Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
(x-3)^2
-
x^2+x+1
-
x^3-3*x^2+4
-
(x^2+4)/x
-
Expresiones idénticas
- (sin(pi*x)/(pi*x))^ dos
- ( seno de ( número pi multiplicar por x) dividir por ( número pi multiplicar por x)) al cuadrado
- ( seno de ( número pi multiplicar por x) dividir por ( número pi multiplicar por x)) en el grado dos
- (sin(pi*x)/(pi*x))2
- sinpi*x/pi*x2
- (sin(pi*x)/(pi*x))²
- (sin(pi*x)/(pi*x)) en el grado 2
- (sin(pix)/(pix))^2
- (sin(pix)/(pix))2
- sinpix/pix2
- sinpix/pix^2
- (sin(pi*x) dividir por (pi*x))^2
-
Expresiones con funciones
- Seno sin
- sin(x)+x
- sin(x)/sin(x+pi/4)
- sin(x)+cos(x)^2
- sin(x)*cos(x)^(2)
- sin(x)+9
- Número Pi pi
- pi/2+(4*cos(x)+4*cos(3*x))/(9*pi)
- pi*(1-x)*tg(pi*x/2)
- pi+x
- pi-arctg((x*10^(-5))/3)
- pi*70*cos(500*pi*x)
- Número Pi pi
- pi/2+(4*cos(x)+4*cos(3*x))/(9*pi)
- pi*(1-x)*tg(pi*x/2)
- pi+x
- pi-arctg((x*10^(-5))/3)
- pi*70*cos(500*pi*x)
Gráfico de la función y = (sin(pi*x)/(pi*x))^2
Solución
Ha introducido
[src]
2
/sin(pi*x)\
f(x) = |---------|
\ pi*x /
$$f{\left(x \right)} = \left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2}$$
f = (sin(pi*x)/((pi*x)))^2
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
Solución numérica
$$x_{1} = 28$$
$$x_{2} = -22$$
$$x_{3} = -54$$
$$x_{4} = -32$$
$$x_{5} = -38$$
$$x_{6} = 32$$
$$x_{7} = 82$$
$$x_{8} = 76$$
$$x_{9} = -58$$
$$x_{10} = -86$$
$$x_{11} = -50$$
$$x_{12} = 86$$
$$x_{13} = 80$$
$$x_{14} = -64$$
$$x_{15} = -100$$
$$x_{16} = 36$$
$$x_{17} = -12$$
$$x_{18} = 38$$
$$x_{19} = -20$$
$$x_{20} = -8$$
$$x_{21} = -10$$
$$x_{22} = -44$$
$$x_{23} = 66$$
$$x_{24} = -62$$
$$x_{25} = -46$$
$$x_{26} = -48$$
$$x_{27} = 50$$
$$x_{28} = -74$$
$$x_{29} = 4$$
$$x_{30} = 98$$
$$x_{31} = -2$$
$$x_{32} = -66$$
$$x_{33} = 2$$
$$x_{34} = -28$$
$$x_{35} = 78$$
$$x_{36} = -92$$
$$x_{37} = 20$$
$$x_{38} = 54$$
$$x_{39} = 40$$
$$x_{40} = -40$$
$$x_{41} = 90$$
$$x_{42} = 74$$
$$x_{43} = 10$$
$$x_{44} = -76$$
$$x_{45} = 60$$
$$x_{46} = -18$$
$$x_{47} = -98$$
$$x_{48} = -36$$
$$x_{49} = 58$$
$$x_{50} = -30$$
$$x_{51} = 34$$
$$x_{52} = 18$$
$$x_{53} = -60$$
$$x_{54} = 70$$
$$x_{55} = 14$$
$$x_{56} = 30$$
$$x_{57} = 24$$
$$x_{58} = 64$$
$$x_{59} = -84$$
$$x_{60} = 26$$
$$x_{61} = 84$$
$$x_{62} = 52$$
$$x_{63} = 56$$
$$x_{64} = 68$$
$$x_{65} = 44$$
$$x_{66} = 94$$
$$x_{67} = 96$$
$$x_{68} = -26$$
$$x_{69} = 48$$
$$x_{70} = -14$$
$$x_{71} = -78$$
$$x_{72} = 6$$
$$x_{73} = -90$$
$$x_{74} = 16$$
$$x_{75} = -82$$
$$x_{76} = -34$$
$$x_{77} = 92$$
$$x_{78} = 42$$
$$x_{79} = -4$$
$$x_{80} = -56$$
$$x_{81} = 72$$
$$x_{82} = -72$$
$$x_{83} = -52$$
$$x_{84} = -16$$
$$x_{85} = -42$$
$$x_{86} = -6$$
$$x_{87} = 8$$
$$x_{88} = -24$$
$$x_{89} = -68$$
$$x_{90} = 88$$
$$x_{91} = -94$$
$$x_{92} = 46$$
$$x_{93} = -88$$
$$x_{94} = 22$$
$$x_{95} = -96$$
$$x_{96} = -70$$
$$x_{97} = -80$$
$$x_{98} = 100$$
$$x_{99} = 12$$
$$x_{100} = 62$$
o sea hay que resolver la ecuación:
$$\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
Solución numérica
$$x_{1} = 28$$
$$x_{2} = -22$$
$$x_{3} = -54$$
$$x_{4} = -32$$
$$x_{5} = -38$$
$$x_{6} = 32$$
$$x_{7} = 82$$
$$x_{8} = 76$$
$$x_{9} = -58$$
$$x_{10} = -86$$
$$x_{11} = -50$$
$$x_{12} = 86$$
$$x_{13} = 80$$
$$x_{14} = -64$$
$$x_{15} = -100$$
$$x_{16} = 36$$
$$x_{17} = -12$$
$$x_{18} = 38$$
$$x_{19} = -20$$
$$x_{20} = -8$$
$$x_{21} = -10$$
$$x_{22} = -44$$
$$x_{23} = 66$$
$$x_{24} = -62$$
$$x_{25} = -46$$
$$x_{26} = -48$$
$$x_{27} = 50$$
$$x_{28} = -74$$
$$x_{29} = 4$$
$$x_{30} = 98$$
$$x_{31} = -2$$
$$x_{32} = -66$$
$$x_{33} = 2$$
$$x_{34} = -28$$
$$x_{35} = 78$$
$$x_{36} = -92$$
$$x_{37} = 20$$
$$x_{38} = 54$$
$$x_{39} = 40$$
$$x_{40} = -40$$
$$x_{41} = 90$$
$$x_{42} = 74$$
$$x_{43} = 10$$
$$x_{44} = -76$$
$$x_{45} = 60$$
$$x_{46} = -18$$
$$x_{47} = -98$$
$$x_{48} = -36$$
$$x_{49} = 58$$
$$x_{50} = -30$$
$$x_{51} = 34$$
$$x_{52} = 18$$
$$x_{53} = -60$$
$$x_{54} = 70$$
$$x_{55} = 14$$
$$x_{56} = 30$$
$$x_{57} = 24$$
$$x_{58} = 64$$
$$x_{59} = -84$$
$$x_{60} = 26$$
$$x_{61} = 84$$
$$x_{62} = 52$$
$$x_{63} = 56$$
$$x_{64} = 68$$
$$x_{65} = 44$$
$$x_{66} = 94$$
$$x_{67} = 96$$
$$x_{68} = -26$$
$$x_{69} = 48$$
$$x_{70} = -14$$
$$x_{71} = -78$$
$$x_{72} = 6$$
$$x_{73} = -90$$
$$x_{74} = 16$$
$$x_{75} = -82$$
$$x_{76} = -34$$
$$x_{77} = 92$$
$$x_{78} = 42$$
$$x_{79} = -4$$
$$x_{80} = -56$$
$$x_{81} = 72$$
$$x_{82} = -72$$
$$x_{83} = -52$$
$$x_{84} = -16$$
$$x_{85} = -42$$
$$x_{86} = -6$$
$$x_{87} = 8$$
$$x_{88} = -24$$
$$x_{89} = -68$$
$$x_{90} = 88$$
$$x_{91} = -94$$
$$x_{92} = 46$$
$$x_{93} = -88$$
$$x_{94} = 22$$
$$x_{95} = -96$$
$$x_{96} = -70$$
$$x_{97} = -80$$
$$x_{98} = 100$$
$$x_{99} = 12$$
$$x_{100} = 62$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{\pi x \frac{\sin^{2}{\left(\pi x \right)}}{\pi^{2} x^{2}} \left(2 \pi \frac{1}{\pi x} \cos{\left(\pi x \right)} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right)}{\sin{\left(\pi x \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 46$$
$$x_{2} = -26$$
$$x_{3} = 58$$
$$x_{4} = -20$$
$$x_{5} = 50$$
$$x_{6} = 80$$
$$x_{7} = -100$$
$$x_{8} = 60$$
$$x_{9} = 26$$
$$x_{10} = 74$$
$$x_{11} = 72$$
$$x_{12} = -52$$
$$x_{13} = 44$$
$$x_{14} = -98$$
$$x_{15} = -46$$
$$x_{16} = -62$$
$$x_{17} = -94$$
$$x_{18} = 14$$
$$x_{19} = -2$$
$$x_{20} = 34$$
$$x_{21} = -30$$
$$x_{22} = 86$$
$$x_{23} = -88$$
$$x_{24} = 36$$
$$x_{25} = -76$$
$$x_{26} = -44$$
$$x_{27} = -70$$
$$x_{28} = -78$$
$$x_{29} = 78$$
$$x_{30} = 48$$
$$x_{31} = -38$$
$$x_{32} = -80$$
$$x_{33} = -92$$
$$x_{34} = 30$$
$$x_{35} = 16$$
$$x_{36} = 68$$
$$x_{37} = 90$$
$$x_{38} = 38$$
$$x_{39} = -56$$
$$x_{40} = -72$$
$$x_{41} = 32$$
$$x_{42} = 70$$
$$x_{43} = -58$$
$$x_{44} = 64$$
$$x_{45} = -60$$
$$x_{46} = -68$$
$$x_{47} = 8$$
$$x_{48} = 24$$
$$x_{49} = 62$$
$$x_{50} = -24$$
$$x_{51} = -6$$
$$x_{52} = -4$$
$$x_{53} = -54$$
$$x_{54} = 18$$
$$x_{55} = 52$$
$$x_{56} = 20$$
$$x_{57} = 96$$
$$x_{58} = -64$$
$$x_{59} = 10$$
$$x_{60} = 54$$
$$x_{61} = -84$$
$$x_{62} = -48$$
$$x_{63} = -8$$
$$x_{64} = 98$$
$$x_{65} = 4$$
$$x_{66} = 82$$
$$x_{67} = -10$$
$$x_{68} = -12$$
$$x_{69} = 56$$
$$x_{70} = -66$$
$$x_{71} = -90$$
$$x_{72} = 28$$
$$x_{73} = -22$$
$$x_{74} = -28$$
$$x_{75} = -40$$
$$x_{76} = 40$$
$$x_{77} = 88$$
$$x_{78} = 94$$
$$x_{79} = 2$$
$$x_{80} = 22$$
$$x_{81} = 84$$
$$x_{82} = -16$$
$$x_{83} = -32$$
$$x_{84} = -86$$
$$x_{85} = -82$$
$$x_{86} = 92$$
$$x_{87} = -36$$
$$x_{88} = -74$$
$$x_{89} = 76$$
$$x_{90} = -18$$
$$x_{91} = 6$$
$$x_{92} = -34$$
$$x_{93} = -14$$
$$x_{94} = -96$$
$$x_{95} = 42$$
$$x_{96} = 66$$
$$x_{97} = -42$$
$$x_{98} = 12$$
$$x_{99} = 100$$
$$x_{100} = -50$$
Signos de extremos en los puntos:
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:
$$x_{1} = 46$$
$$x_{2} = -26$$
$$x_{3} = 58$$
$$x_{4} = -20$$
$$x_{5} = 50$$
$$x_{6} = 80$$
$$x_{7} = -100$$
$$x_{8} = 60$$
$$x_{9} = 26$$
$$x_{10} = 74$$
$$x_{11} = 72$$
$$x_{12} = -52$$
$$x_{13} = 44$$
$$x_{14} = -98$$
$$x_{15} = -46$$
$$x_{16} = -62$$
$$x_{17} = -94$$
$$x_{18} = 14$$
$$x_{19} = -2$$
$$x_{20} = 34$$
$$x_{21} = -30$$
$$x_{22} = 86$$
$$x_{23} = -88$$
$$x_{24} = 36$$
$$x_{25} = -76$$
$$x_{26} = -44$$
$$x_{27} = -70$$
$$x_{28} = -78$$
$$x_{29} = 78$$
$$x_{30} = 48$$
$$x_{31} = -38$$
$$x_{32} = -80$$
$$x_{33} = -92$$
$$x_{34} = 30$$
$$x_{35} = 16$$
$$x_{36} = 68$$
$$x_{37} = 90$$
$$x_{38} = 38$$
$$x_{39} = -56$$
$$x_{40} = -72$$
$$x_{41} = 32$$
$$x_{42} = 70$$
$$x_{43} = -58$$
$$x_{44} = 64$$
$$x_{45} = -60$$
$$x_{46} = -68$$
$$x_{47} = 8$$
$$x_{48} = 24$$
$$x_{49} = 62$$
$$x_{50} = -24$$
$$x_{51} = -6$$
$$x_{52} = -4$$
$$x_{53} = -54$$
$$x_{54} = 18$$
$$x_{55} = 52$$
$$x_{56} = 20$$
$$x_{57} = 96$$
$$x_{58} = -64$$
$$x_{59} = 10$$
$$x_{60} = 54$$
$$x_{61} = -84$$
$$x_{62} = -48$$
$$x_{63} = -8$$
$$x_{64} = 98$$
$$x_{65} = 4$$
$$x_{66} = 82$$
$$x_{67} = -10$$
$$x_{68} = -12$$
$$x_{69} = 56$$
$$x_{70} = -66$$
$$x_{71} = -90$$
$$x_{72} = 28$$
$$x_{73} = -22$$
$$x_{74} = -28$$
$$x_{75} = -40$$
$$x_{76} = 40$$
$$x_{77} = 88$$
$$x_{78} = 94$$
$$x_{79} = 2$$
$$x_{80} = 22$$
$$x_{81} = 84$$
$$x_{82} = -16$$
$$x_{83} = -32$$
$$x_{84} = -86$$
$$x_{85} = -82$$
$$x_{86} = 92$$
$$x_{87} = -36$$
$$x_{88} = -74$$
$$x_{89} = 76$$
$$x_{90} = -18$$
$$x_{91} = 6$$
$$x_{92} = -34$$
$$x_{93} = -14$$
$$x_{94} = -96$$
$$x_{95} = 42$$
$$x_{96} = 66$$
$$x_{97} = -42$$
$$x_{98} = 12$$
$$x_{99} = 100$$
$$x_{100} = -50$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[100, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{\pi x \frac{\sin^{2}{\left(\pi x \right)}}{\pi^{2} x^{2}} \left(2 \pi \frac{1}{\pi x} \cos{\left(\pi x \right)} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right)}{\sin{\left(\pi x \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 46$$
$$x_{2} = -26$$
$$x_{3} = 58$$
$$x_{4} = -20$$
$$x_{5} = 50$$
$$x_{6} = 80$$
$$x_{7} = -100$$
$$x_{8} = 60$$
$$x_{9} = 26$$
$$x_{10} = 74$$
$$x_{11} = 72$$
$$x_{12} = -52$$
$$x_{13} = 44$$
$$x_{14} = -98$$
$$x_{15} = -46$$
$$x_{16} = -62$$
$$x_{17} = -94$$
$$x_{18} = 14$$
$$x_{19} = -2$$
$$x_{20} = 34$$
$$x_{21} = -30$$
$$x_{22} = 86$$
$$x_{23} = -88$$
$$x_{24} = 36$$
$$x_{25} = -76$$
$$x_{26} = -44$$
$$x_{27} = -70$$
$$x_{28} = -78$$
$$x_{29} = 78$$
$$x_{30} = 48$$
$$x_{31} = -38$$
$$x_{32} = -80$$
$$x_{33} = -92$$
$$x_{34} = 30$$
$$x_{35} = 16$$
$$x_{36} = 68$$
$$x_{37} = 90$$
$$x_{38} = 38$$
$$x_{39} = -56$$
$$x_{40} = -72$$
$$x_{41} = 32$$
$$x_{42} = 70$$
$$x_{43} = -58$$
$$x_{44} = 64$$
$$x_{45} = -60$$
$$x_{46} = -68$$
$$x_{47} = 8$$
$$x_{48} = 24$$
$$x_{49} = 62$$
$$x_{50} = -24$$
$$x_{51} = -6$$
$$x_{52} = -4$$
$$x_{53} = -54$$
$$x_{54} = 18$$
$$x_{55} = 52$$
$$x_{56} = 20$$
$$x_{57} = 96$$
$$x_{58} = -64$$
$$x_{59} = 10$$
$$x_{60} = 54$$
$$x_{61} = -84$$
$$x_{62} = -48$$
$$x_{63} = -8$$
$$x_{64} = 98$$
$$x_{65} = 4$$
$$x_{66} = 82$$
$$x_{67} = -10$$
$$x_{68} = -12$$
$$x_{69} = 56$$
$$x_{70} = -66$$
$$x_{71} = -90$$
$$x_{72} = 28$$
$$x_{73} = -22$$
$$x_{74} = -28$$
$$x_{75} = -40$$
$$x_{76} = 40$$
$$x_{77} = 88$$
$$x_{78} = 94$$
$$x_{79} = 2$$
$$x_{80} = 22$$
$$x_{81} = 84$$
$$x_{82} = -16$$
$$x_{83} = -32$$
$$x_{84} = -86$$
$$x_{85} = -82$$
$$x_{86} = 92$$
$$x_{87} = -36$$
$$x_{88} = -74$$
$$x_{89} = 76$$
$$x_{90} = -18$$
$$x_{91} = 6$$
$$x_{92} = -34$$
$$x_{93} = -14$$
$$x_{94} = -96$$
$$x_{95} = 42$$
$$x_{96} = 66$$
$$x_{97} = -42$$
$$x_{98} = 12$$
$$x_{99} = 100$$
$$x_{100} = -50$$
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:
$$x_{1} = 46$$
$$x_{2} = -26$$
$$x_{3} = 58$$
$$x_{4} = -20$$
$$x_{5} = 50$$
$$x_{6} = 80$$
$$x_{7} = -100$$
$$x_{8} = 60$$
$$x_{9} = 26$$
$$x_{10} = 74$$
$$x_{11} = 72$$
$$x_{12} = -52$$
$$x_{13} = 44$$
$$x_{14} = -98$$
$$x_{15} = -46$$
$$x_{16} = -62$$
$$x_{17} = -94$$
$$x_{18} = 14$$
$$x_{19} = -2$$
$$x_{20} = 34$$
$$x_{21} = -30$$
$$x_{22} = 86$$
$$x_{23} = -88$$
$$x_{24} = 36$$
$$x_{25} = -76$$
$$x_{26} = -44$$
$$x_{27} = -70$$
$$x_{28} = -78$$
$$x_{29} = 78$$
$$x_{30} = 48$$
$$x_{31} = -38$$
$$x_{32} = -80$$
$$x_{33} = -92$$
$$x_{34} = 30$$
$$x_{35} = 16$$
$$x_{36} = 68$$
$$x_{37} = 90$$
$$x_{38} = 38$$
$$x_{39} = -56$$
$$x_{40} = -72$$
$$x_{41} = 32$$
$$x_{42} = 70$$
$$x_{43} = -58$$
$$x_{44} = 64$$
$$x_{45} = -60$$
$$x_{46} = -68$$
$$x_{47} = 8$$
$$x_{48} = 24$$
$$x_{49} = 62$$
$$x_{50} = -24$$
$$x_{51} = -6$$
$$x_{52} = -4$$
$$x_{53} = -54$$
$$x_{54} = 18$$
$$x_{55} = 52$$
$$x_{56} = 20$$
$$x_{57} = 96$$
$$x_{58} = -64$$
$$x_{59} = 10$$
$$x_{60} = 54$$
$$x_{61} = -84$$
$$x_{62} = -48$$
$$x_{63} = -8$$
$$x_{64} = 98$$
$$x_{65} = 4$$
$$x_{66} = 82$$
$$x_{67} = -10$$
$$x_{68} = -12$$
$$x_{69} = 56$$
$$x_{70} = -66$$
$$x_{71} = -90$$
$$x_{72} = 28$$
$$x_{73} = -22$$
$$x_{74} = -28$$
$$x_{75} = -40$$
$$x_{76} = 40$$
$$x_{77} = 88$$
$$x_{78} = 94$$
$$x_{79} = 2$$
$$x_{80} = 22$$
$$x_{81} = 84$$
$$x_{82} = -16$$
$$x_{83} = -32$$
$$x_{84} = -86$$
$$x_{85} = -82$$
$$x_{86} = 92$$
$$x_{87} = -36$$
$$x_{88} = -74$$
$$x_{89} = 76$$
$$x_{90} = -18$$
$$x_{91} = 6$$
$$x_{92} = -34$$
$$x_{93} = -14$$
$$x_{94} = -96$$
$$x_{95} = 42$$
$$x_{96} = 66$$
$$x_{97} = -42$$
$$x_{98} = 12$$
$$x_{99} = 100$$
$$x_{100} = -50$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[100, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{2 \left(2 \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} - \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \cos{\left(\pi x \right)} - \frac{\left(\pi \sin{\left(\pi x \right)} + \frac{2 \cos{\left(\pi x \right)}}{x} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right) \sin{\left(\pi x \right)}}{\pi} + \frac{\left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \sin{\left(\pi x \right)}}{\pi x}\right)}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -61.7483527761947$$
$$x_{2} = -81.7487569556855$$
$$x_{3} = -83.7487867257144$$
$$x_{4} = -99.7489818056511$$
$$x_{5} = -7.73649601515403$$
$$x_{6} = -77.7486928066096$$
$$x_{7} = -95.7489391632373$$
$$x_{8} = 90.2488802811355$$
$$x_{9} = -87.748842183449$$
$$x_{10} = -69.7485423590771$$
$$x_{11} = 100.248991709865$$
$$x_{12} = 52.2480696155498$$
$$x_{13} = -89.7488680535879$$
$$x_{14} = -29.7465664532108$$
$$x_{15} = 80.2487411636708$$
$$x_{16} = 74.2486397633915$$
$$x_{17} = 4.22735950767312$$
$$x_{18} = 70.2485625726606$$
$$x_{19} = 18.2445188624812$$
$$x_{20} = -57.7482382050547$$
$$x_{21} = 2.20888752467459$$
$$x_{22} = 24.2458621171043$$
$$x_{23} = -9.7393399929161$$
$$x_{24} = 36.2472230924613$$
$$x_{25} = -79.7487256879633$$
$$x_{26} = 98.2489712343711$$
$$x_{27} = -31.7467844060814$$
$$x_{28} = 30.2466765477645$$
$$x_{29} = -75.7486581834026$$
$$x_{30} = -17.744212784795$$
$$x_{31} = 96.2489499100364$$
$$x_{32} = -67.7484991757802$$
$$x_{33} = -15.7434662299526$$
$$x_{34} = 28.2464431968423$$
$$x_{35} = -41.7475591015791$$
$$x_{36} = 40.2474974528301$$
$$x_{37} = 12.2418838185292$$
$$x_{38} = 72.2486022329285$$
$$x_{39} = 10.2403346426075$$
$$x_{40} = 22.2454940358069$$
$$x_{41} = -37.7472987956897$$
$$x_{42} = -63.7484046496571$$
$$x_{43} = -73.7486216761659$$
$$x_{44} = 58.2482676493225$$
$$x_{45} = 92.2489044945726$$
$$x_{46} = -55.7481747273539$$
$$x_{47} = -25.746028000833$$
$$x_{48} = -27.746316803873$$
$$x_{49} = 32.2468811632873$$
$$x_{50} = 42.2476152591584$$
$$x_{51} = -13.7424984608719$$
$$x_{52} = 26.2461745995811$$
$$x_{53} = 44.2477224724436$$
$$x_{54} = -23.7456900482187$$
$$x_{55} = -19.7448062051182$$
$$x_{56} = 6.23436247448923$$
$$x_{57} = -21.7452892305997$$
$$x_{58} = -65.7484533556792$$
$$x_{59} = -33.7469763389274$$
$$x_{60} = 14.2430048647492$$
$$x_{61} = 64.2484288313613$$
$$x_{62} = -45.7477736462445$$
$$x_{63} = 38.2473674018583$$
$$x_{64} = 50.2479931440213$$
$$x_{65} = -49.7479535220692$$
$$x_{66} = -85.748815103141$$
$$x_{67} = 88.2488549731412$$
$$x_{68} = 94.2489276829586$$
$$x_{69} = -91.7488927929056$$
$$x_{70} = 84.2488007625499$$
$$x_{71} = -51.7480329833019$$
$$x_{72} = 20.2450540655959$$
$$x_{73} = 8.23805394372117$$
$$x_{74} = -59.7482974159256$$
$$x_{75} = 48.2479103636241$$
$$x_{76} = 54.2481404729683$$
$$x_{77} = -11.7411939995098$$
$$x_{78} = 66.2484760938889$$
$$x_{79} = -47.7478673704273$$
$$x_{80} = 60.248324929728$$
$$x_{81} = 82.2487716856391$$
$$x_{82} = -39.747435537556$$
$$x_{83} = 78.2487090861744$$
$$x_{84} = -71.7485831268165$$
$$x_{85} = 46.2478204600269$$
$$x_{86} = 16.2438537460501$$
$$x_{87} = -5.73157811771419$$
$$x_{88} = 62.2483785433487$$
$$x_{89} = 56.2482063126328$$
$$x_{90} = -93.7489164739615$$
$$x_{91} = 68.2485205959957$$
$$x_{92} = -97.7489609217602$$
$$x_{93} = -43.747671305176$$
$$x_{94} = 76.2486753311251$$
$$x_{95} = -35.7471466490446$$
$$x_{96} = 34.2470620436327$$
$$x_{97} = -53.7481065043619$$
$$x_{98} = -1.68100482418311$$
$$x_{99} = -3.72099743965763$$
$$x_{100} = 86.2488284946546$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(2 \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} - \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \cos{\left(\pi x \right)} - \frac{\left(\pi \sin{\left(\pi x \right)} + \frac{2 \cos{\left(\pi x \right)}}{x} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right) \sin{\left(\pi x \right)}}{\pi} + \frac{\left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \sin{\left(\pi x \right)}}{\pi x}\right)}{x^{2}}\right) = - \frac{2 \pi^{2}}{3}$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(2 \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} - \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \cos{\left(\pi x \right)} - \frac{\left(\pi \sin{\left(\pi x \right)} + \frac{2 \cos{\left(\pi x \right)}}{x} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right) \sin{\left(\pi x \right)}}{\pi} + \frac{\left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \sin{\left(\pi x \right)}}{\pi x}\right)}{x^{2}}\right) = - \frac{2 \pi^{2}}{3}$$
- 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
$$\left(-\infty, -99.7489818056511\right]$$
Convexa en los intervalos
$$\left[100.248991709865, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{2 \left(2 \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} - \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \cos{\left(\pi x \right)} - \frac{\left(\pi \sin{\left(\pi x \right)} + \frac{2 \cos{\left(\pi x \right)}}{x} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right) \sin{\left(\pi x \right)}}{\pi} + \frac{\left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \sin{\left(\pi x \right)}}{\pi x}\right)}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -61.7483527761947$$
$$x_{2} = -81.7487569556855$$
$$x_{3} = -83.7487867257144$$
$$x_{4} = -99.7489818056511$$
$$x_{5} = -7.73649601515403$$
$$x_{6} = -77.7486928066096$$
$$x_{7} = -95.7489391632373$$
$$x_{8} = 90.2488802811355$$
$$x_{9} = -87.748842183449$$
$$x_{10} = -69.7485423590771$$
$$x_{11} = 100.248991709865$$
$$x_{12} = 52.2480696155498$$
$$x_{13} = -89.7488680535879$$
$$x_{14} = -29.7465664532108$$
$$x_{15} = 80.2487411636708$$
$$x_{16} = 74.2486397633915$$
$$x_{17} = 4.22735950767312$$
$$x_{18} = 70.2485625726606$$
$$x_{19} = 18.2445188624812$$
$$x_{20} = -57.7482382050547$$
$$x_{21} = 2.20888752467459$$
$$x_{22} = 24.2458621171043$$
$$x_{23} = -9.7393399929161$$
$$x_{24} = 36.2472230924613$$
$$x_{25} = -79.7487256879633$$
$$x_{26} = 98.2489712343711$$
$$x_{27} = -31.7467844060814$$
$$x_{28} = 30.2466765477645$$
$$x_{29} = -75.7486581834026$$
$$x_{30} = -17.744212784795$$
$$x_{31} = 96.2489499100364$$
$$x_{32} = -67.7484991757802$$
$$x_{33} = -15.7434662299526$$
$$x_{34} = 28.2464431968423$$
$$x_{35} = -41.7475591015791$$
$$x_{36} = 40.2474974528301$$
$$x_{37} = 12.2418838185292$$
$$x_{38} = 72.2486022329285$$
$$x_{39} = 10.2403346426075$$
$$x_{40} = 22.2454940358069$$
$$x_{41} = -37.7472987956897$$
$$x_{42} = -63.7484046496571$$
$$x_{43} = -73.7486216761659$$
$$x_{44} = 58.2482676493225$$
$$x_{45} = 92.2489044945726$$
$$x_{46} = -55.7481747273539$$
$$x_{47} = -25.746028000833$$
$$x_{48} = -27.746316803873$$
$$x_{49} = 32.2468811632873$$
$$x_{50} = 42.2476152591584$$
$$x_{51} = -13.7424984608719$$
$$x_{52} = 26.2461745995811$$
$$x_{53} = 44.2477224724436$$
$$x_{54} = -23.7456900482187$$
$$x_{55} = -19.7448062051182$$
$$x_{56} = 6.23436247448923$$
$$x_{57} = -21.7452892305997$$
$$x_{58} = -65.7484533556792$$
$$x_{59} = -33.7469763389274$$
$$x_{60} = 14.2430048647492$$
$$x_{61} = 64.2484288313613$$
$$x_{62} = -45.7477736462445$$
$$x_{63} = 38.2473674018583$$
$$x_{64} = 50.2479931440213$$
$$x_{65} = -49.7479535220692$$
$$x_{66} = -85.748815103141$$
$$x_{67} = 88.2488549731412$$
$$x_{68} = 94.2489276829586$$
$$x_{69} = -91.7488927929056$$
$$x_{70} = 84.2488007625499$$
$$x_{71} = -51.7480329833019$$
$$x_{72} = 20.2450540655959$$
$$x_{73} = 8.23805394372117$$
$$x_{74} = -59.7482974159256$$
$$x_{75} = 48.2479103636241$$
$$x_{76} = 54.2481404729683$$
$$x_{77} = -11.7411939995098$$
$$x_{78} = 66.2484760938889$$
$$x_{79} = -47.7478673704273$$
$$x_{80} = 60.248324929728$$
$$x_{81} = 82.2487716856391$$
$$x_{82} = -39.747435537556$$
$$x_{83} = 78.2487090861744$$
$$x_{84} = -71.7485831268165$$
$$x_{85} = 46.2478204600269$$
$$x_{86} = 16.2438537460501$$
$$x_{87} = -5.73157811771419$$
$$x_{88} = 62.2483785433487$$
$$x_{89} = 56.2482063126328$$
$$x_{90} = -93.7489164739615$$
$$x_{91} = 68.2485205959957$$
$$x_{92} = -97.7489609217602$$
$$x_{93} = -43.747671305176$$
$$x_{94} = 76.2486753311251$$
$$x_{95} = -35.7471466490446$$
$$x_{96} = 34.2470620436327$$
$$x_{97} = -53.7481065043619$$
$$x_{98} = -1.68100482418311$$
$$x_{99} = -3.72099743965763$$
$$x_{100} = 86.2488284946546$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(2 \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} - \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \cos{\left(\pi x \right)} - \frac{\left(\pi \sin{\left(\pi x \right)} + \frac{2 \cos{\left(\pi x \right)}}{x} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right) \sin{\left(\pi x \right)}}{\pi} + \frac{\left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \sin{\left(\pi x \right)}}{\pi x}\right)}{x^{2}}\right) = - \frac{2 \pi^{2}}{3}$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(2 \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} - \left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \cos{\left(\pi x \right)} - \frac{\left(\pi \sin{\left(\pi x \right)} + \frac{2 \cos{\left(\pi x \right)}}{x} - \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}\right) \sin{\left(\pi x \right)}}{\pi} + \frac{\left(\cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x}\right) \sin{\left(\pi x \right)}}{\pi x}\right)}{x^{2}}\right) = - \frac{2 \pi^{2}}{3}$$
- 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
$$\left(-\infty, -99.7489818056511\right]$$
Convexa en los intervalos
$$\left[100.248991709865, \infty\right)$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty} \left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty} \left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty} \left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty} \left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\pi^{2}} \frac{1}{x^{2}} \sin^{2}{\left(\pi x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\pi^{2}} \frac{1}{x^{2}} \sin^{2}{\left(\pi x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\pi^{2}} \frac{1}{x^{2}} \sin^{2}{\left(\pi x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\pi^{2}} \frac{1}{x^{2}} \sin^{2}{\left(\pi x \right)}}{x}\right) = 0$$
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:
$$\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = \frac{\sin^{2}{\left(\pi x \right)}}{\pi^{2} x^{2}}$$
- No
$$\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = - \frac{\sin^{2}{\left(\pi x \right)}}{\pi^{2} x^{2}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = \frac{\sin^{2}{\left(\pi x \right)}}{\pi^{2} x^{2}}$$
- No
$$\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right)^{2} = - \frac{\sin^{2}{\left(\pi x \right)}}{\pi^{2} x^{2}}$$
- No
es decir, función
no es
par ni impar