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sin(x/2)/x
Trazado el gráfico de la función/ sin(x/2)/x

Gráfico de la función y = sin(x/2)/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /x\
       sin|-|
          \2/
f(x) = ------
         x
f(x)=sin⁡(x2)xf{\left(x \right)} = \frac{\sin{\left(\frac{x}{2} \right)}}{x} 
f = sin(x/2)/x
Gráfico de la función
02468-8-6-4-2-10101.0-0.5
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:
sin⁡(x2)x=0\frac{\sin{\left(\frac{x}{2} \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=2πx_{1} = 2 \pi
Solución numérica
x1=−62.8318530717959x_{1} = -62.8318530717959
x2=87.9645943005142x_{2} = 87.9645943005142
x3=31.4159265358979x_{3} = 31.4159265358979
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=69.1150383789755x_{5} = 69.1150383789755
x6=−37.6991118430775x_{6} = -37.6991118430775
x7=−81.6814089933346x_{7} = -81.6814089933346
x8=−12.5663706143592x_{8} = -12.5663706143592
x9=12.5663706143592x_{9} = 12.5663706143592
x10=−87.9645943005142x_{10} = -87.9645943005142
x11=−100.530964914873x_{11} = -100.530964914873
x12=−471.238898038469x_{12} = -471.238898038469
x13=−94.2477796076938x_{13} = -94.2477796076938
x14=6.28318530717959x_{14} = 6.28318530717959
x15=−69.1150383789755x_{15} = -69.1150383789755
x16=−25.1327412287183x_{16} = -25.1327412287183
x17=−50.2654824574367x_{17} = -50.2654824574367
x18=−18.8495559215388x_{18} = -18.8495559215388
x19=18.8495559215388x_{19} = 18.8495559215388
x20=−420.973415581032x_{20} = -420.973415581032
x21=37.6991118430775x_{21} = 37.6991118430775
x22=−43.9822971502571x_{22} = -43.9822971502571
x23=−6.28318530717959x_{23} = -6.28318530717959
x24=43.9822971502571x_{24} = 43.9822971502571
x25=−169.646003293849x_{25} = -169.646003293849
x26=56.5486677646163x_{26} = 56.5486677646163
x27=25.1327412287183x_{27} = 25.1327412287183
x28=75.398223686155x_{28} = 75.398223686155
x29=125.663706143592x_{29} = 125.663706143592
x30=81.6814089933346x_{30} = 81.6814089933346
x31=100.530964914873x_{31} = 100.530964914873
x32=−75.398223686155x_{32} = -75.398223686155
x33=−31.4159265358979x_{33} = -31.4159265358979
x34=3769.91118430775x_{34} = 3769.91118430775
x35=62.8318530717959x_{35} = 62.8318530717959
x36=50.2654824574367x_{36} = 50.2654824574367
x37=94.2477796076938x_{37} = 94.2477796076938
x38=−119.380520836412x_{38} = -119.380520836412
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 sin(x/2)/x.
sin⁡(02)0\frac{\sin{\left(\frac{0}{2} \right)}}{0}
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⁡(x2)2x−sin⁡(x2)x2=0\frac{\cos{\left(\frac{x}{2} \right)}}{2 x} - \frac{\sin{\left(\frac{x}{2} \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−97.3482884639088x_{1} = -97.3482884639088
x2=−72.2012444887512x_{2} = -72.2012444887512
x3=84.7758271362638x_{3} = 84.7758271362638
x4=−47.038904997378x_{4} = -47.038904997378
x5=1152.96103454293x_{5} = 1152.96103454293
x6=−191.616277573723x_{6} = -191.616277573723
x7=392.688895606224x_{7} = 392.688895606224
x8=235.602471676449x_{8} = 235.602471676449
x9=−179.048441860834x_{9} = -179.048441860834
x10=34.4415105438615x_{10} = 34.4415105438615
x11=21.8082433188578x_{11} = 21.8082433188578
x12=72.2012444887512x_{12} = 72.2012444887512
x13=−40.7426059185751x_{13} = -40.7426059185751
x14=109.919356575778x_{14} = 109.919356575778
x15=−78.4888647223284x_{15} = -78.4888647223284
x16=103.633964974559x_{16} = 103.633964974559
x17=398.972241329727x_{17} = 398.972241329727
x18=91.0622680279826x_{18} = 91.0622680279826
x19=−21.8082433188578x_{19} = -21.8082433188578
x20=−34.4415105438615x_{20} = -34.4415105438615
x21=−15.4505036738754x_{21} = -15.4505036738754
x22=2.10405653221875⋅10−16x_{22} = 2.10405653221875 \cdot 10^{-16}
x23=675.436498442796x_{23} = 675.436498442796
x24=210.467703071275x_{24} = 210.467703071275
x25=−26945.4400413915x_{25} = -26945.4400413915
x26=40.7426059185751x_{26} = 40.7426059185751
x27=15.4505036738754x_{27} = 15.4505036738754
x28=−8.98681891581813x_{28} = -8.98681891581813
x29=−2786.59124828885x_{29} = -2786.59124828885
x30=8.98681891581813x_{30} = 8.98681891581813
x31=−59.6231975817859x_{31} = -59.6231975817859
x32=59.6231975817859x_{32} = 59.6231975817859
x33=−28.1323878256629x_{33} = -28.1323878256629
x34=97.3482884639088x_{34} = 97.3482884639088
x35=28.1323878256629x_{35} = 28.1323878256629
x36=47.038904997378x_{36} = 47.038904997378
x37=−84.7758271362638x_{37} = -84.7758271362638
x38=−53.3321085176254x_{38} = -53.3321085176254
x39=53.3321085176254x_{39} = 53.3321085176254
x40=2.35332600942599⋅10−18x_{40} = 2.35332600942599 \cdot 10^{-18}
x41=−91.0622680279826x_{41} = -91.0622680279826
x42=65.912778079645x_{42} = 65.912778079645
x43=78.4888647223284x_{43} = 78.4888647223284
x44=−65.912778079645x_{44} = -65.912778079645
Signos de extremos en los puntos:
(-97.34828846390877, -0.0102702270208769)

(-72.20124448875121, -0.0138448661505746)

(84.77582713626384, -0.0117925341145082)

(-47.03890499737801, -0.0212398084888063)

(1152.961034542934, -0.000867330695065707)

(-191.6162775737234, 0.00521847906728291)

(392.688895606224, 0.00254651211596963)

(235.60247167644877, -0.00424428472388261)

(-179.04844186083437, 0.00558473231708678)

(34.44151054386154, -0.0289859011730769)

(21.808243318857798, -0.0456626014115288)

(72.20124448875121, -0.0138448661505746)

(-40.74260591857512, 0.0245148120070371)

(109.91935657577787, -0.00909607316090157)

(-78.48886472232839, 0.0127365265464404)

(103.63396497455933, 0.00964754974379398)

(398.97224132972667, -0.00250640854717266)

(91.06226802798255, 0.0109788491142412)

(-21.808243318857798, -0.0456626014115288)

(-34.44151054386154, -0.0289859011730769)

(-15.450503673875414, 0.0641872767629496)

(2.1040565322187507e-16, 0.5)

(675.436498442796, -0.00148051758889429)

(210.4677030712752, -0.00475110830939177)

(-26945.440041391535, 3.71120306704686e-5)

(40.74260591857512, 0.0245148120070371)

(15.450503673875414, 0.0641872767629496)

(-8.986818915818128, -0.108616814105611)

(-2786.5912482888457, -0.000358861294440359)

(8.986818915818128, -0.108616814105611)

(-59.62319758178592, -0.0167625675106994)

(59.62319758178592, -0.0167625675106994)

(-28.132387825662946, 0.0354567297252311)

(97.34828846390877, -0.0102702270208769)

(28.132387825662946, 0.0354567297252311)

(47.03890499737801, -0.0212398084888063)

(-84.77582713626384, -0.0117925341145082)

(-53.33210851762535, 0.0187372599969656)

(53.33210851762535, 0.0187372599969656)

(2.3533260094259877e-18, 0.5)

(-91.06226802798255, 0.0109788491142412)

(65.91277807964495, 0.0151645855931551)

(78.48886472232839, 0.0127365265464404)

(-65.91277807964495, 0.0151645855931551)


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=−97.3482884639088x_{1} = -97.3482884639088
x2=−72.2012444887512x_{2} = -72.2012444887512
x3=84.7758271362638x_{3} = 84.7758271362638
x4=−47.038904997378x_{4} = -47.038904997378
x5=1152.96103454293x_{5} = 1152.96103454293
x6=235.602471676449x_{6} = 235.602471676449
x7=34.4415105438615x_{7} = 34.4415105438615
x8=21.8082433188578x_{8} = 21.8082433188578
x9=72.2012444887512x_{9} = 72.2012444887512
x10=109.919356575778x_{10} = 109.919356575778
x11=398.972241329727x_{11} = 398.972241329727
x12=−21.8082433188578x_{12} = -21.8082433188578
x13=−34.4415105438615x_{13} = -34.4415105438615
x14=675.436498442796x_{14} = 675.436498442796
x15=210.467703071275x_{15} = 210.467703071275
x16=−8.98681891581813x_{16} = -8.98681891581813
x17=−2786.59124828885x_{17} = -2786.59124828885
x18=8.98681891581813x_{18} = 8.98681891581813
x19=−59.6231975817859x_{19} = -59.6231975817859
x20=59.6231975817859x_{20} = 59.6231975817859
x21=97.3482884639088x_{21} = 97.3482884639088
x22=47.038904997378x_{22} = 47.038904997378
x23=−84.7758271362638x_{23} = -84.7758271362638
Puntos máximos de la función:
x23=−191.616277573723x_{23} = -191.616277573723
x23=392.688895606224x_{23} = 392.688895606224
x23=−179.048441860834x_{23} = -179.048441860834
x23=−40.7426059185751x_{23} = -40.7426059185751
x23=−78.4888647223284x_{23} = -78.4888647223284
x23=103.633964974559x_{23} = 103.633964974559
x23=91.0622680279826x_{23} = 91.0622680279826
x23=−15.4505036738754x_{23} = -15.4505036738754
x23=2.10405653221875⋅10−16x_{23} = 2.10405653221875 \cdot 10^{-16}
x23=−26945.4400413915x_{23} = -26945.4400413915
x23=40.7426059185751x_{23} = 40.7426059185751
x23=15.4505036738754x_{23} = 15.4505036738754
x23=−28.1323878256629x_{23} = -28.1323878256629
x23=28.1323878256629x_{23} = 28.1323878256629
x23=−53.3321085176254x_{23} = -53.3321085176254
x23=53.3321085176254x_{23} = 53.3321085176254
x23=2.35332600942599⋅10−18x_{23} = 2.35332600942599 \cdot 10^{-18}
x23=−91.0622680279826x_{23} = -91.0622680279826
x23=65.912778079645x_{23} = 65.912778079645
x23=78.4888647223284x_{23} = 78.4888647223284
x23=−65.912778079645x_{23} = -65.912778079645
Decrece en los intervalos
[1152.96103454293,∞)\left[1152.96103454293, \infty\right)
Crece en los intervalos
(−∞,−2786.59124828885]\left(-\infty, -2786.59124828885\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⁡(x2)4−cos⁡(x2)x+2sin⁡(x2)x2x=0\frac{- \frac{\sin{\left(\frac{x}{2} \right)}}{4} - \frac{\cos{\left(\frac{x}{2} \right)}}{x} + \frac{2 \sin{\left(\frac{x}{2} \right)}}{x^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=75.2919206461728x_{1} = 75.2919206461728
x2=50.1056505619859x_{2} = 50.1056505619859
x3=18.4116802858733x_{3} = 18.4116802858733
x4=56.4067220079047x_{4} = 56.4067220079047
x5=62.704183453129x_{5} = 62.704183453129
x6=−1017.86816017881x_{6} = -1017.86816017881
x7=−43.7993929589856x_{7} = -43.7993929589856
x8=−119.313458007056x_{8} = -119.313458007056
x9=37.4852911695495x_{9} = 37.4852911695495
x10=4.1631519556362x_{10} = 4.1631519556362
x11=−100.451303298366x_{11} = -100.451303298366
x12=−18.4116802858733x_{12} = -18.4116802858733
x13=68.9990298427339x_{13} = 68.9990298427339
x14=−2808.58098389481x_{14} = -2808.58098389481
x15=11.8807399811454x_{15} = 11.8807399811454
x16=−24.8088900438039x_{16} = -24.8088900438039
x17=923.619578553596x_{17} = 923.619578553596
x18=−31.1584728207744x_{18} = -31.1584728207744
x19=−62.704183453129x_{19} = -62.704183453129
x20=−87.8735229428396x_{20} = -87.8735229428396
x21=31.1584728207744x_{21} = 31.1584728207744
x22=81.5833104625438x_{22} = 81.5833104625438
x23=−11.8807399811454x_{23} = -11.8807399811454
x24=94.1627948243084x_{24} = 94.1627948243084
x25=−81.5833104625438x_{25} = -81.5833104625438
x26=−37.4852911695495x_{26} = -37.4852911695495
x27=87.8735229428396x_{27} = 87.8735229428396
x28=−138.17216989329x_{28} = -138.17216989329
x29=−56.4067220079047x_{29} = -56.4067220079047
x30=138.17216989329x_{30} = 138.17216989329
x31=43.7993929589856x_{31} = 43.7993929589856
x32=−68.9990298427339x_{32} = -68.9990298427339
x33=−94.1627948243084x_{33} = -94.1627948243084
x34=−75.2919206461728x_{34} = -75.2919206461728
x35=24.8088900438039x_{35} = 24.8088900438039
x36=100.451303298366x_{36} = 100.451303298366
x37=−50.1056505619859x_{37} = -50.1056505619859
x38=−4.1631519556362x_{38} = -4.1631519556362
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⁡(x2)4−cos⁡(x2)x+2sin⁡(x2)x2x)=−124\lim_{x \to 0^-}\left(\frac{- \frac{\sin{\left(\frac{x}{2} \right)}}{4} - \frac{\cos{\left(\frac{x}{2} \right)}}{x} + \frac{2 \sin{\left(\frac{x}{2} \right)}}{x^{2}}}{x}\right) = - \frac{1}{24}
lim⁡x→0+(−sin⁡(x2)4−cos⁡(x2)x+2sin⁡(x2)x2x)=−124\lim_{x \to 0^+}\left(\frac{- \frac{\sin{\left(\frac{x}{2} \right)}}{4} - \frac{\cos{\left(\frac{x}{2} \right)}}{x} + \frac{2 \sin{\left(\frac{x}{2} \right)}}{x^{2}}}{x}\right) = - \frac{1}{24}
- 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
[923.619578553596,∞)\left[923.619578553596, \infty\right)
Convexa en los intervalos
(−∞,−1017.86816017881]\left(-\infty, -1017.86816017881\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→−∞(sin⁡(x2)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{x}{2} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(x2)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{x}{2} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x/2)/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x2)x2)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{x}{2} \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x2)x2)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{x}{2} \right)}}{x^{2}}\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:
sin⁡(x2)x=sin⁡(x2)x\frac{\sin{\left(\frac{x}{2} \right)}}{x} = \frac{\sin{\left(\frac{x}{2} \right)}}{x}
- No
sin⁡(x2)x=−sin⁡(x2)x\frac{\sin{\left(\frac{x}{2} \right)}}{x} = - \frac{\sin{\left(\frac{x}{2} \right)}}{x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sin(x/2)/x
    © Sr Examen – Калькулятор