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- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
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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+27/x^3
-
(x^2-8)/(x-3)
-
x-2+4/(x-2)
-
x^2-9*x+14
-
Expresiones idénticas
- e- dos *x*(x^ dos *cos(tres *x)+(uno -x)*sin(tres *x))
- e menos 2 multiplicar por x multiplicar por (x al cuadrado multiplicar por coseno de (3 multiplicar por x) más (1 menos x) multiplicar por seno de (3 multiplicar por x))
- e menos dos multiplicar por x multiplicar por (x en el grado dos multiplicar por coseno de (tres multiplicar por x) más (uno menos x) multiplicar por seno de (tres multiplicar por x))
- e-2*x*(x2*cos(3*x)+(1-x)*sin(3*x))
- e-2*x*x2*cos3*x+1-x*sin3*x
- e-2*x*(x²*cos(3*x)+(1-x)*sin(3*x))
- e-2*x*(x en el grado 2*cos(3*x)+(1-x)*sin(3*x))
- e-2x(x^2cos(3x)+(1-x)sin(3x))
- e-2x(x2cos(3x)+(1-x)sin(3x))
- e-2xx2cos3x+1-xsin3x
- e-2xx^2cos3x+1-xsin3x
-
Expresiones semejantes
- e-2*x*(x^2*cos(3*x)+(1+x)*sin(3*x))
- e-2*x*(x^2*cos(3*x)-(1-x)*sin(3*x))
- e+2*x*(x^2*cos(3*x)+(1-x)*sin(3*x))
-
Expresiones con funciones
- Coseno cos
- cos(x-pi)-2
- cos(x)/(2*sqrt(x))-sin(x)*sqrt(x)
- cos(x)/18+cos(x*sqrt(19))+sin(x*sqrt(19))
- cos(x^2)+2
- cos(x)*x^3+9
- Seno sin
- sin(x-pi/4)*(-2)
- sin(pi*x)/asin(x)
- sin(x)*e^((1/2)*cot(x)^(2))
- sin(697*t)+sin(1209*t)
- sin(3x-29)
Gráfico de la función y = e-2*x*(x^2*cos(3*x)+(1-x)*sin(3*x))
Solución
Ha introducido
[src]
/ 2 \ f(x) = E - 2*x*\x *cos(3*x) + (1 - x)*sin(3*x)/
$$f{\left(x \right)} = - 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e$$
f = -2*x*(x^2*cos(3*x) + (1 - x)*sin(3*x)) + E
Gráfico de la función
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:
$$- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -7.80519181113787$$
$$x_{2} = 27.7391350406061$$
$$x_{3} = 22.5006395913317$$
$$x_{4} = 84.295494861077$$
$$x_{5} = 18.3088343819918$$
$$x_{6} = -36.1188433803924$$
$$x_{7} = 14.1154174315877$$
$$x_{8} = 68.5866521645177$$
$$x_{9} = 82.201001377025$$
$$x_{10} = 75.9174887741924$$
$$x_{11} = 2.51176326196438$$
$$x_{12} = -91.6261076121408$$
$$x_{13} = 71.7284489496945$$
$$x_{14} = -38.2137697610044$$
$$x_{15} = -25.6428682414677$$
$$x_{16} = 34.0244025417973$$
$$x_{17} = -32.9762988205827$$
$$x_{18} = 100.004066832881$$
$$x_{19} = -3.53971692338044$$
$$x_{20} = 39.2616435443518$$
$$x_{21} = -71.7283218450639$$
$$x_{22} = -58.113627788602$$
$$x_{23} = -21.4513451015863$$
$$x_{24} = -0.809176790168203$$
$$x_{25} = 49.7353208244535$$
$$x_{26} = -100.003999270066$$
$$x_{27} = 7.8178935923669$$
$$x_{28} = 40.3090360852991$$
$$x_{29} = 12.0176523103964$$
$$x_{30} = -65.4446777932686$$
$$x_{31} = -23.5472320794165$$
$$x_{32} = 26.6915460988748$$
$$x_{33} = 38.2142098418389$$
$$x_{34} = 78.0119987597878$$
$$x_{35} = -19.355123121769$$
$$x_{36} = -0.945142375621512$$
$$x_{37} = 66.4921086711227$$
$$x_{38} = 8.86729125025727$$
$$x_{39} = -89.5316254929708$$
$$x_{40} = 5.71419446010969$$
$$x_{41} = 73.8229723561137$$
$$x_{42} = 95.8151339590404$$
$$x_{43} = -51.8297209739997$$
$$x_{44} = -12.012533835301$$
$$x_{45} = 87.4372277043971$$
$$x_{46} = -14.111759765425$$
$$x_{47} = -31.9287760870061$$
$$x_{48} = 86.3899836289104$$
$$x_{49} = -62.3028168177213$$
$$x_{50} = 58.1138297680325$$
$$x_{51} = -47.6403417439802$$
$$x_{52} = 31.9294017642354$$
$$x_{53} = 56.0192280461156$$
$$x_{54} = -67.5392354219787$$
$$x_{55} = -78.011891136746$$
$$x_{56} = -29.8336042806827$$
$$x_{57} = -88.4843841787317$$
$$x_{58} = 42.4038206949034$$
$$x_{59} = 16.2123974730076$$
$$x_{60} = -41.356043573484$$
$$x_{61} = 36.1193349092579$$
$$x_{62} = -45.5456105782197$$
$$x_{63} = 93.7206627942325$$
$$x_{64} = 44.498569425187$$
$$x_{65} = -93.720585799908$$
$$x_{66} = -75.9173751870339$$
$$x_{67} = 60.2084173815303$$
$$x_{68} = -97.909531400917$$
$$x_{69} = 9.9187031499951$$
$$x_{70} = 64.3975556903502$$
$$x_{71} = -84.2954025614789$$
$$x_{72} = 80.1065028113244$$
$$x_{73} = 20.4048813808435$$
$$x_{74} = -82.2009043548885$$
$$x_{75} = -69.6337831465115$$
$$x_{76} = -80.1064006947394$$
$$x_{77} = 91.6261881927581$$
$$x_{78} = -73.8228522950252$$
$$x_{79} = 29.8343186124685$$
$$x_{80} = 97.9096019052536$$
$$x_{81} = -56.0190104973322$$
$$x_{82} = -16.2096545445765$$
$$x_{83} = -34.0238499908542$$
$$x_{84} = 62.302992282943$$
$$x_{85} = -95.8150603168249$$
$$x_{86} = 51.8299755895623$$
$$x_{87} = 88.4844680117351$$
$$x_{88} = -60.2082293581498$$
$$x_{89} = -9.91103941461928$$
$$x_{90} = 53.9246106073803$$
$$x_{91} = -49.7350440184369$$
$$x_{92} = -5.68924506450496$$
$$x_{93} = 1.6029805704892$$
$$x_{94} = -43.4508462434361$$
$$x_{95} = -27.7383118008235$$
$$x_{96} = -53.9243756173855$$
o sea hay que resolver la ecuación:
$$- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -7.80519181113787$$
$$x_{2} = 27.7391350406061$$
$$x_{3} = 22.5006395913317$$
$$x_{4} = 84.295494861077$$
$$x_{5} = 18.3088343819918$$
$$x_{6} = -36.1188433803924$$
$$x_{7} = 14.1154174315877$$
$$x_{8} = 68.5866521645177$$
$$x_{9} = 82.201001377025$$
$$x_{10} = 75.9174887741924$$
$$x_{11} = 2.51176326196438$$
$$x_{12} = -91.6261076121408$$
$$x_{13} = 71.7284489496945$$
$$x_{14} = -38.2137697610044$$
$$x_{15} = -25.6428682414677$$
$$x_{16} = 34.0244025417973$$
$$x_{17} = -32.9762988205827$$
$$x_{18} = 100.004066832881$$
$$x_{19} = -3.53971692338044$$
$$x_{20} = 39.2616435443518$$
$$x_{21} = -71.7283218450639$$
$$x_{22} = -58.113627788602$$
$$x_{23} = -21.4513451015863$$
$$x_{24} = -0.809176790168203$$
$$x_{25} = 49.7353208244535$$
$$x_{26} = -100.003999270066$$
$$x_{27} = 7.8178935923669$$
$$x_{28} = 40.3090360852991$$
$$x_{29} = 12.0176523103964$$
$$x_{30} = -65.4446777932686$$
$$x_{31} = -23.5472320794165$$
$$x_{32} = 26.6915460988748$$
$$x_{33} = 38.2142098418389$$
$$x_{34} = 78.0119987597878$$
$$x_{35} = -19.355123121769$$
$$x_{36} = -0.945142375621512$$
$$x_{37} = 66.4921086711227$$
$$x_{38} = 8.86729125025727$$
$$x_{39} = -89.5316254929708$$
$$x_{40} = 5.71419446010969$$
$$x_{41} = 73.8229723561137$$
$$x_{42} = 95.8151339590404$$
$$x_{43} = -51.8297209739997$$
$$x_{44} = -12.012533835301$$
$$x_{45} = 87.4372277043971$$
$$x_{46} = -14.111759765425$$
$$x_{47} = -31.9287760870061$$
$$x_{48} = 86.3899836289104$$
$$x_{49} = -62.3028168177213$$
$$x_{50} = 58.1138297680325$$
$$x_{51} = -47.6403417439802$$
$$x_{52} = 31.9294017642354$$
$$x_{53} = 56.0192280461156$$
$$x_{54} = -67.5392354219787$$
$$x_{55} = -78.011891136746$$
$$x_{56} = -29.8336042806827$$
$$x_{57} = -88.4843841787317$$
$$x_{58} = 42.4038206949034$$
$$x_{59} = 16.2123974730076$$
$$x_{60} = -41.356043573484$$
$$x_{61} = 36.1193349092579$$
$$x_{62} = -45.5456105782197$$
$$x_{63} = 93.7206627942325$$
$$x_{64} = 44.498569425187$$
$$x_{65} = -93.720585799908$$
$$x_{66} = -75.9173751870339$$
$$x_{67} = 60.2084173815303$$
$$x_{68} = -97.909531400917$$
$$x_{69} = 9.9187031499951$$
$$x_{70} = 64.3975556903502$$
$$x_{71} = -84.2954025614789$$
$$x_{72} = 80.1065028113244$$
$$x_{73} = 20.4048813808435$$
$$x_{74} = -82.2009043548885$$
$$x_{75} = -69.6337831465115$$
$$x_{76} = -80.1064006947394$$
$$x_{77} = 91.6261881927581$$
$$x_{78} = -73.8228522950252$$
$$x_{79} = 29.8343186124685$$
$$x_{80} = 97.9096019052536$$
$$x_{81} = -56.0190104973322$$
$$x_{82} = -16.2096545445765$$
$$x_{83} = -34.0238499908542$$
$$x_{84} = 62.302992282943$$
$$x_{85} = -95.8150603168249$$
$$x_{86} = 51.8299755895623$$
$$x_{87} = 88.4844680117351$$
$$x_{88} = -60.2082293581498$$
$$x_{89} = -9.91103941461928$$
$$x_{90} = 53.9246106073803$$
$$x_{91} = -49.7350440184369$$
$$x_{92} = -5.68924506450496$$
$$x_{93} = 1.6029805704892$$
$$x_{94} = -43.4508462434361$$
$$x_{95} = -27.7383118008235$$
$$x_{96} = -53.9243756173855$$
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
$$- 2 x^{2} \cos{\left(3 x \right)} - 2 x \left(- 3 x^{2} \sin{\left(3 x \right)} + 2 x \cos{\left(3 x \right)} + 3 \left(1 - x\right) \cos{\left(3 x \right)} - \sin{\left(3 x \right)}\right) - 2 \left(1 - x\right) \sin{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -87.9645512254701$$
$$x_{2} = 65.9735222979839$$
$$x_{3} = -13.6117757944157$$
$$x_{4} = 74.3510864259024$$
$$x_{5} = -68.0677688940187$$
$$x_{6} = 21.9918368635693$$
$$x_{7} = -79.5869612711768$$
$$x_{8} = -72.2565671961847$$
$$x_{9} = 39.7937173568211$$
$$x_{10} = 28.2747504881455$$
$$x_{11} = 59.6903539565878$$
$$x_{12} = -48.170943745812$$
$$x_{13} = 52.359999115404$$
$$x_{14} = 96.3422106200121$$
$$x_{15} = 72.256694868833$$
$$x_{16} = 8.38228224782493$$
$$x_{17} = -90.0589483078959$$
$$x_{18} = -33.5100249724637$$
$$x_{19} = 78.5398703718322$$
$$x_{20} = 17.8034078744262$$
$$x_{21} = 56.5487719825944$$
$$x_{22} = -4.17043227881737$$
$$x_{23} = 100.530997894843$$
$$x_{24} = 11.5216720168374$$
$$x_{25} = 0$$
$$x_{26} = 48.1712309634192$$
$$x_{27} = -35.6044539327594$$
$$x_{28} = -39.7932965318971$$
$$x_{29} = -2.02786118086652$$
$$x_{30} = -11.5166730356884$$
$$x_{31} = -55.5013620260354$$
$$x_{32} = -52.3597560036917$$
$$x_{33} = -41.8877121419601$$
$$x_{34} = 92.1534237536942$$
$$x_{35} = -85.8701539964212$$
$$x_{36} = 32.4634401831824$$
$$x_{37} = 43.9824694051414$$
$$x_{38} = -21.9904602437192$$
$$x_{39} = 15.7093104545096$$
$$x_{40} = -59.6901668795313$$
$$x_{41} = -19.8959128503852$$
$$x_{42} = 90.0590304978814$$
$$x_{43} = -43.9821248940245$$
$$x_{44} = -46.0765352954475$$
$$x_{45} = 68.0679127613871$$
$$x_{46} = -83.7757566059074$$
$$x_{47} = 19.8975940242995$$
$$x_{48} = -30.3683678105346$$
$$x_{49} = 87.9646373755162$$
$$x_{50} = -77.4925632863524$$
$$x_{51} = 54.4543850492753$$
$$x_{52} = 61.7847428259826$$
$$x_{53} = -92.1533452568736$$
$$x_{54} = -26.1794529021746$$
$$x_{55} = 37.6993462713932$$
$$x_{56} = -61.7845682149693$$
$$x_{57} = -28.273917264214$$
$$x_{58} = 26.1804246396561$$
$$x_{59} = -24.0849697237312$$
$$x_{60} = -95.2949404554302$$
$$x_{61} = 76.4454782703127$$
$$x_{62} = 4.20700066076947$$
$$x_{63} = -57.5957648520122$$
$$x_{64} = -62.8317686516205$$
$$x_{65} = 6.29147622227546$$
$$x_{66} = 41.8880919520471$$
$$x_{67} = 63.8791322982152$$
$$x_{68} = -17.8013087430214$$
$$x_{69} = 24.0861176040221$$
$$x_{70} = 2.15808850992003$$
$$x_{71} = 30.3690901502898$$
$$x_{72} = 70.1623036339133$$
$$x_{73} = 46.0768492087845$$
$$x_{74} = -15.7066158515165$$
$$x_{75} = 10.4749958368593$$
$$x_{76} = 16.7563452754841$$
$$x_{77} = 85.8702443997733$$
$$x_{78} = -63.8789689475609$$
$$x_{79} = 50.2656143509073$$
$$x_{80} = 58.6431597752133$$
$$x_{81} = -81.6813590371603$$
$$x_{82} = -9.42105198379253$$
$$x_{83} = 20.9447097509249$$
$$x_{84} = -50.2653505632743$$
$$x_{85} = 83.7758515854944$$
$$x_{86} = -70.1621682263002$$
$$x_{87} = -99.4837336857622$$
$$x_{88} = -94.2477420841309$$
$$x_{89} = 98.4366042106844$$
$$x_{90} = -65.9733691526097$$
$$x_{91} = -37.698877411849$$
$$x_{92} = 94.2478171312268$$
$$x_{93} = -76.4453642043055$$
$$x_{94} = -7.32425123309633$$
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:
$$x_{1} = -13.6117757944157$$
$$x_{2} = -68.0677688940187$$
$$x_{3} = -72.2565671961847$$
$$x_{4} = 39.7937173568211$$
$$x_{5} = 52.359999115404$$
$$x_{6} = 96.3422106200121$$
$$x_{7} = 8.38228224782493$$
$$x_{8} = 56.5487719825944$$
$$x_{9} = 100.530997894843$$
$$x_{10} = 48.1712309634192$$
$$x_{11} = -11.5166730356884$$
$$x_{12} = -55.5013620260354$$
$$x_{13} = 92.1534237536942$$
$$x_{14} = 43.9824694051414$$
$$x_{15} = -21.9904602437192$$
$$x_{16} = -59.6901668795313$$
$$x_{17} = -19.8959128503852$$
$$x_{18} = 90.0590304978814$$
$$x_{19} = -30.3683678105346$$
$$x_{20} = 87.9646373755162$$
$$x_{21} = 54.4543850492753$$
$$x_{22} = -26.1794529021746$$
$$x_{23} = 37.6993462713932$$
$$x_{24} = -61.7845682149693$$
$$x_{25} = -28.273917264214$$
$$x_{26} = -24.0849697237312$$
$$x_{27} = -95.2949404554302$$
$$x_{28} = 4.20700066076947$$
$$x_{29} = -57.5957648520122$$
$$x_{30} = 6.29147622227546$$
$$x_{31} = 41.8880919520471$$
$$x_{32} = -17.8013087430214$$
$$x_{33} = 2.15808850992003$$
$$x_{34} = 46.0768492087845$$
$$x_{35} = -15.7066158515165$$
$$x_{36} = 10.4749958368593$$
$$x_{37} = 16.7563452754841$$
$$x_{38} = 85.8702443997733$$
$$x_{39} = -63.8789689475609$$
$$x_{40} = 50.2656143509073$$
$$x_{41} = 58.6431597752133$$
$$x_{42} = -9.42105198379253$$
$$x_{43} = 20.9447097509249$$
$$x_{44} = 83.7758515854944$$
$$x_{45} = -70.1621682263002$$
$$x_{46} = -99.4837336857622$$
$$x_{47} = 98.4366042106844$$
$$x_{48} = -65.9733691526097$$
$$x_{49} = 94.2478171312268$$
$$x_{50} = -76.4453642043055$$
$$x_{51} = -7.32425123309633$$
Puntos máximos de la función:
$$x_{51} = -87.9645512254701$$
$$x_{51} = 65.9735222979839$$
$$x_{51} = 74.3510864259024$$
$$x_{51} = 21.9918368635693$$
$$x_{51} = -79.5869612711768$$
$$x_{51} = 28.2747504881455$$
$$x_{51} = 59.6903539565878$$
$$x_{51} = -48.170943745812$$
$$x_{51} = 72.256694868833$$
$$x_{51} = -90.0589483078959$$
$$x_{51} = -33.5100249724637$$
$$x_{51} = 78.5398703718322$$
$$x_{51} = 17.8034078744262$$
$$x_{51} = -4.17043227881737$$
$$x_{51} = 11.5216720168374$$
$$x_{51} = 0$$
$$x_{51} = -35.6044539327594$$
$$x_{51} = -39.7932965318971$$
$$x_{51} = -2.02786118086652$$
$$x_{51} = -52.3597560036917$$
$$x_{51} = -41.8877121419601$$
$$x_{51} = -85.8701539964212$$
$$x_{51} = 32.4634401831824$$
$$x_{51} = 15.7093104545096$$
$$x_{51} = -43.9821248940245$$
$$x_{51} = -46.0765352954475$$
$$x_{51} = 68.0679127613871$$
$$x_{51} = -83.7757566059074$$
$$x_{51} = 19.8975940242995$$
$$x_{51} = -77.4925632863524$$
$$x_{51} = 61.7847428259826$$
$$x_{51} = -92.1533452568736$$
$$x_{51} = 26.1804246396561$$
$$x_{51} = 76.4454782703127$$
$$x_{51} = -62.8317686516205$$
$$x_{51} = 63.8791322982152$$
$$x_{51} = 24.0861176040221$$
$$x_{51} = 30.3690901502898$$
$$x_{51} = 70.1623036339133$$
$$x_{51} = -81.6813590371603$$
$$x_{51} = -50.2653505632743$$
$$x_{51} = -94.2477420841309$$
$$x_{51} = -37.698877411849$$
Decrece en los intervalos
$$\left[100.530997894843, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.4837336857622\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- 2 x^{2} \cos{\left(3 x \right)} - 2 x \left(- 3 x^{2} \sin{\left(3 x \right)} + 2 x \cos{\left(3 x \right)} + 3 \left(1 - x\right) \cos{\left(3 x \right)} - \sin{\left(3 x \right)}\right) - 2 \left(1 - x\right) \sin{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -87.9645512254701$$
$$x_{2} = 65.9735222979839$$
$$x_{3} = -13.6117757944157$$
$$x_{4} = 74.3510864259024$$
$$x_{5} = -68.0677688940187$$
$$x_{6} = 21.9918368635693$$
$$x_{7} = -79.5869612711768$$
$$x_{8} = -72.2565671961847$$
$$x_{9} = 39.7937173568211$$
$$x_{10} = 28.2747504881455$$
$$x_{11} = 59.6903539565878$$
$$x_{12} = -48.170943745812$$
$$x_{13} = 52.359999115404$$
$$x_{14} = 96.3422106200121$$
$$x_{15} = 72.256694868833$$
$$x_{16} = 8.38228224782493$$
$$x_{17} = -90.0589483078959$$
$$x_{18} = -33.5100249724637$$
$$x_{19} = 78.5398703718322$$
$$x_{20} = 17.8034078744262$$
$$x_{21} = 56.5487719825944$$
$$x_{22} = -4.17043227881737$$
$$x_{23} = 100.530997894843$$
$$x_{24} = 11.5216720168374$$
$$x_{25} = 0$$
$$x_{26} = 48.1712309634192$$
$$x_{27} = -35.6044539327594$$
$$x_{28} = -39.7932965318971$$
$$x_{29} = -2.02786118086652$$
$$x_{30} = -11.5166730356884$$
$$x_{31} = -55.5013620260354$$
$$x_{32} = -52.3597560036917$$
$$x_{33} = -41.8877121419601$$
$$x_{34} = 92.1534237536942$$
$$x_{35} = -85.8701539964212$$
$$x_{36} = 32.4634401831824$$
$$x_{37} = 43.9824694051414$$
$$x_{38} = -21.9904602437192$$
$$x_{39} = 15.7093104545096$$
$$x_{40} = -59.6901668795313$$
$$x_{41} = -19.8959128503852$$
$$x_{42} = 90.0590304978814$$
$$x_{43} = -43.9821248940245$$
$$x_{44} = -46.0765352954475$$
$$x_{45} = 68.0679127613871$$
$$x_{46} = -83.7757566059074$$
$$x_{47} = 19.8975940242995$$
$$x_{48} = -30.3683678105346$$
$$x_{49} = 87.9646373755162$$
$$x_{50} = -77.4925632863524$$
$$x_{51} = 54.4543850492753$$
$$x_{52} = 61.7847428259826$$
$$x_{53} = -92.1533452568736$$
$$x_{54} = -26.1794529021746$$
$$x_{55} = 37.6993462713932$$
$$x_{56} = -61.7845682149693$$
$$x_{57} = -28.273917264214$$
$$x_{58} = 26.1804246396561$$
$$x_{59} = -24.0849697237312$$
$$x_{60} = -95.2949404554302$$
$$x_{61} = 76.4454782703127$$
$$x_{62} = 4.20700066076947$$
$$x_{63} = -57.5957648520122$$
$$x_{64} = -62.8317686516205$$
$$x_{65} = 6.29147622227546$$
$$x_{66} = 41.8880919520471$$
$$x_{67} = 63.8791322982152$$
$$x_{68} = -17.8013087430214$$
$$x_{69} = 24.0861176040221$$
$$x_{70} = 2.15808850992003$$
$$x_{71} = 30.3690901502898$$
$$x_{72} = 70.1623036339133$$
$$x_{73} = 46.0768492087845$$
$$x_{74} = -15.7066158515165$$
$$x_{75} = 10.4749958368593$$
$$x_{76} = 16.7563452754841$$
$$x_{77} = 85.8702443997733$$
$$x_{78} = -63.8789689475609$$
$$x_{79} = 50.2656143509073$$
$$x_{80} = 58.6431597752133$$
$$x_{81} = -81.6813590371603$$
$$x_{82} = -9.42105198379253$$
$$x_{83} = 20.9447097509249$$
$$x_{84} = -50.2653505632743$$
$$x_{85} = 83.7758515854944$$
$$x_{86} = -70.1621682263002$$
$$x_{87} = -99.4837336857622$$
$$x_{88} = -94.2477420841309$$
$$x_{89} = 98.4366042106844$$
$$x_{90} = -65.9733691526097$$
$$x_{91} = -37.698877411849$$
$$x_{92} = 94.2478171312268$$
$$x_{93} = -76.4453642043055$$
$$x_{94} = -7.32425123309633$$
Signos de extremos en los puntos:
(-87.96455122547006, 1361299.58273911 + E)
(65.97352229798388, 574298.271827806 + E)
(-13.611775794415664, -5046.05762682675 + E)
(74.35108642590244, 822036.124034965 + E)
(-68.06776889401866, -630748.069009655 + E)
(21.99183686356929, 21270.3512128818 + E)
(-79.58696127117683, 1008223.07271253 + E)
(-72.25656719618469, -754506.750576209 + E)
(39.79371735682107, -126027.907637773 + E)
(28.27475048814548, 45207.1867381598 + E)
(59.69035395658781, 425344.12025037 + E)
(-48.170943745812004, 223557.572379574 + E)
(52.359999115403966, -287095.17354117 + E)
(96.34221062001207, -1788460.4233885 + E)
(72.25669486883304, 754506.750576209 + E)
(8.38228224782493, -1176.05999058124 + E)
(-90.05894830789588, 1460868.77555957 + E)
(-33.510024972463725, 75260.3019912521 + E)
(78.53987037183224, 968946.158990389 + E)
(17.803407874426203, 11284.0440046831 + E)
(56.548771982594374, -361657.228879213 + E)
(-4.170432278817372, 147.222486590438 + E)
(100.5309978948431, -2032027.35846666 + E)
(11.521672016837417, 3057.07558127174 + E)
(0, E)
(48.17123096341922, -223557.572379574 + E)
(-35.60445393275944, 90271.9316745479 + E)
(-39.79329653189708, 126027.907637773 + E)
(-2.0278611808665152, 18.7817935340658 + E)
(-11.51667303568837, -3057.07558102077 + E)
(-55.501362026035395, -341934.940482753 + E)
(-52.359756003691686, 287095.17354117 + E)
(-41.88771214196009, 146992.742941149 + E)
(92.15342375369418, -1565178.48358227 + E)
(-85.87015399642118, 1266360.66058247 + E)
(32.46344018318242, 68422.8447537987 + E)
(43.98246940514135, -170162.469150071 + E)
(-21.99046024371924, -21270.3512128766 + E)
(15.70931045450963, 7751.63266026757 + E)
(-59.690166879531276, -425344.12025037 + E)
(-19.89591285038517, -15753.535489704 + E)
(90.05903049788144, -1460868.77555957 + E)
(-43.982124894024516, 170162.469150071 + E)
(-46.07653529544746, 195647.330787589 + E)
(68.06791276138706, 630748.069009655 + E)
(-83.77575660590739, 1175941.7645513 + E)
(19.89759402429951, 15753.5354897135 + E)
(-30.368367810534615, -56015.7426793193 + E)
(87.96463737551619, -1361299.58273911 + E)
(-77.49256328635238, 930702.787828786 + E)
(54.45438504927531, -322943.022172022 + E)
(61.784742825982576, 471706.541984148 + E)
(-92.15334525687359, 1565178.48358227 + E)
(-26.17945290217456, -35886.93246595 + E)
(37.699346271393175, -107157.7187205 + E)
(-61.784568214969305, -471706.541984148 + E)
(-28.273917264214028, -45207.1867381586 + E)
(26.180424639656117, 35886.9324659519 + E)
(-24.084969723731177, -27944.7354242647 + E)
(-95.29494045543022, -1730772.67154853 + E)
(76.44547827031268, 893479.178660491 + E)
(4.20700066076947, -147.222587641142 + E)
(-57.5957648520122, -382123.667928119 + E)
(-62.83176865162048, 496100.442797604 + E)
(6.291476222275462, -496.256919058581 + E)
(41.88809195204712, -146992.742941149 + E)
(63.87913229821518, 521321.177664704 + E)
(-17.801308743021362, -11284.0440046646 + E)
(24.086117604022085, 27944.7354242677 + E)
(2.1580885099200318, -18.7867519326205 + E)
(30.369090150289818, 56015.7426793201 + E)
(70.16230363391333, 690780.813746827 + E)
(46.076849208784495, -195647.330787589 + E)
(-15.706615851516478, -7751.63266022834 + E)
(10.474995836859318, -2296.85614979228 + E)
(16.756345275484108, -9407.59356246078 + E)
(85.87024439977327, -1266360.66058247 + E)
(-63.87896894756095, -521321.177664704 + E)
(50.265614350907306, -254003.438454135 + E)
(58.64315977521328, -403348.038196235 + E)
(-81.68135903716026, 1089932.65010736 + E)
(-9.421051983792534, -1674.44424991783 + E)
(20.94470975092492, -18374.1375585909 + E)
(-50.26535056327428, 254003.438454135 + E)
(83.77585158549442, -1175941.7645513 + E)
(-70.16216822630015, -690780.813746826 + E)
(-99.4837336857622, -1969185.6744047 + E)
(-94.24774208413095, 1674338.95134572 + E)
(98.43660421068435, -1907653.14424925 + E)
(-65.97336915260975, -574298.271827805 + E)
(-37.69887741184902, 107157.7187205 + E)
(94.24781713122677, -1674338.95134572 + E)
(-76.44536420430552, -893479.178660491 + E)
(-7.32425123309633, -787.923837275761 + E)
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:
$$x_{1} = -13.6117757944157$$
$$x_{2} = -68.0677688940187$$
$$x_{3} = -72.2565671961847$$
$$x_{4} = 39.7937173568211$$
$$x_{5} = 52.359999115404$$
$$x_{6} = 96.3422106200121$$
$$x_{7} = 8.38228224782493$$
$$x_{8} = 56.5487719825944$$
$$x_{9} = 100.530997894843$$
$$x_{10} = 48.1712309634192$$
$$x_{11} = -11.5166730356884$$
$$x_{12} = -55.5013620260354$$
$$x_{13} = 92.1534237536942$$
$$x_{14} = 43.9824694051414$$
$$x_{15} = -21.9904602437192$$
$$x_{16} = -59.6901668795313$$
$$x_{17} = -19.8959128503852$$
$$x_{18} = 90.0590304978814$$
$$x_{19} = -30.3683678105346$$
$$x_{20} = 87.9646373755162$$
$$x_{21} = 54.4543850492753$$
$$x_{22} = -26.1794529021746$$
$$x_{23} = 37.6993462713932$$
$$x_{24} = -61.7845682149693$$
$$x_{25} = -28.273917264214$$
$$x_{26} = -24.0849697237312$$
$$x_{27} = -95.2949404554302$$
$$x_{28} = 4.20700066076947$$
$$x_{29} = -57.5957648520122$$
$$x_{30} = 6.29147622227546$$
$$x_{31} = 41.8880919520471$$
$$x_{32} = -17.8013087430214$$
$$x_{33} = 2.15808850992003$$
$$x_{34} = 46.0768492087845$$
$$x_{35} = -15.7066158515165$$
$$x_{36} = 10.4749958368593$$
$$x_{37} = 16.7563452754841$$
$$x_{38} = 85.8702443997733$$
$$x_{39} = -63.8789689475609$$
$$x_{40} = 50.2656143509073$$
$$x_{41} = 58.6431597752133$$
$$x_{42} = -9.42105198379253$$
$$x_{43} = 20.9447097509249$$
$$x_{44} = 83.7758515854944$$
$$x_{45} = -70.1621682263002$$
$$x_{46} = -99.4837336857622$$
$$x_{47} = 98.4366042106844$$
$$x_{48} = -65.9733691526097$$
$$x_{49} = 94.2478171312268$$
$$x_{50} = -76.4453642043055$$
$$x_{51} = -7.32425123309633$$
Puntos máximos de la función:
$$x_{51} = -87.9645512254701$$
$$x_{51} = 65.9735222979839$$
$$x_{51} = 74.3510864259024$$
$$x_{51} = 21.9918368635693$$
$$x_{51} = -79.5869612711768$$
$$x_{51} = 28.2747504881455$$
$$x_{51} = 59.6903539565878$$
$$x_{51} = -48.170943745812$$
$$x_{51} = 72.256694868833$$
$$x_{51} = -90.0589483078959$$
$$x_{51} = -33.5100249724637$$
$$x_{51} = 78.5398703718322$$
$$x_{51} = 17.8034078744262$$
$$x_{51} = -4.17043227881737$$
$$x_{51} = 11.5216720168374$$
$$x_{51} = 0$$
$$x_{51} = -35.6044539327594$$
$$x_{51} = -39.7932965318971$$
$$x_{51} = -2.02786118086652$$
$$x_{51} = -52.3597560036917$$
$$x_{51} = -41.8877121419601$$
$$x_{51} = -85.8701539964212$$
$$x_{51} = 32.4634401831824$$
$$x_{51} = 15.7093104545096$$
$$x_{51} = -43.9821248940245$$
$$x_{51} = -46.0765352954475$$
$$x_{51} = 68.0679127613871$$
$$x_{51} = -83.7757566059074$$
$$x_{51} = 19.8975940242995$$
$$x_{51} = -77.4925632863524$$
$$x_{51} = 61.7847428259826$$
$$x_{51} = -92.1533452568736$$
$$x_{51} = 26.1804246396561$$
$$x_{51} = 76.4454782703127$$
$$x_{51} = -62.8317686516205$$
$$x_{51} = 63.8791322982152$$
$$x_{51} = 24.0861176040221$$
$$x_{51} = 30.3690901502898$$
$$x_{51} = 70.1623036339133$$
$$x_{51} = -81.6813590371603$$
$$x_{51} = -50.2653505632743$$
$$x_{51} = -94.2477420841309$$
$$x_{51} = -37.698877411849$$
Decrece en los intervalos
$$\left[100.530997894843, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.4837336857622\right]$$
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
$$2 \left(6 x^{2} \sin{\left(3 x \right)} + x \left(9 x^{2} \cos{\left(3 x \right)} + 12 x \sin{\left(3 x \right)} - 9 \left(x - 1\right) \sin{\left(3 x \right)} + 4 \cos{\left(3 x \right)}\right) - 4 x \cos{\left(3 x \right)} + 6 \left(x - 1\right) \cos{\left(3 x \right)} + 2 \sin{\left(3 x \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 93.7277748555153$$
$$x_{2} = -51.8425826670991$$
$$x_{3} = -34.0434178975265$$
$$x_{4} = 20.4374280177627$$
$$x_{5} = -38.2311969636889$$
$$x_{6} = 59.1723888231602$$
$$x_{7} = 51.8428305908029$$
$$x_{8} = 66.502131344851$$
$$x_{9} = 34.0439924583135$$
$$x_{10} = 75.9262699056966$$
$$x_{11} = -75.9261542878579$$
$$x_{12} = -21.482319591543$$
$$x_{13} = 78.0205441479829$$
$$x_{14} = 43.4665402880797$$
$$x_{15} = 42.4195405844178$$
$$x_{16} = 82.2091113174536$$
$$x_{17} = -5.80609832230541$$
$$x_{18} = -23.5754654798559$$
$$x_{19} = -97.9163402639376$$
$$x_{20} = 50.7957706200407$$
$$x_{21} = -1.63851006885487$$
$$x_{22} = -49.7484473226791$$
$$x_{23} = 36.1377890296509$$
$$x_{24} = -69.6433539662408$$
$$x_{25} = -3.72748547685904$$
$$x_{26} = -56.0309104712877$$
$$x_{27} = -37.184231495257$$
$$x_{28} = 24.6232113474275$$
$$x_{29} = 7.9009077475202$$
$$x_{30} = -45.5602465481582$$
$$x_{31} = 3.77094199555594$$
$$x_{32} = -53.9367377736279$$
$$x_{33} = -31.9496242303655$$
$$x_{34} = 73.8320026071376$$
$$x_{35} = -60.2193014365752$$
$$x_{36} = 12.0725077599022$$
$$x_{37} = 14.1622664448762$$
$$x_{38} = -95.8220180119334$$
$$x_{39} = 22.5301750724685$$
$$x_{40} = -8.93400505945408$$
$$x_{41} = 100.0107321786$$
$$x_{42} = 44.5135494858579$$
$$x_{43} = 60.2194852073142$$
$$x_{44} = 29.8566584482407$$
$$x_{45} = -41.3721619307587$$
$$x_{46} = 38.2316526562371$$
$$x_{47} = 86.3977003485997$$
$$x_{48} = -27.7622977721478$$
$$x_{49} = 97.9164097884094$$
$$x_{50} = -91.633383383705$$
$$x_{51} = -58.1250988956428$$
$$x_{52} = -82.2090126932091$$
$$x_{53} = -89.5390714639324$$
$$x_{54} = 56.0311227308978$$
$$x_{55} = 90.5863081780552$$
$$x_{56} = 31.9502764618634$$
$$x_{57} = 84.3034033061984$$
$$x_{58} = -65.4548608089156$$
$$x_{59} = -73.8318803388134$$
$$x_{60} = 58.1252961417881$$
$$x_{61} = -100.01066553508$$
$$x_{62} = -7.89043605658187$$
$$x_{63} = 64.4079041100163$$
$$x_{64} = -36.1372790548606$$
$$x_{65} = -71.737613358671$$
$$x_{66} = -78.0204346524777$$
$$x_{67} = 88.4920020821134$$
$$x_{68} = -12.0679732051585$$
$$x_{69} = 27.7631612164057$$
$$x_{70} = -93.7276989791406$$
$$x_{71} = -14.1589637088364$$
$$x_{72} = -29.8559116754859$$
$$x_{73} = 1.80094897098405$$
$$x_{74} = 95.8220906082739$$
$$x_{75} = 46.607593757678$$
$$x_{76} = -25.6688054873748$$
$$x_{77} = 16.2532676019953$$
$$x_{78} = 80.1148247832435$$
$$x_{79} = 71.7377428681676$$
$$x_{80} = -87.4447634806711$$
$$x_{81} = 53.9369668279978$$
$$x_{82} = -16.2507562244632$$
$$x_{83} = -67.5491028557929$$
$$x_{84} = -80.1147209363935$$
$$x_{85} = 0.247351595221807$$
$$x_{86} = -84.3033095202993$$
$$x_{87} = 62.3136883266699$$
$$x_{88} = 9.98481273077832$$
$$x_{89} = -43.4661876771094$$
$$x_{90} = 18.3450729680688$$
$$x_{91} = -47.6543342835173$$
$$x_{92} = 40.3255726796291$$
$$x_{93} = 49.7487165464924$$
$$x_{94} = 5.82507346352407$$
$$x_{95} = -9.9782092489233$$
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
$$\left[100.0107321786, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.01066553508\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(6 x^{2} \sin{\left(3 x \right)} + x \left(9 x^{2} \cos{\left(3 x \right)} + 12 x \sin{\left(3 x \right)} - 9 \left(x - 1\right) \sin{\left(3 x \right)} + 4 \cos{\left(3 x \right)}\right) - 4 x \cos{\left(3 x \right)} + 6 \left(x - 1\right) \cos{\left(3 x \right)} + 2 \sin{\left(3 x \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 93.7277748555153$$
$$x_{2} = -51.8425826670991$$
$$x_{3} = -34.0434178975265$$
$$x_{4} = 20.4374280177627$$
$$x_{5} = -38.2311969636889$$
$$x_{6} = 59.1723888231602$$
$$x_{7} = 51.8428305908029$$
$$x_{8} = 66.502131344851$$
$$x_{9} = 34.0439924583135$$
$$x_{10} = 75.9262699056966$$
$$x_{11} = -75.9261542878579$$
$$x_{12} = -21.482319591543$$
$$x_{13} = 78.0205441479829$$
$$x_{14} = 43.4665402880797$$
$$x_{15} = 42.4195405844178$$
$$x_{16} = 82.2091113174536$$
$$x_{17} = -5.80609832230541$$
$$x_{18} = -23.5754654798559$$
$$x_{19} = -97.9163402639376$$
$$x_{20} = 50.7957706200407$$
$$x_{21} = -1.63851006885487$$
$$x_{22} = -49.7484473226791$$
$$x_{23} = 36.1377890296509$$
$$x_{24} = -69.6433539662408$$
$$x_{25} = -3.72748547685904$$
$$x_{26} = -56.0309104712877$$
$$x_{27} = -37.184231495257$$
$$x_{28} = 24.6232113474275$$
$$x_{29} = 7.9009077475202$$
$$x_{30} = -45.5602465481582$$
$$x_{31} = 3.77094199555594$$
$$x_{32} = -53.9367377736279$$
$$x_{33} = -31.9496242303655$$
$$x_{34} = 73.8320026071376$$
$$x_{35} = -60.2193014365752$$
$$x_{36} = 12.0725077599022$$
$$x_{37} = 14.1622664448762$$
$$x_{38} = -95.8220180119334$$
$$x_{39} = 22.5301750724685$$
$$x_{40} = -8.93400505945408$$
$$x_{41} = 100.0107321786$$
$$x_{42} = 44.5135494858579$$
$$x_{43} = 60.2194852073142$$
$$x_{44} = 29.8566584482407$$
$$x_{45} = -41.3721619307587$$
$$x_{46} = 38.2316526562371$$
$$x_{47} = 86.3977003485997$$
$$x_{48} = -27.7622977721478$$
$$x_{49} = 97.9164097884094$$
$$x_{50} = -91.633383383705$$
$$x_{51} = -58.1250988956428$$
$$x_{52} = -82.2090126932091$$
$$x_{53} = -89.5390714639324$$
$$x_{54} = 56.0311227308978$$
$$x_{55} = 90.5863081780552$$
$$x_{56} = 31.9502764618634$$
$$x_{57} = 84.3034033061984$$
$$x_{58} = -65.4548608089156$$
$$x_{59} = -73.8318803388134$$
$$x_{60} = 58.1252961417881$$
$$x_{61} = -100.01066553508$$
$$x_{62} = -7.89043605658187$$
$$x_{63} = 64.4079041100163$$
$$x_{64} = -36.1372790548606$$
$$x_{65} = -71.737613358671$$
$$x_{66} = -78.0204346524777$$
$$x_{67} = 88.4920020821134$$
$$x_{68} = -12.0679732051585$$
$$x_{69} = 27.7631612164057$$
$$x_{70} = -93.7276989791406$$
$$x_{71} = -14.1589637088364$$
$$x_{72} = -29.8559116754859$$
$$x_{73} = 1.80094897098405$$
$$x_{74} = 95.8220906082739$$
$$x_{75} = 46.607593757678$$
$$x_{76} = -25.6688054873748$$
$$x_{77} = 16.2532676019953$$
$$x_{78} = 80.1148247832435$$
$$x_{79} = 71.7377428681676$$
$$x_{80} = -87.4447634806711$$
$$x_{81} = 53.9369668279978$$
$$x_{82} = -16.2507562244632$$
$$x_{83} = -67.5491028557929$$
$$x_{84} = -80.1147209363935$$
$$x_{85} = 0.247351595221807$$
$$x_{86} = -84.3033095202993$$
$$x_{87} = 62.3136883266699$$
$$x_{88} = 9.98481273077832$$
$$x_{89} = -43.4661876771094$$
$$x_{90} = 18.3450729680688$$
$$x_{91} = -47.6543342835173$$
$$x_{92} = 40.3255726796291$$
$$x_{93} = 49.7487165464924$$
$$x_{94} = 5.82507346352407$$
$$x_{95} = -9.9782092489233$$
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
$$\left[100.0107321786, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.01066553508\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función E - 2*x*(x^2*cos(3*x) + (1 - x)*sin(3*x)), dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e}{x}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e}{x}\right)$$
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:
$$- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e = 2 x \left(x^{2} \cos{\left(3 x \right)} - \left(x + 1\right) \sin{\left(3 x \right)}\right) + e$$
- No
$$- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e = - 2 x \left(x^{2} \cos{\left(3 x \right)} - \left(x + 1\right) \sin{\left(3 x \right)}\right) - e$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e = 2 x \left(x^{2} \cos{\left(3 x \right)} - \left(x + 1\right) \sin{\left(3 x \right)}\right) + e$$
- No
$$- 2 x \left(x^{2} \cos{\left(3 x \right)} + \left(1 - x\right) \sin{\left(3 x \right)}\right) + e = - 2 x \left(x^{2} \cos{\left(3 x \right)} - \left(x + 1\right) \sin{\left(3 x \right)}\right) - e$$
- No
es decir, función
no es
par ni impar