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- Gráfico de la función implícita
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- 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=3x^3-x
-
y=x²-2x-8
-
-x^4+x^2
-
x*e^(-x^1)
-
Expresiones idénticas
- sin(dos *z)/(z+ uno)
- seno de (2 multiplicar por z) dividir por (z más 1)
- seno de (dos multiplicar por z) dividir por (z más uno)
- sin(2z)/(z+1)
- sin2z/z+1
- sin(2*z) dividir por (z+1)
-
Expresiones semejantes
- sin(2*z)/(z-1)
-
Expresiones con funciones
- Seno sin
- sin(5+6*x)^(2)+3^tan(x)
- sin(x)+arcsin(x)
- sin(2*x^3)^2/(x^3)
- sin(x)*sin(1.5*pi+x)-6*x
- sin(x)+sin(10/3*x)+log(x)-(0,84*x+3)
Gráfico de la función y = sin(2*z)/(z+1)
Solución
Ha introducido
[src]
sin(2*z)
f(z) = --------
z + 1
$$f{\left(z \right)} = \frac{\sin{\left(2 z \right)}}{z + 1}$$
f = sin(2*z)/(z + 1)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$z_{1} = -1$$
$$z_{1} = -1$$
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:
$$\frac{\sin{\left(2 z \right)}}{z + 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = \frac{\pi}{2}$$
Solución numérica
$$z_{1} = -86.3937979737193$$
$$z_{2} = -298.45130209103$$
$$z_{3} = 29.845130209103$$
$$z_{4} = 14.1371669411541$$
$$z_{5} = 48.6946861306418$$
$$z_{6} = -72.2566310325652$$
$$z_{7} = -80.1106126665397$$
$$z_{8} = -7.85398163397448$$
$$z_{9} = -59.6902604182061$$
$$z_{10} = -370.707933123596$$
$$z_{11} = -6.28318530717959$$
$$z_{12} = 12.5663706143592$$
$$z_{13} = 67.5442420521806$$
$$z_{14} = 56.5486677646163$$
$$z_{15} = -58.1194640914112$$
$$z_{16} = 95.8185759344887$$
$$z_{17} = 72.2566310325652$$
$$z_{18} = 37.6991118430775$$
$$z_{19} = 80.1106126665397$$
$$z_{20} = 100.530964914873$$
$$z_{21} = 94.2477796076938$$
$$z_{22} = -14.1371669411541$$
$$z_{23} = 7.85398163397448$$
$$z_{24} = -29.845130209103$$
$$z_{25} = 70.6858347057703$$
$$z_{26} = -20.4203522483337$$
$$z_{27} = -87.9645943005142$$
$$z_{28} = -31.4159265358979$$
$$z_{29} = 59.6902604182061$$
$$z_{30} = 42.4115008234622$$
$$z_{31} = 0$$
$$z_{32} = -17.2787595947439$$
$$z_{33} = -67.5442420521806$$
$$z_{34} = 50.2654824574367$$
$$z_{35} = -53.4070751110265$$
$$z_{36} = 45.553093477052$$
$$z_{37} = -45.553093477052$$
$$z_{38} = -21.9911485751286$$
$$z_{39} = -23.5619449019235$$
$$z_{40} = 58.1194640914112$$
$$z_{41} = -109.955742875643$$
$$z_{42} = -36.1283155162826$$
$$z_{43} = 87.9645943005142$$
$$z_{44} = -51.8362787842316$$
$$z_{45} = -73.8274273593601$$
$$z_{46} = 4.71238898038469$$
$$z_{47} = 64.4026493985908$$
$$z_{48} = -42.4115008234622$$
$$z_{49} = -95.8185759344887$$
$$z_{50} = -75.398223686155$$
$$z_{51} = 81.6814089933346$$
$$z_{52} = -65.9734457253857$$
$$z_{53} = -37.6991118430775$$
$$z_{54} = 1.5707963267949$$
$$z_{55} = -43.9822971502571$$
$$z_{56} = 20.4203522483337$$
$$z_{57} = 23.5619449019235$$
$$z_{58} = -10.9955742875643$$
$$z_{59} = -1.5707963267949$$
$$z_{60} = 92.6769832808989$$
$$z_{61} = 6.28318530717959$$
$$z_{62} = 28.2743338823081$$
$$z_{63} = -83.2522053201295$$
$$z_{64} = -94.2477796076938$$
$$z_{65} = 86.3937979737193$$
$$z_{66} = 43.9822971502571$$
$$z_{67} = -9.42477796076938$$
$$z_{68} = 65.9734457253857$$
$$z_{69} = 89.5353906273091$$
$$z_{70} = -15.707963267949$$
$$z_{71} = -64.4026493985908$$
$$z_{72} = -50.2654824574367$$
$$z_{73} = 36.1283155162826$$
$$z_{74} = 15.707963267949$$
$$z_{75} = 51.8362787842316$$
$$z_{76} = 26.7035375555132$$
$$z_{77} = 73.8274273593601$$
$$z_{78} = -39.2699081698724$$
$$z_{79} = 21.9911485751286$$
$$z_{80} = 34.5575191894877$$
$$z_{81} = -97.3893722612836$$
$$z_{82} = -28.2743338823081$$
$$z_{83} = 78.5398163397448$$
$$z_{84} = -89.5353906273091$$
$$z_{85} = -854.513201776528$$
$$z_{86} = -81.6814089933346$$
$$z_{87} = -61.261056745001$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(2 z \right)}}{z + 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = \frac{\pi}{2}$$
Solución numérica
$$z_{1} = -86.3937979737193$$
$$z_{2} = -298.45130209103$$
$$z_{3} = 29.845130209103$$
$$z_{4} = 14.1371669411541$$
$$z_{5} = 48.6946861306418$$
$$z_{6} = -72.2566310325652$$
$$z_{7} = -80.1106126665397$$
$$z_{8} = -7.85398163397448$$
$$z_{9} = -59.6902604182061$$
$$z_{10} = -370.707933123596$$
$$z_{11} = -6.28318530717959$$
$$z_{12} = 12.5663706143592$$
$$z_{13} = 67.5442420521806$$
$$z_{14} = 56.5486677646163$$
$$z_{15} = -58.1194640914112$$
$$z_{16} = 95.8185759344887$$
$$z_{17} = 72.2566310325652$$
$$z_{18} = 37.6991118430775$$
$$z_{19} = 80.1106126665397$$
$$z_{20} = 100.530964914873$$
$$z_{21} = 94.2477796076938$$
$$z_{22} = -14.1371669411541$$
$$z_{23} = 7.85398163397448$$
$$z_{24} = -29.845130209103$$
$$z_{25} = 70.6858347057703$$
$$z_{26} = -20.4203522483337$$
$$z_{27} = -87.9645943005142$$
$$z_{28} = -31.4159265358979$$
$$z_{29} = 59.6902604182061$$
$$z_{30} = 42.4115008234622$$
$$z_{31} = 0$$
$$z_{32} = -17.2787595947439$$
$$z_{33} = -67.5442420521806$$
$$z_{34} = 50.2654824574367$$
$$z_{35} = -53.4070751110265$$
$$z_{36} = 45.553093477052$$
$$z_{37} = -45.553093477052$$
$$z_{38} = -21.9911485751286$$
$$z_{39} = -23.5619449019235$$
$$z_{40} = 58.1194640914112$$
$$z_{41} = -109.955742875643$$
$$z_{42} = -36.1283155162826$$
$$z_{43} = 87.9645943005142$$
$$z_{44} = -51.8362787842316$$
$$z_{45} = -73.8274273593601$$
$$z_{46} = 4.71238898038469$$
$$z_{47} = 64.4026493985908$$
$$z_{48} = -42.4115008234622$$
$$z_{49} = -95.8185759344887$$
$$z_{50} = -75.398223686155$$
$$z_{51} = 81.6814089933346$$
$$z_{52} = -65.9734457253857$$
$$z_{53} = -37.6991118430775$$
$$z_{54} = 1.5707963267949$$
$$z_{55} = -43.9822971502571$$
$$z_{56} = 20.4203522483337$$
$$z_{57} = 23.5619449019235$$
$$z_{58} = -10.9955742875643$$
$$z_{59} = -1.5707963267949$$
$$z_{60} = 92.6769832808989$$
$$z_{61} = 6.28318530717959$$
$$z_{62} = 28.2743338823081$$
$$z_{63} = -83.2522053201295$$
$$z_{64} = -94.2477796076938$$
$$z_{65} = 86.3937979737193$$
$$z_{66} = 43.9822971502571$$
$$z_{67} = -9.42477796076938$$
$$z_{68} = 65.9734457253857$$
$$z_{69} = 89.5353906273091$$
$$z_{70} = -15.707963267949$$
$$z_{71} = -64.4026493985908$$
$$z_{72} = -50.2654824574367$$
$$z_{73} = 36.1283155162826$$
$$z_{74} = 15.707963267949$$
$$z_{75} = 51.8362787842316$$
$$z_{76} = 26.7035375555132$$
$$z_{77} = 73.8274273593601$$
$$z_{78} = -39.2699081698724$$
$$z_{79} = 21.9911485751286$$
$$z_{80} = 34.5575191894877$$
$$z_{81} = -97.3893722612836$$
$$z_{82} = -28.2743338823081$$
$$z_{83} = 78.5398163397448$$
$$z_{84} = -89.5353906273091$$
$$z_{85} = -854.513201776528$$
$$z_{86} = -81.6814089933346$$
$$z_{87} = -61.261056745001$$
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
$$\frac{2 \cos{\left(2 z \right)}}{z + 1} - \frac{\sin{\left(2 z \right)}}{\left(z + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -80.8928816808707$$
$$z_{2} = -99.7430349489701$$
$$z_{3} = -63.6132585554971$$
$$z_{4} = -47.9039581285518$$
$$z_{5} = -98.172197695036$$
$$z_{6} = 84.0346635398793$$
$$z_{7} = 46.3332101330021$$
$$z_{8} = 38.478177732588$$
$$z_{9} = 19.6228339741551$$
$$z_{10} = 82.4638118824473$$
$$z_{11} = 16.4790625040945$$
$$z_{12} = 41.6202371710741$$
$$z_{13} = -46.33297711484$$
$$z_{14} = 66.7551541995631$$
$$z_{15} = -57.3296278828154$$
$$z_{16} = 62.0424894121024$$
$$z_{17} = 33.7649303669424$$
$$z_{18} = 8.61339783522102$$
$$z_{19} = -5.44173882723211$$
$$z_{20} = 18.0510381254578$$
$$z_{21} = -33.76449140679$$
$$z_{22} = -62.042359483332$$
$$z_{23} = 98.1722495795572$$
$$z_{24} = 32.1937937404494$$
$$z_{25} = 55.7588651168628$$
$$z_{26} = -25.9081038458568$$
$$z_{27} = 24.3374775388136$$
$$z_{28} = -82.4637383451864$$
$$z_{29} = 2.28057021563236$$
$$z_{30} = -60.471454983256$$
$$z_{31} = 85.6055131901373$$
$$z_{32} = 74.6095191089778$$
$$z_{33} = 40.049216384194$$
$$z_{34} = -27.4794955733025$$
$$z_{35} = -11.7577501031099$$
$$z_{36} = 187.70883626286$$
$$z_{37} = 25.9088498373569$$
$$z_{38} = 27.4801585795776$$
$$z_{39} = -13.3315066037056$$
$$z_{40} = 63.6133821448946$$
$$z_{41} = 99.743085211956$$
$$z_{42} = 44.762232510841$$
$$z_{43} = -19.6215319912886$$
$$z_{44} = -3.83985112537054$$
$$z_{45} = 60.4715917520353$$
$$z_{46} = -16.4772142695041$$
$$z_{47} = 11.7613921271159$$
$$z_{48} = -49.4749271716005$$
$$z_{49} = 0.637196330969125$$
$$z_{50} = 30.6226232987428$$
$$z_{51} = -38.4778397936073$$
$$z_{52} = 68.3260341292281$$
$$z_{53} = -79.322022596248$$
$$z_{54} = -90.317989831739$$
$$z_{55} = -41.6199483594113$$
$$z_{56} = -84.0345927265613$$
$$z_{57} = -85.6054449521595$$
$$z_{58} = 88.7472068907202$$
$$z_{59} = 76.1803827297937$$
$$z_{60} = -69.8968079909424$$
$$z_{61} = 22.7660290738215$$
$$z_{62} = 96.6014126819581$$
$$z_{63} = 54.1879434234347$$
$$z_{64} = -93.4596775892801$$
$$z_{65} = -54.1877730844831$$
$$z_{66} = 3.87589679173726$$
$$z_{67} = -68.3259270041136$$
$$z_{68} = 91.8888937560759$$
$$z_{69} = 69.8969103540797$$
$$z_{70} = 10.1878453044909$$
$$z_{71} = -10.1829786980484$$
$$z_{72} = -40.0489044548563$$
$$z_{73} = -77.7511609427056$$
$$z_{74} = -71.4676852032561$$
$$z_{75} = -18.0494987719381$$
$$z_{76} = 52.6170143837707$$
$$z_{77} = 90.3180511337085$$
$$z_{78} = -24.3366319328506$$
$$z_{79} = -76.1802965592094$$
$$z_{80} = 47.9041761074919$$
$$z_{81} = -55.7587042438767$$
$$z_{82} = 77.7512436659628$$
$$z_{83} = -35.3356368025816$$
$$z_{84} = -44.7619828411789$$
$$z_{85} = -91.8888345323567$$
$$z_{86} = -2.15134433588925$$
$$z_{87} = -32.1933108467876$$
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:
$$z_{1} = -80.8928816808707$$
$$z_{2} = -99.7430349489701$$
$$z_{3} = 84.0346635398793$$
$$z_{4} = 46.3332101330021$$
$$z_{5} = -46.33297711484$$
$$z_{6} = 62.0424894121024$$
$$z_{7} = 33.7649303669424$$
$$z_{8} = 8.61339783522102$$
$$z_{9} = -5.44173882723211$$
$$z_{10} = 18.0510381254578$$
$$z_{11} = -33.76449140679$$
$$z_{12} = -62.042359483332$$
$$z_{13} = 55.7588651168628$$
$$z_{14} = 24.3374775388136$$
$$z_{15} = 2.28057021563236$$
$$z_{16} = 74.6095191089778$$
$$z_{17} = 40.049216384194$$
$$z_{18} = -27.4794955733025$$
$$z_{19} = -11.7577501031099$$
$$z_{20} = 187.70883626286$$
$$z_{21} = 27.4801585795776$$
$$z_{22} = 99.743085211956$$
$$z_{23} = 11.7613921271159$$
$$z_{24} = -49.4749271716005$$
$$z_{25} = 30.6226232987428$$
$$z_{26} = 68.3260341292281$$
$$z_{27} = -90.317989831739$$
$$z_{28} = -84.0345927265613$$
$$z_{29} = 96.6014126819581$$
$$z_{30} = -93.4596775892801$$
$$z_{31} = -68.3259270041136$$
$$z_{32} = -40.0489044548563$$
$$z_{33} = -77.7511609427056$$
$$z_{34} = -71.4676852032561$$
$$z_{35} = -18.0494987719381$$
$$z_{36} = 52.6170143837707$$
$$z_{37} = 90.3180511337085$$
$$z_{38} = -24.3366319328506$$
$$z_{39} = -55.7587042438767$$
$$z_{40} = 77.7512436659628$$
$$z_{41} = -2.15134433588925$$
Puntos máximos de la función:
$$z_{41} = -63.6132585554971$$
$$z_{41} = -47.9039581285518$$
$$z_{41} = -98.172197695036$$
$$z_{41} = 38.478177732588$$
$$z_{41} = 19.6228339741551$$
$$z_{41} = 82.4638118824473$$
$$z_{41} = 16.4790625040945$$
$$z_{41} = 41.6202371710741$$
$$z_{41} = 66.7551541995631$$
$$z_{41} = -57.3296278828154$$
$$z_{41} = 98.1722495795572$$
$$z_{41} = 32.1937937404494$$
$$z_{41} = -25.9081038458568$$
$$z_{41} = -82.4637383451864$$
$$z_{41} = -60.471454983256$$
$$z_{41} = 85.6055131901373$$
$$z_{41} = 25.9088498373569$$
$$z_{41} = -13.3315066037056$$
$$z_{41} = 63.6133821448946$$
$$z_{41} = 44.762232510841$$
$$z_{41} = -19.6215319912886$$
$$z_{41} = -3.83985112537054$$
$$z_{41} = 60.4715917520353$$
$$z_{41} = -16.4772142695041$$
$$z_{41} = 0.637196330969125$$
$$z_{41} = -38.4778397936073$$
$$z_{41} = -79.322022596248$$
$$z_{41} = -41.6199483594113$$
$$z_{41} = -85.6054449521595$$
$$z_{41} = 88.7472068907202$$
$$z_{41} = 76.1803827297937$$
$$z_{41} = -69.8968079909424$$
$$z_{41} = 22.7660290738215$$
$$z_{41} = 54.1879434234347$$
$$z_{41} = -54.1877730844831$$
$$z_{41} = 3.87589679173726$$
$$z_{41} = 91.8888937560759$$
$$z_{41} = 69.8969103540797$$
$$z_{41} = 10.1878453044909$$
$$z_{41} = -10.1829786980484$$
$$z_{41} = -76.1802965592094$$
$$z_{41} = 47.9041761074919$$
$$z_{41} = -35.3356368025816$$
$$z_{41} = -44.7619828411789$$
$$z_{41} = -91.8888345323567$$
$$z_{41} = -32.1933108467876$$
Decrece en los intervalos
$$\left[187.70883626286, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7430349489701\right]$$
$$\frac{d}{d z} f{\left(z \right)} = 0$$
(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
$$\frac{2 \cos{\left(2 z \right)}}{z + 1} - \frac{\sin{\left(2 z \right)}}{\left(z + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -80.8928816808707$$
$$z_{2} = -99.7430349489701$$
$$z_{3} = -63.6132585554971$$
$$z_{4} = -47.9039581285518$$
$$z_{5} = -98.172197695036$$
$$z_{6} = 84.0346635398793$$
$$z_{7} = 46.3332101330021$$
$$z_{8} = 38.478177732588$$
$$z_{9} = 19.6228339741551$$
$$z_{10} = 82.4638118824473$$
$$z_{11} = 16.4790625040945$$
$$z_{12} = 41.6202371710741$$
$$z_{13} = -46.33297711484$$
$$z_{14} = 66.7551541995631$$
$$z_{15} = -57.3296278828154$$
$$z_{16} = 62.0424894121024$$
$$z_{17} = 33.7649303669424$$
$$z_{18} = 8.61339783522102$$
$$z_{19} = -5.44173882723211$$
$$z_{20} = 18.0510381254578$$
$$z_{21} = -33.76449140679$$
$$z_{22} = -62.042359483332$$
$$z_{23} = 98.1722495795572$$
$$z_{24} = 32.1937937404494$$
$$z_{25} = 55.7588651168628$$
$$z_{26} = -25.9081038458568$$
$$z_{27} = 24.3374775388136$$
$$z_{28} = -82.4637383451864$$
$$z_{29} = 2.28057021563236$$
$$z_{30} = -60.471454983256$$
$$z_{31} = 85.6055131901373$$
$$z_{32} = 74.6095191089778$$
$$z_{33} = 40.049216384194$$
$$z_{34} = -27.4794955733025$$
$$z_{35} = -11.7577501031099$$
$$z_{36} = 187.70883626286$$
$$z_{37} = 25.9088498373569$$
$$z_{38} = 27.4801585795776$$
$$z_{39} = -13.3315066037056$$
$$z_{40} = 63.6133821448946$$
$$z_{41} = 99.743085211956$$
$$z_{42} = 44.762232510841$$
$$z_{43} = -19.6215319912886$$
$$z_{44} = -3.83985112537054$$
$$z_{45} = 60.4715917520353$$
$$z_{46} = -16.4772142695041$$
$$z_{47} = 11.7613921271159$$
$$z_{48} = -49.4749271716005$$
$$z_{49} = 0.637196330969125$$
$$z_{50} = 30.6226232987428$$
$$z_{51} = -38.4778397936073$$
$$z_{52} = 68.3260341292281$$
$$z_{53} = -79.322022596248$$
$$z_{54} = -90.317989831739$$
$$z_{55} = -41.6199483594113$$
$$z_{56} = -84.0345927265613$$
$$z_{57} = -85.6054449521595$$
$$z_{58} = 88.7472068907202$$
$$z_{59} = 76.1803827297937$$
$$z_{60} = -69.8968079909424$$
$$z_{61} = 22.7660290738215$$
$$z_{62} = 96.6014126819581$$
$$z_{63} = 54.1879434234347$$
$$z_{64} = -93.4596775892801$$
$$z_{65} = -54.1877730844831$$
$$z_{66} = 3.87589679173726$$
$$z_{67} = -68.3259270041136$$
$$z_{68} = 91.8888937560759$$
$$z_{69} = 69.8969103540797$$
$$z_{70} = 10.1878453044909$$
$$z_{71} = -10.1829786980484$$
$$z_{72} = -40.0489044548563$$
$$z_{73} = -77.7511609427056$$
$$z_{74} = -71.4676852032561$$
$$z_{75} = -18.0494987719381$$
$$z_{76} = 52.6170143837707$$
$$z_{77} = 90.3180511337085$$
$$z_{78} = -24.3366319328506$$
$$z_{79} = -76.1802965592094$$
$$z_{80} = 47.9041761074919$$
$$z_{81} = -55.7587042438767$$
$$z_{82} = 77.7512436659628$$
$$z_{83} = -35.3356368025816$$
$$z_{84} = -44.7619828411789$$
$$z_{85} = -91.8888345323567$$
$$z_{86} = -2.15134433588925$$
$$z_{87} = -32.1933108467876$$
Signos de extremos en los puntos:
(-80.89288168087066, -0.0125165145614667)
(-99.74303494897006, -0.0101271667464578)
(-63.61325855549712, 0.0159705490539322)
(-47.90395812855178, 0.0213189510240566)
(-98.172197695036, 0.0102908731702446)
(84.03466353987932, -0.011759706828775)
(46.33321013300211, -0.0211256369273738)
(38.47817773258804, 0.0253284184609493)
(19.622833974155125, 0.0484756955066268)
(82.46381188244735, 0.0119810254787546)
(16.479062504094543, 0.0571879131957477)
(41.620237171074066, 0.02346141784367)
(-46.33297711483998, -0.0220576552494669)
(66.75515419956312, 0.014758622931699)
(-57.32962788281537, 0.0177519476934681)
(62.04248941210236, -0.0158618188965606)
(33.76493036694237, -0.0287616451426903)
(8.61339783522102, -0.103881083065715)
(-5.441738827232107, -0.223724038559713)
(18.05103812545779, -0.0524725090383133)
(-33.76449140678997, -0.0305172928724435)
(-62.04235948333203, -0.0163815170913838)
(98.17224957955716, 0.0100833377779178)
(32.19379374044943, 0.0301226964076625)
(55.758865116862815, -0.0176177095723664)
(-25.908103845856804, 0.040139489885367)
(24.337477538813616, -0.0394595455509357)
(-82.46373834518636, 0.0122751691140578)
(2.2805702156323617, -0.301345089868793)
(-60.471454983255995, 0.0168141953145683)
(85.60551319013732, 0.0115464165970439)
(74.60951910897779, -0.0132255587275224)
(40.04921638419398, -0.0243591940123493)
(-27.479495573302465, -0.0377583390827228)
(-11.757750103109899, -0.0928559995735437)
(187.70883626286036, -0.00529915031894235)
(25.908849837356932, 0.0371560813395048)
(27.480158579577623, -0.0351067542272663)
(-13.331506603705582, 0.0810265146895922)
(63.61338214489455, 0.0154762067428617)
(99.74308521195597, -0.0099261173294882)
(44.762232510840974, 0.0218507765312293)
(-19.621531991288602, 0.0536819267559723)
(-3.8398511253705365, 0.346796963988786)
(60.47159175203532, 0.0162671389088241)
(-16.47721426950408, 0.0645774211200931)
(11.761392127115943, -0.0783012785650106)
(-49.4749271716005, -0.0206281240012608)
(0.637196330969125, 0.584165185611747)
(30.622623298742763, -0.0316189777418261)
(-38.47783979360729, 0.0266800600927598)
(68.32603412922812, -0.0144242203517829)
(-79.32202259624798, 0.0127675408573764)
(-90.317989831739, -0.0111957773969401)
(-41.619948359411254, 0.0246165810275241)
(-84.03459272656133, -0.0120429550853916)
(-85.6054449521595, 0.0118193638511416)
(88.74720689072021, 0.0111422351274035)
(76.18038272979369, 0.0129563884100013)
(-69.89680799094245, 0.0145140783224103)
(22.766029073821525, 0.0420675562800483)
(96.60141268195807, -0.0102456188988178)
(54.18794342343466, 0.0181191560900984)
(-93.45967758928009, -0.0108153673519588)
(-54.18777308448308, 0.0188004828697393)
(3.8758967917372615, 0.204020590583108)
(-68.3259270041136, -0.014852709331234)
(91.88889375607587, 0.0107653937165865)
(69.89691035407965, 0.0141046362542619)
(10.187845304490947, 0.0892935867062234)
(-10.182978698048352, 0.108736064581312)
(-40.048904454856334, -0.0256068139934971)
(-77.75116094270562, -0.0130288424129444)
(-71.46768520325605, -0.014190544578759)
(-18.049498771938094, -0.0586275451496126)
(52.61701438377071, -0.0186499851363279)
(90.31805113370845, -0.0109505732775833)
(-24.336631932850587, -0.0428412529680419)
(-76.18029655920942, 0.0133010633202575)
(47.90417610749193, 0.0204470828020304)
(-55.7587042438767, -0.0182611756271808)
(77.75124366596276, -0.0126979562288175)
(-35.33563680258161, 0.0291211721162102)
(-44.76198284117895, 0.0228493928087149)
(-91.8888345323567, 0.0110022850853576)
(-2.1513443358892483, -0.796668913740646)
(-32.19331084678758, 0.0320540376107124)
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:
$$z_{1} = -80.8928816808707$$
$$z_{2} = -99.7430349489701$$
$$z_{3} = 84.0346635398793$$
$$z_{4} = 46.3332101330021$$
$$z_{5} = -46.33297711484$$
$$z_{6} = 62.0424894121024$$
$$z_{7} = 33.7649303669424$$
$$z_{8} = 8.61339783522102$$
$$z_{9} = -5.44173882723211$$
$$z_{10} = 18.0510381254578$$
$$z_{11} = -33.76449140679$$
$$z_{12} = -62.042359483332$$
$$z_{13} = 55.7588651168628$$
$$z_{14} = 24.3374775388136$$
$$z_{15} = 2.28057021563236$$
$$z_{16} = 74.6095191089778$$
$$z_{17} = 40.049216384194$$
$$z_{18} = -27.4794955733025$$
$$z_{19} = -11.7577501031099$$
$$z_{20} = 187.70883626286$$
$$z_{21} = 27.4801585795776$$
$$z_{22} = 99.743085211956$$
$$z_{23} = 11.7613921271159$$
$$z_{24} = -49.4749271716005$$
$$z_{25} = 30.6226232987428$$
$$z_{26} = 68.3260341292281$$
$$z_{27} = -90.317989831739$$
$$z_{28} = -84.0345927265613$$
$$z_{29} = 96.6014126819581$$
$$z_{30} = -93.4596775892801$$
$$z_{31} = -68.3259270041136$$
$$z_{32} = -40.0489044548563$$
$$z_{33} = -77.7511609427056$$
$$z_{34} = -71.4676852032561$$
$$z_{35} = -18.0494987719381$$
$$z_{36} = 52.6170143837707$$
$$z_{37} = 90.3180511337085$$
$$z_{38} = -24.3366319328506$$
$$z_{39} = -55.7587042438767$$
$$z_{40} = 77.7512436659628$$
$$z_{41} = -2.15134433588925$$
Puntos máximos de la función:
$$z_{41} = -63.6132585554971$$
$$z_{41} = -47.9039581285518$$
$$z_{41} = -98.172197695036$$
$$z_{41} = 38.478177732588$$
$$z_{41} = 19.6228339741551$$
$$z_{41} = 82.4638118824473$$
$$z_{41} = 16.4790625040945$$
$$z_{41} = 41.6202371710741$$
$$z_{41} = 66.7551541995631$$
$$z_{41} = -57.3296278828154$$
$$z_{41} = 98.1722495795572$$
$$z_{41} = 32.1937937404494$$
$$z_{41} = -25.9081038458568$$
$$z_{41} = -82.4637383451864$$
$$z_{41} = -60.471454983256$$
$$z_{41} = 85.6055131901373$$
$$z_{41} = 25.9088498373569$$
$$z_{41} = -13.3315066037056$$
$$z_{41} = 63.6133821448946$$
$$z_{41} = 44.762232510841$$
$$z_{41} = -19.6215319912886$$
$$z_{41} = -3.83985112537054$$
$$z_{41} = 60.4715917520353$$
$$z_{41} = -16.4772142695041$$
$$z_{41} = 0.637196330969125$$
$$z_{41} = -38.4778397936073$$
$$z_{41} = -79.322022596248$$
$$z_{41} = -41.6199483594113$$
$$z_{41} = -85.6054449521595$$
$$z_{41} = 88.7472068907202$$
$$z_{41} = 76.1803827297937$$
$$z_{41} = -69.8968079909424$$
$$z_{41} = 22.7660290738215$$
$$z_{41} = 54.1879434234347$$
$$z_{41} = -54.1877730844831$$
$$z_{41} = 3.87589679173726$$
$$z_{41} = 91.8888937560759$$
$$z_{41} = 69.8969103540797$$
$$z_{41} = 10.1878453044909$$
$$z_{41} = -10.1829786980484$$
$$z_{41} = -76.1802965592094$$
$$z_{41} = 47.9041761074919$$
$$z_{41} = -35.3356368025816$$
$$z_{41} = -44.7619828411789$$
$$z_{41} = -91.8888345323567$$
$$z_{41} = -32.1933108467876$$
Decrece en los intervalos
$$\left[187.70883626286, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7430349489701\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = 0$$
(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:
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = $$
segunda derivada
$$\frac{2 \left(- 2 \sin{\left(2 z \right)} - \frac{2 \cos{\left(2 z \right)}}{z + 1} + \frac{\sin{\left(2 z \right)}}{\left(z + 1\right)^{2}}\right)}{z + 1} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = 0$$
(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:
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = $$
segunda derivada
$$\frac{2 \left(- 2 \sin{\left(2 z \right)} - \frac{2 \cos{\left(2 z \right)}}{z + 1} + \frac{\sin{\left(2 z \right)}}{\left(z + 1\right)^{2}}\right)}{z + 1} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
$$z_{1} = -1$$
$$z_{1} = -1$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con z->+oo y z->-oo
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(2 z \right)}}{z + 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{z \to \infty}\left(\frac{\sin{\left(2 z \right)}}{z + 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(2 z \right)}}{z + 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{z \to \infty}\left(\frac{\sin{\left(2 z \right)}}{z + 1}\right) = 0$$
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 sin(2*z)/(z + 1), dividida por z con z->+oo y z ->-oo
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(2 z \right)}}{z \left(z + 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{z \to \infty}\left(\frac{\sin{\left(2 z \right)}}{z \left(z + 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(2 z \right)}}{z \left(z + 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{z \to \infty}\left(\frac{\sin{\left(2 z \right)}}{z \left(z + 1\right)}\right) = 0$$
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(-z) и f = -f(-z).
Pues, comprobamos:
$$\frac{\sin{\left(2 z \right)}}{z + 1} = - \frac{\sin{\left(2 z \right)}}{1 - z}$$
- No
$$\frac{\sin{\left(2 z \right)}}{z + 1} = \frac{\sin{\left(2 z \right)}}{1 - z}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(2 z \right)}}{z + 1} = - \frac{\sin{\left(2 z \right)}}{1 - z}$$
- No
$$\frac{\sin{\left(2 z \right)}}{z + 1} = \frac{\sin{\left(2 z \right)}}{1 - z}$$
- No
es decir, función
no es
par ni impar