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 =:
-
y=(x+1)^3
-
y=2x
-
2*x^3-3*x
-
3-x^2
-
Expresiones idénticas
- - catorce *exp(dos *t)/ nueve + cincuenta *cos(t*sqrt(dos))/ nueve + cuatro *t*exp(dos *t)/ tres + siete *sqrt(dos)*sin(t*sqrt(dos))/ dieciocho
- menos 14 multiplicar por exponente de (2 multiplicar por t) dividir por 9 más 50 multiplicar por coseno de (t multiplicar por raíz cuadrada de (2)) dividir por 9 más 4 multiplicar por t multiplicar por exponente de (2 multiplicar por t) dividir por 3 más 7 multiplicar por raíz cuadrada de (2) multiplicar por seno de (t multiplicar por raíz cuadrada de (2)) dividir por 18
- menos cotangente de angente de orce multiplicar por exponente de (dos multiplicar por t) dividir por nueve más cincuenta multiplicar por coseno de (t multiplicar por raíz cuadrada de (dos)) dividir por nueve más cuatro multiplicar por t multiplicar por exponente de (dos multiplicar por t) dividir por tres más siete multiplicar por raíz cuadrada de (dos) multiplicar por seno de (t multiplicar por raíz cuadrada de (dos)) dividir por dieciocho
- -14*exp(2*t)/9+50*cos(t*√(2))/9+4*t*exp(2*t)/3+7*√(2)*sin(t*√(2))/18
- -14exp(2t)/9+50cos(tsqrt(2))/9+4texp(2t)/3+7sqrt(2)sin(tsqrt(2))/18
- -14exp2t/9+50costsqrt2/9+4texp2t/3+7sqrt2sintsqrt2/18
- -14*exp(2*t) dividir por 9+50*cos(t*sqrt(2)) dividir por 9+4*t*exp(2*t) dividir por 3+7*sqrt(2)*sin(t*sqrt(2)) dividir por 18
-
Expresiones semejantes
- -14*exp(2*t)/9+50*cos(t*sqrt(2))/9+4*t*exp(2*t)/3-7*sqrt(2)*sin(t*sqrt(2))/18
- -14*exp(2*t)/9+50*cos(t*sqrt(2))/9-4*t*exp(2*t)/3+7*sqrt(2)*sin(t*sqrt(2))/18
- 14*exp(2*t)/9+50*cos(t*sqrt(2))/9+4*t*exp(2*t)/3+7*sqrt(2)*sin(t*sqrt(2))/18
- -14*exp(2*t)/9-50*cos(t*sqrt(2))/9+4*t*exp(2*t)/3+7*sqrt(2)*sin(t*sqrt(2))/18
-
Expresiones con funciones
- Exponente exp
- exp(-exp(-1/tan(x)))
- exp((2*x-2)/x)
- exp(-2,81*x)*(-3*10^(-5)*sin(2,81*x))*947532000
- exp(4*x)/(1+exp(x*(4+x^3)))
- exp(-100x)*(-50*cos(200x)+25*sin(200x))
- Coseno cos
- cos[x]/x^2
- cos(x-pi/3)-1
- cos(x)/(1+cos^2*2x)
- cos(1)/((3*x))
- cos(sqrt(x))+1
- Raíz cuadrada sqrt
- sqrt(x^2+4*x+4)
- sqrt(x^2+1)-sqrt(x^2-1)
- sqrt(x^2-6*x+11)
- sqrt[x^2+10x+26]-2
- sqrt(9*x)^2+5
- Exponente exp
- exp(-exp(-1/tan(x)))
- exp((2*x-2)/x)
- exp(-2,81*x)*(-3*10^(-5)*sin(2,81*x))*947532000
- exp(4*x)/(1+exp(x*(4+x^3)))
- exp(-100x)*(-50*cos(200x)+25*sin(200x))
- Raíz cuadrada sqrt
- sqrt(x^2+4*x+4)
- sqrt(x^2+1)-sqrt(x^2-1)
- sqrt(x^2-6*x+11)
- sqrt[x^2+10x+26]-2
- sqrt(9*x)^2+5
- Seno sin
- sin(0,5x+pi/4)+0,5
- sin(x×4)
- sin(x^4-3*x^2+1)
- sin(x)+cos(x)+sin(x)/x!
- sin(2*x+x-1)
- Raíz cuadrada sqrt
- sqrt(x^2+4*x+4)
- sqrt(x^2+1)-sqrt(x^2-1)
- sqrt(x^2-6*x+11)
- sqrt[x^2+10x+26]-2
- sqrt(9*x)^2+5
Trazado el gráfico de la función/
-14*exp(2*t)/9+50*cos(t*sqrt(2))/9+4*t*exp(2*t)/3+7*sqrt(2)*sin(t*sqrt(2))/18
Gráfico de la función y = -14*exp(2*t)/9+50*cos(t*sqrt(2))/9+4*t*exp(2*t)/3+7*sqrt(2)*sin(t*sqrt(2))/18
Solución
Ha introducido
[src]
2*t / ___\ 2*t ___ / ___\
-14*e 50*cos\t*\/ 2 / 4*t*e 7*\/ 2 *sin\t*\/ 2 /
f(t) = -------- + --------------- + -------- + --------------------
9 9 3 18
$$f{\left(t \right)} = \frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right)$$
f = ((7*sqrt(2))*sin(sqrt(2)*t))/18 + ((4*t)*exp(2*t))/3 + (-14*exp(2*t))/9 + (50*cos(sqrt(2)*t))/9
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:
$$\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right) = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = -47.6912189166499$$
$$t_{2} = -12.148155411446$$
$$t_{3} = 1.14609061132884$$
$$t_{4} = -67.6841921383625$$
$$t_{5} = -85.455723890996$$
$$t_{6} = -74.3485165456$$
$$t_{7} = -14.3695968804612$$
$$t_{8} = -32.1411286330956$$
$$t_{9} = -34.3625701021748$$
$$t_{10} = -23.2553627567788$$
$$t_{11} = -29.9196871640164$$
$$t_{12} = -7.70527277723061$$
$$t_{13} = -27.6982456949372$$
$$t_{14} = -3.26348442564007$$
$$t_{15} = -83.2342824219168$$
$$t_{16} = -5.48381162863367$$
$$t_{17} = -61.019867731125$$
$$t_{18} = -76.5699580146792$$
$$t_{19} = -69.9056336074417$$
$$t_{20} = -0.990670137807717$$
$$t_{21} = -92.1200482982335$$
$$t_{22} = -96.5629312363919$$
$$t_{23} = -52.1341018548082$$
$$t_{24} = -49.912660385729$$
$$t_{25} = -56.5769847929666$$
$$t_{26} = -36.5840115712539$$
$$t_{27} = -43.2483359784915$$
$$t_{28} = -81.0128409528376$$
$$t_{29} = -41.0268945094123$$
$$t_{30} = -89.8986068291543$$
$$t_{31} = -72.1270750765209$$
$$t_{32} = -94.3414897673127$$
$$t_{33} = -63.2413092002041$$
$$t_{34} = -9.9267139378327$$
$$t_{35} = -54.3555433238874$$
$$t_{36} = -21.0339212876997$$
$$t_{37} = -45.4697774475707$$
$$t_{38} = -65.4627506692833$$
$$t_{39} = -98.7843727054711$$
$$t_{40} = -25.476804225858$$
$$t_{41} = -16.5910383495413$$
$$t_{42} = -87.6771653600752$$
o sea hay que resolver la ecuación:
$$\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right) = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = -47.6912189166499$$
$$t_{2} = -12.148155411446$$
$$t_{3} = 1.14609061132884$$
$$t_{4} = -67.6841921383625$$
$$t_{5} = -85.455723890996$$
$$t_{6} = -74.3485165456$$
$$t_{7} = -14.3695968804612$$
$$t_{8} = -32.1411286330956$$
$$t_{9} = -34.3625701021748$$
$$t_{10} = -23.2553627567788$$
$$t_{11} = -29.9196871640164$$
$$t_{12} = -7.70527277723061$$
$$t_{13} = -27.6982456949372$$
$$t_{14} = -3.26348442564007$$
$$t_{15} = -83.2342824219168$$
$$t_{16} = -5.48381162863367$$
$$t_{17} = -61.019867731125$$
$$t_{18} = -76.5699580146792$$
$$t_{19} = -69.9056336074417$$
$$t_{20} = -0.990670137807717$$
$$t_{21} = -92.1200482982335$$
$$t_{22} = -96.5629312363919$$
$$t_{23} = -52.1341018548082$$
$$t_{24} = -49.912660385729$$
$$t_{25} = -56.5769847929666$$
$$t_{26} = -36.5840115712539$$
$$t_{27} = -43.2483359784915$$
$$t_{28} = -81.0128409528376$$
$$t_{29} = -41.0268945094123$$
$$t_{30} = -89.8986068291543$$
$$t_{31} = -72.1270750765209$$
$$t_{32} = -94.3414897673127$$
$$t_{33} = -63.2413092002041$$
$$t_{34} = -9.9267139378327$$
$$t_{35} = -54.3555433238874$$
$$t_{36} = -21.0339212876997$$
$$t_{37} = -45.4697774475707$$
$$t_{38} = -65.4627506692833$$
$$t_{39} = -98.7843727054711$$
$$t_{40} = -25.476804225858$$
$$t_{41} = -16.5910383495413$$
$$t_{42} = -87.6771653600752$$
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
$$\frac{8 t e^{2 t}}{3} - \frac{16 e^{2 t}}{9} - \frac{50 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{9} + \frac{7 \cos{\left(\sqrt{2} t \right)}}{9} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -82.1235616873772$$
$$t_{2} = -66.5734714038229$$
$$t_{3} = -4.37330165985759$$
$$t_{4} = -91.0093275636939$$
$$t_{5} = -53.2448225893478$$
$$t_{6} = -51.0233811202686$$
$$t_{7} = -57.6877055275062$$
$$t_{8} = -55.466264058427$$
$$t_{9} = -95.4522105018523$$
$$t_{10} = -17.7017590840809$$
$$t_{11} = -75.4592372801396$$
$$t_{12} = -86.5664446255356$$
$$t_{13} = -46.5804981821103$$
$$t_{14} = -24.3660834913184$$
$$t_{15} = -93.2307690327731$$
$$t_{16} = -0.0830671258322571$$
$$t_{17} = 1.03338755252582$$
$$t_{18} = -26.5875249603976$$
$$t_{19} = -84.3450031564564$$
$$t_{20} = -59.9091469965854$$
$$t_{21} = -19.9232005531601$$
$$t_{22} = -37.6947323057935$$
$$t_{23} = -44.3590567130311$$
$$t_{24} = -77.6806787492188$$
$$t_{25} = -31.030407898556$$
$$t_{26} = -13.2588761459326$$
$$t_{27} = -2.14242641424928$$
$$t_{28} = -73.2377958110605$$
$$t_{29} = -42.1376152439519$$
$$t_{30} = -88.7878860946147$$
$$t_{31} = -15.4803176150016$$
$$t_{32} = -11.0374346761198$$
$$t_{33} = -97.6736519709315$$
$$t_{34} = -33.2518493676352$$
$$t_{35} = -64.3520299347437$$
$$t_{36} = -62.1305884656645$$
$$t_{37} = -22.1446420222392$$
$$t_{38} = -6.59454849416224$$
$$t_{39} = -35.4732908367143$$
$$t_{40} = -79.902120218298$$
$$t_{41} = -71.0163543419813$$
$$t_{42} = -39.9161737748727$$
$$t_{43} = -99.8950934400107$$
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:
$$t_{1} = -82.1235616873772$$
$$t_{2} = -91.0093275636939$$
$$t_{3} = -51.0233811202686$$
$$t_{4} = -55.466264058427$$
$$t_{5} = -95.4522105018523$$
$$t_{6} = -86.5664446255356$$
$$t_{7} = -46.5804981821103$$
$$t_{8} = -24.3660834913184$$
$$t_{9} = 1.03338755252582$$
$$t_{10} = -59.9091469965854$$
$$t_{11} = -19.9232005531601$$
$$t_{12} = -37.6947323057935$$
$$t_{13} = -77.6806787492188$$
$$t_{14} = -2.14242641424928$$
$$t_{15} = -73.2377958110605$$
$$t_{16} = -42.1376152439519$$
$$t_{17} = -15.4803176150016$$
$$t_{18} = -11.0374346761198$$
$$t_{19} = -33.2518493676352$$
$$t_{20} = -64.3520299347437$$
$$t_{21} = -6.59454849416224$$
$$t_{22} = -99.8950934400107$$
Puntos máximos de la función:
$$t_{22} = -66.5734714038229$$
$$t_{22} = -4.37330165985759$$
$$t_{22} = -53.2448225893478$$
$$t_{22} = -57.6877055275062$$
$$t_{22} = -17.7017590840809$$
$$t_{22} = -75.4592372801396$$
$$t_{22} = -93.2307690327731$$
$$t_{22} = -0.0830671258322571$$
$$t_{22} = -26.5875249603976$$
$$t_{22} = -84.3450031564564$$
$$t_{22} = -44.3590567130311$$
$$t_{22} = -31.030407898556$$
$$t_{22} = -13.2588761459326$$
$$t_{22} = -88.7878860946147$$
$$t_{22} = -97.6736519709315$$
$$t_{22} = -62.1305884656645$$
$$t_{22} = -22.1446420222392$$
$$t_{22} = -35.4732908367143$$
$$t_{22} = -79.902120218298$$
$$t_{22} = -71.0163543419813$$
$$t_{22} = -39.9161737748727$$
Decrece en los intervalos
$$\left[1.03338755252582, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.8950934400107\right]$$
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(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
$$\frac{8 t e^{2 t}}{3} - \frac{16 e^{2 t}}{9} - \frac{50 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{9} + \frac{7 \cos{\left(\sqrt{2} t \right)}}{9} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = -82.1235616873772$$
$$t_{2} = -66.5734714038229$$
$$t_{3} = -4.37330165985759$$
$$t_{4} = -91.0093275636939$$
$$t_{5} = -53.2448225893478$$
$$t_{6} = -51.0233811202686$$
$$t_{7} = -57.6877055275062$$
$$t_{8} = -55.466264058427$$
$$t_{9} = -95.4522105018523$$
$$t_{10} = -17.7017590840809$$
$$t_{11} = -75.4592372801396$$
$$t_{12} = -86.5664446255356$$
$$t_{13} = -46.5804981821103$$
$$t_{14} = -24.3660834913184$$
$$t_{15} = -93.2307690327731$$
$$t_{16} = -0.0830671258322571$$
$$t_{17} = 1.03338755252582$$
$$t_{18} = -26.5875249603976$$
$$t_{19} = -84.3450031564564$$
$$t_{20} = -59.9091469965854$$
$$t_{21} = -19.9232005531601$$
$$t_{22} = -37.6947323057935$$
$$t_{23} = -44.3590567130311$$
$$t_{24} = -77.6806787492188$$
$$t_{25} = -31.030407898556$$
$$t_{26} = -13.2588761459326$$
$$t_{27} = -2.14242641424928$$
$$t_{28} = -73.2377958110605$$
$$t_{29} = -42.1376152439519$$
$$t_{30} = -88.7878860946147$$
$$t_{31} = -15.4803176150016$$
$$t_{32} = -11.0374346761198$$
$$t_{33} = -97.6736519709315$$
$$t_{34} = -33.2518493676352$$
$$t_{35} = -64.3520299347437$$
$$t_{36} = -62.1305884656645$$
$$t_{37} = -22.1446420222392$$
$$t_{38} = -6.59454849416224$$
$$t_{39} = -35.4732908367143$$
$$t_{40} = -79.902120218298$$
$$t_{41} = -71.0163543419813$$
$$t_{42} = -39.9161737748727$$
$$t_{43} = -99.8950934400107$$
Signos de extremos en los puntos:
/ ___\ ___ / ___\
50*cos\82.1235616873772*\/ 2 / 7*\/ 2 *sin\82.1235616873772*\/ 2 /
(-82.12356168737719, -5.17503676202054e-70 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\66.5734714038229*\/ 2 / 7*\/ 2 *sin\66.5734714038229*\/ 2 /
(-66.57347140382291, -1.35145711943326e-56 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\4.37330165985759*\/ 2 / 7*\/ 2 *sin\4.37330165985759*\/ 2 /
(-4.373301659857594, -0.00117447685096334 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\91.0093275636939*\/ 2 / 7*\/ 2 *sin\91.0093275636939*\/ 2 /
(-91.00932756369393, -1.09612237246476e-77 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\53.2448225893478*\/ 2 / 7*\/ 2 *sin\53.2448225893478*\/ 2 /
(-53.24482258934781, -4.09981319144366e-45 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\51.0233811202686*\/ 2 / 7*\/ 2 *sin\51.0233811202686*\/ 2 /
(-51.02338112026863, -3.34334115220086e-43 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\57.6877055275062*\/ 2 / 7*\/ 2 *sin\57.6877055275062*\/ 2 /
(-57.68770552750618, -6.13497926864371e-49 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\55.466264058427*\/ 2 / 7*\/ 2 *sin\55.466264058427*\/ 2 /
(-55.466264058426994, -5.01906648510858e-47 + ----------------------------- - ----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\95.4522105018523*\/ 2 / 7*\/ 2 *sin\95.4522105018523*\/ 2 /
(-95.45221050185229, -1.58951309411117e-81 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\17.7017590840809*\/ 2 / 7*\/ 2 *sin\17.7017590840809*\/ 2 /
(-17.701759084080884, -1.05955089679186e-14 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\75.4592372801396*\/ 2 / 7*\/ 2 *sin\75.4592372801396*\/ 2 /
(-75.45923728013965, -2.92586058645026e-64 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\86.5664446255356*\/ 2 / 7*\/ 2 *sin\86.5664446255356*\/ 2 /
(-86.56644462553555, -7.54125853732662e-74 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\46.5804981821103*\/ 2 / 7*\/ 2 *sin\46.5804981821103*\/ 2 /
(-46.58049818211026, -2.2109538423037e-39 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\24.3660834913184*\/ 2 / 7*\/ 2 *sin\24.3660834913184*\/ 2 /
(-24.36608349131843, -2.33305577035163e-20 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\93.2307690327731*\/ 2 / 7*\/ 2 *sin\93.2307690327731*\/ 2 /
(-93.23076903277311, -1.32032809432067e-79 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\0.0830671258322571*\/ 2 / 7*\/ 2 *sin\0.0830671258322571*\/ 2 /
(-0.08306712583225709, -1.4112535943714 + -------------------------------- - -------------------------------------)
9 18 / ___\ ___ / ___\
50*cos\1.03338755252582*\/ 2 / 7*\/ 2 *sin\1.03338755252582*\/ 2 /
(1.0333875525258183, -1.4037502853051 + ------------------------------ + -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\26.5875249603976*\/ 2 / 7*\/ 2 *sin\26.5875249603976*\/ 2 /
(-26.587524960397616, -2.98288536671326e-22 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\84.3450031564564*\/ 2 / 7*\/ 2 *sin\84.3450031564564*\/ 2 /
(-84.34500315645637, -6.24921159442002e-72 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\59.9091469965854*\/ 2 / 7*\/ 2 *sin\59.9091469965854*\/ 2 /
(-59.90914699658536, -7.48831466898766e-51 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\19.9232005531601*\/ 2 / 7*\/ 2 *sin\19.9232005531601*\/ 2 /
(-19.923200553160065, -1.39296581317188e-16 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\37.6947323057935*\/ 2 / 7*\/ 2 *sin\37.6947323057935*\/ 2 /
(-37.69473230579353, -9.40218284899873e-32 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\44.3590567130311*\/ 2 / 7*\/ 2 *sin\44.3590567130311*\/ 2 /
(-44.35905671303108, -1.79229084338781e-37 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\77.6806787492188*\/ 2 / 7*\/ 2 *sin\77.6806787492188*\/ 2 /
(-77.68067874921883, -3.54116000398269e-66 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\31.030407898556*\/ 2 / 7*\/ 2 *sin\31.030407898556*\/ 2 /
(-31.030407898555982, -4.78724117123825e-26 + ----------------------------- - ----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\13.2588761459326*\/ 2 / 7*\/ 2 *sin\13.2588761459326*\/ 2 /
(-13.258876145932641, -5.85541131734793e-11 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\2.14242641424928*\/ 2 / 7*\/ 2 *sin\2.14242641424928*\/ 2 /
(-2.1424264142492846, -0.0607798719738458 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\73.2377958110605*\/ 2 / 7*\/ 2 *sin\73.2377958110605*\/ 2 /
(-73.23779581106047, -2.41544161934843e-62 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\42.1376152439519*\/ 2 / 7*\/ 2 *sin\42.1376152439519*\/ 2 /
(-42.1376152439519, -1.44944591939558e-35 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\88.7878860946147*\/ 2 / 7*\/ 2 *sin\88.7878860946147*\/ 2 /
(-88.78788609461475, -9.09460630662491e-76 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\15.4803176150016*\/ 2 / 7*\/ 2 *sin\15.4803176150016*\/ 2 /
(-15.480317615001562, -7.94769641190152e-13 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\11.0374346761198*\/ 2 / 7*\/ 2 *sin\11.0374346761198*\/ 2 /
(-11.037434676119835, -4.21163371416618e-9 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\97.6736519709315*\/ 2 / 7*\/ 2 *sin\97.6736519709315*\/ 2 /
(-97.67365197093147, -1.91256725857317e-83 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\33.2518493676352*\/ 2 / 7*\/ 2 *sin\33.2518493676352*\/ 2 /
(-33.25184936763517, -6.01923766389167e-28 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\64.3520299347437*\/ 2 / 7*\/ 2 *sin\64.3520299347437*\/ 2 /
(-64.35202993474373, -1.11132474852584e-54 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\62.1305884656645*\/ 2 / 7*\/ 2 *sin\62.1305884656645*\/ 2 /
(-62.130588465664545, -9.12809513816115e-53 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\22.1446420222392*\/ 2 / 7*\/ 2 *sin\22.1446420222392*\/ 2 /
(-22.14464202223925, -1.81098029144111e-18 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\6.59454849416224*\/ 2 / 7*\/ 2 *sin\6.59454849416224*\/ 2 /
(-6.594548494162237, -1.93604932036221e-5 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\35.4732908367143*\/ 2 / 7*\/ 2 *sin\35.4732908367143*\/ 2 /
(-35.47329083671435, -7.53676141209859e-30 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\79.902120218298*\/ 2 / 7*\/ 2 *sin\79.902120218298*\/ 2 /
(-79.90212021829801, -4.28245299264831e-68 + ----------------------------- - ----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\71.0163543419813*\/ 2 / 7*\/ 2 *sin\71.0163543419813*\/ 2 /
(-71.01635434198128, -1.9922881858096e-60 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\39.9161737748727*\/ 2 / 7*\/ 2 *sin\39.9161737748727*\/ 2 /
(-39.91617377487271, -1.1690987096601e-33 + ------------------------------ - -----------------------------------)
9 18 / ___\ ___ / ___\
50*cos\99.8950934400107*\/ 2 / 7*\/ 2 *sin\99.8950934400107*\/ 2 /
(-99.89509344001065, -2.30011682032021e-85 + ------------------------------ - -----------------------------------)
9 18 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:
$$t_{1} = -82.1235616873772$$
$$t_{2} = -91.0093275636939$$
$$t_{3} = -51.0233811202686$$
$$t_{4} = -55.466264058427$$
$$t_{5} = -95.4522105018523$$
$$t_{6} = -86.5664446255356$$
$$t_{7} = -46.5804981821103$$
$$t_{8} = -24.3660834913184$$
$$t_{9} = 1.03338755252582$$
$$t_{10} = -59.9091469965854$$
$$t_{11} = -19.9232005531601$$
$$t_{12} = -37.6947323057935$$
$$t_{13} = -77.6806787492188$$
$$t_{14} = -2.14242641424928$$
$$t_{15} = -73.2377958110605$$
$$t_{16} = -42.1376152439519$$
$$t_{17} = -15.4803176150016$$
$$t_{18} = -11.0374346761198$$
$$t_{19} = -33.2518493676352$$
$$t_{20} = -64.3520299347437$$
$$t_{21} = -6.59454849416224$$
$$t_{22} = -99.8950934400107$$
Puntos máximos de la función:
$$t_{22} = -66.5734714038229$$
$$t_{22} = -4.37330165985759$$
$$t_{22} = -53.2448225893478$$
$$t_{22} = -57.6877055275062$$
$$t_{22} = -17.7017590840809$$
$$t_{22} = -75.4592372801396$$
$$t_{22} = -93.2307690327731$$
$$t_{22} = -0.0830671258322571$$
$$t_{22} = -26.5875249603976$$
$$t_{22} = -84.3450031564564$$
$$t_{22} = -44.3590567130311$$
$$t_{22} = -31.030407898556$$
$$t_{22} = -13.2588761459326$$
$$t_{22} = -88.7878860946147$$
$$t_{22} = -97.6736519709315$$
$$t_{22} = -62.1305884656645$$
$$t_{22} = -22.1446420222392$$
$$t_{22} = -35.4732908367143$$
$$t_{22} = -79.902120218298$$
$$t_{22} = -71.0163543419813$$
$$t_{22} = -39.9161737748727$$
Decrece en los intervalos
$$\left[1.03338755252582, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.8950934400107\right]$$
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
$$\frac{48 t e^{2 t} - 8 e^{2 t} - 7 \sqrt{2} \sin{\left(\sqrt{2} t \right)} - 100 \cos{\left(\sqrt{2} t \right)}}{9} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 0.611186271916722$$
$$t_{2} = -47.6912189166499$$
$$t_{3} = -67.6841921383625$$
$$t_{4} = -16.5910383495413$$
$$t_{5} = -85.455723890996$$
$$t_{6} = -74.3485165456$$
$$t_{7} = -18.8124798186205$$
$$t_{8} = -32.1411286330956$$
$$t_{9} = -34.3625701021748$$
$$t_{10} = -29.9196871640164$$
$$t_{11} = -23.2553627567788$$
$$t_{12} = -27.6982456949372$$
$$t_{13} = -83.2342824219168$$
$$t_{14} = -9.92671395043974$$
$$t_{15} = -1.0890223138877$$
$$t_{16} = -61.019867731125$$
$$t_{17} = -76.5699580146792$$
$$t_{18} = -14.3695968804637$$
$$t_{19} = -69.9056336074417$$
$$t_{20} = -7.70527193374364$$
$$t_{21} = -92.1200482982335$$
$$t_{22} = -96.5629312363919$$
$$t_{23} = -52.1341018548082$$
$$t_{24} = -49.912660385729$$
$$t_{25} = -56.5769847929666$$
$$t_{26} = -36.5840115712539$$
$$t_{27} = -43.2483359784915$$
$$t_{28} = -81.0128409528376$$
$$t_{29} = -3.26068591809474$$
$$t_{30} = -41.0268945094123$$
$$t_{31} = -89.8986068291543$$
$$t_{32} = -5.48386392522753$$
$$t_{33} = -72.1270750765209$$
$$t_{34} = -12.1481554112662$$
$$t_{35} = -94.3414897673127$$
$$t_{36} = -63.2413092002041$$
$$t_{37} = -54.3555433238874$$
$$t_{38} = -21.0339212876997$$
$$t_{39} = -45.4697774475707$$
$$t_{40} = -65.4627506692833$$
$$t_{41} = -98.7843727054711$$
$$t_{42} = -25.476804225858$$
$$t_{43} = -87.6771653600752$$
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
$$\left[0.611186271916722, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -96.5629312363919\right]$$
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(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
$$\frac{48 t e^{2 t} - 8 e^{2 t} - 7 \sqrt{2} \sin{\left(\sqrt{2} t \right)} - 100 \cos{\left(\sqrt{2} t \right)}}{9} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$t_{1} = 0.611186271916722$$
$$t_{2} = -47.6912189166499$$
$$t_{3} = -67.6841921383625$$
$$t_{4} = -16.5910383495413$$
$$t_{5} = -85.455723890996$$
$$t_{6} = -74.3485165456$$
$$t_{7} = -18.8124798186205$$
$$t_{8} = -32.1411286330956$$
$$t_{9} = -34.3625701021748$$
$$t_{10} = -29.9196871640164$$
$$t_{11} = -23.2553627567788$$
$$t_{12} = -27.6982456949372$$
$$t_{13} = -83.2342824219168$$
$$t_{14} = -9.92671395043974$$
$$t_{15} = -1.0890223138877$$
$$t_{16} = -61.019867731125$$
$$t_{17} = -76.5699580146792$$
$$t_{18} = -14.3695968804637$$
$$t_{19} = -69.9056336074417$$
$$t_{20} = -7.70527193374364$$
$$t_{21} = -92.1200482982335$$
$$t_{22} = -96.5629312363919$$
$$t_{23} = -52.1341018548082$$
$$t_{24} = -49.912660385729$$
$$t_{25} = -56.5769847929666$$
$$t_{26} = -36.5840115712539$$
$$t_{27} = -43.2483359784915$$
$$t_{28} = -81.0128409528376$$
$$t_{29} = -3.26068591809474$$
$$t_{30} = -41.0268945094123$$
$$t_{31} = -89.8986068291543$$
$$t_{32} = -5.48386392522753$$
$$t_{33} = -72.1270750765209$$
$$t_{34} = -12.1481554112662$$
$$t_{35} = -94.3414897673127$$
$$t_{36} = -63.2413092002041$$
$$t_{37} = -54.3555433238874$$
$$t_{38} = -21.0339212876997$$
$$t_{39} = -45.4697774475707$$
$$t_{40} = -65.4627506692833$$
$$t_{41} = -98.7843727054711$$
$$t_{42} = -25.476804225858$$
$$t_{43} = -87.6771653600752$$
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
$$\left[0.611186271916722, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -96.5629312363919\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con t->+oo y t->-oo
$$\lim_{t \to -\infty}\left(\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right)\right) = \left\langle - \frac{50}{9}, \frac{50}{9}\right\rangle + \sqrt{2} \left\langle - \frac{7}{18}, \frac{7}{18}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{50}{9}, \frac{50}{9}\right\rangle + \sqrt{2} \left\langle - \frac{7}{18}, \frac{7}{18}\right\rangle$$
$$\lim_{t \to \infty}\left(\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right)\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
$$\lim_{t \to -\infty}\left(\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right)\right) = \left\langle - \frac{50}{9}, \frac{50}{9}\right\rangle + \sqrt{2} \left\langle - \frac{7}{18}, \frac{7}{18}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{50}{9}, \frac{50}{9}\right\rangle + \sqrt{2} \left\langle - \frac{7}{18}, \frac{7}{18}\right\rangle$$
$$\lim_{t \to \infty}\left(\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right)\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-14*exp(2*t))/9 + (50*cos(t*sqrt(2)))/9 + ((4*t)*exp(2*t))/3 + ((7*sqrt(2))*sin(t*sqrt(2)))/18, dividida por t con t->+oo y t ->-oo
$$\lim_{t \to -\infty}\left(\frac{\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right)}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right)}{t}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
$$\lim_{t \to -\infty}\left(\frac{\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right)}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right)}{t}\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
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:
$$\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right) = - \frac{4 t e^{- 2 t}}{3} - \frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9} - \frac{14 e^{- 2 t}}{9}$$
- No
$$\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right) = \frac{4 t e^{- 2 t}}{3} + \frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} - \frac{50 \cos{\left(\sqrt{2} t \right)}}{9} + \frac{14 e^{- 2 t}}{9}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right) = - \frac{4 t e^{- 2 t}}{3} - \frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9} - \frac{14 e^{- 2 t}}{9}$$
- No
$$\frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} + \left(\frac{4 t e^{2 t}}{3} + \left(\frac{\left(-1\right) 14 e^{2 t}}{9} + \frac{50 \cos{\left(\sqrt{2} t \right)}}{9}\right)\right) = \frac{4 t e^{- 2 t}}{3} + \frac{7 \sqrt{2} \sin{\left(\sqrt{2} t \right)}}{18} - \frac{50 \cos{\left(\sqrt{2} t \right)}}{9} + \frac{14 e^{- 2 t}}{9}$$
- No
es decir, función
no es
par ni impar