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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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

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
$$t_{1} = -47.6912189166499$$
$$t_{2} = -12.148155411446$$
$$t_{3} = 1.14609061132884$$
$$t_{4} = -67.6841921383625$$
$$t_{5} = -85.455723890996$$
$$t_{6} = -74.3485165456$$
$$t_{7} = -14.3695968804612$$
$$t_{8} = -32.1411286330956$$
$$t_{9} = -34.3625701021748$$
$$t_{10} = -23.2553627567788$$
$$t_{11} = -29.9196871640164$$
$$t_{12} = -7.70527277723061$$
$$t_{13} = -27.6982456949372$$
$$t_{14} = -3.26348442564007$$
$$t_{15} = -83.2342824219168$$
$$t_{16} = -5.48381162863367$$
$$t_{17} = -61.019867731125$$
$$t_{18} = -76.5699580146792$$
$$t_{19} = -69.9056336074417$$
$$t_{20} = -0.990670137807717$$
$$t_{21} = -92.1200482982335$$
$$t_{22} = -96.5629312363919$$
$$t_{23} = -52.1341018548082$$
$$t_{24} = -49.912660385729$$
$$t_{25} = -56.5769847929666$$
$$t_{26} = -36.5840115712539$$
$$t_{27} = -43.2483359784915$$
$$t_{28} = -81.0128409528376$$
$$t_{29} = -41.0268945094123$$
$$t_{30} = -89.8986068291543$$
$$t_{31} = -72.1270750765209$$
$$t_{32} = -94.3414897673127$$
$$t_{33} = -63.2413092002041$$
$$t_{34} = -9.9267139378327$$
$$t_{35} = -54.3555433238874$$
$$t_{36} = -21.0339212876997$$
$$t_{37} = -45.4697774475707$$
$$t_{38} = -65.4627506692833$$
$$t_{39} = -98.7843727054711$$
$$t_{40} = -25.476804225858$$
$$t_{41} = -16.5910383495413$$
$$t_{42} = -87.6771653600752$$
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando t es igual a 0:
sustituimos t = 0 en (-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.
$$\frac{7 \sqrt{2} \sin{\left(0 \sqrt{2} \right)}}{18} + \left(\frac{0 \cdot 4 e^{0 \cdot 2}}{3} + \left(\frac{\left(-1\right) 14 e^{0 \cdot 2}}{9} + \frac{50 \cos{\left(0 \sqrt{2} \right)}}{9}\right)\right)$$
Resultado:
$$f{\left(0 \right)} = 4$$
Punto:
(0, 4)
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:
                                                   /                   ___\       ___    /                   ___\ 
                                             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
$$\left(-\infty, -96.5629312363919\right]$$
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
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
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