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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                  /     2\                           
                  \2 - x /*cos(x)   sin(x) - x*cos(x)
f(x) = x*sin(x) + --------------- - -----------------
                         2                  2
f(x)=(xsin⁡(x)+(2−x2)cos⁡(x)2)−−xcos⁡(x)+sin⁡(x)2f{\left(x \right)} = \left(x \sin{\left(x \right)} + \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2}\right) - \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2} 
f = x*sin(x) + ((2 - x^2)*cos(x))/2 - (-x*cos(x) + sin(x))/2
Gráfico de la función
02468-8-6-4-2-1010-100100
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⁡(x)+(2−x2)cos⁡(x)2)−−xcos⁡(x)+sin⁡(x)2=0\left(x \sin{\left(x \right)} + \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2}\right) - \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−54.9417868092552x_{1} = -54.9417868092552
x2=−29.7790118722013x_{2} = -29.7790118722013
x3=−80.0857908927295x_{3} = -80.0857908927295
x4=58.0847230588184x_{4} = 58.0847230588184
x5=−51.7980236080864x_{5} = -51.7980236080864
x6=−45.509605477046x_{6} = -45.509605477046
x7=−42.3648192466304x_{7} = -42.3648192466304
x8=86.3705041940733x_{8} = 86.3705041940733
x9=4.12505955787644x_{9} = 4.12505955787644
x10=−61.2286490856077x_{10} = -61.2286490856077
x11=20.3192050176938x_{11} = 20.3192050176938
x12=76.9428525737738x_{12} = 76.9428525737738
x13=98.939849690994x_{13} = 98.939849690994
x14=10.7997296308575x_{14} = 10.7997296308575
x15=−32.9268416565236x_{15} = -32.9268416565236
x16=−92.655511537458x_{16} = -92.655511537458
x17=−58.0853167292879x_{17} = -58.0853167292879
x18=−76.943190684033x_{18} = -76.943190684033
x19=54.9411231438086x_{19} = 54.9411231438086
x20=−48.6539822354487x_{20} = -48.6539822354487
x21=89.5129193795901x_{21} = 89.5129193795901
x22=−64.3718126533781x_{22} = -64.3718126533781
x23=61.2281148857664x_{23} = 61.2281148857664
x24=−10.8176018645304x_{24} = -10.8176018645304
x25=51.7972767821084x_{25} = 51.7972767821084
x26=70.6573220380329x_{26} = 70.6573220380329
x27=−89.5131691427441x_{27} = -89.5131691427441
x28=13.9881448296044x_{28} = 13.9881448296044
x29=29.7767436105645x_{29} = 29.7767436105645
x30=−26.6297259683358x_{30} = -26.6297259683358
x31=17.1583039185054x_{31} = 17.1583039185054
x32=−17.1652110133848x_{32} = -17.1652110133848
x33=32.9249883021957x_{33} = 32.9249883021957
x34=95.7975869260642x_{34} = 95.7975869260642
x35=−4.2776293912398x_{35} = -4.2776293912398
x36=7.56582067978772x_{36} = 7.56582067978772
x37=73.800137626604x_{37} = 73.800137626604
x38=−98.9400541039626x_{38} = -98.9400541039626
x39=26.626885381147x_{39} = 26.626885381147
x40=−36.0735938017531x_{40} = -36.0735938017531
x41=−23.4784037227872x_{41} = -23.4784037227872
x42=−692.718295016965x_{42} = -692.718295016965
x43=64.3713294100119x_{43} = 64.3713294100119
x44=−20.324107072595x_{44} = -20.324107072595
x45=−39.2195258595176x_{45} = -39.2195258595176
x46=36.0720508962562x_{46} = 36.0720508962562
x47=−7.60378254104491x_{47} = -7.60378254104491
x48=23.474741834632x_{48} = 23.474741834632
x49=−83.228315415321x_{49} = -83.228315415321
x50=45.5086374624916x_{50} = 45.5086374624916
x51=−95.7978049762696x_{51} = -95.7978049762696
x52=83.228026478556x_{52} = 83.228026478556
x53=48.653135557315x_{53} = 48.653135557315
x54=92.6552784378581x_{54} = 92.6552784378581
x55=−73.8005051730503x_{55} = -73.8005051730503
x56=67.5143916258972x_{56} = 67.5143916258972
x57=42.3637017632323x_{57} = 42.3637017632323
x58=−70.6577230415554x_{58} = -70.6577230415554
x59=−86.3707724744134x_{59} = -86.3707724744134
x60=−67.5148308750222x_{60} = -67.5148308750222
x61=39.2182213353859x_{61} = 39.2182213353859
x62=−13.9986236129207x_{62} = -13.9986236129207
x63=80.0854788174622x_{63} = 80.0854788174622
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) + ((2 - x^2)*cos(x))/2 - (sin(x) - x*cos(x))/2.
−sin⁡(0)−0cos⁡(0)2+(0sin⁡(0)+(2−02)cos⁡(0)2)- \frac{\sin{\left(0 \right)} - 0 \cos{\left(0 \right)}}{2} + \left(0 \sin{\left(0 \right)} + \frac{\left(2 - 0^{2}\right) \cos{\left(0 \right)}}{2}\right)
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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
−xsin⁡(x)2−(2−x2)sin⁡(x)2+sin⁡(x)=0- \frac{x \sin{\left(x \right)}}{2} - \frac{\left(2 - x^{2}\right) \sin{\left(x \right)}}{2} + \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=1x_{2} = 1
x3=πx_{3} = \pi
Signos de extremos en los puntos:
(0, 1)

    sin(1)          
(1, ------ + cos(1))
      2             

            2      
          pi    pi 
(pi, -1 + --- - --)
           2    2


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=1x_{1} = 1
Puntos máximos de la función:
x1=0x_{1} = 0
x1=πx_{1} = \pi
Decrece en los intervalos
(−∞,0]∪[1,∞)\left(-\infty, 0\right] \cup \left[1, \infty\right)
Crece en los intervalos
(−∞,1]∪[π,∞)\left(-\infty, 1\right] \cup \left[\pi, \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
xsin⁡(x)−xcos⁡(x)2+(x2−2)cos⁡(x)2−sin⁡(x)2+cos⁡(x)=0x \sin{\left(x \right)} - \frac{x \cos{\left(x \right)}}{2} + \frac{\left(x^{2} - 2\right) \cos{\left(x \right)}}{2} - \frac{\sin{\left(x \right)}}{2} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−39.3200997911221x_{1} = -39.3200997911221
x2=26.779523413937x_{2} = 26.779523413937
x3=61.2939451205206x_{3} = 61.2939451205206
x4=58.1541424076718x_{4} = 58.1541424076718
x5=73.854686660561x_{5} = 73.854686660561
x6=−48.7352892631046x_{6} = -48.7352892631046
x7=23.6481708154477x_{7} = 23.6481708154477
x8=2.41352739608161x_{8} = 2.41352739608161
x9=−42.4580305922627x_{9} = -42.4580305922627
x10=−70.7139131086823x_{10} = -70.7139131086823
x11=−76.9948235344069x_{11} = -76.9948235344069
x12=83.2763629879876x_{12} = 83.2763629879876
x13=−58.1535518920027x_{13} = -58.1535518920027
x14=−11.1656732464925x_{14} = -11.1656732464925
x15=−33.046284696792x_{15} = -33.046284696792
x16=67.5740526651947x_{16} = 67.5740526651947
x17=11.1810751482291x_{17} = 11.1810751482291
x18=−51.8744502729041x_{18} = -51.8744502729041
x19=−73.8543203258377x_{19} = -73.8543203258377
x20=51.8751921133361x_{20} = 51.8751921133361
x21=−17.3901768408315x_{21} = -17.3901768408315
x22=−8.08365076227732x_{22} = -8.08365076227732
x23=−98.9802708766234x_{23} = -98.9802708766234
x24=20.519983804286x_{24} = 20.519983804286
x25=−86.4168051847116x_{25} = -86.4168051847116
x26=−45.5964586594414x_{26} = -45.5964586594414
x27=86.4170728188749x_{27} = 86.4170728188749
x28=92.6986728032932x_{28} = 92.6986728032932
x29=29.9130423127956x_{29} = 29.9130423127956
x30=33.0481076541221x_{30} = 33.0481076541221
x31=−196.359700085069x_{31} = -196.359700085069
x32=−26.7767536014234x_{32} = -26.7767536014234
x33=−5.06026836518236x_{33} = -5.06026836518236
x34=−67.5736151455506x_{34} = -67.5736151455506
x35=5.12485035716986x_{35} = 5.12485035716986
x36=0x_{36} = 0
x37=−95.8393334645179x_{37} = -95.8393334645179
x38=48.7361295392441x_{38} = 48.7361295392441
x39=17.3966801527861x_{39} = 17.3966801527861
x40=42.4591369509051x_{40} = 42.4591369509051
x41=−29.910819411552x_{41} = -29.910819411552
x42=80.1357227352106x_{42} = 80.1357227352106
x43=8.1119826164815x_{43} = 8.1119826164815
x44=76.9951606189843x_{44} = 76.9951606189843
x45=−83.276074800594x_{45} = -83.276074800594
x46=64.4339234092994x_{46} = 64.4339234092994
x47=−80.1354115339569x_{47} = -80.1354115339569
x48=89.5578448748316x_{48} = 89.5578448748316
x49=−89.5575956718273x_{49} = -89.5575956718273
x50=−61.2934134764651x_{50} = -61.2934134764651
x51=70.7143126702811x_{51} = 70.7143126702811
x52=−2.22109820882798x_{52} = -2.22109820882798
x53=98.9804749142348x_{53} = 98.9804749142348
x54=14.2814705578331x_{54} = 14.2814705578331
x55=−14.2718933831386x_{55} = -14.2718933831386
x56=−36.1827931655659x_{56} = -36.1827931655659
x57=−23.6446256872296x_{57} = -23.6446256872296
x58=39.3213891805148x_{58} = 39.3213891805148
x59=−64.4334422583637x_{59} = -64.4334422583637
x60=−20.5152887094394x_{60} = -20.5152887094394
x61=−55.0138858041882x_{61} = -55.0138858041882
x62=−92.6984401916674x_{62} = -92.6984401916674
x63=36.1843149453611x_{63} = 36.1843149453611
x64=95.8395510876656x_{64} = 95.8395510876656
x65=55.0145455295839x_{65} = 55.0145455295839
x66=45.5974183148387x_{66} = 45.5974183148387

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
[98.9804749142348,∞)\left[98.9804749142348, \infty\right)
Convexa en los intervalos
(−∞,−196.359700085069]\left(-\infty, -196.359700085069\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((xsin⁡(x)+(2−x2)cos⁡(x)2)−−xcos⁡(x)+sin⁡(x)2)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(x \sin{\left(x \right)} + \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2}\right) - \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞((xsin⁡(x)+(2−x2)cos⁡(x)2)−−xcos⁡(x)+sin⁡(x)2)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(x \sin{\left(x \right)} + \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2}\right) - \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x*sin(x) + ((2 - x^2)*cos(x))/2 - (sin(x) - x*cos(x))/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→−∞((xsin⁡(x)+(2−x2)cos⁡(x)2)−−xcos⁡(x)+sin⁡(x)2x)y = x \lim_{x \to -\infty}\left(\frac{\left(x \sin{\left(x \right)} + \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2}\right) - \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((xsin⁡(x)+(2−x2)cos⁡(x)2)−−xcos⁡(x)+sin⁡(x)2x)y = x \lim_{x \to \infty}\left(\frac{\left(x \sin{\left(x \right)} + \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2}\right) - \frac{- x \cos{\left(x \right)} + \sin{\left(x \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:
(xsin⁡(x)+(2−x2)cos⁡(x)2)−−xcos⁡(x)+sin⁡(x)2=xsin⁡(x)−xcos⁡(x)2+(2−x2)cos⁡(x)2+sin⁡(x)2\left(x \sin{\left(x \right)} + \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2}\right) - \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2} = x \sin{\left(x \right)} - \frac{x \cos{\left(x \right)}}{2} + \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}
- No
(xsin⁡(x)+(2−x2)cos⁡(x)2)−−xcos⁡(x)+sin⁡(x)2=−xsin⁡(x)+xcos⁡(x)2−(2−x2)cos⁡(x)2−sin⁡(x)2\left(x \sin{\left(x \right)} + \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2}\right) - \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{2} = - x \sin{\left(x \right)} + \frac{x \cos{\left(x \right)}}{2} - \frac{\left(2 - x^{2}\right) \cos{\left(x \right)}}{2} - \frac{\sin{\left(x \right)}}{2}
- No
es decir, función
no es
par ni impar
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