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(-cos(4*x*sqrt(2)/3)-sin(4*x*sqrt(2)/3))*exp(x/3)
Trazado el gráfico de la función/ (-cos(4*x*sqrt(2)/3)-sin(4*x*sqrt(2)/3))*exp(x/3)

Gráfico de la función y = (-cos(4*x*sqrt(2)/3)-sin(4*x*sqrt(2)/3))*exp(x/3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                            x
       /     /      ___\      /      ___\\  -
       |     |4*x*\/ 2 |      |4*x*\/ 2 ||  3
f(x) = |- cos|---------| - sin|---------||*e 
       \     \    3    /      \    3    //
f(x)=(−sin⁡(2⋅4x3)−cos⁡(2⋅4x3))ex3f{\left(x \right)} = \left(- \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)} - \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}\right) e^{\frac{x}{3}} 
f = (-sin((sqrt(2)*(4*x))/3) - cos((sqrt(2)*(4*x))/3))*exp(x/3)
Gráfico de la función
02468-8-6-4-2-1010-5050
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⁡(2⋅4x3)−cos⁡(2⋅4x3))ex3=0\left(- \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)} - \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}\right) e^{\frac{x}{3}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−32π32x_{1} = - \frac{3 \sqrt{2} \pi}{32}
x2=452π32x_{2} = \frac{45 \sqrt{2} \pi}{32}
x3=−32ilog⁡(−(−1)316)2x_{3} = - \frac{3 \sqrt{2} i \log{\left(- \left(-1\right)^{\frac{3}{16}} \right)}}{2}
Solución numérica
x1=−45.4007100243058x_{1} = -45.4007100243058
x2=47.8998316770199x_{2} = 47.8998316770199
x3=74.5571293059701x_{3} = 74.5571293059701
x4=−13.7451690899274x_{4} = -13.7451690899274
x5=67.8928048987325x_{5} = 67.8928048987325
x6=−22.0755745989744x_{6} = -22.0755745989744
x7=17.9103718444509x_{7} = 17.9103718444509
x8=19.5764529462603x_{8} = 19.5764529462603
x9=−53.7311155333527x_{9} = -53.7311155333527
x10=29.5729395571166x_{10} = 29.5729395571166
x11=89.5518592222546x_{11} = 89.5518592222546
x12=44.5676694734011x_{12} = 44.5676694734011
x13=36.2372639643542x_{13} = 36.2372639643542
x14=34.5711828625448x_{14} = 34.5711828625448
x15=−102.047467485825x_{15} = -102.047467485825
x16=57.8963182878762x_{16} = 57.8963182878762
x17=−3.74868247907112x_{17} = -3.74868247907112
x18=42.9015883715917x_{18} = 42.9015883715917
x19=−92.0509808749686x_{19} = -92.0509808749686
x20=−60.3954399405903x_{20} = -60.3954399405903
x21=−42.068547820687x_{21} = -42.068547820687
x22=−80.3884131623029x_{22} = -80.3884131623029
x23=56.2302371860668x_{23} = 56.2302371860668
x24=79.5553726113982x_{24} = 79.5553726113982
x25=−10.4130068863087x_{25} = -10.4130068863087
x26=−40.4024667188776x_{26} = -40.4024667188776
x27=6.2478041317852x_{27} = 6.2478041317852
x28=−55.3971966351621x_{28} = -55.3971966351621
x29=−63.7276021442091x_{29} = -63.7276021442091
x30=54.5641560842574x_{30} = 54.5641560842574
x31=−85.3866564677311x_{31} = -85.3866564677311
x32=−5.41476358088051x_{32} = -5.41476358088051
x33=−70.3919265514466x_{33} = -70.3919265514466
x34=51.2319938806387x_{34} = 51.2319938806387
x35=−90.3848997731593x_{35} = -90.3848997731593
x36=16.2442907426415x_{36} = 16.2442907426415
x37=26.2407773534979x_{37} = 26.2407773534979
x38=69.5588860005419x_{38} = 69.5588860005419
x39=66.2267237969232x_{39} = 66.2267237969232
x40=−62.0615210423997x_{40} = -62.0615210423997
x41=61.228480491495x_{41} = 61.228480491495
x42=−82.0544942641123x_{42} = -82.0544942641123
x43=−15.4112501917368x_{43} = -15.4112501917368
x44=−20.409493497165x_{44} = -20.409493497165
x45=−95.3831430785874x_{45} = -95.3831430785874
x46=−73.7240887550654x_{46} = -73.7240887550654
x47=7.91388523359459x_{47} = 7.91388523359459
x48=84.5536159168264x_{48} = 84.5536159168264
x49=87.8857781204452x_{49} = 87.8857781204452
x50=99.5483458331109x_{50} = 99.5483458331109
x51=46.2337505752105x_{51} = 46.2337505752105
x52=49.5659127788293x_{52} = 49.5659127788293
x53=59.5623993896856x_{53} = 59.5623993896856
x54=9.57996633540398x_{54} = 9.57996633540398
x55=−43.7346289224964x_{55} = -43.7346289224964
x56=−50.398953329734x_{56} = -50.398953329734
x57=−30.4059801080213x_{57} = -30.4059801080213
x58=−35.4042234134495x_{58} = -35.4042234134495
x59=14.5782096408321x_{59} = 14.5782096408321
x60=39.5694261679729x_{60} = 39.5694261679729
x61=−52.0650344315434x_{61} = -52.0650344315434
x62=−0.416520275452347x_{62} = -0.416520275452347
x63=64.5606426951138x_{63} = 64.5606426951138
x64=86.2196970186358x_{64} = 86.2196970186358
x65=−33.7381423116401x_{65} = -33.7381423116401
x66=−65.3936832460185x_{66} = -65.3936832460185
x67=37.9033450661636x_{67} = 37.9033450661636
x68=−83.7205753659217x_{68} = -83.7205753659217
x69=−23.7416557007838x_{69} = -23.7416557007838
x70=−32.0720612098307x_{70} = -32.0720612098307
x71=−2.08260137726173x_{71} = -2.08260137726173
x72=−72.058007653256x_{72} = -72.058007653256
x73=4.58172302997581x_{73} = 4.58172302997581
x74=−75.3901698568748x_{74} = -75.3901698568748
x75=77.8892915095889x_{75} = 77.8892915095889
x76=−25.4077368025932x_{76} = -25.4077368025932
x77=76.2232104077795x_{77} = 76.2232104077795
x78=27.9068584553072x_{78} = 27.9068584553072
x79=24.5746962516885x_{79} = 24.5746962516885
x80=−100.381386384016x_{80} = -100.381386384016
x81=−12.0790879881181x_{81} = -12.0790879881181
x82=−93.717061976778x_{82} = -93.717061976778
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(((4*x)*sqrt(2))/3) - sin(((4*x)*sqrt(2))/3))*exp(x/3).
(−cos⁡(0⋅423)−sin⁡(0⋅423))e03\left(- \cos{\left(\frac{0 \cdot 4 \sqrt{2}}{3} \right)} - \sin{\left(\frac{0 \cdot 4 \sqrt{2}}{3} \right)}\right) e^{\frac{0}{3}}
Resultado:
f(0)=−1f{\left(0 \right)} = -1
Punto:
(0, -1)
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
(42sin⁡(2⋅4x3)3−42cos⁡(2⋅4x3)3)ex3+(−sin⁡(2⋅4x3)−cos⁡(2⋅4x3))ex33=0\left(\frac{4 \sqrt{2} \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}}{3} - \frac{4 \sqrt{2} \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}}{3}\right) e^{\frac{x}{3}} + \frac{\left(- \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)} - \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}\right) e^{\frac{x}{3}}}{3} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=32atan⁡(−83331−3331+6631+8231)4x_{1} = \frac{3 \sqrt{2} \operatorname{atan}{\left(- \frac{8 \sqrt{33}}{31} - \frac{33}{31} + \frac{\sqrt{66}}{31} + \frac{8 \sqrt{2}}{31} \right)}}{4}
x2=−32atan⁡(−83331−8231+6631+3331)4x_{2} = - \frac{3 \sqrt{2} \operatorname{atan}{\left(- \frac{8 \sqrt{33}}{31} - \frac{8 \sqrt{2}}{31} + \frac{\sqrt{66}}{31} + \frac{33}{31} \right)}}{4}
Signos de extremos en los puntos:
                                                                                                                                                                    /           ____     ____       ___\ 
                                                                                                                                                            ___     |  33   8*\/ 33    \/ 66    8*\/ 2 | 
             /           ____     ____       ___\                                                                                                         \/ 2 *atan|- -- - -------- + ------ + -------| 
     ___     |  33   8*\/ 33    \/ 66    8*\/ 2 |                                                                                                                   \  31      31        31        31  / 
 3*\/ 2 *atan|- -- - -------- + ------ + -------|  /     /      /           ____     ____       ___\\      /      /           ____     ____       ___\\\  ---------------------------------------------- 
             \  31      31        31        31  /  |     |      |  33   8*\/ 33    \/ 66    8*\/ 2 ||      |      |  33   8*\/ 33    \/ 66    8*\/ 2 |||                        4                        
(------------------------------------------------, |- cos|2*atan|- -- - -------- + ------ + -------|| - sin|2*atan|- -- - -------- + ------ + -------|||*e                                              )
                        4                          \     \      \  31      31        31        31  //      \      \  31      31        31        31  ///                                                 

                                                                                                                                                                /         ___       ____     ____\  
                                                                                                                                                        ___     |33   8*\/ 2    8*\/ 33    \/ 66 |  
              /         ___       ____     ____\                                                                                                     -\/ 2 *atan|-- - ------- - -------- + ------|  
      ___     |33   8*\/ 2    8*\/ 33    \/ 66 |                                                                                                                \31      31        31        31  /  
 -3*\/ 2 *atan|-- - ------- - -------- + ------|  /     /      /         ___       ____     ____\\      /      /         ___       ____     ____\\\  ---------------------------------------------- 
              \31      31        31        31  /  |     |      |33   8*\/ 2    8*\/ 33    \/ 66 ||      |      |33   8*\/ 2    8*\/ 33    \/ 66 |||                        4                        
(-----------------------------------------------, |- cos|2*atan|-- - ------- - -------- + ------|| + sin|2*atan|-- - ------- - -------- + ------|||*e                                              )
                        4                         \     \      \31      31        31        31  //      \      \31      31        31        31  ///


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=−32atan⁡(−83331−8231+6631+3331)4x_{1} = - \frac{3 \sqrt{2} \operatorname{atan}{\left(- \frac{8 \sqrt{33}}{31} - \frac{8 \sqrt{2}}{31} + \frac{\sqrt{66}}{31} + \frac{33}{31} \right)}}{4}
Puntos máximos de la función:
x1=32atan⁡(−83331−3331+6631+8231)4x_{1} = \frac{3 \sqrt{2} \operatorname{atan}{\left(- \frac{8 \sqrt{33}}{31} - \frac{33}{31} + \frac{\sqrt{66}}{31} + \frac{8 \sqrt{2}}{31} \right)}}{4}
Decrece en los intervalos
(−∞,32atan⁡(−83331−3331+6631+8231)4]∪[−32atan⁡(−83331−8231+6631+3331)4,∞)\left(-\infty, \frac{3 \sqrt{2} \operatorname{atan}{\left(- \frac{8 \sqrt{33}}{31} - \frac{33}{31} + \frac{\sqrt{66}}{31} + \frac{8 \sqrt{2}}{31} \right)}}{4}\right] \cup \left[- \frac{3 \sqrt{2} \operatorname{atan}{\left(- \frac{8 \sqrt{33}}{31} - \frac{8 \sqrt{2}}{31} + \frac{\sqrt{66}}{31} + \frac{33}{31} \right)}}{4}, \infty\right)
Crece en los intervalos
[32atan⁡(−83331−3331+6631+8231)4,−32atan⁡(−83331−8231+6631+3331)4]\left[\frac{3 \sqrt{2} \operatorname{atan}{\left(- \frac{8 \sqrt{33}}{31} - \frac{33}{31} + \frac{\sqrt{66}}{31} + \frac{8 \sqrt{2}}{31} \right)}}{4}, - \frac{3 \sqrt{2} \operatorname{atan}{\left(- \frac{8 \sqrt{33}}{31} - \frac{8 \sqrt{2}}{31} + \frac{\sqrt{66}}{31} + \frac{33}{31} \right)}}{4}\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((−sin⁡(2⋅4x3)−cos⁡(2⋅4x3))ex3)=0\lim_{x \to -\infty}\left(\left(- \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)} - \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}\right) e^{\frac{x}{3}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((−sin⁡(2⋅4x3)−cos⁡(2⋅4x3))ex3)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(- \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)} - \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}\right) e^{\frac{x}{3}}\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(((4*x)*sqrt(2))/3) - sin(((4*x)*sqrt(2))/3))*exp(x/3), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−sin⁡(2⋅4x3)−cos⁡(2⋅4x3))ex3x)=0\lim_{x \to -\infty}\left(\frac{\left(- \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)} - \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}\right) e^{\frac{x}{3}}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((−sin⁡(2⋅4x3)−cos⁡(2⋅4x3))ex3x)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{\left(- \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)} - \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}\right) e^{\frac{x}{3}}}{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⁡(2⋅4x3)−cos⁡(2⋅4x3))ex3=(sin⁡(42x3)−cos⁡(42x3))e−x3\left(- \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)} - \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}\right) e^{\frac{x}{3}} = \left(\sin{\left(\frac{4 \sqrt{2} x}{3} \right)} - \cos{\left(\frac{4 \sqrt{2} x}{3} \right)}\right) e^{- \frac{x}{3}}
- No
(−sin⁡(2⋅4x3)−cos⁡(2⋅4x3))ex3=−(sin⁡(42x3)−cos⁡(42x3))e−x3\left(- \sin{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)} - \cos{\left(\frac{\sqrt{2} \cdot 4 x}{3} \right)}\right) e^{\frac{x}{3}} = - \left(\sin{\left(\frac{4 \sqrt{2} x}{3} \right)} - \cos{\left(\frac{4 \sqrt{2} x}{3} \right)}\right) e^{- \frac{x}{3}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (-cos(4*x*sqrt(2)/3)-sin(4*x*sqrt(2)/3))*exp(x/3)
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