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- Integrales paso a paso
- Gráfico de la función paramétrica
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- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
x-x^3
x^2/(x-3)
x^2/(x^2+1)
-x^3+x
z*cosz
Gráfico de la función y = z*cosz
Solución
Gráfico de la función
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje Z con f = 0
o sea hay que resolver la ecuación:
$$z \cos{\left(z \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = - \frac{\pi}{2}$$
$$z_{3} = \frac{\pi}{2}$$
Solución numérica
$$z_{1} = 26.7035375555132$$
$$z_{2} = 114.668131856027$$
$$z_{3} = 64.4026493985908$$
$$z_{4} = 17.2787595947439$$
$$z_{5} = 80.1106126665397$$
$$z_{6} = -48.6946861306418$$
$$z_{7} = 23.5619449019235$$
$$z_{8} = -114.668131856027$$
$$z_{9} = -39.2699081698724$$
$$z_{10} = -86.3937979737193$$
$$z_{11} = -98.9601685880785$$
$$z_{12} = 36.1283155162826$$
$$z_{13} = 0$$
$$z_{14} = -51.8362787842316$$
$$z_{15} = -45.553093477052$$
$$z_{16} = -73.8274273593601$$
$$z_{17} = -10.9955742875643$$
$$z_{18} = 14.1371669411541$$
$$z_{19} = -4.71238898038469$$
$$z_{20} = -14.1371669411541$$
$$z_{21} = -42.4115008234622$$
$$z_{22} = -17.2787595947439$$
$$z_{23} = 51.8362787842316$$
$$z_{24} = -29.845130209103$$
$$z_{25} = 70.6858347057703$$
$$z_{26} = -95.8185759344887$$
$$z_{27} = 92.6769832808989$$
$$z_{28} = 67.5442420521806$$
$$z_{29} = 61.261056745001$$
$$z_{30} = -76.9690200129499$$
$$z_{31} = 10.9955742875643$$
$$z_{32} = -70.6858347057703$$
$$z_{33} = 58.1194640914112$$
$$z_{34} = -83.2522053201295$$
$$z_{35} = -7.85398163397448$$
$$z_{36} = 39.2699081698724$$
$$z_{37} = 54.9778714378214$$
$$z_{38} = 73.8274273593601$$
$$z_{39} = -80.1106126665397$$
$$z_{40} = -26.7035375555132$$
$$z_{41} = 7.85398163397448$$
$$z_{42} = 29.845130209103$$
$$z_{43} = -58.1194640914112$$
$$z_{44} = 48.6946861306418$$
$$z_{45} = 76.9690200129499$$
$$z_{46} = -67.5442420521806$$
$$z_{47} = 83.2522053201295$$
$$z_{48} = 95.8185759344887$$
$$z_{49} = 20.4203522483337$$
$$z_{50} = 32.9867228626928$$
$$z_{51} = 42.4115008234622$$
$$z_{52} = 89.5353906273091$$
$$z_{53} = 86.3937979737193$$
$$z_{54} = -54.9778714378214$$
$$z_{55} = -92.6769832808989$$
$$z_{56} = -36.1283155162826$$
$$z_{57} = -1.5707963267949$$
$$z_{58} = -23.5619449019235$$
$$z_{59} = -64.4026493985908$$
$$z_{60} = 4.71238898038469$$
$$z_{61} = -20.4203522483337$$
$$z_{62} = 1.5707963267949$$
$$z_{63} = 45.553093477052$$
$$z_{64} = 98.9601685880785$$
$$z_{65} = -32.9867228626928$$
$$z_{66} = -61.261056745001$$
$$z_{67} = -89.5353906273091$$
o sea hay que resolver la ecuación:
$$z \cos{\left(z \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = - \frac{\pi}{2}$$
$$z_{3} = \frac{\pi}{2}$$
Solución numérica
$$z_{1} = 26.7035375555132$$
$$z_{2} = 114.668131856027$$
$$z_{3} = 64.4026493985908$$
$$z_{4} = 17.2787595947439$$
$$z_{5} = 80.1106126665397$$
$$z_{6} = -48.6946861306418$$
$$z_{7} = 23.5619449019235$$
$$z_{8} = -114.668131856027$$
$$z_{9} = -39.2699081698724$$
$$z_{10} = -86.3937979737193$$
$$z_{11} = -98.9601685880785$$
$$z_{12} = 36.1283155162826$$
$$z_{13} = 0$$
$$z_{14} = -51.8362787842316$$
$$z_{15} = -45.553093477052$$
$$z_{16} = -73.8274273593601$$
$$z_{17} = -10.9955742875643$$
$$z_{18} = 14.1371669411541$$
$$z_{19} = -4.71238898038469$$
$$z_{20} = -14.1371669411541$$
$$z_{21} = -42.4115008234622$$
$$z_{22} = -17.2787595947439$$
$$z_{23} = 51.8362787842316$$
$$z_{24} = -29.845130209103$$
$$z_{25} = 70.6858347057703$$
$$z_{26} = -95.8185759344887$$
$$z_{27} = 92.6769832808989$$
$$z_{28} = 67.5442420521806$$
$$z_{29} = 61.261056745001$$
$$z_{30} = -76.9690200129499$$
$$z_{31} = 10.9955742875643$$
$$z_{32} = -70.6858347057703$$
$$z_{33} = 58.1194640914112$$
$$z_{34} = -83.2522053201295$$
$$z_{35} = -7.85398163397448$$
$$z_{36} = 39.2699081698724$$
$$z_{37} = 54.9778714378214$$
$$z_{38} = 73.8274273593601$$
$$z_{39} = -80.1106126665397$$
$$z_{40} = -26.7035375555132$$
$$z_{41} = 7.85398163397448$$
$$z_{42} = 29.845130209103$$
$$z_{43} = -58.1194640914112$$
$$z_{44} = 48.6946861306418$$
$$z_{45} = 76.9690200129499$$
$$z_{46} = -67.5442420521806$$
$$z_{47} = 83.2522053201295$$
$$z_{48} = 95.8185759344887$$
$$z_{49} = 20.4203522483337$$
$$z_{50} = 32.9867228626928$$
$$z_{51} = 42.4115008234622$$
$$z_{52} = 89.5353906273091$$
$$z_{53} = 86.3937979737193$$
$$z_{54} = -54.9778714378214$$
$$z_{55} = -92.6769832808989$$
$$z_{56} = -36.1283155162826$$
$$z_{57} = -1.5707963267949$$
$$z_{58} = -23.5619449019235$$
$$z_{59} = -64.4026493985908$$
$$z_{60} = 4.71238898038469$$
$$z_{61} = -20.4203522483337$$
$$z_{62} = 1.5707963267949$$
$$z_{63} = 45.553093477052$$
$$z_{64} = 98.9601685880785$$
$$z_{65} = -32.9867228626928$$
$$z_{66} = -61.261056745001$$
$$z_{67} = -89.5353906273091$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d z} f{\left(z \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d z} f{\left(z \right)} = $$
primera derivada
$$- z \sin{\left(z \right)} + \cos{\left(z \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = 62.8477631944545$$
$$z_{2} = 47.145097736761$$
$$z_{3} = 37.7256128277765$$
$$z_{4} = -22.0364967279386$$
$$z_{5} = -91.1171613944647$$
$$z_{6} = -9.52933440536196$$
$$z_{7} = -53.4257904773947$$
$$z_{8} = 84.8347887180423$$
$$z_{9} = -78.5525459842429$$
$$z_{10} = 40.8651703304881$$
$$z_{11} = 6.43729817917195$$
$$z_{12} = -81.6936492356017$$
$$z_{13} = 9.52933440536196$$
$$z_{14} = 87.9759605524932$$
$$z_{15} = 22.0364967279386$$
$$z_{16} = 56.5663442798215$$
$$z_{17} = 97.3996388790738$$
$$z_{18} = 100.540910786842$$
$$z_{19} = 0.86033358901938$$
$$z_{20} = -75.4114834888481$$
$$z_{21} = 59.7070073053355$$
$$z_{22} = -44.0050179208308$$
$$z_{23} = 15.7712848748159$$
$$z_{24} = -50.2853663377737$$
$$z_{25} = -28.309642854452$$
$$z_{26} = -97.3996388790738$$
$$z_{27} = 34.5864242152889$$
$$z_{28} = -37.7256128277765$$
$$z_{29} = 28.309642854452$$
$$z_{30} = 44.0050179208308$$
$$z_{31} = 94.2583883450399$$
$$z_{32} = 81.6936492356017$$
$$z_{33} = -34.5864242152889$$
$$z_{34} = -69.1295029738953$$
$$z_{35} = 31.4477146375462$$
$$z_{36} = 50.2853663377737$$
$$z_{37} = 78.5525459842429$$
$$z_{38} = -31.4477146375462$$
$$z_{39} = 3.42561845948173$$
$$z_{40} = -116.247530303932$$
$$z_{41} = 72.270467060309$$
$$z_{42} = -18.90240995686$$
$$z_{43} = -0.86033358901938$$
$$z_{44} = -25.1724463266467$$
$$z_{45} = -62.8477631944545$$
$$z_{46} = -65.9885986984904$$
$$z_{47} = -40.8651703304881$$
$$z_{48} = -59.7070073053355$$
$$z_{49} = 18.90240995686$$
$$z_{50} = -147.661626855354$$
$$z_{51} = -56.5663442798215$$
$$z_{52} = -94.2583883450399$$
$$z_{53} = 91.1171613944647$$
$$z_{54} = -47.145097736761$$
$$z_{55} = 75.4114834888481$$
$$z_{56} = 12.6452872238566$$
$$z_{57} = -87.9759605524932$$
$$z_{58} = -100.540910786842$$
$$z_{59} = 69.1295029738953$$
$$z_{60} = -12.6452872238566$$
$$z_{61} = -3.42561845948173$$
$$z_{62} = -15.7712848748159$$
$$z_{63} = 25.1724463266467$$
$$z_{64} = 53.4257904773947$$
$$z_{65} = -84.8347887180423$$
$$z_{66} = 65.9885986984904$$
$$z_{67} = -6.43729817917195$$
$$z_{68} = -72.270467060309$$
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:
$$z_{1} = 47.145097736761$$
$$z_{2} = 84.8347887180423$$
$$z_{3} = 40.8651703304881$$
$$z_{4} = -81.6936492356017$$
$$z_{5} = 9.52933440536196$$
$$z_{6} = 22.0364967279386$$
$$z_{7} = 97.3996388790738$$
$$z_{8} = -75.4114834888481$$
$$z_{9} = 59.7070073053355$$
$$z_{10} = -44.0050179208308$$
$$z_{11} = 15.7712848748159$$
$$z_{12} = -50.2853663377737$$
$$z_{13} = 34.5864242152889$$
$$z_{14} = -37.7256128277765$$
$$z_{15} = 28.309642854452$$
$$z_{16} = -69.1295029738953$$
$$z_{17} = 78.5525459842429$$
$$z_{18} = -31.4477146375462$$
$$z_{19} = 3.42561845948173$$
$$z_{20} = 72.270467060309$$
$$z_{21} = -18.90240995686$$
$$z_{22} = -0.86033358901938$$
$$z_{23} = -25.1724463266467$$
$$z_{24} = -62.8477631944545$$
$$z_{25} = -56.5663442798215$$
$$z_{26} = -94.2583883450399$$
$$z_{27} = 91.1171613944647$$
$$z_{28} = -87.9759605524932$$
$$z_{29} = -100.540910786842$$
$$z_{30} = -12.6452872238566$$
$$z_{31} = 53.4257904773947$$
$$z_{32} = 65.9885986984904$$
$$z_{33} = -6.43729817917195$$
Puntos máximos de la función:
$$z_{33} = 62.8477631944545$$
$$z_{33} = 37.7256128277765$$
$$z_{33} = -22.0364967279386$$
$$z_{33} = -91.1171613944647$$
$$z_{33} = -9.52933440536196$$
$$z_{33} = -53.4257904773947$$
$$z_{33} = -78.5525459842429$$
$$z_{33} = 6.43729817917195$$
$$z_{33} = 87.9759605524932$$
$$z_{33} = 56.5663442798215$$
$$z_{33} = 100.540910786842$$
$$z_{33} = 0.86033358901938$$
$$z_{33} = -28.309642854452$$
$$z_{33} = -97.3996388790738$$
$$z_{33} = 44.0050179208308$$
$$z_{33} = 94.2583883450399$$
$$z_{33} = 81.6936492356017$$
$$z_{33} = -34.5864242152889$$
$$z_{33} = 31.4477146375462$$
$$z_{33} = 50.2853663377737$$
$$z_{33} = -116.247530303932$$
$$z_{33} = -65.9885986984904$$
$$z_{33} = -40.8651703304881$$
$$z_{33} = -59.7070073053355$$
$$z_{33} = 18.90240995686$$
$$z_{33} = -147.661626855354$$
$$z_{33} = -47.145097736761$$
$$z_{33} = 75.4114834888481$$
$$z_{33} = 12.6452872238566$$
$$z_{33} = 69.1295029738953$$
$$z_{33} = -3.42561845948173$$
$$z_{33} = -15.7712848748159$$
$$z_{33} = 25.1724463266467$$
$$z_{33} = -84.8347887180423$$
$$z_{33} = -72.270467060309$$
Decrece en los intervalos
$$\left[97.3996388790738, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.540910786842\right]$$
$$\frac{d}{d z} f{\left(z \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d z} f{\left(z \right)} = $$
primera derivada
$$- z \sin{\left(z \right)} + \cos{\left(z \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = 62.8477631944545$$
$$z_{2} = 47.145097736761$$
$$z_{3} = 37.7256128277765$$
$$z_{4} = -22.0364967279386$$
$$z_{5} = -91.1171613944647$$
$$z_{6} = -9.52933440536196$$
$$z_{7} = -53.4257904773947$$
$$z_{8} = 84.8347887180423$$
$$z_{9} = -78.5525459842429$$
$$z_{10} = 40.8651703304881$$
$$z_{11} = 6.43729817917195$$
$$z_{12} = -81.6936492356017$$
$$z_{13} = 9.52933440536196$$
$$z_{14} = 87.9759605524932$$
$$z_{15} = 22.0364967279386$$
$$z_{16} = 56.5663442798215$$
$$z_{17} = 97.3996388790738$$
$$z_{18} = 100.540910786842$$
$$z_{19} = 0.86033358901938$$
$$z_{20} = -75.4114834888481$$
$$z_{21} = 59.7070073053355$$
$$z_{22} = -44.0050179208308$$
$$z_{23} = 15.7712848748159$$
$$z_{24} = -50.2853663377737$$
$$z_{25} = -28.309642854452$$
$$z_{26} = -97.3996388790738$$
$$z_{27} = 34.5864242152889$$
$$z_{28} = -37.7256128277765$$
$$z_{29} = 28.309642854452$$
$$z_{30} = 44.0050179208308$$
$$z_{31} = 94.2583883450399$$
$$z_{32} = 81.6936492356017$$
$$z_{33} = -34.5864242152889$$
$$z_{34} = -69.1295029738953$$
$$z_{35} = 31.4477146375462$$
$$z_{36} = 50.2853663377737$$
$$z_{37} = 78.5525459842429$$
$$z_{38} = -31.4477146375462$$
$$z_{39} = 3.42561845948173$$
$$z_{40} = -116.247530303932$$
$$z_{41} = 72.270467060309$$
$$z_{42} = -18.90240995686$$
$$z_{43} = -0.86033358901938$$
$$z_{44} = -25.1724463266467$$
$$z_{45} = -62.8477631944545$$
$$z_{46} = -65.9885986984904$$
$$z_{47} = -40.8651703304881$$
$$z_{48} = -59.7070073053355$$
$$z_{49} = 18.90240995686$$
$$z_{50} = -147.661626855354$$
$$z_{51} = -56.5663442798215$$
$$z_{52} = -94.2583883450399$$
$$z_{53} = 91.1171613944647$$
$$z_{54} = -47.145097736761$$
$$z_{55} = 75.4114834888481$$
$$z_{56} = 12.6452872238566$$
$$z_{57} = -87.9759605524932$$
$$z_{58} = -100.540910786842$$
$$z_{59} = 69.1295029738953$$
$$z_{60} = -12.6452872238566$$
$$z_{61} = -3.42561845948173$$
$$z_{62} = -15.7712848748159$$
$$z_{63} = 25.1724463266467$$
$$z_{64} = 53.4257904773947$$
$$z_{65} = -84.8347887180423$$
$$z_{66} = 65.9885986984904$$
$$z_{67} = -6.43729817917195$$
$$z_{68} = -72.270467060309$$
Signos de extremos en los puntos:
(62.84776319445445, 62.8398089721545)
(47.14509773676103, -47.1344957575419)
(37.7256128277765, 37.71236621281)
(-22.036496727938566, 22.0138420791585)
(-91.11716139446474, 91.1116744496469)
(-9.529334405361963, 9.47729425947979)
(-53.42579047739466, 53.4164341598961)
(84.83478871804229, -84.8288955236568)
(-78.55254598424293, 78.5461815917343)
(40.86517033048807, -40.8529404645174)
(6.437298179171947, 6.36100394483385)
(-81.69364923560168, -81.6875294965246)
(9.529334405361963, -9.47729425947979)
(87.97596055249322, 87.9702777324248)
(22.036496727938566, -22.0138420791585)
(56.56634427982152, 56.5575071728762)
(97.39963887907376, -97.3945057956234)
(100.54091078684232, 100.535938055826)
(0.8603335890193797, 0.561096338191045)
(-75.41148348884815, -75.4048540732019)
(59.70700730533546, -59.6986348402658)
(-44.005017920830845, -43.9936599791065)
(15.771284874815882, -15.7396769621337)
(-50.28536633777365, -50.2754260353972)
(-28.30964285445201, 28.2919975390943)
(-97.39963887907376, 97.3945057956234)
(34.58642421528892, -34.5719767335884)
(-37.7256128277765, -37.71236621281)
(28.30964285445201, -28.2919975390943)
(44.005017920830845, 43.9936599791065)
(94.25838834503986, 94.2530842251087)
(81.69364923560168, 81.6875294965246)
(-34.58642421528892, 34.5719767335884)
(-69.12950297389526, -69.1222713069218)
(31.447714637546234, 31.4318272785346)
(50.28536633777365, 50.2754260353972)
(78.55254598424293, -78.5461815917343)
(-31.447714637546234, -31.4318272785346)
(3.4256184594817283, -3.2883713955909)
(-116.2475303039321, 116.243229375987)
(72.27046706030896, -72.2635495982494)
(-18.902409956860023, -18.876013697969)
(-0.8603335890193797, -0.561096338191045)
(-25.172446326646664, -25.1526068178715)
(-62.84776319445445, -62.8398089721545)
(-65.98859869849039, 65.9810229367917)
(-40.86517033048807, 40.8529404645174)
(-59.70700730533546, 59.6986348402658)
(18.902409956860023, 18.876013697969)
(-147.66162685535437, 147.658240851742)
(-56.56634427982152, -56.5575071728762)
(-94.25838834503986, -94.2530842251087)
(91.11716139446474, -91.1116744496469)
(-47.14509773676103, 47.1344957575419)
(75.41148348884815, 75.4048540732019)
(12.645287223856643, 12.6059312978927)
(-87.97596055249322, -87.9702777324248)
(-100.54091078684232, -100.535938055826)
(69.12950297389526, 69.1222713069218)
(-12.645287223856643, -12.6059312978927)
(-3.4256184594817283, 3.2883713955909)
(-15.771284874815882, 15.7396769621337)
(25.172446326646664, 25.1526068178715)
(53.42579047739466, -53.4164341598961)
(-84.83478871804229, 84.8288955236568)
(65.98859869849039, -65.9810229367917)
(-6.437298179171947, -6.36100394483385)
(-72.27046706030896, 72.2635495982494)
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:
$$z_{1} = 47.145097736761$$
$$z_{2} = 84.8347887180423$$
$$z_{3} = 40.8651703304881$$
$$z_{4} = -81.6936492356017$$
$$z_{5} = 9.52933440536196$$
$$z_{6} = 22.0364967279386$$
$$z_{7} = 97.3996388790738$$
$$z_{8} = -75.4114834888481$$
$$z_{9} = 59.7070073053355$$
$$z_{10} = -44.0050179208308$$
$$z_{11} = 15.7712848748159$$
$$z_{12} = -50.2853663377737$$
$$z_{13} = 34.5864242152889$$
$$z_{14} = -37.7256128277765$$
$$z_{15} = 28.309642854452$$
$$z_{16} = -69.1295029738953$$
$$z_{17} = 78.5525459842429$$
$$z_{18} = -31.4477146375462$$
$$z_{19} = 3.42561845948173$$
$$z_{20} = 72.270467060309$$
$$z_{21} = -18.90240995686$$
$$z_{22} = -0.86033358901938$$
$$z_{23} = -25.1724463266467$$
$$z_{24} = -62.8477631944545$$
$$z_{25} = -56.5663442798215$$
$$z_{26} = -94.2583883450399$$
$$z_{27} = 91.1171613944647$$
$$z_{28} = -87.9759605524932$$
$$z_{29} = -100.540910786842$$
$$z_{30} = -12.6452872238566$$
$$z_{31} = 53.4257904773947$$
$$z_{32} = 65.9885986984904$$
$$z_{33} = -6.43729817917195$$
Puntos máximos de la función:
$$z_{33} = 62.8477631944545$$
$$z_{33} = 37.7256128277765$$
$$z_{33} = -22.0364967279386$$
$$z_{33} = -91.1171613944647$$
$$z_{33} = -9.52933440536196$$
$$z_{33} = -53.4257904773947$$
$$z_{33} = -78.5525459842429$$
$$z_{33} = 6.43729817917195$$
$$z_{33} = 87.9759605524932$$
$$z_{33} = 56.5663442798215$$
$$z_{33} = 100.540910786842$$
$$z_{33} = 0.86033358901938$$
$$z_{33} = -28.309642854452$$
$$z_{33} = -97.3996388790738$$
$$z_{33} = 44.0050179208308$$
$$z_{33} = 94.2583883450399$$
$$z_{33} = 81.6936492356017$$
$$z_{33} = -34.5864242152889$$
$$z_{33} = 31.4477146375462$$
$$z_{33} = 50.2853663377737$$
$$z_{33} = -116.247530303932$$
$$z_{33} = -65.9885986984904$$
$$z_{33} = -40.8651703304881$$
$$z_{33} = -59.7070073053355$$
$$z_{33} = 18.90240995686$$
$$z_{33} = -147.661626855354$$
$$z_{33} = -47.145097736761$$
$$z_{33} = 75.4114834888481$$
$$z_{33} = 12.6452872238566$$
$$z_{33} = 69.1295029738953$$
$$z_{33} = -3.42561845948173$$
$$z_{33} = -15.7712848748159$$
$$z_{33} = 25.1724463266467$$
$$z_{33} = -84.8347887180423$$
$$z_{33} = -72.270467060309$$
Decrece en los intervalos
$$\left[97.3996388790738, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.540910786842\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con z->+oo y z->-oo
$$\lim_{z \to -\infty}\left(z \cos{\left(z \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{z \to \infty}\left(z \cos{\left(z \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{z \to -\infty}\left(z \cos{\left(z \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{z \to \infty}\left(z \cos{\left(z \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función z*cos(z), dividida por z con z->+oo y z ->-oo
$$\lim_{z \to -\infty} \cos{\left(z \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -1, 1\right\rangle z$$
$$\lim_{z \to \infty} \cos{\left(z \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -1, 1\right\rangle z$$
$$\lim_{z \to -\infty} \cos{\left(z \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -1, 1\right\rangle z$$
$$\lim_{z \to \infty} \cos{\left(z \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -1, 1\right\rangle z$$
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-z) и f = -f(-z).
Pues, comprobamos:
$$z \cos{\left(z \right)} = - z \cos{\left(z \right)}$$
- No
$$z \cos{\left(z \right)} = z \cos{\left(z \right)}$$
- Sí
es decir, función
es
impar
Pues, comprobamos:
$$z \cos{\left(z \right)} = - z \cos{\left(z \right)}$$
- No
$$z \cos{\left(z \right)} = z \cos{\left(z \right)}$$
- Sí
es decir, función
es
impar