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

Gráfico de la función y = (sin(pi*x)/(pi*x))^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                  2
       /sin(pi*x)\ 
f(x) = |---------| 
       \   pi*x  /
f(x)=(sin⁡(πx)πx)2f{\left(x \right)} = \left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} 
f = (sin(pi*x)/((pi*x)))^2
Gráfico de la función
02468-8-6-4-2-101002
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⁡(πx)πx)2=0\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1x_{1} = 1
Solución numérica
x1=28x_{1} = 28
x2=−22x_{2} = -22
x3=−54x_{3} = -54
x4=−32x_{4} = -32
x5=−38x_{5} = -38
x6=32x_{6} = 32
x7=82x_{7} = 82
x8=76x_{8} = 76
x9=−58x_{9} = -58
x10=−86x_{10} = -86
x11=−50x_{11} = -50
x12=86x_{12} = 86
x13=80x_{13} = 80
x14=−64x_{14} = -64
x15=−100x_{15} = -100
x16=36x_{16} = 36
x17=−12x_{17} = -12
x18=38x_{18} = 38
x19=−20x_{19} = -20
x20=−8x_{20} = -8
x21=−10x_{21} = -10
x22=−44x_{22} = -44
x23=66x_{23} = 66
x24=−62x_{24} = -62
x25=−46x_{25} = -46
x26=−48x_{26} = -48
x27=50x_{27} = 50
x28=−74x_{28} = -74
x29=4x_{29} = 4
x30=98x_{30} = 98
x31=−2x_{31} = -2
x32=−66x_{32} = -66
x33=2x_{33} = 2
x34=−28x_{34} = -28
x35=78x_{35} = 78
x36=−92x_{36} = -92
x37=20x_{37} = 20
x38=54x_{38} = 54
x39=40x_{39} = 40
x40=−40x_{40} = -40
x41=90x_{41} = 90
x42=74x_{42} = 74
x43=10x_{43} = 10
x44=−76x_{44} = -76
x45=60x_{45} = 60
x46=−18x_{46} = -18
x47=−98x_{47} = -98
x48=−36x_{48} = -36
x49=58x_{49} = 58
x50=−30x_{50} = -30
x51=34x_{51} = 34
x52=18x_{52} = 18
x53=−60x_{53} = -60
x54=70x_{54} = 70
x55=14x_{55} = 14
x56=30x_{56} = 30
x57=24x_{57} = 24
x58=64x_{58} = 64
x59=−84x_{59} = -84
x60=26x_{60} = 26
x61=84x_{61} = 84
x62=52x_{62} = 52
x63=56x_{63} = 56
x64=68x_{64} = 68
x65=44x_{65} = 44
x66=94x_{66} = 94
x67=96x_{67} = 96
x68=−26x_{68} = -26
x69=48x_{69} = 48
x70=−14x_{70} = -14
x71=−78x_{71} = -78
x72=6x_{72} = 6
x73=−90x_{73} = -90
x74=16x_{74} = 16
x75=−82x_{75} = -82
x76=−34x_{76} = -34
x77=92x_{77} = 92
x78=42x_{78} = 42
x79=−4x_{79} = -4
x80=−56x_{80} = -56
x81=72x_{81} = 72
x82=−72x_{82} = -72
x83=−52x_{83} = -52
x84=−16x_{84} = -16
x85=−42x_{85} = -42
x86=−6x_{86} = -6
x87=8x_{87} = 8
x88=−24x_{88} = -24
x89=−68x_{89} = -68
x90=88x_{90} = 88
x91=−94x_{91} = -94
x92=46x_{92} = 46
x93=−88x_{93} = -88
x94=22x_{94} = 22
x95=−96x_{95} = -96
x96=−70x_{96} = -70
x97=−80x_{97} = -80
x98=100x_{98} = 100
x99=12x_{99} = 12
x100=62x_{100} = 62
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(pi*x)/((pi*x)))^2.
(sin⁡(0π)0π)2\left(\frac{\sin{\left(0 \pi \right)}}{0 \pi}\right)^{2}
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
πxsin⁡2(πx)π2x2(2π1πxcos⁡(πx)−2sin⁡(πx)πx2)sin⁡(πx)=0\frac{\pi x \frac{\sin^{2}{\left(\pi x \right)}}{\pi^{2} x^{2}} \left(2 \pi \frac{1}{\pi x} \cos{\left(\pi x \right)} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right)}{\sin{\left(\pi x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=46x_{1} = 46
x2=−26x_{2} = -26
x3=58x_{3} = 58
x4=−20x_{4} = -20
x5=50x_{5} = 50
x6=80x_{6} = 80
x7=−100x_{7} = -100
x8=60x_{8} = 60
x9=26x_{9} = 26
x10=74x_{10} = 74
x11=72x_{11} = 72
x12=−52x_{12} = -52
x13=44x_{13} = 44
x14=−98x_{14} = -98
x15=−46x_{15} = -46
x16=−62x_{16} = -62
x17=−94x_{17} = -94
x18=14x_{18} = 14
x19=−2x_{19} = -2
x20=34x_{20} = 34
x21=−30x_{21} = -30
x22=86x_{22} = 86
x23=−88x_{23} = -88
x24=36x_{24} = 36
x25=−76x_{25} = -76
x26=−44x_{26} = -44
x27=−70x_{27} = -70
x28=−78x_{28} = -78
x29=78x_{29} = 78
x30=48x_{30} = 48
x31=−38x_{31} = -38
x32=−80x_{32} = -80
x33=−92x_{33} = -92
x34=30x_{34} = 30
x35=16x_{35} = 16
x36=68x_{36} = 68
x37=90x_{37} = 90
x38=38x_{38} = 38
x39=−56x_{39} = -56
x40=−72x_{40} = -72
x41=32x_{41} = 32
x42=70x_{42} = 70
x43=−58x_{43} = -58
x44=64x_{44} = 64
x45=−60x_{45} = -60
x46=−68x_{46} = -68
x47=8x_{47} = 8
x48=24x_{48} = 24
x49=62x_{49} = 62
x50=−24x_{50} = -24
x51=−6x_{51} = -6
x52=−4x_{52} = -4
x53=−54x_{53} = -54
x54=18x_{54} = 18
x55=52x_{55} = 52
x56=20x_{56} = 20
x57=96x_{57} = 96
x58=−64x_{58} = -64
x59=10x_{59} = 10
x60=54x_{60} = 54
x61=−84x_{61} = -84
x62=−48x_{62} = -48
x63=−8x_{63} = -8
x64=98x_{64} = 98
x65=4x_{65} = 4
x66=82x_{66} = 82
x67=−10x_{67} = -10
x68=−12x_{68} = -12
x69=56x_{69} = 56
x70=−66x_{70} = -66
x71=−90x_{71} = -90
x72=28x_{72} = 28
x73=−22x_{73} = -22
x74=−28x_{74} = -28
x75=−40x_{75} = -40
x76=40x_{76} = 40
x77=88x_{77} = 88
x78=94x_{78} = 94
x79=2x_{79} = 2
x80=22x_{80} = 22
x81=84x_{81} = 84
x82=−16x_{82} = -16
x83=−32x_{83} = -32
x84=−86x_{84} = -86
x85=−82x_{85} = -82
x86=92x_{86} = 92
x87=−36x_{87} = -36
x88=−74x_{88} = -74
x89=76x_{89} = 76
x90=−18x_{90} = -18
x91=6x_{91} = 6
x92=−34x_{92} = -34
x93=−14x_{93} = -14
x94=−96x_{94} = -96
x95=42x_{95} = 42
x96=66x_{96} = 66
x97=−42x_{97} = -42
x98=12x_{98} = 12
x99=100x_{99} = 100
x100=−50x_{100} = -50
Signos de extremos en los puntos:
(46, 0)

(-26, 0)

(58, 0)

(-20, 0)

(50, 0)

(80, 0)

(-100, 0)

(60, 0)

(26, 0)

(74, 0)

(72, 0)

(-52, 0)

(44, 0)

(-98, 0)

(-46, 0)

(-62, 0)

(-94, 0)

(14, 0)

(-2, 0)

(34, 0)

(-30, 0)

(86, 0)

(-88, 0)

(36, 0)

(-76, 0)

(-44, 0)

(-70, 0)

(-78, 0)

(78, 0)

(48, 0)

(-38, 0)

(-80, 0)

(-92, 0)

(30, 0)

(16, 0)

(68, 0)

(90, 0)

(38, 0)

(-56, 0)

(-72, 0)

(32, 0)

(70, 0)

(-58, 0)

(64, 0)

(-60, 0)

(-68, 0)

(8, 0)

(24, 0)

(62, 0)

(-24, 0)

(-6, 0)

(-4, 0)

(-54, 0)

(18, 0)

(52, 0)

(20, 0)

(96, 0)

(-64, 0)

(10, 0)

(54, 0)

(-84, 0)

(-48, 0)

(-8, 0)

(98, 0)

(4, 0)

(82, 0)

(-10, 0)

(-12, 0)

(56, 0)

(-66, 0)

(-90, 0)

(28, 0)

(-22, 0)

(-28, 0)

(-40, 0)

(40, 0)

(88, 0)

(94, 0)

(2, 0)

(22, 0)

(84, 0)

(-16, 0)

(-32, 0)

(-86, 0)

(-82, 0)

(92, 0)

(-36, 0)

(-74, 0)

(76, 0)

(-18, 0)

(6, 0)

(-34, 0)

(-14, 0)

(-96, 0)

(42, 0)

(66, 0)

(-42, 0)

(12, 0)

(100, 0)

(-50, 0)


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=46x_{1} = 46
x2=−26x_{2} = -26
x3=58x_{3} = 58
x4=−20x_{4} = -20
x5=50x_{5} = 50
x6=80x_{6} = 80
x7=−100x_{7} = -100
x8=60x_{8} = 60
x9=26x_{9} = 26
x10=74x_{10} = 74
x11=72x_{11} = 72
x12=−52x_{12} = -52
x13=44x_{13} = 44
x14=−98x_{14} = -98
x15=−46x_{15} = -46
x16=−62x_{16} = -62
x17=−94x_{17} = -94
x18=14x_{18} = 14
x19=−2x_{19} = -2
x20=34x_{20} = 34
x21=−30x_{21} = -30
x22=86x_{22} = 86
x23=−88x_{23} = -88
x24=36x_{24} = 36
x25=−76x_{25} = -76
x26=−44x_{26} = -44
x27=−70x_{27} = -70
x28=−78x_{28} = -78
x29=78x_{29} = 78
x30=48x_{30} = 48
x31=−38x_{31} = -38
x32=−80x_{32} = -80
x33=−92x_{33} = -92
x34=30x_{34} = 30
x35=16x_{35} = 16
x36=68x_{36} = 68
x37=90x_{37} = 90
x38=38x_{38} = 38
x39=−56x_{39} = -56
x40=−72x_{40} = -72
x41=32x_{41} = 32
x42=70x_{42} = 70
x43=−58x_{43} = -58
x44=64x_{44} = 64
x45=−60x_{45} = -60
x46=−68x_{46} = -68
x47=8x_{47} = 8
x48=24x_{48} = 24
x49=62x_{49} = 62
x50=−24x_{50} = -24
x51=−6x_{51} = -6
x52=−4x_{52} = -4
x53=−54x_{53} = -54
x54=18x_{54} = 18
x55=52x_{55} = 52
x56=20x_{56} = 20
x57=96x_{57} = 96
x58=−64x_{58} = -64
x59=10x_{59} = 10
x60=54x_{60} = 54
x61=−84x_{61} = -84
x62=−48x_{62} = -48
x63=−8x_{63} = -8
x64=98x_{64} = 98
x65=4x_{65} = 4
x66=82x_{66} = 82
x67=−10x_{67} = -10
x68=−12x_{68} = -12
x69=56x_{69} = 56
x70=−66x_{70} = -66
x71=−90x_{71} = -90
x72=28x_{72} = 28
x73=−22x_{73} = -22
x74=−28x_{74} = -28
x75=−40x_{75} = -40
x76=40x_{76} = 40
x77=88x_{77} = 88
x78=94x_{78} = 94
x79=2x_{79} = 2
x80=22x_{80} = 22
x81=84x_{81} = 84
x82=−16x_{82} = -16
x83=−32x_{83} = -32
x84=−86x_{84} = -86
x85=−82x_{85} = -82
x86=92x_{86} = 92
x87=−36x_{87} = -36
x88=−74x_{88} = -74
x89=76x_{89} = 76
x90=−18x_{90} = -18
x91=6x_{91} = 6
x92=−34x_{92} = -34
x93=−14x_{93} = -14
x94=−96x_{94} = -96
x95=42x_{95} = 42
x96=66x_{96} = 66
x97=−42x_{97} = -42
x98=12x_{98} = 12
x99=100x_{99} = 100
x100=−50x_{100} = -50
La función no tiene puntos máximos
Decrece en los intervalos
[100,∞)\left[100, \infty\right)
Crece en los intervalos
(−∞,−100]\left(-\infty, -100\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(2(cos⁡(πx)−sin⁡(πx)πx)2−(cos⁡(πx)−sin⁡(πx)πx)cos⁡(πx)−(πsin⁡(πx)+2cos⁡(πx)x−2sin⁡(πx)πx2)sin⁡(πx)π+(cos⁡(πx)−sin⁡(πx)πx)sin⁡(πx)πx)x2=0\frac{2 \left(2 \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} - \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \cos{\left(\pi x \right)} - \frac{\left(\pi \sin{\left(\pi x \right)} + \frac{2 \cos{\left(\pi x \right)}}{x} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right) \sin{\left(\pi x \right)}}{\pi} + \frac{\left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \sin{\left(\pi x \right)}}{\pi x}\right)}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−61.7483527761947x_{1} = -61.7483527761947
x2=−81.7487569556855x_{2} = -81.7487569556855
x3=−83.7487867257144x_{3} = -83.7487867257144
x4=−99.7489818056511x_{4} = -99.7489818056511
x5=−7.73649601515403x_{5} = -7.73649601515403
x6=−77.7486928066096x_{6} = -77.7486928066096
x7=−95.7489391632373x_{7} = -95.7489391632373
x8=90.2488802811355x_{8} = 90.2488802811355
x9=−87.748842183449x_{9} = -87.748842183449
x10=−69.7485423590771x_{10} = -69.7485423590771
x11=100.248991709865x_{11} = 100.248991709865
x12=52.2480696155498x_{12} = 52.2480696155498
x13=−89.7488680535879x_{13} = -89.7488680535879
x14=−29.7465664532108x_{14} = -29.7465664532108
x15=80.2487411636708x_{15} = 80.2487411636708
x16=74.2486397633915x_{16} = 74.2486397633915
x17=4.22735950767312x_{17} = 4.22735950767312
x18=70.2485625726606x_{18} = 70.2485625726606
x19=18.2445188624812x_{19} = 18.2445188624812
x20=−57.7482382050547x_{20} = -57.7482382050547
x21=2.20888752467459x_{21} = 2.20888752467459
x22=24.2458621171043x_{22} = 24.2458621171043
x23=−9.7393399929161x_{23} = -9.7393399929161
x24=36.2472230924613x_{24} = 36.2472230924613
x25=−79.7487256879633x_{25} = -79.7487256879633
x26=98.2489712343711x_{26} = 98.2489712343711
x27=−31.7467844060814x_{27} = -31.7467844060814
x28=30.2466765477645x_{28} = 30.2466765477645
x29=−75.7486581834026x_{29} = -75.7486581834026
x30=−17.744212784795x_{30} = -17.744212784795
x31=96.2489499100364x_{31} = 96.2489499100364
x32=−67.7484991757802x_{32} = -67.7484991757802
x33=−15.7434662299526x_{33} = -15.7434662299526
x34=28.2464431968423x_{34} = 28.2464431968423
x35=−41.7475591015791x_{35} = -41.7475591015791
x36=40.2474974528301x_{36} = 40.2474974528301
x37=12.2418838185292x_{37} = 12.2418838185292
x38=72.2486022329285x_{38} = 72.2486022329285
x39=10.2403346426075x_{39} = 10.2403346426075
x40=22.2454940358069x_{40} = 22.2454940358069
x41=−37.7472987956897x_{41} = -37.7472987956897
x42=−63.7484046496571x_{42} = -63.7484046496571
x43=−73.7486216761659x_{43} = -73.7486216761659
x44=58.2482676493225x_{44} = 58.2482676493225
x45=92.2489044945726x_{45} = 92.2489044945726
x46=−55.7481747273539x_{46} = -55.7481747273539
x47=−25.746028000833x_{47} = -25.746028000833
x48=−27.746316803873x_{48} = -27.746316803873
x49=32.2468811632873x_{49} = 32.2468811632873
x50=42.2476152591584x_{50} = 42.2476152591584
x51=−13.7424984608719x_{51} = -13.7424984608719
x52=26.2461745995811x_{52} = 26.2461745995811
x53=44.2477224724436x_{53} = 44.2477224724436
x54=−23.7456900482187x_{54} = -23.7456900482187
x55=−19.7448062051182x_{55} = -19.7448062051182
x56=6.23436247448923x_{56} = 6.23436247448923
x57=−21.7452892305997x_{57} = -21.7452892305997
x58=−65.7484533556792x_{58} = -65.7484533556792
x59=−33.7469763389274x_{59} = -33.7469763389274
x60=14.2430048647492x_{60} = 14.2430048647492
x61=64.2484288313613x_{61} = 64.2484288313613
x62=−45.7477736462445x_{62} = -45.7477736462445
x63=38.2473674018583x_{63} = 38.2473674018583
x64=50.2479931440213x_{64} = 50.2479931440213
x65=−49.7479535220692x_{65} = -49.7479535220692
x66=−85.748815103141x_{66} = -85.748815103141
x67=88.2488549731412x_{67} = 88.2488549731412
x68=94.2489276829586x_{68} = 94.2489276829586
x69=−91.7488927929056x_{69} = -91.7488927929056
x70=84.2488007625499x_{70} = 84.2488007625499
x71=−51.7480329833019x_{71} = -51.7480329833019
x72=20.2450540655959x_{72} = 20.2450540655959
x73=8.23805394372117x_{73} = 8.23805394372117
x74=−59.7482974159256x_{74} = -59.7482974159256
x75=48.2479103636241x_{75} = 48.2479103636241
x76=54.2481404729683x_{76} = 54.2481404729683
x77=−11.7411939995098x_{77} = -11.7411939995098
x78=66.2484760938889x_{78} = 66.2484760938889
x79=−47.7478673704273x_{79} = -47.7478673704273
x80=60.248324929728x_{80} = 60.248324929728
x81=82.2487716856391x_{81} = 82.2487716856391
x82=−39.747435537556x_{82} = -39.747435537556
x83=78.2487090861744x_{83} = 78.2487090861744
x84=−71.7485831268165x_{84} = -71.7485831268165
x85=46.2478204600269x_{85} = 46.2478204600269
x86=16.2438537460501x_{86} = 16.2438537460501
x87=−5.73157811771419x_{87} = -5.73157811771419
x88=62.2483785433487x_{88} = 62.2483785433487
x89=56.2482063126328x_{89} = 56.2482063126328
x90=−93.7489164739615x_{90} = -93.7489164739615
x91=68.2485205959957x_{91} = 68.2485205959957
x92=−97.7489609217602x_{92} = -97.7489609217602
x93=−43.747671305176x_{93} = -43.747671305176
x94=76.2486753311251x_{94} = 76.2486753311251
x95=−35.7471466490446x_{95} = -35.7471466490446
x96=34.2470620436327x_{96} = 34.2470620436327
x97=−53.7481065043619x_{97} = -53.7481065043619
x98=−1.68100482418311x_{98} = -1.68100482418311
x99=−3.72099743965763x_{99} = -3.72099743965763
x100=86.2488284946546x_{100} = 86.2488284946546
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−(2(2(cos⁡(πx)−sin⁡(πx)πx)2−(cos⁡(πx)−sin⁡(πx)πx)cos⁡(πx)−(πsin⁡(πx)+2cos⁡(πx)x−2sin⁡(πx)πx2)sin⁡(πx)π+(cos⁡(πx)−sin⁡(πx)πx)sin⁡(πx)πx)x2)=−2π23\lim_{x \to 0^-}\left(\frac{2 \left(2 \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} - \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \cos{\left(\pi x \right)} - \frac{\left(\pi \sin{\left(\pi x \right)} + \frac{2 \cos{\left(\pi x \right)}}{x} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right) \sin{\left(\pi x \right)}}{\pi} + \frac{\left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \sin{\left(\pi x \right)}}{\pi x}\right)}{x^{2}}\right) = - \frac{2 \pi^{2}}{3}
lim⁡x→0+(2(2(cos⁡(πx)−sin⁡(πx)πx)2−(cos⁡(πx)−sin⁡(πx)πx)cos⁡(πx)−(πsin⁡(πx)+2cos⁡(πx)x−2sin⁡(πx)πx2)sin⁡(πx)π+(cos⁡(πx)−sin⁡(πx)πx)sin⁡(πx)πx)x2)=−2π23\lim_{x \to 0^+}\left(\frac{2 \left(2 \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} - \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \cos{\left(\pi x \right)} - \frac{\left(\pi \sin{\left(\pi x \right)} + \frac{2 \cos{\left(\pi x \right)}}{x} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right) \sin{\left(\pi x \right)}}{\pi} + \frac{\left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \sin{\left(\pi x \right)}}{\pi x}\right)}{x^{2}}\right) = - \frac{2 \pi^{2}}{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
(−∞,−99.7489818056511]\left(-\infty, -99.7489818056511\right]
Convexa en los intervalos
[100.248991709865,∞)\left[100.248991709865, \infty\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⁡(πx)πx)2=0\lim_{x \to -\infty} \left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(πx)πx)2=0\lim_{x \to \infty} \left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = 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(pi*x)/((pi*x)))^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(1π21x2sin⁡2(πx)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{\pi^{2}} \frac{1}{x^{2}} \sin^{2}{\left(\pi x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(1π21x2sin⁡2(πx)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{\pi^{2}} \frac{1}{x^{2}} \sin^{2}{\left(\pi x \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:
(sin⁡(πx)πx)2=sin⁡2(πx)π2x2\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = \frac{\sin^{2}{\left(\pi x \right)}}{\pi^{2} x^{2}}
- No
(sin⁡(πx)πx)2=−sin⁡2(πx)π2x2\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = - \frac{\sin^{2}{\left(\pi x \right)}}{\pi^{2} x^{2}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор