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(-cos(x*sqrt(2)/2)-sin(x*sqrt(2)/2))*exp(x*sqrt(2)/2)+(-cos(x*sqrt(2)/2)-sin(x*sqrt(2)/2))*exp(-x*sqrt(2)/2)
Trazado el gráfico de la función/ (-cos(x*sqrt(2)/2)-sin(x*sqrt(2)/2))*exp(x*sqrt(2)/2)+(-cos(x*sqrt(2)/2)-sin(x*sqrt(2)/2))*exp(-x*sqrt(2)/2)

Gráfico de la función y = (-cos(x*sqrt(2)/2)-sin(x*sqrt(2)/2))*exp(x*sqrt(2)/2)+(-cos(x*sqrt(2)/2)-sin(x*sqrt(2)/2))*exp(-x*sqrt(2)/2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                            ___                                         ___
                                        x*\/ 2                                     -x*\/ 2 
       /     /    ___\      /    ___\\  -------   /     /    ___\      /    ___\\  --------
       |     |x*\/ 2 |      |x*\/ 2 ||     2      |     |x*\/ 2 |      |x*\/ 2 ||     2    
f(x) = |- cos|-------| - sin|-------||*e        + |- cos|-------| - sin|-------||*e        
       \     \   2   /      \   2   //            \     \   2   /      \   2   //
f(x)=(−sin⁡(2x2)−cos⁡(2x2))e2(−x)2+(−sin⁡(2x2)−cos⁡(2x2))e2x2f{\left(x \right)} = \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}} + \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}} 
f = (-sin((sqrt(2)*x)/2) - cos((sqrt(2)*x)/2))*exp((sqrt(2)*(-x))/2) + (-sin((sqrt(2)*x)/2) - cos((sqrt(2)*x)/2))*exp((sqrt(2)*x)/2)
Gráfico de la función
02468-8-6-4-2-1010-20002000
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
(−sin⁡(2x2)−cos⁡(2x2))e2(−x)2+(−sin⁡(2x2)−cos⁡(2x2))e2x2=0\left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}} + \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−2π4x_{1} = - \frac{\sqrt{2} \pi}{4}
Solución numérica
x1=−23.3251354253314x_{1} = -23.3251354253314
x2=−5.55360367269796x_{2} = -5.55360367269796
x3=43.3181086470441x_{3} = 43.3181086470441
x4=16.6608110180939x_{4} = 16.6608110180939
x5=−41.0966671779649x_{5} = -41.0966671779649
x6=−27.7680183634898x_{6} = -27.7680183634898
x7=−32.2109013016482x_{7} = -32.2109013016482
x8=−9.99648661085632x_{8} = -9.99648661085632
x9=−18.8822524871731x_{9} = -18.8822524871731
x10=38.8752257088857x_{10} = 38.8752257088857
x11=7.77504514177714x_{11} = 7.77504514177714
x12=3.33216220361877x_{12} = 3.33216220361877
x13=−1.11072073453959x_{13} = -1.11072073453959
x14=−14.4393695490147x_{14} = -14.4393695490147
x15=−36.6537842398065x_{15} = -36.6537842398065
x16=25.5465768944106x_{16} = 25.5465768944106
x17=12.2179280799355x_{17} = 12.2179280799355
x18=34.4323427707273x_{18} = 34.4323427707273
x19=29.989459832569x_{19} = 29.989459832569
x20=21.1036939562522x_{20} = 21.1036939562522
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (-cos((x*sqrt(2))/2) - sin((x*sqrt(2))/2))*exp((x*sqrt(2))/2) + (-cos((x*sqrt(2))/2) - sin((x*sqrt(2))/2))*exp(((-x)*sqrt(2))/2).
(−cos⁡(022)−sin⁡(022))e022+(−cos⁡(022)−sin⁡(022))e−022\left(- \cos{\left(\frac{0 \sqrt{2}}{2} \right)} - \sin{\left(\frac{0 \sqrt{2}}{2} \right)}\right) e^{\frac{0 \sqrt{2}}{2}} + \left(- \cos{\left(\frac{0 \sqrt{2}}{2} \right)} - \sin{\left(\frac{0 \sqrt{2}}{2} \right)}\right) e^{\frac{- 0 \sqrt{2}}{2}}
Resultado:
f(0)=−2f{\left(0 \right)} = -2
Punto:
(0, -2)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
(2sin⁡(2x2)2−2cos⁡(2x2)2)e2(−x)2+(2sin⁡(2x2)2−2cos⁡(2x2)2)e2x2−2(−sin⁡(2x2)−cos⁡(2x2))e2(−x)22+2(−sin⁡(2x2)−cos⁡(2x2))e2x22=0\left(\frac{\sqrt{2} \sin{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - \frac{\sqrt{2} \cos{\left(\frac{\sqrt{2} x}{2} \right)}}{2}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}} + \left(\frac{\sqrt{2} \sin{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - \frac{\sqrt{2} \cos{\left(\frac{\sqrt{2} x}{2} \right)}}{2}\right) e^{\frac{\sqrt{2} x}{2}} - \frac{\sqrt{2} \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}}}{2} + \frac{\sqrt{2} \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}}}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−8.88576094443548x_{1} = -8.88576094443548
x2=−39.9859464434253x_{2} = -39.9859464434253
x3=2.15429000113047x_{3} = 2.15429000113047
x4=33.3216220361877x_{4} = 33.3216220361877
x5=37.7645049743461x_{5} = 37.7645049743461
x6=−4.44023205911364x_{6} = -4.44023205911364
x7=6.66421026246123x_{7} = 6.66421026246123
x8=−97.7434246394841x_{8} = -97.7434246394841
x9=42.2073879125045x_{9} = 42.2073879125045
x10=11.1072071322714x_{10} = 11.1072071322714
x11=−26.6572976289502x_{11} = -26.6572976289502
x12=24.435856159871x_{12} = 24.435856159871
x13=−17.7715317526163x_{13} = -17.7715317526163
x14=28.8787390980294x_{14} = 28.8787390980294
x15=−13.3286488052652x_{15} = -13.3286488052652
x16=15.5500902831563x_{16} = 15.5500902831563
x17=−48.871712319742x_{17} = -48.871712319742
x18=−44.4288293815837x_{18} = -44.4288293815837
x19=−22.2144146907918x_{19} = -22.2144146907918
x20=−31.1001805671086x_{20} = -31.1001805671086
x21=−35.5430635052669x_{21} = -35.5430635052669
x22=19.9929732217119x_{22} = 19.9929732217119
Signos de extremos en los puntos:
                                                                                                       ___                                                                                      ___ 
                     /     /                   ___\      /                   ___\\  4.44288047221774*\/ 2    /     /                   ___\      /                   ___\\  -4.44288047221774*\/ 2  
(-8.885760944435477, \- cos\4.44288047221774*\/ 2 / + sin\4.44288047221774*\/ 2 //*e                       + \- cos\4.44288047221774*\/ 2 / + sin\4.44288047221774*\/ 2 //*e                       )

                                                                                                      ___                                                                                      ___ 
                    /     /                   ___\      /                   ___\\  19.9929732217126*\/ 2    /     /                   ___\      /                   ___\\  -19.9929732217126*\/ 2  
(-39.9859464434253, \- cos\19.9929732217126*\/ 2 / + sin\19.9929732217126*\/ 2 //*e                       + \- cos\19.9929732217126*\/ 2 / + sin\19.9929732217126*\/ 2 //*e                       )

                                                                                                      ___                                                                                      ___ 
                    /     /                   ___\      /                   ___\\  1.07714500056524*\/ 2    /     /                   ___\      /                   ___\\  -1.07714500056524*\/ 2  
(2.154290001130474, \- cos\1.07714500056524*\/ 2 / - sin\1.07714500056524*\/ 2 //*e                       + \- cos\1.07714500056524*\/ 2 / - sin\1.07714500056524*\/ 2 //*e                       )

                                                                                                      ___                                                                                      ___ 
                    /     /                   ___\      /                   ___\\  16.6608110180939*\/ 2    /     /                   ___\      /                   ___\\  -16.6608110180939*\/ 2  
(33.32162203618775, \- cos\16.6608110180939*\/ 2 / - sin\16.6608110180939*\/ 2 //*e                       + \- cos\16.6608110180939*\/ 2 / - sin\16.6608110180939*\/ 2 //*e                       )

                                                                                                      ___                                                                                      ___ 
                    /     /                   ___\      /                   ___\\  18.8822524871731*\/ 2    /     /                   ___\      /                   ___\\  -18.8822524871731*\/ 2  
(37.76450497434611, \- cos\18.8822524871731*\/ 2 / - sin\18.8822524871731*\/ 2 //*e                       + \- cos\18.8822524871731*\/ 2 / - sin\18.8822524871731*\/ 2 //*e                       )

                                                                                                       ___                                                                                      ___ 
                     /     /                   ___\      /                   ___\\  2.22011602955682*\/ 2    /     /                   ___\      /                   ___\\  -2.22011602955682*\/ 2  
(-4.440232059113636, \- cos\2.22011602955682*\/ 2 / + sin\2.22011602955682*\/ 2 //*e                       + \- cos\2.22011602955682*\/ 2 / + sin\2.22011602955682*\/ 2 //*e                       )

                                                                                                       ___                                                                                      ___ 
                     /     /                   ___\      /                   ___\\  3.33210513123061*\/ 2    /     /                   ___\      /                   ___\\  -3.33210513123061*\/ 2  
(6.6642102624612285, \- cos\3.33210513123061*\/ 2 / - sin\3.33210513123061*\/ 2 //*e                       + \- cos\3.33210513123061*\/ 2 / - sin\3.33210513123061*\/ 2 //*e                       )

                                                                                                    ___                                                                                   ___ 
                     /     /                  ___\      /                  ___\\  48.871712319742*\/ 2    /     /                  ___\      /                  ___\\  -48.871712319742*\/ 2  
(-97.74342463948406, \- cos\48.871712319742*\/ 2 / + sin\48.871712319742*\/ 2 //*e                      + \- cos\48.871712319742*\/ 2 / + sin\48.871712319742*\/ 2 //*e                      )

                                                                                                      ___                                                                                      ___ 
                    /     /                   ___\      /                   ___\\  21.1036939562522*\/ 2    /     /                   ___\      /                   ___\\  -21.1036939562522*\/ 2  
(42.20738791250448, \- cos\21.1036939562522*\/ 2 / - sin\21.1036939562522*\/ 2 //*e                       + \- cos\21.1036939562522*\/ 2 / - sin\21.1036939562522*\/ 2 //*e                       )

                                                                                                       ___                                                                                      ___ 
                     /     /                   ___\      /                   ___\\  5.55360356613571*\/ 2    /     /                   ___\      /                   ___\\  -5.55360356613571*\/ 2  
(11.107207132271425, \- cos\5.55360356613571*\/ 2 / - sin\5.55360356613571*\/ 2 //*e                       + \- cos\5.55360356613571*\/ 2 / - sin\5.55360356613571*\/ 2 //*e                       )

                                                                                                        ___                                                                                      ___ 
                      /     /                   ___\      /                   ___\\  13.3286488144751*\/ 2    /     /                   ___\      /                   ___\\  -13.3286488144751*\/ 2  
(-26.657297628950197, \- cos\13.3286488144751*\/ 2 / + sin\13.3286488144751*\/ 2 //*e                       + \- cos\13.3286488144751*\/ 2 / + sin\13.3286488144751*\/ 2 //*e                       )

                                                                                                       ___                                                                                      ___ 
                     /     /                   ___\      /                   ___\\  12.2179280799355*\/ 2    /     /                   ___\      /                   ___\\  -12.2179280799355*\/ 2  
(24.435856159871012, \- cos\12.2179280799355*\/ 2 / - sin\12.2179280799355*\/ 2 //*e                       + \- cos\12.2179280799355*\/ 2 / - sin\12.2179280799355*\/ 2 //*e                       )

                                                                                                        ___                                                                                      ___ 
                      /     /                   ___\      /                   ___\\  8.88576587630813*\/ 2    /     /                   ___\      /                   ___\\  -8.88576587630813*\/ 2  
(-17.771531752616266, \- cos\8.88576587630813*\/ 2 / + sin\8.88576587630813*\/ 2 //*e                       + \- cos\8.88576587630813*\/ 2 / + sin\8.88576587630813*\/ 2 //*e                       )

                                                                                                      ___                                                                                      ___ 
                    /     /                   ___\      /                   ___\\  14.4393695490147*\/ 2    /     /                   ___\      /                   ___\\  -14.4393695490147*\/ 2  
(28.87873909802938, \- cos\14.4393695490147*\/ 2 / - sin\14.4393695490147*\/ 2 //*e                       + \- cos\14.4393695490147*\/ 2 / - sin\14.4393695490147*\/ 2 //*e                       )

                                                                                                        ___                                                                                      ___ 
                      /     /                   ___\      /                   ___\\  6.66432440263258*\/ 2    /     /                   ___\      /                   ___\\  -6.66432440263258*\/ 2  
(-13.328648805265157, \- cos\6.66432440263258*\/ 2 / + sin\6.66432440263258*\/ 2 //*e                       + \- cos\6.66432440263258*\/ 2 / + sin\6.66432440263258*\/ 2 //*e                       )

                                                                                                       ___                                                                                      ___ 
                     /     /                   ___\      /                   ___\\  7.77504514157814*\/ 2    /     /                   ___\      /                   ___\\  -7.77504514157814*\/ 2  
(15.550090283156285, \- cos\7.77504514157814*\/ 2 / - sin\7.77504514157814*\/ 2 //*e                       + \- cos\7.77504514157814*\/ 2 / - sin\7.77504514157814*\/ 2 //*e                       )

                                                                                                    ___                                                                                   ___ 
                     /     /                  ___\      /                  ___\\  24.435856159871*\/ 2    /     /                  ___\      /                  ___\\  -24.435856159871*\/ 2  
(-48.87171231974203, \- cos\24.435856159871*\/ 2 / + sin\24.435856159871*\/ 2 //*e                      + \- cos\24.435856159871*\/ 2 / + sin\24.435856159871*\/ 2 //*e                      )

                                                                                                       ___                                                                                      ___ 
                     /     /                   ___\      /                   ___\\  22.2144146907918*\/ 2    /     /                   ___\      /                   ___\\  -22.2144146907918*\/ 2  
(-44.42882938158366, \- cos\22.2144146907918*\/ 2 / + sin\22.2144146907918*\/ 2 //*e                       + \- cos\22.2144146907918*\/ 2 / + sin\22.2144146907918*\/ 2 //*e                       )

                                                                                                      ___                                                                                      ___ 
                    /     /                   ___\      /                   ___\\  11.1072073453959*\/ 2    /     /                   ___\      /                   ___\\  -11.1072073453959*\/ 2  
(-22.2144146907918, \- cos\11.1072073453959*\/ 2 / + sin\11.1072073453959*\/ 2 //*e                       + \- cos\11.1072073453959*\/ 2 / + sin\11.1072073453959*\/ 2 //*e                       )

                                                                                                        ___                                                                                      ___ 
                      /     /                   ___\      /                   ___\\  15.5500902835543*\/ 2    /     /                   ___\      /                   ___\\  -15.5500902835543*\/ 2  
(-31.100180567108563, \- cos\15.5500902835543*\/ 2 / + sin\15.5500902835543*\/ 2 //*e                       + \- cos\15.5500902835543*\/ 2 / + sin\15.5500902835543*\/ 2 //*e                       )

                                                                                                       ___                                                                                      ___ 
                     /     /                   ___\      /                   ___\\  17.7715317526335*\/ 2    /     /                   ___\      /                   ___\\  -17.7715317526335*\/ 2  
(-35.54306350526693, \- cos\17.7715317526335*\/ 2 / + sin\17.7715317526335*\/ 2 //*e                       + \- cos\17.7715317526335*\/ 2 / + sin\17.7715317526335*\/ 2 //*e                       )

                                                                                                       ___                                                                                      ___ 
                     /     /                   ___\      /                   ___\\  9.99648661085595*\/ 2    /     /                   ___\      /                   ___\\  -9.99648661085595*\/ 2  
(19.992973221711903, \- cos\9.99648661085595*\/ 2 / - sin\9.99648661085595*\/ 2 //*e                       + \- cos\9.99648661085595*\/ 2 / - sin\9.99648661085595*\/ 2 //*e                       )


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:
x1=−8.88576094443548x_{1} = -8.88576094443548
x2=2.15429000113047x_{2} = 2.15429000113047
x3=37.7645049743461x_{3} = 37.7645049743461
x4=−97.7434246394841x_{4} = -97.7434246394841
x5=11.1072071322714x_{5} = 11.1072071322714
x6=−26.6572976289502x_{6} = -26.6572976289502
x7=−17.7715317526163x_{7} = -17.7715317526163
x8=28.8787390980294x_{8} = 28.8787390980294
x9=−44.4288293815837x_{9} = -44.4288293815837
x10=−35.5430635052669x_{10} = -35.5430635052669
x11=19.9929732217119x_{11} = 19.9929732217119
Puntos máximos de la función:
x11=−39.9859464434253x_{11} = -39.9859464434253
x11=33.3216220361877x_{11} = 33.3216220361877
x11=−4.44023205911364x_{11} = -4.44023205911364
x11=6.66421026246123x_{11} = 6.66421026246123
x11=42.2073879125045x_{11} = 42.2073879125045
x11=24.435856159871x_{11} = 24.435856159871
x11=−13.3286488052652x_{11} = -13.3286488052652
x11=15.5500902831563x_{11} = 15.5500902831563
x11=−48.871712319742x_{11} = -48.871712319742
x11=−22.2144146907918x_{11} = -22.2144146907918
x11=−31.1001805671086x_{11} = -31.1001805671086
Decrece en los intervalos
[37.7645049743461,∞)\left[37.7645049743461, \infty\right)
Crece en los intervalos
(−∞,−97.7434246394841]\left(-\infty, -97.7434246394841\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
(e2x2−e−2x2)(2sin⁡(2x2)−2cos⁡(2x2))2=0\frac{\left(e^{\frac{\sqrt{2} x}{2}} - e^{- \frac{\sqrt{2} x}{2}}\right) \left(2 \sin{\left(\frac{\sqrt{2} x}{2} \right)} - 2 \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right)}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=2π4x_{2} = \frac{\sqrt{2} \pi}{4}

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
(−∞,0]∪[2π4,∞)\left(-\infty, 0\right] \cup \left[\frac{\sqrt{2} \pi}{4}, \infty\right)
Convexa en los intervalos
[0,2π4]\left[0, \frac{\sqrt{2} \pi}{4}\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((−sin⁡(2x2)−cos⁡(2x2))e2(−x)2+(−sin⁡(2x2)−cos⁡(2x2))e2x2)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}} + \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞((−sin⁡(2x2)−cos⁡(2x2))e2(−x)2+(−sin⁡(2x2)−cos⁡(2x2))e2x2)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}} + \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-cos((x*sqrt(2))/2) - sin((x*sqrt(2))/2))*exp((x*sqrt(2))/2) + (-cos((x*sqrt(2))/2) - sin((x*sqrt(2))/2))*exp(((-x)*sqrt(2))/2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−sin⁡(2x2)−cos⁡(2x2))e2(−x)2+(−sin⁡(2x2)−cos⁡(2x2))e2x2x)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{\left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}} + \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}}}{x}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
lim⁡x→∞((−sin⁡(2x2)−cos⁡(2x2))e2(−x)2+(−sin⁡(2x2)−cos⁡(2x2))e2x2x)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{\left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}} + \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}}}{x}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
(−sin⁡(2x2)−cos⁡(2x2))e2(−x)2+(−sin⁡(2x2)−cos⁡(2x2))e2x2=(sin⁡(2x2)−cos⁡(2x2))e2x2+(sin⁡(2x2)−cos⁡(2x2))e−2x2\left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}} + \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}} = \left(\sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}} + \left(\sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{- \frac{\sqrt{2} x}{2}}
- No
(−sin⁡(2x2)−cos⁡(2x2))e2(−x)2+(−sin⁡(2x2)−cos⁡(2x2))e2x2=−(sin⁡(2x2)−cos⁡(2x2))e2x2−(sin⁡(2x2)−cos⁡(2x2))e−2x2\left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} \left(- x\right)}{2}} + \left(- \sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}} = - \left(\sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{\frac{\sqrt{2} x}{2}} - \left(\sin{\left(\frac{\sqrt{2} x}{2} \right)} - \cos{\left(\frac{\sqrt{2} x}{2} \right)}\right) e^{- \frac{\sqrt{2} x}{2}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (-cos(x*sqrt(2)/2)-sin(x*sqrt(2)/2))*exp(x*sqrt(2)/2)+(-cos(x*sqrt(2)/2)-sin(x*sqrt(2)/2))*exp(-x*sqrt(2)/2)
    © Sr Examen – Калькулятор