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Trazado el gráfico de la función/ -cos(x/2)/2-(x/8-1/8)sin(x/2)/2

Gráfico de la función y = -cos(x/2)/2-(x/8-1/8)sin(x/2)/2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           /x\    /x   1\    /x\
       -cos|-|    |- - -|*sin|-|
           \2/    \8   8/    \2/
f(x) = -------- - --------------
          2             2
f(x)=−(x8−18)sin⁡(x2)2+(−1)cos⁡(x2)2f{\left(x \right)} = - \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\left(-1\right) \cos{\left(\frac{x}{2} \right)}}{2} 
f = -(x/8 - 1/8)*sin(x/2)/2 + (-cos(x/2))/2
Gráfico de la función
02468-8-6-4-2-10101-1
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:
−(x8−18)sin⁡(x2)2+(−1)cos⁡(x2)2=0- \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\left(-1\right) \cos{\left(\frac{x}{2} \right)}}{2} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−100.373459104023x_{1} = -100.373459104023
x2=−87.7848687610232x_{2} = -87.7848687610232
x3=100.370297523311x_{3} = 100.370297523311
x4=17.9684380064521x_{4} = 17.9684380064521
x5=81.4832607664107x_{5} = 81.4832607664107
x6=37.2648683515728x_{6} = 37.2648683515728
x7=−11.4220569372309x_{7} = -11.4220569372309
x8=49.9414269616741x_{8} = 49.9414269616741
x9=−43.6275421109017x_{9} = -43.6275421109017
x10=43.6111291113x_{10} = 43.6111291113
x11=11.2399609265578x_{11} = 11.2399609265578
x12=−56.2710907943176x_{12} = -56.2710907943176
x13=−62.5815231271754x_{13} = -62.5815231271754
x14=75.1833722454018x_{14} = 75.1833722454018
x15=−94.0798955123631x_{15} = -94.0798955123631
x16=−18.0545600572773x_{16} = -18.0545600572773
x17=−49.954016484692x_{17} = -49.954016484692
x18=−24.5253057146884x_{18} = -24.5253057146884
x19=62.573448345104x_{19} = 62.573448345104
x20=30.8929388643776x_{20} = 30.8929388643776
x21=3.81921962088753x_{21} = 3.81921962088753
x22=−4.31430001404567x_{22} = -4.31430001404567
x23=−75.1889863091062x_{23} = -75.1889863091062
x24=−68.8870898515208x_{24} = -68.8870898515208
x25=87.7807412347801x_{25} = 87.7807412347801
x26=94.0762990750652x_{26} = 94.0762990750652
x27=68.8804120305583x_{27} = 68.8804120305583
x28=56.2611309199693x_{28} = 56.2611309199693
x29=−81.4880462283736x_{29} = -81.4880462283736
x30=−30.9248617887199x_{30} = -30.9248617887199
x31=24.4758744792161x_{31} = 24.4758744792161
x32=−37.28714412374x_{32} = -37.28714412374
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 (-cos(x/2))/2 - (x/8 - 1/8)*sin(x/2)/2.
(−1)cos⁡(02)2−(−18+08)sin⁡(02)2\frac{\left(-1\right) \cos{\left(\frac{0}{2} \right)}}{2} - \frac{\left(- \frac{1}{8} + \frac{0}{8}\right) \sin{\left(\frac{0}{2} \right)}}{2}
Resultado:
f(0)=−12f{\left(0 \right)} = - \frac{1}{2}
Punto:
(0, -1/2)
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
−(x8−18)cos⁡(x2)4+3sin⁡(x2)16=0- \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \cos{\left(\frac{x}{2} \right)}}{4} + \frac{3 \sin{\left(\frac{x}{2} \right)}}{16} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−72.0928234997439x_{1} = -72.0928234997439
x2=40.5395084538546x_{2} = 40.5395084538546
x3=21.4195644604296x_{3} = 21.4195644604296
x4=97.2648772499214x_{4} = 97.2648772499214
x5=−84.6831790656061x_{5} = -84.6831790656061
x6=53.1780992694566x_{6} = 53.1780992694566
x7=−53.1865162868496x_{7} = -53.1865162868496
x8=−46.8745347430149x_{8} = -46.8745347430149
x9=34.1999323668518x_{9} = 34.1999323668518
x10=−0.485362558276441x_{10} = -0.485362558276441
x11=14.8925876523829x_{11} = 14.8925876523829
x12=−14.9900112041524x_{12} = -14.9900112041524
x13=46.8637225983274x_{13} = 46.8637225983274
x14=90.9730108075959x_{14} = 90.9730108075959
x15=179.003392323252x_{15} = 179.003392323252
x16=65.7887551882277x_{16} = 65.7887551882277
x17=−78.3889486119439x_{17} = -78.3889486119439
x18=72.0882258411896x_{18} = 72.0882258411896
x19=−21.469262706264x_{19} = -21.469262706264
x20=84.6798429343937x_{20} = 84.6798429343937
x21=59.4857976886522x_{21} = 59.4857976886522
x22=−65.7942704913902x_{22} = -65.7942704913902
x23=27.834382800265x_{23} = 27.834382800265
x24=−8.2765436209072x_{24} = -8.2765436209072
x25=8.00876143267673x_{25} = 8.00876143267673
x26=−59.4925355374796x_{26} = -59.4925355374796
x27=103.555681188122x_{27} = 103.555681188122
x28=−34.2200439239142x_{28} = -34.2200439239142
x29=−90.9759025986992x_{29} = -90.9759025986992
x30=78.3850572624003x_{30} = 78.3850572624003
x31=−40.5539051427507x_{31} = -40.5539051427507
x32=−97.2674079064454x_{32} = -97.2674079064454
x33=1416.84981134451x_{33} = 1416.84981134451
x34=−27.8644346285563x_{34} = -27.8644346285563
Signos de extremos en los puntos:
(-72.09282349974393, 4.59389354372914)

(40.53950845385463, -2.51826357281154)

(21.419564460429562, 1.36541620628308)

(97.26487724992135, 6.03600591986278)

(-84.68317906560608, 5.37704419460649)

(53.17809926945661, -3.29690091496593)

(-53.18651628684961, -3.42111264145565)

(-46.87453474301491, 3.03111013420797)

(34.19993236685178, 2.13083942229892)

(-0.4853625582764406, -0.507657432693553)

(14.89258765238292, -0.995365828004467)

(-14.990011204152419, -1.11133040564379)

(46.863722598327406, 2.90712242054539)

(90.97301080759586, -5.64412041401817)

(179.00339232325229, -11.1357413416032)

(65.78875518822768, -4.07815101169087)

(-78.38894861194393, -4.98538014145867)

(72.08822584118958, 4.46932432786362)

(-21.46926270626403, 1.4857842065916)

(84.67984293439375, 5.25235682700065)

(59.48579768865222, 3.68730412130719)

(-65.79427049139015, -4.20263430613656)

(27.834382800265022, -1.74583718643573)

(-8.276543620907198, 0.758375279900881)

(8.008761432676732, 0.657927652486149)

(-59.49253553747962, 3.81167297570818)

(103.55568118812153, -6.42799097104595)

(-34.220043923914226, 2.25395858951351)

(-90.97590259869924, -5.76884940626111)

(78.38505726240027, -4.86074477885135)

(-40.55390514275072, -2.64191627024295)

(-97.2674079064454, 6.16076874563619)

(1416.8498113445055, 88.491937493712)

(-27.86443462855629, -1.86803002902737)


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=40.5395084538546x_{1} = 40.5395084538546
x2=53.1780992694566x_{2} = 53.1780992694566
x3=−53.1865162868496x_{3} = -53.1865162868496
x4=−0.485362558276441x_{4} = -0.485362558276441
x5=14.8925876523829x_{5} = 14.8925876523829
x6=−14.9900112041524x_{6} = -14.9900112041524
x7=90.9730108075959x_{7} = 90.9730108075959
x8=179.003392323252x_{8} = 179.003392323252
x9=65.7887551882277x_{9} = 65.7887551882277
x10=−78.3889486119439x_{10} = -78.3889486119439
x11=−65.7942704913902x_{11} = -65.7942704913902
x12=27.834382800265x_{12} = 27.834382800265
x13=103.555681188122x_{13} = 103.555681188122
x14=−90.9759025986992x_{14} = -90.9759025986992
x15=78.3850572624003x_{15} = 78.3850572624003
x16=−40.5539051427507x_{16} = -40.5539051427507
x17=−27.8644346285563x_{17} = -27.8644346285563
Puntos máximos de la función:
x17=−72.0928234997439x_{17} = -72.0928234997439
x17=21.4195644604296x_{17} = 21.4195644604296
x17=97.2648772499214x_{17} = 97.2648772499214
x17=−84.6831790656061x_{17} = -84.6831790656061
x17=−46.8745347430149x_{17} = -46.8745347430149
x17=34.1999323668518x_{17} = 34.1999323668518
x17=46.8637225983274x_{17} = 46.8637225983274
x17=72.0882258411896x_{17} = 72.0882258411896
x17=−21.469262706264x_{17} = -21.469262706264
x17=84.6798429343937x_{17} = 84.6798429343937
x17=59.4857976886522x_{17} = 59.4857976886522
x17=−8.2765436209072x_{17} = -8.2765436209072
x17=8.00876143267673x_{17} = 8.00876143267673
x17=−59.4925355374796x_{17} = -59.4925355374796
x17=−34.2200439239142x_{17} = -34.2200439239142
x17=−97.2674079064454x_{17} = -97.2674079064454
x17=1416.84981134451x_{17} = 1416.84981134451
Decrece en los intervalos
[179.003392323252,∞)\left[179.003392323252, \infty\right)
Crece en los intervalos
(−∞,−90.9759025986992]\left(-\infty, -90.9759025986992\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
(x−1)sin⁡(x2)+4cos⁡(x2)64=0\frac{\left(x - 1\right) \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{2} \right)}}{64} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−50.1092735086132x_{1} = -50.1092735086132
x2=18.3975766589136x_{2} = 18.3975766589136
x3=62.702379600244x_{3} = 62.702379600244
x4=4.60997495542275x_{4} = 4.60997495542275
x5=56.4045253179118x_{5} = 56.4045253179118
x6=−75.2934613414557x_{6} = -75.2934613414557
x7=75.2906424022808x_{7} = 75.2906424022808
x8=31.152145665807x_{8} = 31.152145665807
x9=43.7959048957786x_{9} = 43.7959048957786
x10=37.4806902808426x_{10} = 37.4806902808426
x11=−62.7064417052844x_{11} = -62.7064417052844
x12=87.872570390469x_{12} = 87.872570390469
x13=−11.9679852063568x_{13} = -11.9679852063568
x14=−100.452150829621x_{14} = -100.452150829621
x15=100.450566275218x_{15} = 100.450566275218
x16=50.1029183076511x_{16} = 50.1029183076511
x17=94.1619603591067x_{17} = 94.1619603591067
x18=−37.4920198749377x_{18} = -37.4920198749377
x19=−69.000878243345x_{19} = -69.000878243345
x20=−18.4437743604163x_{20} = -18.4437743604163
x21=−24.8254108738778x_{21} = -24.8254108738778
x22=68.9975224617713x_{22} = 68.9975224617713
x23=−5.12633666448475x_{23} = -5.12633666448475
x24=−81.5846142943435x_{24} = -81.5846142943435
x25=−56.4095429498975x_{25} = -56.4095429498975
x26=−87.874640577245x_{26} = -87.874640577245
x27=24.7997151676252x_{27} = 24.7997151676252
x28=11.8605961353608x_{28} = 11.8605961353608
x29=81.5822129210625x_{29} = 81.5822129210625
x30=−94.1637634512637x_{30} = -94.1637634512637
x31=−43.8042146500336x_{31} = -43.8042146500336
x32=−31.1685060612189x_{32} = -31.1685060612189

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
[100.450566275218,∞)\left[100.450566275218, \infty\right)
Convexa en los intervalos
(−∞,−94.1637634512637]\left(-\infty, -94.1637634512637\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→−∞(−(x8−18)sin⁡(x2)2+(−1)cos⁡(x2)2)y = \lim_{x \to -\infty}\left(- \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\left(-1\right) \cos{\left(\frac{x}{2} \right)}}{2}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(−(x8−18)sin⁡(x2)2+(−1)cos⁡(x2)2)y = \lim_{x \to \infty}\left(- \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\left(-1\right) \cos{\left(\frac{x}{2} \right)}}{2}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-cos(x/2))/2 - (x/8 - 1/8)*sin(x/2)/2, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(−(x8−18)sin⁡(x2)2+(−1)cos⁡(x2)2x)y = x \lim_{x \to -\infty}\left(\frac{- \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\left(-1\right) \cos{\left(\frac{x}{2} \right)}}{2}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(−(x8−18)sin⁡(x2)2+(−1)cos⁡(x2)2x)y = x \lim_{x \to \infty}\left(\frac{- \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\left(-1\right) \cos{\left(\frac{x}{2} \right)}}{2}}{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:
−(x8−18)sin⁡(x2)2+(−1)cos⁡(x2)2=(−x8−18)sin⁡(x2)2−cos⁡(x2)2- \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\left(-1\right) \cos{\left(\frac{x}{2} \right)}}{2} = \frac{\left(- \frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} - \frac{\cos{\left(\frac{x}{2} \right)}}{2}
- No
−(x8−18)sin⁡(x2)2+(−1)cos⁡(x2)2=−(−x8−18)sin⁡(x2)2+cos⁡(x2)2- \frac{\left(\frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\left(-1\right) \cos{\left(\frac{x}{2} \right)}}{2} = - \frac{\left(- \frac{x}{8} - \frac{1}{8}\right) \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\cos{\left(\frac{x}{2} \right)}}{2}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор