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Trazado el gráfico de la función/ sqrt(e^abs(sinx))+2*ln(3x)-1/9

Gráfico de la función y = sqrt(e^abs(sinx))+2*ln(3x)-1/9

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          ___________                 
         /  |sin(x)|                 1
f(x) = \/  E          + 2*log(3*x) - -
                                     9
f(x)=(e∣sin⁡(x)∣+2log⁡(3x))−19f{\left(x \right)} = \left(\sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} + 2 \log{\left(3 x \right)}\right) - \frac{1}{9} 
f = sqrt(E^Abs(sin(x))) + 2*log(3*x) - 1/9
Gráfico de la función
02468-8-6-4-2-1010-1010
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:
(e∣sin⁡(x)∣+2log⁡(3x))−19=0\left(\sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} + 2 \log{\left(3 x \right)}\right) - \frac{1}{9} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 sqrt(E^Abs(sin(x))) + 2*log(3*x) - 1/9.
(2log⁡(0⋅3)+e∣sin⁡(0)∣)−19\left(2 \log{\left(0 \cdot 3 \right)} + \sqrt{e^{\left|{\sin{\left(0 \right)}}\right|}}\right) - \frac{1}{9}
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
e∣sin⁡(x)∣2cos⁡(x)sign⁡(sin⁡(x))2+2x=0\frac{e^{\frac{\left|{\sin{\left(x \right)}}\right|}{2}} \cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{2} + \frac{2}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=42.4687060642419x_{1} = 42.4687060642419
x2=39.3316899979173x_{2} = 39.3316899979173
x3=−14.3088133810527x_{3} = -14.3088133810527
x4=−61.3006600471x_{4} = -61.3006600471
x5=26.7943953424806x_{5} = 26.7943953424806
x6=111.548293018578x_{6} = 111.548293018578
x7=−17.4191866243435x_{7} = -17.4191866243435
x8=−102.12552311228x_{8} = -102.12552311228
x9=67.580161279298x_{9} = 67.580161279298
x10=83.2813472749682x_{10} = 83.2813472749682
x11=73.8602895988072x_{11} = 73.8602895988072
x12=61.3006600471x_{12} = 61.3006600471
x13=−8.16322158318824x_{13} = -8.16322158318824
x14=5.23057200681925x_{14} = 5.23057200681925
x15=70.7201575048469x_{15} = 70.7201575048469
x16=55.0220009108274x_{16} = 55.0220009108274
x17=33.0602732608648x_{17} = 33.0602732608648
x18=−67.580161279298x_{18} = -67.580161279298
x19=−1908.51880826374x_{19} = -1908.51880826374
x20=17.4191866243435x_{20} = 17.4191866243435
x21=23.6649184719595x_{21} = 23.6649184719595
x22=−11.216305532298x_{22} = -11.216305532298
x23=86.4218802132698x_{23} = 86.4218802132698
x24=64.4403208058624x_{24} = 64.4403208058624
x25=−51.8830828251914x_{25} = -51.8830828251914
x26=−8373.91550786741x_{26} = -8373.91550786741
x27=−45.6063534145312x_{27} = -45.6063534145312
x28=89.5624875183509x_{28} = 89.5624875183509
x29=−77.0005409238851x_{29} = -77.0005409238851
x30=−64.4403208058624x_{30} = -64.4403208058624
x31=−73.8602895988072x_{31} = -73.8602895988072
x32=92.703161627384x_{32} = 92.703161627384
x33=−33.0602732608648x_{33} = -33.0602732608648
x34=−5.23057200681925x_{34} = -5.23057200681925
x35=−661.308922268859x_{35} = -661.308922268859
x36=−387.9929458273x_{36} = -387.9929458273
x37=−20.5391705188037x_{37} = -20.5391705188037
x38=98.9846848078687x_{38} = 98.9846848078687
x39=−70.7201575048469x_{39} = -70.7201575048469
x40=−92.703161627384x_{40} = -92.703161627384
x41=−281.181171089351x_{41} = -281.181171089351
x42=−39.3316899979173x_{42} = -39.3316899979173
x43=80.1408974526241x_{43} = 80.1408974526241
x44=36.1954699136195x_{44} = 36.1954699136195
x45=−26.7943953424806x_{45} = -26.7943953424806
x46=−83.2813472749682x_{46} = -83.2813472749682
x47=−42.4687060642419x_{47} = -42.4687060642419
x48=−23.6649184719595x_{48} = -23.6649184719595
x49=−80.1408974526241x_{49} = -80.1408974526241
x50=8.16322158318824x_{50} = 8.16322158318824
x51=14.3088133810527x_{51} = 14.3088133810527
x52=58.1612081468779x_{52} = 58.1612081468779
x53=−86.4218802132698x_{53} = -86.4218802132698
x54=77.0005409238851x_{54} = 77.0005409238851
x55=−58.1612081468779x_{55} = -58.1612081468779
x56=−55.0220009108274x_{56} = -55.0220009108274
x57=95.8438959694044x_{57} = 95.8438959694044
x58=−29.9264232755379x_{58} = -29.9264232755379
x59=−36.1954699136195x_{59} = -36.1954699136195
x60=−105.266406452805x_{60} = -105.266406452805
x61=−95.8438959694044x_{61} = -95.8438959694044
x62=−7685.90674266605x_{62} = -7685.90674266605
x63=20.5391705188037x_{63} = 20.5391705188037
x64=45.6063534145312x_{64} = 45.6063534145312
x65=51.8830828251914x_{65} = 51.8830828251914
x66=−89.5624875183509x_{66} = -89.5624875183509
x67=−48.7445098572293x_{67} = -48.7445098572293
x68=29.9264232755379x_{68} = 29.9264232755379
x69=48.7445098572293x_{69} = 48.7445098572293
Signos de extremos en los puntos:
(42.46870606424195, 11.2310217744461)

(39.33168999791728, 11.0773238077452)

(-14.308813381052687, 9.04451639181416 + 2*pi*I)

(-61.300660047100024, 11.9657696963687 + 2*pi*I)

(26.794395342480584, 10.3078234490048)

(111.54829301857764, 13.16355493742)

(-17.41918662434352, 9.44188448563534 + 2*pi*I)

(-102.12552311227964, 12.9870073872536 + 2*pi*I)

(67.58016127929798, 12.1609320241053)

(83.28134727496824, 12.5789339569173)

(73.86028959880723, 12.3387403731541)

(61.300660047100024, 11.9657696963687)

(-8.163221583188243, 7.89546917542736 + 2*pi*I)

(5.230572006819245, 6.93912978599036)

(70.72015750484694, 12.2518105761584)

(55.02200091082742, 11.7494986218622)

(33.06027326086482, 10.7292722077761)

(-67.58016127929798, 12.1609320241053 + 2*pi*I)

(-1908.5188082637444, 18.8429995251241 + 2*pi*I)

(17.41918662434352, 9.44188448563534)

(23.664918471959545, 10.0584612345515)

(-11.216305532297994, 8.54969174703785 + 2*pi*I)

(86.42188021326984, 12.6529915138211)

(64.44032080586238, 12.0657290436632)

(-51.8830828251914, 11.6319177789513 + 2*pi*I)

(-8373.915507867407, 21.800588415748 + 2*pi*I)

(-45.60635341453117, 11.3737603004 + 2*pi*I)

(89.56248751835092, 12.724405275769)

(-77.00054092388514, 12.4220501862081 + 2*pi*I)

(-64.44032080586238, 12.0657290436632 + 2*pi*I)

(-73.86028959880723, 12.3387403731541 + 2*pi*I)

(92.70316162738399, 12.7933574636829)

(-33.06027326086482, 10.7292722077761 + 2*pi*I)

(-5.230572006819245, 6.93912978599036 + 2*pi*I)

(-661.308922268859, 16.7232713625005 + 2*pi*I)

(-387.99294582729976, 15.6567929374418 + 2*pi*I)

(-20.53917051880369, 9.77370038096726 + 2*pi*I)

(98.98468480786867, 12.9245173075251)

(-70.72015750484694, 12.2518105761584 + 2*pi*I)

(-92.70316162738399, 12.7933574636829 + 2*pi*I)

(-281.18117108935127, 15.0128024466033 + 2*pi*I)

(-39.33168999791728, 11.0773238077452 + 2*pi*I)

(80.14089745262413, 12.5020293778519)

(36.19546991361951, 10.910845608056)

(-26.794395342480584, 10.3078234490048 + 2*pi*I)

(-83.28134727496824, 12.5789339569173 + 2*pi*I)

(-42.46870606424195, 11.2310217744461 + 2*pi*I)

(-23.664918471959545, 10.0584612345515 + 2*pi*I)

(-80.14089745262413, 12.5020293778519 + 2*pi*I)

(8.163221583188243, 7.89546917542736)

(14.308813381052687, 9.04451639181416)

(58.16120814687788, 11.8605539577725)

(-86.42188021326984, 12.6529915138211 + 2*pi*I)

(77.00054092388514, 12.4220501862081)

(-58.16120814687788, 11.8605539577725 + 2*pi*I)

(-55.02200091082742, 11.7494986218622 + 2*pi*I)

(95.84389596940441, 12.8600120903097)

(-29.926423275537896, 10.5295981871459 + 2*pi*I)

(-36.19546991361951, 10.910845608056 + 2*pi*I)

(-105.2664064528047, 13.0476044025427 + 2*pi*I)

(-95.84389596940441, 12.8600120903097 + 2*pi*I)

(-7685.906742666052, 21.6291219711522 + 2*pi*I)

(20.53917051880369, 9.77370038096726)

(45.60635341453117, 11.3737603004)

(51.8830828251914, 11.6319177789513)

(-89.56248751835092, 12.724405275769 + 2*pi*I)

(-48.74450985722931, 11.5069972134943 + 2*pi*I)

(29.926423275537896, 10.5295981871459)

(48.74450985722931, 11.5069972134943)


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:
x69=42.4687060642419x_{69} = 42.4687060642419
x69=39.3316899979173x_{69} = 39.3316899979173
x69=26.7943953424806x_{69} = 26.7943953424806
x69=111.548293018578x_{69} = 111.548293018578
x69=67.580161279298x_{69} = 67.580161279298
x69=83.2813472749682x_{69} = 83.2813472749682
x69=73.8602895988072x_{69} = 73.8602895988072
x69=61.3006600471x_{69} = 61.3006600471
x69=5.23057200681925x_{69} = 5.23057200681925
x69=70.7201575048469x_{69} = 70.7201575048469
x69=55.0220009108274x_{69} = 55.0220009108274
x69=33.0602732608648x_{69} = 33.0602732608648
x69=17.4191866243435x_{69} = 17.4191866243435
x69=23.6649184719595x_{69} = 23.6649184719595
x69=86.4218802132698x_{69} = 86.4218802132698
x69=64.4403208058624x_{69} = 64.4403208058624
x69=89.5624875183509x_{69} = 89.5624875183509
x69=92.703161627384x_{69} = 92.703161627384
x69=98.9846848078687x_{69} = 98.9846848078687
x69=80.1408974526241x_{69} = 80.1408974526241
x69=36.1954699136195x_{69} = 36.1954699136195
x69=8.16322158318824x_{69} = 8.16322158318824
x69=14.3088133810527x_{69} = 14.3088133810527
x69=58.1612081468779x_{69} = 58.1612081468779
x69=77.0005409238851x_{69} = 77.0005409238851
x69=95.8438959694044x_{69} = 95.8438959694044
x69=20.5391705188037x_{69} = 20.5391705188037
x69=45.6063534145312x_{69} = 45.6063534145312
x69=51.8830828251914x_{69} = 51.8830828251914
x69=29.9264232755379x_{69} = 29.9264232755379
x69=48.7445098572293x_{69} = 48.7445098572293
Decrece en los intervalos
(−∞,5.23057200681925]\left(-\infty, 5.23057200681925\right]
Crece en los intervalos
[111.548293018578,∞)\left[111.548293018578, \infty\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
−e∣sin⁡(x)∣2sin⁡(x)sign⁡(sin⁡(x))2+e∣sin⁡(x)∣2cos⁡2(x)δ(sin⁡(x))+e∣sin⁡(x)∣2cos⁡2(x)sign⁡2(sin⁡(x))4−2x2=0- \frac{e^{\frac{\left|{\sin{\left(x \right)}}\right|}{2}} \sin{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{2} + e^{\frac{\left|{\sin{\left(x \right)}}\right|}{2}} \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\right) + \frac{e^{\frac{\left|{\sin{\left(x \right)}}\right|}{2}} \cos^{2}{\left(x \right)} \operatorname{sign}^{2}{\left(\sin{\left(x \right)} \right)}}{4} - \frac{2}{x^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((e∣sin⁡(x)∣+2log⁡(3x))−19)=∞\lim_{x \to -\infty}\left(\left(\sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} + 2 \log{\left(3 x \right)}\right) - \frac{1}{9}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞((e∣sin⁡(x)∣+2log⁡(3x))−19)=∞\lim_{x \to \infty}\left(\left(\sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} + 2 \log{\left(3 x \right)}\right) - \frac{1}{9}\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 sqrt(E^Abs(sin(x))) + 2*log(3*x) - 1/9, dividida por x con x->+oo y x ->-oo
No se ha logrado calcular el límite a la izquierda
lim⁡x→−∞((e∣sin⁡(x)∣+2log⁡(3x))−19x)\lim_{x \to -\infty}\left(\frac{\left(\sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} + 2 \log{\left(3 x \right)}\right) - \frac{1}{9}}{x}\right)
No se ha logrado calcular el límite a la derecha
lim⁡x→∞((e∣sin⁡(x)∣+2log⁡(3x))−19x)\lim_{x \to \infty}\left(\frac{\left(\sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} + 2 \log{\left(3 x \right)}\right) - \frac{1}{9}}{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:
(e∣sin⁡(x)∣+2log⁡(3x))−19=e∣sin⁡(x)∣+2log⁡(−3x)−19\left(\sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} + 2 \log{\left(3 x \right)}\right) - \frac{1}{9} = \sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} + 2 \log{\left(- 3 x \right)} - \frac{1}{9}
- No
(e∣sin⁡(x)∣+2log⁡(3x))−19=−e∣sin⁡(x)∣−2log⁡(−3x)+19\left(\sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} + 2 \log{\left(3 x \right)}\right) - \frac{1}{9} = - \sqrt{e^{\left|{\sin{\left(x \right)}}\right|}} - 2 \log{\left(- 3 x \right)} + \frac{1}{9}
- No
es decir, función
no es
par ni impar
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