766, -1.86685146053908)
(-97.76271007085826, 0.267915527280075)
(68.41969891997607, -1.21939794095157)
(44.41609357008084, 0.914915940591104)
(-99.83562545587401, -1.21939794095161)
(-25.828080687717726, -1.21939794095159)
(48.034701049094, 1.86685146053909)
(-36.13698976052575, 0.588108263645588)
(-48.034701049094, 1.86685146053909)
(25.828080687717726, -1.21939794095159)
(56.342879014087806, 1.70486332547014)
(-22.19693732565702, -1.70486332547014)
(-77.62900509449773, 1.86685146053907)
(24.01756786051659, -1.96641492229032)
(90.20240969596122, -0.0353698741883275)
(-10.120117419768759, 1.21939794095159)
(-85.93817501512618, 1.96641492229031)
(-96.06511147761026, 1.48631817948337)
(-27.900996072733466, 0.267915527280059)
(-94.2477796076938, 2)
(-37.90490059360599, 1.70486332547013)
(-19.753333179681537, 0.0353698741883297)
(-49.83168603761296, 0.914915940591105)
(86.84942093231246, -1.96641492229032)
(-24.01756786051659, -1.96641492229032)
(-53.612863861554956, -1.70486332547013)
(-15.707963267948966, -2)
(32.326737781145034, -1.86685146053909)
(75.83202010597877, 0.914915940591078)
(22.19693732565702, -1.70486332547014)
(-67.55291629642369, 0.588108263645561)
(-63.742664317042966, -1.86685146053909)
(-34.12372276966399, -0.914915940591114)
(-60.1240568380298, -0.914915940591091)
(-58.11078984716804, 0.588108263645612)
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:
$$t_{1} = 80.3571482096613$$
$$t_{2} = 42.4028265792191$$
$$t_{3} = 51.8449530284747$$
$$t_{4} = 4.04536991173257$$
$$t_{5} = -43.6089593406824$$
$$t_{6} = 17.5252951378654$$
$$t_{7} = -13.8906313980325$$
$$t_{8} = -72.0508422820368$$
$$t_{9} = -70.2302117471772$$
$$t_{10} = -5.58784584818021$$
$$t_{11} = -129.709076055324$$
$$t_{12} = 98.2931495194264$$
$$t_{13} = 70.2302117471772$$
$$t_{14} = 61.9210418265488$$
$$t_{15} = 99.835625455874$$
$$t_{16} = 7.39835867538134$$
$$t_{17} = 72.0508422820368$$
$$t_{18} = -91.5399833739277$$
$$t_{19} = 34.123722769664$$
$$t_{20} = 60.1240568380298$$
$$t_{21} = -51.8449530284747$$
$$t_{22} = -40.6349157461388$$
$$t_{23} = -73.818753115117$$
$$t_{24} = -82.0547468029093$$
$$t_{25} = 13.8906313980325$$
$$t_{26} = 12.1930328047845$$
$$t_{27} = 82.0547468029093$$
$$t_{28} = -4.04536991173257$$
$$t_{29} = -61.9210418265488$$
$$t_{30} = 68.4196989199761$$
$$t_{31} = -99.835625455874$$
$$t_{32} = -25.8280806877177$$
$$t_{33} = 25.8280806877177$$
$$t_{34} = -22.196937325657$$
$$t_{35} = 24.0175678605166$$
$$t_{36} = 90.2024096959612$$
$$t_{37} = 86.8494209323125$$
$$t_{38} = -24.0175678605166$$
$$t_{39} = -53.612863861555$$
$$t_{40} = -15.707963267949$$
$$t_{41} = 32.326737781145$$
$$t_{42} = 22.196937325657$$
$$t_{43} = -63.742664317043$$
$$t_{44} = -34.123722769664$$
$$t_{45} = -60.1240568380298$$
Puntos máximos de la función:
$$t_{45} = 46.2130785585998$$
$$t_{45} = -11.6625933562164$$
$$t_{45} = 8.30960459256762$$
$$t_{45} = -66.3467835349603$$
$$t_{45} = -64.6491849417123$$
$$t_{45} = -39.7255311284656$$
$$t_{45} = 10.1201174197688$$
$$t_{45} = -46.2130785585998$$
$$t_{45} = 94.2477796076938$$
$$t_{45} = 31.4159265358979$$
$$t_{45} = 36.1369897605258$$
$$t_{45} = -29.5985946659815$$
$$t_{45} = 37.904900593606$$
$$t_{45} = 58.110789847168$$
$$t_{45} = 1.81733186991646$$
$$t_{45} = 43.0785198921143$$
$$t_{45} = 0$$
$$t_{45} = 18.415759501715$$
$$t_{45} = -23.1063219433303$$
$$t_{45} = 49.831686037613$$
$$t_{45} = -1.81733186991646$$
$$t_{45} = 84.127662187925$$
$$t_{45} = 92.4304477377773$$
$$t_{45} = 27.9009960727335$$
$$t_{45} = 85.9381750151262$$
$$t_{45} = -84.127662187925$$
$$t_{45} = -75.8320201059788$$
$$t_{45} = -87.7588055499857$$
$$t_{45} = 66.3467835349603$$
$$t_{45} = 96.0651114776103$$
$$t_{45} = -97.7627100708583$$
$$t_{45} = 44.4160935700808$$
$$t_{45} = 48.034701049094$$
$$t_{45} = -36.1369897605258$$
$$t_{45} = -48.034701049094$$
$$t_{45} = 56.3428790140878$$
$$t_{45} = -77.6290050944977$$
$$t_{45} = -10.1201174197688$$
$$t_{45} = -85.9381750151262$$
$$t_{45} = -96.0651114776103$$
$$t_{45} = -27.9009960727335$$
$$t_{45} = -94.2477796076938$$
$$t_{45} = -37.904900593606$$
$$t_{45} = -19.7533331796815$$
$$t_{45} = -49.831686037613$$
$$t_{45} = 75.8320201059788$$
$$t_{45} = -67.5529162964237$$
$$t_{45} = -58.110789847168$$
Decrece en los intervalos
$$\left[99.835625455874, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -129.709076055324\right]$$
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
$$- (9 \cos{\left(3 t \right)} + \frac{361 \cos{\left(\frac{19 t}{5} \right)}}{25}) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 59.7666348013518$$
$$t_{2} = 26.2178783740958$$
$$t_{3} = -83.737864501547$$
$$t_{4} = 64.1773575858188$$
$$t_{5} = -54.0794526621184$$
$$t_{6} = 79.8853208537677$$
$$t_{7} = 35.8289758205111$$
$$t_{8} = 88.1832213664235$$
$$t_{9} = -33.643671951768$$
$$t_{10} = 3.77258356308965$$
$$t_{11} = 78.0902198112852$$
$$t_{12} = -21.7725215092193$$
$$t_{13} = -44.0586715334028$$
$$t_{14} = 20.1210125525621$$
$$t_{15} = 0.449596528459635$$
$$t_{16} = 93.7981830792342$$
$$t_{17} = 7.85398163397448$$
$$t_{18} = 41.9258416420448$$
$$t_{19} = -92.0200341918237$$
$$t_{20} = -23.5619449019235$$
$$t_{21} = 44.0586715334028$$
$$t_{22} = 76.3120709238748$$
$$t_{23} = 96.4755250235639$$
$$t_{24} = -45.778385289824$$
$$t_{25} = -65.89707134224$$
$$t_{26} = -81.6050346101889$$
$$t_{27} = 69.7874159300674$$
$$t_{28} = 98.0203631707835$$
$$t_{29} = -59.7666348013518$$
$$t_{30} = 6.06455824127032$$
$$t_{31} = 100.312337848964$$
$$t_{32} = 62.3822565433362$$
$$t_{33} = 34.481144806342$$
$$t_{34} = -9.64340502667865$$
$$t_{35} = -27.6433429728083$$
$$t_{36} = -75.4745980693007$$
$$t_{37} = 74.1267670551317$$
$$t_{38} = -3.77258356308965$$
$$t_{39} = -36.6139746977001$$
$$t_{40} = 11.9353797048593$$
$$t_{41} = -57.6338049099937$$
$$t_{42} = -16.1575597964086$$
$$t_{43} = 55.8762902135243$$
$$t_{44} = 72.4752580984745$$
$$t_{45} = -12.6427449975049$$
$$t_{46} = 21.7725215092193$$
$$t_{47} = -98.0203631707835$$
$$t_{48} = -60.6041076559258$$
$$t_{49} = 82.3123999028345$$
$$t_{50} = 86.3937979737193$$
$$t_{51} = -89.8347303230806$$
$$t_{52} = -74.1267670551317$$
$$t_{53} = -88.1832213664235$$
$$t_{54} = -78.0902198112852$$
$$t_{55} = -93.7981830792342$$
$$t_{56} = -30.070422021875$$
$$t_{57} = -79.8853208537677$$
$$t_{58} = -2.22774541587006$$
$$t_{59} = -7.85398163397448$$
$$t_{60} = 54.0794526621184$$
$$t_{61} = -64.1773575858188$$
$$t_{62} = 30.070422021875$$
$$t_{63} = 40.1683269455754$$
$$t_{64} = 2.22774541587006$$
$$t_{65} = -35.8289758205111$$
$$t_{66} = -41.9258416420448$$
$$t_{67} = -26.2178783740958$$
$$t_{68} = -69.7874159300674$$
$$t_{69} = 16.1575597964086$$
$$t_{70} = -11.9353797048593$$
$$t_{71} = 24.4603636776264$$
$$t_{72} = -40.1683269455754$$
$$t_{73} = 52.321937965649$$
$$t_{74} = -17.935708683819$$
$$t_{75} = 38.3714893941695$$
$$t_{76} = 92.0200341918237$$
$$t_{77} = -68.029901233598$$
$$t_{78} = 31.8655230643576$$
$$t_{79} = -51.53693908846$$
$$t_{80} = 68.029901233598$$
$$t_{81} = 28.3507082654538$$
$$t_{82} = -50.189108074291$$
$$t_{83} = 48.4693943178698$$
$$t_{84} = 14.3624587539261$$
$$t_{85} = -31.8655230643576$$
$$t_{86} = 45.778385289824$$
$$t_{87} = 65.89707134224$$
$$t_{88} = -55.8762902135243$$
$$t_{89} = 17.935708683819$$
$$t_{90} = -95.5932841217167$$
$$t_{91} = 58.4188037871827$$
$$t_{92} = 90.4751960446041$$
$$t_{93} = -71.5842534814733$$
$$t_{94} = -47.5734863323065$$
$$t_{95} = 50.189108074291$$
$$t_{96} = -20.1210125525621$$
$$t_{97} = -6.06455824127032$$
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
$$\left[98.0203631707835, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.5932841217167\right]$$
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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
$$- (9 \cos{\left(3 t \right)} + \frac{361 \cos{\left(\frac{19 t}{5} \right)}}{25}) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 59.7666348013518$$
$$t_{2} = 26.2178783740958$$
$$t_{3} = -83.737864501547$$
$$t_{4} = 64.1773575858188$$
$$t_{5} = -54.0794526621184$$
$$t_{6} = 79.8853208537677$$
$$t_{7} = 35.8289758205111$$
$$t_{8} = 88.1832213664235$$
$$t_{9} = -33.643671951768$$
$$t_{10} = 3.77258356308965$$
$$t_{11} = 78.0902198112852$$
$$t_{12} = -21.7725215092193$$
$$t_{13} = -44.0586715334028$$
$$t_{14} = 20.1210125525621$$
$$t_{15} = 0.449596528459635$$
$$t_{16} = 93.7981830792342$$
$$t_{17} = 7.85398163397448$$
$$t_{18} = 41.9258416420448$$
$$t_{19} = -92.0200341918237$$
$$t_{20} = -23.5619449019235$$
$$t_{21} = 44.0586715334028$$
$$t_{22} = 76.3120709238748$$
$$t_{23} = 96.4755250235639$$
$$t_{24} = -45.778385289824$$
$$t_{25} = -65.89707134224$$
$$t_{26} = -81.6050346101889$$
$$t_{27} = 69.7874159300674$$
$$t_{28} = 98.0203631707835$$
$$t_{29} = -59.7666348013518$$
$$t_{30} = 6.06455824127032$$
$$t_{31} = 100.312337848964$$
$$t_{32} = 62.3822565433362$$
$$t_{33} = 34.481144806342$$
$$t_{34} = -9.64340502667865$$
$$t_{35} = -27.6433429728083$$
$$t_{36} = -75.4745980693007$$
$$t_{37} = 74.1267670551317$$
$$t_{38} = -3.77258356308965$$
$$t_{39} = -36.6139746977001$$
$$t_{40} = 11.9353797048593$$
$$t_{41} = -57.6338049099937$$
$$t_{42} = -16.1575597964086$$
$$t_{43} = 55.8762902135243$$
$$t_{44} = 72.4752580984745$$
$$t_{45} = -12.6427449975049$$
$$t_{46} = 21.7725215092193$$
$$t_{47} = -98.0203631707835$$
$$t_{48} = -60.6041076559258$$
$$t_{49} = 82.3123999028345$$
$$t_{50} = 86.3937979737193$$
$$t_{51} = -89.8347303230806$$
$$t_{52} = -74.1267670551317$$
$$t_{53} = -88.1832213664235$$
$$t_{54} = -78.0902198112852$$
$$t_{55} = -93.7981830792342$$
$$t_{56} = -30.070422021875$$
$$t_{57} = -79.8853208537677$$
$$t_{58} = -2.22774541587006$$
$$t_{59} = -7.85398163397448$$
$$t_{60} = 54.0794526621184$$
$$t_{61} = -64.1773575858188$$
$$t_{62} = 30.070422021875$$
$$t_{63} = 40.1683269455754$$
$$t_{64} = 2.22774541587006$$
$$t_{65} = -35.8289758205111$$
$$t_{66} = -41.9258416420448$$
$$t_{67} = -26.2178783740958$$
$$t_{68} = -69.7874159300674$$
$$t_{69} = 16.1575597964086$$
$$t_{70} = -11.9353797048593$$
$$t_{71} = 24.4603636776264$$
$$t_{72} = -40.1683269455754$$
$$t_{73} = 52.321937965649$$
$$t_{74} = -17.935708683819$$
$$t_{75} = 38.3714893941695$$
$$t_{76} = 92.0200341918237$$
$$t_{77} = -68.029901233598$$
$$t_{78} = 31.8655230643576$$
$$t_{79} = -51.53693908846$$
$$t_{80} = 68.029901233598$$
$$t_{81} = 28.3507082654538$$
$$t_{82} = -50.189108074291$$
$$t_{83} = 48.4693943178698$$
$$t_{84} = 14.3624587539261$$
$$t_{85} = -31.8655230643576$$
$$t_{86} = 45.778385289824$$
$$t_{87} = 65.89707134224$$
$$t_{88} = -55.8762902135243$$
$$t_{89} = 17.935708683819$$
$$t_{90} = -95.5932841217167$$
$$t_{91} = 58.4188037871827$$
$$t_{92} = 90.4751960446041$$
$$t_{93} = -71.5842534814733$$
$$t_{94} = -47.5734863323065$$
$$t_{95} = 50.189108074291$$
$$t_{96} = -20.1210125525621$$
$$t_{97} = -6.06455824127032$$
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
$$\left[98.0203631707835, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.5932841217167\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con t->+oo y t->-oo
$$\lim_{t \to -\infty}\left(\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}\right) = \left\langle -2, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{t \to \infty}\left(\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}\right) = \left\langle -2, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{t \to -\infty}\left(\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}\right) = \left\langle -2, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{t \to \infty}\left(\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}\right) = \left\langle -2, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -2, 2\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(19*t/5) + cos(3*t), dividida por t con t->+oo y t ->-oo
$$\lim_{t \to -\infty}\left(\frac{\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{t \to -\infty}\left(\frac{\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}}{t}\right) = 0$$
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(-t) и f = -f(-t).
Pues, comprobamos:
$$\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)} = \cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}$$
- Sí
$$\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)} = - \cos{\left(3 t \right)} - \cos{\left(\frac{19 t}{5} \right)}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)} = \cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)}$$
- Sí
$$\cos{\left(3 t \right)} + \cos{\left(\frac{19 t}{5} \right)} = - \cos{\left(3 t \right)} - \cos{\left(\frac{19 t}{5} \right)}$$
- No
es decir, función
es
par