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Trazado el gráfico de la función/ 4/13+x-3*cos(x)+(-cos(3*x)-sin(3*x))*exp(2*x)+sin(x)

Gráfico de la función y = 4/13+x-3*cos(x)+(-cos(3*x)-sin(3*x))*exp(2*x)+sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                                     2*x         
f(x) = 4/13 + x - 3*cos(x) + (-cos(3*x) - sin(3*x))*e    + sin(x)
f(x)=(((x+413)−3cos⁡(x))+(−sin⁡(3x)−cos⁡(3x))e2x)+sin⁡(x)f{\left(x \right)} = \left(\left(\left(x + \frac{4}{13}\right) - 3 \cos{\left(x \right)}\right) + \left(- \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x}\right) + \sin{\left(x \right)} 
f = x + 4/13 - 3*cos(x) + (-sin(3*x) - cos(3*x))*exp(2*x) + sin(x)
Gráfico de la función
02468-8-6-4-2-1010-500000000500000000
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:
(((x+413)−3cos⁡(x))+(−sin⁡(3x)−cos⁡(3x))e2x)+sin⁡(x)=0\left(\left(\left(x + \frac{4}{13}\right) - 3 \cos{\left(x \right)}\right) + \left(- \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x}\right) + \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=1.85547044610061x_{1} = 1.85547044610061
x2=6.02139032138421x_{2} = 6.02139032138421
x3=10.2101761279743x_{3} = 10.2101761279743
x4=3.92750707364079x_{4} = 3.92750707364079
x5=14.3989663289544x_{5} = 14.3989663289544
x6=12.3045712266058x_{6} = 12.3045712266058
x7=−2.79529770343991x_{7} = -2.79529770343991
x8=8.11578123568296x_{8} = 8.11578123568296
x9=−3.46963679690032x_{9} = -3.46963679690032
x10=13.3517687777493x_{10} = 13.3517687777493
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 4/13 + x - 3*cos(x) + (-cos(3*x) - sin(3*x))*exp(2*x) + sin(x).
((413−3cos⁡(0))+(−cos⁡(0⋅3)−sin⁡(0⋅3))e0⋅2)+sin⁡(0)\left(\left(\frac{4}{13} - 3 \cos{\left(0 \right)}\right) + \left(- \cos{\left(0 \cdot 3 \right)} - \sin{\left(0 \cdot 3 \right)}\right) e^{0 \cdot 2}\right) + \sin{\left(0 \right)}
Resultado:
f(0)=−4813f{\left(0 \right)} = - \frac{48}{13}
Punto:
(0, -48/13)
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
2(−sin⁡(3x)−cos⁡(3x))e2x+(3sin⁡(3x)−3cos⁡(3x))e2x+3sin⁡(x)+cos⁡(x)+1=02 \left(- \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x} + \left(3 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)}\right) e^{2 x} + 3 \sin{\left(x \right)} + \cos{\left(x \right)} + 1 = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−333.652322389311x_{1} = -333.652322389311
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=0.361389404971151x_{3} = 0.361389404971151
x4=−50.90898356623x_{4} = -50.90898356623
x5=−25.7762423375116x_{5} = -25.7762423375116
x6=−59.6902604182061x_{6} = -59.6902604182061
x7=−65.9734457253857x_{7} = -65.9734457253857
x8=11.9769733188113x_{8} = 11.9769733188113
x9=−3.13846509917659x_{9} = -3.13846509917659
x10=−15.7079632679489x_{10} = -15.7079632679489
x11=−21.9911485751286x_{11} = -21.9911485751286
x12=−34.5575191894877x_{12} = -34.5575191894877
x13=7.78818315963965x_{13} = 7.78818315963965
x14=−32.0594276446912x_{14} = -32.0594276446912
x15=−44.6257982590504x_{15} = -44.6257982590504
x16=−88.6080954093075x_{16} = -88.6080954093075
x17=−40.8407044966673x_{17} = -40.8407044966673
x18=−101.174466023667x_{18} = -101.174466023667
x19=14.0713684212043x_{19} = 14.0713684212043
x20=−28.2743338823081x_{20} = -28.2743338823081
x21=−63.4753541805892x_{21} = -63.4753541805892
x22=1.51771971157501x_{22} = 1.51771971157501
x23=−82.3249101021279x_{23} = -82.3249101021279
x24=−38.3426129518708x_{24} = -38.3426129518708
x25=−53.4070751110265x_{25} = -53.4070751110265
x26=5.69378813293342x_{26} = 5.69378813293342
x27=−57.1921688734096x_{27} = -57.1921688734096
x28=−91.106186954104x_{28} = -91.106186954104
x29=−76.0417247949483x_{29} = -76.0417247949483
x30=−69.7585394877687x_{30} = -69.7585394877687
x31=3.59933314881342x_{31} = 3.59933314881342
x32=−9.42477794991536x_{32} = -9.42477794991536
x33=−94.8912807164871x_{33} = -94.8912807164871
x34=−47.1238898038469x_{34} = -47.1238898038469
x35=−13.2098717231534x_{35} = -13.2098717231534
x36=−6.9266866804356x_{36} = -6.9266866804356
x37=−72.2566310325652x_{37} = -72.2566310325652
x38=−84.8230016469244x_{38} = -84.8230016469244
x39=9.88257821620932x_{39} = 9.88257821620932
x40=−19.493057030332x_{40} = -19.493057030332
x41=−0.518418390323355x_{41} = -0.518418390323355
x42=−78.5398163397448x_{42} = -78.5398163397448
Signos de extremos en los puntos:
(-333.6523223893114, -336.344630081619)

(-97.3893722612836, -94.0816799535913)

(0.361389404971151, -4.56798566188401)

(-50.90898356622998, -53.6012912585377)

(-25.77624233751163, -28.4685500298193)

(-59.69026041820607, -56.3825681105138)

(-65.97344572538566, -62.6657534176934)

(11.976973318811337, 29766750072.6451)

(-3.138465099176588, 0.167981695772718)

(-15.707963267948928, -12.4002709602566)

(-21.991148575128552, -18.6834562674362)

(-34.55751918948773, -31.2498268817954)

(7.788183159639653, 6845340.28161944)

(-32.05942764469122, -34.7517353369989)

(-44.62579825905039, -47.3181059513581)

(-88.6080954093075, -91.3004031016152)

(-40.840704496667314, -37.533012188975)

(-101.17446602366667, -103.866773715974)

(14.071368421204268, 1962907763565.65)

(-28.274333882308138, -24.9666415746158)

(-63.47535418058915, -66.1676618728968)

(1.5177197115750125, 26.5113223358438)

(-82.3249101021279, -85.0172177944356)

(-38.342612951870805, -41.0349206441785)

(-53.40707511102649, -50.0993828033342)

(5.693788132933424, 103809.800095488)

(-57.192168873409564, -59.8844765657173)

(-91.106186954104, -87.7984946464117)

(-76.04172479494832, -78.734032487256)

(-69.75853948776873, -72.4508471800764)

(3.5993331488134177, 1580.34762036187)

(-9.42477794991536, -6.11708564656466)

(-94.89128071648709, -97.5835884087948)

(-47.1238898038469, -43.8161974961546)

(-13.20987172315338, -15.9021794154558)

(-6.9266866804355995, -9.61899286813561)

(-72.25663103256524, -68.9489387248729)

(-84.82300164692441, -81.5153093392321)

(9.882578216209316, 451401461.753958)

(-19.493057030332043, -22.1853647226397)

(-0.5184183903233555, -2.96302352934797)

(-78.53981633974483, -75.2321240320525)


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=−333.652322389311x_{1} = -333.652322389311
x2=0.361389404971151x_{2} = 0.361389404971151
x3=−50.90898356623x_{3} = -50.90898356623
x4=−25.7762423375116x_{4} = -25.7762423375116
x5=−32.0594276446912x_{5} = -32.0594276446912
x6=−44.6257982590504x_{6} = -44.6257982590504
x7=−88.6080954093075x_{7} = -88.6080954093075
x8=−101.174466023667x_{8} = -101.174466023667
x9=−63.4753541805892x_{9} = -63.4753541805892
x10=−82.3249101021279x_{10} = -82.3249101021279
x11=−38.3426129518708x_{11} = -38.3426129518708
x12=−57.1921688734096x_{12} = -57.1921688734096
x13=−76.0417247949483x_{13} = -76.0417247949483
x14=−69.7585394877687x_{14} = -69.7585394877687
x15=−94.8912807164871x_{15} = -94.8912807164871
x16=−13.2098717231534x_{16} = -13.2098717231534
x17=−6.9266866804356x_{17} = -6.9266866804356
x18=−19.493057030332x_{18} = -19.493057030332
Puntos máximos de la función:
x18=−97.3893722612836x_{18} = -97.3893722612836
x18=−59.6902604182061x_{18} = -59.6902604182061
x18=−65.9734457253857x_{18} = -65.9734457253857
x18=11.9769733188113x_{18} = 11.9769733188113
x18=−3.13846509917659x_{18} = -3.13846509917659
x18=−15.7079632679489x_{18} = -15.7079632679489
x18=−21.9911485751286x_{18} = -21.9911485751286
x18=−34.5575191894877x_{18} = -34.5575191894877
x18=7.78818315963965x_{18} = 7.78818315963965
x18=−40.8407044966673x_{18} = -40.8407044966673
x18=14.0713684212043x_{18} = 14.0713684212043
x18=−28.2743338823081x_{18} = -28.2743338823081
x18=1.51771971157501x_{18} = 1.51771971157501
x18=−53.4070751110265x_{18} = -53.4070751110265
x18=5.69378813293342x_{18} = 5.69378813293342
x18=−91.106186954104x_{18} = -91.106186954104
x18=3.59933314881342x_{18} = 3.59933314881342
x18=−9.42477794991536x_{18} = -9.42477794991536
x18=−47.1238898038469x_{18} = -47.1238898038469
x18=−72.2566310325652x_{18} = -72.2566310325652
x18=−84.8230016469244x_{18} = -84.8230016469244
x18=9.88257821620932x_{18} = 9.88257821620932
x18=−0.518418390323355x_{18} = -0.518418390323355
x18=−78.5398163397448x_{18} = -78.5398163397448
Decrece en los intervalos
[0.361389404971151,∞)\left[0.361389404971151, \infty\right)
Crece en los intervalos
(−∞,−333.652322389311]\left(-\infty, -333.652322389311\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
12(sin⁡(3x)−cos⁡(3x))e2x+5(sin⁡(3x)+cos⁡(3x))e2x−sin⁡(x)+3cos⁡(x)=012 \left(\sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x} + 5 \left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{2 x} - \sin{\left(x \right)} + 3 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−77.2907705673466x_{1} = -77.2907705673466
x2=−52.1580293386282x_{2} = -52.1580293386282
x3=−83.5739558745262x_{3} = -83.5739558745262
x4=−30.1668807634997x_{4} = -30.1668807634997
x5=−36.4500660706793x_{5} = -36.4500660706793
x6=−45.8748440314486x_{6} = -45.8748440314486
x7=−86.715548528116x_{7} = -86.715548528116
x8=13.743770513455x_{8} = 13.743770513455
x9=−67.8659926065772x_{9} = -67.8659926065772
x10=−71.007585260167x_{10} = -71.007585260167
x11=−33.3084734170895x_{11} = -33.3084734170895
x12=−42.7332513778588x_{12} = -42.7332513778588
x13=9.55498030840972x_{13} = 9.55498030840972
x14=−20.7421028027303x_{14} = -20.7421028027303
x15=−74.1491779137568x_{15} = -74.1491779137568
x16=−14.4589174955511x_{16} = -14.4589174955511
x17=−8.1757322865618x_{17} = -8.1757322865618
x18=−92.9987338352955x_{18} = -92.9987338352955
x19=6.41338751644464x_{19} = 6.41338751644464
x20=−23.8836954563201x_{20} = -23.8836954563201
x21=−27.0252881099099x_{21} = -27.0252881099099
x22=−49.0164366850384x_{22} = -49.0164366850384
x23=−11.3173248421443x_{23} = -11.3173248421443
x24=−17.6005101491405x_{24} = -17.6005101491405
x25=−89.8571411817058x_{25} = -89.8571411817058
x26=−58.4412146458078x_{26} = -58.4412146458078
x27=11.6493754110654x_{27} = 11.6493754110654
x28=−64.7243999529874x_{28} = -64.7243999529874
x29=−80.4323632209364x_{29} = -80.4323632209364
x30=4.31899328013983x_{30} = 4.31899328013983
x31=−5.03419207166994x_{31} = -5.03419207166994
x32=2.2251519195253x_{32} = 2.2251519195253
x33=−39.5916587242691x_{33} = -39.5916587242691
x34=−96.1403264888853x_{34} = -96.1403264888853
x35=−1.912246135343x_{35} = -1.912246135343
x36=−55.299621992218x_{36} = -55.299621992218
x37=0.085723541961364x_{37} = 0.085723541961364
x38=−61.5828072993976x_{38} = -61.5828072993976
x39=−99.2819191424751x_{39} = -99.2819191424751

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
[6.41338751644464,∞)\left[6.41338751644464, \infty\right)
Convexa en los intervalos
(−∞,−96.1403264888853]\left(-\infty, -96.1403264888853\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((((x+413)−3cos⁡(x))+(−sin⁡(3x)−cos⁡(3x))e2x)+sin⁡(x))=−∞\lim_{x \to -\infty}\left(\left(\left(\left(x + \frac{4}{13}\right) - 3 \cos{\left(x \right)}\right) + \left(- \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x}\right) + \sin{\left(x \right)}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((((x+413)−3cos⁡(x))+(−sin⁡(3x)−cos⁡(3x))e2x)+sin⁡(x))y = \lim_{x \to \infty}\left(\left(\left(\left(x + \frac{4}{13}\right) - 3 \cos{\left(x \right)}\right) + \left(- \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x}\right) + \sin{\left(x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 4/13 + x - 3*cos(x) + (-cos(3*x) - sin(3*x))*exp(2*x) + sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((((x+413)−3cos⁡(x))+(−sin⁡(3x)−cos⁡(3x))e2x)+sin⁡(x)x)=1\lim_{x \to -\infty}\left(\frac{\left(\left(\left(x + \frac{4}{13}\right) - 3 \cos{\left(x \right)}\right) + \left(- \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x}\right) + \sin{\left(x \right)}}{x}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xy = x
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((((x+413)−3cos⁡(x))+(−sin⁡(3x)−cos⁡(3x))e2x)+sin⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\left(\left(\left(x + \frac{4}{13}\right) - 3 \cos{\left(x \right)}\right) + \left(- \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x}\right) + \sin{\left(x \right)}}{x}\right)
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:
(((x+413)−3cos⁡(x))+(−sin⁡(3x)−cos⁡(3x))e2x)+sin⁡(x)=−x+(sin⁡(3x)−cos⁡(3x))e−2x−sin⁡(x)−3cos⁡(x)+413\left(\left(\left(x + \frac{4}{13}\right) - 3 \cos{\left(x \right)}\right) + \left(- \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x}\right) + \sin{\left(x \right)} = - x + \left(\sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{- 2 x} - \sin{\left(x \right)} - 3 \cos{\left(x \right)} + \frac{4}{13}
- No
(((x+413)−3cos⁡(x))+(−sin⁡(3x)−cos⁡(3x))e2x)+sin⁡(x)=x−(sin⁡(3x)−cos⁡(3x))e−2x+sin⁡(x)+3cos⁡(x)−413\left(\left(\left(x + \frac{4}{13}\right) - 3 \cos{\left(x \right)}\right) + \left(- \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{2 x}\right) + \sin{\left(x \right)} = x - \left(\sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{- 2 x} + \sin{\left(x \right)} + 3 \cos{\left(x \right)} - \frac{4}{13}
- No
es decir, función
no es
par ni impar
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