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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x-x^3
-
x^4-x^3
-
x*2
-
x/(x-2)
-
Expresiones idénticas
- (sin(tres *x))/(dos *x)+(x- uno)/(x^ dos - cuatro)
- ( seno de (3 multiplicar por x)) dividir por (2 multiplicar por x) más (x menos 1) dividir por (x al cuadrado menos 4)
- ( seno de (tres multiplicar por x)) dividir por (dos multiplicar por x) más (x menos uno) dividir por (x en el grado dos menos cuatro)
- (sin(3*x))/(2*x)+(x-1)/(x2-4)
- sin3*x/2*x+x-1/x2-4
- (sin(3*x))/(2*x)+(x-1)/(x²-4)
- (sin(3*x))/(2*x)+(x-1)/(x en el grado 2-4)
- (sin(3x))/(2x)+(x-1)/(x^2-4)
- (sin(3x))/(2x)+(x-1)/(x2-4)
- sin3x/2x+x-1/x2-4
- sin3x/2x+x-1/x^2-4
- (sin(3*x)) dividir por (2*x)+(x-1) dividir por (x^2-4)
-
Expresiones semejantes
- (sin(3*x))/(2*x)-(x-1)/(x^2-4)
- (sin(3*x))/(2*x)+(x-1)/(x^2+4)
- (sin(3*x))/(2*x)+(x+1)/(x^2-4)
-
Expresiones con funciones
- Seno sin
- sin(x)/(2*x+16)
- sin(10*x)
- sinx/(2+cosx)
- sin(x^x)
- sin(x)*cos(x)+x
Gráfico de la función y = (sin(3*x))/(2*x)+(x-1)/(x^2-4)
Solución
Ha introducido
[src]
sin(3*x) x - 1
f(x) = -------- + ------
2*x 2
x - 4
$$f{\left(x \right)} = \frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}$$
f = (x - 1)/(x^2 - 4) + sin(3*x)/((2*x))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
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:
$$\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x} = 0$$
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
o sea hay que resolver la ecuación:
$$\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x} = 0$$
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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
$$3 \frac{1}{2 x} \cos{\left(3 x \right)} - \frac{2 x \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} + \frac{1}{x^{2} - 4} - \frac{\sin{\left(3 x \right)}}{2 x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -6.8442016972462$$
$$x_{2} = 29.8343601469208$$
$$x_{3} = -45.5455313755881$$
$$x_{4} = -36.1317902652531$$
$$x_{5} = -100.003985793389$$
$$x_{6} = 97.9140622726068$$
$$x_{7} = -49.7349794816344$$
$$x_{8} = -84.3007879836724$$
$$x_{9} = -14.1102331426194$$
$$x_{10} = 73.8229870916504$$
$$x_{11} = -25.6615132195048$$
$$x_{12} = -97.9095172876736$$
$$x_{13} = -93.7205702713482$$
$$x_{14} = -9.90684805563754$$
$$x_{15} = 53.932597935885$$
$$x_{16} = -62.3027782930547$$
$$x_{17} = 42.4038530963323$$
$$x_{18} = 91.630948817728$$
$$x_{19} = 49.7439589271831$$
$$x_{20} = 86.3899949788424$$
$$x_{21} = 84.2955066914144$$
$$x_{22} = -38.2259712921274$$
$$x_{23} = -27.7554476509841$$
$$x_{24} = -43.4507577981096$$
$$x_{25} = -34.0376393480567$$
$$x_{26} = 9.95756673500463$$
$$x_{27} = 31.9294426102745$$
$$x_{28} = -89.5316083266597$$
$$x_{29} = 100.008435359475$$
$$x_{30} = -67.5459933026873$$
$$x_{31} = -56.0189614233548$$
$$x_{32} = -65.4516580560656$$
$$x_{33} = 22.5190281723597$$
$$x_{34} = 34.0244420673533$$
$$x_{35} = -21.4739793725166$$
$$x_{36} = -31.9435253678456$$
$$x_{37} = -78.0177204974878$$
$$x_{38} = -12.0101369031188$$
$$x_{39} = 71.7284643872573$$
$$x_{40} = 18.3311063703045$$
$$x_{41} = 56.026925449142$$
$$x_{42} = 88.4844789091251$$
$$x_{43} = -53.9243220629616$$
$$x_{44} = 64.4042772811497$$
$$x_{45} = 58.1212575693941$$
$$x_{46} = 16.2373072260106$$
$$x_{47} = 60.2155938389814$$
$$x_{48} = 78.0120122217482$$
$$x_{49} = 9287.07144689999$$
$$x_{50} = -16.2086080610791$$
$$x_{51} = -95.8150455211189$$
$$x_{52} = 3.72465876314914$$
$$x_{53} = 27.7391761293701$$
$$x_{54} = 95.8196901125633$$
$$x_{55} = -18.3059450316161$$
$$x_{56} = 44.4986000648908$$
$$x_{57} = -69.6403323820584$$
$$x_{58} = 40.3090703092807$$
$$x_{59} = 51.8382755576786$$
$$x_{60} = -60.2081877256431$$
$$x_{61} = -82.2064300459562$$
$$x_{62} = 66.4986237979768$$
$$x_{63} = -184.830982372558$$
$$x_{64} = 38.2142459113972$$
$$x_{65} = -51.8296623064097$$
$$x_{66} = -58.1135826616926$$
$$x_{67} = 80.1065156951404$$
$$x_{68} = -23.5676784432488$$
$$x_{69} = -75.9233692564601$$
$$x_{70} = 82.2010137178493$$
$$x_{71} = 7.86603628726821$$
$$x_{72} = -29.8494573795171$$
$$x_{73} = -73.8290206544931$$
$$x_{74} = -91.6260912952803$$
$$x_{75} = 208.916433417059$$
$$x_{76} = 93.7253189392189$$
$$x_{77} = -7.79646958415439$$
$$x_{78} = 14.1436862283443$$
$$x_{79} = -8495.39014716725$$
$$x_{80} = -71.7346749372935$$
$$x_{81} = -47.6402704273772$$
$$x_{82} = 36.1193727823579$$
$$x_{83} = 12.0503556217359$$
$$x_{84} = -374.372230951242$$
$$x_{85} = 75.9175028523412$$
$$x_{86} = -80.1120741590105$$
$$x_{87} = 62.3099338592203$$
$$x_{88} = -5.66377961266401$$
$$x_{89} = -320.96640001395$$
$$x_{90} = 68.5929731388662$$
$$x_{91} = 5.7788018906052$$
$$x_{92} = 20.4250248320047$$
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:
$$x_{1} = -45.5455313755881$$
$$x_{2} = -100.003985793389$$
$$x_{3} = 97.9140622726068$$
$$x_{4} = -49.7349794816344$$
$$x_{5} = -14.1102331426194$$
$$x_{6} = -97.9095172876736$$
$$x_{7} = -93.7205702713482$$
$$x_{8} = -9.90684805563754$$
$$x_{9} = 53.932597935885$$
$$x_{10} = -62.3027782930547$$
$$x_{11} = 91.630948817728$$
$$x_{12} = 49.7439589271831$$
$$x_{13} = -43.4507577981096$$
$$x_{14} = 9.95756673500463$$
$$x_{15} = -89.5316083266597$$
$$x_{16} = 100.008435359475$$
$$x_{17} = -56.0189614233548$$
$$x_{18} = 22.5190281723597$$
$$x_{19} = -12.0101369031188$$
$$x_{20} = 18.3311063703045$$
$$x_{21} = 56.026925449142$$
$$x_{22} = -53.9243220629616$$
$$x_{23} = 64.4042772811497$$
$$x_{24} = 58.1212575693941$$
$$x_{25} = 16.2373072260106$$
$$x_{26} = 60.2155938389814$$
$$x_{27} = -16.2086080610791$$
$$x_{28} = -95.8150455211189$$
$$x_{29} = 3.72465876314914$$
$$x_{30} = 95.8196901125633$$
$$x_{31} = -18.3059450316161$$
$$x_{32} = 51.8382755576786$$
$$x_{33} = -60.2081877256431$$
$$x_{34} = 66.4986237979768$$
$$x_{35} = -51.8296623064097$$
$$x_{36} = -58.1135826616926$$
$$x_{37} = 7.86603628726821$$
$$x_{38} = -91.6260912952803$$
$$x_{39} = 208.916433417059$$
$$x_{40} = 93.7253189392189$$
$$x_{41} = -7.79646958415439$$
$$x_{42} = 14.1436862283443$$
$$x_{43} = -47.6402704273772$$
$$x_{44} = 12.0503556217359$$
$$x_{45} = -374.372230951242$$
$$x_{46} = 62.3099338592203$$
$$x_{47} = -5.66377961266401$$
$$x_{48} = 68.5929731388662$$
$$x_{49} = 5.7788018906052$$
$$x_{50} = 20.4250248320047$$
Puntos máximos de la función:
$$x_{50} = -6.8442016972462$$
$$x_{50} = 29.8343601469208$$
$$x_{50} = -36.1317902652531$$
$$x_{50} = -84.3007879836724$$
$$x_{50} = 73.8229870916504$$
$$x_{50} = -25.6615132195048$$
$$x_{50} = 42.4038530963323$$
$$x_{50} = 86.3899949788424$$
$$x_{50} = 84.2955066914144$$
$$x_{50} = -38.2259712921274$$
$$x_{50} = -27.7554476509841$$
$$x_{50} = -34.0376393480567$$
$$x_{50} = 31.9294426102745$$
$$x_{50} = -67.5459933026873$$
$$x_{50} = -65.4516580560656$$
$$x_{50} = 34.0244420673533$$
$$x_{50} = -21.4739793725166$$
$$x_{50} = -31.9435253678456$$
$$x_{50} = -78.0177204974878$$
$$x_{50} = 71.7284643872573$$
$$x_{50} = 88.4844789091251$$
$$x_{50} = 78.0120122217482$$
$$x_{50} = 9287.07144689999$$
$$x_{50} = 27.7391761293701$$
$$x_{50} = 44.4986000648908$$
$$x_{50} = -69.6403323820584$$
$$x_{50} = 40.3090703092807$$
$$x_{50} = -82.2064300459562$$
$$x_{50} = -184.830982372558$$
$$x_{50} = 38.2142459113972$$
$$x_{50} = 80.1065156951404$$
$$x_{50} = -23.5676784432488$$
$$x_{50} = -75.9233692564601$$
$$x_{50} = 82.2010137178493$$
$$x_{50} = -29.8494573795171$$
$$x_{50} = -73.8290206544931$$
$$x_{50} = -8495.39014716725$$
$$x_{50} = -71.7346749372935$$
$$x_{50} = 36.1193727823579$$
$$x_{50} = 75.9175028523412$$
$$x_{50} = -80.1120741590105$$
$$x_{50} = -320.96640001395$$
Decrece en los intervalos
$$\left[208.916433417059, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -374.372230951242\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$3 \frac{1}{2 x} \cos{\left(3 x \right)} - \frac{2 x \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} + \frac{1}{x^{2} - 4} - \frac{\sin{\left(3 x \right)}}{2 x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -6.8442016972462$$
$$x_{2} = 29.8343601469208$$
$$x_{3} = -45.5455313755881$$
$$x_{4} = -36.1317902652531$$
$$x_{5} = -100.003985793389$$
$$x_{6} = 97.9140622726068$$
$$x_{7} = -49.7349794816344$$
$$x_{8} = -84.3007879836724$$
$$x_{9} = -14.1102331426194$$
$$x_{10} = 73.8229870916504$$
$$x_{11} = -25.6615132195048$$
$$x_{12} = -97.9095172876736$$
$$x_{13} = -93.7205702713482$$
$$x_{14} = -9.90684805563754$$
$$x_{15} = 53.932597935885$$
$$x_{16} = -62.3027782930547$$
$$x_{17} = 42.4038530963323$$
$$x_{18} = 91.630948817728$$
$$x_{19} = 49.7439589271831$$
$$x_{20} = 86.3899949788424$$
$$x_{21} = 84.2955066914144$$
$$x_{22} = -38.2259712921274$$
$$x_{23} = -27.7554476509841$$
$$x_{24} = -43.4507577981096$$
$$x_{25} = -34.0376393480567$$
$$x_{26} = 9.95756673500463$$
$$x_{27} = 31.9294426102745$$
$$x_{28} = -89.5316083266597$$
$$x_{29} = 100.008435359475$$
$$x_{30} = -67.5459933026873$$
$$x_{31} = -56.0189614233548$$
$$x_{32} = -65.4516580560656$$
$$x_{33} = 22.5190281723597$$
$$x_{34} = 34.0244420673533$$
$$x_{35} = -21.4739793725166$$
$$x_{36} = -31.9435253678456$$
$$x_{37} = -78.0177204974878$$
$$x_{38} = -12.0101369031188$$
$$x_{39} = 71.7284643872573$$
$$x_{40} = 18.3311063703045$$
$$x_{41} = 56.026925449142$$
$$x_{42} = 88.4844789091251$$
$$x_{43} = -53.9243220629616$$
$$x_{44} = 64.4042772811497$$
$$x_{45} = 58.1212575693941$$
$$x_{46} = 16.2373072260106$$
$$x_{47} = 60.2155938389814$$
$$x_{48} = 78.0120122217482$$
$$x_{49} = 9287.07144689999$$
$$x_{50} = -16.2086080610791$$
$$x_{51} = -95.8150455211189$$
$$x_{52} = 3.72465876314914$$
$$x_{53} = 27.7391761293701$$
$$x_{54} = 95.8196901125633$$
$$x_{55} = -18.3059450316161$$
$$x_{56} = 44.4986000648908$$
$$x_{57} = -69.6403323820584$$
$$x_{58} = 40.3090703092807$$
$$x_{59} = 51.8382755576786$$
$$x_{60} = -60.2081877256431$$
$$x_{61} = -82.2064300459562$$
$$x_{62} = 66.4986237979768$$
$$x_{63} = -184.830982372558$$
$$x_{64} = 38.2142459113972$$
$$x_{65} = -51.8296623064097$$
$$x_{66} = -58.1135826616926$$
$$x_{67} = 80.1065156951404$$
$$x_{68} = -23.5676784432488$$
$$x_{69} = -75.9233692564601$$
$$x_{70} = 82.2010137178493$$
$$x_{71} = 7.86603628726821$$
$$x_{72} = -29.8494573795171$$
$$x_{73} = -73.8290206544931$$
$$x_{74} = -91.6260912952803$$
$$x_{75} = 208.916433417059$$
$$x_{76} = 93.7253189392189$$
$$x_{77} = -7.79646958415439$$
$$x_{78} = 14.1436862283443$$
$$x_{79} = -8495.39014716725$$
$$x_{80} = -71.7346749372935$$
$$x_{81} = -47.6402704273772$$
$$x_{82} = 36.1193727823579$$
$$x_{83} = 12.0503556217359$$
$$x_{84} = -374.372230951242$$
$$x_{85} = 75.9175028523412$$
$$x_{86} = -80.1120741590105$$
$$x_{87} = 62.3099338592203$$
$$x_{88} = -5.66377961266401$$
$$x_{89} = -320.96640001395$$
$$x_{90} = 68.5929731388662$$
$$x_{91} = 5.7788018906052$$
$$x_{92} = 20.4250248320047$$
Signos de extremos en los puntos:
(-6.844201697246203, -0.110496626300929)
(29.83436014692076, 0.0492916066775978)
(-45.54553137558814, -0.0334566698491087)
(-36.13179026525313, -0.014692381490361)
(-100.00398579338876, -0.0151031782262389)
(97.91406227260678, 0.00500645940015039)
(-49.73497948163444, -0.0305951992568579)
(-84.30078798367242, -0.00607866743628605)
(-14.110233142619435, -0.112768844529317)
(73.82298709165045, 0.0201445964052492)
(-25.661513219504762, -0.0212527857853032)
(-97.90951728767358, -0.015428616296855)
(-93.72057027134824, -0.0161234737103264)
(-9.906848055637544, -0.16592976045388)
(53.93259793588496, 0.00895225187409264)
(-62.30277829305467, -0.0243493361809332)
(42.40385309633234, 0.0348662327962823)
(91.63094881772804, 0.00534274867218842)
(49.74395892718314, 0.00967943320418455)
(86.38999497884242, 0.017234890290084)
(84.29550669141443, 0.0176600086515727)
(-38.225971292127404, -0.0138387823909699)
(-27.75544765098412, -0.0195091945333374)
(-43.45075779810961, -0.0350982310905786)
(-34.03763934805667, -0.0156584497267228)
(9.957566735004626, 0.0439444416179827)
(31.92944261027449, 0.0461100390926758)
(-89.53160832665971, -0.0168838906553467)
(100.00843535947489, 0.00490358144209832)
(-67.54599330268731, -0.00763482898780914)
(-56.01896142335479, -0.0271170008964228)
(-65.45165805606563, -0.00788726708402482)
(22.51902817235966, 0.0205706870571991)
(34.02444206735335, 0.0433151130269109)
(-21.473979372516613, -0.0258833577183852)
(-31.943525367845556, -0.0167608285744374)
(-78.01772049748784, -0.00658169329213978)
(-12.01013690311882, -0.134200404556965)
(71.72846438725732, 0.0207278772106654)
(18.33110637030448, 0.0249247096319015)
(56.026925449142, 0.00862821450183353)
(88.48447890912506, 0.0168297642176638)
(-53.92432206296164, -0.0281849953646293)
(64.40427728114967, 0.0075372218732648)
(58.1212575693941, 0.00832684906723533)
(16.237307226010618, 0.027895282996978)
(60.21559383898144, 0.00804585311187564)
(78.01201222174818, 0.0190713056842914)
(9287.071446899987, 0.00016150326563976)
(-16.20860806107913, -0.0972893685144118)
(-95.81504552111886, -0.0157683917696909)
(3.7246587631491357, 0.14385849613445)
(27.73917612937014, 0.0529462923869054)
(95.81969011256334, 0.00511374883592063)
(-18.3059450316161, -0.0855714815578505)
(44.49860006489079, 0.0332456817464416)
(-69.64033238205843, -0.00739805882234463)
(40.309070309280735, 0.0366531171939254)
(51.83827555767863, 0.00930162503199509)
(-60.20818772564306, -0.0252068652113071)
(-82.20643004595622, -0.00623757174245023)
(66.49862379797683, 0.00730630857246911)
(-184.83098237255774, -0.00273508762690104)
(38.21424591139722, 0.0386333751941908)
(-51.82966230640966, -0.0293406406378047)
(-58.11358266169263, -0.0261270431159949)
(80.10651569514039, 0.0185764514089803)
(-23.567678443248848, -0.0233398694294342)
(-75.92336925646015, -0.00676840491806044)
(82.20101371784929, 0.0181066371679824)
(7.866036287268207, 0.0551137316660185)
(-29.849457379517137, -0.0180306221782676)
(-73.82902065449306, -0.00696602568961221)
(-91.6260912952803, -0.0164949205542315)
(208.9164334170588, 0.00237082941710456)
(93.72531893921894, 0.00522573987564745)
(-7.7964695841543925, -0.218087899388977)
(14.143686228344317, 0.0316999185816176)
(-8495.39014716725, -5.88693112440019e-5)
(-71.73467493729352, -0.00717554103572266)
(-47.640270427377175, -0.0319619516666842)
(36.11937278235789, 0.0408402560702292)
(12.050355621735926, 0.0367723604210944)
(-374.3722309512423, -0.00401391405223129)
(75.9175028523412, 0.0195932575409922)
(-80.11207415901049, -0.0064050113545272)
(62.30993385922029, 0.00778322619509438)
(-5.663779612664006, -0.321986473907335)
(-320.96640001395, -0.00156762457855618)
(68.59297313886621, 0.00708913748775013)
(5.7788018906052, 0.0761950080632073)
(20.425024832004656, 0.0225359124294104)
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:
$$x_{1} = -45.5455313755881$$
$$x_{2} = -100.003985793389$$
$$x_{3} = 97.9140622726068$$
$$x_{4} = -49.7349794816344$$
$$x_{5} = -14.1102331426194$$
$$x_{6} = -97.9095172876736$$
$$x_{7} = -93.7205702713482$$
$$x_{8} = -9.90684805563754$$
$$x_{9} = 53.932597935885$$
$$x_{10} = -62.3027782930547$$
$$x_{11} = 91.630948817728$$
$$x_{12} = 49.7439589271831$$
$$x_{13} = -43.4507577981096$$
$$x_{14} = 9.95756673500463$$
$$x_{15} = -89.5316083266597$$
$$x_{16} = 100.008435359475$$
$$x_{17} = -56.0189614233548$$
$$x_{18} = 22.5190281723597$$
$$x_{19} = -12.0101369031188$$
$$x_{20} = 18.3311063703045$$
$$x_{21} = 56.026925449142$$
$$x_{22} = -53.9243220629616$$
$$x_{23} = 64.4042772811497$$
$$x_{24} = 58.1212575693941$$
$$x_{25} = 16.2373072260106$$
$$x_{26} = 60.2155938389814$$
$$x_{27} = -16.2086080610791$$
$$x_{28} = -95.8150455211189$$
$$x_{29} = 3.72465876314914$$
$$x_{30} = 95.8196901125633$$
$$x_{31} = -18.3059450316161$$
$$x_{32} = 51.8382755576786$$
$$x_{33} = -60.2081877256431$$
$$x_{34} = 66.4986237979768$$
$$x_{35} = -51.8296623064097$$
$$x_{36} = -58.1135826616926$$
$$x_{37} = 7.86603628726821$$
$$x_{38} = -91.6260912952803$$
$$x_{39} = 208.916433417059$$
$$x_{40} = 93.7253189392189$$
$$x_{41} = -7.79646958415439$$
$$x_{42} = 14.1436862283443$$
$$x_{43} = -47.6402704273772$$
$$x_{44} = 12.0503556217359$$
$$x_{45} = -374.372230951242$$
$$x_{46} = 62.3099338592203$$
$$x_{47} = -5.66377961266401$$
$$x_{48} = 68.5929731388662$$
$$x_{49} = 5.7788018906052$$
$$x_{50} = 20.4250248320047$$
Puntos máximos de la función:
$$x_{50} = -6.8442016972462$$
$$x_{50} = 29.8343601469208$$
$$x_{50} = -36.1317902652531$$
$$x_{50} = -84.3007879836724$$
$$x_{50} = 73.8229870916504$$
$$x_{50} = -25.6615132195048$$
$$x_{50} = 42.4038530963323$$
$$x_{50} = 86.3899949788424$$
$$x_{50} = 84.2955066914144$$
$$x_{50} = -38.2259712921274$$
$$x_{50} = -27.7554476509841$$
$$x_{50} = -34.0376393480567$$
$$x_{50} = 31.9294426102745$$
$$x_{50} = -67.5459933026873$$
$$x_{50} = -65.4516580560656$$
$$x_{50} = 34.0244420673533$$
$$x_{50} = -21.4739793725166$$
$$x_{50} = -31.9435253678456$$
$$x_{50} = -78.0177204974878$$
$$x_{50} = 71.7284643872573$$
$$x_{50} = 88.4844789091251$$
$$x_{50} = 78.0120122217482$$
$$x_{50} = 9287.07144689999$$
$$x_{50} = 27.7391761293701$$
$$x_{50} = 44.4986000648908$$
$$x_{50} = -69.6403323820584$$
$$x_{50} = 40.3090703092807$$
$$x_{50} = -82.2064300459562$$
$$x_{50} = -184.830982372558$$
$$x_{50} = 38.2142459113972$$
$$x_{50} = 80.1065156951404$$
$$x_{50} = -23.5676784432488$$
$$x_{50} = -75.9233692564601$$
$$x_{50} = 82.2010137178493$$
$$x_{50} = -29.8494573795171$$
$$x_{50} = -73.8290206544931$$
$$x_{50} = -8495.39014716725$$
$$x_{50} = -71.7346749372935$$
$$x_{50} = 36.1193727823579$$
$$x_{50} = 75.9175028523412$$
$$x_{50} = -80.1120741590105$$
$$x_{50} = -320.96640001395$$
Decrece en los intervalos
$$\left[208.916433417059, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -374.372230951242\right]$$
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
$$\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -50.2609982389496$$
$$x_{2} = -26.1716961844179$$
$$x_{3} = 32.4561460526875$$
$$x_{4} = 19.8852366300728$$
$$x_{5} = 83.7731718267936$$
$$x_{6} = 6.25182173288128$$
$$x_{7} = -490.087999905919$$
$$x_{8} = 46.071934723584$$
$$x_{9} = 100.528768621236$$
$$x_{10} = -4.08306625181193$$
$$x_{11} = 26.1712480756893$$
$$x_{12} = 85.8676306643428$$
$$x_{13} = -99.4815490015694$$
$$x_{14} = -32.4564338558956$$
$$x_{15} = -156.031017113967$$
$$x_{16} = 65.9700444593934$$
$$x_{17} = -72.2535850911322$$
$$x_{18} = -85.867590346822$$
$$x_{19} = -81.6786651707097$$
$$x_{20} = 10.4519424473123$$
$$x_{21} = 68.0645451700063$$
$$x_{22} = 98.434327087986$$
$$x_{23} = 94.24543787104$$
$$x_{24} = -19.8860330040126$$
$$x_{25} = 37.6933136699573$$
$$x_{26} = 74.3480113145759$$
$$x_{27} = -77.4897252611726$$
$$x_{28} = -92.1509549179101$$
$$x_{29} = -17.7904514264001$$
$$x_{30} = -7.3056940002017$$
$$x_{31} = 80.6314333643326$$
$$x_{32} = 58.6393143127934$$
$$x_{33} = -58.6392275353824$$
$$x_{34} = 78.5369635890159$$
$$x_{35} = -87.9620480734275$$
$$x_{36} = 52.3556843829651$$
$$x_{37} = 34.5509713485856$$
$$x_{38} = -6.23797325191036$$
$$x_{39} = -46.0717935328677$$
$$x_{40} = 76.4424896895273$$
$$x_{41} = 4.15245578417923$$
$$x_{42} = -94.2454044215276$$
$$x_{43} = 92.1509899095978$$
$$x_{44} = -83.7731294610269$$
$$x_{45} = -96.3398515417059$$
$$x_{46} = 21.9807580188912$$
$$x_{47} = 61.7810213502029$$
$$x_{48} = -13.5983208492852$$
$$x_{49} = -48.1664050519949$$
$$x_{50} = 41.8826755189078$$
$$x_{51} = -33.5035416139575$$
$$x_{52} = -28.2666825886918$$
$$x_{53} = 30.3612602948113$$
$$x_{54} = 54.4502389708762$$
$$x_{55} = 17.7894412187581$$
$$x_{56} = -243.996121186448$$
$$x_{57} = -11.5015040182966$$
$$x_{58} = -24.0766116228193$$
$$x_{59} = -79.5841972044492$$
$$x_{60} = -10.4485671344656$$
$$x_{61} = 24.0760787417329$$
$$x_{62} = 56.5447818673297$$
$$x_{63} = 50.2611166586781$$
$$x_{64} = 12.5495032551421$$
$$x_{65} = -61.781099471617$$
$$x_{66} = -39.7878193754871$$
$$x_{67} = 72.2535280718162$$
$$x_{68} = -21.9814026495338$$
$$x_{69} = 43.9773161421544$$
$$x_{70} = 96.3398835488781$$
$$x_{71} = 90.0565395062887$$
$$x_{72} = -59.6865811532257$$
$$x_{73} = -43.9771610091387$$
$$x_{74} = -41.882504261406$$
$$x_{75} = -90.05650286315$$
$$x_{76} = -15.694583753881$$
$$x_{77} = -55.497517021885$$
$$x_{78} = 210.485648731234$$
$$x_{79} = 87.9620864878774$$
$$x_{80} = 20.9336407381412$$
$$x_{81} = 63.875536758646$$
$$x_{82} = -37.6931015438093$$
$$x_{83} = -63.8756098108952$$
$$x_{84} = -68.0646094615132$$
$$x_{85} = -65.9701129211828$$
$$x_{86} = 48.1665341047083$$
$$x_{87} = -35.598344794322$$
$$x_{88} = -57.5920539982314$$
$$x_{89} = -70.1591000132373$$
$$x_{90} = 8.35299123429176$$
$$x_{91} = 70.1590395216714$$
$$x_{92} = 28.2663003669026$$
$$x_{93} = 39.7880094224301$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
$$\lim_{x \to -2^-}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -2$$
- es el punto de flexión
$$\lim_{x \to 0^-}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = - \frac{35}{8}$$
$$\lim_{x \to 0^+}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = - \frac{35}{8}$$
- los límites son iguales, es decir omitimos el punto correspondiente
$$\lim_{x \to 2^-}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = \infty$$
- los límites no son iguales, signo
$$x_{3} = 2$$
- es el punto de flexión
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
$$\left[210.485648731234, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -490.087999905919\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -50.2609982389496$$
$$x_{2} = -26.1716961844179$$
$$x_{3} = 32.4561460526875$$
$$x_{4} = 19.8852366300728$$
$$x_{5} = 83.7731718267936$$
$$x_{6} = 6.25182173288128$$
$$x_{7} = -490.087999905919$$
$$x_{8} = 46.071934723584$$
$$x_{9} = 100.528768621236$$
$$x_{10} = -4.08306625181193$$
$$x_{11} = 26.1712480756893$$
$$x_{12} = 85.8676306643428$$
$$x_{13} = -99.4815490015694$$
$$x_{14} = -32.4564338558956$$
$$x_{15} = -156.031017113967$$
$$x_{16} = 65.9700444593934$$
$$x_{17} = -72.2535850911322$$
$$x_{18} = -85.867590346822$$
$$x_{19} = -81.6786651707097$$
$$x_{20} = 10.4519424473123$$
$$x_{21} = 68.0645451700063$$
$$x_{22} = 98.434327087986$$
$$x_{23} = 94.24543787104$$
$$x_{24} = -19.8860330040126$$
$$x_{25} = 37.6933136699573$$
$$x_{26} = 74.3480113145759$$
$$x_{27} = -77.4897252611726$$
$$x_{28} = -92.1509549179101$$
$$x_{29} = -17.7904514264001$$
$$x_{30} = -7.3056940002017$$
$$x_{31} = 80.6314333643326$$
$$x_{32} = 58.6393143127934$$
$$x_{33} = -58.6392275353824$$
$$x_{34} = 78.5369635890159$$
$$x_{35} = -87.9620480734275$$
$$x_{36} = 52.3556843829651$$
$$x_{37} = 34.5509713485856$$
$$x_{38} = -6.23797325191036$$
$$x_{39} = -46.0717935328677$$
$$x_{40} = 76.4424896895273$$
$$x_{41} = 4.15245578417923$$
$$x_{42} = -94.2454044215276$$
$$x_{43} = 92.1509899095978$$
$$x_{44} = -83.7731294610269$$
$$x_{45} = -96.3398515417059$$
$$x_{46} = 21.9807580188912$$
$$x_{47} = 61.7810213502029$$
$$x_{48} = -13.5983208492852$$
$$x_{49} = -48.1664050519949$$
$$x_{50} = 41.8826755189078$$
$$x_{51} = -33.5035416139575$$
$$x_{52} = -28.2666825886918$$
$$x_{53} = 30.3612602948113$$
$$x_{54} = 54.4502389708762$$
$$x_{55} = 17.7894412187581$$
$$x_{56} = -243.996121186448$$
$$x_{57} = -11.5015040182966$$
$$x_{58} = -24.0766116228193$$
$$x_{59} = -79.5841972044492$$
$$x_{60} = -10.4485671344656$$
$$x_{61} = 24.0760787417329$$
$$x_{62} = 56.5447818673297$$
$$x_{63} = 50.2611166586781$$
$$x_{64} = 12.5495032551421$$
$$x_{65} = -61.781099471617$$
$$x_{66} = -39.7878193754871$$
$$x_{67} = 72.2535280718162$$
$$x_{68} = -21.9814026495338$$
$$x_{69} = 43.9773161421544$$
$$x_{70} = 96.3398835488781$$
$$x_{71} = 90.0565395062887$$
$$x_{72} = -59.6865811532257$$
$$x_{73} = -43.9771610091387$$
$$x_{74} = -41.882504261406$$
$$x_{75} = -90.05650286315$$
$$x_{76} = -15.694583753881$$
$$x_{77} = -55.497517021885$$
$$x_{78} = 210.485648731234$$
$$x_{79} = 87.9620864878774$$
$$x_{80} = 20.9336407381412$$
$$x_{81} = 63.875536758646$$
$$x_{82} = -37.6931015438093$$
$$x_{83} = -63.8756098108952$$
$$x_{84} = -68.0646094615132$$
$$x_{85} = -65.9701129211828$$
$$x_{86} = 48.1665341047083$$
$$x_{87} = -35.598344794322$$
$$x_{88} = -57.5920539982314$$
$$x_{89} = -70.1591000132373$$
$$x_{90} = 8.35299123429176$$
$$x_{91} = 70.1590395216714$$
$$x_{92} = 28.2663003669026$$
$$x_{93} = 39.7880094224301$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
$$\lim_{x \to -2^-}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -2$$
- es el punto de flexión
$$\lim_{x \to 0^-}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = - \frac{35}{8}$$
$$\lim_{x \to 0^+}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = - \frac{35}{8}$$
- los límites son iguales, es decir omitimos el punto correspondiente
$$\lim_{x \to 2^-}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{8 x^{2} \left(x - 1\right)}{\left(x^{2} - 4\right)^{3}} - \frac{4 x}{\left(x^{2} - 4\right)^{2}} - \frac{2 \left(x - 1\right)}{\left(x^{2} - 4\right)^{2}} - \frac{9 \sin{\left(3 x \right)}}{2 x} - \frac{3 \cos{\left(3 x \right)}}{x^{2}} + \frac{\sin{\left(3 x \right)}}{x^{3}}\right) = \infty$$
- los límites no son iguales, signo
$$x_{3} = 2$$
- es el punto de flexión
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
$$\left[210.485648731234, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -490.087999905919\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}\right) = 0$$
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(3*x)/((2*x)) + (x - 1)/(x^2 - 4), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}}{x}\right) = 0$$
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:
$$\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x} = \frac{- x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}$$
- No
$$\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x} = - \frac{- x - 1}{x^{2} - 4} - \frac{\sin{\left(3 x \right)}}{2 x}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x} = \frac{- x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x}$$
- No
$$\frac{x - 1}{x^{2} - 4} + \frac{\sin{\left(3 x \right)}}{2 x} = - \frac{- x - 1}{x^{2} - 4} - \frac{\sin{\left(3 x \right)}}{2 x}$$
- No
es decir, función
no es
par ni impar