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Trazado el gráfico de la función/ x+sin(x)/x-sin(x)

Gráfico de la función y = x+sin(x)/x-sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           sin(x)         
f(x) = x + ------ - sin(x)
             x
f(x)=(x+sin⁡(x)x)−sin⁡(x)f{\left(x \right)} = \left(x + \frac{\sin{\left(x \right)}}{x}\right) - \sin{\left(x \right)} 
f = x + sin(x)/x - sin(x)
Gráfico de la función
02468-8-6-4-2-1010-2525
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
(x+sin⁡(x)x)−sin⁡(x)=0\left(x + \frac{\sin{\left(x \right)}}{x}\right) - \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−1.61679542746632x_{1} = -1.61679542746632
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(x)/x - sin(x).
sin⁡(0)0−sin⁡(0)\frac{\sin{\left(0 \right)}}{0} - \sin{\left(0 \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
−cos⁡(x)+1+cos⁡(x)x−sin⁡(x)x2=0- \cos{\left(x \right)} + 1 + \frac{\cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−5.70322424793371x_{1} = -5.70322424793371
x2=−63.008601057135x_{2} = -63.008601057135
x3=−100.390302545598x_{3} = -100.390302545598
x4=−69.283723128387x_{4} = -69.283723128387
x5=−75.2359014311085x_{5} = -75.2359014311085
x6=−81.5254268580108x_{6} = -81.5254268580108
x7=−24.8521339160568x_{7} = -24.8521339160568
x8=−62.6541129417412x_{8} = -62.6541129417412
x9=1.61331929451625⋅10−15x_{9} = 1.61331929451625 \cdot 10^{-15}
x10=−326.647477867129x_{10} = -326.647477867129
x11=−12.1678484681202x_{11} = -12.1678484681202
x12=−43.7700315569785x_{12} = -43.7700315569785
x13=−18.5253495518224x_{13} = -18.5253495518224
x14=−94.3925917532253x_{14} = -94.3925917532253
x15=−68.9455321239561x_{15} = -68.9455321239561
x16=−6.77724197085065x_{16} = -6.77724197085065
x17=−1017.92032662177x_{17} = -1017.92032662177
x18=−50.4625559219722x_{18} = -50.4625559219722
x19=−56.3613596480526x_{19} = -56.3613596480526
x20=−25.4073333632489x_{20} = -25.4073333632489
x21=−50.0668667662435x_{21} = -50.0668667662435
x22=−3600.2887474904x_{22} = -3600.2887474904
x23=−37.4699099712197x_{23} = -37.4699099712197
x24=−37.9255952832049x_{24} = -37.9255952832049
x25=−87.8142618608307x_{25} = -87.8142618608307
x26=−81.8368012320981x_{26} = -81.8368012320981
x27=−56.7347538298523x_{27} = -56.7347538298523
x28=−31.6630467098109x_{28} = -31.6630467098109
x29=−88.1144175203513x_{29} = -88.1144175203513
x30=−12.9419189831345x_{30} = -12.9419189831345
x31=−19.1632396027265x_{31} = -19.1632396027265
x32=−44.1925557901646x_{32} = -44.1925557901646
x33=−31.1649181283879x_{33} = -31.1649181283879
x34=−94.1025234327673x_{34} = -94.1025234327673
x35=−75.5598545577739x_{35} = -75.5598545577739
x36=0.661724133774237x_{36} = 0.661724133774237
x37=−112.964687016262x_{37} = -112.964687016262
x38=−226.288491106564x_{38} = -226.288491106564
Signos de extremos en los puntos:
(-5.70322424793371, -6.34730009668128)

(-63.008601057134975, -62.8299813408142)

(-100.39030254559772, -100.531898057234)

(-69.28372312838695, -69.1134140509466)

(-75.23590143110847, -75.3996598481559)

(-81.52542685801083, -81.6826827889124)

(-24.852133916056836, -25.1402166548156)

(-62.654112941741175, -62.83374063492)

(1.6133192945162522e-15, 1)

(-326.6474778671288, -326.725795453735)

(-12.167848468120207, -12.5877971676219)

(-43.77003155697847, -43.9855199691468)

(-18.525349551822423, -18.8611018662603)

(-94.39259175322535, -94.2467564181765)

(-68.94553212395613, -69.1166746215958)

(-6.777241970850652, -6.2330709747409)

(-1017.920326621766, -1017.87599074552)

(-50.462555921972246, -50.2628755271123)

(-56.36135964805263, -56.5508783649746)

(-25.407333363248917, -25.1255067010541)

(-50.06686676624353, -50.2681201704331)

(-3600.2887474903982, -3600.26517665012)

(-37.469909971219685, -37.7031738528102)

(-37.92559528320494, -37.6951222577405)

(-87.81426186083073, -87.9657341873134)

(-81.83680123209813, -81.6801424343559)

(-56.734753829852345, -56.5464788381907)

(-31.66304670981087, -31.4107085791225)

(-88.11441752035134, -87.963460213563)

(-12.941918983134485, -12.546795571619)

(-19.16323960272653, -18.8385730463094)

(-44.192555790164555, -43.9791201338519)

(-31.16491812838795, -31.4212689077625)

(-94.10252343276733, -94.2488075168137)

(-75.55985455777392, -75.3967967140964)

(0.6617241337742369, 0.975847594764275)

(-112.96468701626199, -113.098117672709)

(-226.28849110656378, -226.194394639322)


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=−5.70322424793371x_{1} = -5.70322424793371
x2=−100.390302545598x_{2} = -100.390302545598
x3=−75.2359014311085x_{3} = -75.2359014311085
x4=−81.5254268580108x_{4} = -81.5254268580108
x5=−24.8521339160568x_{5} = -24.8521339160568
x6=−62.6541129417412x_{6} = -62.6541129417412
x7=−326.647477867129x_{7} = -326.647477867129
x8=−12.1678484681202x_{8} = -12.1678484681202
x9=−43.7700315569785x_{9} = -43.7700315569785
x10=−18.5253495518224x_{10} = -18.5253495518224
x11=−68.9455321239561x_{11} = -68.9455321239561
x12=−56.3613596480526x_{12} = -56.3613596480526
x13=−50.0668667662435x_{13} = -50.0668667662435
x14=−37.4699099712197x_{14} = -37.4699099712197
x15=−87.8142618608307x_{15} = -87.8142618608307
x16=−31.1649181283879x_{16} = -31.1649181283879
x17=−94.1025234327673x_{17} = -94.1025234327673
x18=0.661724133774237x_{18} = 0.661724133774237
x19=−112.964687016262x_{19} = -112.964687016262
Puntos máximos de la función:
x19=−63.008601057135x_{19} = -63.008601057135
x19=−69.283723128387x_{19} = -69.283723128387
x19=1.61331929451625⋅10−15x_{19} = 1.61331929451625 \cdot 10^{-15}
x19=−94.3925917532253x_{19} = -94.3925917532253
x19=−6.77724197085065x_{19} = -6.77724197085065
x19=−50.4625559219722x_{19} = -50.4625559219722
x19=−25.4073333632489x_{19} = -25.4073333632489
x19=−3600.2887474904x_{19} = -3600.2887474904
x19=−37.9255952832049x_{19} = -37.9255952832049
x19=−81.8368012320981x_{19} = -81.8368012320981
x19=−56.7347538298523x_{19} = -56.7347538298523
x19=−31.6630467098109x_{19} = -31.6630467098109
x19=−88.1144175203513x_{19} = -88.1144175203513
x19=−12.9419189831345x_{19} = -12.9419189831345
x19=−19.1632396027265x_{19} = -19.1632396027265
x19=−44.1925557901646x_{19} = -44.1925557901646
x19=−75.5598545577739x_{19} = -75.5598545577739
x19=−226.288491106564x_{19} = -226.288491106564
Decrece en los intervalos
[0.661724133774237,∞)\left[0.661724133774237, \infty\right)
Crece en los intervalos
(−∞,−326.647477867129]\left(-\infty, -326.647477867129\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
sin⁡(x)−sin⁡(x)x−2cos⁡(x)x2+2sin⁡(x)x3=0\sin{\left(x \right)} - \frac{\sin{\left(x \right)}}{x} - \frac{2 \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=40.8419335479559x_{1} = 40.8419335479559
x2=21.9954785221928x_{2} = 21.9954785221928
x3=−59.6897083160264x_{3} = -59.6897083160264
x4=12.5800836577711x_{4} = 12.5800836577711
x5=−50.2647062907623x_{5} = -50.2647062907623
x6=−91.1059486142278x_{6} = -91.1059486142278
x7=−25.129695019447x_{7} = -25.129695019447
x8=−84.8227269101142x_{8} = -84.8227269101142
x9=−109.955578943458x_{9} = -109.955578943458
x10=3.3705194763986x_{10} = 3.3705194763986
x11=25.1360373917548x_{11} = 25.1360373917548
x12=87.9648557427322x_{12} = 87.9648557427322
x13=−62.8313543906765x_{13} = -62.8313543906765
x14=69.115463200353x_{14} = 69.115463200353
x15=521.504387863872x_{15} = 521.504387863872
x16=−94.2475568118577x_{16} = -94.2475568118577
x17=−72.2562531889342x_{17} = -72.2562531889342
x18=−18.8442061214293x_{18} = -18.8442061214293
x19=−40.8395339880157x_{19} = -40.8395339880157
x20=53.4077896462847x_{20} = 53.4077896462847
x21=75.3985802194623x_{21} = 75.3985802194623
x22=−69.1146256604297x_{22} = -69.1146256604297
x23=−12.554607383177x_{23} = -12.554607383177
x24=−81.6811128488846x_{24} = -81.6811128488846
x25=−100.530768969887x_{25} = -100.530768969887
x26=50.2662900572897x_{26} = 50.2662900572897
x27=−47.1230078370339x_{27} = -47.1230078370339
x28=56.5493044416065x_{28} = 56.5493044416065
x29=−65.9729930706693x_{29} = -65.9729930706693
x30=62.832367858231x_{30} = 62.832367858231
x31=−6.23861011154471x_{31} = -6.23861011154471
x32=81.6817124721827x_{32} = 81.6817124721827
x33=47.1248099084275x_{33} = 47.1248099084275
x34=−75.3978764766734x_{34} = -75.3978764766734
x35=78.5401447448195x_{35} = 78.5401447448195
x36=15.7166052611986x_{36} = 15.7166052611986
x37=−53.4063867877706x_{37} = -53.4063867877706
x38=43.9833550164136x_{38} = 43.9833550164136
x39=−34.5558913319288x_{39} = -34.5558913319288
x40=−28.2719169757845x_{40} = -28.2719169757845
x41=−56.5480531733744x_{41} = -56.5480531733744
x42=−43.9812861761432x_{42} = -43.9812861761432
x43=59.6908313022525x_{43} = 59.6908313022525
x44=65.9739122941239x_{44} = 65.9739122941239
x45=−15.7003319654029x_{45} = -15.7003319654029
x46=−31.4139622633267x_{46} = -31.4139622633267
x47=100.531164794866x_{47} = 100.531164794866
x48=−97.3891635369823x_{48} = -97.3891635369823
x49=28.2769266377753x_{49} = 28.2769266377753
x50=97.3895853136736x_{50} = 97.3895853136736
x51=−78.5394961843648x_{51} = -78.5394961843648
x52=−9.40429582399773x_{52} = -9.40429582399773
x53=9.44975410985695x_{53} = 9.44975410985695
x54=94.2480071788438x_{54} = 94.2480071788438
x55=−21.9871908078841x_{55} = -21.9871908078841
x56=84.8232829333673x_{56} = 84.8232829333673
x57=−37.6977408205772x_{57} = -37.6977408205772
x58=6.3416156136378x_{58} = 6.3416156136378
x59=34.5592435636382x_{59} = 34.5592435636382
x60=37.7005572582663x_{60} = 37.7005572582663
x61=18.8554944423173x_{61} = 18.8554944423173
x62=−87.9643387312335x_{62} = -87.9643387312335
x63=−2.96278670908593x_{63} = -2.96278670908593
x64=72.2570194689854x_{64} = 72.2570194689854
x65=31.4180191575035x_{65} = 31.4180191575035
x66=91.1064305803132x_{66} = 91.1064305803132
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(sin⁡(x)−sin⁡(x)x−2cos⁡(x)x2+2sin⁡(x)x3)=−13\lim_{x \to 0^-}\left(\sin{\left(x \right)} - \frac{\sin{\left(x \right)}}{x} - \frac{2 \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{3}}\right) = - \frac{1}{3}
lim⁡x→0+(sin⁡(x)−sin⁡(x)x−2cos⁡(x)x2+2sin⁡(x)x3)=−13\lim_{x \to 0^+}\left(\sin{\left(x \right)} - \frac{\sin{\left(x \right)}}{x} - \frac{2 \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{3}}\right) = - \frac{1}{3}
- los límites son iguales, es decir omitimos el punto correspondiente

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
[521.504387863872,∞)\left[521.504387863872, \infty\right)
Convexa en los intervalos
(−∞,−100.530768969887]\left(-\infty, -100.530768969887\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((x+sin⁡(x)x)−sin⁡(x))=−∞\lim_{x \to -\infty}\left(\left(x + \frac{\sin{\left(x \right)}}{x}\right) - \sin{\left(x \right)}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞((x+sin⁡(x)x)−sin⁡(x))=∞\lim_{x \to \infty}\left(\left(x + \frac{\sin{\left(x \right)}}{x}\right) - \sin{\left(x \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 x + sin(x)/x - sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x+sin⁡(x)x)−sin⁡(x)x)=1\lim_{x \to -\infty}\left(\frac{\left(x + \frac{\sin{\left(x \right)}}{x}\right) - \sin{\left(x \right)}}{x}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xy = x
lim⁡x→∞((x+sin⁡(x)x)−sin⁡(x)x)=1\lim_{x \to \infty}\left(\frac{\left(x + \frac{\sin{\left(x \right)}}{x}\right) - \sin{\left(x \right)}}{x}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xy = 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:
(x+sin⁡(x)x)−sin⁡(x)=−x+sin⁡(x)+sin⁡(x)x\left(x + \frac{\sin{\left(x \right)}}{x}\right) - \sin{\left(x \right)} = - x + \sin{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}
- No
(x+sin⁡(x)x)−sin⁡(x)=x−sin⁡(x)−sin⁡(x)x\left(x + \frac{\sin{\left(x \right)}}{x}\right) - \sin{\left(x \right)} = x - \sin{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор