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- Integrales paso a paso
- Gráfico de la función paramétrica
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- Superficie dada por la ecuación
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- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- 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
- cosx/(x+ uno)
- coseno de x dividir por (x más 1)
- coseno de x dividir por (x más uno)
- cosx/x+1
- cosx dividir por (x+1)
-
Expresiones semejantes
- (cos(x))/x+1
- (-cos(x)+(2*sin(x)/(x+1))+(2*cos(x)/(x+1)^2))/(x+1)
- cosx/(x-1)
- cos(x)/(x+1)
-
Expresiones con funciones
- cosx
- cosx-logcosx/loge
- cosx+x*sinx
- cosx-(cosx)^2
- cosx+2x^(2)+x
- cosx/(x^2)
Gráfico de la función y = cosx/(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
$$x_{1} = 1.5707963267949$$
$$x_{2} = 39.2699081698724$$
$$x_{3} = -39.2699081698724$$
$$x_{4} = 14.1371669411541$$
$$x_{5} = -70.6858347057703$$
$$x_{6} = 32.9867228626928$$
$$x_{7} = 61.261056745001$$
$$x_{8} = 36.1283155162826$$
$$x_{9} = -61.261056745001$$
$$x_{10} = -36.1283155162826$$
$$x_{11} = -48.6946861306418$$
$$x_{12} = 7.85398163397448$$
$$x_{13} = 70.6858347057703$$
$$x_{14} = -80.1106126665397$$
$$x_{15} = 42.4115008234622$$
$$x_{16} = 51.8362787842316$$
$$x_{17} = -98.9601685880785$$
$$x_{18} = 80.1106126665397$$
$$x_{19} = -64.4026493985908$$
$$x_{20} = -29.845130209103$$
$$x_{21} = 98.9601685880785$$
$$x_{22} = -256.039801267568$$
$$x_{23} = 83.2522053201295$$
$$x_{24} = -89.5353906273091$$
$$x_{25} = -86.3937979737193$$
$$x_{26} = -10.9955742875643$$
$$x_{27} = 17.2787595947439$$
$$x_{28} = -26.7035375555132$$
$$x_{29} = -54.9778714378214$$
$$x_{30} = -92.6769832808989$$
$$x_{31} = 89.5353906273091$$
$$x_{32} = 58.1194640914112$$
$$x_{33} = 4.71238898038469$$
$$x_{34} = 20.4203522483337$$
$$x_{35} = 73.8274273593601$$
$$x_{36} = -67.5442420521806$$
$$x_{37} = 45.553093477052$$
$$x_{38} = -58.1194640914112$$
$$x_{39} = -51.8362787842316$$
$$x_{40} = -83.2522053201295$$
$$x_{41} = -76.9690200129499$$
$$x_{42} = -14.1371669411541$$
$$x_{43} = -73.8274273593601$$
$$x_{44} = 92.6769832808989$$
$$x_{45} = -20.4203522483337$$
$$x_{46} = 64.4026493985908$$
$$x_{47} = 95.8185759344887$$
$$x_{48} = 67.5442420521806$$
$$x_{49} = 54.9778714378214$$
$$x_{50} = 48.6946861306418$$
$$x_{51} = -4.71238898038469$$
$$x_{52} = 76.9690200129499$$
$$x_{53} = -45.553093477052$$
$$x_{54} = -7.85398163397448$$
$$x_{55} = -161.792021659874$$
$$x_{56} = -95.8185759344887$$
$$x_{57} = 29.845130209103$$
$$x_{58} = -17.2787595947439$$
$$x_{59} = -32.9867228626928$$
$$x_{60} = 26.7035375555132$$
$$x_{61} = -1.5707963267949$$
$$x_{62} = 10.9955742875643$$
$$x_{63} = -127.234502470387$$
$$x_{64} = 230.90706003885$$
$$x_{65} = 86.3937979737193$$
$$x_{66} = 23.5619449019235$$
$$x_{67} = -23.5619449019235$$
$$x_{68} = -42.4115008234622$$
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
$$x_{1} = 1.5707963267949$$
$$x_{2} = 39.2699081698724$$
$$x_{3} = -39.2699081698724$$
$$x_{4} = 14.1371669411541$$
$$x_{5} = -70.6858347057703$$
$$x_{6} = 32.9867228626928$$
$$x_{7} = 61.261056745001$$
$$x_{8} = 36.1283155162826$$
$$x_{9} = -61.261056745001$$
$$x_{10} = -36.1283155162826$$
$$x_{11} = -48.6946861306418$$
$$x_{12} = 7.85398163397448$$
$$x_{13} = 70.6858347057703$$
$$x_{14} = -80.1106126665397$$
$$x_{15} = 42.4115008234622$$
$$x_{16} = 51.8362787842316$$
$$x_{17} = -98.9601685880785$$
$$x_{18} = 80.1106126665397$$
$$x_{19} = -64.4026493985908$$
$$x_{20} = -29.845130209103$$
$$x_{21} = 98.9601685880785$$
$$x_{22} = -256.039801267568$$
$$x_{23} = 83.2522053201295$$
$$x_{24} = -89.5353906273091$$
$$x_{25} = -86.3937979737193$$
$$x_{26} = -10.9955742875643$$
$$x_{27} = 17.2787595947439$$
$$x_{28} = -26.7035375555132$$
$$x_{29} = -54.9778714378214$$
$$x_{30} = -92.6769832808989$$
$$x_{31} = 89.5353906273091$$
$$x_{32} = 58.1194640914112$$
$$x_{33} = 4.71238898038469$$
$$x_{34} = 20.4203522483337$$
$$x_{35} = 73.8274273593601$$
$$x_{36} = -67.5442420521806$$
$$x_{37} = 45.553093477052$$
$$x_{38} = -58.1194640914112$$
$$x_{39} = -51.8362787842316$$
$$x_{40} = -83.2522053201295$$
$$x_{41} = -76.9690200129499$$
$$x_{42} = -14.1371669411541$$
$$x_{43} = -73.8274273593601$$
$$x_{44} = 92.6769832808989$$
$$x_{45} = -20.4203522483337$$
$$x_{46} = 64.4026493985908$$
$$x_{47} = 95.8185759344887$$
$$x_{48} = 67.5442420521806$$
$$x_{49} = 54.9778714378214$$
$$x_{50} = 48.6946861306418$$
$$x_{51} = -4.71238898038469$$
$$x_{52} = 76.9690200129499$$
$$x_{53} = -45.553093477052$$
$$x_{54} = -7.85398163397448$$
$$x_{55} = -161.792021659874$$
$$x_{56} = -95.8185759344887$$
$$x_{57} = 29.845130209103$$
$$x_{58} = -17.2787595947439$$
$$x_{59} = -32.9867228626928$$
$$x_{60} = 26.7035375555132$$
$$x_{61} = -1.5707963267949$$
$$x_{62} = 10.9955742875643$$
$$x_{63} = -127.234502470387$$
$$x_{64} = 230.90706003885$$
$$x_{65} = 86.3937979737193$$
$$x_{66} = 23.5619449019235$$
$$x_{67} = -23.5619449019235$$
$$x_{68} = -42.4115008234622$$
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
$$x_{1} = 9.32825706323943$$
$$x_{2} = 43.9600588531378$$
$$x_{3} = -94.2370546693974$$
$$x_{4} = 2.88996969767843$$
$$x_{5} = 72.242978694986$$
$$x_{6} = 18.7990914357831$$
$$x_{7} = 69.1007741687956$$
$$x_{8} = -31.3830252979972$$
$$x_{9} = -25.0912562079058$$
$$x_{10} = -91.0950880256329$$
$$x_{11} = -28.2376364595748$$
$$x_{12} = -637.741738184573$$
$$x_{13} = -53.3879890840753$$
$$x_{14} = 40.8167952172419$$
$$x_{15} = 1313.18496827279$$
$$x_{16} = -81.6690132946536$$
$$x_{17} = 15.6479679638982$$
$$x_{18} = -72.24259540785$$
$$x_{19} = -2.57625015820118$$
$$x_{20} = 6.14411351301787$$
$$x_{21} = -12.4794779911025$$
$$x_{22} = 94.2372799036618$$
$$x_{23} = 84.8113487041494$$
$$x_{24} = 81.6693131963402$$
$$x_{25} = 56.5312876685112$$
$$x_{26} = -47.1022022669651$$
$$x_{27} = 100.521115065812$$
$$x_{28} = -100.520917114109$$
$$x_{29} = -62.815677356778$$
$$x_{30} = 37.673259943911$$
$$x_{31} = -97.378996929011$$
$$x_{32} = 47.1031041186137$$
$$x_{33} = 65.9585122146304$$
$$x_{34} = 78.5272426949571$$
$$x_{35} = -87.9530943542027$$
$$x_{36} = -15.6397620877646$$
$$x_{37} = 53.388691007263$$
$$x_{38} = -21.9434371567881$$
$$x_{39} = -59.6732185170696$$
$$x_{40} = 12.492390025579$$
$$x_{41} = 172.781841669816$$
$$x_{42} = 50.2459712046114$$
$$x_{43} = 31.38505790634$$
$$x_{44} = -75.3847808857452$$
$$x_{45} = 34.5293808983144$$
$$x_{46} = -37.6718497263809$$
$$x_{47} = -34.527701946778$$
$$x_{48} = 75.3851328811964$$
$$x_{49} = -6.08916120309943$$
$$x_{50} = -43.9590233567938$$
$$x_{51} = 59.6737803264459$$
$$x_{52} = -69.1003552230555$$
$$x_{53} = -40.8155939881502$$
$$x_{54} = 28.2401476526276$$
$$x_{55} = -9.30494468339504$$
$$x_{56} = -78.5269183093816$$
$$x_{57} = 21.9475985837942$$
$$x_{58} = -18.7934144113698$$
$$x_{59} = -56.5306616416093$$
$$x_{60} = 91.0953290668266$$
$$x_{61} = 87.9533529268738$$
$$x_{62} = 197.915309953386$$
$$x_{63} = -84.8110706151124$$
$$x_{64} = 25.0944376288815$$
$$x_{65} = 62.8161843480611$$
$$x_{66} = -50.2451786914948$$
$$x_{67} = -65.9580523911179$$
$$x_{68} = 97.379207861883$$
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:
$$x_{1} = 9.32825706323943$$
$$x_{2} = -94.2370546693974$$
$$x_{3} = 2.88996969767843$$
$$x_{4} = 72.242978694986$$
$$x_{5} = -31.3830252979972$$
$$x_{6} = -25.0912562079058$$
$$x_{7} = 40.8167952172419$$
$$x_{8} = -81.6690132946536$$
$$x_{9} = 15.6479679638982$$
$$x_{10} = -12.4794779911025$$
$$x_{11} = 84.8113487041494$$
$$x_{12} = -100.520917114109$$
$$x_{13} = -62.815677356778$$
$$x_{14} = 47.1031041186137$$
$$x_{15} = 65.9585122146304$$
$$x_{16} = 78.5272426949571$$
$$x_{17} = -87.9530943542027$$
$$x_{18} = 53.388691007263$$
$$x_{19} = 172.781841669816$$
$$x_{20} = -75.3847808857452$$
$$x_{21} = 34.5293808983144$$
$$x_{22} = -37.6718497263809$$
$$x_{23} = -6.08916120309943$$
$$x_{24} = -43.9590233567938$$
$$x_{25} = 59.6737803264459$$
$$x_{26} = -69.1003552230555$$
$$x_{27} = 28.2401476526276$$
$$x_{28} = 21.9475985837942$$
$$x_{29} = -18.7934144113698$$
$$x_{30} = -56.5306616416093$$
$$x_{31} = 91.0953290668266$$
$$x_{32} = 197.915309953386$$
$$x_{33} = -50.2451786914948$$
$$x_{34} = 97.379207861883$$
Puntos máximos de la función:
$$x_{34} = 43.9600588531378$$
$$x_{34} = 18.7990914357831$$
$$x_{34} = 69.1007741687956$$
$$x_{34} = -91.0950880256329$$
$$x_{34} = -28.2376364595748$$
$$x_{34} = -637.741738184573$$
$$x_{34} = -53.3879890840753$$
$$x_{34} = 1313.18496827279$$
$$x_{34} = -72.24259540785$$
$$x_{34} = -2.57625015820118$$
$$x_{34} = 6.14411351301787$$
$$x_{34} = 94.2372799036618$$
$$x_{34} = 81.6693131963402$$
$$x_{34} = 56.5312876685112$$
$$x_{34} = -47.1022022669651$$
$$x_{34} = 100.521115065812$$
$$x_{34} = 37.673259943911$$
$$x_{34} = -97.378996929011$$
$$x_{34} = -15.6397620877646$$
$$x_{34} = -21.9434371567881$$
$$x_{34} = -59.6732185170696$$
$$x_{34} = 12.492390025579$$
$$x_{34} = 50.2459712046114$$
$$x_{34} = 31.38505790634$$
$$x_{34} = -34.527701946778$$
$$x_{34} = 75.3851328811964$$
$$x_{34} = -40.8155939881502$$
$$x_{34} = -9.30494468339504$$
$$x_{34} = -78.5269183093816$$
$$x_{34} = 87.9533529268738$$
$$x_{34} = -84.8110706151124$$
$$x_{34} = 25.0944376288815$$
$$x_{34} = 62.8161843480611$$
$$x_{34} = -65.9580523911179$$
Decrece en los intervalos
$$\left[197.915309953386, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.520917114109\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$x_{1} = 9.32825706323943$$
$$x_{2} = 43.9600588531378$$
$$x_{3} = -94.2370546693974$$
$$x_{4} = 2.88996969767843$$
$$x_{5} = 72.242978694986$$
$$x_{6} = 18.7990914357831$$
$$x_{7} = 69.1007741687956$$
$$x_{8} = -31.3830252979972$$
$$x_{9} = -25.0912562079058$$
$$x_{10} = -91.0950880256329$$
$$x_{11} = -28.2376364595748$$
$$x_{12} = -637.741738184573$$
$$x_{13} = -53.3879890840753$$
$$x_{14} = 40.8167952172419$$
$$x_{15} = 1313.18496827279$$
$$x_{16} = -81.6690132946536$$
$$x_{17} = 15.6479679638982$$
$$x_{18} = -72.24259540785$$
$$x_{19} = -2.57625015820118$$
$$x_{20} = 6.14411351301787$$
$$x_{21} = -12.4794779911025$$
$$x_{22} = 94.2372799036618$$
$$x_{23} = 84.8113487041494$$
$$x_{24} = 81.6693131963402$$
$$x_{25} = 56.5312876685112$$
$$x_{26} = -47.1022022669651$$
$$x_{27} = 100.521115065812$$
$$x_{28} = -100.520917114109$$
$$x_{29} = -62.815677356778$$
$$x_{30} = 37.673259943911$$
$$x_{31} = -97.378996929011$$
$$x_{32} = 47.1031041186137$$
$$x_{33} = 65.9585122146304$$
$$x_{34} = 78.5272426949571$$
$$x_{35} = -87.9530943542027$$
$$x_{36} = -15.6397620877646$$
$$x_{37} = 53.388691007263$$
$$x_{38} = -21.9434371567881$$
$$x_{39} = -59.6732185170696$$
$$x_{40} = 12.492390025579$$
$$x_{41} = 172.781841669816$$
$$x_{42} = 50.2459712046114$$
$$x_{43} = 31.38505790634$$
$$x_{44} = -75.3847808857452$$
$$x_{45} = 34.5293808983144$$
$$x_{46} = -37.6718497263809$$
$$x_{47} = -34.527701946778$$
$$x_{48} = 75.3851328811964$$
$$x_{49} = -6.08916120309943$$
$$x_{50} = -43.9590233567938$$
$$x_{51} = 59.6737803264459$$
$$x_{52} = -69.1003552230555$$
$$x_{53} = -40.8155939881502$$
$$x_{54} = 28.2401476526276$$
$$x_{55} = -9.30494468339504$$
$$x_{56} = -78.5269183093816$$
$$x_{57} = 21.9475985837942$$
$$x_{58} = -18.7934144113698$$
$$x_{59} = -56.5306616416093$$
$$x_{60} = 91.0953290668266$$
$$x_{61} = 87.9533529268738$$
$$x_{62} = 197.915309953386$$
$$x_{63} = -84.8110706151124$$
$$x_{64} = 25.0944376288815$$
$$x_{65} = 62.8161843480611$$
$$x_{66} = -50.2451786914948$$
$$x_{67} = -65.9580523911179$$
$$x_{68} = 97.379207861883$$
Signos de extremos en los puntos:
(9.328257063239425, -0.0963710979823201)
(43.960058853137774, 0.022236464203186)
(-94.23705466939735, -0.010724732692878)
(2.8899696976784344, -0.248976134877405)
(72.242978694986, -0.0136519134817116)
(18.79909143578314, 0.0504430691319447)
(69.10077416879557, 0.0142637264671467)
(-31.38302529799723, -0.0328953023371544)
(-25.091256207905772, -0.0414731225016059)
(-91.09508802563293, 0.0110987005999837)
(-28.237636459574798, 0.0366891865463047)
(-637.7417381845734, 0.00157049350907233)
(-53.387989084075315, 0.0190848682073296)
(40.81679521724192, -0.023907001519389)
(1313.1849682727866, 0.000760927673528925)
(-81.66901329465364, -0.0123953812433342)
(15.647967963898166, -0.0599593189797558)
(-72.24259540785, 0.0140351638863266)
(-2.5762501582011796, 0.535705052303484)
(6.1441135130178655, 0.138623930394573)
(-12.479477991102517, -0.0867833198945747)
(94.23727990366179, 0.0104995111118831)
(84.81134870414938, -0.0116526790492257)
(81.66931319634023, 0.0120955020439642)
(56.53128766851124, 0.0173792211238612)
(-47.10220226696507, 0.0216858368023364)
(100.52111506581193, 0.00984968979094353)
(-100.52091711410945, -0.0100476316966419)
(-62.815677356778, -0.0161750096209984)
(37.673259943911006, 0.0258490197028825)
(-97.37899692901101, 0.0103751461271118)
(47.10310411861372, -0.0207841885412821)
(65.95851221463039, -0.0149329557083856)
(78.52724269495707, -0.0125733134820883)
(-87.95309435420273, -0.0114996928375307)
(-15.63976208776456, 0.0681483206400774)
(53.388691007263, -0.0183830682189117)
(-21.94343715678808, 0.0476933188520339)
(-59.673218517069586, 0.0170410762454831)
(12.492390025578958, 0.0739131230459364)
(172.781841669816, -0.0057542458670116)
(50.24597120461141, 0.0195100148956696)
(31.385057906339963, 0.0308637274812354)
(-75.38478088574516, -0.0134423955413013)
(34.5293808983144, -0.0281345781753277)
(-37.67184972638089, -0.0272587398500595)
(-34.52770194677802, 0.0298128246468963)
(75.38513288119637, 0.0130904310684593)
(-6.089161203099427, -0.192809042427521)
(-43.95902335679378, -0.0232716924030311)
(59.67378032644585, -0.0164793457895915)
(-69.1003552230555, -0.0146826283229769)
(-40.81559398815024, 0.0251078697468112)
(28.240147652627645, -0.0341795711715136)
(-9.304944683395044, 0.119546681963348)
(-78.5269183093816, 0.0128976727485698)
(21.947598583794207, -0.0435362264748061)
(-18.793414411369753, -0.0561120230339157)
(-56.53066164160934, -0.0180051500304447)
(91.09532906682657, -0.0108576739325778)
(87.9533529268738, 0.0112411368826843)
(197.91530995338616, -0.00502720159537522)
(-84.81107061511238, 0.0119307487512748)
(25.094437628881476, 0.0382942342355763)
(62.81618434806106, 0.0156680826074814)
(-50.245178691494786, -0.0203023709567303)
(-65.9580523911179, 0.0153927263543733)
(97.37920786188297, -0.0101642243790071)
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:
$$x_{1} = 9.32825706323943$$
$$x_{2} = -94.2370546693974$$
$$x_{3} = 2.88996969767843$$
$$x_{4} = 72.242978694986$$
$$x_{5} = -31.3830252979972$$
$$x_{6} = -25.0912562079058$$
$$x_{7} = 40.8167952172419$$
$$x_{8} = -81.6690132946536$$
$$x_{9} = 15.6479679638982$$
$$x_{10} = -12.4794779911025$$
$$x_{11} = 84.8113487041494$$
$$x_{12} = -100.520917114109$$
$$x_{13} = -62.815677356778$$
$$x_{14} = 47.1031041186137$$
$$x_{15} = 65.9585122146304$$
$$x_{16} = 78.5272426949571$$
$$x_{17} = -87.9530943542027$$
$$x_{18} = 53.388691007263$$
$$x_{19} = 172.781841669816$$
$$x_{20} = -75.3847808857452$$
$$x_{21} = 34.5293808983144$$
$$x_{22} = -37.6718497263809$$
$$x_{23} = -6.08916120309943$$
$$x_{24} = -43.9590233567938$$
$$x_{25} = 59.6737803264459$$
$$x_{26} = -69.1003552230555$$
$$x_{27} = 28.2401476526276$$
$$x_{28} = 21.9475985837942$$
$$x_{29} = -18.7934144113698$$
$$x_{30} = -56.5306616416093$$
$$x_{31} = 91.0953290668266$$
$$x_{32} = 197.915309953386$$
$$x_{33} = -50.2451786914948$$
$$x_{34} = 97.379207861883$$
Puntos máximos de la función:
$$x_{34} = 43.9600588531378$$
$$x_{34} = 18.7990914357831$$
$$x_{34} = 69.1007741687956$$
$$x_{34} = -91.0950880256329$$
$$x_{34} = -28.2376364595748$$
$$x_{34} = -637.741738184573$$
$$x_{34} = -53.3879890840753$$
$$x_{34} = 1313.18496827279$$
$$x_{34} = -72.24259540785$$
$$x_{34} = -2.57625015820118$$
$$x_{34} = 6.14411351301787$$
$$x_{34} = 94.2372799036618$$
$$x_{34} = 81.6693131963402$$
$$x_{34} = 56.5312876685112$$
$$x_{34} = -47.1022022669651$$
$$x_{34} = 100.521115065812$$
$$x_{34} = 37.673259943911$$
$$x_{34} = -97.378996929011$$
$$x_{34} = -15.6397620877646$$
$$x_{34} = -21.9434371567881$$
$$x_{34} = -59.6732185170696$$
$$x_{34} = 12.492390025579$$
$$x_{34} = 50.2459712046114$$
$$x_{34} = 31.38505790634$$
$$x_{34} = -34.527701946778$$
$$x_{34} = 75.3851328811964$$
$$x_{34} = -40.8155939881502$$
$$x_{34} = -9.30494468339504$$
$$x_{34} = -78.5269183093816$$
$$x_{34} = 87.9533529268738$$
$$x_{34} = -84.8110706151124$$
$$x_{34} = 25.0944376288815$$
$$x_{34} = 62.8161843480611$$
$$x_{34} = -65.9580523911179$$
Decrece en los intervalos
$$\left[197.915309953386, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.520917114109\right]$$
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
$$x_{1} = 23.4801553706306$$
$$x_{2} = 95.797912862081$$
$$x_{3} = 86.3709050594723$$
$$x_{4} = -54.9407852505616$$
$$x_{5} = 4.32863617605124$$
$$x_{6} = -86.3703684986956$$
$$x_{7} = 83.228458145445$$
$$x_{8} = 42.3653647291314$$
$$x_{9} = -26.6254109350763$$
$$x_{10} = 32.9277399444348$$
$$x_{11} = -58.0844210975337$$
$$x_{12} = 64.3720505127272$$
$$x_{13} = -10.7898786754269$$
$$x_{14} = 26.6310922236611$$
$$x_{15} = 36.0743437126941$$
$$x_{16} = 92.6556268279389$$
$$x_{17} = 89.5132926274963$$
$$x_{18} = -89.5127931011103$$
$$x_{19} = -80.0853208283276$$
$$x_{20} = 45.5100787997204$$
$$x_{21} = 98.9401552763972$$
$$x_{22} = -45.5081427660817$$
$$x_{23} = 51.7983897861238$$
$$x_{24} = -95.7974767616183$$
$$x_{25} = 70.657920700132$$
$$x_{26} = -64.3710840254309$$
$$x_{27} = 10.825651157762$$
$$x_{28} = -67.5141687409854$$
$$x_{29} = -32.9240332040206$$
$$x_{30} = -39.2175523279643$$
$$x_{31} = -29.7755709323142$$
$$x_{32} = 136.64474976163$$
$$x_{33} = 29.7801075137773$$
$$x_{34} = -76.9426813176863$$
$$x_{35} = -98.9397464504172$$
$$x_{36} = -17.1546413657741$$
$$x_{37} = 39.22016138731$$
$$x_{38} = -73.7999513585394$$
$$x_{39} = -48.6527034788051$$
$$x_{40} = 48.654396838104$$
$$x_{41} = 61.2289118119026$$
$$x_{42} = -4.00507341668955$$
$$x_{43} = -42.3631297553676$$
$$x_{44} = -83.2278802717944$$
$$x_{45} = -36.0712578833702$$
$$x_{46} = 17.1684571899007$$
$$x_{47} = 14.0034717913284$$
$$x_{48} = -70.6571186927646$$
$$x_{49} = -13.9825085391948$$
$$x_{50} = 76.9433575383977$$
$$x_{51} = -7.54372449628009$$
$$x_{52} = 102.082358062481$$
$$x_{53} = 80.0859449790141$$
$$x_{54} = -61.2278434114583$$
$$x_{55} = 58.0856084395179$$
$$x_{56} = 7.61991323310644$$
$$x_{57} = 20.3264348242219$$
$$x_{58} = -20.3166301288662$$
$$x_{59} = 67.5150472396589$$
$$x_{60} = 54.9421125829153$$
$$x_{61} = -92.6551606286758$$
$$x_{62} = -23.4728313498836$$
$$x_{63} = 73.80068645168$$
$$x_{64} = -51.7968961320869$$
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
$$\left[102.082358062481, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.7974767616183\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$x_{1} = 23.4801553706306$$
$$x_{2} = 95.797912862081$$
$$x_{3} = 86.3709050594723$$
$$x_{4} = -54.9407852505616$$
$$x_{5} = 4.32863617605124$$
$$x_{6} = -86.3703684986956$$
$$x_{7} = 83.228458145445$$
$$x_{8} = 42.3653647291314$$
$$x_{9} = -26.6254109350763$$
$$x_{10} = 32.9277399444348$$
$$x_{11} = -58.0844210975337$$
$$x_{12} = 64.3720505127272$$
$$x_{13} = -10.7898786754269$$
$$x_{14} = 26.6310922236611$$
$$x_{15} = 36.0743437126941$$
$$x_{16} = 92.6556268279389$$
$$x_{17} = 89.5132926274963$$
$$x_{18} = -89.5127931011103$$
$$x_{19} = -80.0853208283276$$
$$x_{20} = 45.5100787997204$$
$$x_{21} = 98.9401552763972$$
$$x_{22} = -45.5081427660817$$
$$x_{23} = 51.7983897861238$$
$$x_{24} = -95.7974767616183$$
$$x_{25} = 70.657920700132$$
$$x_{26} = -64.3710840254309$$
$$x_{27} = 10.825651157762$$
$$x_{28} = -67.5141687409854$$
$$x_{29} = -32.9240332040206$$
$$x_{30} = -39.2175523279643$$
$$x_{31} = -29.7755709323142$$
$$x_{32} = 136.64474976163$$
$$x_{33} = 29.7801075137773$$
$$x_{34} = -76.9426813176863$$
$$x_{35} = -98.9397464504172$$
$$x_{36} = -17.1546413657741$$
$$x_{37} = 39.22016138731$$
$$x_{38} = -73.7999513585394$$
$$x_{39} = -48.6527034788051$$
$$x_{40} = 48.654396838104$$
$$x_{41} = 61.2289118119026$$
$$x_{42} = -4.00507341668955$$
$$x_{43} = -42.3631297553676$$
$$x_{44} = -83.2278802717944$$
$$x_{45} = -36.0712578833702$$
$$x_{46} = 17.1684571899007$$
$$x_{47} = 14.0034717913284$$
$$x_{48} = -70.6571186927646$$
$$x_{49} = -13.9825085391948$$
$$x_{50} = 76.9433575383977$$
$$x_{51} = -7.54372449628009$$
$$x_{52} = 102.082358062481$$
$$x_{53} = 80.0859449790141$$
$$x_{54} = -61.2278434114583$$
$$x_{55} = 58.0856084395179$$
$$x_{56} = 7.61991323310644$$
$$x_{57} = 20.3264348242219$$
$$x_{58} = -20.3166301288662$$
$$x_{59} = 67.5150472396589$$
$$x_{60} = 54.9421125829153$$
$$x_{61} = -92.6551606286758$$
$$x_{62} = -23.4728313498836$$
$$x_{63} = 73.80068645168$$
$$x_{64} = -51.7968961320869$$
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
$$\left[102.082358062481, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.7974767616183\right]$$
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:
$$\frac{\cos{\left(x \right)}}{x + 1} = \frac{\cos{\left(x \right)}}{1 - x}$$
- No
$$\frac{\cos{\left(x \right)}}{x + 1} = - \frac{\cos{\left(x \right)}}{1 - x}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\cos{\left(x \right)}}{x + 1} = \frac{\cos{\left(x \right)}}{1 - x}$$
- No
$$\frac{\cos{\left(x \right)}}{x + 1} = - \frac{\cos{\left(x \right)}}{1 - x}$$
- No
es decir, función
no es
par ni impar