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- Puntos de cruce de las funciones
-
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- Gráfico de la función y =:
-
(x-1)/(x+2)
-
-1-x
-
-exp(-x)+(-cos(x*sqrt(3)/2)-sin(x*sqrt(3)/2))*exp(x/2)
-
3/(x^2+1)
-
Expresiones idénticas
- sin(pi*x)/((pi*x))
- seno de ( número pi multiplicar por x) dividir por (( número pi multiplicar por x))
- sin(pix)/((pix))
- sinpix/pix
- sin(pi*x) dividir por ((pi*x))
-
Expresiones semejantes
- (sin(pi*x)/(pi*x))^2
-
Expresiones con funciones
- Seno sin
- sin(x)+x
- sin(x)*sin(3*x)
- sin(x)/sin(x+pi/4)
- sinx+cos^2x
- sin(x)*cos(x)^(2)
- Número Pi pi
- pi-2*arcsin(2.752x)
- pix
- pi/2+(4*cos(x)+4*cos(3*x))/(9*pi)
- pi/8-cosx/pi+sinx/2
- pi*(1-x)*tg(pi*x/2)
- Número Pi pi
- pi-2*arcsin(2.752x)
- pix
- pi/2+(4*cos(x)+4*cos(3*x))/(9*pi)
- pi/8-cosx/pi+sinx/2
- pi*(1-x)*tg(pi*x/2)
Gráfico de la función y = sin(pi*x)/((pi*x))
Solución
Ha introducido
[src]
sin(pi*x)
f(x) = ---------
pi*x
$$f{\left(x \right)} = \frac{\sin{\left(\pi x \right)}}{\pi x}$$
f = sin(pi*x)/((pi*x))
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:
$$\frac{\sin{\left(\pi x \right)}}{\pi x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
Solución numérica
$$x_{1} = 66$$
$$x_{2} = 94$$
$$x_{3} = 98$$
$$x_{4} = 46$$
$$x_{5} = 24$$
$$x_{6} = -24$$
$$x_{7} = -46$$
$$x_{8} = 18$$
$$x_{9} = 22$$
$$x_{10} = -82$$
$$x_{11} = -74$$
$$x_{12} = -42$$
$$x_{13} = 50$$
$$x_{14} = 64$$
$$x_{15} = 36$$
$$x_{16} = -60$$
$$x_{17} = -66$$
$$x_{18} = -8$$
$$x_{19} = -26$$
$$x_{20} = -2$$
$$x_{21} = 86$$
$$x_{22} = 42$$
$$x_{23} = -36$$
$$x_{24} = 100$$
$$x_{25} = -18$$
$$x_{26} = 12$$
$$x_{27} = 90$$
$$x_{28} = -88$$
$$x_{29} = -92$$
$$x_{30} = 52$$
$$x_{31} = -44$$
$$x_{32} = 6$$
$$x_{33} = 38$$
$$x_{34} = 96$$
$$x_{35} = -84$$
$$x_{36} = 10$$
$$x_{37} = -98$$
$$x_{38} = 62$$
$$x_{39} = -28$$
$$x_{40} = 14$$
$$x_{41} = 30$$
$$x_{42} = -14$$
$$x_{43} = -72$$
$$x_{44} = -20$$
$$x_{45} = 54$$
$$x_{46} = -52$$
$$x_{47} = -12$$
$$x_{48} = 68$$
$$x_{49} = 32$$
$$x_{50} = 28$$
$$x_{51} = -16$$
$$x_{52} = -40$$
$$x_{53} = -48$$
$$x_{54} = 44$$
$$x_{55} = -6$$
$$x_{56} = -50$$
$$x_{57} = 16$$
$$x_{58} = -78$$
$$x_{59} = 4$$
$$x_{60} = -56$$
$$x_{61} = -10$$
$$x_{62} = 74$$
$$x_{63} = -34$$
$$x_{64} = -62$$
$$x_{65} = -58$$
$$x_{66} = 84$$
$$x_{67} = -68$$
$$x_{68} = 76$$
$$x_{69} = -94$$
$$x_{70} = -86$$
$$x_{71} = -4$$
$$x_{72} = 40$$
$$x_{73} = -96$$
$$x_{74} = -70$$
$$x_{75} = 8$$
$$x_{76} = 82$$
$$x_{77} = -80$$
$$x_{78} = -76$$
$$x_{79} = -38$$
$$x_{80} = -32$$
$$x_{81} = -30$$
$$x_{82} = -54$$
$$x_{83} = 48$$
$$x_{84} = 20$$
$$x_{85} = 56$$
$$x_{86} = 58$$
$$x_{87} = 78$$
$$x_{88} = 70$$
$$x_{89} = 60$$
$$x_{90} = 92$$
$$x_{91} = -90$$
$$x_{92} = 72$$
$$x_{93} = 88$$
$$x_{94} = 2$$
$$x_{95} = -64$$
$$x_{96} = -100$$
$$x_{97} = 80$$
$$x_{98} = -22$$
$$x_{99} = 34$$
$$x_{100} = 26$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(\pi x \right)}}{\pi x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 1$$
Solución numérica
$$x_{1} = 66$$
$$x_{2} = 94$$
$$x_{3} = 98$$
$$x_{4} = 46$$
$$x_{5} = 24$$
$$x_{6} = -24$$
$$x_{7} = -46$$
$$x_{8} = 18$$
$$x_{9} = 22$$
$$x_{10} = -82$$
$$x_{11} = -74$$
$$x_{12} = -42$$
$$x_{13} = 50$$
$$x_{14} = 64$$
$$x_{15} = 36$$
$$x_{16} = -60$$
$$x_{17} = -66$$
$$x_{18} = -8$$
$$x_{19} = -26$$
$$x_{20} = -2$$
$$x_{21} = 86$$
$$x_{22} = 42$$
$$x_{23} = -36$$
$$x_{24} = 100$$
$$x_{25} = -18$$
$$x_{26} = 12$$
$$x_{27} = 90$$
$$x_{28} = -88$$
$$x_{29} = -92$$
$$x_{30} = 52$$
$$x_{31} = -44$$
$$x_{32} = 6$$
$$x_{33} = 38$$
$$x_{34} = 96$$
$$x_{35} = -84$$
$$x_{36} = 10$$
$$x_{37} = -98$$
$$x_{38} = 62$$
$$x_{39} = -28$$
$$x_{40} = 14$$
$$x_{41} = 30$$
$$x_{42} = -14$$
$$x_{43} = -72$$
$$x_{44} = -20$$
$$x_{45} = 54$$
$$x_{46} = -52$$
$$x_{47} = -12$$
$$x_{48} = 68$$
$$x_{49} = 32$$
$$x_{50} = 28$$
$$x_{51} = -16$$
$$x_{52} = -40$$
$$x_{53} = -48$$
$$x_{54} = 44$$
$$x_{55} = -6$$
$$x_{56} = -50$$
$$x_{57} = 16$$
$$x_{58} = -78$$
$$x_{59} = 4$$
$$x_{60} = -56$$
$$x_{61} = -10$$
$$x_{62} = 74$$
$$x_{63} = -34$$
$$x_{64} = -62$$
$$x_{65} = -58$$
$$x_{66} = 84$$
$$x_{67} = -68$$
$$x_{68} = 76$$
$$x_{69} = -94$$
$$x_{70} = -86$$
$$x_{71} = -4$$
$$x_{72} = 40$$
$$x_{73} = -96$$
$$x_{74} = -70$$
$$x_{75} = 8$$
$$x_{76} = 82$$
$$x_{77} = -80$$
$$x_{78} = -76$$
$$x_{79} = -38$$
$$x_{80} = -32$$
$$x_{81} = -30$$
$$x_{82} = -54$$
$$x_{83} = 48$$
$$x_{84} = 20$$
$$x_{85} = 56$$
$$x_{86} = 58$$
$$x_{87} = 78$$
$$x_{88} = 70$$
$$x_{89} = 60$$
$$x_{90} = 92$$
$$x_{91} = -90$$
$$x_{92} = 72$$
$$x_{93} = 88$$
$$x_{94} = 2$$
$$x_{95} = -64$$
$$x_{96} = -100$$
$$x_{97} = 80$$
$$x_{98} = -22$$
$$x_{99} = 34$$
$$x_{100} = 26$$
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
$$\pi \frac{1}{\pi x} \cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -93.4989163426041$$
$$x_{2} = -77.4986926151852$$
$$x_{3} = 62.4983788330278$$
$$x_{4} = -17.4942089405157$$
$$x_{5} = -43.4976706942357$$
$$x_{6} = 92.498904627205$$
$$x_{7} = -65.4984530873445$$
$$x_{8} = -37.4972979719751$$
$$x_{9} = -59.4982970904721$$
$$x_{10} = 98.4989713513802$$
$$x_{11} = -69.4985421208549$$
$$x_{12} = 14.4930100858125$$
$$x_{13} = 38.4973681610039$$
$$x_{14} = -5.4815366506072$$
$$x_{15} = -71.4985829017843$$
$$x_{16} = -7.48647424796277$$
$$x_{17} = 20.4950567085724$$
$$x_{18} = 76.4986755247957$$
$$x_{19} = -63.4984043640786$$
$$x_{20} = 40.4974981393456$$
$$x_{21} = -89.4988679101893$$
$$x_{22} = 44.4977230417979$$
$$x_{23} = 34.497062987411$$
$$x_{24} = -51.4980325482676$$
$$x_{25} = 36.4972239364023$$
$$x_{26} = 96.4989500319316$$
$$x_{27} = 28.4964445750536$$
$$x_{28} = 2.45902403295676$$
$$x_{29} = 86.4988286462618$$
$$x_{30} = 88.4988551179946$$
$$x_{31} = -75.4986579816647$$
$$x_{32} = -73.4986214632578$$
$$x_{33} = -97.4989608009887$$
$$x_{34} = -83.498786560895$$
$$x_{35} = -27.4963152641774$$
$$x_{36} = -53.4981061014042$$
$$x_{37} = -45.4977730881378$$
$$x_{38} = 82.498771852252$$
$$x_{39} = 72.4986024484725$$
$$x_{40} = 52.4980700253935$$
$$x_{41} = -23.4956879327866$$
$$x_{42} = 42.4976158829888$$
$$x_{43} = 50.4979935867835$$
$$x_{44} = 22.4954962352822$$
$$x_{45} = -57.4982378564567$$
$$x_{46} = 32.4968822257241$$
$$x_{47} = 46.4978209817408$$
$$x_{48} = -47.497866858586$$
$$x_{49} = -49.4979530509703$$
$$x_{50} = -95.4989390373396$$
$$x_{51} = -87.4988420333992$$
$$x_{52} = -29.4965651169754$$
$$x_{53} = 48.497910843436$$
$$x_{54} = 66.498476349905$$
$$x_{55} = 68.4985208373393$$
$$x_{56} = -11.4911849577712$$
$$x_{57} = 8.48806869609345$$
$$x_{58} = 6.48438712638634$$
$$x_{59} = 21.4952866979685$$
$$x_{60} = -91.4988926557255$$
$$x_{61} = -79.4987255060814$$
$$x_{62} = -99.498981689699$$
$$x_{63} = -33.4969753049907$$
$$x_{64} = 100.498991822276$$
$$x_{65} = 60.4983252387847$$
$$x_{66} = 74.4986399675602$$
$$x_{67} = 10.4903444442946$$
$$x_{68} = 90.4988804196764$$
$$x_{69} = 3.47088972434702$$
$$x_{70} = -61.4983524716552$$
$$x_{71} = 12.4918907982296$$
$$x_{72} = 84.4988009213945$$
$$x_{73} = -41.4975584299323$$
$$x_{74} = -85.4988149459663$$
$$x_{75} = -55.4981743530517$$
$$x_{76} = -81.4987567826499$$
$$x_{77} = 54.4981408534326$$
$$x_{78} = 70.4985628005579$$
$$x_{79} = 56.4982066667669$$
$$x_{80} = -67.4984989231734$$
$$x_{81} = 94.4989278100525$$
$$x_{82} = 4.47740858343252$$
$$x_{83} = 78.4987092701363$$
$$x_{84} = 24.4958639759983$$
$$x_{85} = 64.4984291034297$$
$$x_{86} = -15.4934613105668$$
$$x_{87} = -31.496783235492$$
$$x_{88} = -13.4924919434404$$
$$x_{89} = 26.4961761912728$$
$$x_{90} = 16.4938577979135$$
$$x_{91} = -39.497434795684$$
$$x_{92} = -19.4948031185367$$
$$x_{93} = 58.498267979768$$
$$x_{94} = 80.4987413386361$$
$$x_{95} = -21.4952866979685$$
$$x_{96} = 30.4966777527268$$
$$x_{97} = 18.4945220979251$$
$$x_{98} = -35.4971457291586$$
$$x_{99} = -9.48932661808597$$
$$x_{100} = -25.4960262074141$$
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:
$$x_{1} = -93.4989163426041$$
$$x_{2} = -77.4986926151852$$
$$x_{3} = -17.4942089405157$$
$$x_{4} = -43.4976706942357$$
$$x_{5} = -65.4984530873445$$
$$x_{6} = -37.4972979719751$$
$$x_{7} = -59.4982970904721$$
$$x_{8} = -69.4985421208549$$
$$x_{9} = -5.4815366506072$$
$$x_{10} = -71.4985829017843$$
$$x_{11} = -7.48647424796277$$
$$x_{12} = -63.4984043640786$$
$$x_{13} = -89.4988679101893$$
$$x_{14} = -51.4980325482676$$
$$x_{15} = -75.4986579816647$$
$$x_{16} = -73.4986214632578$$
$$x_{17} = -97.4989608009887$$
$$x_{18} = -83.498786560895$$
$$x_{19} = -27.4963152641774$$
$$x_{20} = -53.4981061014042$$
$$x_{21} = -45.4977730881378$$
$$x_{22} = -23.4956879327866$$
$$x_{23} = -57.4982378564567$$
$$x_{24} = -47.497866858586$$
$$x_{25} = -49.4979530509703$$
$$x_{26} = -95.4989390373396$$
$$x_{27} = -87.4988420333992$$
$$x_{28} = -29.4965651169754$$
$$x_{29} = -11.4911849577712$$
$$x_{30} = 21.4952866979685$$
$$x_{31} = -91.4988926557255$$
$$x_{32} = -79.4987255060814$$
$$x_{33} = -99.498981689699$$
$$x_{34} = -33.4969753049907$$
$$x_{35} = 3.47088972434702$$
$$x_{36} = -61.4983524716552$$
$$x_{37} = -41.4975584299323$$
$$x_{38} = -85.4988149459663$$
$$x_{39} = -55.4981743530517$$
$$x_{40} = -81.4987567826499$$
$$x_{41} = -67.4984989231734$$
$$x_{42} = -15.4934613105668$$
$$x_{43} = -31.496783235492$$
$$x_{44} = -13.4924919434404$$
$$x_{45} = -39.497434795684$$
$$x_{46} = -19.4948031185367$$
$$x_{47} = -21.4952866979685$$
$$x_{48} = -35.4971457291586$$
$$x_{49} = -9.48932661808597$$
$$x_{50} = -25.4960262074141$$
Puntos máximos de la función:
$$x_{50} = 62.4983788330278$$
$$x_{50} = 92.498904627205$$
$$x_{50} = 98.4989713513802$$
$$x_{50} = 14.4930100858125$$
$$x_{50} = 38.4973681610039$$
$$x_{50} = 20.4950567085724$$
$$x_{50} = 76.4986755247957$$
$$x_{50} = 40.4974981393456$$
$$x_{50} = 44.4977230417979$$
$$x_{50} = 34.497062987411$$
$$x_{50} = 36.4972239364023$$
$$x_{50} = 96.4989500319316$$
$$x_{50} = 28.4964445750536$$
$$x_{50} = 2.45902403295676$$
$$x_{50} = 86.4988286462618$$
$$x_{50} = 88.4988551179946$$
$$x_{50} = 82.498771852252$$
$$x_{50} = 72.4986024484725$$
$$x_{50} = 52.4980700253935$$
$$x_{50} = 42.4976158829888$$
$$x_{50} = 50.4979935867835$$
$$x_{50} = 22.4954962352822$$
$$x_{50} = 32.4968822257241$$
$$x_{50} = 46.4978209817408$$
$$x_{50} = 48.497910843436$$
$$x_{50} = 66.498476349905$$
$$x_{50} = 68.4985208373393$$
$$x_{50} = 8.48806869609345$$
$$x_{50} = 6.48438712638634$$
$$x_{50} = 100.498991822276$$
$$x_{50} = 60.4983252387847$$
$$x_{50} = 74.4986399675602$$
$$x_{50} = 10.4903444442946$$
$$x_{50} = 90.4988804196764$$
$$x_{50} = 12.4918907982296$$
$$x_{50} = 84.4988009213945$$
$$x_{50} = 54.4981408534326$$
$$x_{50} = 70.4985628005579$$
$$x_{50} = 56.4982066667669$$
$$x_{50} = 94.4989278100525$$
$$x_{50} = 4.47740858343252$$
$$x_{50} = 78.4987092701363$$
$$x_{50} = 24.4958639759983$$
$$x_{50} = 64.4984291034297$$
$$x_{50} = 26.4961761912728$$
$$x_{50} = 16.4938577979135$$
$$x_{50} = 58.498267979768$$
$$x_{50} = 80.4987413386361$$
$$x_{50} = 30.4966777527268$$
$$x_{50} = 18.4945220979251$$
Decrece en los intervalos
$$\left[21.4952866979685, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.498981689699\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\pi \frac{1}{\pi x} \cos{\left(\pi x \right)} - \frac{\sin{\left(\pi x \right)}}{\pi x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -93.4989163426041$$
$$x_{2} = -77.4986926151852$$
$$x_{3} = 62.4983788330278$$
$$x_{4} = -17.4942089405157$$
$$x_{5} = -43.4976706942357$$
$$x_{6} = 92.498904627205$$
$$x_{7} = -65.4984530873445$$
$$x_{8} = -37.4972979719751$$
$$x_{9} = -59.4982970904721$$
$$x_{10} = 98.4989713513802$$
$$x_{11} = -69.4985421208549$$
$$x_{12} = 14.4930100858125$$
$$x_{13} = 38.4973681610039$$
$$x_{14} = -5.4815366506072$$
$$x_{15} = -71.4985829017843$$
$$x_{16} = -7.48647424796277$$
$$x_{17} = 20.4950567085724$$
$$x_{18} = 76.4986755247957$$
$$x_{19} = -63.4984043640786$$
$$x_{20} = 40.4974981393456$$
$$x_{21} = -89.4988679101893$$
$$x_{22} = 44.4977230417979$$
$$x_{23} = 34.497062987411$$
$$x_{24} = -51.4980325482676$$
$$x_{25} = 36.4972239364023$$
$$x_{26} = 96.4989500319316$$
$$x_{27} = 28.4964445750536$$
$$x_{28} = 2.45902403295676$$
$$x_{29} = 86.4988286462618$$
$$x_{30} = 88.4988551179946$$
$$x_{31} = -75.4986579816647$$
$$x_{32} = -73.4986214632578$$
$$x_{33} = -97.4989608009887$$
$$x_{34} = -83.498786560895$$
$$x_{35} = -27.4963152641774$$
$$x_{36} = -53.4981061014042$$
$$x_{37} = -45.4977730881378$$
$$x_{38} = 82.498771852252$$
$$x_{39} = 72.4986024484725$$
$$x_{40} = 52.4980700253935$$
$$x_{41} = -23.4956879327866$$
$$x_{42} = 42.4976158829888$$
$$x_{43} = 50.4979935867835$$
$$x_{44} = 22.4954962352822$$
$$x_{45} = -57.4982378564567$$
$$x_{46} = 32.4968822257241$$
$$x_{47} = 46.4978209817408$$
$$x_{48} = -47.497866858586$$
$$x_{49} = -49.4979530509703$$
$$x_{50} = -95.4989390373396$$
$$x_{51} = -87.4988420333992$$
$$x_{52} = -29.4965651169754$$
$$x_{53} = 48.497910843436$$
$$x_{54} = 66.498476349905$$
$$x_{55} = 68.4985208373393$$
$$x_{56} = -11.4911849577712$$
$$x_{57} = 8.48806869609345$$
$$x_{58} = 6.48438712638634$$
$$x_{59} = 21.4952866979685$$
$$x_{60} = -91.4988926557255$$
$$x_{61} = -79.4987255060814$$
$$x_{62} = -99.498981689699$$
$$x_{63} = -33.4969753049907$$
$$x_{64} = 100.498991822276$$
$$x_{65} = 60.4983252387847$$
$$x_{66} = 74.4986399675602$$
$$x_{67} = 10.4903444442946$$
$$x_{68} = 90.4988804196764$$
$$x_{69} = 3.47088972434702$$
$$x_{70} = -61.4983524716552$$
$$x_{71} = 12.4918907982296$$
$$x_{72} = 84.4988009213945$$
$$x_{73} = -41.4975584299323$$
$$x_{74} = -85.4988149459663$$
$$x_{75} = -55.4981743530517$$
$$x_{76} = -81.4987567826499$$
$$x_{77} = 54.4981408534326$$
$$x_{78} = 70.4985628005579$$
$$x_{79} = 56.4982066667669$$
$$x_{80} = -67.4984989231734$$
$$x_{81} = 94.4989278100525$$
$$x_{82} = 4.47740858343252$$
$$x_{83} = 78.4987092701363$$
$$x_{84} = 24.4958639759983$$
$$x_{85} = 64.4984291034297$$
$$x_{86} = -15.4934613105668$$
$$x_{87} = -31.496783235492$$
$$x_{88} = -13.4924919434404$$
$$x_{89} = 26.4961761912728$$
$$x_{90} = 16.4938577979135$$
$$x_{91} = -39.497434795684$$
$$x_{92} = -19.4948031185367$$
$$x_{93} = 58.498267979768$$
$$x_{94} = 80.4987413386361$$
$$x_{95} = -21.4952866979685$$
$$x_{96} = 30.4966777527268$$
$$x_{97} = 18.4945220979251$$
$$x_{98} = -35.4971457291586$$
$$x_{99} = -9.48932661808597$$
$$x_{100} = -25.4960262074141$$
Signos de extremos en los puntos:
0.0106953111235615*sin(1.49891634260409*pi)
(-93.49891634260409, -------------------------------------------)
pi 0.0129034434808525*sin(1.49869261518522*pi)
(-77.49869261518522, -------------------------------------------)
pi 0.0160004150295102*sin(0.4983788330278*pi)
(62.4983788330278, ------------------------------------------)
pi 0.0571617729844333*sin(1.49420894051568*pi)
(-17.49420894051568, -------------------------------------------)
pi 0.0229897367845152*sin(1.49767069423572*pi)
(-43.49767069423572, -------------------------------------------)
pi 0.010810938832522*sin(0.498904627205008*pi)
(92.49890462720501, -------------------------------------------)
pi 0.0152675361457234*sin(1.49845308734453*pi)
(-65.49845308734453, -------------------------------------------)
pi 0.0266685882472754*sin(1.49729797197509*pi)
(-37.49729797197509, -------------------------------------------)
pi 0.0168072037167621*sin(1.49829709047214*pi)
(-59.498297090472136, -------------------------------------------)
pi 0.0101523902867234*sin(0.498971351380163*pi)
(98.49897135138016, --------------------------------------------)
pi 0.0143887910376745*sin(1.49854212085494*pi)
(-69.49854212085494, -------------------------------------------)
pi 0.0689987790030534*sin(0.493010085812484*pi)
(14.493010085812484, --------------------------------------------)
pi 0.0259758016656566*sin(0.497368161003912*pi)
(38.49736816100391, --------------------------------------------)
pi 0.182430596334557*sin(1.4815366506072*pi)
(-5.481536650607196, -----------------------------------------)
pi 0.0139862911880879*sin(1.49858290178435*pi)
(-71.49858290178435, -------------------------------------------)
pi 0.133574225580502*sin(1.48647424796277*pi)
(-7.486474247962769, ------------------------------------------)
pi 0.0487922533818475*sin(0.495056708572449*pi)
(20.49505670857245, --------------------------------------------)
pi 0.0130721217477271*sin(0.498675524795672*pi)
(76.49867552479567, --------------------------------------------)
pi 0.0157484272245069*sin(1.49840436407858*pi)
(-63.49840436407858, -------------------------------------------)
pi 0.0246928834111963*sin(0.497498139345559*pi)
(40.49749813934556, --------------------------------------------)
pi 0.0111733256894767*sin(1.49886791018928*pi)
(-89.49886791018928, -------------------------------------------)
pi 0.0224730600049057*sin(0.497723041797855*pi)
(44.497723041797855, --------------------------------------------)
pi 0.0289879750158711*sin(0.497062987410978*pi)
(34.49706298741098, --------------------------------------------)
pi 0.0194182175612773*sin(1.49803254826764*pi)
(-51.49803254826764, -------------------------------------------)
pi 0.0273993441732044*sin(0.497223936402321*pi)
(36.49722393640232, --------------------------------------------)
pi 0.0103628070530208*sin(0.498950031931628*pi)
(96.49895003193163, --------------------------------------------)
pi 0.0350920970988578*sin(0.496444575053626*pi)
(28.496444575053626, --------------------------------------------)
pi 0.40666540326472*sin(0.459024032956761*pi)
(2.459024032956761, ------------------------------------------)
pi 0.011560850194741*sin(0.498828646261799*pi)
(86.4988286462618, -------------------------------------------)
pi 0.011299581205505*sin(0.498855117994594*pi)
(88.4988551179946, -------------------------------------------)
pi 0.0132452685482549*sin(1.49865798166468*pi)
(-75.49865798166468, -------------------------------------------)
pi 0.0136056973599145*sin(1.49862146325782*pi)
(-73.49862146325782, -------------------------------------------)
pi 0.0102565195750257*sin(1.49896080098874*pi)
(-97.49896080098874, -------------------------------------------)
pi 0.0119762219451023*sin(1.49878656089503*pi)
(-83.49878656089503, -------------------------------------------)
pi 0.0363685093945229*sin(1.49631526417743*pi)
(-27.496315264177433, -------------------------------------------)
pi 0.0186922504902235*sin(1.49810610140423*pi)
(-53.498106101404225, -------------------------------------------)
pi 0.0219790977035911*sin(1.49777308813785*pi)
(-45.49777308813785, -------------------------------------------)
pi 0.0121213925680119*sin(0.498771852252048*pi)
(82.49877185225205, --------------------------------------------)
pi 0.0137933693371639*sin(0.498602448472525*pi)
(72.49860244847252, --------------------------------------------)
pi 0.0190483192909053*sin(0.49807002539351*pi)
(52.49807002539351, -------------------------------------------)
pi 0.042561001102018*sin(1.49568793278657*pi)
(-23.49568793278657, ------------------------------------------)
pi 0.0235307317651268*sin(0.497615882988754*pi)
(42.497615882988754, --------------------------------------------)
pi 0.0198027669808593*sin(0.497993586783473*pi)
(50.49799358678347, --------------------------------------------)
pi 0.044453342550923*sin(0.495496235282229*pi)
(22.49549623528223, -------------------------------------------)
pi 0.0173918373376325*sin(1.49823785645669*pi)
(-57.49823785645669, -------------------------------------------)
pi 0.0307721827913821*sin(0.496882225724129*pi)
(32.49688222572413, --------------------------------------------)
pi 0.0215063841463171*sin(0.497820981740837*pi)
(46.49782098174084, --------------------------------------------)
pi 0.021053577058045*sin(1.49786685858598*pi)
(-47.49786685858598, ------------------------------------------)
pi 0.0202028556407223*sin(1.49795305097026*pi)
(-49.49795305097026, -------------------------------------------)
pi 0.0104713205202102*sin(1.49893903733958*pi)
(-95.49893903733958, -------------------------------------------)
pi 0.0114287226751903*sin(1.49884203339924*pi)
(-87.49884203339924, -------------------------------------------)
pi 0.033902252551586*sin(1.49656511697539*pi)
(-29.496565116975386, ------------------------------------------)
pi 0.0206194448917246*sin(0.497910843435982*pi)
(48.49791084343598, --------------------------------------------)
pi 0.0150379385346839*sin(0.498476349905047*pi)
(66.49847634990505, --------------------------------------------)
pi 0.0145988553880552*sin(0.498520837339328*pi)
(68.49852083733933, --------------------------------------------)
pi 0.0870232272541852*sin(1.49118495777122*pi)
(-11.491184957771223, -------------------------------------------)
pi 0.117812430106773*sin(0.488068696093448*pi)
(8.488068696093448, -------------------------------------------)
pi 0.154216579070486*sin(0.484387126386342*pi)
(6.484387126386342, -------------------------------------------)
pi 0.0465218265776614*sin(1.49528669796846*pi)
(21.49528669796846, -------------------------------------------)
pi 0.0109290940138763*sin(1.49889265572551*pi)
(-91.49889265572551, -------------------------------------------)
pi 0.0125788180078875*sin(1.49872550608136*pi)
(-79.49872550608136, -------------------------------------------)
pi 0.010050354114363*sin(1.49898168969904*pi)
(-99.49898168969904, ------------------------------------------)
pi 0.0298534417180948*sin(1.49697530499073*pi)
(-33.49697530499073, -------------------------------------------)
pi 0.00995034857432615*sin(0.498991822276054*pi)
(100.49899182227605, ---------------------------------------------)
pi 0.0165293831862789*sin(0.49832523878473*pi)
(60.49832523878473, -------------------------------------------)
pi 0.0134230638362719*sin(0.49863996756018*pi)
(74.49863996756018, -------------------------------------------)
pi 0.0953257545841471*sin(0.490344444294619*pi)
(10.49034444429462, --------------------------------------------)
pi 0.0110498604553187*sin(0.498880419676439*pi)
(90.49888041967644, --------------------------------------------)
pi 0.288110565134169*sin(1.47088972434702*pi)
(3.47088972434702, ------------------------------------------)
pi 0.0162605982080724*sin(1.49835247165517*pi)
(-61.49835247165517, -------------------------------------------)
pi 0.0800519325818731*sin(0.491890798229646*pi)
(12.491890798229646, --------------------------------------------)
pi 0.0118344874613103*sin(0.498800921394462*pi)
(84.49880092139446, --------------------------------------------)
pi 0.0240978032885592*sin(1.49755842993227*pi)
(-41.49755842993227, -------------------------------------------)
pi 0.0116960685435463*sin(1.49881494596634*pi)
(-85.49881494596634, -------------------------------------------)
pi 0.0180186107319224*sin(1.4981743530517*pi)
(-55.498174353051695, ------------------------------------------)
pi 0.0122701258212676*sin(1.49875678264988*pi)
(-81.49875678264988, -------------------------------------------)
pi 0.0183492497971518*sin(0.498140853432595*pi)
(54.498140853432595, --------------------------------------------)
pi 0.0141846863294082*sin(0.498562800557877*pi)
(70.49856280055788, --------------------------------------------)
pi 0.0176996768392689*sin(0.498206666766912*pi)
(56.49820666676691, --------------------------------------------)
pi 0.014815144276589*sin(1.4984989231734*pi)
(-67.4984989231734, -----------------------------------------)
pi 0.0105821306460752*sin(0.498927810052521*pi)
(94.49892781005252, --------------------------------------------)
pi 0.223343476782583*sin(0.477408583432515*pi)
(4.477408583432515, -------------------------------------------)
pi 0.0127390629641911*sin(0.498709270136345*pi)
(78.49870927013635, --------------------------------------------)
pi 0.0408232181963382*sin(0.495863975998319*pi)
(24.49586397599832, --------------------------------------------)
pi 0.0155042535748646*sin(0.498429103429686*pi)
(64.49842910342969, --------------------------------------------)
pi 0.0645433567073861*sin(1.49346131056681*pi)
(-15.493461310566811, -------------------------------------------)
pi 0.0317492739662746*sin(1.49678323549202*pi)
(-31.496783235492018, -------------------------------------------)
pi 0.0741152934677989*sin(1.49249194344043*pi)
(-13.492491943440427, -------------------------------------------)
pi 0.0377412949242607*sin(0.496176191272795*pi)
(26.496176191272795, --------------------------------------------)
pi 0.0606286298967913*sin(0.493857797913467*pi)
(16.493857797913467, --------------------------------------------)
pi 0.025318099901244*sin(1.49743479568401*pi)
(-39.49743479568401, ------------------------------------------)
pi 0.0512957219377684*sin(1.49480311853674*pi)
(-19.494803118536737, -------------------------------------------)
pi 0.0170945232147019*sin(0.498267979768038*pi)
(58.49826797976804, --------------------------------------------)
pi 0.0124225544818555*sin(0.498741338636094*pi)
(80.4987413386361, --------------------------------------------)
pi 0.0465218265776614*sin(1.49528669796846*pi)
(-21.49528669796846, -------------------------------------------)
pi 0.0327904569838132*sin(0.496677752726768*pi)
(30.496677752726768, --------------------------------------------)
pi 0.0540700643523084*sin(0.494522097925103*pi)
(18.494522097925103, --------------------------------------------)
pi 0.0281712791115643*sin(1.49714572915862*pi)
(-35.49714572915862, -------------------------------------------)
pi 0.105381555535677*sin(1.48932661808597*pi)
(-9.48932661808597, ------------------------------------------)
pi 0.0392217984035961*sin(1.49602620741414*pi)
(-25.496026207414143, -------------------------------------------)
pi 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:
$$x_{1} = -93.4989163426041$$
$$x_{2} = -77.4986926151852$$
$$x_{3} = -17.4942089405157$$
$$x_{4} = -43.4976706942357$$
$$x_{5} = -65.4984530873445$$
$$x_{6} = -37.4972979719751$$
$$x_{7} = -59.4982970904721$$
$$x_{8} = -69.4985421208549$$
$$x_{9} = -5.4815366506072$$
$$x_{10} = -71.4985829017843$$
$$x_{11} = -7.48647424796277$$
$$x_{12} = -63.4984043640786$$
$$x_{13} = -89.4988679101893$$
$$x_{14} = -51.4980325482676$$
$$x_{15} = -75.4986579816647$$
$$x_{16} = -73.4986214632578$$
$$x_{17} = -97.4989608009887$$
$$x_{18} = -83.498786560895$$
$$x_{19} = -27.4963152641774$$
$$x_{20} = -53.4981061014042$$
$$x_{21} = -45.4977730881378$$
$$x_{22} = -23.4956879327866$$
$$x_{23} = -57.4982378564567$$
$$x_{24} = -47.497866858586$$
$$x_{25} = -49.4979530509703$$
$$x_{26} = -95.4989390373396$$
$$x_{27} = -87.4988420333992$$
$$x_{28} = -29.4965651169754$$
$$x_{29} = -11.4911849577712$$
$$x_{30} = 21.4952866979685$$
$$x_{31} = -91.4988926557255$$
$$x_{32} = -79.4987255060814$$
$$x_{33} = -99.498981689699$$
$$x_{34} = -33.4969753049907$$
$$x_{35} = 3.47088972434702$$
$$x_{36} = -61.4983524716552$$
$$x_{37} = -41.4975584299323$$
$$x_{38} = -85.4988149459663$$
$$x_{39} = -55.4981743530517$$
$$x_{40} = -81.4987567826499$$
$$x_{41} = -67.4984989231734$$
$$x_{42} = -15.4934613105668$$
$$x_{43} = -31.496783235492$$
$$x_{44} = -13.4924919434404$$
$$x_{45} = -39.497434795684$$
$$x_{46} = -19.4948031185367$$
$$x_{47} = -21.4952866979685$$
$$x_{48} = -35.4971457291586$$
$$x_{49} = -9.48932661808597$$
$$x_{50} = -25.4960262074141$$
Puntos máximos de la función:
$$x_{50} = 62.4983788330278$$
$$x_{50} = 92.498904627205$$
$$x_{50} = 98.4989713513802$$
$$x_{50} = 14.4930100858125$$
$$x_{50} = 38.4973681610039$$
$$x_{50} = 20.4950567085724$$
$$x_{50} = 76.4986755247957$$
$$x_{50} = 40.4974981393456$$
$$x_{50} = 44.4977230417979$$
$$x_{50} = 34.497062987411$$
$$x_{50} = 36.4972239364023$$
$$x_{50} = 96.4989500319316$$
$$x_{50} = 28.4964445750536$$
$$x_{50} = 2.45902403295676$$
$$x_{50} = 86.4988286462618$$
$$x_{50} = 88.4988551179946$$
$$x_{50} = 82.498771852252$$
$$x_{50} = 72.4986024484725$$
$$x_{50} = 52.4980700253935$$
$$x_{50} = 42.4976158829888$$
$$x_{50} = 50.4979935867835$$
$$x_{50} = 22.4954962352822$$
$$x_{50} = 32.4968822257241$$
$$x_{50} = 46.4978209817408$$
$$x_{50} = 48.497910843436$$
$$x_{50} = 66.498476349905$$
$$x_{50} = 68.4985208373393$$
$$x_{50} = 8.48806869609345$$
$$x_{50} = 6.48438712638634$$
$$x_{50} = 100.498991822276$$
$$x_{50} = 60.4983252387847$$
$$x_{50} = 74.4986399675602$$
$$x_{50} = 10.4903444442946$$
$$x_{50} = 90.4988804196764$$
$$x_{50} = 12.4918907982296$$
$$x_{50} = 84.4988009213945$$
$$x_{50} = 54.4981408534326$$
$$x_{50} = 70.4985628005579$$
$$x_{50} = 56.4982066667669$$
$$x_{50} = 94.4989278100525$$
$$x_{50} = 4.47740858343252$$
$$x_{50} = 78.4987092701363$$
$$x_{50} = 24.4958639759983$$
$$x_{50} = 64.4984291034297$$
$$x_{50} = 26.4961761912728$$
$$x_{50} = 16.4938577979135$$
$$x_{50} = 58.498267979768$$
$$x_{50} = 80.4987413386361$$
$$x_{50} = 30.4966777527268$$
$$x_{50} = 18.4945220979251$$
Decrece en los intervalos
$$\left[21.4952866979685, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.498981689699\right]$$
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
$$\frac{- \pi \sin{\left(\pi x \right)} - \frac{2 \cos{\left(\pi x \right)}}{x} + \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -79.9974668634637$$
$$x_{2} = 43.9953938487458$$
$$x_{3} = -91.997797295256$$
$$x_{4} = 81.9975286523095$$
$$x_{5} = 97.9979321625662$$
$$x_{6} = -27.9927602769161$$
$$x_{7} = 1.89087849558881$$
$$x_{8} = 27.9927602769161$$
$$x_{9} = 51.9961026419358$$
$$x_{10} = 11.9830813457211$$
$$x_{11} = -77.9974019055472$$
$$x_{12} = -19.9898610294878$$
$$x_{13} = 47.9957777888324$$
$$x_{14} = 79.9974668634637$$
$$x_{15} = -59.9966223736973$$
$$x_{16} = 31.9936657543493$$
$$x_{17} = -53.9962470158014$$
$$x_{18} = 3.94845748309471$$
$$x_{19} = 63.996833504125$$
$$x_{20} = 65.9969294706369$$
$$x_{21} = 53.9962470158014$$
$$x_{22} = -97.9979321625662$$
$$x_{23} = 5.96596938287278$$
$$x_{24} = 9.9796807491065$$
$$x_{25} = -63.996833504125$$
$$x_{26} = -15.9873214599582$$
$$x_{27} = -7.97456196377526$$
$$x_{28} = 25.9922029454487$$
$$x_{29} = 67.9970197910395$$
$$x_{30} = 69.9971049493967$$
$$x_{31} = 23.9915526041107$$
$$x_{32} = -67.9970197910395$$
$$x_{33} = 45.9955941685152$$
$$x_{34} = -83.9975874984829$$
$$x_{35} = 95.9978890801192$$
$$x_{36} = -29.9932432255018$$
$$x_{37} = 71.9971853759727$$
$$x_{38} = -49.9959467145556$$
$$x_{39} = -39.9949330850646$$
$$x_{40} = -43.9953938487458$$
$$x_{41} = 73.9972614545044$$
$$x_{42} = -57.9965058854237$$
$$x_{43} = 91.997797295256$$
$$x_{44} = -89.9977483430314$$
$$x_{45} = -87.9976971654778$$
$$x_{46} = -13.985505543252$$
$$x_{47} = -1.89087849558881$$
$$x_{48} = -3.94845748309471$$
$$x_{49} = -81.9975286523095$$
$$x_{50} = 59.9966223736973$$
$$x_{51} = -71.9971853759727$$
$$x_{52} = -65.9969294706369$$
$$x_{53} = -51.9961026419358$$
$$x_{54} = 83.9975874984829$$
$$x_{55} = 61.9967313449593$$
$$x_{56} = -9.9796807491065$$
$$x_{57} = 75.9973335283253$$
$$x_{58} = 57.9965058854237$$
$$x_{59} = -99.9979735215726$$
$$x_{60} = -95.9978890801192$$
$$x_{61} = -31.9936657543493$$
$$x_{62} = -75.9973335283253$$
$$x_{63} = 19.9898610294878$$
$$x_{64} = 13.985505543252$$
$$x_{65} = 17.9887326886962$$
$$x_{66} = -85.9976436073178$$
$$x_{67} = 77.9974019055472$$
$$x_{68} = 85.9976436073178$$
$$x_{69} = -61.9967313449593$$
$$x_{70} = 93.9978441642119$$
$$x_{71} = -33.9940385367634$$
$$x_{72} = -73.9972614545044$$
$$x_{73} = 21.9907838362503$$
$$x_{74} = 15.9873214599582$$
$$x_{75} = 33.9940385367634$$
$$x_{76} = -45.9955941685152$$
$$x_{77} = -23.9915526041107$$
$$x_{78} = -21.9907838362503$$
$$x_{79} = 55.9963810744805$$
$$x_{80} = -69.9971049493967$$
$$x_{81} = -37.9946663079835$$
$$x_{82} = -35.9943698713985$$
$$x_{83} = -55.9963810744805$$
$$x_{84} = 49.9959467145556$$
$$x_{85} = 99.9979735215726$$
$$x_{86} = 37.9946663079835$$
$$x_{87} = -5.96596938287278$$
$$x_{88} = -93.9978441642119$$
$$x_{89} = 35.9943698713985$$
$$x_{90} = -47.9957777888324$$
$$x_{91} = 29.9932432255018$$
$$x_{92} = -41.99517444252$$
$$x_{93} = 7.97456196377526$$
$$x_{94} = 89.9977483430314$$
$$x_{95} = 41.99517444252$$
$$x_{96} = -17.9887326886962$$
$$x_{97} = 39.9949330850646$$
$$x_{98} = 87.9976971654778$$
$$x_{99} = -25.9922029454487$$
$$x_{100} = -11.9830813457211$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \pi \sin{\left(\pi x \right)} - \frac{2 \cos{\left(\pi x \right)}}{x} + \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}}{x}\right) = - \frac{\pi^{2}}{3}$$
$$\lim_{x \to 0^+}\left(\frac{- \pi \sin{\left(\pi x \right)} - \frac{2 \cos{\left(\pi x \right)}}{x} + \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}}{x}\right) = - \frac{\pi^{2}}{3}$$
- 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
$$\left[-1.89087849558881, 1.89087849558881\right]$$
Convexa en los intervalos
$$\left(-\infty, -99.9979735215726\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \pi \sin{\left(\pi x \right)} - \frac{2 \cos{\left(\pi x \right)}}{x} + \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -79.9974668634637$$
$$x_{2} = 43.9953938487458$$
$$x_{3} = -91.997797295256$$
$$x_{4} = 81.9975286523095$$
$$x_{5} = 97.9979321625662$$
$$x_{6} = -27.9927602769161$$
$$x_{7} = 1.89087849558881$$
$$x_{8} = 27.9927602769161$$
$$x_{9} = 51.9961026419358$$
$$x_{10} = 11.9830813457211$$
$$x_{11} = -77.9974019055472$$
$$x_{12} = -19.9898610294878$$
$$x_{13} = 47.9957777888324$$
$$x_{14} = 79.9974668634637$$
$$x_{15} = -59.9966223736973$$
$$x_{16} = 31.9936657543493$$
$$x_{17} = -53.9962470158014$$
$$x_{18} = 3.94845748309471$$
$$x_{19} = 63.996833504125$$
$$x_{20} = 65.9969294706369$$
$$x_{21} = 53.9962470158014$$
$$x_{22} = -97.9979321625662$$
$$x_{23} = 5.96596938287278$$
$$x_{24} = 9.9796807491065$$
$$x_{25} = -63.996833504125$$
$$x_{26} = -15.9873214599582$$
$$x_{27} = -7.97456196377526$$
$$x_{28} = 25.9922029454487$$
$$x_{29} = 67.9970197910395$$
$$x_{30} = 69.9971049493967$$
$$x_{31} = 23.9915526041107$$
$$x_{32} = -67.9970197910395$$
$$x_{33} = 45.9955941685152$$
$$x_{34} = -83.9975874984829$$
$$x_{35} = 95.9978890801192$$
$$x_{36} = -29.9932432255018$$
$$x_{37} = 71.9971853759727$$
$$x_{38} = -49.9959467145556$$
$$x_{39} = -39.9949330850646$$
$$x_{40} = -43.9953938487458$$
$$x_{41} = 73.9972614545044$$
$$x_{42} = -57.9965058854237$$
$$x_{43} = 91.997797295256$$
$$x_{44} = -89.9977483430314$$
$$x_{45} = -87.9976971654778$$
$$x_{46} = -13.985505543252$$
$$x_{47} = -1.89087849558881$$
$$x_{48} = -3.94845748309471$$
$$x_{49} = -81.9975286523095$$
$$x_{50} = 59.9966223736973$$
$$x_{51} = -71.9971853759727$$
$$x_{52} = -65.9969294706369$$
$$x_{53} = -51.9961026419358$$
$$x_{54} = 83.9975874984829$$
$$x_{55} = 61.9967313449593$$
$$x_{56} = -9.9796807491065$$
$$x_{57} = 75.9973335283253$$
$$x_{58} = 57.9965058854237$$
$$x_{59} = -99.9979735215726$$
$$x_{60} = -95.9978890801192$$
$$x_{61} = -31.9936657543493$$
$$x_{62} = -75.9973335283253$$
$$x_{63} = 19.9898610294878$$
$$x_{64} = 13.985505543252$$
$$x_{65} = 17.9887326886962$$
$$x_{66} = -85.9976436073178$$
$$x_{67} = 77.9974019055472$$
$$x_{68} = 85.9976436073178$$
$$x_{69} = -61.9967313449593$$
$$x_{70} = 93.9978441642119$$
$$x_{71} = -33.9940385367634$$
$$x_{72} = -73.9972614545044$$
$$x_{73} = 21.9907838362503$$
$$x_{74} = 15.9873214599582$$
$$x_{75} = 33.9940385367634$$
$$x_{76} = -45.9955941685152$$
$$x_{77} = -23.9915526041107$$
$$x_{78} = -21.9907838362503$$
$$x_{79} = 55.9963810744805$$
$$x_{80} = -69.9971049493967$$
$$x_{81} = -37.9946663079835$$
$$x_{82} = -35.9943698713985$$
$$x_{83} = -55.9963810744805$$
$$x_{84} = 49.9959467145556$$
$$x_{85} = 99.9979735215726$$
$$x_{86} = 37.9946663079835$$
$$x_{87} = -5.96596938287278$$
$$x_{88} = -93.9978441642119$$
$$x_{89} = 35.9943698713985$$
$$x_{90} = -47.9957777888324$$
$$x_{91} = 29.9932432255018$$
$$x_{92} = -41.99517444252$$
$$x_{93} = 7.97456196377526$$
$$x_{94} = 89.9977483430314$$
$$x_{95} = 41.99517444252$$
$$x_{96} = -17.9887326886962$$
$$x_{97} = 39.9949330850646$$
$$x_{98} = 87.9976971654778$$
$$x_{99} = -25.9922029454487$$
$$x_{100} = -11.9830813457211$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \pi \sin{\left(\pi x \right)} - \frac{2 \cos{\left(\pi x \right)}}{x} + \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}}{x}\right) = - \frac{\pi^{2}}{3}$$
$$\lim_{x \to 0^+}\left(\frac{- \pi \sin{\left(\pi x \right)} - \frac{2 \cos{\left(\pi x \right)}}{x} + \frac{2 \sin{\left(\pi x \right)}}{\pi x^{2}}}{x}\right) = - \frac{\pi^{2}}{3}$$
- 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
$$\left[-1.89087849558881, 1.89087849558881\right]$$
Convexa en los intervalos
$$\left(-\infty, -99.9979735215726\right]$$
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
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\pi x \right)}}{\pi x}\right) = 0$$
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)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\pi x} \sin{\left(\pi x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\pi x} \sin{\left(\pi x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\pi x} \sin{\left(\pi x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\pi x} \sin{\left(\pi x \right)}}{x}\right) = 0$$
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:
$$\frac{\sin{\left(\pi x \right)}}{\pi x} = \frac{\sin{\left(\pi x \right)}}{\pi x}$$
- No
$$\frac{\sin{\left(\pi x \right)}}{\pi x} = - \frac{\sin{\left(\pi x \right)}}{\pi x}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(\pi x \right)}}{\pi x} = \frac{\sin{\left(\pi x \right)}}{\pi x}$$
- No
$$\frac{\sin{\left(\pi x \right)}}{\pi x} = - \frac{\sin{\left(\pi x \right)}}{\pi x}$$
- No
es decir, función
no es
par ni impar