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Trazado el gráfico de la función/ -exp*(sin(x))+2*x*sin(exp*(x-5))+4*sin(x-1)

Gráfico de la función y = -exp*(sin(x))+2*x*sin(exp*(x-5))+4*sin(x-1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          sin(x)          / x - 5\               
f(x) = - e       + 2*x*sin\e     / + 4*sin(x - 1)
f(x)=(2xsin⁡(ex−5)−esin⁡(x))+4sin⁡(x−1)f{\left(x \right)} = \left(2 x \sin{\left(e^{x - 5} \right)} - e^{\sin{\left(x \right)}}\right) + 4 \sin{\left(x - 1 \right)} 
f = (2*x)*sin(exp(x - 5)) - exp(sin(x)) + 4*sin(x - 1)
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:
(2xsin⁡(ex−5)−esin⁡(x))+4sin⁡(x−1)=0\left(2 x \sin{\left(e^{x - 5} \right)} - e^{\sin{\left(x \right)}}\right) + 4 \sin{\left(x - 1 \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−54.8137811083669x_{1} = -54.8137811083669
x2=−27.3899361769736x_{2} = -27.3899361769736
x3=−48.5305958011873x_{3} = -48.5305958011873
x4=−21.1067508698466x_{4} = -21.1067508698466
x5=−40987.4749536807x_{5} = -40987.4749536807
x6=−33.6731214841531x_{6} = -33.6731214841531
x7=−105.079263565804x_{7} = -105.079263565804
x8=−67.380151722726x_{8} = -67.380151722726
x9=−58.8058627128714x_{9} = -58.8058627128714
x10=−10.8314831109218x_{10} = -10.8314831109218
x11=14.4827948035331x_{11} = 14.4827948035331
x12=−2.25805937346031x_{12} = -2.25805937346031
x13=−42.2474104940077x_{13} = -42.2474104940077
x14=−4.54810812283131x_{14} = -4.54810812283131
x15=−83.9386039415898x_{15} = -83.9386039415898
x16=−79.9465223370852x_{16} = -79.9465223370852
x17=−35.9642251868281x_{17} = -35.9642251868281
x18=−8.54038636540066x_{18} = -8.54038636540066
x19=12.1824559329007x_{19} = 12.1824559329007
x20=−404.381054607749x_{20} = -404.381054607749
x21=−764.81372081966x_{21} = -764.81372081966
x22=−29.6810398796485x_{22} = -29.6810398796485
x23=−14.8235655824188x_{23} = -14.8235655824188
x24=−98.796078258624x_{24} = -98.796078258624
x25=6.01166223909158x_{25} = 6.01166223909158
x26=−65.089048020051x_{26} = -65.089048020051
x27=−61.0969664155465x_{27} = -61.0969664155465
x28=−23.3978545724626x_{28} = -23.3978545724626
x29=−71.3722333272306x_{29} = -71.3722333272306
x30=−86.2297076442648x_{30} = -86.2297076442648
x31=−90.2217892487694x_{31} = -90.2217892487694
x32=−96.5049745559489x_{32} = -96.5049745559489
x33=−77.6554186344102x_{33} = -77.6554186344102
x34=1.69818983629885x_{34} = 1.69818983629885
x35=−1041.27387433556x_{35} = -1041.27387433556
x36=−205.610228480677x_{36} = -205.610228480677
x37=−92.5128929514444x_{37} = -92.5128929514444
x38=−17.1146692627895x_{38} = -17.1146692627895
x39=−46.2394920985123x_{39} = -46.2394920985123
x40=−73.6633370299056x_{40} = -73.6633370299056
x41=−39.9563067913327x_{41} = -39.9563067913327
x42=−4616.40631412075x_{42} = -4616.40631412075
x43=−52.5226774056918x_{43} = -52.5226774056918
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 -exp(sin(x)) + (2*x)*sin(exp(x - 5)) + 4*sin(x - 1).
4sin⁡(−1)+(−esin⁡(0)+0⋅2sin⁡(e−5))4 \sin{\left(-1 \right)} + \left(- e^{\sin{\left(0 \right)}} + 0 \cdot 2 \sin{\left(e^{-5} \right)}\right)
Resultado:
f(0)=−4sin⁡(1)−1f{\left(0 \right)} = - 4 \sin{\left(1 \right)} - 1
Punto:
(0, -1 - 4*sin(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
2xex−5cos⁡(ex−5)−esin⁡(x)cos⁡(x)+2sin⁡(ex−5)+4cos⁡(x−1)=02 x e^{x - 5} \cos{\left(e^{x - 5} \right)} - e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 2 \sin{\left(e^{x - 5} \right)} + 4 \cos{\left(x - 1 \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−47.3782484291623x_{1} = -47.3782484291623
x2=−72.5109896578807x_{2} = -72.5109896578807
x3=−34.8118778148031x_{3} = -34.8118778148031
x4=−78.7941749650602x_{4} = -78.7941749650602
x5=12.9631073720759x_{5} = 12.9631073720759
x6=−31.8338711190445x_{6} = -31.8338711190445
x7=−9.67913814910049x_{7} = -9.67913814910049
x8=9.98417957147691x_{8} = 9.98417957147691
x9=−63.2497976549425x_{9} = -63.2497976549425
x10=−19.2675005043477x_{10} = -19.2675005043477
x11=−113.515280112379x_{11} = -113.515280112379
x12=−56.9666123477629x_{12} = -56.9666123477629
x13=−53.6614337363419x_{13} = -53.6614337363419
x14=−88.3825388836608x_{14} = -88.3825388836608
x15=4.06583439570859x_{15} = 4.06583439570859
x16=14.8595436717617x_{16} = 14.8595436717617
x17=5.47759667841792x_{17} = 5.47759667841792
x18=−22.2455072004573x_{18} = -22.2455072004573
x19=−82.0993535764812x_{19} = -82.0993535764812
x20=−75.8161682693017x_{20} = -75.8161682693017
x21=−28.5286925076236x_{21} = -28.5286925076236
x22=−41.0950631219827x_{22} = -41.0950631219827
x23=−44.4002417334037x_{23} = -44.4002417334037
x24=−126.081650726738x_{24} = -126.081650726738
x25=−91.3605455794194x_{25} = -91.3605455794194
x26=−100.94890949802x_{26} = -100.94890949802
x27=−94.6657241908404x_{27} = -94.6657241908404
x28=−38.1170564262241x_{28} = -38.1170564262241
x29=−97.643730886599x_{29} = -97.643730886599
x30=−12.9843150788944x_{30} = -12.9843150788944
x31=−50.6834270405833x_{31} = -50.6834270405833
x32=−59.9446190435215x_{32} = -59.9446190435215
x33=12.1596668379616x_{33} = 12.1596668379616
x34=−6.70109967410176x_{34} = -6.70109967410176
x35=−85.0773602722398x_{35} = -85.0773602722398
x36=−25.5506858118641x_{36} = -25.5506858118641
x37=−69.5329829621221x_{37} = -69.5329829621221
x38=−0.419588571054977x_{38} = -0.419588571054977
x39=−66.2278043507011x_{39} = -66.2278043507011
x40=−3.3961823029777x_{40} = -3.3961823029777
x41=−15.9623218982963x_{41} = -15.9623218982963
Signos de extremos en los puntos:
(-47.378248429162305, 2.51528659455668)

(-72.51098965788066, 2.51528659455668)

(-34.81187781480313, 2.51528659455668)

(-78.79417496506024, 2.51528659455668)

(12.963107372075855, 22.185194913677)

(-31.833871119044545, -4.61975184863277)

(-9.67913814910049, 2.51527843252472)

(9.984179571476915, 21.0860754272896)

(-63.24979765494248, -4.61975184863276)

(-19.267500504347748, -4.61975184974607)

(-113.51528011237917, -4.61975184863276)

(-56.9666123477629, -4.61975184863276)

(-53.66143373634189, 2.51528659455668)

(-88.38253888366083, -4.61975184863276)

(4.065834395708586, 2.96605490877545)

(14.859543671761742, 31.4484249346342)

(5.477596678417921, 6.56940635326971)

(-22.245507200457304, 2.51528659449127)

(-82.09935357648123, -4.61975184863276)

(-75.81616826930166, -4.61975184863276)

(-28.528692507623578, 2.51528659455653)

(-41.09506312198272, 2.51528659455668)

(-44.40024173340372, -4.61975184863276)

(-126.08165072673835, -4.61975184863276)

(-91.36054557941941, 2.51528659455668)

(-100.94890949802, -4.61975184863276)

(-94.66572419084041, -4.61975184863276)

(-38.11705642622413, -4.61975184863276)

(-97.643730886599, 2.51528659455668)

(-12.9843150788944, -4.61975225038677)

(-50.68342704058331, -4.61975184863276)

(-59.94461904352148, 2.51528659455668)

(12.159666837961602, -28.9388882631752)

(-6.701099674101763, -4.61986288045579)

(-85.07736027223983, 2.51528659455668)

(-25.550685811864113, -4.61975184863552)

(-69.53298296212206, -4.61975184863276)

(-0.41958857105497743, -4.62346429997344)

(-66.22780435070106, 2.51528659455668)

(-3.396182302977698, 2.51375324653675)

(-15.962321898296294, 2.51528656942016)


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=−31.8338711190445x_{1} = -31.8338711190445
x2=−63.2497976549425x_{2} = -63.2497976549425
x3=−19.2675005043477x_{3} = -19.2675005043477
x4=−113.515280112379x_{4} = -113.515280112379
x5=−56.9666123477629x_{5} = -56.9666123477629
x6=−88.3825388836608x_{6} = -88.3825388836608
x7=4.06583439570859x_{7} = 4.06583439570859
x8=−82.0993535764812x_{8} = -82.0993535764812
x9=−75.8161682693017x_{9} = -75.8161682693017
x10=−44.4002417334037x_{10} = -44.4002417334037
x11=−126.081650726738x_{11} = -126.081650726738
x12=−100.94890949802x_{12} = -100.94890949802
x13=−94.6657241908404x_{13} = -94.6657241908404
x14=−38.1170564262241x_{14} = -38.1170564262241
x15=−12.9843150788944x_{15} = -12.9843150788944
x16=−50.6834270405833x_{16} = -50.6834270405833
x17=12.1596668379616x_{17} = 12.1596668379616
x18=−6.70109967410176x_{18} = -6.70109967410176
x19=−25.5506858118641x_{19} = -25.5506858118641
x20=−69.5329829621221x_{20} = -69.5329829621221
x21=−0.419588571054977x_{21} = -0.419588571054977
Puntos máximos de la función:
x21=−47.3782484291623x_{21} = -47.3782484291623
x21=−72.5109896578807x_{21} = -72.5109896578807
x21=−34.8118778148031x_{21} = -34.8118778148031
x21=−78.7941749650602x_{21} = -78.7941749650602
x21=12.9631073720759x_{21} = 12.9631073720759
x21=−9.67913814910049x_{21} = -9.67913814910049
x21=9.98417957147691x_{21} = 9.98417957147691
x21=−53.6614337363419x_{21} = -53.6614337363419
x21=14.8595436717617x_{21} = 14.8595436717617
x21=5.47759667841792x_{21} = 5.47759667841792
x21=−22.2455072004573x_{21} = -22.2455072004573
x21=−28.5286925076236x_{21} = -28.5286925076236
x21=−41.0950631219827x_{21} = -41.0950631219827
x21=−91.3605455794194x_{21} = -91.3605455794194
x21=−97.643730886599x_{21} = -97.643730886599
x21=−59.9446190435215x_{21} = -59.9446190435215
x21=−85.0773602722398x_{21} = -85.0773602722398
x21=−66.2278043507011x_{21} = -66.2278043507011
x21=−3.3961823029777x_{21} = -3.3961823029777
x21=−15.9623218982963x_{21} = -15.9623218982963
Decrece en los intervalos
[12.1596668379616,∞)\left[12.1596668379616, \infty\right)
Crece en los intervalos
(−∞,−126.081650726738]\left(-\infty, -126.081650726738\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((2xsin⁡(ex−5)−esin⁡(x))+4sin⁡(x−1))=⟨−4−e,4−e−1⟩\lim_{x \to -\infty}\left(\left(2 x \sin{\left(e^{x - 5} \right)} - e^{\sin{\left(x \right)}}\right) + 4 \sin{\left(x - 1 \right)}\right) = \left\langle -4 - e, 4 - e^{-1}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−4−e,4−e−1⟩y = \left\langle -4 - e, 4 - e^{-1}\right\rangle
lim⁡x→∞((2xsin⁡(ex−5)−esin⁡(x))+4sin⁡(x−1))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(2 x \sin{\left(e^{x - 5} \right)} - e^{\sin{\left(x \right)}}\right) + 4 \sin{\left(x - 1 \right)}\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 -exp(sin(x)) + (2*x)*sin(exp(x - 5)) + 4*sin(x - 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((2xsin⁡(ex−5)−esin⁡(x))+4sin⁡(x−1)x)=0\lim_{x \to -\infty}\left(\frac{\left(2 x \sin{\left(e^{x - 5} \right)} - e^{\sin{\left(x \right)}}\right) + 4 \sin{\left(x - 1 \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((2xsin⁡(ex−5)−esin⁡(x))+4sin⁡(x−1)x)y = x \lim_{x \to \infty}\left(\frac{\left(2 x \sin{\left(e^{x - 5} \right)} - e^{\sin{\left(x \right)}}\right) + 4 \sin{\left(x - 1 \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:
(2xsin⁡(ex−5)−esin⁡(x))+4sin⁡(x−1)=−2xsin⁡(e−x−5)−4sin⁡(x+1)−e−sin⁡(x)\left(2 x \sin{\left(e^{x - 5} \right)} - e^{\sin{\left(x \right)}}\right) + 4 \sin{\left(x - 1 \right)} = - 2 x \sin{\left(e^{- x - 5} \right)} - 4 \sin{\left(x + 1 \right)} - e^{- \sin{\left(x \right)}}
- No
(2xsin⁡(ex−5)−esin⁡(x))+4sin⁡(x−1)=2xsin⁡(e−x−5)+4sin⁡(x+1)+e−sin⁡(x)\left(2 x \sin{\left(e^{x - 5} \right)} - e^{\sin{\left(x \right)}}\right) + 4 \sin{\left(x - 1 \right)} = 2 x \sin{\left(e^{- x - 5} \right)} + 4 \sin{\left(x + 1 \right)} + e^{- \sin{\left(x \right)}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор