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Trazado el gráfico de la función/ ln(1+x^2)+sin^2(x/3)

Gráfico de la función y = ln(1+x^2)+sin^2(x/3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /     2\      2/x\
f(x) = log\1 + x / + sin |-|
                         \3/
f(x)=log⁡(x2+1)+sin⁡2(x3)f{\left(x \right)} = \log{\left(x^{2} + 1 \right)} + \sin^{2}{\left(\frac{x}{3} \right)} 
f = log(x^2 + 1) + sin(x/3)^2
Gráfico de la función
02468-8-6-4-2-101005
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:
log⁡(x2+1)+sin⁡2(x3)=0\log{\left(x^{2} + 1 \right)} + \sin^{2}{\left(\frac{x}{3} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
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 log(1 + x^2) + sin(x/3)^2.
log⁡(02+1)+sin⁡2(03)\log{\left(0^{2} + 1 \right)} + \sin^{2}{\left(\frac{0}{3} \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
2xx2+1+2sin⁡(x3)cos⁡(x3)3=0\frac{2 x}{x^{2} + 1} + \frac{2 \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{3} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−70.8130570457361x_{1} = -70.8130570457361
x2=−6.43085088987635x_{2} = -6.43085088987635
x3=75.2785619494359x_{3} = 75.2785619494359
x4=−23.9412396826903x_{4} = -23.9412396826903
x5=−42.6232395580237x_{5} = -42.6232395580237
x6=99.0510771819535x_{6} = 99.0510771819535
x7=−89.63585948037x_{7} = -89.63585948037
x8=−61.4078130617512x_{8} = -61.4078130617512
x9=−94.1521356214567x_{9} = -94.1521356214567
x10=52.0096453813026x_{10} = 52.0096453813026
x11=94.1521356214567x_{11} = 94.1521356214567
x12=−8.22076669163097x_{12} = -8.22076669163097
x13=−56.388809377123x_{13} = -56.388809377123
x14=−46.9316831723948x_{14} = -46.9316831723948
x15=−65.8365853387199x_{15} = -65.8365853387199
x16=27.950228589841x_{16} = 27.950228589841
x17=23.9412396826903x_{17} = 23.9412396826903
x18=80.2228874798627x_{18} = 80.2228874798627
x19=18.3514818786207x_{19} = 18.3514818786207
x20=−75.2785619494359x_{20} = -75.2785619494359
x21=33.2585710813262x_{21} = 33.2585710813262
x22=−27.950228589841x_{22} = -27.950228589841
x23=−99.0510771819535x_{23} = -99.0510771819535
x24=0x_{24} = 0
x25=−80.2228874798627x_{25} = -80.2228874798627
x26=−18.3514818786207x_{26} = -18.3514818786207
x27=−150.736727474924x_{27} = -150.736727474924
x28=42.6232395580237x_{28} = 42.6232395580237
x29=8.22076669163097x_{29} = 8.22076669163097
x30=65.8365853387199x_{30} = 65.8365853387199
x31=56.388809377123x_{31} = 56.388809377123
x32=−52.0096453813026x_{32} = -52.0096453813026
x33=84.7166910244793x_{33} = 84.7166910244793
x34=14.7619679085848x_{34} = 14.7619679085848
x35=−33.2585710813262x_{35} = -33.2585710813262
x36=−278.063318664902x_{36} = -278.063318664902
x37=6.43085088987635x_{37} = 6.43085088987635
x38=−14.7619679085848x_{38} = -14.7619679085848
x39=−37.457976449678x_{39} = -37.457976449678
x40=−84.7166910244793x_{40} = -84.7166910244793
x41=89.63585948037x_{41} = 89.63585948037
x42=37.457976449678x_{42} = 37.457976449678
x43=46.9316831723948x_{43} = 46.9316831723948
x44=70.8130570457361x_{44} = 70.8130570457361
x45=61.4078130617512x_{45} = 61.4078130617512
Signos de extremos en los puntos:
(-70.81305704573606, 9.51848889900326)

(-6.430850889876349, 4.45233740324899)

(75.27856194943593, 8.64415738253468)

(-23.94123968269029, 7.33704813056943)

(-42.62323955802366, 8.50037634236811)

(99.05107718195352, 10.190455233879)

(-89.63585948037003, 9.99051423355457)

(-61.4078130617512, 9.23294816539768)

(-94.15213562145671, 9.09095275242567)

(52.00964538130263, 8.89989215886184)

(94.15213562145671, 9.09095275242567)

(-8.220766691630969, 4.38062266053026)

(-56.388809377123046, 8.06769261804512)

(-46.93168317239477, 7.70193910648502)

(-65.83658533871993, 8.37666181962594)

(27.95022858984101, 6.6737562387914)

(23.94123968269029, 7.33704813056943)

(80.22288747986269, 9.76837310714338)

(18.351481878620657, 5.84969744235365)

(-75.27856194943593, 8.64415738253468)

(33.25857108132624, 8.00133984165076)

(-27.95022858984101, 6.6737562387914)

(-99.05107718195352, 10.190455233879)

(0, 0)

(-80.22288747986269, 9.76837310714338)

(-18.351481878620657, 5.84969744235365)

(-150.73672747492415, 10.0315098086609)

(42.62323955802366, 8.50037634236811)

(8.220766691630969, 4.38062266053026)

(65.83658533871993, 8.37666181962594)

(56.388809377123046, 8.06769261804512)

(-52.00964538130263, 8.89989215886184)

(84.71669102447932, 8.88001985784857)

(14.76196790858475, 6.34593508810399)

(-33.25857108132624, 8.00133984165076)

(-278.0633186649018, 12.2555942276782)

(6.430850889876349, 4.45233740324899)

(-14.76196790858475, 6.34593508810399)

(-37.45797644967802, 7.25359860290258)

(-84.71669102447932, 8.88001985784857)

(89.63585948037003, 9.99051423355457)

(37.45797644967802, 7.25359860290258)

(46.93168317239477, 7.70193910648502)

(70.81305704573606, 9.51848889900326)

(61.4078130617512, 9.23294816539768)


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=75.2785619494359x_{1} = 75.2785619494359
x2=−94.1521356214567x_{2} = -94.1521356214567
x3=94.1521356214567x_{3} = 94.1521356214567
x4=−8.22076669163097x_{4} = -8.22076669163097
x5=−56.388809377123x_{5} = -56.388809377123
x6=−46.9316831723948x_{6} = -46.9316831723948
x7=−65.8365853387199x_{7} = -65.8365853387199
x8=27.950228589841x_{8} = 27.950228589841
x9=18.3514818786207x_{9} = 18.3514818786207
x10=−75.2785619494359x_{10} = -75.2785619494359
x11=−27.950228589841x_{11} = -27.950228589841
x12=0x_{12} = 0
x13=−18.3514818786207x_{13} = -18.3514818786207
x14=−150.736727474924x_{14} = -150.736727474924
x15=8.22076669163097x_{15} = 8.22076669163097
x16=65.8365853387199x_{16} = 65.8365853387199
x17=56.388809377123x_{17} = 56.388809377123
x18=84.7166910244793x_{18} = 84.7166910244793
x19=−37.457976449678x_{19} = -37.457976449678
x20=−84.7166910244793x_{20} = -84.7166910244793
x21=37.457976449678x_{21} = 37.457976449678
x22=46.9316831723948x_{22} = 46.9316831723948
Puntos máximos de la función:
x22=−70.8130570457361x_{22} = -70.8130570457361
x22=−6.43085088987635x_{22} = -6.43085088987635
x22=−23.9412396826903x_{22} = -23.9412396826903
x22=−42.6232395580237x_{22} = -42.6232395580237
x22=99.0510771819535x_{22} = 99.0510771819535
x22=−89.63585948037x_{22} = -89.63585948037
x22=−61.4078130617512x_{22} = -61.4078130617512
x22=52.0096453813026x_{22} = 52.0096453813026
x22=23.9412396826903x_{22} = 23.9412396826903
x22=80.2228874798627x_{22} = 80.2228874798627
x22=33.2585710813262x_{22} = 33.2585710813262
x22=−99.0510771819535x_{22} = -99.0510771819535
x22=−80.2228874798627x_{22} = -80.2228874798627
x22=42.6232395580237x_{22} = 42.6232395580237
x22=−52.0096453813026x_{22} = -52.0096453813026
x22=14.7619679085848x_{22} = 14.7619679085848
x22=−33.2585710813262x_{22} = -33.2585710813262
x22=−278.063318664902x_{22} = -278.063318664902
x22=6.43085088987635x_{22} = 6.43085088987635
x22=−14.7619679085848x_{22} = -14.7619679085848
x22=89.63585948037x_{22} = 89.63585948037
x22=70.8130570457361x_{22} = 70.8130570457361
x22=61.4078130617512x_{22} = 61.4078130617512
Decrece en los intervalos
[94.1521356214567,∞)\left[94.1521356214567, \infty\right)
Crece en los intervalos
(−∞,−150.736727474924]\left(-\infty, -150.736727474924\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
2(−2x2(x2+1)2−sin⁡2(x3)9+cos⁡2(x3)9+1x2+1)=02 \left(- \frac{2 x^{2}}{\left(x^{2} + 1\right)^{2}} - \frac{\sin^{2}{\left(\frac{x}{3} \right)}}{9} + \frac{\cos^{2}{\left(\frac{x}{3} \right)}}{9} + \frac{1}{x^{2} + 1}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1.21147148596966x_{1} = 1.21147148596966
x2=58.9009743665833x_{2} = 58.9009743665833
x3=101.317677807899x_{3} = 101.317677807899
x4=−7.30857948991694x_{4} = -7.30857948991694
x5=54.1970646136329x_{5} = 54.1970646136329
x6=−16.5421667598838x_{6} = -16.5421667598838
x7=−25.9381165481115x_{7} = -25.9381165481115
x8=−68.3267503779755x_{8} = -68.3267503779755
x9=−21.1758431484799x_{9} = -21.1758431484799
x10=157.865572477788x_{10} = 157.865572477788
x11=−44.7744192788177x_{11} = -44.7744192788177
x12=25.9381165481115x_{12} = 25.9381165481115
x13=−40.0469042722987x_{13} = -40.0469042722987
x14=35.3536925612401x_{14} = 35.3536925612401
x15=−49.4745757307675x_{15} = -49.4745757307675
x16=−54.1970646136329x_{16} = -54.1970646136329
x17=82.468791254516x_{17} = 82.468791254516
x18=−91.8931832506918x_{18} = -91.8931832506918
x19=91.8931832506918x_{19} = 91.8931832506918
x20=77.7521861814358x_{20} = 77.7521861814358
x21=49.4745757307675x_{21} = 49.4745757307675
x22=11.6841652099542x_{22} = 11.6841652099542
x23=−35.3536925612401x_{23} = -35.3536925612401
x24=63.6205840942105x_{24} = 63.6205840942105
x25=40.0469042722987x_{25} = 40.0469042722987
x26=68.3267503779755x_{26} = 68.3267503779755
x27=21.1758431484799x_{27} = 21.1758431484799
x28=87.1774204991994x_{28} = 87.1774204991994
x29=7.30857948991694x_{29} = 7.30857948991694
x30=44.7744192788177x_{30} = 44.7744192788177
x31=−393.484567052778x_{31} = -393.484567052778
x32=−11.6841652099542x_{32} = -11.6841652099542
x33=−87.1774204991994x_{33} = -87.1774204991994
x34=16.5421667598838x_{34} = 16.5421667598838
x35=−101.317677807899x_{35} = -101.317677807899
x36=−1.21147148596966x_{36} = -1.21147148596966
x37=−506.581867996643x_{37} = -506.581867996643
x38=30.6161718615451x_{38} = 30.6161718615451
x39=−73.0445579935x_{39} = -73.0445579935
x40=−30.6161718615451x_{40} = -30.6161718615451
x41=−77.7521861814358x_{41} = -77.7521861814358
x42=−58.9009743665833x_{42} = -58.9009743665833
x43=73.0445579935x_{43} = 73.0445579935
x44=−96.6025279349004x_{44} = -96.6025279349004
x45=96.6025279349004x_{45} = 96.6025279349004
x46=−82.468791254516x_{46} = -82.468791254516
x47=−63.6205840942105x_{47} = -63.6205840942105

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
[157.865572477788,∞)\left[157.865572477788, \infty\right)
Convexa en los intervalos
(−∞,−96.6025279349004]\left(-\infty, -96.6025279349004\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x2+1)+sin⁡2(x3))=∞\lim_{x \to -\infty}\left(\log{\left(x^{2} + 1 \right)} + \sin^{2}{\left(\frac{x}{3} \right)}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(log⁡(x2+1)+sin⁡2(x3))=∞\lim_{x \to \infty}\left(\log{\left(x^{2} + 1 \right)} + \sin^{2}{\left(\frac{x}{3} \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(1 + x^2) + sin(x/3)^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x2+1)+sin⁡2(x3)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x^{2} + 1 \right)} + \sin^{2}{\left(\frac{x}{3} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(log⁡(x2+1)+sin⁡2(x3)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x^{2} + 1 \right)} + \sin^{2}{\left(\frac{x}{3} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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:
log⁡(x2+1)+sin⁡2(x3)=log⁡(x2+1)+sin⁡2(x3)\log{\left(x^{2} + 1 \right)} + \sin^{2}{\left(\frac{x}{3} \right)} = \log{\left(x^{2} + 1 \right)} + \sin^{2}{\left(\frac{x}{3} \right)}
- No
log⁡(x2+1)+sin⁡2(x3)=−log⁡(x2+1)−sin⁡2(x3)\log{\left(x^{2} + 1 \right)} + \sin^{2}{\left(\frac{x}{3} \right)} = - \log{\left(x^{2} + 1 \right)} - \sin^{2}{\left(\frac{x}{3} \right)}
- No
es decir, función
no es
par ni impar
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