Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^2*e^((-1)/x)
-
x^2*e^(2-x)
-
x+27/x^3
-
(x^2-8)/(x-3)
-
Expresiones idénticas
- - ochenta *cos(dos *t)- veinte *sin(dos *t)+(-cos(ocho *t/ cinco)-sin(ocho *t/ cinco))*exp(- cuatro *t/ cinco)
- menos 80 multiplicar por coseno de (2 multiplicar por t) menos 20 multiplicar por seno de (2 multiplicar por t) más ( menos coseno de (8 multiplicar por t dividir por 5) menos seno de (8 multiplicar por t dividir por 5)) multiplicar por exponente de ( menos 4 multiplicar por t dividir por 5)
- menos ochenta multiplicar por coseno de (dos multiplicar por t) menos veinte multiplicar por seno de (dos multiplicar por t) más ( menos coseno de (ocho multiplicar por t dividir por cinco) menos seno de (ocho multiplicar por t dividir por cinco)) multiplicar por exponente de ( menos cuatro multiplicar por t dividir por cinco)
- -80cos(2t)-20sin(2t)+(-cos(8t/5)-sin(8t/5))exp(-4t/5)
- -80cos2t-20sin2t+-cos8t/5-sin8t/5exp-4t/5
- -80*cos(2*t)-20*sin(2*t)+(-cos(8*t dividir por 5)-sin(8*t dividir por 5))*exp(-4*t dividir por 5)
-
Expresiones semejantes
- -80*cos(2*t)-20*sin(2*t)+(cos(8*t/5)-sin(8*t/5))*exp(-4*t/5)
- 80*cos(2*t)-20*sin(2*t)+(-cos(8*t/5)-sin(8*t/5))*exp(-4*t/5)
- -80*cos(2*t)-20*sin(2*t)-(-cos(8*t/5)-sin(8*t/5))*exp(-4*t/5)
- -80*cos(2*t)+20*sin(2*t)+(-cos(8*t/5)-sin(8*t/5))*exp(-4*t/5)
- -80*cos(2*t)-20*sin(2*t)+(-cos(8*t/5)-sin(8*t/5))*exp(4*t/5)
- -80*cos(2*t)-20*sin(2*t)+(-cos(8*t/5)+sin(8*t/5))*exp(-4*t/5)
-
Expresiones con funciones
- Coseno cos
- cos(x-pi)-2
- cos(x)/(2*sqrt(x))-sin(x)*sqrt(x)
- cos(x)/18+cos(x*sqrt(19))+sin(x*sqrt(19))
- cos(((4-x))/(2))
- cos(2*x+(pi/2))
- Seno sin
- sin(x-pi/4)*(-2)
- sin(pi*x)/asin(x)
- sin(x)*e^((1/2)*cot(x)^(2))
- sin(697*t)+sin(1209*t)
- sin(x)/1+sin(x)
- Coseno cos
- cos(x-pi)-2
- cos(x)/(2*sqrt(x))-sin(x)*sqrt(x)
- cos(x)/18+cos(x*sqrt(19))+sin(x*sqrt(19))
- cos(((4-x))/(2))
- cos(2*x+(pi/2))
- Seno sin
- sin(x-pi/4)*(-2)
- sin(pi*x)/asin(x)
- sin(x)*e^((1/2)*cot(x)^(2))
- sin(697*t)+sin(1209*t)
- sin(x)/1+sin(x)
- Exponente exp
- exp(2/(x-1))
- exp(1/2-3*x/2-lambertw(exp(3/2-9*x/2)/(2*x^3))/3)/x
- exp(-10/(x+13))
- exp*(1/(x-1))
- exp-exp^(-1/x)
Gráfico de la función y = -80*cos(2*t)-20*sin(2*t)+(-cos(8*t/5)-sin(8*t/5))*exp(-4*t/5)
Solución
Ha introducido
[src]
-4*t
----
/ /8*t\ /8*t\\ 5
f(t) = -80*cos(2*t) - 20*sin(2*t) + |- cos|---| - sin|---||*e
\ \ 5 / \ 5 //
$$f{\left(t \right)} = \left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}$$
f = -20*sin(2*t) - 80*cos(2*t) + (-sin((8*t)/5) - cos((8*t)/5))*exp((-4*t)/5)
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:
$$\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = 71.5937222007312$$
$$t_{2} = -20.1258244459766$$
$$t_{3} = -3.94456671584687$$
$$t_{4} = 41.7485919916282$$
$$t_{5} = 96.7264634294496$$
$$t_{6} = 76.3061111811159$$
$$t_{7} = 40.1777956648333$$
$$t_{8} = -14.2357147295654$$
$$t_{9} = 93.5848707758598$$
$$t_{10} = 46.4609809720129$$
$$t_{11} = 62.1689442399618$$
$$t_{12} = 74.735314854321$$
$$t_{13} = 33.8946103576537$$
$$t_{14} = -29.9433049808294$$
$$t_{15} = 84.1600928150904$$
$$t_{16} = 73.1645185275261$$
$$t_{17} = -35.8337912049964$$
$$t_{18} = 32.3238140308589$$
$$t_{19} = -24.0528186999891$$
$$t_{20} = 11.9034612609892$$
$$t_{21} = 99.8680560830394$$
$$t_{22} = -26.0163141767702$$
$$t_{23} = 92.0140744490649$$
$$t_{24} = 5.62030954186923$$
$$t_{25} = 82.5892964882955$$
$$t_{26} = -2.25034760639695$$
$$t_{27} = 18.1866470938183$$
$$t_{28} = -27.9798095775535$$
$$t_{29} = 8.76186304448589$$
$$t_{30} = -18.1623249088547$$
$$t_{31} = 30.753017704064$$
$$t_{32} = 24.4698323968631$$
$$t_{33} = -12.2737154451895$$
$$t_{34} = 21.3282397434098$$
$$t_{35} = 85.7308891418853$$
$$t_{36} = -22.0893226485089$$
$$t_{37} = 77.8769075079108$$
$$t_{38} = -6.09616227189675$$
$$t_{39} = 2.47986217962915$$
$$t_{40} = 49.6025736256027$$
$$t_{41} = 48.0317772988078$$
$$t_{42} = 15.0450544499278$$
$$t_{43} = 4.04975944775334$$
$$t_{44} = 38.6069993380384$$
$$t_{45} = 26.0406287236715$$
$$t_{46} = 55.8857589327823$$
$$t_{47} = -8.32763630241458$$
$$t_{48} = 90.44327812227$$
$$t_{49} = 19.7574434174736$$
$$t_{50} = 98.2972597562445$$
$$t_{51} = 27.6114250504723$$
$$t_{52} = 60.598147913167$$
$$t_{53} = 52.7441662791925$$
$$t_{54} = -37.7972866134998$$
$$t_{55} = -16.1988660622066$$
$$t_{56} = -33.8702957964884$$
$$t_{57} = 16.6158507742721$$
$$t_{58} = 54.3149626059874$$
$$t_{59} = 70.0229258739363$$
$$t_{60} = 10.3326632529713$$
$$t_{61} = 63.7397405667567$$
$$t_{62} = -31.9068003881223$$
$$t_{63} = -10.3124889464504$$
$$t_{64} = 68.4521295471414$$
o sea hay que resolver la ecuación:
$$\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = 71.5937222007312$$
$$t_{2} = -20.1258244459766$$
$$t_{3} = -3.94456671584687$$
$$t_{4} = 41.7485919916282$$
$$t_{5} = 96.7264634294496$$
$$t_{6} = 76.3061111811159$$
$$t_{7} = 40.1777956648333$$
$$t_{8} = -14.2357147295654$$
$$t_{9} = 93.5848707758598$$
$$t_{10} = 46.4609809720129$$
$$t_{11} = 62.1689442399618$$
$$t_{12} = 74.735314854321$$
$$t_{13} = 33.8946103576537$$
$$t_{14} = -29.9433049808294$$
$$t_{15} = 84.1600928150904$$
$$t_{16} = 73.1645185275261$$
$$t_{17} = -35.8337912049964$$
$$t_{18} = 32.3238140308589$$
$$t_{19} = -24.0528186999891$$
$$t_{20} = 11.9034612609892$$
$$t_{21} = 99.8680560830394$$
$$t_{22} = -26.0163141767702$$
$$t_{23} = 92.0140744490649$$
$$t_{24} = 5.62030954186923$$
$$t_{25} = 82.5892964882955$$
$$t_{26} = -2.25034760639695$$
$$t_{27} = 18.1866470938183$$
$$t_{28} = -27.9798095775535$$
$$t_{29} = 8.76186304448589$$
$$t_{30} = -18.1623249088547$$
$$t_{31} = 30.753017704064$$
$$t_{32} = 24.4698323968631$$
$$t_{33} = -12.2737154451895$$
$$t_{34} = 21.3282397434098$$
$$t_{35} = 85.7308891418853$$
$$t_{36} = -22.0893226485089$$
$$t_{37} = 77.8769075079108$$
$$t_{38} = -6.09616227189675$$
$$t_{39} = 2.47986217962915$$
$$t_{40} = 49.6025736256027$$
$$t_{41} = 48.0317772988078$$
$$t_{42} = 15.0450544499278$$
$$t_{43} = 4.04975944775334$$
$$t_{44} = 38.6069993380384$$
$$t_{45} = 26.0406287236715$$
$$t_{46} = 55.8857589327823$$
$$t_{47} = -8.32763630241458$$
$$t_{48} = 90.44327812227$$
$$t_{49} = 19.7574434174736$$
$$t_{50} = 98.2972597562445$$
$$t_{51} = 27.6114250504723$$
$$t_{52} = 60.598147913167$$
$$t_{53} = 52.7441662791925$$
$$t_{54} = -37.7972866134998$$
$$t_{55} = -16.1988660622066$$
$$t_{56} = -33.8702957964884$$
$$t_{57} = 16.6158507742721$$
$$t_{58} = 54.3149626059874$$
$$t_{59} = 70.0229258739363$$
$$t_{60} = 10.3326632529713$$
$$t_{61} = 63.7397405667567$$
$$t_{62} = -31.9068003881223$$
$$t_{63} = -10.3124889464504$$
$$t_{64} = 68.4521295471414$$
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
$$- \frac{4 \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}}{5} + \left(\frac{8 \sin{\left(\frac{8 t}{5} \right)}}{5} - \frac{8 \cos{\left(\frac{8 t}{5} \right)}}{5}\right) e^{\frac{\left(-1\right) 4 t}{5}} + 160 \sin{\left(2 t \right)} - 40 \cos{\left(2 t \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 22.1136379067686$$
$$t_{2} = 31.5384158674614$$
$$t_{3} = 29.9676195406663$$
$$t_{4} = 50.3879717890001$$
$$t_{5} = 72.3791203641287$$
$$t_{6} = 100.653454246437$$
$$t_{7} = 45.6755828086154$$
$$t_{8} = 77.0915093445134$$
$$t_{9} = 14.2596562316776$$
$$t_{10} = 64.5251387301542$$
$$t_{11} = 67.666731383744$$
$$t_{12} = 37.821601174641$$
$$t_{13} = -33.1783278478226$$
$$t_{14} = 231.029549370413$$
$$t_{15} = 36.2508048478461$$
$$t_{16} = 56.6711570961797$$
$$t_{17} = -25.3243462519939$$
$$t_{18} = -7.63561646197875$$
$$t_{19} = 97.511861592847$$
$$t_{20} = 95.9410652660521$$
$$t_{21} = 94.3702689392572$$
$$t_{22} = -39.0688140733743$$
$$t_{23} = 9.54726476204874$$
$$t_{24} = -29.2513370331717$$
$$t_{25} = 59.8127497497695$$
$$t_{26} = -23.3608507566309$$
$$t_{27} = -13.5440223795782$$
$$t_{28} = 34.6800085210512$$
$$t_{29} = 88.0870836320776$$
$$t_{30} = 86.5162873052827$$
$$t_{31} = -19.4338528652826$$
$$t_{32} = -11.5836856349292$$
$$t_{33} = 6.4056966162943$$
$$t_{34} = 80.2331019981032$$
$$t_{35} = 28.3968232138706$$
$$t_{36} = 58.2419534229746$$
$$t_{37} = -31.2148324394912$$
$$t_{38} = -9.62753408038416$$
$$t_{39} = 253.020697945542$$
$$t_{40} = 42.5339901550256$$
$$t_{41} = -27.2878416357148$$
$$t_{42} = 1.6946360179382$$
$$t_{43} = -5.01188415768002$$
$$t_{44} = 0.123351471449465$$
$$t_{45} = -1.45978595831549$$
$$t_{46} = 73.9499166909236$$
$$t_{47} = 51.958768115795$$
$$t_{48} = 89.6578799588725$$
$$t_{49} = 20.5428415804029$$
$$t_{50} = 53.5295644425899$$
$$t_{51} = 78.6623056713083$$
$$t_{52} = -35.1418232563624$$
$$t_{53} = 66.0959350569491$$
$$t_{54} = -37.1053186648767$$
$$t_{55} = -15.5068952914144$$
$$t_{56} = 75.5207130177185$$
$$t_{57} = 23.6844342334812$$
$$t_{58} = 15.8304526025542$$
$$t_{59} = 81.8038983248981$$
$$t_{60} = 7.97646933662247$$
$$t_{61} = -17.4703441920742$$
$$t_{62} = 44.1047864818205$$
$$t_{63} = 12.6888596750542$$
$$t_{64} = -21.3973541839018$$
$$t_{65} = 55.1003607693848$$
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:
$$t_{1} = 22.1136379067686$$
$$t_{2} = 31.5384158674614$$
$$t_{3} = 50.3879717890001$$
$$t_{4} = 72.3791203641287$$
$$t_{5} = 100.653454246437$$
$$t_{6} = 37.821601174641$$
$$t_{7} = 56.6711570961797$$
$$t_{8} = -7.63561646197875$$
$$t_{9} = 97.511861592847$$
$$t_{10} = 94.3702689392572$$
$$t_{11} = -39.0688140733743$$
$$t_{12} = 9.54726476204874$$
$$t_{13} = 59.8127497497695$$
$$t_{14} = -23.3608507566309$$
$$t_{15} = 34.6800085210512$$
$$t_{16} = 88.0870836320776$$
$$t_{17} = -19.4338528652826$$
$$t_{18} = -11.5836856349292$$
$$t_{19} = 6.4056966162943$$
$$t_{20} = 28.3968232138706$$
$$t_{21} = -31.2148324394912$$
$$t_{22} = -27.2878416357148$$
$$t_{23} = 0.123351471449465$$
$$t_{24} = 53.5295644425899$$
$$t_{25} = 78.6623056713083$$
$$t_{26} = -35.1418232563624$$
$$t_{27} = 66.0959350569491$$
$$t_{28} = -15.5068952914144$$
$$t_{29} = 75.5207130177185$$
$$t_{30} = 15.8304526025542$$
$$t_{31} = 81.8038983248981$$
$$t_{32} = 44.1047864818205$$
$$t_{33} = 12.6888596750542$$
Puntos máximos de la función:
$$t_{33} = 29.9676195406663$$
$$t_{33} = 45.6755828086154$$
$$t_{33} = 77.0915093445134$$
$$t_{33} = 14.2596562316776$$
$$t_{33} = 64.5251387301542$$
$$t_{33} = 67.666731383744$$
$$t_{33} = -33.1783278478226$$
$$t_{33} = 231.029549370413$$
$$t_{33} = 36.2508048478461$$
$$t_{33} = -25.3243462519939$$
$$t_{33} = 95.9410652660521$$
$$t_{33} = -29.2513370331717$$
$$t_{33} = -13.5440223795782$$
$$t_{33} = 86.5162873052827$$
$$t_{33} = 80.2331019981032$$
$$t_{33} = 58.2419534229746$$
$$t_{33} = -9.62753408038416$$
$$t_{33} = 253.020697945542$$
$$t_{33} = 42.5339901550256$$
$$t_{33} = 1.6946360179382$$
$$t_{33} = -5.01188415768002$$
$$t_{33} = -1.45978595831549$$
$$t_{33} = 73.9499166909236$$
$$t_{33} = 51.958768115795$$
$$t_{33} = 89.6578799588725$$
$$t_{33} = 20.5428415804029$$
$$t_{33} = -37.1053186648767$$
$$t_{33} = 23.6844342334812$$
$$t_{33} = 7.97646933662247$$
$$t_{33} = -17.4703441920742$$
$$t_{33} = -21.3973541839018$$
$$t_{33} = 55.1003607693848$$
Decrece en los intervalos
$$\left[100.653454246437, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -39.0688140733743\right]$$
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(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
$$- \frac{4 \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}}{5} + \left(\frac{8 \sin{\left(\frac{8 t}{5} \right)}}{5} - \frac{8 \cos{\left(\frac{8 t}{5} \right)}}{5}\right) e^{\frac{\left(-1\right) 4 t}{5}} + 160 \sin{\left(2 t \right)} - 40 \cos{\left(2 t \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 22.1136379067686$$
$$t_{2} = 31.5384158674614$$
$$t_{3} = 29.9676195406663$$
$$t_{4} = 50.3879717890001$$
$$t_{5} = 72.3791203641287$$
$$t_{6} = 100.653454246437$$
$$t_{7} = 45.6755828086154$$
$$t_{8} = 77.0915093445134$$
$$t_{9} = 14.2596562316776$$
$$t_{10} = 64.5251387301542$$
$$t_{11} = 67.666731383744$$
$$t_{12} = 37.821601174641$$
$$t_{13} = -33.1783278478226$$
$$t_{14} = 231.029549370413$$
$$t_{15} = 36.2508048478461$$
$$t_{16} = 56.6711570961797$$
$$t_{17} = -25.3243462519939$$
$$t_{18} = -7.63561646197875$$
$$t_{19} = 97.511861592847$$
$$t_{20} = 95.9410652660521$$
$$t_{21} = 94.3702689392572$$
$$t_{22} = -39.0688140733743$$
$$t_{23} = 9.54726476204874$$
$$t_{24} = -29.2513370331717$$
$$t_{25} = 59.8127497497695$$
$$t_{26} = -23.3608507566309$$
$$t_{27} = -13.5440223795782$$
$$t_{28} = 34.6800085210512$$
$$t_{29} = 88.0870836320776$$
$$t_{30} = 86.5162873052827$$
$$t_{31} = -19.4338528652826$$
$$t_{32} = -11.5836856349292$$
$$t_{33} = 6.4056966162943$$
$$t_{34} = 80.2331019981032$$
$$t_{35} = 28.3968232138706$$
$$t_{36} = 58.2419534229746$$
$$t_{37} = -31.2148324394912$$
$$t_{38} = -9.62753408038416$$
$$t_{39} = 253.020697945542$$
$$t_{40} = 42.5339901550256$$
$$t_{41} = -27.2878416357148$$
$$t_{42} = 1.6946360179382$$
$$t_{43} = -5.01188415768002$$
$$t_{44} = 0.123351471449465$$
$$t_{45} = -1.45978595831549$$
$$t_{46} = 73.9499166909236$$
$$t_{47} = 51.958768115795$$
$$t_{48} = 89.6578799588725$$
$$t_{49} = 20.5428415804029$$
$$t_{50} = 53.5295644425899$$
$$t_{51} = 78.6623056713083$$
$$t_{52} = -35.1418232563624$$
$$t_{53} = 66.0959350569491$$
$$t_{54} = -37.1053186648767$$
$$t_{55} = -15.5068952914144$$
$$t_{56} = 75.5207130177185$$
$$t_{57} = 23.6844342334812$$
$$t_{58} = 15.8304526025542$$
$$t_{59} = 81.8038983248981$$
$$t_{60} = 7.97646933662247$$
$$t_{61} = -17.4703441920742$$
$$t_{62} = 44.1047864818205$$
$$t_{63} = 12.6888596750542$$
$$t_{64} = -21.3973541839018$$
$$t_{65} = 55.1003607693848$$
Signos de extremos en los puntos:
(22.113637906768624, -82.4621124830361)
(31.538415867461374, -82.4621125123662)
(29.967619540666323, 82.462112512408)
(50.387971789000126, -82.4621125123532)
(72.37912036412868, -82.4621125123532)
(100.65345424643681, -82.4621125123532)
(45.67558280861543, 82.4621125123532)
(77.09150934451337, 82.4621125123532)
(14.25965623167759, 82.462128211435)
(64.5251387301542, 82.4621125123532)
(67.666731383744, 82.4621125123532)
(37.82160117464095, -82.4621125123531)
(-33.17832784782255, 425982495967.921)
(231.02954937041324, 82.4621125123532)
(36.250804847846055, 82.4621125123529)
(56.67115709617971, -82.4621125123532)
(-25.32434625199387, 795497849.085983)
(-7.63561646197875, -495.470592521725)
(97.51186159284703, -82.4621125123532)
(95.94106526605212, 82.4621125123532)
(94.37026893925723, -82.4621125123532)
(-39.06881407337434, -47419425119204.6)
(9.547264762048739, -82.4618768751647)
(-29.25133703317167, 18408372808.1336)
(59.8127497497695, -82.4621125123532)
(-23.36085075663088, -165367688.280377)
(-13.54402237957821, 64244.8026639157)
(34.680008521051164, -82.4621125123529)
(88.08708363207764, -82.4621125123532)
(86.51628730528275, 82.4621125123532)
(-19.433852865282553, -7146202.19545918)
(-11.583685634929193, -13332.4991441387)
(6.405696616294296, -82.4537058648811)
(80.23310199810317, 82.4621125123532)
(28.396823213870626, -82.4621125125044)
(58.24195342297461, 82.4621125123532)
(-31.21483243949115, -88553060862.2272)
(-9.627534080384159, 2707.99611206471)
(253.0206979455418, 82.4621125123532)
(42.53399015502564, 82.4621125123532)
(-27.28784163571483, -3826724717.67707)
(1.694636017938203, 82.5885965797596)
(-5.011884157680023, 118.222373155984)
(0.1233514714494654, -83.5280907318408)
(-1.4597859583154935, 86.9861261826466)
(73.94991669092357, 82.4621125123532)
(51.95876811579502, 82.4621125123532)
(89.65787995887254, 82.4621125123532)
(20.542841580402918, 82.4621124313744)
(53.529564442589916, -82.4621125123532)
(78.66230567130826, -82.4621125123532)
(-35.14182325636243, -2049179161232.81)
(66.09593505694909, -82.4621125123532)
(-37.10531866487666, 9857530004545.41)
(-15.50689529141443, -308896.285160714)
(75.52071301771846, -82.4621125123532)
(23.684434233481202, 82.4621125054119)
(15.830452602554237, -82.4621162293517)
(81.80389832489806, -82.4621125123532)
(7.976469336622474, 82.4601220911188)
(-17.470344192074243, 1485613.52149387)
(44.10478648182054, -82.4621125123532)
(12.688859675054172, -82.4621558758268)
(-21.397354183901783, 34376533.4513548)
(55.100360769384814, 82.4621125123532)
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:
$$t_{1} = 22.1136379067686$$
$$t_{2} = 31.5384158674614$$
$$t_{3} = 50.3879717890001$$
$$t_{4} = 72.3791203641287$$
$$t_{5} = 100.653454246437$$
$$t_{6} = 37.821601174641$$
$$t_{7} = 56.6711570961797$$
$$t_{8} = -7.63561646197875$$
$$t_{9} = 97.511861592847$$
$$t_{10} = 94.3702689392572$$
$$t_{11} = -39.0688140733743$$
$$t_{12} = 9.54726476204874$$
$$t_{13} = 59.8127497497695$$
$$t_{14} = -23.3608507566309$$
$$t_{15} = 34.6800085210512$$
$$t_{16} = 88.0870836320776$$
$$t_{17} = -19.4338528652826$$
$$t_{18} = -11.5836856349292$$
$$t_{19} = 6.4056966162943$$
$$t_{20} = 28.3968232138706$$
$$t_{21} = -31.2148324394912$$
$$t_{22} = -27.2878416357148$$
$$t_{23} = 0.123351471449465$$
$$t_{24} = 53.5295644425899$$
$$t_{25} = 78.6623056713083$$
$$t_{26} = -35.1418232563624$$
$$t_{27} = 66.0959350569491$$
$$t_{28} = -15.5068952914144$$
$$t_{29} = 75.5207130177185$$
$$t_{30} = 15.8304526025542$$
$$t_{31} = 81.8038983248981$$
$$t_{32} = 44.1047864818205$$
$$t_{33} = 12.6888596750542$$
Puntos máximos de la función:
$$t_{33} = 29.9676195406663$$
$$t_{33} = 45.6755828086154$$
$$t_{33} = 77.0915093445134$$
$$t_{33} = 14.2596562316776$$
$$t_{33} = 64.5251387301542$$
$$t_{33} = 67.666731383744$$
$$t_{33} = -33.1783278478226$$
$$t_{33} = 231.029549370413$$
$$t_{33} = 36.2508048478461$$
$$t_{33} = -25.3243462519939$$
$$t_{33} = 95.9410652660521$$
$$t_{33} = -29.2513370331717$$
$$t_{33} = -13.5440223795782$$
$$t_{33} = 86.5162873052827$$
$$t_{33} = 80.2331019981032$$
$$t_{33} = 58.2419534229746$$
$$t_{33} = -9.62753408038416$$
$$t_{33} = 253.020697945542$$
$$t_{33} = 42.5339901550256$$
$$t_{33} = 1.6946360179382$$
$$t_{33} = -5.01188415768002$$
$$t_{33} = -1.45978595831549$$
$$t_{33} = 73.9499166909236$$
$$t_{33} = 51.958768115795$$
$$t_{33} = 89.6578799588725$$
$$t_{33} = 20.5428415804029$$
$$t_{33} = -37.1053186648767$$
$$t_{33} = 23.6844342334812$$
$$t_{33} = 7.97646933662247$$
$$t_{33} = -17.4703441920742$$
$$t_{33} = -21.3973541839018$$
$$t_{33} = 55.1003607693848$$
Decrece en los intervalos
$$\left[100.653454246437, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -39.0688140733743\right]$$
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
$$16 \left(- \frac{4 \left(\sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{- \frac{4 t}{5}}}{25} + \frac{3 \left(\sin{\left(\frac{8 t}{5} \right)} + \cos{\left(\frac{8 t}{5} \right)}\right) e^{- \frac{4 t}{5}}}{25} + 5 \sin{\left(2 t \right)} + 20 \cos{\left(2 t \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 24.4698323968848$$
$$t_{2} = -34.4498553077171$$
$$t_{3} = 41.7485919916282$$
$$t_{4} = 62.1689442399618$$
$$t_{5} = -10.8953692123701$$
$$t_{6} = 96.7264634294496$$
$$t_{7} = 63.7397405667567$$
$$t_{8} = 15.0450544114441$$
$$t_{9} = 92.0140744490649$$
$$t_{10} = -18.7418780199612$$
$$t_{11} = -6.96873981791682$$
$$t_{12} = -14.8148916066741$$
$$t_{13} = 30.7530177040638$$
$$t_{14} = 48.0317772988078$$
$$t_{15} = -8.95142725490189$$
$$t_{16} = 10.332664453277$$
$$t_{17} = 49.6025736256027$$
$$t_{18} = -16.7783475312627$$
$$t_{19} = 2.47921972934951$$
$$t_{20} = 82.5892964882955$$
$$t_{21} = 40.1777956648333$$
$$t_{22} = 84.1600928150904$$
$$t_{23} = 8.76186927590472$$
$$t_{24} = 32.3238140308588$$
$$t_{25} = 77.8769075079108$$
$$t_{26} = 52.7441662791925$$
$$t_{27} = 5.6203497654622$$
$$t_{28} = 70.0229258739363$$
$$t_{29} = 19.7574434173839$$
$$t_{30} = -30.5228644907452$$
$$t_{31} = 76.3061111811159$$
$$t_{32} = 71.5937222007312$$
$$t_{33} = 21.3282397435501$$
$$t_{34} = -36.4133507162526$$
$$t_{35} = 46.4609809720129$$
$$t_{36} = 55.8857589327823$$
$$t_{37} = -2.27341022393598$$
$$t_{38} = 18.1866470915768$$
$$t_{39} = 99.8680560830394$$
$$t_{40} = 11.9034613089838$$
$$t_{41} = 60.598147913167$$
$$t_{42} = 54.3149626059874$$
$$t_{43} = 73.1645185275261$$
$$t_{44} = 85.7308891418853$$
$$t_{45} = 27.6114250504725$$
$$t_{46} = 16.6158507626354$$
$$t_{47} = 74.735314854321$$
$$t_{48} = 90.44327812227$$
$$t_{49} = 98.2972597562445$$
$$t_{50} = 4.04973363825643$$
$$t_{51} = 93.5848707758598$$
$$t_{52} = 26.0406287236757$$
$$t_{53} = 68.4521295471414$$
$$t_{54} = 38.6069993380384$$
$$t_{55} = -3.97172204320893$$
$$t_{56} = -24.6323783569943$$
$$t_{57} = 33.8946103576537$$
$$t_{58} = -12.8524732834532$$
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
$$\left[99.8680560830394, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -36.4133507162526\right]$$
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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
$$16 \left(- \frac{4 \left(\sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{- \frac{4 t}{5}}}{25} + \frac{3 \left(\sin{\left(\frac{8 t}{5} \right)} + \cos{\left(\frac{8 t}{5} \right)}\right) e^{- \frac{4 t}{5}}}{25} + 5 \sin{\left(2 t \right)} + 20 \cos{\left(2 t \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 24.4698323968848$$
$$t_{2} = -34.4498553077171$$
$$t_{3} = 41.7485919916282$$
$$t_{4} = 62.1689442399618$$
$$t_{5} = -10.8953692123701$$
$$t_{6} = 96.7264634294496$$
$$t_{7} = 63.7397405667567$$
$$t_{8} = 15.0450544114441$$
$$t_{9} = 92.0140744490649$$
$$t_{10} = -18.7418780199612$$
$$t_{11} = -6.96873981791682$$
$$t_{12} = -14.8148916066741$$
$$t_{13} = 30.7530177040638$$
$$t_{14} = 48.0317772988078$$
$$t_{15} = -8.95142725490189$$
$$t_{16} = 10.332664453277$$
$$t_{17} = 49.6025736256027$$
$$t_{18} = -16.7783475312627$$
$$t_{19} = 2.47921972934951$$
$$t_{20} = 82.5892964882955$$
$$t_{21} = 40.1777956648333$$
$$t_{22} = 84.1600928150904$$
$$t_{23} = 8.76186927590472$$
$$t_{24} = 32.3238140308588$$
$$t_{25} = 77.8769075079108$$
$$t_{26} = 52.7441662791925$$
$$t_{27} = 5.6203497654622$$
$$t_{28} = 70.0229258739363$$
$$t_{29} = 19.7574434173839$$
$$t_{30} = -30.5228644907452$$
$$t_{31} = 76.3061111811159$$
$$t_{32} = 71.5937222007312$$
$$t_{33} = 21.3282397435501$$
$$t_{34} = -36.4133507162526$$
$$t_{35} = 46.4609809720129$$
$$t_{36} = 55.8857589327823$$
$$t_{37} = -2.27341022393598$$
$$t_{38} = 18.1866470915768$$
$$t_{39} = 99.8680560830394$$
$$t_{40} = 11.9034613089838$$
$$t_{41} = 60.598147913167$$
$$t_{42} = 54.3149626059874$$
$$t_{43} = 73.1645185275261$$
$$t_{44} = 85.7308891418853$$
$$t_{45} = 27.6114250504725$$
$$t_{46} = 16.6158507626354$$
$$t_{47} = 74.735314854321$$
$$t_{48} = 90.44327812227$$
$$t_{49} = 98.2972597562445$$
$$t_{50} = 4.04973363825643$$
$$t_{51} = 93.5848707758598$$
$$t_{52} = 26.0406287236757$$
$$t_{53} = 68.4521295471414$$
$$t_{54} = 38.6069993380384$$
$$t_{55} = -3.97172204320893$$
$$t_{56} = -24.6323783569943$$
$$t_{57} = 33.8946103576537$$
$$t_{58} = -12.8524732834532$$
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
$$\left[99.8680560830394, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -36.4133507162526\right]$$
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:
$$y = \lim_{t \to -\infty}\left(\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}\right)$$
$$\lim_{t \to \infty}\left(\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}\right) = \left\langle -100, 100\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -100, 100\right\rangle$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{t \to -\infty}\left(\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}\right)$$
$$\lim_{t \to \infty}\left(\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}\right) = \left\langle -100, 100\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -100, 100\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función -80*cos(2*t) - 20*sin(2*t) + (-cos((8*t)/5) - sin((8*t)/5))*exp((-4*t)/5), dividida por t con t->+oo y t ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = t \lim_{t \to -\infty}\left(\frac{\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}}{t}\right)$$
$$\lim_{t \to \infty}\left(\frac{\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = t \lim_{t \to -\infty}\left(\frac{\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}}{t}\right)$$
$$\lim_{t \to \infty}\left(\frac{\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}}}{t}\right) = 0$$
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:
$$\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}} = \left(\sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{4 t}{5}} + 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}$$
- No
$$\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}} = - \left(\sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{4 t}{5}} - 20 \sin{\left(2 t \right)} + 80 \cos{\left(2 t \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}} = \left(\sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{4 t}{5}} + 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}$$
- No
$$\left(- 20 \sin{\left(2 t \right)} - 80 \cos{\left(2 t \right)}\right) + \left(- \sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{\left(-1\right) 4 t}{5}} = - \left(\sin{\left(\frac{8 t}{5} \right)} - \cos{\left(\frac{8 t}{5} \right)}\right) e^{\frac{4 t}{5}} - 20 \sin{\left(2 t \right)} + 80 \cos{\left(2 t \right)}$$
- No
es decir, función
no es
par ni impar
Gráfico