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

Gráfico de la función y = cos(2*x)+exp(3*x)+sin(2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   3*x           
f(x) = cos(2*x) + e    + sin(2*x)
f(x)=(e3x+cos⁡(2x))+sin⁡(2x)f{\left(x \right)} = \left(e^{3 x} + \cos{\left(2 x \right)}\right) + \sin{\left(2 x \right)} 
f = exp(3*x) + cos(2*x) + sin(2*x)
Gráfico de la función
02468-8-6-4-2-101020000000000000-10000000000000
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:
(e3x+cos⁡(2x))+sin⁡(2x)=0\left(e^{3 x} + \cos{\left(2 x \right)}\right) + \sin{\left(2 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−22.3838476568273x_{1} = -22.3838476568273
x2=−38.0918109247762x_{2} = -38.0918109247762
x3=−17.6714586764426x_{3} = -17.6714586764426
x4=−77.3617190946487x_{4} = -77.3617190946487
x5=−66.3661448070844x_{5} = -66.3661448070844
x6=−25.5254403104171x_{6} = -25.5254403104171
x7=−9.81747704246816x_{7} = -9.81747704246816
x8=−41.233403578366x_{8} = -41.233403578366
x9=−3.53430051890507x_{9} = -3.53430051890507
x10=−75.7909227678538x_{10} = -75.7909227678538
x11=−99.3528676697772x_{11} = -99.3528676697772
x12=−184.175869316702x_{12} = -184.175869316702
x13=−8.24668071566684x_{13} = -8.24668071566684
x14=−61.6537558266997x_{14} = -61.6537558266997
x15=−45.9457925587507x_{15} = -45.9457925587507
x16=−39.6626072515711x_{16} = -39.6626072515711
x17=−31.8086256175967x_{17} = -31.8086256175967
x18=−97.7820713429823x_{18} = -97.7820713429823
x19=−58.5121631731099x_{19} = -58.5121631731099
x20=−47.5165888855456x_{20} = -47.5165888855456
x21=−53.7997741927252x_{21} = -53.7997741927252
x22=−52.2289778659303x_{22} = -52.2289778659303
x23=−33.3794219443916x_{23} = -33.3794219443916
x24=−11.388273369263x_{24} = -11.388273369263
x25=−44.3749962319558x_{25} = -44.3749962319558
x26=−55.3705705195201x_{26} = -55.3705705195201
x27=−89.9280897090078x_{27} = -89.9280897090078
x28=−91.4988860358027x_{28} = -91.4988860358027
x29=−69.5077374606742x_{29} = -69.5077374606742
x30=−83.6449044018282x_{30} = -83.6449044018282
x31=−30.2378292908018x_{31} = -30.2378292908018
x32=−80.5033117482384x_{32} = -80.5033117482384
x33=−16.1006623496477x_{33} = -16.1006623496477
x34=−23.9546439836222x_{34} = -23.9546439836222
x35=−74.2201264410589x_{35} = -74.2201264410589
x36=−19.2422550032375x_{36} = -19.2422550032375
x37=−67.9369411338793x_{37} = -67.9369411338793
x38=−88.3572933822129x_{38} = -88.3572933822129
x39=−36.5210145979813x_{39} = -36.5210145979813
x40=−96.2112750161874x_{40} = -96.2112750161874
x41=−1.96251472845998x_{41} = -1.96251472845998
x42=−0.477501830655232x_{42} = -0.477501830655232
x43=−60.0829594999048x_{43} = -60.0829594999048
x44=−82.0741080750334x_{44} = -82.0741080750334
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(2*x) + exp(3*x) + sin(2*x).
sin⁡(0⋅2)+(cos⁡(0⋅2)+e0⋅3)\sin{\left(0 \cdot 2 \right)} + \left(\cos{\left(0 \cdot 2 \right)} + e^{0 \cdot 3}\right)
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
3e3x−2sin⁡(2x)+2cos⁡(2x)=03 e^{3 x} - 2 \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−29.4524311274043x_{1} = -29.4524311274043
x2=−92.2842841992002x_{2} = -92.2842841992002
x3=−21.5984494934298x_{3} = -21.5984494934298
x4=−95.42587685279x_{4} = -95.42587685279
x5=−84.4303025652257x_{5} = -84.4303025652257
x6=−100.138265833175x_{6} = -100.138265833175
x7=−73.4347282776614x_{7} = -73.4347282776614
x8=−81.2887099116359x_{8} = -81.2887099116359
x9=−13.7444678594553x_{9} = -13.7444678594553
x10=−18.45685683984x_{10} = -18.45685683984
x11=−27.8816348006094x_{11} = -27.8816348006094
x12=−93.8550805259951x_{12} = -93.8550805259951
x13=−79.717913584841x_{13} = -79.717913584841
x14=−42.0188017417635x_{14} = -42.0188017417635
x15=−70.2931356240716x_{15} = -70.2931356240716
x16=−40.4480054149686x_{16} = -40.4480054149686
x17=−65.5807466436869x_{17} = -65.5807466436869
x18=−12.1736715326604x_{18} = -12.1736715326604
x19=−87.5718952188155x_{19} = -87.5718952188155
x20=−71.8639319508665x_{20} = -71.8639319508665
x21=−4.31969114748392x_{21} = -4.31969114748392
x22=−64.009950316892x_{22} = -64.009950316892
x23=−1.1929018771322x_{23} = -1.1929018771322
x24=−32.5940237809941x_{24} = -32.5940237809941
x25=−34.164820107789x_{25} = -34.164820107789
x26=−59.2975613365073x_{26} = -59.2975613365073
x27=−15.3152641862502x_{27} = -15.3152641862502
x28=−20.0276531666349x_{28} = -20.0276531666349
x29=−49.872783375738x_{29} = -49.872783375738
x30=−5.8904862142625x_{30} = -5.8904862142625
x31=−48.3019870489431x_{31} = -48.3019870489431
x32=−43.5895980685584x_{32} = -43.5895980685584
x33=−26.3108384738145x_{33} = -26.3108384738145
x34=−57.7267650097125x_{34} = -57.7267650097125
x35=−37.3064127613788x_{35} = -37.3064127613788
x36=−118.987821754713x_{36} = -118.987821754713
x37=−35.7356164345839x_{37} = -35.7356164345839
x38=−56.1559686829176x_{38} = -56.1559686829176
x39=−86.0010988920206x_{39} = -86.0010988920206
x40=−62.4391539900971x_{40} = -62.4391539900971
x41=−7.46128255237654x_{41} = -7.46128255237654
x42=−51.4435797025329x_{42} = -51.4435797025329
x43=−78.1471172580461x_{43} = -78.1471172580461
Signos de extremos en los puntos:
(-29.45243112740431, -1.41421356237309)

(-92.28428419920017, -1.41421356237309)

(-21.59844949342983, 1.41421356237309)

(-95.42587685278997, -1.41421356237309)

(-84.43030256522569, 1.41421356237309)

(-100.13826583317466, 1.41421356237309)

(-73.43472827766142, -1.41421356237309)

(-81.2887099116359, 1.41421356237309)

(-13.744467859455346, -1.41421356237309)

(-18.456856839840036, 1.41421356237309)

(-27.881634800609415, 1.41421356237309)

(-93.85508052599508, 1.41421356237309)

(-79.717913584841, -1.41421356237309)

(-42.01880174176348, -1.41421356237309)

(-70.29313562407162, -1.41421356237309)

(-40.44800541496859, 1.41421356237309)

(-65.58074664368694, 1.41421356237309)

(-12.173671532660448, 1.4142135623731)

(-87.57189521881548, 1.41421356237309)

(-71.86393195086652, 1.4142135623731)

(-4.319691147483921, -1.41421120761268)

(-64.00995031689203, -1.41421356237309)

(-1.192901877132198, -1.3856818780789)

(-32.59402378099411, -1.41421356237309)

(-34.164820107789, 1.41421356237309)

(-59.29756133650735, 1.41421356237309)

(-15.315264186250243, 1.41421356237309)

(-20.02765316663493, -1.41421356237309)

(-49.87278337573797, 1.41421356237309)

(-5.890486214262505, 1.41421358352663)

(-48.30198704894307, -1.41421356237309)

(-43.58959806855838, 1.41421356237309)

(-26.310838473814517, -1.41421356237309)

(-57.72676500971245, -1.4142135623731)

(-37.306412761378795, 1.41421356237309)

(-118.98782175471342, 1.41421356237309)

(-35.735616434583896, -1.41421356237309)

(-56.15596868291755, 1.4142135623731)

(-86.0010988920206, -1.41421356237309)

(-62.43915399009714, 1.41421356237309)

(-7.461282552376537, -1.41421356218307)

(-51.443579702532865, -1.41421356237309)

(-78.14711725804611, 1.41421356237309)


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=−29.4524311274043x_{1} = -29.4524311274043
x2=−92.2842841992002x_{2} = -92.2842841992002
x3=−95.42587685279x_{3} = -95.42587685279
x4=−73.4347282776614x_{4} = -73.4347282776614
x5=−13.7444678594553x_{5} = -13.7444678594553
x6=−79.717913584841x_{6} = -79.717913584841
x7=−42.0188017417635x_{7} = -42.0188017417635
x8=−70.2931356240716x_{8} = -70.2931356240716
x9=−4.31969114748392x_{9} = -4.31969114748392
x10=−64.009950316892x_{10} = -64.009950316892
x11=−1.1929018771322x_{11} = -1.1929018771322
x12=−32.5940237809941x_{12} = -32.5940237809941
x13=−20.0276531666349x_{13} = -20.0276531666349
x14=−48.3019870489431x_{14} = -48.3019870489431
x15=−26.3108384738145x_{15} = -26.3108384738145
x16=−57.7267650097125x_{16} = -57.7267650097125
x17=−35.7356164345839x_{17} = -35.7356164345839
x18=−86.0010988920206x_{18} = -86.0010988920206
x19=−7.46128255237654x_{19} = -7.46128255237654
x20=−51.4435797025329x_{20} = -51.4435797025329
Puntos máximos de la función:
x20=−21.5984494934298x_{20} = -21.5984494934298
x20=−84.4303025652257x_{20} = -84.4303025652257
x20=−100.138265833175x_{20} = -100.138265833175
x20=−81.2887099116359x_{20} = -81.2887099116359
x20=−18.45685683984x_{20} = -18.45685683984
x20=−27.8816348006094x_{20} = -27.8816348006094
x20=−93.8550805259951x_{20} = -93.8550805259951
x20=−40.4480054149686x_{20} = -40.4480054149686
x20=−65.5807466436869x_{20} = -65.5807466436869
x20=−12.1736715326604x_{20} = -12.1736715326604
x20=−87.5718952188155x_{20} = -87.5718952188155
x20=−71.8639319508665x_{20} = -71.8639319508665
x20=−34.164820107789x_{20} = -34.164820107789
x20=−59.2975613365073x_{20} = -59.2975613365073
x20=−15.3152641862502x_{20} = -15.3152641862502
x20=−49.872783375738x_{20} = -49.872783375738
x20=−5.8904862142625x_{20} = -5.8904862142625
x20=−43.5895980685584x_{20} = -43.5895980685584
x20=−37.3064127613788x_{20} = -37.3064127613788
x20=−118.987821754713x_{20} = -118.987821754713
x20=−56.1559686829176x_{20} = -56.1559686829176
x20=−62.4391539900971x_{20} = -62.4391539900971
x20=−78.1471172580461x_{20} = -78.1471172580461
Decrece en los intervalos
[−1.1929018771322,∞)\left[-1.1929018771322, \infty\right)
Crece en los intervalos
(−∞,−95.42587685279]\left(-\infty, -95.42587685279\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
9e3x−4sin⁡(2x)−4cos⁡(2x)=09 e^{3 x} - 4 \sin{\left(2 x \right)} - 4 \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−67.9369411338793x_{1} = -67.9369411338793
x2=−25.5254403104171x_{2} = -25.5254403104171
x3=−36.5210145979813x_{3} = -36.5210145979813
x4=−33.3794219443916x_{4} = -33.3794219443916
x5=−6.67588438728339x_{5} = -6.67588438728339
x6=−16.1006623496477x_{6} = -16.1006623496477
x7=−99.3528676697772x_{7} = -99.3528676697772
x8=−83.6449044018282x_{8} = -83.6449044018282
x9=−1.9656810833923x_{9} = -1.9656810833923
x10=−89.9280897090078x_{10} = -89.9280897090078
x11=−60.0829594999048x_{11} = -60.0829594999048
x12=−88.3572933822129x_{12} = -88.3572933822129
x13=−55.3705705195201x_{13} = -55.3705705195201
x14=−45.9457925587507x_{14} = -45.9457925587507
x15=−3.53427197045858x_{15} = -3.53427197045858
x16=−44.3749962319558x_{16} = -44.3749962319558
x17=−22.3838476568273x_{17} = -22.3838476568273
x18=−97.7820713429823x_{18} = -97.7820713429823
x19=−96.2112750161874x_{19} = -96.2112750161874
x20=−11.388273369263x_{20} = -11.388273369263
x21=−23.9546439836222x_{21} = -23.9546439836222
x22=−80.5033117482384x_{22} = -80.5033117482384
x23=−184.175869316702x_{23} = -184.175869316702
x24=−58.5121631731099x_{24} = -58.5121631731099
x25=−66.3661448070844x_{25} = -66.3661448070844
x26=−41.233403578366x_{26} = -41.233403578366
x27=−61.6537558266997x_{27} = -61.6537558266997
x28=−30.2378292908018x_{28} = -30.2378292908018
x29=−75.7909227678538x_{29} = -75.7909227678538
x30=−19.2422550032375x_{30} = -19.2422550032375
x31=−31.8086256175967x_{31} = -31.8086256175967
x32=−69.5077374606742x_{32} = -69.5077374606742
x33=−91.4988860358027x_{33} = -91.4988860358027
x34=−53.7997741927252x_{34} = -53.7997741927252
x35=−47.5165888855456x_{35} = -47.5165888855456
x36=−52.2289778659303x_{36} = -52.2289778659303
x37=−38.0918109247762x_{37} = -38.0918109247762
x38=−74.2201264410589x_{38} = -74.2201264410589
x39=−8.24668071568754x_{39} = -8.24668071568754
x40=−17.6714586764426x_{40} = -17.6714586764426
x41=−77.3617190946487x_{41} = -77.3617190946487
x42=−39.6626072515711x_{42} = -39.6626072515711
x43=−9.81747704246798x_{43} = -9.81747704246798
x44=−82.0741080750334x_{44} = -82.0741080750334

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
[−1.9656810833923,∞)\left[-1.9656810833923, \infty\right)
Convexa en los intervalos
(−∞,−184.175869316702]\left(-\infty, -184.175869316702\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((e3x+cos⁡(2x))+sin⁡(2x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\left(e^{3 x} + \cos{\left(2 x \right)}\right) + \sin{\left(2 x \right)}\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞((e3x+cos⁡(2x))+sin⁡(2x))=∞\lim_{x \to \infty}\left(\left(e^{3 x} + \cos{\left(2 x \right)}\right) + \sin{\left(2 x \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 cos(2*x) + exp(3*x) + sin(2*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((e3x+cos⁡(2x))+sin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\left(e^{3 x} + \cos{\left(2 x \right)}\right) + \sin{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((e3x+cos⁡(2x))+sin⁡(2x)x)=∞\lim_{x \to \infty}\left(\frac{\left(e^{3 x} + \cos{\left(2 x \right)}\right) + \sin{\left(2 x \right)}}{x}\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(-x) и f = -f(-x).
Pues, comprobamos:
(e3x+cos⁡(2x))+sin⁡(2x)=−sin⁡(2x)+cos⁡(2x)+e−3x\left(e^{3 x} + \cos{\left(2 x \right)}\right) + \sin{\left(2 x \right)} = - \sin{\left(2 x \right)} + \cos{\left(2 x \right)} + e^{- 3 x}
- No
(e3x+cos⁡(2x))+sin⁡(2x)=sin⁡(2x)−cos⁡(2x)−e−3x\left(e^{3 x} + \cos{\left(2 x \right)}\right) + \sin{\left(2 x \right)} = \sin{\left(2 x \right)} - \cos{\left(2 x \right)} - e^{- 3 x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор