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¿Cómo usar?
- Gráfico de la función y =:
-
x-x^3
-
x/(x^2-1)
-
x^2/(x-3)
-
x^2/(x^2+1)
-
Expresiones idénticas
- arctg((quince)/(x− dieciocho)^ tres)+(sin (x− siete)/(x^ dos − cuarenta y nueve))
- arctg((15) dividir por (x−18) al cubo ) más ( seno de (x−7) dividir por (x al cuadrado −49))
- arctg((quince) dividir por (x− dieciocho) en el grado tres) más ( seno de (x− siete) dividir por (x en el grado dos − cuarenta y nueve))
- arctg((15)/(x−18)3)+(sin (x−7)/(x2−49))
- arctg15/x−183+sin x−7/x2−49
- arctg((15)/(x−18)³)+(sin (x−7)/(x²−49))
- arctg((15)/(x−18) en el grado 3)+(sin (x−7)/(x en el grado 2−49))
- arctg15/x−18^3+sin x−7/x^2−49
- arctg((15) dividir por (x−18)^3)+(sin (x−7) dividir por (x^2−49))
-
Expresiones semejantes
- arctg((15)/(x−18)^3)-(sin (x−7)/(x^2−49))
-
Expresiones con funciones
- arctg
- arctg(2/(x-1))
- arctg(1/((x^2)-4))
- arctg(x)-0,5*ln(1+x^2)
- arctg(x/(x^2-1))
- arctgx-ln(1+x^2)
- Seno sin
- sin(x)/(2*x+16)
- sinx/(2+cosx)
- sin(x)/2
- sin(x^x)
- sin(x)*e^x
Gráfico de la función y = arctg((15)/(x−18)^3)+(sin (x−7)/(x^2−49))
Solución
Ha introducido
[src]
/ 15 \ sin(x - 7)
f(x) = atan|---------| + ----------
| 3| 2
\(x - 18) / x - 49
$$f{\left(x \right)} = \operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49}$$
f = atan(15/(x - 18)^3) + sin(x - 7)/(x^2 - 49)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -7$$
$$x_{2} = 7$$
$$x_{3} = 18$$
$$x_{1} = -7$$
$$x_{2} = 7$$
$$x_{3} = 18$$
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:
$$\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -43.1462724853472$$
$$x_{2} = 82.0063808676567$$
$$x_{3} = -74.5769999942575$$
$$x_{4} = -62.0036283542186$$
$$x_{5} = -99.7230428169496$$
$$x_{6} = -11.7983972831639$$
$$x_{7} = -65.3662000466304$$
$$x_{8} = 4.03957291079201$$
$$x_{9} = 9.21897295681234$$
$$x_{10} = 55.1873823274237$$
$$x_{11} = -84.2052813852368$$
$$x_{12} = 104.643495610971$$
$$x_{13} = -52.8062776358602$$
$$x_{14} = -80.8636786168181$$
$$x_{15} = -30.5829118060154$$
$$x_{16} = 280.383847415247$$
$$x_{17} = -96.7651213038375$$
$$x_{18} = -55.7171947049145$$
$$x_{19} = -5.58670652172326$$
$$x_{20} = 100.97810510363$$
$$x_{21} = -18.0440271057062$$
$$x_{22} = 62.8426751788344$$
$$x_{23} = 75.6548199609954$$
$$x_{24} = 69.27498397012$$
$$x_{25} = -71.6458714359388$$
$$x_{26} = 56.2793060884911$$
$$x_{27} = -59.0864040799867$$
$$x_{28} = -93.4367238611107$$
$$x_{29} = 79.6711966386377$$
$$x_{30} = -15.065056556323$$
$$x_{31} = -87.1502637636866$$
$$x_{32} = 67.2859807257816$$
$$x_{33} = -8.7293304973432$$
$$x_{34} = 167.347445652567$$
$$x_{35} = -677.84674357799$$
$$x_{36} = -24.3083873996827$$
$$x_{37} = -40.2434074736969$$
$$x_{38} = 98.3880004205129$$
$$x_{39} = -36.8630317517713$$
$$x_{40} = 73.4629330462982$$
$$x_{41} = -46.5254747618831$$
$$x_{42} = 94.6633643346979$$
$$x_{43} = -49.4312560574656$$
$$x_{44} = -27.670608536181$$
$$x_{45} = 85.8986963205016$$
$$x_{46} = -90.4851340134267$$
$$x_{47} = 92.1389186736838$$
$$x_{48} = -68.2902843406367$$
$$x_{49} = 61.1665966783263$$
$$x_{50} = -2.34726021899064$$
$$x_{51} = 0.578320721476488$$
$$x_{52} = -33.95905673455$$
$$x_{53} = 117.167725773897$$
$$x_{54} = 88.3406104239676$$
$$x_{55} = -194.002723168149$$
$$x_{56} = -77.9255410502588$$
$$x_{57} = -21.3747260022445$$
$$x_{58} = -256.846138701193$$
$$x_{59} = 258.255263580182$$
o sea hay que resolver la ecuación:
$$\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -43.1462724853472$$
$$x_{2} = 82.0063808676567$$
$$x_{3} = -74.5769999942575$$
$$x_{4} = -62.0036283542186$$
$$x_{5} = -99.7230428169496$$
$$x_{6} = -11.7983972831639$$
$$x_{7} = -65.3662000466304$$
$$x_{8} = 4.03957291079201$$
$$x_{9} = 9.21897295681234$$
$$x_{10} = 55.1873823274237$$
$$x_{11} = -84.2052813852368$$
$$x_{12} = 104.643495610971$$
$$x_{13} = -52.8062776358602$$
$$x_{14} = -80.8636786168181$$
$$x_{15} = -30.5829118060154$$
$$x_{16} = 280.383847415247$$
$$x_{17} = -96.7651213038375$$
$$x_{18} = -55.7171947049145$$
$$x_{19} = -5.58670652172326$$
$$x_{20} = 100.97810510363$$
$$x_{21} = -18.0440271057062$$
$$x_{22} = 62.8426751788344$$
$$x_{23} = 75.6548199609954$$
$$x_{24} = 69.27498397012$$
$$x_{25} = -71.6458714359388$$
$$x_{26} = 56.2793060884911$$
$$x_{27} = -59.0864040799867$$
$$x_{28} = -93.4367238611107$$
$$x_{29} = 79.6711966386377$$
$$x_{30} = -15.065056556323$$
$$x_{31} = -87.1502637636866$$
$$x_{32} = 67.2859807257816$$
$$x_{33} = -8.7293304973432$$
$$x_{34} = 167.347445652567$$
$$x_{35} = -677.84674357799$$
$$x_{36} = -24.3083873996827$$
$$x_{37} = -40.2434074736969$$
$$x_{38} = 98.3880004205129$$
$$x_{39} = -36.8630317517713$$
$$x_{40} = 73.4629330462982$$
$$x_{41} = -46.5254747618831$$
$$x_{42} = 94.6633643346979$$
$$x_{43} = -49.4312560574656$$
$$x_{44} = -27.670608536181$$
$$x_{45} = 85.8986963205016$$
$$x_{46} = -90.4851340134267$$
$$x_{47} = 92.1389186736838$$
$$x_{48} = -68.2902843406367$$
$$x_{49} = 61.1665966783263$$
$$x_{50} = -2.34726021899064$$
$$x_{51} = 0.578320721476488$$
$$x_{52} = -33.95905673455$$
$$x_{53} = 117.167725773897$$
$$x_{54} = 88.3406104239676$$
$$x_{55} = -194.002723168149$$
$$x_{56} = -77.9255410502588$$
$$x_{57} = -21.3747260022445$$
$$x_{58} = -256.846138701193$$
$$x_{59} = 258.255263580182$$
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
$$- \frac{2 x \sin{\left(x - 7 \right)}}{\left(x^{2} - 49\right)^{2}} + \frac{\cos{\left(x - 7 \right)}}{x^{2} - 49} - \frac{45}{\left(1 + \frac{225}{\left(x - 18\right)^{6}}\right) \left(x - 18\right)^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 39.616368299213$$
$$x_{2} = -38.4930899959304$$
$$x_{3} = 58.746221112162$$
$$x_{4} = 37.243363203005$$
$$x_{5} = -44.7848738612109$$
$$x_{6} = -4.25949442481788$$
$$x_{7} = -85.6563215028264$$
$$x_{8} = 83.9283705614661$$
$$x_{9} = 55.7261416332986$$
$$x_{10} = 112.232625653239$$
$$x_{11} = 77.6386072660827$$
$$x_{12} = 322.72331829161$$
$$x_{13} = 65.0499704112954$$
$$x_{14} = -88.7932046455853$$
$$x_{15} = 61.9907825636782$$
$$x_{16} = 2.43288745650569$$
$$x_{17} = -22.7559695846311$$
$$x_{18} = -0.908633644730496$$
$$x_{19} = -91.9408801415187$$
$$x_{20} = -13.2069648676888$$
$$x_{21} = 93.3843443037553$$
$$x_{22} = 74.5414217354957$$
$$x_{23} = -47.9406189375993$$
$$x_{24} = -60.5150326953888$$
$$x_{25} = 49.4803433095141$$
$$x_{26} = -79.3715360768524$$
$$x_{27} = -9.89796933281585$$
$$x_{28} = -57.3627521094044$$
$$x_{29} = -29.0626691039527$$
$$x_{30} = 68.2639545683044$$
$$x_{31} = -76.2224613346636$$
$$x_{32} = 80.821226078285$$
$$x_{33} = -176.77086336586$$
$$x_{34} = -25.896087058162$$
$$x_{35} = -98.2252582179878$$
$$x_{36} = -63.6499886395614$$
$$x_{37} = -51.0745215831412$$
$$x_{38} = 96.5035854057696$$
$$x_{39} = -54.2283461483598$$
$$x_{40} = 6.66896819664144$$
$$x_{41} = -73.0864601175605$$
$$x_{42} = -120.216035010594$$
$$x_{43} = 17.9585083116315$$
$$x_{44} = 43.2829691732838$$
$$x_{45} = -16.4211857976036$$
$$x_{46} = -66.8010031985272$$
$$x_{47} = -35.3593562575238$$
$$x_{48} = 99.6668174334976$$
$$x_{49} = -95.0781404091631$$
$$x_{50} = 71.3463482720541$$
$$x_{51} = 46.0748630292243$$
$$x_{52} = 87.1023786218781$$
$$x_{53} = -107.64739294356$$
$$x_{54} = 90.2165238619289$$
$$x_{55} = -69.9364988549245$$
$$x_{56} = -19.5794578059975$$
$$x_{57} = 52.4277555891037$$
$$x_{58} = -82.5080002121975$$
$$x_{59} = -41.6512972943426$$
$$x_{60} = -32.1978174872384$$
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:
$$x_{1} = -38.4930899959304$$
$$x_{2} = 37.243363203005$$
$$x_{3} = -44.7848738612109$$
$$x_{4} = -4.25949442481788$$
$$x_{5} = 55.7261416332986$$
$$x_{6} = 112.232625653239$$
$$x_{7} = -88.7932046455853$$
$$x_{8} = 61.9907825636782$$
$$x_{9} = 2.43288745650569$$
$$x_{10} = -13.2069648676888$$
$$x_{11} = 93.3843443037553$$
$$x_{12} = 74.5414217354957$$
$$x_{13} = 49.4803433095141$$
$$x_{14} = -57.3627521094044$$
$$x_{15} = 68.2639545683044$$
$$x_{16} = -76.2224613346636$$
$$x_{17} = 80.821226078285$$
$$x_{18} = -176.77086336586$$
$$x_{19} = -25.896087058162$$
$$x_{20} = -63.6499886395614$$
$$x_{21} = -51.0745215831412$$
$$x_{22} = -120.216035010594$$
$$x_{23} = 17.9585083116315$$
$$x_{24} = 43.2829691732838$$
$$x_{25} = 99.6668174334976$$
$$x_{26} = -95.0781404091631$$
$$x_{27} = 87.1023786218781$$
$$x_{28} = -107.64739294356$$
$$x_{29} = -69.9364988549245$$
$$x_{30} = -19.5794578059975$$
$$x_{31} = -82.5080002121975$$
$$x_{32} = -32.1978174872384$$
Puntos máximos de la función:
$$x_{32} = 39.616368299213$$
$$x_{32} = 58.746221112162$$
$$x_{32} = -85.6563215028264$$
$$x_{32} = 83.9283705614661$$
$$x_{32} = 77.6386072660827$$
$$x_{32} = 322.72331829161$$
$$x_{32} = 65.0499704112954$$
$$x_{32} = -22.7559695846311$$
$$x_{32} = -0.908633644730496$$
$$x_{32} = -91.9408801415187$$
$$x_{32} = -47.9406189375993$$
$$x_{32} = -60.5150326953888$$
$$x_{32} = -79.3715360768524$$
$$x_{32} = -9.89796933281585$$
$$x_{32} = -29.0626691039527$$
$$x_{32} = -98.2252582179878$$
$$x_{32} = 96.5035854057696$$
$$x_{32} = -54.2283461483598$$
$$x_{32} = 6.66896819664144$$
$$x_{32} = -73.0864601175605$$
$$x_{32} = -16.4211857976036$$
$$x_{32} = -66.8010031985272$$
$$x_{32} = -35.3593562575238$$
$$x_{32} = 71.3463482720541$$
$$x_{32} = 46.0748630292243$$
$$x_{32} = 90.2165238619289$$
$$x_{32} = 52.4277555891037$$
$$x_{32} = -41.6512972943426$$
Decrece en los intervalos
$$\left[112.232625653239, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -176.77086336586\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{2 x \sin{\left(x - 7 \right)}}{\left(x^{2} - 49\right)^{2}} + \frac{\cos{\left(x - 7 \right)}}{x^{2} - 49} - \frac{45}{\left(1 + \frac{225}{\left(x - 18\right)^{6}}\right) \left(x - 18\right)^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 39.616368299213$$
$$x_{2} = -38.4930899959304$$
$$x_{3} = 58.746221112162$$
$$x_{4} = 37.243363203005$$
$$x_{5} = -44.7848738612109$$
$$x_{6} = -4.25949442481788$$
$$x_{7} = -85.6563215028264$$
$$x_{8} = 83.9283705614661$$
$$x_{9} = 55.7261416332986$$
$$x_{10} = 112.232625653239$$
$$x_{11} = 77.6386072660827$$
$$x_{12} = 322.72331829161$$
$$x_{13} = 65.0499704112954$$
$$x_{14} = -88.7932046455853$$
$$x_{15} = 61.9907825636782$$
$$x_{16} = 2.43288745650569$$
$$x_{17} = -22.7559695846311$$
$$x_{18} = -0.908633644730496$$
$$x_{19} = -91.9408801415187$$
$$x_{20} = -13.2069648676888$$
$$x_{21} = 93.3843443037553$$
$$x_{22} = 74.5414217354957$$
$$x_{23} = -47.9406189375993$$
$$x_{24} = -60.5150326953888$$
$$x_{25} = 49.4803433095141$$
$$x_{26} = -79.3715360768524$$
$$x_{27} = -9.89796933281585$$
$$x_{28} = -57.3627521094044$$
$$x_{29} = -29.0626691039527$$
$$x_{30} = 68.2639545683044$$
$$x_{31} = -76.2224613346636$$
$$x_{32} = 80.821226078285$$
$$x_{33} = -176.77086336586$$
$$x_{34} = -25.896087058162$$
$$x_{35} = -98.2252582179878$$
$$x_{36} = -63.6499886395614$$
$$x_{37} = -51.0745215831412$$
$$x_{38} = 96.5035854057696$$
$$x_{39} = -54.2283461483598$$
$$x_{40} = 6.66896819664144$$
$$x_{41} = -73.0864601175605$$
$$x_{42} = -120.216035010594$$
$$x_{43} = 17.9585083116315$$
$$x_{44} = 43.2829691732838$$
$$x_{45} = -16.4211857976036$$
$$x_{46} = -66.8010031985272$$
$$x_{47} = -35.3593562575238$$
$$x_{48} = 99.6668174334976$$
$$x_{49} = -95.0781404091631$$
$$x_{50} = 71.3463482720541$$
$$x_{51} = 46.0748630292243$$
$$x_{52} = 87.1023786218781$$
$$x_{53} = -107.64739294356$$
$$x_{54} = 90.2165238619289$$
$$x_{55} = -69.9364988549245$$
$$x_{56} = -19.5794578059975$$
$$x_{57} = 52.4277555891037$$
$$x_{58} = -82.5080002121975$$
$$x_{59} = -41.6512972943426$$
$$x_{60} = -32.1978174872384$$
Signos de extremos en los puntos:
(39.616368299212986, 0.00209816038465741)
(-38.493089995930376, -0.000779914489914381)
(58.74622111216198, 0.000514475786442175)
(37.24336320300504, 0.00141611407515111)
(-44.78487386121087, -0.000571000866027867)
(-4.259494424817876, -0.0326457543455179)
(-85.65632150282642, 0.000123714441785647)
(83.9283705614661, 0.00019518662869738)
(55.72614163329864, -4.76602425941437e-5)
(112.23262565323878, -6.17685449231973e-5)
(77.63860726608272, 0.000237787003567778)
(322.72331829160976, 1.01359258633345e-5)
(65.04997041129539, 0.000382531242715372)
(-88.79320464558532, -0.000139903281725586)
(61.990782563678195, -8.73619050529521e-5)
(2.43288745650569, -0.0269437081659349)
(-22.75596958463108, 0.0019029045934998)
(-0.9086336447304962, 0.0185081679658252)
(-91.94088014151873, 0.000107679134272738)
(-13.206964867688798, -0.0082856836208053)
(93.38434430375531, -8.02988561507707e-5)
(74.54142173549572, -9.85890105920133e-5)
(-47.940618937599346, 0.000391958009347993)
(-60.515032695388804, 0.000245664138086798)
(49.48034330951412, 6.50090628392347e-5)
(-79.37153607685242, 0.000143691015978932)
(-9.897969332815853, 0.0182672034537524)
(-57.36275210940439, -0.000343299637928573)
(-29.0626691039527, 0.00111024436554193)
(68.26395456830441, -9.87538971234401e-5)
(-76.22246133466363, -0.000191440271656197)
(80.82122607828502, -9.37426097836467e-5)
(-176.77086336585987, -3.40800543392824e-5)
(-25.89608705816199, -0.00177947233512727)
(-98.2252582179878, 9.46041792684594e-5)
(-63.64998863956135, -0.00027725093596408)
(-51.07452158314125, -0.000435803745729241)
(96.50358540576956, 0.000138895176287741)
(-54.228346148359755, 0.000305822923305202)
(6.668968196641439, 0.0615193621598029)
(-73.08646011756048, 0.000169038256038131)
(-120.21603501059386, -7.50995310873934e-5)
(17.958508311631466, -1.57444525335767)
(43.2829691732838, 0.000386547012086254)
(-16.42118579760359, 0.00411933088008017)
(-66.80100319852721, 0.000201907565224935)
(-35.35935625752378, 0.00073257873644391)
(99.66681743349764, -7.36241365157881e-5)
(-95.07814040916305, -0.000121567343619895)
(71.3463482720541, 0.000296851777700753)
(46.07486302922431, 0.00115089947000696)
(87.10237862187809, -8.7201573719003e-5)
(-107.64739294356033, -9.42061754333146e-5)
(90.21652386192893, 0.000163357751582608)
(-69.9364988549245, -0.000228471065125669)
(-19.579457805997492, -0.0032504819838296)
(52.42775558910369, 0.000735101172630324)
(-82.5080002121975, -0.000162678495160853)
(-41.65129729434258, 0.000521951723441617)
(-32.19781748723837, -0.00112841089097065)
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:
$$x_{1} = -38.4930899959304$$
$$x_{2} = 37.243363203005$$
$$x_{3} = -44.7848738612109$$
$$x_{4} = -4.25949442481788$$
$$x_{5} = 55.7261416332986$$
$$x_{6} = 112.232625653239$$
$$x_{7} = -88.7932046455853$$
$$x_{8} = 61.9907825636782$$
$$x_{9} = 2.43288745650569$$
$$x_{10} = -13.2069648676888$$
$$x_{11} = 93.3843443037553$$
$$x_{12} = 74.5414217354957$$
$$x_{13} = 49.4803433095141$$
$$x_{14} = -57.3627521094044$$
$$x_{15} = 68.2639545683044$$
$$x_{16} = -76.2224613346636$$
$$x_{17} = 80.821226078285$$
$$x_{18} = -176.77086336586$$
$$x_{19} = -25.896087058162$$
$$x_{20} = -63.6499886395614$$
$$x_{21} = -51.0745215831412$$
$$x_{22} = -120.216035010594$$
$$x_{23} = 17.9585083116315$$
$$x_{24} = 43.2829691732838$$
$$x_{25} = 99.6668174334976$$
$$x_{26} = -95.0781404091631$$
$$x_{27} = 87.1023786218781$$
$$x_{28} = -107.64739294356$$
$$x_{29} = -69.9364988549245$$
$$x_{30} = -19.5794578059975$$
$$x_{31} = -82.5080002121975$$
$$x_{32} = -32.1978174872384$$
Puntos máximos de la función:
$$x_{32} = 39.616368299213$$
$$x_{32} = 58.746221112162$$
$$x_{32} = -85.6563215028264$$
$$x_{32} = 83.9283705614661$$
$$x_{32} = 77.6386072660827$$
$$x_{32} = 322.72331829161$$
$$x_{32} = 65.0499704112954$$
$$x_{32} = -22.7559695846311$$
$$x_{32} = -0.908633644730496$$
$$x_{32} = -91.9408801415187$$
$$x_{32} = -47.9406189375993$$
$$x_{32} = -60.5150326953888$$
$$x_{32} = -79.3715360768524$$
$$x_{32} = -9.89796933281585$$
$$x_{32} = -29.0626691039527$$
$$x_{32} = -98.2252582179878$$
$$x_{32} = 96.5035854057696$$
$$x_{32} = -54.2283461483598$$
$$x_{32} = 6.66896819664144$$
$$x_{32} = -73.0864601175605$$
$$x_{32} = -16.4211857976036$$
$$x_{32} = -66.8010031985272$$
$$x_{32} = -35.3593562575238$$
$$x_{32} = 71.3463482720541$$
$$x_{32} = 46.0748630292243$$
$$x_{32} = 90.2165238619289$$
$$x_{32} = 52.4277555891037$$
$$x_{32} = -41.6512972943426$$
Decrece en los intervalos
$$\left[112.232625653239, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -176.77086336586\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -7$$
$$x_{2} = 7$$
$$x_{3} = 18$$
$$x_{1} = -7$$
$$x_{2} = 7$$
$$x_{3} = 18$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49}\right) = 0$$
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 atan(15/(x - 18)^3) + sin(x - 7)/(x^2 - 49), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49}}{x}\right) = 0$$
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:
$$\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49} = \operatorname{atan}{\left(\frac{15}{\left(- x - 18\right)^{3}} \right)} - \frac{\sin{\left(x + 7 \right)}}{x^{2} - 49}$$
- No
$$\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49} = - \operatorname{atan}{\left(\frac{15}{\left(- x - 18\right)^{3}} \right)} + \frac{\sin{\left(x + 7 \right)}}{x^{2} - 49}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49} = \operatorname{atan}{\left(\frac{15}{\left(- x - 18\right)^{3}} \right)} - \frac{\sin{\left(x + 7 \right)}}{x^{2} - 49}$$
- No
$$\operatorname{atan}{\left(\frac{15}{\left(x - 18\right)^{3}} \right)} + \frac{\sin{\left(x - 7 \right)}}{x^{2} - 49} = - \operatorname{atan}{\left(\frac{15}{\left(- x - 18\right)^{3}} \right)} + \frac{\sin{\left(x + 7 \right)}}{x^{2} - 49}$$
- No
es decir, función
no es
par ni impar