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^3+3*x^2+2
-
-sqrt(3*x+1)
-
x*(-1-log(x))
-
-exp(-2*x)-exp(2*x)
-
Expresiones idénticas
- - cuatro . setenta y nueve *exp(- dos *t)- seis mil seiscientos ochenta y uno . setenta y siete *exp(- cero . dos *t)+ tres mil ciento cincuenta y tres . setenta y siete *exp(- cero . tres *t)- dos . treinta y nueve *t*cos(t^ dos)+ uno . cuarenta y dos *t- cuatrocientos dos . veintiocho + cuatro . setenta y ocho *t*sin(t)*cos(t)+ cero . tres *sin(t)*cos(t)+ cuatro . cincuenta y siete *cos(t^ dos)
- menos 4.79 multiplicar por exponente de ( menos 2 multiplicar por t) menos 6681.77 multiplicar por exponente de ( menos 0.2 multiplicar por t) más 3153.77 multiplicar por exponente de ( menos 0.3 multiplicar por t) menos 2.39 multiplicar por t multiplicar por coseno de (t al cuadrado ) más 1.42 multiplicar por t menos 402.28 más 4.78 multiplicar por t multiplicar por seno de (t) multiplicar por coseno de (t) más 0.3 multiplicar por seno de (t) multiplicar por coseno de (t) más 4.57 multiplicar por coseno de (t al cuadrado )
- menos cuatro . setenta y nueve multiplicar por exponente de ( menos dos multiplicar por t) menos seis mil seiscientos ochenta y uno . setenta y siete multiplicar por exponente de ( menos cero . dos multiplicar por t) más tres mil ciento cincuenta y tres . setenta y siete multiplicar por exponente de ( menos cero . tres multiplicar por t) menos dos . treinta y nueve multiplicar por t multiplicar por coseno de (t en el grado dos) más uno . cuarenta y dos multiplicar por t menos cuatrocientos dos . veintiocho más cuatro . setenta y ocho multiplicar por t multiplicar por seno de (t) multiplicar por coseno de (t) más cero . tres multiplicar por seno de (t) multiplicar por coseno de (t) más cuatro . cincuenta y siete multiplicar por coseno de (t en el grado dos)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t2)
- -4.79*exp-2*t-6681.77*exp-0.2*t+3153.77*exp-0.3*t-2.39*t*cost2+1.42*t-402.28+4.78*t*sint*cost+0.3*sint*cost+4.57*cost2
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t²)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t²)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t en el grado 2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t en el grado 2)
- -4.79exp(-2t)-6681.77exp(-0.2t)+3153.77exp(-0.3t)-2.39tcos(t^2)+1.42t-402.28+4.78tsin(t)cos(t)+0.3sin(t)cos(t)+4.57cos(t^2)
- -4.79exp(-2t)-6681.77exp(-0.2t)+3153.77exp(-0.3t)-2.39tcos(t2)+1.42t-402.28+4.78tsin(t)cos(t)+0.3sin(t)cos(t)+4.57cos(t2)
- -4.79exp-2t-6681.77exp-0.2t+3153.77exp-0.3t-2.39tcost2+1.42t-402.28+4.78tsintcost+0.3sintcost+4.57cost2
- -4.79exp-2t-6681.77exp-0.2t+3153.77exp-0.3t-2.39tcost^2+1.42t-402.28+4.78tsintcost+0.3sintcost+4.57cost^2
-
Expresiones semejantes
- -4.79*exp(-2*t)+6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
- 4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t+402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)+2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28-4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)-0.3*sin(t)*cos(t)+4.57*cos(t^2)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)-1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)-3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
- -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)-4.57*cos(t^2)
- -4.79*exp(-2*t)-6681.77*exp(0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
- -4.79*exp(2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
-
Expresiones con funciones
- Exponente exp
- exp(1+x)/(-1+exp(1+x))
- exp(-x)*sin(x)
- exp(2-x)*(2-x)^-1
- exp(2*x)/20+(-cos(x)-sin(x))*exp(x)+(-cos(x)-sin(x))*exp(-x)
- exp(-2*x-log(2)*log(x))
- Exponente exp
- exp(1+x)/(-1+exp(1+x))
- exp(-x)*sin(x)
- exp(2-x)*(2-x)^-1
- exp(2*x)/20+(-cos(x)-sin(x))*exp(x)+(-cos(x)-sin(x))*exp(-x)
- exp(-2*x-log(2)*log(x))
- Exponente exp
- exp(1+x)/(-1+exp(1+x))
- exp(-x)*sin(x)
- exp(2-x)*(2-x)^-1
- exp(2*x)/20+(-cos(x)-sin(x))*exp(x)+(-cos(x)-sin(x))*exp(-x)
- exp(-2*x-log(2)*log(x))
- Coseno cos
- cos(x/2)^(2)
- cos(2*x)-sqrt(3)/2
- cos(2*x*sqrt(2))+sin(2*x*sqrt(2))
- cos(2/x)-2*sin(1/x)+1/x
- cos(4*x)+sin(4*x)
- Seno sin
- sin(x)/2+(1+x)*exp(-x)
- sin(x)^1
- sin(3*x)+x^4
- sin(4*x)*cos(x)+2*sin(3*x)-cos(4*x)*sin(x)
- sin(asin(x))
- Coseno cos
- cos(x/2)^(2)
- cos(2*x)-sqrt(3)/2
- cos(2*x*sqrt(2))+sin(2*x*sqrt(2))
- cos(2/x)-2*sin(1/x)+1/x
- cos(4*x)+sin(4*x)
- Seno sin
- sin(x)/2+(1+x)*exp(-x)
- sin(x)^1
- sin(3*x)+x^4
- sin(4*x)*cos(x)+2*sin(3*x)-cos(4*x)*sin(x)
- sin(asin(x))
- Coseno cos
- cos(x/2)^(2)
- cos(2*x)-sqrt(3)/2
- cos(2*x*sqrt(2))+sin(2*x*sqrt(2))
- cos(2/x)-2*sin(1/x)+1/x
- cos(4*x)+sin(4*x)
- Coseno cos
- cos(x/2)^(2)
- cos(2*x)-sqrt(3)/2
- cos(2*x*sqrt(2))+sin(2*x*sqrt(2))
- cos(2/x)-2*sin(1/x)+1/x
- cos(4*x)+sin(4*x)
Trazado el gráfico de la función/
-4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
Gráfico de la función y = -4.79*exp(-2*t)-6681.77*exp(-0.2*t)+3153.77*exp(-0.3*t)-2.39*t*cos(t^2)+1.42*t-402.28+4.78*t*sin(t)*cos(t)+0.3*sin(t)*cos(t)+4.57*cos(t^2)
Solución
Ha introducido
[src]
-t -3*t
--- ----
-2*t 5 10 / 2\
479*e 668177*e 315377*e 239*t / 2\ 71*t 10057 239*t 3*sin(t) 457*cos\t /
f(t) = - --------- - ----------- + ------------ - -----*cos\t / + ---- - ----- + -----*sin(t)*cos(t) + --------*cos(t) + -----------
100 100 100 100 50 25 50 10 100
$$f{\left(t \right)} = \left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100}$$
f = ((239*t/50)*sin(t))*cos(t) + 71*t/50 - 239*t/100*cos(t^2) - 479*exp(-2*t)/100 - 668177*exp(-t/5)/100 + 315377*exp(-3*t/10)/100 - 10057/25 + (3*sin(t)/10)*cos(t) + 457*cos(t^2)/100
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:
$$\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = 354.762692608311$$
$$t_{2} = 79.7574270111063$$
$$t_{3} = 75.8891458116847$$
$$t_{4} = 101.656166152599$$
$$t_{5} = 189.281537589827$$
$$t_{6} = 233.629900303766$$
$$t_{7} = 108.325398986805$$
$$t_{8} = 81.9772102867905$$
$$t_{9} = 72.7117669302286$$
$$t_{10} = 194.576897567704$$
$$t_{11} = 88.1419996080917$$
$$t_{12} = 117.163607344832$$
$$t_{13} = 353.786930910009$$
$$t_{14} = 103.977464215842$$
$$t_{15} = 107.014171189609$$
$$t_{16} = 426.247510770376$$
$$t_{17} = 293.384019868464$$
$$t_{18} = 82.2367537264362$$
o sea hay que resolver la ecuación:
$$\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = 354.762692608311$$
$$t_{2} = 79.7574270111063$$
$$t_{3} = 75.8891458116847$$
$$t_{4} = 101.656166152599$$
$$t_{5} = 189.281537589827$$
$$t_{6} = 233.629900303766$$
$$t_{7} = 108.325398986805$$
$$t_{8} = 81.9772102867905$$
$$t_{9} = 72.7117669302286$$
$$t_{10} = 194.576897567704$$
$$t_{11} = 88.1419996080917$$
$$t_{12} = 117.163607344832$$
$$t_{13} = 353.786930910009$$
$$t_{14} = 103.977464215842$$
$$t_{15} = 107.014171189609$$
$$t_{16} = 426.247510770376$$
$$t_{17} = 293.384019868464$$
$$t_{18} = 82.2367537264362$$
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
$$\frac{239 t^{2} \sin{\left(t^{2} \right)}}{50} - \frac{239 t \sin^{2}{\left(t \right)}}{50} - \frac{457 t \sin{\left(t^{2} \right)}}{50} + \left(\frac{239 t \cos{\left(t \right)}}{50} + \frac{239 \sin{\left(t \right)}}{50}\right) \cos{\left(t \right)} - \frac{3 \sin^{2}{\left(t \right)}}{10} + \frac{3 \cos^{2}{\left(t \right)}}{10} - \frac{239 \cos{\left(t^{2} \right)}}{100} + \frac{71}{50} + \frac{479 e^{- 2 t}}{50} + \frac{668177 e^{- \frac{t}{5}}}{500} - \frac{946131 e^{- \frac{3 t}{10}}}{1000} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 25.5620939701186$$
$$t_{2} = 60.0284976028315$$
$$t_{3} = 46.1858721311232$$
$$t_{4} = 11.3550769228536$$
$$t_{5} = 80.5832776004928$$
$$t_{6} = 28.2475762469396$$
$$t_{7} = 16.2425019772975$$
$$t_{8} = 64.2500433893311$$
$$t_{9} = 83.399627922718$$
$$t_{10} = 12.1607346004456$$
$$t_{11} = 9.89223778713139$$
$$t_{12} = 42.2427277419535$$
$$t_{13} = 15.3535638720642$$
$$t_{14} = 16.1506470944167$$
$$t_{15} = 32.0022742879839$$
$$t_{16} = 74.1047625388337$$
$$t_{17} = 22.1367155965153$$
$$t_{18} = 12.0128475826591$$
$$t_{19} = 56.3574966932718$$
$$t_{20} = 70.1408641990001$$
$$t_{21} = 31.5581819037535$$
$$t_{22} = 18.2473953741883$$
$$t_{23} = 66.2007249189901$$
$$t_{24} = 58.2755948247452$$
$$t_{25} = 19.9740733044362$$
$$t_{26} = 13.3856413987408$$
$$t_{27} = 52.2496958105707$$
$$t_{28} = 78.2292266849166$$
$$t_{29} = 48.3462657435853$$
$$t_{30} = 82.204453822137$$
$$t_{31} = 30.1842561918235$$
$$t_{32} = 20.4399367237379$$
$$t_{33} = 31.8051618149549$$
$$t_{34} = 94.1405521380375$$
$$t_{35} = 33.9089191283485$$
$$t_{36} = 30.2880727021071$$
$$t_{37} = 19.8165650233114$$
$$t_{38} = 14.2900354282217$$
$$t_{39} = 24.0432127233219$$
$$t_{40} = 8.27580500489127$$
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:
$$t_{1} = 25.5620939701186$$
$$t_{2} = 28.2475762469396$$
$$t_{3} = 16.2425019772975$$
$$t_{4} = 64.2500433893311$$
$$t_{5} = 83.399627922718$$
$$t_{6} = 42.2427277419535$$
$$t_{7} = 32.0022742879839$$
$$t_{8} = 74.1047625388337$$
$$t_{9} = 22.1367155965153$$
$$t_{10} = 12.0128475826591$$
$$t_{11} = 70.1408641990001$$
$$t_{12} = 18.2473953741883$$
$$t_{13} = 78.2292266849166$$
$$t_{14} = 48.3462657435853$$
$$t_{15} = 30.1842561918235$$
$$t_{16} = 31.8051618149549$$
$$t_{17} = 33.9089191283485$$
$$t_{18} = 30.2880727021071$$
$$t_{19} = 24.0432127233219$$
$$t_{20} = 8.27580500489127$$
Puntos máximos de la función:
$$t_{20} = 60.0284976028315$$
$$t_{20} = 46.1858721311232$$
$$t_{20} = 11.3550769228536$$
$$t_{20} = 80.5832776004928$$
$$t_{20} = 12.1607346004456$$
$$t_{20} = 9.89223778713139$$
$$t_{20} = 15.3535638720642$$
$$t_{20} = 16.1506470944167$$
$$t_{20} = 56.3574966932718$$
$$t_{20} = 31.5581819037535$$
$$t_{20} = 66.2007249189901$$
$$t_{20} = 58.2755948247452$$
$$t_{20} = 19.9740733044362$$
$$t_{20} = 13.3856413987408$$
$$t_{20} = 52.2496958105707$$
$$t_{20} = 82.204453822137$$
$$t_{20} = 20.4399367237379$$
$$t_{20} = 94.1405521380375$$
$$t_{20} = 19.8165650233114$$
$$t_{20} = 14.2900354282217$$
Decrece en los intervalos
$$\left[83.399627922718, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 8.27580500489127\right]$$
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(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
$$\frac{239 t^{2} \sin{\left(t^{2} \right)}}{50} - \frac{239 t \sin^{2}{\left(t \right)}}{50} - \frac{457 t \sin{\left(t^{2} \right)}}{50} + \left(\frac{239 t \cos{\left(t \right)}}{50} + \frac{239 \sin{\left(t \right)}}{50}\right) \cos{\left(t \right)} - \frac{3 \sin^{2}{\left(t \right)}}{10} + \frac{3 \cos^{2}{\left(t \right)}}{10} - \frac{239 \cos{\left(t^{2} \right)}}{100} + \frac{71}{50} + \frac{479 e^{- 2 t}}{50} + \frac{668177 e^{- \frac{t}{5}}}{500} - \frac{946131 e^{- \frac{3 t}{10}}}{1000} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 25.5620939701186$$
$$t_{2} = 60.0284976028315$$
$$t_{3} = 46.1858721311232$$
$$t_{4} = 11.3550769228536$$
$$t_{5} = 80.5832776004928$$
$$t_{6} = 28.2475762469396$$
$$t_{7} = 16.2425019772975$$
$$t_{8} = 64.2500433893311$$
$$t_{9} = 83.399627922718$$
$$t_{10} = 12.1607346004456$$
$$t_{11} = 9.89223778713139$$
$$t_{12} = 42.2427277419535$$
$$t_{13} = 15.3535638720642$$
$$t_{14} = 16.1506470944167$$
$$t_{15} = 32.0022742879839$$
$$t_{16} = 74.1047625388337$$
$$t_{17} = 22.1367155965153$$
$$t_{18} = 12.0128475826591$$
$$t_{19} = 56.3574966932718$$
$$t_{20} = 70.1408641990001$$
$$t_{21} = 31.5581819037535$$
$$t_{22} = 18.2473953741883$$
$$t_{23} = 66.2007249189901$$
$$t_{24} = 58.2755948247452$$
$$t_{25} = 19.9740733044362$$
$$t_{26} = 13.3856413987408$$
$$t_{27} = 52.2496958105707$$
$$t_{28} = 78.2292266849166$$
$$t_{29} = 48.3462657435853$$
$$t_{30} = 82.204453822137$$
$$t_{31} = 30.1842561918235$$
$$t_{32} = 20.4399367237379$$
$$t_{33} = 31.8051618149549$$
$$t_{34} = 94.1405521380375$$
$$t_{35} = 33.9089191283485$$
$$t_{36} = 30.2880727021071$$
$$t_{37} = 19.8165650233114$$
$$t_{38} = 14.2900354282217$$
$$t_{39} = 24.0432127233219$$
$$t_{40} = 8.27580500489127$$
Signos de extremos en los puntos:
(25.562093970118603, -414.87829086283)
(60.02849760283145, -88.2835437217465)
(46.185872131123155, -336.956413026074)
(11.355076922853568, -966.798553103985)
(80.58327760049276, -256.089191382472)
(28.247576246939648, -451.53778599988)
(16.24250197729749, -614.537902927669)
(64.25004338933105, -413.844144156691)
(83.39962792271795, -536.560274584493)
(12.160734600445558, -887.204198378458)
(9.89223778713139, -1113.86466225692)
(42.24272774195354, -406.595524955307)
(15.353563872064166, -650.984264595503)
(16.15064709441668, -554.931295198481)
(32.002274287983944, -368.986689780612)
(74.1047625388337, -562.938407829749)
(22.13671559651531, -479.585882644763)
(12.012847582659134, -953.430384547783)
(56.357496693271784, -242.541953666734)
(70.14086419900013, -316.988743375628)
(31.558181903753546, -277.325435122802)
(18.247395374188265, -616.763794244041)
(66.20072491899013, -85.1179615968724)
(58.27559482474522, -227.73075755387)
(19.974073304436192, -408.605548347614)
(13.385641398740821, -726.509445629229)
(52.24969581057071, -299.975832678507)
(78.22922668491661, -582.484783813349)
(48.34626574358535, -370.757005197777)
(82.20445382213697, 76.5122382448092)
(30.184256191823458, -487.915560544215)
(20.439936723737866, -436.151749895781)
(31.80516181495487, -386.374452123016)
(94.14055213803755, -96.1015174199105)
(33.90891912834853, -516.22986069639)
(30.288072702107087, -498.530038443833)
(19.816565023311412, -405.631648770265)
(14.29003542822171, -702.798589597198)
(24.043212723321915, -520.489678718798)
(8.275805004891268, -1430.98505800608)
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:
$$t_{1} = 25.5620939701186$$
$$t_{2} = 28.2475762469396$$
$$t_{3} = 16.2425019772975$$
$$t_{4} = 64.2500433893311$$
$$t_{5} = 83.399627922718$$
$$t_{6} = 42.2427277419535$$
$$t_{7} = 32.0022742879839$$
$$t_{8} = 74.1047625388337$$
$$t_{9} = 22.1367155965153$$
$$t_{10} = 12.0128475826591$$
$$t_{11} = 70.1408641990001$$
$$t_{12} = 18.2473953741883$$
$$t_{13} = 78.2292266849166$$
$$t_{14} = 48.3462657435853$$
$$t_{15} = 30.1842561918235$$
$$t_{16} = 31.8051618149549$$
$$t_{17} = 33.9089191283485$$
$$t_{18} = 30.2880727021071$$
$$t_{19} = 24.0432127233219$$
$$t_{20} = 8.27580500489127$$
Puntos máximos de la función:
$$t_{20} = 60.0284976028315$$
$$t_{20} = 46.1858721311232$$
$$t_{20} = 11.3550769228536$$
$$t_{20} = 80.5832776004928$$
$$t_{20} = 12.1607346004456$$
$$t_{20} = 9.89223778713139$$
$$t_{20} = 15.3535638720642$$
$$t_{20} = 16.1506470944167$$
$$t_{20} = 56.3574966932718$$
$$t_{20} = 31.5581819037535$$
$$t_{20} = 66.2007249189901$$
$$t_{20} = 58.2755948247452$$
$$t_{20} = 19.9740733044362$$
$$t_{20} = 13.3856413987408$$
$$t_{20} = 52.2496958105707$$
$$t_{20} = 82.204453822137$$
$$t_{20} = 20.4399367237379$$
$$t_{20} = 94.1405521380375$$
$$t_{20} = 19.8165650233114$$
$$t_{20} = 14.2900354282217$$
Decrece en los intervalos
$$\left[83.399627922718, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 8.27580500489127\right]$$
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
$$\frac{95600 t^{3} \cos{\left(t^{2} \right)} - 182800 t^{2} \cos{\left(t^{2} \right)} - 95600 t \sin{\left(t \right)} \cos{\left(t \right)} + 143400 t \sin{\left(t^{2} \right)} - 47800 \left(t \sin{\left(t \right)} - 2 \cos{\left(t \right)}\right) \cos{\left(t \right)} - 47800 \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \sin{\left(t \right)} - 47800 \sin^{2}{\left(t \right)} - 12000 \sin{\left(t \right)} \cos{\left(t \right)} - 91400 \sin{\left(t^{2} \right)} - 191600 e^{- 2 t} - 2672708 e^{- \frac{t}{5}} + 2838393 e^{- \frac{3 t}{10}}}{10000} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 27.1423414899827$$
$$t_{2} = 92.1079314197503$$
$$t_{3} = 74.68532551653$$
$$t_{4} = 4.19579863508362$$
$$t_{5} = 60.2504023869576$$
$$t_{6} = 28.1647734733153$$
$$t_{7} = 59.0655514178733$$
$$t_{8} = 15.9029232862203$$
$$t_{9} = 27.7716012671948$$
$$t_{10} = 35.204527298822$$
$$t_{11} = 54.1251819748878$$
$$t_{12} = 67.9222297482101$$
$$t_{13} = 32.1249698004877$$
$$t_{14} = 34.2088467749407$$
$$t_{15} = 47.7412460615829$$
$$t_{16} = 44.4705946492161$$
$$t_{17} = 56.0919358391114$$
$$t_{18} = 22.3146668988395$$
$$t_{19} = 20.2479791661075$$
$$t_{20} = 44.0803330127687$$
$$t_{21} = 62.2255059636583$$
$$t_{22} = 94.2155518234989$$
$$t_{23} = 70.1520136399282$$
$$t_{24} = 45.6213319854102$$
$$t_{25} = 85.4370579478017$$
$$t_{26} = 7.62422539951845$$
$$t_{27} = -0.230512041259116$$
$$t_{28} = 10.2603171216896$$
$$t_{29} = 58.154320228599$$
$$t_{30} = 2.85795539283023$$
$$t_{31} = 80.1827264666901$$
$$t_{32} = 84.1775641473314$$
$$t_{33} = 40.164762525446$$
$$t_{34} = 46.0326554812457$$
$$t_{35} = 0.332823648912459$$
$$t_{36} = 77.3915161721166$$
$$t_{37} = 49.3271177838759$$
$$t_{38} = 96.0483578210583$$
$$t_{39} = 18.11917938914$$
$$t_{40} = 91.885964965925$$
$$t_{41} = 6.26944233760186$$
$$t_{42} = 76.1020794385702$$
$$t_{43} = 16.5800964075703$$
$$t_{44} = 6.74962666827121$$
$$t_{45} = 90.212505111484$$
$$t_{46} = 37.5367242232826$$
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
$$\left[94.2155518234989, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -0.230512041259116\right]$$
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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
$$\frac{95600 t^{3} \cos{\left(t^{2} \right)} - 182800 t^{2} \cos{\left(t^{2} \right)} - 95600 t \sin{\left(t \right)} \cos{\left(t \right)} + 143400 t \sin{\left(t^{2} \right)} - 47800 \left(t \sin{\left(t \right)} - 2 \cos{\left(t \right)}\right) \cos{\left(t \right)} - 47800 \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \sin{\left(t \right)} - 47800 \sin^{2}{\left(t \right)} - 12000 \sin{\left(t \right)} \cos{\left(t \right)} - 91400 \sin{\left(t^{2} \right)} - 191600 e^{- 2 t} - 2672708 e^{- \frac{t}{5}} + 2838393 e^{- \frac{3 t}{10}}}{10000} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 27.1423414899827$$
$$t_{2} = 92.1079314197503$$
$$t_{3} = 74.68532551653$$
$$t_{4} = 4.19579863508362$$
$$t_{5} = 60.2504023869576$$
$$t_{6} = 28.1647734733153$$
$$t_{7} = 59.0655514178733$$
$$t_{8} = 15.9029232862203$$
$$t_{9} = 27.7716012671948$$
$$t_{10} = 35.204527298822$$
$$t_{11} = 54.1251819748878$$
$$t_{12} = 67.9222297482101$$
$$t_{13} = 32.1249698004877$$
$$t_{14} = 34.2088467749407$$
$$t_{15} = 47.7412460615829$$
$$t_{16} = 44.4705946492161$$
$$t_{17} = 56.0919358391114$$
$$t_{18} = 22.3146668988395$$
$$t_{19} = 20.2479791661075$$
$$t_{20} = 44.0803330127687$$
$$t_{21} = 62.2255059636583$$
$$t_{22} = 94.2155518234989$$
$$t_{23} = 70.1520136399282$$
$$t_{24} = 45.6213319854102$$
$$t_{25} = 85.4370579478017$$
$$t_{26} = 7.62422539951845$$
$$t_{27} = -0.230512041259116$$
$$t_{28} = 10.2603171216896$$
$$t_{29} = 58.154320228599$$
$$t_{30} = 2.85795539283023$$
$$t_{31} = 80.1827264666901$$
$$t_{32} = 84.1775641473314$$
$$t_{33} = 40.164762525446$$
$$t_{34} = 46.0326554812457$$
$$t_{35} = 0.332823648912459$$
$$t_{36} = 77.3915161721166$$
$$t_{37} = 49.3271177838759$$
$$t_{38} = 96.0483578210583$$
$$t_{39} = 18.11917938914$$
$$t_{40} = 91.885964965925$$
$$t_{41} = 6.26944233760186$$
$$t_{42} = 76.1020794385702$$
$$t_{43} = 16.5800964075703$$
$$t_{44} = 6.74962666827121$$
$$t_{45} = 90.212505111484$$
$$t_{46} = 37.5367242232826$$
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
$$\left[94.2155518234989, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -0.230512041259116\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con t->+oo y t->-oo
$$\lim_{t \to -\infty}\left(\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100}\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{t \to \infty}\left(\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100}\right)$$
$$\lim_{t \to -\infty}\left(\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100}\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{t \to \infty}\left(\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función -479*exp(-2*t)/100 - 668177*exp(-t/5)/100 + 315377*exp(-3*t/10)/100 - 239*t/100*cos(t^2) + 71*t/50 - 10057/25 + ((239*t/50)*sin(t))*cos(t) + (3*sin(t)/10)*cos(t) + 457*cos(t^2)/100, dividida por t con t->+oo y t ->-oo
$$\lim_{t \to -\infty}\left(\frac{\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100}}{t}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = t \lim_{t \to \infty}\left(\frac{\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100}}{t}\right)$$
$$\lim_{t \to -\infty}\left(\frac{\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100}}{t}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = t \lim_{t \to \infty}\left(\frac{\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100}}{t}\right)$$
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:
$$\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100} = \frac{239 t \sin{\left(t \right)} \cos{\left(t \right)}}{50} + \frac{239 t \cos{\left(t^{2} \right)}}{100} - \frac{71 t}{50} + \frac{315377 e^{\frac{3 t}{10}}}{100} - \frac{668177 e^{\frac{t}{5}}}{100} - \frac{479 e^{2 t}}{100} - \frac{3 \sin{\left(t \right)} \cos{\left(t \right)}}{10} + \frac{457 \cos{\left(t^{2} \right)}}{100} - \frac{10057}{25}$$
- No
$$\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100} = - \frac{239 t \sin{\left(t \right)} \cos{\left(t \right)}}{50} - \frac{239 t \cos{\left(t^{2} \right)}}{100} + \frac{71 t}{50} - \frac{315377 e^{\frac{3 t}{10}}}{100} + \frac{668177 e^{\frac{t}{5}}}{100} + \frac{479 e^{2 t}}{100} + \frac{3 \sin{\left(t \right)} \cos{\left(t \right)}}{10} - \frac{457 \cos{\left(t^{2} \right)}}{100} + \frac{10057}{25}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100} = \frac{239 t \sin{\left(t \right)} \cos{\left(t \right)}}{50} + \frac{239 t \cos{\left(t^{2} \right)}}{100} - \frac{71 t}{50} + \frac{315377 e^{\frac{3 t}{10}}}{100} - \frac{668177 e^{\frac{t}{5}}}{100} - \frac{479 e^{2 t}}{100} - \frac{3 \sin{\left(t \right)} \cos{\left(t \right)}}{10} + \frac{457 \cos{\left(t^{2} \right)}}{100} - \frac{10057}{25}$$
- No
$$\left(\left(\frac{239 t}{50} \sin{\left(t \right)} \cos{\left(t \right)} + \left(\left(\frac{71 t}{50} + \left(- \frac{239 t}{100} \cos{\left(t^{2} \right)} + \left(\left(- \frac{479 e^{- 2 t}}{100} - \frac{668177 e^{- \frac{t}{5}}}{100}\right) + \frac{315377 e^{- \frac{3 t}{10}}}{100}\right)\right)\right) - \frac{10057}{25}\right)\right) + \frac{3 \sin{\left(t \right)}}{10} \cos{\left(t \right)}\right) + \frac{457 \cos{\left(t^{2} \right)}}{100} = - \frac{239 t \sin{\left(t \right)} \cos{\left(t \right)}}{50} - \frac{239 t \cos{\left(t^{2} \right)}}{100} + \frac{71 t}{50} - \frac{315377 e^{\frac{3 t}{10}}}{100} + \frac{668177 e^{\frac{t}{5}}}{100} + \frac{479 e^{2 t}}{100} + \frac{3 \sin{\left(t \right)} \cos{\left(t \right)}}{10} - \frac{457 \cos{\left(t^{2} \right)}}{100} + \frac{10057}{25}$$
- No
es decir, función
no es
par ni impar