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-
¿Cómo usar?
- Gráfico de la función y =:
-
-1+(-1-x)*exp(-6*x)
-
y=3x-2
-
y=2^x-3^x
-
x*ln
-
Expresiones idénticas
- cos(dos , ocho *t)+cos(dos *t)
- coseno de (2,8 multiplicar por t) más coseno de (2 multiplicar por t)
- coseno de (dos , ocho multiplicar por t) más coseno de (dos multiplicar por t)
- cos(2,8t)+cos(2t)
- cos2,8t+cos2t
-
Expresiones semejantes
- cos(2,8*t)-cos(2*t)
-
Expresiones con funciones
- Coseno cos
- cos(2x)/sin(x)
- cos2x+x^3-4
- cos(x)^cos(1)*exp(-x*sin(1))
- cos(3*x-2)
- cos(x+pi/4)+2
- Coseno cos
- cos(2x)/sin(x)
- cos2x+x^3-4
- cos(x)^cos(1)*exp(-x*sin(1))
- cos(3*x-2)
- cos(x+pi/4)+2
Gráfico de la función y = cos(2,8*t)+cos(2*t)
Solución
Ha introducido
[src]
/14*t\
f(t) = cos|----| + cos(2*t)
\ 5 /
$$f{\left(t \right)} = \cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}$$
f = cos(2*t) + cos(14*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
$$- 2 \sin{\left(2 t \right)} - \frac{14 \sin{\left(\frac{14 t}{5} \right)}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -64.102282685171$$
$$t_{2} = -82.8243833292252$$
$$t_{3} = 64.102282685171$$
$$t_{4} = 58.5472860823155$$
$$t_{5} = -74.2552493502645$$
$$t_{6} = 6.58355202059937$$
$$t_{7} = -82.109230984239$$
$$t_{8} = -30.1454969225228$$
$$t_{9} = -41.7810815606359$$
$$t_{10} = 66.40126771629$$
$$t_{11} = 12.1385486234548$$
$$t_{12} = -7.85398163397448$$
$$t_{13} = 86.3937979737193$$
$$t_{14} = 35.7004935253783$$
$$t_{15} = -43.5544751593528$$
$$t_{16} = 82.109230984239$$
$$t_{17} = 18.2191366587125$$
$$t_{18} = 33.9270999266614$$
$$t_{19} = -61.5614234584208$$
$$t_{20} = 97.8171942521879$$
$$t_{21} = 15.707963267949$$
$$t_{22} = -35.7004935253783$$
$$t_{23} = -15.707963267949$$
$$t_{24} = 48.394319417222$$
$$t_{25} = 1818.55432443759$$
$$t_{26} = 91.7366062169303$$
$$t_{27} = 27.8465118914038$$
$$t_{28} = -49.6350631946104$$
$$t_{29} = -33.9270999266614$$
$$t_{30} = -57.4890448285849$$
$$t_{31} = 68.1746613150069$$
$$t_{32} = -27.8465118914038$$
$$t_{33} = 71.9562643191455$$
$$t_{34} = 78.5398163397448$$
$$t_{35} = -91.7366062169303$$
$$t_{36} = 4.28456698948035$$
$$t_{37} = -83.8826245829558$$
$$t_{38} = 108.685313262268$$
$$t_{39} = 79.8102459531199$$
$$t_{40} = -23.5619449019235$$
$$t_{41} = -19.9925302574293$$
$$t_{42} = 45.8534601904718$$
$$t_{43} = 53.7074418244463$$
$$t_{44} = -3.56941464449414$$
$$t_{45} = -48.394319417222$$
$$t_{46} = 19.9925302574293$$
$$t_{47} = 7.85398163397448$$
$$t_{48} = 74.2552493502645$$
$$t_{49} = -73.1970080965338$$
$$t_{50} = 30.1454969225228$$
$$t_{51} = -5.342808243211$$
$$t_{52} = 14.4375336545739$$
$$t_{53} = 89.9632126182134$$
$$t_{54} = -79.8102459531199$$
$$t_{55} = 76.0286429489813$$
$$t_{56} = -94.2477796076938$$
$$t_{57} = -53.7074418244463$$
$$t_{58} = 19.2773779124431$$
$$t_{59} = -87.6642275870944$$
$$t_{60} = 57.4890448285849$$
$$t_{61} = -71.9562643191455$$
$$t_{62} = 56.2483010511965$$
$$t_{63} = 83.8826245829558$$
$$t_{64} = -18.2191366587125$$
$$t_{65} = 10.365155024738$$
$$t_{66} = 87.6642275870944$$
$$t_{67} = 40.5403377832475$$
$$t_{68} = -10.365155024738$$
$$t_{69} = 3.56941464449414$$
$$t_{70} = -26.0731182926869$$
$$t_{71} = -12.1385486234548$$
$$t_{72} = -99.5905878509048$$
$$t_{73} = -68.1746613150069$$
$$t_{74} = 2.51117339076349$$
$$t_{75} = 22.2915152885483$$
$$t_{76} = 0$$
$$t_{77} = 26.0731182926869$$
$$t_{78} = 60.3206796810324$$
$$t_{79} = -76.0286429489813$$
$$t_{80} = -45.8534601904718$$
$$t_{81} = -22.2915152885483$$
$$t_{82} = 34.9853411803921$$
$$t_{83} = -95.5182092210689$$
$$t_{84} = -31.4159265358979$$
$$t_{85} = 41.7810815606359$$
$$t_{86} = 37.9994785564973$$
$$t_{87} = 99.5905878509048$$
$$t_{88} = -56.2483010511965$$
$$t_{89} = 94.2477796076938$$
$$t_{90} = -69.4154050923952$$
$$t_{91} = -89.9632126182134$$
$$t_{92} = -37.9994785564973$$
$$t_{93} = -97.8171942521879$$
$$t_{94} = 52.4666980470579$$
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:
$$t_{1} = -64.102282685171$$
$$t_{2} = 64.102282685171$$
$$t_{3} = -82.109230984239$$
$$t_{4} = -30.1454969225228$$
$$t_{5} = -41.7810815606359$$
$$t_{6} = 66.40126771629$$
$$t_{7} = 12.1385486234548$$
$$t_{8} = -7.85398163397448$$
$$t_{9} = 86.3937979737193$$
$$t_{10} = -43.5544751593528$$
$$t_{11} = 82.109230984239$$
$$t_{12} = -61.5614234584208$$
$$t_{13} = 97.8171942521879$$
$$t_{14} = 48.394319417222$$
$$t_{15} = 1818.55432443759$$
$$t_{16} = 27.8465118914038$$
$$t_{17} = -57.4890448285849$$
$$t_{18} = 68.1746613150069$$
$$t_{19} = -27.8465118914038$$
$$t_{20} = -83.8826245829558$$
$$t_{21} = 108.685313262268$$
$$t_{22} = 79.8102459531199$$
$$t_{23} = -23.5619449019235$$
$$t_{24} = 45.8534601904718$$
$$t_{25} = -3.56941464449414$$
$$t_{26} = -48.394319417222$$
$$t_{27} = 7.85398163397448$$
$$t_{28} = -73.1970080965338$$
$$t_{29} = 30.1454969225228$$
$$t_{30} = -5.342808243211$$
$$t_{31} = 14.4375336545739$$
$$t_{32} = -79.8102459531199$$
$$t_{33} = 19.2773779124431$$
$$t_{34} = 57.4890448285849$$
$$t_{35} = 83.8826245829558$$
$$t_{36} = 10.365155024738$$
$$t_{37} = -10.365155024738$$
$$t_{38} = 3.56941464449414$$
$$t_{39} = -26.0731182926869$$
$$t_{40} = -12.1385486234548$$
$$t_{41} = -99.5905878509048$$
$$t_{42} = -68.1746613150069$$
$$t_{43} = 26.0731182926869$$
$$t_{44} = -45.8534601904718$$
$$t_{45} = 34.9853411803921$$
$$t_{46} = -95.5182092210689$$
$$t_{47} = 41.7810815606359$$
$$t_{48} = 99.5905878509048$$
$$t_{49} = -97.8171942521879$$
$$t_{50} = 52.4666980470579$$
Puntos máximos de la función:
$$t_{50} = -82.8243833292252$$
$$t_{50} = 58.5472860823155$$
$$t_{50} = -74.2552493502645$$
$$t_{50} = 6.58355202059937$$
$$t_{50} = 35.7004935253783$$
$$t_{50} = 18.2191366587125$$
$$t_{50} = 33.9270999266614$$
$$t_{50} = 15.707963267949$$
$$t_{50} = -35.7004935253783$$
$$t_{50} = -15.707963267949$$
$$t_{50} = 91.7366062169303$$
$$t_{50} = -49.6350631946104$$
$$t_{50} = -33.9270999266614$$
$$t_{50} = 71.9562643191455$$
$$t_{50} = 78.5398163397448$$
$$t_{50} = -91.7366062169303$$
$$t_{50} = 4.28456698948035$$
$$t_{50} = -19.9925302574293$$
$$t_{50} = 53.7074418244463$$
$$t_{50} = 19.9925302574293$$
$$t_{50} = 74.2552493502645$$
$$t_{50} = 89.9632126182134$$
$$t_{50} = 76.0286429489813$$
$$t_{50} = -94.2477796076938$$
$$t_{50} = -53.7074418244463$$
$$t_{50} = -87.6642275870944$$
$$t_{50} = -71.9562643191455$$
$$t_{50} = 56.2483010511965$$
$$t_{50} = -18.2191366587125$$
$$t_{50} = 87.6642275870944$$
$$t_{50} = 40.5403377832475$$
$$t_{50} = 2.51117339076349$$
$$t_{50} = 22.2915152885483$$
$$t_{50} = 0$$
$$t_{50} = 60.3206796810324$$
$$t_{50} = -76.0286429489813$$
$$t_{50} = -22.2915152885483$$
$$t_{50} = -31.4159265358979$$
$$t_{50} = 37.9994785564973$$
$$t_{50} = -56.2483010511965$$
$$t_{50} = 94.2477796076938$$
$$t_{50} = -69.4154050923952$$
$$t_{50} = -89.9632126182134$$
$$t_{50} = -37.9994785564973$$
Decrece en los intervalos
$$\left[1818.55432443759, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.5905878509048\right]$$
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(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
$$- 2 \sin{\left(2 t \right)} - \frac{14 \sin{\left(\frac{14 t}{5} \right)}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -64.102282685171$$
$$t_{2} = -82.8243833292252$$
$$t_{3} = 64.102282685171$$
$$t_{4} = 58.5472860823155$$
$$t_{5} = -74.2552493502645$$
$$t_{6} = 6.58355202059937$$
$$t_{7} = -82.109230984239$$
$$t_{8} = -30.1454969225228$$
$$t_{9} = -41.7810815606359$$
$$t_{10} = 66.40126771629$$
$$t_{11} = 12.1385486234548$$
$$t_{12} = -7.85398163397448$$
$$t_{13} = 86.3937979737193$$
$$t_{14} = 35.7004935253783$$
$$t_{15} = -43.5544751593528$$
$$t_{16} = 82.109230984239$$
$$t_{17} = 18.2191366587125$$
$$t_{18} = 33.9270999266614$$
$$t_{19} = -61.5614234584208$$
$$t_{20} = 97.8171942521879$$
$$t_{21} = 15.707963267949$$
$$t_{22} = -35.7004935253783$$
$$t_{23} = -15.707963267949$$
$$t_{24} = 48.394319417222$$
$$t_{25} = 1818.55432443759$$
$$t_{26} = 91.7366062169303$$
$$t_{27} = 27.8465118914038$$
$$t_{28} = -49.6350631946104$$
$$t_{29} = -33.9270999266614$$
$$t_{30} = -57.4890448285849$$
$$t_{31} = 68.1746613150069$$
$$t_{32} = -27.8465118914038$$
$$t_{33} = 71.9562643191455$$
$$t_{34} = 78.5398163397448$$
$$t_{35} = -91.7366062169303$$
$$t_{36} = 4.28456698948035$$
$$t_{37} = -83.8826245829558$$
$$t_{38} = 108.685313262268$$
$$t_{39} = 79.8102459531199$$
$$t_{40} = -23.5619449019235$$
$$t_{41} = -19.9925302574293$$
$$t_{42} = 45.8534601904718$$
$$t_{43} = 53.7074418244463$$
$$t_{44} = -3.56941464449414$$
$$t_{45} = -48.394319417222$$
$$t_{46} = 19.9925302574293$$
$$t_{47} = 7.85398163397448$$
$$t_{48} = 74.2552493502645$$
$$t_{49} = -73.1970080965338$$
$$t_{50} = 30.1454969225228$$
$$t_{51} = -5.342808243211$$
$$t_{52} = 14.4375336545739$$
$$t_{53} = 89.9632126182134$$
$$t_{54} = -79.8102459531199$$
$$t_{55} = 76.0286429489813$$
$$t_{56} = -94.2477796076938$$
$$t_{57} = -53.7074418244463$$
$$t_{58} = 19.2773779124431$$
$$t_{59} = -87.6642275870944$$
$$t_{60} = 57.4890448285849$$
$$t_{61} = -71.9562643191455$$
$$t_{62} = 56.2483010511965$$
$$t_{63} = 83.8826245829558$$
$$t_{64} = -18.2191366587125$$
$$t_{65} = 10.365155024738$$
$$t_{66} = 87.6642275870944$$
$$t_{67} = 40.5403377832475$$
$$t_{68} = -10.365155024738$$
$$t_{69} = 3.56941464449414$$
$$t_{70} = -26.0731182926869$$
$$t_{71} = -12.1385486234548$$
$$t_{72} = -99.5905878509048$$
$$t_{73} = -68.1746613150069$$
$$t_{74} = 2.51117339076349$$
$$t_{75} = 22.2915152885483$$
$$t_{76} = 0$$
$$t_{77} = 26.0731182926869$$
$$t_{78} = 60.3206796810324$$
$$t_{79} = -76.0286429489813$$
$$t_{80} = -45.8534601904718$$
$$t_{81} = -22.2915152885483$$
$$t_{82} = 34.9853411803921$$
$$t_{83} = -95.5182092210689$$
$$t_{84} = -31.4159265358979$$
$$t_{85} = 41.7810815606359$$
$$t_{86} = 37.9994785564973$$
$$t_{87} = 99.5905878509048$$
$$t_{88} = -56.2483010511965$$
$$t_{89} = 94.2477796076938$$
$$t_{90} = -69.4154050923952$$
$$t_{91} = -89.9632126182134$$
$$t_{92} = -37.9994785564973$$
$$t_{93} = -97.8171942521879$$
$$t_{94} = 52.4666980470579$$
Signos de extremos en los puntos:
(-64.10228268517098, -1.73979136688366)
(-82.82438332922517, 0.186393461135227)
(64.10228268517098, -1.73979136688366)
(58.547286082315516, 0.18639346113524)
(-74.25524935026449, 0.186393461135245)
(6.583552020599371, 1.73979136688365)
(-82.10923098423896, -0.186393461135251)
(-30.14549692252282, -1.73979136688365)
(-41.7810815606359, -1.03800098264585)
(66.40126771629001, -0.186393461135248)
(12.13854862345483, -0.186393461135242)
(-7.853981633974483, -2)
(86.39379797371932, -2)
(35.70049352537828, 0.18639346113524)
(-43.55447515935276, -0.186393461135236)
(82.10923098423896, -0.186393461135251)
(18.21913665871245, 1.03800098264584)
(33.92709992666142, 1.03800098264585)
(-61.561423458420755, -1.73979136688365)
(97.81719425218793, -0.18639346113526)
(15.707963267948966, 2)
(-35.70049352537828, 0.18639346113524)
(-15.707963267948966, 2)
(48.39431941722201, -1.73979136688365)
(1818.554324437586, -0.186393461135291)
(91.73660621693031, 1.03800098264584)
(27.846511891403797, -0.18639346113524)
(-49.63506319461038, 1.03800098264585)
(-33.92709992666142, 1.03800098264585)
(-57.489044828584866, -1.03800098264585)
(68.17466131500686, -1.03800098264583)
(-27.846511891403797, -0.18639346113524)
(71.95626431914546, 1.73979136688365)
(78.53981633974483, 2)
(-91.73660621693031, 1.03800098264584)
(4.2845669894803455, 0.186393461135242)
(-83.88262458295583, -1.03800098264584)
(108.68531326226766, -1.73979136688363)
(79.81024595311995, -1.73979136688365)
(-23.56194490192345, -2)
(-19.992530257429312, 0.18639346113524)
(45.853460190471786, -1.73979136688364)
(53.70744182444627, 1.73979136688365)
(-3.5694146444941373, -0.186393461135243)
(-48.39431941722201, -1.73979136688365)
(19.992530257429312, 0.18639346113524)
(7.853981633974483, -2)
(74.25524935026449, 0.186393461135245)
(-73.19700809653384, -1.03800098264584)
(30.14549692252282, -1.73979136688365)
(-5.342808243210999, -1.03800098264584)
(14.437533654573855, -1.73979136688365)
(89.96321261821345, 0.186393461135248)
(-79.81024595311995, -1.73979136688365)
(76.02864294898134, 1.03800098264583)
(-94.2477796076938, 2)
(-53.70744182444627, 1.73979136688365)
(19.277377912443104, -0.186393461135245)
(-87.66422758709443, 1.73979136688365)
(57.489044828584866, -1.03800098264585)
(-71.95626431914546, 1.73979136688365)
(56.24830105119649, 1.73979136688365)
(83.88262458295583, -1.03800098264584)
(-18.21913665871245, 1.03800098264584)
(10.365155024737968, -1.03800098264584)
(87.66422758709443, 1.73979136688365)
(40.54033778324753, 1.73979136688365)
(-10.365155024737968, -1.03800098264584)
(3.5694146444941373, -0.186393461135243)
(-26.073118292686935, -1.03800098264585)
(-12.13854862345483, -0.186393461135242)
(-99.5905878509048, -1.03800098264583)
(-68.17466131500686, -1.03800098264583)
(2.511173390763485, 1.03800098264584)
(22.29151528854834, 1.73979136688365)
(0, 2)
(26.073118292686935, -1.03800098264585)
(60.32067968103238, 1.03800098264583)
(-76.02864294898134, 1.03800098264583)
(-45.853460190471786, -1.73979136688364)
(-22.29151528854834, 1.73979136688365)
(34.98534118039207, -0.186393461135245)
(-95.51820922106891, -1.73979136688366)
(-31.41592653589793, 2)
(41.7810815606359, -1.03800098264585)
(37.9994785564973, 1.73979136688365)
(99.5905878509048, -1.03800098264583)
(-56.24830105119649, 1.73979136688365)
(94.2477796076938, 2)
(-69.41540509239523, 1.73979136688365)
(-89.96321261821345, 0.186393461135248)
(-37.9994785564973, 1.73979136688365)
(-97.81719425218793, -0.18639346113526)
(52.466698047057896, -1.03800098264583)
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:
$$t_{1} = -64.102282685171$$
$$t_{2} = 64.102282685171$$
$$t_{3} = -82.109230984239$$
$$t_{4} = -30.1454969225228$$
$$t_{5} = -41.7810815606359$$
$$t_{6} = 66.40126771629$$
$$t_{7} = 12.1385486234548$$
$$t_{8} = -7.85398163397448$$
$$t_{9} = 86.3937979737193$$
$$t_{10} = -43.5544751593528$$
$$t_{11} = 82.109230984239$$
$$t_{12} = -61.5614234584208$$
$$t_{13} = 97.8171942521879$$
$$t_{14} = 48.394319417222$$
$$t_{15} = 1818.55432443759$$
$$t_{16} = 27.8465118914038$$
$$t_{17} = -57.4890448285849$$
$$t_{18} = 68.1746613150069$$
$$t_{19} = -27.8465118914038$$
$$t_{20} = -83.8826245829558$$
$$t_{21} = 108.685313262268$$
$$t_{22} = 79.8102459531199$$
$$t_{23} = -23.5619449019235$$
$$t_{24} = 45.8534601904718$$
$$t_{25} = -3.56941464449414$$
$$t_{26} = -48.394319417222$$
$$t_{27} = 7.85398163397448$$
$$t_{28} = -73.1970080965338$$
$$t_{29} = 30.1454969225228$$
$$t_{30} = -5.342808243211$$
$$t_{31} = 14.4375336545739$$
$$t_{32} = -79.8102459531199$$
$$t_{33} = 19.2773779124431$$
$$t_{34} = 57.4890448285849$$
$$t_{35} = 83.8826245829558$$
$$t_{36} = 10.365155024738$$
$$t_{37} = -10.365155024738$$
$$t_{38} = 3.56941464449414$$
$$t_{39} = -26.0731182926869$$
$$t_{40} = -12.1385486234548$$
$$t_{41} = -99.5905878509048$$
$$t_{42} = -68.1746613150069$$
$$t_{43} = 26.0731182926869$$
$$t_{44} = -45.8534601904718$$
$$t_{45} = 34.9853411803921$$
$$t_{46} = -95.5182092210689$$
$$t_{47} = 41.7810815606359$$
$$t_{48} = 99.5905878509048$$
$$t_{49} = -97.8171942521879$$
$$t_{50} = 52.4666980470579$$
Puntos máximos de la función:
$$t_{50} = -82.8243833292252$$
$$t_{50} = 58.5472860823155$$
$$t_{50} = -74.2552493502645$$
$$t_{50} = 6.58355202059937$$
$$t_{50} = 35.7004935253783$$
$$t_{50} = 18.2191366587125$$
$$t_{50} = 33.9270999266614$$
$$t_{50} = 15.707963267949$$
$$t_{50} = -35.7004935253783$$
$$t_{50} = -15.707963267949$$
$$t_{50} = 91.7366062169303$$
$$t_{50} = -49.6350631946104$$
$$t_{50} = -33.9270999266614$$
$$t_{50} = 71.9562643191455$$
$$t_{50} = 78.5398163397448$$
$$t_{50} = -91.7366062169303$$
$$t_{50} = 4.28456698948035$$
$$t_{50} = -19.9925302574293$$
$$t_{50} = 53.7074418244463$$
$$t_{50} = 19.9925302574293$$
$$t_{50} = 74.2552493502645$$
$$t_{50} = 89.9632126182134$$
$$t_{50} = 76.0286429489813$$
$$t_{50} = -94.2477796076938$$
$$t_{50} = -53.7074418244463$$
$$t_{50} = -87.6642275870944$$
$$t_{50} = -71.9562643191455$$
$$t_{50} = 56.2483010511965$$
$$t_{50} = -18.2191366587125$$
$$t_{50} = 87.6642275870944$$
$$t_{50} = 40.5403377832475$$
$$t_{50} = 2.51117339076349$$
$$t_{50} = 22.2915152885483$$
$$t_{50} = 0$$
$$t_{50} = 60.3206796810324$$
$$t_{50} = -76.0286429489813$$
$$t_{50} = -22.2915152885483$$
$$t_{50} = -31.4159265358979$$
$$t_{50} = 37.9994785564973$$
$$t_{50} = -56.2483010511965$$
$$t_{50} = 94.2477796076938$$
$$t_{50} = -69.4154050923952$$
$$t_{50} = -89.9632126182134$$
$$t_{50} = -37.9994785564973$$
Decrece en los intervalos
$$\left[1818.55432443759, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.5905878509048\right]$$
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
$$- 4 \left(\cos{\left(2 t \right)} + \frac{49 \cos{\left(\frac{14 t}{5} \right)}}{25}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 28.4297852908531$$
$$t_{2} = -88.2378704517834$$
$$t_{3} = 51.9917301927765$$
$$t_{4} = 3.92699081698724$$
$$t_{5} = 59.845711826751$$
$$t_{6} = -47.7442276637183$$
$$t_{7} = 70.0654968458989$$
$$t_{8} = 90.3207887907066$$
$$t_{9} = 663.661448070844$$
$$t_{10} = -51.9917301927765$$
$$t_{11} = -96.0918520857579$$
$$t_{12} = -98.174770424681$$
$$t_{13} = 47.7442276637183$$
$$t_{14} = -9.69805411203856$$
$$t_{15} = 24.1822827617949$$
$$t_{16} = -44.137748558802$$
$$t_{17} = -93.6274417478224$$
$$t_{18} = -73.6719759508152$$
$$t_{19} = 85.7734601138479$$
$$t_{20} = 0.620337859871442$$
$$t_{21} = -13.8638907898849$$
$$t_{22} = 34.4020677809428$$
$$t_{23} = 73.6719759508152$$
$$t_{24} = 39.8902460297439$$
$$t_{25} = 13.8638907898849$$
$$t_{26} = -85.7734601138479$$
$$t_{27} = -1.84407247806408$$
$$t_{28} = -57.9640126828662$$
$$t_{29} = -16.3283011278204$$
$$t_{30} = -81.5259575847897$$
$$t_{31} = -3.92699081698724$$
$$t_{32} = 84.5497254956552$$
$$t_{33} = -37.4258356918083$$
$$t_{34} = 96.0918520857579$$
$$t_{35} = 723.186648185524$$
$$t_{36} = -70.0654968458989$$
$$t_{37} = 21.7178724238594$$
$$t_{38} = -11.7809724509617$$
$$t_{39} = -21.7178724238594$$
$$t_{40} = -83.4076567286745$$
$$t_{41} = 82.4668071567321$$
$$t_{42} = -63.4521909316673$$
$$t_{43} = 100.257688763604$$
$$t_{44} = 32.0362643957694$$
$$t_{45} = 16.3283011278204$$
$$t_{46} = 93.6274417478224$$
$$t_{47} = -6.00990915591041$$
$$t_{48} = -65.8179943168407$$
$$t_{49} = -55.5982092976928$$
$$t_{50} = -29.5718540578339$$
$$t_{51} = -54.3575335779499$$
$$t_{52} = -139.527596933477$$
$$t_{53} = 80.3838888178089$$
$$t_{54} = 54.3575335779499$$
$$t_{55} = 44.137748558802$$
$$t_{56} = 19.6349540849362$$
$$t_{57} = -50.1100310488918$$
$$t_{58} = -42.2560494149173$$
$$t_{59} = 46.5035519439755$$
$$t_{60} = -33.259999013962$$
$$t_{61} = 26.5480861469683$$
$$t_{62} = -105.087902486713$$
$$t_{63} = 62.2115152119244$$
$$t_{64} = -59.845711826751$$
$$t_{65} = 92.4037071296297$$
$$t_{66} = 98.174770424681$$
$$t_{67} = -19.6349540849362$$
$$t_{68} = -20.5758036568786$$
$$t_{69} = 72.5299071838344$$
$$t_{70} = -62.2115152119244$$
$$t_{71} = -32.0362643957694$$
$$t_{72} = -27.4889357189107$$
$$t_{73} = 88.2378704517834$$
$$t_{74} = 67.6996934607255$$
$$t_{75} = 6.00990915591041$$
$$t_{76} = -77.9194784798734$$
$$t_{77} = 9.69805411203856$$
$$t_{78} = 1.84407247806408$$
$$t_{79} = 42.2560494149173$$
$$t_{80} = -17.552035746013$$
$$t_{81} = 547.93464190015$$
$$t_{82} = -24.1822827617949$$
$$t_{83} = -100.257688763604$$
$$t_{84} = 57.9640126828662$$
$$t_{85} = 50.1100310488918$$
$$t_{86} = -67.6996934607255$$
$$t_{87} = 11.7809724509617$$
$$t_{88} = 8.47431949384593$$
$$t_{89} = -75.5536750947$$
$$t_{90} = -80.3838888178089$$
$$t_{91} = -39.8902460297439$$
$$t_{92} = 65.8179943168407$$
$$t_{93} = 77.9194784798734$$
$$t_{94} = -35.3429173528852$$
$$t_{95} = 36.2837669248275$$
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
$$\left[723.186648185524, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -105.087902486713\right]$$
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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
$$- 4 \left(\cos{\left(2 t \right)} + \frac{49 \cos{\left(\frac{14 t}{5} \right)}}{25}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 28.4297852908531$$
$$t_{2} = -88.2378704517834$$
$$t_{3} = 51.9917301927765$$
$$t_{4} = 3.92699081698724$$
$$t_{5} = 59.845711826751$$
$$t_{6} = -47.7442276637183$$
$$t_{7} = 70.0654968458989$$
$$t_{8} = 90.3207887907066$$
$$t_{9} = 663.661448070844$$
$$t_{10} = -51.9917301927765$$
$$t_{11} = -96.0918520857579$$
$$t_{12} = -98.174770424681$$
$$t_{13} = 47.7442276637183$$
$$t_{14} = -9.69805411203856$$
$$t_{15} = 24.1822827617949$$
$$t_{16} = -44.137748558802$$
$$t_{17} = -93.6274417478224$$
$$t_{18} = -73.6719759508152$$
$$t_{19} = 85.7734601138479$$
$$t_{20} = 0.620337859871442$$
$$t_{21} = -13.8638907898849$$
$$t_{22} = 34.4020677809428$$
$$t_{23} = 73.6719759508152$$
$$t_{24} = 39.8902460297439$$
$$t_{25} = 13.8638907898849$$
$$t_{26} = -85.7734601138479$$
$$t_{27} = -1.84407247806408$$
$$t_{28} = -57.9640126828662$$
$$t_{29} = -16.3283011278204$$
$$t_{30} = -81.5259575847897$$
$$t_{31} = -3.92699081698724$$
$$t_{32} = 84.5497254956552$$
$$t_{33} = -37.4258356918083$$
$$t_{34} = 96.0918520857579$$
$$t_{35} = 723.186648185524$$
$$t_{36} = -70.0654968458989$$
$$t_{37} = 21.7178724238594$$
$$t_{38} = -11.7809724509617$$
$$t_{39} = -21.7178724238594$$
$$t_{40} = -83.4076567286745$$
$$t_{41} = 82.4668071567321$$
$$t_{42} = -63.4521909316673$$
$$t_{43} = 100.257688763604$$
$$t_{44} = 32.0362643957694$$
$$t_{45} = 16.3283011278204$$
$$t_{46} = 93.6274417478224$$
$$t_{47} = -6.00990915591041$$
$$t_{48} = -65.8179943168407$$
$$t_{49} = -55.5982092976928$$
$$t_{50} = -29.5718540578339$$
$$t_{51} = -54.3575335779499$$
$$t_{52} = -139.527596933477$$
$$t_{53} = 80.3838888178089$$
$$t_{54} = 54.3575335779499$$
$$t_{55} = 44.137748558802$$
$$t_{56} = 19.6349540849362$$
$$t_{57} = -50.1100310488918$$
$$t_{58} = -42.2560494149173$$
$$t_{59} = 46.5035519439755$$
$$t_{60} = -33.259999013962$$
$$t_{61} = 26.5480861469683$$
$$t_{62} = -105.087902486713$$
$$t_{63} = 62.2115152119244$$
$$t_{64} = -59.845711826751$$
$$t_{65} = 92.4037071296297$$
$$t_{66} = 98.174770424681$$
$$t_{67} = -19.6349540849362$$
$$t_{68} = -20.5758036568786$$
$$t_{69} = 72.5299071838344$$
$$t_{70} = -62.2115152119244$$
$$t_{71} = -32.0362643957694$$
$$t_{72} = -27.4889357189107$$
$$t_{73} = 88.2378704517834$$
$$t_{74} = 67.6996934607255$$
$$t_{75} = 6.00990915591041$$
$$t_{76} = -77.9194784798734$$
$$t_{77} = 9.69805411203856$$
$$t_{78} = 1.84407247806408$$
$$t_{79} = 42.2560494149173$$
$$t_{80} = -17.552035746013$$
$$t_{81} = 547.93464190015$$
$$t_{82} = -24.1822827617949$$
$$t_{83} = -100.257688763604$$
$$t_{84} = 57.9640126828662$$
$$t_{85} = 50.1100310488918$$
$$t_{86} = -67.6996934607255$$
$$t_{87} = 11.7809724509617$$
$$t_{88} = 8.47431949384593$$
$$t_{89} = -75.5536750947$$
$$t_{90} = -80.3838888178089$$
$$t_{91} = -39.8902460297439$$
$$t_{92} = 65.8179943168407$$
$$t_{93} = 77.9194784798734$$
$$t_{94} = -35.3429173528852$$
$$t_{95} = 36.2837669248275$$
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
$$\left[723.186648185524, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -105.087902486713\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con t->+oo y t->-oo
$$\lim_{t \to -\infty}\left(\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}\right) = \left\langle -2, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{t \to \infty}\left(\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}\right) = \left\langle -2, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{t \to -\infty}\left(\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}\right) = \left\langle -2, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{t \to \infty}\left(\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}\right) = \left\langle -2, 2\right\rangle$$
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(14*t/5) + cos(2*t), dividida por t con t->+oo y t ->-oo
$$\lim_{t \to -\infty}\left(\frac{\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{t \to -\infty}\left(\frac{\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}}{t}\right) = 0$$
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:
$$\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)} = \cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}$$
- Sí
$$\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)} = - \cos{\left(2 t \right)} - \cos{\left(\frac{14 t}{5} \right)}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)} = \cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)}$$
- Sí
$$\cos{\left(2 t \right)} + \cos{\left(\frac{14 t}{5} \right)} = - \cos{\left(2 t \right)} - \cos{\left(\frac{14 t}{5} \right)}$$
- No
es decir, función
es
par