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- Integrales paso a paso
- Gráfico de la función paramétrica
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- 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 =:
-
y=(x+1)^3
-
y=2x
-
2*x^3-3*x
-
3-x^2
-
Expresiones idénticas
- cos(x)/(x- uno)
- coseno de (x) dividir por (x menos 1)
- coseno de (x) dividir por (x menos uno)
- cosx/x-1
- cos(x) dividir por (x-1)
-
Expresiones semejantes
- cos(x)/(x+1)
- cosx/(x-1)
-
Expresiones con funciones
- Coseno cos
- cos[x]/x^2
- cos(x-pi/3)-1
- cos(x)/(1+cos^2*2x)
- cos(1)/((3*x))
- cos(sqrt(x))+1
Gráfico de la función y = cos(x)/(x-1)
Solución
Ha introducido
[src]
cos(x)
f(x) = ------
x - 1
$$f{\left(x \right)} = \frac{\cos{\left(x \right)}}{x - 1}$$
f = cos(x)/(x - 1)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 1$$
$$x_{1} = 1$$
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:
$$\frac{\cos{\left(x \right)}}{x - 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = \frac{3 \pi}{2}$$
Solución numérica
$$x_{1} = -1.5707963267949$$
$$x_{2} = -64.4026493985908$$
$$x_{3} = 76.9690200129499$$
$$x_{4} = -23.5619449019235$$
$$x_{5} = -58.1194640914112$$
$$x_{6} = 199.491133502952$$
$$x_{7} = 61.261056745001$$
$$x_{8} = 80.1106126665397$$
$$x_{9} = -29.845130209103$$
$$x_{10} = -48.6946861306418$$
$$x_{11} = 1173.38485611579$$
$$x_{12} = -4.71238898038469$$
$$x_{13} = -86.3937979737193$$
$$x_{14} = -36.1283155162826$$
$$x_{15} = -98.9601685880785$$
$$x_{16} = 391.128285371929$$
$$x_{17} = -92.6769832808989$$
$$x_{18} = -39.2699081698724$$
$$x_{19} = 73.8274273593601$$
$$x_{20} = 42.4115008234622$$
$$x_{21} = 67.5442420521806$$
$$x_{22} = -32.9867228626928$$
$$x_{23} = 14.1371669411541$$
$$x_{24} = 4.71238898038469$$
$$x_{25} = 32.9867228626928$$
$$x_{26} = -10.9955742875643$$
$$x_{27} = 70.6858347057703$$
$$x_{28} = 36.1283155162826$$
$$x_{29} = 20.4203522483337$$
$$x_{30} = -70.6858347057703$$
$$x_{31} = -26.7035375555132$$
$$x_{32} = 10.9955742875643$$
$$x_{33} = -714.712328691678$$
$$x_{34} = 23.5619449019235$$
$$x_{35} = 45.553093477052$$
$$x_{36} = 83.2522053201295$$
$$x_{37} = -67.5442420521806$$
$$x_{38} = -89.5353906273091$$
$$x_{39} = -54.9778714378214$$
$$x_{40} = 95.8185759344887$$
$$x_{41} = -17.2787595947439$$
$$x_{42} = 26.7035375555132$$
$$x_{43} = 17.2787595947439$$
$$x_{44} = -42.4115008234622$$
$$x_{45} = 54.9778714378214$$
$$x_{46} = -7.85398163397448$$
$$x_{47} = 48.6946861306418$$
$$x_{48} = -51.8362787842316$$
$$x_{49} = 89.5353906273091$$
$$x_{50} = 92.6769832808989$$
$$x_{51} = 58.1194640914112$$
$$x_{52} = -80.1106126665397$$
$$x_{53} = -73.8274273593601$$
$$x_{54} = 86.3937979737193$$
$$x_{55} = -76.9690200129499$$
$$x_{56} = 51.8362787842316$$
$$x_{57} = 39.2699081698724$$
$$x_{58} = -20.4203522483337$$
$$x_{59} = 64.4026493985908$$
$$x_{60} = -136.659280431156$$
$$x_{61} = -83.2522053201295$$
$$x_{62} = 538.78314009065$$
$$x_{63} = 98.9601685880785$$
$$x_{64} = 7.85398163397448$$
$$x_{65} = -95.8185759344887$$
$$x_{66} = -14.1371669411541$$
$$x_{67} = 29.845130209103$$
$$x_{68} = -45.553093477052$$
$$x_{69} = -61.261056745001$$
o sea hay que resolver la ecuación:
$$\frac{\cos{\left(x \right)}}{x - 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = \frac{3 \pi}{2}$$
Solución numérica
$$x_{1} = -1.5707963267949$$
$$x_{2} = -64.4026493985908$$
$$x_{3} = 76.9690200129499$$
$$x_{4} = -23.5619449019235$$
$$x_{5} = -58.1194640914112$$
$$x_{6} = 199.491133502952$$
$$x_{7} = 61.261056745001$$
$$x_{8} = 80.1106126665397$$
$$x_{9} = -29.845130209103$$
$$x_{10} = -48.6946861306418$$
$$x_{11} = 1173.38485611579$$
$$x_{12} = -4.71238898038469$$
$$x_{13} = -86.3937979737193$$
$$x_{14} = -36.1283155162826$$
$$x_{15} = -98.9601685880785$$
$$x_{16} = 391.128285371929$$
$$x_{17} = -92.6769832808989$$
$$x_{18} = -39.2699081698724$$
$$x_{19} = 73.8274273593601$$
$$x_{20} = 42.4115008234622$$
$$x_{21} = 67.5442420521806$$
$$x_{22} = -32.9867228626928$$
$$x_{23} = 14.1371669411541$$
$$x_{24} = 4.71238898038469$$
$$x_{25} = 32.9867228626928$$
$$x_{26} = -10.9955742875643$$
$$x_{27} = 70.6858347057703$$
$$x_{28} = 36.1283155162826$$
$$x_{29} = 20.4203522483337$$
$$x_{30} = -70.6858347057703$$
$$x_{31} = -26.7035375555132$$
$$x_{32} = 10.9955742875643$$
$$x_{33} = -714.712328691678$$
$$x_{34} = 23.5619449019235$$
$$x_{35} = 45.553093477052$$
$$x_{36} = 83.2522053201295$$
$$x_{37} = -67.5442420521806$$
$$x_{38} = -89.5353906273091$$
$$x_{39} = -54.9778714378214$$
$$x_{40} = 95.8185759344887$$
$$x_{41} = -17.2787595947439$$
$$x_{42} = 26.7035375555132$$
$$x_{43} = 17.2787595947439$$
$$x_{44} = -42.4115008234622$$
$$x_{45} = 54.9778714378214$$
$$x_{46} = -7.85398163397448$$
$$x_{47} = 48.6946861306418$$
$$x_{48} = -51.8362787842316$$
$$x_{49} = 89.5353906273091$$
$$x_{50} = 92.6769832808989$$
$$x_{51} = 58.1194640914112$$
$$x_{52} = -80.1106126665397$$
$$x_{53} = -73.8274273593601$$
$$x_{54} = 86.3937979737193$$
$$x_{55} = -76.9690200129499$$
$$x_{56} = 51.8362787842316$$
$$x_{57} = 39.2699081698724$$
$$x_{58} = -20.4203522483337$$
$$x_{59} = 64.4026493985908$$
$$x_{60} = -136.659280431156$$
$$x_{61} = -83.2522053201295$$
$$x_{62} = 538.78314009065$$
$$x_{63} = 98.9601685880785$$
$$x_{64} = 7.85398163397448$$
$$x_{65} = -95.8185759344887$$
$$x_{66} = -14.1371669411541$$
$$x_{67} = 29.845130209103$$
$$x_{68} = -45.553093477052$$
$$x_{69} = -61.261056745001$$
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
$$- \frac{\sin{\left(x \right)}}{x - 1} - \frac{\cos{\left(x \right)}}{\left(x - 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -18.7990914357831$$
$$x_{2} = -15.6479679638982$$
$$x_{3} = 15.6397620877646$$
$$x_{4} = 188.490225654589$$
$$x_{5} = 100.520917114109$$
$$x_{6} = 2.57625015820118$$
$$x_{7} = -122.514017419839$$
$$x_{8} = 56.5306616416093$$
$$x_{9} = 25.0912562079058$$
$$x_{10} = 6.08916120309943$$
$$x_{11} = -72.242978694986$$
$$x_{12} = -546.635295693876$$
$$x_{13} = -37.673259943911$$
$$x_{14} = -91.0953290668266$$
$$x_{15} = 12.4794779911025$$
$$x_{16} = 91.0950880256329$$
$$x_{17} = 53.3879890840753$$
$$x_{18} = 75.3847808857452$$
$$x_{19} = 43.9590233567938$$
$$x_{20} = -6.14411351301787$$
$$x_{21} = -21.9475985837942$$
$$x_{22} = -100.521115065812$$
$$x_{23} = 40.8155939881502$$
$$x_{24} = 34.527701946778$$
$$x_{25} = -43.9600588531378$$
$$x_{26} = -56.5312876685112$$
$$x_{27} = 191.631906205502$$
$$x_{28} = -65.9585122146304$$
$$x_{29} = -40.8167952172419$$
$$x_{30} = 18.7934144113698$$
$$x_{31} = -81.6693131963402$$
$$x_{32} = 37.6718497263809$$
$$x_{33} = 65.9580523911179$$
$$x_{34} = 28.2376364595748$$
$$x_{35} = -59.6737803264459$$
$$x_{36} = -94.2372799036618$$
$$x_{37} = -50.2459712046114$$
$$x_{38} = -69.1007741687956$$
$$x_{39} = 87.9530943542027$$
$$x_{40} = -78.5272426949571$$
$$x_{41} = -28.2401476526276$$
$$x_{42} = -84.8113487041494$$
$$x_{43} = -9.32825706323943$$
$$x_{44} = 78.5269183093816$$
$$x_{45} = 72.24259540785$$
$$x_{46} = 50.2451786914948$$
$$x_{47} = 31.3830252979972$$
$$x_{48} = 9.30494468339504$$
$$x_{49} = 81.6690132946536$$
$$x_{50} = 47.1022022669651$$
$$x_{51} = -1077.56535302402$$
$$x_{52} = -97.379207861883$$
$$x_{53} = -2.88996969767843$$
$$x_{54} = 69.1003552230555$$
$$x_{55} = -47.1031041186137$$
$$x_{56} = -53.388691007263$$
$$x_{57} = 21.9434371567881$$
$$x_{58} = -62.8161843480611$$
$$x_{59} = 59.6732185170696$$
$$x_{60} = 97.378996929011$$
$$x_{61} = 62.815677356778$$
$$x_{62} = -25.0944376288815$$
$$x_{63} = 94.2370546693974$$
$$x_{64} = -12.492390025579$$
$$x_{65} = -75.3851328811964$$
$$x_{66} = 84.8110706151124$$
$$x_{67} = -87.9533529268738$$
$$x_{68} = -34.5293808983144$$
$$x_{69} = -31.38505790634$$
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:
$$x_{1} = -18.7990914357831$$
$$x_{2} = 15.6397620877646$$
$$x_{3} = 2.57625015820118$$
$$x_{4} = -546.635295693876$$
$$x_{5} = -37.673259943911$$
$$x_{6} = 91.0950880256329$$
$$x_{7} = 53.3879890840753$$
$$x_{8} = -6.14411351301787$$
$$x_{9} = -100.521115065812$$
$$x_{10} = 40.8155939881502$$
$$x_{11} = 34.527701946778$$
$$x_{12} = -43.9600588531378$$
$$x_{13} = -56.5312876685112$$
$$x_{14} = 191.631906205502$$
$$x_{15} = -81.6693131963402$$
$$x_{16} = 65.9580523911179$$
$$x_{17} = 28.2376364595748$$
$$x_{18} = -94.2372799036618$$
$$x_{19} = -50.2459712046114$$
$$x_{20} = -69.1007741687956$$
$$x_{21} = 78.5269183093816$$
$$x_{22} = 72.24259540785$$
$$x_{23} = 9.30494468339504$$
$$x_{24} = 47.1022022669651$$
$$x_{25} = 21.9434371567881$$
$$x_{26} = -62.8161843480611$$
$$x_{27} = 59.6732185170696$$
$$x_{28} = 97.378996929011$$
$$x_{29} = -25.0944376288815$$
$$x_{30} = -12.492390025579$$
$$x_{31} = -75.3851328811964$$
$$x_{32} = 84.8110706151124$$
$$x_{33} = -87.9533529268738$$
$$x_{34} = -31.38505790634$$
Puntos máximos de la función:
$$x_{34} = -15.6479679638982$$
$$x_{34} = 188.490225654589$$
$$x_{34} = 100.520917114109$$
$$x_{34} = -122.514017419839$$
$$x_{34} = 56.5306616416093$$
$$x_{34} = 25.0912562079058$$
$$x_{34} = 6.08916120309943$$
$$x_{34} = -72.242978694986$$
$$x_{34} = -91.0953290668266$$
$$x_{34} = 12.4794779911025$$
$$x_{34} = 75.3847808857452$$
$$x_{34} = 43.9590233567938$$
$$x_{34} = -21.9475985837942$$
$$x_{34} = -65.9585122146304$$
$$x_{34} = -40.8167952172419$$
$$x_{34} = 18.7934144113698$$
$$x_{34} = 37.6718497263809$$
$$x_{34} = -59.6737803264459$$
$$x_{34} = 87.9530943542027$$
$$x_{34} = -78.5272426949571$$
$$x_{34} = -28.2401476526276$$
$$x_{34} = -84.8113487041494$$
$$x_{34} = -9.32825706323943$$
$$x_{34} = 50.2451786914948$$
$$x_{34} = 31.3830252979972$$
$$x_{34} = 81.6690132946536$$
$$x_{34} = -1077.56535302402$$
$$x_{34} = -97.379207861883$$
$$x_{34} = -2.88996969767843$$
$$x_{34} = 69.1003552230555$$
$$x_{34} = -47.1031041186137$$
$$x_{34} = -53.388691007263$$
$$x_{34} = 62.815677356778$$
$$x_{34} = 94.2370546693974$$
$$x_{34} = -34.5293808983144$$
Decrece en los intervalos
$$\left[191.631906205502, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -546.635295693876\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{\sin{\left(x \right)}}{x - 1} - \frac{\cos{\left(x \right)}}{\left(x - 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -18.7990914357831$$
$$x_{2} = -15.6479679638982$$
$$x_{3} = 15.6397620877646$$
$$x_{4} = 188.490225654589$$
$$x_{5} = 100.520917114109$$
$$x_{6} = 2.57625015820118$$
$$x_{7} = -122.514017419839$$
$$x_{8} = 56.5306616416093$$
$$x_{9} = 25.0912562079058$$
$$x_{10} = 6.08916120309943$$
$$x_{11} = -72.242978694986$$
$$x_{12} = -546.635295693876$$
$$x_{13} = -37.673259943911$$
$$x_{14} = -91.0953290668266$$
$$x_{15} = 12.4794779911025$$
$$x_{16} = 91.0950880256329$$
$$x_{17} = 53.3879890840753$$
$$x_{18} = 75.3847808857452$$
$$x_{19} = 43.9590233567938$$
$$x_{20} = -6.14411351301787$$
$$x_{21} = -21.9475985837942$$
$$x_{22} = -100.521115065812$$
$$x_{23} = 40.8155939881502$$
$$x_{24} = 34.527701946778$$
$$x_{25} = -43.9600588531378$$
$$x_{26} = -56.5312876685112$$
$$x_{27} = 191.631906205502$$
$$x_{28} = -65.9585122146304$$
$$x_{29} = -40.8167952172419$$
$$x_{30} = 18.7934144113698$$
$$x_{31} = -81.6693131963402$$
$$x_{32} = 37.6718497263809$$
$$x_{33} = 65.9580523911179$$
$$x_{34} = 28.2376364595748$$
$$x_{35} = -59.6737803264459$$
$$x_{36} = -94.2372799036618$$
$$x_{37} = -50.2459712046114$$
$$x_{38} = -69.1007741687956$$
$$x_{39} = 87.9530943542027$$
$$x_{40} = -78.5272426949571$$
$$x_{41} = -28.2401476526276$$
$$x_{42} = -84.8113487041494$$
$$x_{43} = -9.32825706323943$$
$$x_{44} = 78.5269183093816$$
$$x_{45} = 72.24259540785$$
$$x_{46} = 50.2451786914948$$
$$x_{47} = 31.3830252979972$$
$$x_{48} = 9.30494468339504$$
$$x_{49} = 81.6690132946536$$
$$x_{50} = 47.1022022669651$$
$$x_{51} = -1077.56535302402$$
$$x_{52} = -97.379207861883$$
$$x_{53} = -2.88996969767843$$
$$x_{54} = 69.1003552230555$$
$$x_{55} = -47.1031041186137$$
$$x_{56} = -53.388691007263$$
$$x_{57} = 21.9434371567881$$
$$x_{58} = -62.8161843480611$$
$$x_{59} = 59.6732185170696$$
$$x_{60} = 97.378996929011$$
$$x_{61} = 62.815677356778$$
$$x_{62} = -25.0944376288815$$
$$x_{63} = 94.2370546693974$$
$$x_{64} = -12.492390025579$$
$$x_{65} = -75.3851328811964$$
$$x_{66} = 84.8110706151124$$
$$x_{67} = -87.9533529268738$$
$$x_{68} = -34.5293808983144$$
$$x_{69} = -31.38505790634$$
Signos de extremos en los puntos:
(-18.79909143578314, -0.0504430691319447)
(-15.647967963898166, 0.0599593189797558)
(15.63976208776456, -0.0681483206400774)
(188.4902256545889, 0.00533353551155559)
(100.52091711410945, 0.0100476316966419)
(2.5762501582011796, -0.535705052303484)
(-122.51401741983913, 0.00809598171844709)
(56.53066164160934, 0.0180051500304447)
(25.091256207905772, 0.0414731225016059)
(6.089161203099427, 0.192809042427521)
(-72.242978694986, 0.0136519134817116)
(-546.6352956938762, -0.00182602973305305)
(-37.673259943911006, -0.0258490197028825)
(-91.09532906682657, 0.0108576739325778)
(12.479477991102517, 0.0867833198945747)
(91.09508802563293, -0.0110987005999837)
(53.387989084075315, -0.0190848682073296)
(75.38478088574516, 0.0134423955413013)
(43.95902335679378, 0.0232716924030311)
(-6.1441135130178655, -0.138623930394573)
(-21.947598583794207, 0.0435362264748061)
(-100.52111506581193, -0.00984968979094353)
(40.81559398815024, -0.0251078697468112)
(34.52770194677802, -0.0298128246468963)
(-43.960058853137774, -0.022236464203186)
(-56.53128766851124, -0.0173792211238612)
(191.63190620550185, -0.00524563941809883)
(-65.95851221463039, 0.0149329557083856)
(-40.81679521724192, 0.023907001519389)
(18.793414411369753, 0.0561120230339157)
(-81.66931319634023, -0.0120955020439642)
(37.67184972638089, 0.0272587398500595)
(65.9580523911179, -0.0153927263543733)
(28.237636459574798, -0.0366891865463047)
(-59.67378032644585, 0.0164793457895915)
(-94.23727990366179, -0.0104995111118831)
(-50.24597120461141, -0.0195100148956696)
(-69.10077416879557, -0.0142637264671467)
(87.95309435420273, 0.0114996928375307)
(-78.52724269495707, 0.0125733134820883)
(-28.240147652627645, 0.0341795711715136)
(-84.81134870414938, 0.0116526790492257)
(-9.328257063239425, 0.0963710979823201)
(78.5269183093816, -0.0128976727485698)
(72.24259540785, -0.0140351638863266)
(50.245178691494786, 0.0203023709567303)
(31.38302529799723, 0.0328953023371544)
(9.304944683395044, -0.119546681963348)
(81.66901329465364, 0.0123953812433342)
(47.10220226696507, -0.0216858368023364)
(-1077.5653530240243, 0.000927157142018311)
(-97.37920786188297, 0.0101642243790071)
(-2.8899696976784344, 0.248976134877405)
(69.1003552230555, 0.0146826283229769)
(-47.10310411861372, 0.0207841885412821)
(-53.388691007263, 0.0183830682189117)
(21.94343715678808, -0.0476933188520339)
(-62.81618434806106, -0.0156680826074814)
(59.673218517069586, -0.0170410762454831)
(97.37899692901101, -0.0103751461271118)
(62.815677356778, 0.0161750096209984)
(-25.094437628881476, -0.0382942342355763)
(94.23705466939735, 0.010724732692878)
(-12.492390025578958, -0.0739131230459364)
(-75.38513288119637, -0.0130904310684593)
(84.81107061511238, -0.0119307487512748)
(-87.9533529268738, -0.0112411368826843)
(-34.5293808983144, 0.0281345781753277)
(-31.385057906339963, -0.0308637274812354)
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:
$$x_{1} = -18.7990914357831$$
$$x_{2} = 15.6397620877646$$
$$x_{3} = 2.57625015820118$$
$$x_{4} = -546.635295693876$$
$$x_{5} = -37.673259943911$$
$$x_{6} = 91.0950880256329$$
$$x_{7} = 53.3879890840753$$
$$x_{8} = -6.14411351301787$$
$$x_{9} = -100.521115065812$$
$$x_{10} = 40.8155939881502$$
$$x_{11} = 34.527701946778$$
$$x_{12} = -43.9600588531378$$
$$x_{13} = -56.5312876685112$$
$$x_{14} = 191.631906205502$$
$$x_{15} = -81.6693131963402$$
$$x_{16} = 65.9580523911179$$
$$x_{17} = 28.2376364595748$$
$$x_{18} = -94.2372799036618$$
$$x_{19} = -50.2459712046114$$
$$x_{20} = -69.1007741687956$$
$$x_{21} = 78.5269183093816$$
$$x_{22} = 72.24259540785$$
$$x_{23} = 9.30494468339504$$
$$x_{24} = 47.1022022669651$$
$$x_{25} = 21.9434371567881$$
$$x_{26} = -62.8161843480611$$
$$x_{27} = 59.6732185170696$$
$$x_{28} = 97.378996929011$$
$$x_{29} = -25.0944376288815$$
$$x_{30} = -12.492390025579$$
$$x_{31} = -75.3851328811964$$
$$x_{32} = 84.8110706151124$$
$$x_{33} = -87.9533529268738$$
$$x_{34} = -31.38505790634$$
Puntos máximos de la función:
$$x_{34} = -15.6479679638982$$
$$x_{34} = 188.490225654589$$
$$x_{34} = 100.520917114109$$
$$x_{34} = -122.514017419839$$
$$x_{34} = 56.5306616416093$$
$$x_{34} = 25.0912562079058$$
$$x_{34} = 6.08916120309943$$
$$x_{34} = -72.242978694986$$
$$x_{34} = -91.0953290668266$$
$$x_{34} = 12.4794779911025$$
$$x_{34} = 75.3847808857452$$
$$x_{34} = 43.9590233567938$$
$$x_{34} = -21.9475985837942$$
$$x_{34} = -65.9585122146304$$
$$x_{34} = -40.8167952172419$$
$$x_{34} = 18.7934144113698$$
$$x_{34} = 37.6718497263809$$
$$x_{34} = -59.6737803264459$$
$$x_{34} = 87.9530943542027$$
$$x_{34} = -78.5272426949571$$
$$x_{34} = -28.2401476526276$$
$$x_{34} = -84.8113487041494$$
$$x_{34} = -9.32825706323943$$
$$x_{34} = 50.2451786914948$$
$$x_{34} = 31.3830252979972$$
$$x_{34} = 81.6690132946536$$
$$x_{34} = -1077.56535302402$$
$$x_{34} = -97.379207861883$$
$$x_{34} = -2.88996969767843$$
$$x_{34} = 69.1003552230555$$
$$x_{34} = -47.1031041186137$$
$$x_{34} = -53.388691007263$$
$$x_{34} = 62.815677356778$$
$$x_{34} = 94.2370546693974$$
$$x_{34} = -34.5293808983144$$
Decrece en los intervalos
$$\left[191.631906205502, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -546.635295693876\right]$$
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
$$\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x - 1} + \frac{2 \cos{\left(x \right)}}{\left(x - 1\right)^{2}}}{x - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -61.2289118119026$$
$$x_{2} = -51.7983897861238$$
$$x_{3} = 29.7755709323142$$
$$x_{4} = -89.5132926274963$$
$$x_{5} = 83.2278802717944$$
$$x_{6} = -83.228458145445$$
$$x_{7} = 92.6551606286758$$
$$x_{8} = -54.9421125829153$$
$$x_{9} = -26.6310922236611$$
$$x_{10} = -86.3709050594723$$
$$x_{11} = 98.9397464504172$$
$$x_{12} = -39.22016138731$$
$$x_{13} = 39.2175523279643$$
$$x_{14} = 32.9240332040206$$
$$x_{15} = 26.6254109350763$$
$$x_{16} = 67.5141687409854$$
$$x_{17} = -168.063376807947$$
$$x_{18} = -42.3653647291314$$
$$x_{19} = -48.654396838104$$
$$x_{20} = 51.7968961320869$$
$$x_{21} = 89.5127931011103$$
$$x_{22} = -10.825651157762$$
$$x_{23} = -73.80068645168$$
$$x_{24} = 86.3703684986956$$
$$x_{25} = 45.5081427660817$$
$$x_{26} = -92.6556268279389$$
$$x_{27} = 105.224163309626$$
$$x_{28} = 58.0844210975337$$
$$x_{29} = -4.32863617605124$$
$$x_{30} = -70.657920700132$$
$$x_{31} = -237.181848301852$$
$$x_{32} = 80.0853208283276$$
$$x_{33} = -17.1684571899007$$
$$x_{34} = -80.0859449790141$$
$$x_{35} = 10.7898786754269$$
$$x_{36} = -105.224524739209$$
$$x_{37} = -67.5150472396589$$
$$x_{38} = 64.3710840254309$$
$$x_{39} = 70.6571186927646$$
$$x_{40} = 4.00507341668955$$
$$x_{41} = -45.5100787997204$$
$$x_{42} = 54.9407852505616$$
$$x_{43} = 36.0712578833702$$
$$x_{44} = 7.54372449628009$$
$$x_{45} = 48.6527034788051$$
$$x_{46} = -32.9277399444348$$
$$x_{47} = -58.0856084395179$$
$$x_{48} = -29.7801075137773$$
$$x_{49} = 23.4728313498836$$
$$x_{50} = -36.0743437126941$$
$$x_{51} = -76.9433575383977$$
$$x_{52} = -98.9401552763972$$
$$x_{53} = 95.7974767616183$$
$$x_{54} = 567.053940733425$$
$$x_{55} = 17.1546413657741$$
$$x_{56} = 20.3166301288662$$
$$x_{57} = -20.3264348242219$$
$$x_{58} = -23.4801553706306$$
$$x_{59} = -7.61991323310644$$
$$x_{60} = 76.9426813176863$$
$$x_{61} = 61.2278434114583$$
$$x_{62} = -64.3720505127272$$
$$x_{63} = 13.9825085391948$$
$$x_{64} = 287.448745694871$$
$$x_{65} = 73.7999513585394$$
$$x_{66} = 42.3631297553676$$
$$x_{67} = -95.797912862081$$
$$x_{68} = -14.0034717913284$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x - 1} + \frac{2 \cos{\left(x \right)}}{\left(x - 1\right)^{2}}}{x - 1}\right) = -\infty$$
$$\lim_{x \to 1^+}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x - 1} + \frac{2 \cos{\left(x \right)}}{\left(x - 1\right)^{2}}}{x - 1}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = 1$$
- 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
$$\left[567.053940733425, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.797912862081\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x - 1} + \frac{2 \cos{\left(x \right)}}{\left(x - 1\right)^{2}}}{x - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -61.2289118119026$$
$$x_{2} = -51.7983897861238$$
$$x_{3} = 29.7755709323142$$
$$x_{4} = -89.5132926274963$$
$$x_{5} = 83.2278802717944$$
$$x_{6} = -83.228458145445$$
$$x_{7} = 92.6551606286758$$
$$x_{8} = -54.9421125829153$$
$$x_{9} = -26.6310922236611$$
$$x_{10} = -86.3709050594723$$
$$x_{11} = 98.9397464504172$$
$$x_{12} = -39.22016138731$$
$$x_{13} = 39.2175523279643$$
$$x_{14} = 32.9240332040206$$
$$x_{15} = 26.6254109350763$$
$$x_{16} = 67.5141687409854$$
$$x_{17} = -168.063376807947$$
$$x_{18} = -42.3653647291314$$
$$x_{19} = -48.654396838104$$
$$x_{20} = 51.7968961320869$$
$$x_{21} = 89.5127931011103$$
$$x_{22} = -10.825651157762$$
$$x_{23} = -73.80068645168$$
$$x_{24} = 86.3703684986956$$
$$x_{25} = 45.5081427660817$$
$$x_{26} = -92.6556268279389$$
$$x_{27} = 105.224163309626$$
$$x_{28} = 58.0844210975337$$
$$x_{29} = -4.32863617605124$$
$$x_{30} = -70.657920700132$$
$$x_{31} = -237.181848301852$$
$$x_{32} = 80.0853208283276$$
$$x_{33} = -17.1684571899007$$
$$x_{34} = -80.0859449790141$$
$$x_{35} = 10.7898786754269$$
$$x_{36} = -105.224524739209$$
$$x_{37} = -67.5150472396589$$
$$x_{38} = 64.3710840254309$$
$$x_{39} = 70.6571186927646$$
$$x_{40} = 4.00507341668955$$
$$x_{41} = -45.5100787997204$$
$$x_{42} = 54.9407852505616$$
$$x_{43} = 36.0712578833702$$
$$x_{44} = 7.54372449628009$$
$$x_{45} = 48.6527034788051$$
$$x_{46} = -32.9277399444348$$
$$x_{47} = -58.0856084395179$$
$$x_{48} = -29.7801075137773$$
$$x_{49} = 23.4728313498836$$
$$x_{50} = -36.0743437126941$$
$$x_{51} = -76.9433575383977$$
$$x_{52} = -98.9401552763972$$
$$x_{53} = 95.7974767616183$$
$$x_{54} = 567.053940733425$$
$$x_{55} = 17.1546413657741$$
$$x_{56} = 20.3166301288662$$
$$x_{57} = -20.3264348242219$$
$$x_{58} = -23.4801553706306$$
$$x_{59} = -7.61991323310644$$
$$x_{60} = 76.9426813176863$$
$$x_{61} = 61.2278434114583$$
$$x_{62} = -64.3720505127272$$
$$x_{63} = 13.9825085391948$$
$$x_{64} = 287.448745694871$$
$$x_{65} = 73.7999513585394$$
$$x_{66} = 42.3631297553676$$
$$x_{67} = -95.797912862081$$
$$x_{68} = -14.0034717913284$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x - 1} + \frac{2 \cos{\left(x \right)}}{\left(x - 1\right)^{2}}}{x - 1}\right) = -\infty$$
$$\lim_{x \to 1^+}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x - 1} + \frac{2 \cos{\left(x \right)}}{\left(x - 1\right)^{2}}}{x - 1}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = 1$$
- 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
$$\left[567.053940733425, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.797912862081\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 1$$
$$x_{1} = 1$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x - 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x - 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x - 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x - 1}\right) = 0$$
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 cos(x)/(x - 1), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x \left(x - 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x \left(x - 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x \left(x - 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x \left(x - 1\right)}\right) = 0$$
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:
$$\frac{\cos{\left(x \right)}}{x - 1} = \frac{\cos{\left(x \right)}}{- x - 1}$$
- No
$$\frac{\cos{\left(x \right)}}{x - 1} = - \frac{\cos{\left(x \right)}}{- x - 1}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\cos{\left(x \right)}}{x - 1} = \frac{\cos{\left(x \right)}}{- x - 1}$$
- No
$$\frac{\cos{\left(x \right)}}{x - 1} = - \frac{\cos{\left(x \right)}}{- x - 1}$$
- No
es decir, función
no es
par ni impar