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Trazado el gráfico de la función/ ln(|sinx|)+ln(|cosx|)+2*x*e^x

Gráfico de la función y = ln(|sinx|)+ln(|cosx|)+2*x*e^x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                            x
f(x) = log(|sin(x)|) + log(|cos(x)|) + 2*x*E
f(x)=ex2x+(log⁡(∣sin⁡(x)∣)+log⁡(∣cos⁡(x)∣))f{\left(x \right)} = e^{x} 2 x + \left(\log{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\right) 
f = E^x*(2*x) + log(Abs(sin(x))) + log(Abs(cos(x)))
Gráfico de la función
02468-8-6-4-2-1010-500000500000
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:
ex2x+(log⁡(∣sin⁡(x)∣)+log⁡(∣cos⁡(x)∣))=0e^{x} 2 x + \left(\log{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=0.372586906534774x_{1} = 0.372586906534774
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 log(Abs(sin(x))) + log(Abs(cos(x))) + (2*x)*E^x.
(log⁡(∣sin⁡(0)∣)+log⁡(∣cos⁡(0)∣))+0⋅2e0\left(\log{\left(\left|{\sin{\left(0 \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(0 \right)}}\right| \right)}\right) + 0 \cdot 2 e^{0}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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+2ex−sin⁡(x)sign⁡(cos⁡(x))∣cos⁡(x)∣+cos⁡(x)sign⁡(sin⁡(x))∣sin⁡(x)∣=02 x e^{x} + 2 e^{x} - \frac{\sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\left|{\cos{\left(x \right)}}\right|} + \frac{\cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{\left|{\sin{\left(x \right)}}\right|} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−46.3384916404494x_{1} = -46.3384916404494
x2=−30.6305283725012x_{2} = -30.6305283725012
x3=−98.174770424681x_{3} = -98.174770424681
x4=−16.4933619636257x_{4} = -16.4933619636257
x5=−68.329640215578x_{5} = -68.329640215578
x6=−84.037603483527x_{6} = -84.037603483527
x7=−55.7632696012188x_{7} = -55.7632696012188
x8=−85.6083998103219x_{8} = -85.6083998103219
x9=−19.6349541126022x_{9} = -19.6349541126022
x10=−41.6261026600648x_{10} = -41.6261026600648
x11=−93.4623814442964x_{11} = -93.4623814442964
x12=−71.4712328691678x_{12} = -71.4712328691678
x13=−33.7721210260903x_{13} = -33.7721210260903
x14=−76.1836218495525x_{14} = -76.1836218495525
x15=−54.1924732744239x_{15} = -54.1924732744239
x16=−65.1880475619882x_{16} = -65.1880475619882
x17=−10.2103455381236x_{17} = -10.2103455381236
x18=−96.6039740978861x_{18} = -96.6039740978861
x19=−27.488935718926x_{19} = -27.488935718926
x20=1.52798726886801x_{20} = 1.52798726886801
x21=−60.4756585816035x_{21} = -60.4756585816035
x22=−43.1968989868597x_{22} = -43.1968989868597
x23=−13.3517785974004x_{23} = -13.3517785974004
x24=−38.484510006475x_{24} = -38.484510006475
x25=−49.4800842940392x_{25} = -49.4800842940392
x26=−5.50693190633047x_{26} = -5.50693190633047
x27=−62.0464549083984x_{27} = -62.0464549083984
x28=−87.1791961371168x_{28} = -87.1791961371168
x29=−57.3340659280137x_{29} = -57.3340659280137
x30=−90.3207887907066x_{30} = -90.3207887907066
x31=−79.3252145031423x_{31} = -79.3252145031423
x32=−2.41901457891826x_{32} = -2.41901457891826
x33=−25.9181393921849x_{33} = -25.9181393921849
x34=−35.3429173528852x_{34} = -35.3429173528852
x35=−63.6172512351933x_{35} = -63.6172512351933
x36=−24.3473430656323x_{36} = -24.3473430656323
x37=−0.716639609763256x_{37} = -0.716639609763256
x38=−52.621676947629x_{38} = -52.621676947629
x39=−69.9004365423729x_{39} = -69.9004365423729
x40=−7.07116140316689x_{40} = -7.07116140316689
x41=−21.2057504179671x_{41} = -21.2057504179671
x42=−32.2013246992955x_{42} = -32.2013246992955
x43=−18.0641578800096x_{43} = -18.0641578800096
x44=−11.7810136797307x_{44} = -11.7810136797307
x45=−77.7544181763474x_{45} = -77.7544181763474
x46=−99.7455667514759x_{46} = -99.7455667514759
x47=−40.0553063332699x_{47} = -40.0553063332699
x48=−82.4668071567321x_{48} = -82.4668071567321
x49=−47.9092879672443x_{49} = -47.9092879672443
x50=−91.8915851175014x_{50} = -91.8915851175014
x51=−3.95526266232386x_{51} = -3.95526266232386
x52=−74.6128255227576x_{52} = -74.6128255227576
Signos de extremos en los puntos:
(-46.33849164044945, -0.693147180559945)

(-30.63052837250122, -0.693147180562997)

(-98.17477042468104, -0.693147180559945)

(-16.49336196362572, -0.693149447098994)

(-68.329640215578, -0.693147180559945)

(-84.03760348352696, -0.693147180559945)

(-55.76326960121883, -0.693147180559945)

(-85.60839981032187, -0.693147180559945)

(-19.63495411260219, -0.693147297162393)

(-41.62610266006476, -0.693147180559946)

(-93.46238144429635, -0.693147180559945)

(-71.47123286916779, -0.693147180559945)

(-33.77212102609031, -0.693147180560091)

(-76.18362184955248, -0.693147180559945)

(-54.19247327442393, -0.693147180559945)

(-65.18804756198821, -0.693147180559945)

(-10.210345538123603, -0.693898469323288)

(-96.60397409788614, -0.693147180559945)

(-27.48893571892596, -0.693147180623317)

(1.5279872688680114, 10.9324260034019)

(-60.47565858160352, -0.693147180559945)

(-43.19689898685966, -0.693147180559945)

(-13.351778597400418, -0.693189639321455)

(-38.48451000647497, -0.693147180559947)

(-49.480084294039244, -0.693147180559945)

(-5.506931906330469, -0.738014653199873)

(-62.04645490839842, -0.693147180559945)

(-87.17919613711676, -0.693147180559945)

(-57.33406592801373, -0.693147180559945)

(-90.32078879070656, -0.693147180559945)

(-79.32521450314228, -0.693147180559945)

(-2.4190145789182647, -1.13169028095714)

(-25.91813939218488, -0.693147180847373)

(-35.34291735288518, -0.693147180559977)

(-63.617251235193315, -0.693147180559945)

(-24.347343065632277, -0.693147181858813)

(-0.716639609763256, -1.40263283070348)

(-52.621676947629034, -0.693147180559945)

(-69.9004365423729, -0.693147180559945)

(-7.0711614031668875, -0.705170786262028)

(-21.20575041796708, -0.693147206738355)

(-32.20132469929554, -0.693147180560612)

(-18.064157880009617, -0.693147696600286)

(-11.781013679730744, -0.693327395842177)

(-77.75441817634739, -0.693147180559945)

(-99.74556675147593, -0.693147180559945)

(-40.05530633326986, -0.693147180559946)

(-82.46680715673207, -0.693147180559945)

(-47.909287967244346, -0.693147180559945)

(-91.89158511750145, -0.693147180559945)

(-3.955262662323856, -0.84626195775183)

(-74.61282552275759, -0.693147180559945)


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:
La función no tiene puntos mínimos
Puntos máximos de la función:
x52=−46.3384916404494x_{52} = -46.3384916404494
x52=−30.6305283725012x_{52} = -30.6305283725012
x52=−98.174770424681x_{52} = -98.174770424681
x52=−16.4933619636257x_{52} = -16.4933619636257
x52=−68.329640215578x_{52} = -68.329640215578
x52=−84.037603483527x_{52} = -84.037603483527
x52=−55.7632696012188x_{52} = -55.7632696012188
x52=−85.6083998103219x_{52} = -85.6083998103219
x52=−19.6349541126022x_{52} = -19.6349541126022
x52=−41.6261026600648x_{52} = -41.6261026600648
x52=−93.4623814442964x_{52} = -93.4623814442964
x52=−71.4712328691678x_{52} = -71.4712328691678
x52=−33.7721210260903x_{52} = -33.7721210260903
x52=−76.1836218495525x_{52} = -76.1836218495525
x52=−54.1924732744239x_{52} = -54.1924732744239
x52=−65.1880475619882x_{52} = -65.1880475619882
x52=−10.2103455381236x_{52} = -10.2103455381236
x52=−96.6039740978861x_{52} = -96.6039740978861
x52=−27.488935718926x_{52} = -27.488935718926
x52=1.52798726886801x_{52} = 1.52798726886801
x52=−60.4756585816035x_{52} = -60.4756585816035
x52=−43.1968989868597x_{52} = -43.1968989868597
x52=−13.3517785974004x_{52} = -13.3517785974004
x52=−38.484510006475x_{52} = -38.484510006475
x52=−49.4800842940392x_{52} = -49.4800842940392
x52=−5.50693190633047x_{52} = -5.50693190633047
x52=−62.0464549083984x_{52} = -62.0464549083984
x52=−87.1791961371168x_{52} = -87.1791961371168
x52=−57.3340659280137x_{52} = -57.3340659280137
x52=−90.3207887907066x_{52} = -90.3207887907066
x52=−79.3252145031423x_{52} = -79.3252145031423
x52=−2.41901457891826x_{52} = -2.41901457891826
x52=−25.9181393921849x_{52} = -25.9181393921849
x52=−35.3429173528852x_{52} = -35.3429173528852
x52=−63.6172512351933x_{52} = -63.6172512351933
x52=−24.3473430656323x_{52} = -24.3473430656323
x52=−0.716639609763256x_{52} = -0.716639609763256
x52=−52.621676947629x_{52} = -52.621676947629
x52=−69.9004365423729x_{52} = -69.9004365423729
x52=−7.07116140316689x_{52} = -7.07116140316689
x52=−21.2057504179671x_{52} = -21.2057504179671
x52=−32.2013246992955x_{52} = -32.2013246992955
x52=−18.0641578800096x_{52} = -18.0641578800096
x52=−11.7810136797307x_{52} = -11.7810136797307
x52=−77.7544181763474x_{52} = -77.7544181763474
x52=−99.7455667514759x_{52} = -99.7455667514759
x52=−40.0553063332699x_{52} = -40.0553063332699
x52=−82.4668071567321x_{52} = -82.4668071567321
x52=−47.9092879672443x_{52} = -47.9092879672443
x52=−91.8915851175014x_{52} = -91.8915851175014
x52=−3.95526266232386x_{52} = -3.95526266232386
x52=−74.6128255227576x_{52} = -74.6128255227576
Decrece en los intervalos
(−∞,−99.7455667514759]\left(-\infty, -99.7455667514759\right]
Crece en los intervalos
[1.52798726886801,∞)\left[1.52798726886801, \infty\right)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(ex2x+(log⁡(∣sin⁡(x)∣)+log⁡(∣cos⁡(x)∣)))=2log⁡(∣⟨−1,1⟩∣)\lim_{x \to -\infty}\left(e^{x} 2 x + \left(\log{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\right)\right) = 2 \log{\left(\left|{\left\langle -1, 1\right\rangle}\right| \right)}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=2log⁡(∣⟨−1,1⟩∣)y = 2 \log{\left(\left|{\left\langle -1, 1\right\rangle}\right| \right)}
lim⁡x→∞(ex2x+(log⁡(∣sin⁡(x)∣)+log⁡(∣cos⁡(x)∣)))=∞\lim_{x \to \infty}\left(e^{x} 2 x + \left(\log{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\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 log(Abs(sin(x))) + log(Abs(cos(x))) + (2*x)*E^x, dividida por x con x->+oo y x ->-oo
No se ha logrado calcular el límite a la izquierda
lim⁡x→−∞(ex2x+(log⁡(∣sin⁡(x)∣)+log⁡(∣cos⁡(x)∣))x)\lim_{x \to -\infty}\left(\frac{e^{x} 2 x + \left(\log{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\right)}{x}\right)
No se ha logrado calcular el límite a la derecha
lim⁡x→∞(ex2x+(log⁡(∣sin⁡(x)∣)+log⁡(∣cos⁡(x)∣))x)\lim_{x \to \infty}\left(\frac{e^{x} 2 x + \left(\log{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\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:
ex2x+(log⁡(∣sin⁡(x)∣)+log⁡(∣cos⁡(x)∣))=−2xe−x+log⁡(∣sin⁡(x)∣)+log⁡(∣cos⁡(x)∣)e^{x} 2 x + \left(\log{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\right) = - 2 x e^{- x} + \log{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}
- No
ex2x+(log⁡(∣sin⁡(x)∣)+log⁡(∣cos⁡(x)∣))=2xe−x−log⁡(∣sin⁡(x)∣)−log⁡(∣cos⁡(x)∣)e^{x} 2 x + \left(\log{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}\right) = 2 x e^{- x} - \log{\left(\left|{\sin{\left(x \right)}}\right| \right)} - \log{\left(\left|{\cos{\left(x \right)}}\right| \right)}
- No
es decir, función
no es
par ni impar
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