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Trazado el gráfico de la función/ x/sin(3*x^2-2*x+5*ln(x))

Gráfico de la función y = x/sin(3*x^2-2*x+5*ln(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   x             
f(x) = --------------------------
          /   2                 \
       sin\3*x  - 2*x + 5*log(x)/
f(x)=xsin⁡((3x2−2x)+5log⁡(x))f{\left(x \right)} = \frac{x}{\sin{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}} 
f = x/sin(3*x^2 - 2*x + 5*log(x))
Gráfico de la función
02468-8-6-4-2-1010-1000500
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:
xsin⁡((3x2−2x)+5log⁡(x))=0\frac{x}{\sin{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
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 x/sin(3*x^2 - 2*x + 5*log(x)).
0sin⁡(5log⁡(0)+(3⋅02−0))\frac{0}{\sin{\left(5 \log{\left(0 \right)} + \left(3 \cdot 0^{2} - 0\right) \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
−x(6x−2+5x)cos⁡((3x2−2x)+5log⁡(x))sin⁡2((3x2−2x)+5log⁡(x))+1sin⁡((3x2−2x)+5log⁡(x))=0- \frac{x \left(6 x - 2 + \frac{5}{x}\right) \cos{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}}{\sin^{2}{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}} + \frac{1}{\sin{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=69.3518397128453x_{1} = 69.3518397128453
x2=36.2503760171547x_{2} = 36.2503760171547
x3=26.2500871979717x_{3} = 26.2500871979717
x4=91.9962745506224x_{4} = 91.9962745506224
x5=94.0410900393117x_{5} = 94.0410900393117
x6=86.2471548696151x_{6} = 86.2471548696151
x7=17.9988651650404x_{7} = 17.9988651650404
x8=32.2551815966246x_{8} = 32.2551815966246
x9=68.2508159666692x_{9} = 68.2508159666692
x10=80.2749085387481x_{10} = 80.2749085387481
x11=22.2494582854661x_{11} = 22.2494582854661
x12=98.2486683530012x_{12} = 98.2486683530012
x13=46.1722781004293x_{13} = 46.1722781004293
x14=50.2499888383649x_{14} = 50.2499888383649
x15=60.2489429599525x_{15} = 60.2489429599525
x16=51.9917507808877x_{16} = 51.9917507808877
x17=66.5649833212253x_{17} = 66.5649833212253
x18=56.0007967942266x_{18} = 56.0007967942266
x19=58.2497996227996x_{19} = 58.2497996227996
x20=82.0880490106075x_{20} = 82.0880490106075
x21=88.0025162774933x_{21} = 88.0025162774933
x22=2.20305923308699x_{22} = 2.20305923308699
x23=62.2512048990574x_{23} = 62.2512048990574
x24=48.2490330264814x_{24} = 48.2490330264814
x25=37.930245061061x_{25} = 37.930245061061
x26=44.5567180620131x_{26} = 44.5567180620131
x27=72.5398030316872x_{27} = 72.5398030316872
x28=43.1741149904354x_{28} = 43.1741149904354
x29=54.1651093152856x_{29} = 54.1651093152856
x30=74.252313252773x_{30} = 74.252313252773
x31=82.9226761969356x_{31} = 82.9226761969356
x32=78.2518980838374x_{32} = 78.2518980838374
x33=29.9955679458232x_{33} = 29.9955679458232
x34=10.2496627382046x_{34} = 10.2496627382046
x35=90.214050697212x_{35} = 90.214050697212
x36=100.249397645636x_{36} = 100.249397645636
x37=20.2506020633755x_{37} = 20.2506020633755
x38=64.1251262102081x_{38} = 64.1251262102081
x39=24.21291859674x_{39} = 24.21291859674
x40=16.0933594601266x_{40} = 16.0933594601266
x41=96.2498803648005x_{41} = 96.2498803648005
x42=14.0017545090562x_{42} = 14.0017545090562
x43=28.2513747254368x_{43} = 28.2513747254368
x44=8.59517544141172x_{44} = 8.59517544141172
x45=33.9943109020524x_{45} = 33.9943109020524
x46=6.27236583058133x_{46} = 6.27236583058133
x47=76.2991095408372x_{47} = 76.2991095408372
x48=12.2504993814909x_{48} = 12.2504993814909
x49=40.2394739344955x_{49} = 40.2394739344955
x50=4.33215404732646x_{50} = 4.33215404732646
Signos de extremos en los puntos:
(69.35183971284535, -69.3518397548722)

(36.2503760171547, -36.2503763137731)

(26.250087197971705, 26.2500879837721)

(91.99627455062245, 91.9962745685873)

(94.0410900393117, 94.0410900561275)

(86.24715486961514, -86.2471548914273)

(17.998865165040357, 17.9988676248156)

(32.255181596624574, 32.2551820185049)

(68.25081596666917, -68.2508160107693)

(80.27490853874812, -80.2749085658145)

(22.249458285466115, -22.2494595806678)

(98.24866835300121, 98.2486683677435)

(46.17227810042931, 46.1722782434751)

(50.24998883836493, -50.2499889492191)

(60.248942959952544, 60.2489430241381)

(51.991750780887735, 51.9917508809295)

(66.5649833212253, -66.5649833687727)

(56.00079679422662, -56.0007968742171)

(58.24979962279963, -58.249799693848)

(82.0880490106075, -82.0880490359153)

(88.00251627749334, 88.0025162980231)

(2.2030592330869863, 2.20430641706638)

(62.25120489905736, -62.2512049572275)

(48.249033026481406, 48.2490331517696)

(37.93024506106104, -37.9302453198049)

(44.55671806201312, -44.5567182212645)

(72.53980303168717, -72.5398030683986)

(43.1741149904354, 43.1741151655563)

(54.165109315285555, -54.1651094037202)

(74.25231325277299, 74.2523132869957)

(82.92267619693561, 82.922676221485)

(78.25189808383736, 78.2518981130635)

(29.995567945823172, -29.9955684711016)

(10.24966273820459, -10.2496762951944)

(90.21405069721199, -90.2140507162653)

(100.24939764563622, 100.249397659512)

(20.250602063375528, -20.2506037851539)

(64.12512621020805, 64.1251262634106)

(24.212918596740018, -24.2129195997744)

(16.093359460126642, -16.0933629120228)

(96.24988036480052, 96.2498803804825)

(14.001754509056182, 14.0017597725531)

(28.251374725436804, -28.2513753548572)

(8.595175441411723, 8.59519856870192)

(33.99431090205241, -33.9943112621129)

(6.272365830581334, 6.27242589070157)

(76.2991095408372, -76.2991095723718)

(12.250499381490922, -12.2505072741563)

(40.23947393449545, 40.239474151009)

(4.332154047326465, -4.33233655523216)


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=26.2500871979717x_{1} = 26.2500871979717
x2=91.9962745506224x_{2} = 91.9962745506224
x3=94.0410900393117x_{3} = 94.0410900393117
x4=17.9988651650404x_{4} = 17.9988651650404
x5=32.2551815966246x_{5} = 32.2551815966246
x6=98.2486683530012x_{6} = 98.2486683530012
x7=46.1722781004293x_{7} = 46.1722781004293
x8=60.2489429599525x_{8} = 60.2489429599525
x9=51.9917507808877x_{9} = 51.9917507808877
x10=88.0025162774933x_{10} = 88.0025162774933
x11=2.20305923308699x_{11} = 2.20305923308699
x12=48.2490330264814x_{12} = 48.2490330264814
x13=43.1741149904354x_{13} = 43.1741149904354
x14=74.252313252773x_{14} = 74.252313252773
x15=82.9226761969356x_{15} = 82.9226761969356
x16=78.2518980838374x_{16} = 78.2518980838374
x17=100.249397645636x_{17} = 100.249397645636
x18=64.1251262102081x_{18} = 64.1251262102081
x19=96.2498803648005x_{19} = 96.2498803648005
x20=14.0017545090562x_{20} = 14.0017545090562
x21=8.59517544141172x_{21} = 8.59517544141172
x22=6.27236583058133x_{22} = 6.27236583058133
x23=40.2394739344955x_{23} = 40.2394739344955
Puntos máximos de la función:
x23=69.3518397128453x_{23} = 69.3518397128453
x23=36.2503760171547x_{23} = 36.2503760171547
x23=86.2471548696151x_{23} = 86.2471548696151
x23=68.2508159666692x_{23} = 68.2508159666692
x23=80.2749085387481x_{23} = 80.2749085387481
x23=22.2494582854661x_{23} = 22.2494582854661
x23=50.2499888383649x_{23} = 50.2499888383649
x23=66.5649833212253x_{23} = 66.5649833212253
x23=56.0007967942266x_{23} = 56.0007967942266
x23=58.2497996227996x_{23} = 58.2497996227996
x23=82.0880490106075x_{23} = 82.0880490106075
x23=62.2512048990574x_{23} = 62.2512048990574
x23=37.930245061061x_{23} = 37.930245061061
x23=44.5567180620131x_{23} = 44.5567180620131
x23=72.5398030316872x_{23} = 72.5398030316872
x23=54.1651093152856x_{23} = 54.1651093152856
x23=29.9955679458232x_{23} = 29.9955679458232
x23=10.2496627382046x_{23} = 10.2496627382046
x23=90.214050697212x_{23} = 90.214050697212
x23=20.2506020633755x_{23} = 20.2506020633755
x23=24.21291859674x_{23} = 24.21291859674
x23=16.0933594601266x_{23} = 16.0933594601266
x23=28.2513747254368x_{23} = 28.2513747254368
x23=33.9943109020524x_{23} = 33.9943109020524
x23=76.2991095408372x_{23} = 76.2991095408372
x23=12.2504993814909x_{23} = 12.2504993814909
x23=4.33215404732646x_{23} = 4.33215404732646
Decrece en los intervalos
[100.249397645636,∞)\left[100.249397645636, \infty\right)
Crece en los intervalos
(−∞,2.20305923308699]\left(-\infty, 2.20305923308699\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(xsin⁡((3x2−2x)+5log⁡(x)))y = \lim_{x \to -\infty}\left(\frac{x}{\sin{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(xsin⁡((3x2−2x)+5log⁡(x)))y = \lim_{x \to \infty}\left(\frac{x}{\sin{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x/sin(3*x^2 - 2*x + 5*log(x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞1sin⁡((3x2−2x)+5log⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty} \frac{1}{\sin{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}} = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
lim⁡x→∞1sin⁡((3x2−2x)+5log⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty} \frac{1}{\sin{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}} = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
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:
xsin⁡((3x2−2x)+5log⁡(x))=−xsin⁡(3x2+2x+5log⁡(−x))\frac{x}{\sin{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}} = - \frac{x}{\sin{\left(3 x^{2} + 2 x + 5 \log{\left(- x \right)} \right)}}
- No
xsin⁡((3x2−2x)+5log⁡(x))=xsin⁡(3x2+2x+5log⁡(−x))\frac{x}{\sin{\left(\left(3 x^{2} - 2 x\right) + 5 \log{\left(x \right)} \right)}} = \frac{x}{\sin{\left(3 x^{2} + 2 x + 5 \log{\left(- x \right)} \right)}}
- No
es decir, función
no es
par ni impar
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