Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- 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
-
¿Cómo usar?
- Gráfico de la función y =:
-
(x-1)/(x+2)
-
-1-x
-
-exp(-x)+(-cos(x*sqrt(3)/2)-sin(x*sqrt(3)/2))*exp(x/2)
-
3/(x^2+1)
- Derivada de:
-
x^2*cos(3*x)
-
Expresiones idénticas
- x^ dos *cos(tres *x)
- x al cuadrado multiplicar por coseno de (3 multiplicar por x)
- x en el grado dos multiplicar por coseno de (tres multiplicar por x)
- x2*cos(3*x)
- x2*cos3*x
- x²*cos(3*x)
- x en el grado 2*cos(3*x)
- x^2cos(3x)
- x2cos(3x)
- x2cos3x
- x^2cos3x
-
Expresiones con funciones
- Coseno cos
- cos(x)+exp(x)+exp(-x)+sin(x)
- cos(x)-3
- cos(x)*4*x
- cos(x)/8+cos(3*x)+sin(3*x)
- cos(x)+1/cos(x)
Gráfico de la función y = x^2*cos(3*x)
Solución
Ha introducido
[src]
2 f(x) = x *cos(3*x)
$$f{\left(x \right)} = x^{2} \cos{\left(3 x \right)}$$
f = x^2*cos(3*x)
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:
$$x^{2} \cos{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{5 \pi}{6}$$
$$x_{3} = - \frac{\pi}{2}$$
$$x_{4} = - \frac{\pi}{6}$$
$$x_{5} = \frac{\pi}{6}$$
$$x_{6} = \frac{\pi}{2}$$
$$x_{7} = \frac{5 \pi}{6}$$
Solución numérica
$$x_{1} = -27.7507351067098$$
$$x_{2} = -47.6474885794452$$
$$x_{3} = 27.7507351067098$$
$$x_{4} = 73.8274273593601$$
$$x_{5} = -31.9395253114962$$
$$x_{6} = 34.0339204138894$$
$$x_{7} = -89.5353906273091$$
$$x_{8} = -41.3643032722656$$
$$x_{9} = 42.4115008234622$$
$$x_{10} = -5.75958653158129$$
$$x_{11} = 14.1371669411541$$
$$x_{12} = 31.9395253114962$$
$$x_{13} = -1.5707963267949$$
$$x_{14} = 40.317105721069$$
$$x_{15} = 44.5058959258554$$
$$x_{16} = -67.5442420521806$$
$$x_{17} = 16.2315620435473$$
$$x_{18} = -100.007366139275$$
$$x_{19} = 5.75958653158129$$
$$x_{20} = 89.5353906273091$$
$$x_{21} = -78.0162175641465$$
$$x_{22} = -80.1106126665397$$
$$x_{23} = 36.1283155162826$$
$$x_{24} = -91.6297857297023$$
$$x_{25} = -60.2138591938044$$
$$x_{26} = 95.8185759344887$$
$$x_{27} = -3.66519142918809$$
$$x_{28} = -34.0339204138894$$
$$x_{29} = 71.733032256967$$
$$x_{30} = -69.6386371545737$$
$$x_{31} = 92.6769832808989$$
$$x_{32} = -97.9129710368819$$
$$x_{33} = -12.0427718387609$$
$$x_{34} = 60.2138591938044$$
$$x_{35} = 75.9218224617533$$
$$x_{36} = 78.0162175641465$$
$$x_{37} = -29.845130209103$$
$$x_{38} = 0$$
$$x_{39} = -38.2227106186758$$
$$x_{40} = -75.9218224617533$$
$$x_{41} = 23.5619449019235$$
$$x_{42} = 20.4203522483337$$
$$x_{43} = -25.6563400043166$$
$$x_{44} = 3.66519142918809$$
$$x_{45} = -87.4409955249159$$
$$x_{46} = 53.9306738866248$$
$$x_{47} = 38.2227106186758$$
$$x_{48} = -32.9867228626928$$
$$x_{49} = -71.733032256967$$
$$x_{50} = 7.85398163397448$$
$$x_{51} = -17.2787595947439$$
$$x_{52} = 12.0427718387609$$
$$x_{53} = -63.3554518473942$$
$$x_{54} = 26.7035375555132$$
$$x_{55} = 64.4026493985908$$
$$x_{56} = -9.94837673636768$$
$$x_{57} = -14.1371669411541$$
$$x_{58} = -51.8362787842316$$
$$x_{59} = -16.2315620435473$$
$$x_{60} = 9.94837673636768$$
$$x_{61} = 58.1194640914112$$
$$x_{62} = 0.523598775598299$$
$$x_{63} = 49.7418836818384$$
$$x_{64} = 18.3259571459405$$
$$x_{65} = -95.8185759344887$$
$$x_{66} = -36.1283155162826$$
$$x_{67} = 86.3937979737193$$
$$x_{68} = 62.3082542961976$$
$$x_{69} = -49.7418836818384$$
$$x_{70} = 97.9129710368819$$
$$x_{71} = 84.2994028713261$$
$$x_{72} = 56.025068989018$$
$$x_{73} = -23.5619449019235$$
$$x_{74} = 66.497044500984$$
$$x_{75} = -82.2050077689329$$
$$x_{76} = -7.85398163397448$$
$$x_{77} = 67.5442420521806$$
$$x_{78} = 80.1106126665397$$
$$x_{79} = -93.7241808320955$$
$$x_{80} = -53.9306738866248$$
$$x_{81} = 29.845130209103$$
$$x_{82} = -73.8274273593601$$
$$x_{83} = -121.998514714404$$
$$x_{84} = 51.8362787842316$$
$$x_{85} = 100.007366139275$$
$$x_{86} = -45.553093477052$$
$$x_{87} = 82.2050077689329$$
$$x_{88} = -2.61799387799149$$
$$x_{89} = -43.4586983746588$$
$$x_{90} = 22.5147473507269$$
$$x_{91} = -84.2994028713261$$
$$x_{92} = -65.4498469497874$$
$$x_{93} = 88.4881930761125$$
$$x_{94} = -58.1194640914112$$
o sea hay que resolver la ecuación:
$$x^{2} \cos{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{5 \pi}{6}$$
$$x_{3} = - \frac{\pi}{2}$$
$$x_{4} = - \frac{\pi}{6}$$
$$x_{5} = \frac{\pi}{6}$$
$$x_{6} = \frac{\pi}{2}$$
$$x_{7} = \frac{5 \pi}{6}$$
Solución numérica
$$x_{1} = -27.7507351067098$$
$$x_{2} = -47.6474885794452$$
$$x_{3} = 27.7507351067098$$
$$x_{4} = 73.8274273593601$$
$$x_{5} = -31.9395253114962$$
$$x_{6} = 34.0339204138894$$
$$x_{7} = -89.5353906273091$$
$$x_{8} = -41.3643032722656$$
$$x_{9} = 42.4115008234622$$
$$x_{10} = -5.75958653158129$$
$$x_{11} = 14.1371669411541$$
$$x_{12} = 31.9395253114962$$
$$x_{13} = -1.5707963267949$$
$$x_{14} = 40.317105721069$$
$$x_{15} = 44.5058959258554$$
$$x_{16} = -67.5442420521806$$
$$x_{17} = 16.2315620435473$$
$$x_{18} = -100.007366139275$$
$$x_{19} = 5.75958653158129$$
$$x_{20} = 89.5353906273091$$
$$x_{21} = -78.0162175641465$$
$$x_{22} = -80.1106126665397$$
$$x_{23} = 36.1283155162826$$
$$x_{24} = -91.6297857297023$$
$$x_{25} = -60.2138591938044$$
$$x_{26} = 95.8185759344887$$
$$x_{27} = -3.66519142918809$$
$$x_{28} = -34.0339204138894$$
$$x_{29} = 71.733032256967$$
$$x_{30} = -69.6386371545737$$
$$x_{31} = 92.6769832808989$$
$$x_{32} = -97.9129710368819$$
$$x_{33} = -12.0427718387609$$
$$x_{34} = 60.2138591938044$$
$$x_{35} = 75.9218224617533$$
$$x_{36} = 78.0162175641465$$
$$x_{37} = -29.845130209103$$
$$x_{38} = 0$$
$$x_{39} = -38.2227106186758$$
$$x_{40} = -75.9218224617533$$
$$x_{41} = 23.5619449019235$$
$$x_{42} = 20.4203522483337$$
$$x_{43} = -25.6563400043166$$
$$x_{44} = 3.66519142918809$$
$$x_{45} = -87.4409955249159$$
$$x_{46} = 53.9306738866248$$
$$x_{47} = 38.2227106186758$$
$$x_{48} = -32.9867228626928$$
$$x_{49} = -71.733032256967$$
$$x_{50} = 7.85398163397448$$
$$x_{51} = -17.2787595947439$$
$$x_{52} = 12.0427718387609$$
$$x_{53} = -63.3554518473942$$
$$x_{54} = 26.7035375555132$$
$$x_{55} = 64.4026493985908$$
$$x_{56} = -9.94837673636768$$
$$x_{57} = -14.1371669411541$$
$$x_{58} = -51.8362787842316$$
$$x_{59} = -16.2315620435473$$
$$x_{60} = 9.94837673636768$$
$$x_{61} = 58.1194640914112$$
$$x_{62} = 0.523598775598299$$
$$x_{63} = 49.7418836818384$$
$$x_{64} = 18.3259571459405$$
$$x_{65} = -95.8185759344887$$
$$x_{66} = -36.1283155162826$$
$$x_{67} = 86.3937979737193$$
$$x_{68} = 62.3082542961976$$
$$x_{69} = -49.7418836818384$$
$$x_{70} = 97.9129710368819$$
$$x_{71} = 84.2994028713261$$
$$x_{72} = 56.025068989018$$
$$x_{73} = -23.5619449019235$$
$$x_{74} = 66.497044500984$$
$$x_{75} = -82.2050077689329$$
$$x_{76} = -7.85398163397448$$
$$x_{77} = 67.5442420521806$$
$$x_{78} = 80.1106126665397$$
$$x_{79} = -93.7241808320955$$
$$x_{80} = -53.9306738866248$$
$$x_{81} = 29.845130209103$$
$$x_{82} = -73.8274273593601$$
$$x_{83} = -121.998514714404$$
$$x_{84} = 51.8362787842316$$
$$x_{85} = 100.007366139275$$
$$x_{86} = -45.553093477052$$
$$x_{87} = 82.2050077689329$$
$$x_{88} = -2.61799387799149$$
$$x_{89} = -43.4586983746588$$
$$x_{90} = 22.5147473507269$$
$$x_{91} = -84.2994028713261$$
$$x_{92} = -65.4498469497874$$
$$x_{93} = 88.4881930761125$$
$$x_{94} = -58.1194640914112$$
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
$$- 3 x^{2} \sin{\left(3 x \right)} + 2 x \cos{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -57.5997231867839$$
$$x_{2} = -11.5384110184102$$
$$x_{3} = 94.2501373603978$$
$$x_{4} = 65.9768137973912$$
$$x_{5} = 6.31822725550968$$
$$x_{6} = -35.6109562885689$$
$$x_{7} = -4.24076625725554$$
$$x_{8} = 61.7882518935787$$
$$x_{9} = 17.8148265565878$$
$$x_{10} = -72.2597062722654$$
$$x_{11} = 59.6939829539286$$
$$x_{12} = -87.967120448543$$
$$x_{13} = -81.6841294396421$$
$$x_{14} = -2.19277791090745$$
$$x_{15} = -17.8148265565878$$
$$x_{16} = -51.3170101463671$$
$$x_{17} = -90.0614568087362$$
$$x_{18} = -77.4954862683366$$
$$x_{19} = -70.1654029544622$$
$$x_{20} = 37.7050049352483$$
$$x_{21} = -9.44825895659546$$
$$x_{22} = 78.5426455910908$$
$$x_{23} = 52.3641211184374$$
$$x_{24} = -59.6939829539286$$
$$x_{25} = 54.4583530486096$$
$$x_{26} = 35.6109562885689$$
$$x_{27} = -61.7882518935787$$
$$x_{28} = -43.9873487211332$$
$$x_{29} = 50.2699027802066$$
$$x_{30} = -19.9079118108102$$
$$x_{31} = -22.0012459236092$$
$$x_{32} = 72.2597062722654$$
$$x_{33} = -37.7050049352483$$
$$x_{34} = -15.7220892009256$$
$$x_{35} = -26.1884224615332$$
$$x_{36} = -76.4483279927407$$
$$x_{37} = -28.2821897477697$$
$$x_{38} = 100.533175319193$$
$$x_{39} = 92.1557958386718$$
$$x_{40} = 0$$
$$x_{41} = -5.27787047164924$$
$$x_{42} = 48.1756998057147$$
$$x_{43} = -79.5898059196778$$
$$x_{44} = 39.7990900239077$$
$$x_{45} = 8.40396768807019$$
$$x_{46} = 87.967120448543$$
$$x_{47} = -24.0947641678942$$
$$x_{48} = -54.4583530486096$$
$$x_{49} = 32.4699670568908$$
$$x_{50} = -46.0815142886463$$
$$x_{51} = -92.1557958386718$$
$$x_{52} = -39.7990900239077$$
$$x_{53} = 90.0614568087362$$
$$x_{54} = 63.8825291038655$$
$$x_{55} = -65.9768137973912$$
$$x_{56} = -41.8932060932537$$
$$x_{57} = 2.19277791090745$$
$$x_{58} = 19.9079118108102$$
$$x_{59} = -13.6298592553469$$
$$x_{60} = 26.1884224615332$$
$$x_{61} = 24.0947641678942$$
$$x_{62} = 46.0815142886463$$
$$x_{63} = -99.4860010336778$$
$$x_{64} = -94.2501373603978$$
$$x_{65} = 12.5840132115367$$
$$x_{66} = 15.7220892009256$$
$$x_{67} = 56.5525970612491$$
$$x_{68} = -85.8727869534015$$
$$x_{69} = 98.4388272431212$$
$$x_{70} = 4.24076625725554$$
$$x_{71} = 3.20985344776581$$
$$x_{72} = 41.8932060932537$$
$$x_{73} = -30.3760435170464$$
$$x_{74} = -50.2699027802066$$
$$x_{75} = -83.7784565381466$$
$$x_{76} = 76.4483279927407$$
$$x_{77} = -48.1756998057147$$
$$x_{78} = -63.8825291038655$$
$$x_{79} = 68.0711052836269$$
$$x_{80} = -55.5054736300684$$
$$x_{81} = 28.2821897477697$$
$$x_{82} = 10.493124973438$$
$$x_{83} = 70.1654029544622$$
$$x_{84} = 30.3760435170464$$
$$x_{85} = 96.3444812114328$$
$$x_{86} = 22.0012459236092$$
$$x_{87} = -34.5639476991297$$
$$x_{88} = 83.7784565381466$$
$$x_{89} = 43.9873487211332$$
$$x_{90} = 85.8727869534015$$
$$x_{91} = -33.5169509084747$$
$$x_{92} = -68.0711052836269$$
$$x_{93} = 74.3540147599617$$
$$x_{94} = -7.36049192242691$$
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:
$$x_{1} = -57.5997231867839$$
$$x_{2} = -11.5384110184102$$
$$x_{3} = 65.9768137973912$$
$$x_{4} = 61.7882518935787$$
$$x_{5} = 17.8148265565878$$
$$x_{6} = -72.2597062722654$$
$$x_{7} = 59.6939829539286$$
$$x_{8} = -17.8148265565878$$
$$x_{9} = -51.3170101463671$$
$$x_{10} = -70.1654029544622$$
$$x_{11} = -9.44825895659546$$
$$x_{12} = 78.5426455910908$$
$$x_{13} = -59.6939829539286$$
$$x_{14} = -61.7882518935787$$
$$x_{15} = -19.9079118108102$$
$$x_{16} = -22.0012459236092$$
$$x_{17} = 72.2597062722654$$
$$x_{18} = -15.7220892009256$$
$$x_{19} = -26.1884224615332$$
$$x_{20} = -76.4483279927407$$
$$x_{21} = -28.2821897477697$$
$$x_{22} = 0$$
$$x_{23} = -5.27787047164924$$
$$x_{24} = -24.0947641678942$$
$$x_{25} = 32.4699670568908$$
$$x_{26} = 63.8825291038655$$
$$x_{27} = -65.9768137973912$$
$$x_{28} = 19.9079118108102$$
$$x_{29} = -13.6298592553469$$
$$x_{30} = 26.1884224615332$$
$$x_{31} = 24.0947641678942$$
$$x_{32} = -99.4860010336778$$
$$x_{33} = 15.7220892009256$$
$$x_{34} = 3.20985344776581$$
$$x_{35} = -30.3760435170464$$
$$x_{36} = 76.4483279927407$$
$$x_{37} = -63.8825291038655$$
$$x_{38} = 68.0711052836269$$
$$x_{39} = -55.5054736300684$$
$$x_{40} = 28.2821897477697$$
$$x_{41} = 70.1654029544622$$
$$x_{42} = 30.3760435170464$$
$$x_{43} = 22.0012459236092$$
$$x_{44} = -34.5639476991297$$
$$x_{45} = -68.0711052836269$$
$$x_{46} = 74.3540147599617$$
$$x_{47} = -7.36049192242691$$
Puntos máximos de la función:
$$x_{47} = 94.2501373603978$$
$$x_{47} = 6.31822725550968$$
$$x_{47} = -35.6109562885689$$
$$x_{47} = -4.24076625725554$$
$$x_{47} = -87.967120448543$$
$$x_{47} = -81.6841294396421$$
$$x_{47} = -2.19277791090745$$
$$x_{47} = -90.0614568087362$$
$$x_{47} = -77.4954862683366$$
$$x_{47} = 37.7050049352483$$
$$x_{47} = 52.3641211184374$$
$$x_{47} = 54.4583530486096$$
$$x_{47} = 35.6109562885689$$
$$x_{47} = -43.9873487211332$$
$$x_{47} = 50.2699027802066$$
$$x_{47} = -37.7050049352483$$
$$x_{47} = 100.533175319193$$
$$x_{47} = 92.1557958386718$$
$$x_{47} = 48.1756998057147$$
$$x_{47} = -79.5898059196778$$
$$x_{47} = 39.7990900239077$$
$$x_{47} = 8.40396768807019$$
$$x_{47} = 87.967120448543$$
$$x_{47} = -54.4583530486096$$
$$x_{47} = -46.0815142886463$$
$$x_{47} = -92.1557958386718$$
$$x_{47} = -39.7990900239077$$
$$x_{47} = 90.0614568087362$$
$$x_{47} = -41.8932060932537$$
$$x_{47} = 2.19277791090745$$
$$x_{47} = 46.0815142886463$$
$$x_{47} = -94.2501373603978$$
$$x_{47} = 12.5840132115367$$
$$x_{47} = 56.5525970612491$$
$$x_{47} = -85.8727869534015$$
$$x_{47} = 98.4388272431212$$
$$x_{47} = 4.24076625725554$$
$$x_{47} = 41.8932060932537$$
$$x_{47} = -50.2699027802066$$
$$x_{47} = -83.7784565381466$$
$$x_{47} = -48.1756998057147$$
$$x_{47} = 10.493124973438$$
$$x_{47} = 96.3444812114328$$
$$x_{47} = 83.7784565381466$$
$$x_{47} = 43.9873487211332$$
$$x_{47} = 85.8727869534015$$
$$x_{47} = -33.5169509084747$$
Decrece en los intervalos
$$\left[78.5426455910908, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.4860010336778\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- 3 x^{2} \sin{\left(3 x \right)} + 2 x \cos{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -57.5997231867839$$
$$x_{2} = -11.5384110184102$$
$$x_{3} = 94.2501373603978$$
$$x_{4} = 65.9768137973912$$
$$x_{5} = 6.31822725550968$$
$$x_{6} = -35.6109562885689$$
$$x_{7} = -4.24076625725554$$
$$x_{8} = 61.7882518935787$$
$$x_{9} = 17.8148265565878$$
$$x_{10} = -72.2597062722654$$
$$x_{11} = 59.6939829539286$$
$$x_{12} = -87.967120448543$$
$$x_{13} = -81.6841294396421$$
$$x_{14} = -2.19277791090745$$
$$x_{15} = -17.8148265565878$$
$$x_{16} = -51.3170101463671$$
$$x_{17} = -90.0614568087362$$
$$x_{18} = -77.4954862683366$$
$$x_{19} = -70.1654029544622$$
$$x_{20} = 37.7050049352483$$
$$x_{21} = -9.44825895659546$$
$$x_{22} = 78.5426455910908$$
$$x_{23} = 52.3641211184374$$
$$x_{24} = -59.6939829539286$$
$$x_{25} = 54.4583530486096$$
$$x_{26} = 35.6109562885689$$
$$x_{27} = -61.7882518935787$$
$$x_{28} = -43.9873487211332$$
$$x_{29} = 50.2699027802066$$
$$x_{30} = -19.9079118108102$$
$$x_{31} = -22.0012459236092$$
$$x_{32} = 72.2597062722654$$
$$x_{33} = -37.7050049352483$$
$$x_{34} = -15.7220892009256$$
$$x_{35} = -26.1884224615332$$
$$x_{36} = -76.4483279927407$$
$$x_{37} = -28.2821897477697$$
$$x_{38} = 100.533175319193$$
$$x_{39} = 92.1557958386718$$
$$x_{40} = 0$$
$$x_{41} = -5.27787047164924$$
$$x_{42} = 48.1756998057147$$
$$x_{43} = -79.5898059196778$$
$$x_{44} = 39.7990900239077$$
$$x_{45} = 8.40396768807019$$
$$x_{46} = 87.967120448543$$
$$x_{47} = -24.0947641678942$$
$$x_{48} = -54.4583530486096$$
$$x_{49} = 32.4699670568908$$
$$x_{50} = -46.0815142886463$$
$$x_{51} = -92.1557958386718$$
$$x_{52} = -39.7990900239077$$
$$x_{53} = 90.0614568087362$$
$$x_{54} = 63.8825291038655$$
$$x_{55} = -65.9768137973912$$
$$x_{56} = -41.8932060932537$$
$$x_{57} = 2.19277791090745$$
$$x_{58} = 19.9079118108102$$
$$x_{59} = -13.6298592553469$$
$$x_{60} = 26.1884224615332$$
$$x_{61} = 24.0947641678942$$
$$x_{62} = 46.0815142886463$$
$$x_{63} = -99.4860010336778$$
$$x_{64} = -94.2501373603978$$
$$x_{65} = 12.5840132115367$$
$$x_{66} = 15.7220892009256$$
$$x_{67} = 56.5525970612491$$
$$x_{68} = -85.8727869534015$$
$$x_{69} = 98.4388272431212$$
$$x_{70} = 4.24076625725554$$
$$x_{71} = 3.20985344776581$$
$$x_{72} = 41.8932060932537$$
$$x_{73} = -30.3760435170464$$
$$x_{74} = -50.2699027802066$$
$$x_{75} = -83.7784565381466$$
$$x_{76} = 76.4483279927407$$
$$x_{77} = -48.1756998057147$$
$$x_{78} = -63.8825291038655$$
$$x_{79} = 68.0711052836269$$
$$x_{80} = -55.5054736300684$$
$$x_{81} = 28.2821897477697$$
$$x_{82} = 10.493124973438$$
$$x_{83} = 70.1654029544622$$
$$x_{84} = 30.3760435170464$$
$$x_{85} = 96.3444812114328$$
$$x_{86} = 22.0012459236092$$
$$x_{87} = -34.5639476991297$$
$$x_{88} = 83.7784565381466$$
$$x_{89} = 43.9873487211332$$
$$x_{90} = 85.8727869534015$$
$$x_{91} = -33.5169509084747$$
$$x_{92} = -68.0711052836269$$
$$x_{93} = 74.3540147599617$$
$$x_{94} = -7.36049192242691$$
Signos de extremos en los puntos:
(-57.5997231867839, -3317.50591129616)
(-11.538411018410194, -132.913261447715)
(94.25013736039777, 8882.86617857006)
(65.97681379739119, -4352.71775364898)
(6.318227255509681, 39.6996119436951)
(-35.61095628856888, 1267.91804395867)
(-4.240766257255545, 17.7659120606034)
(61.788251893578696, -3817.56586924258)
(17.81482655658785, -317.146056148211)
(-72.25970627226545, -5221.2429425173)
(59.69398295392857, -3563.14939946717)
(-87.96712044854304, 7737.9920673583)
(-81.68412943964212, 6672.0747911911)
(-2.1927779109074463, 4.60036024565405)
(-17.81482655658785, -317.146056148211)
(-51.317010146367075, -2633.21333626447)
(-90.0614568087362, 8110.84378942169)
(-77.49548626833665, 6005.32818207724)
(-70.16540295446222, -4922.96156458467)
(37.70500493524829, 1421.44522703497)
(-9.448258956595456, -89.0482014403384)
(78.5426455910908, -6168.72496623232)
(52.364121118437446, 2741.77898529512)
(-59.69398295392857, -3563.14939946717)
(54.45835304860955, 2965.49001951849)
(35.61095628856888, 1267.91804395867)
(-61.788251893578696, -3817.56586924258)
(-43.987348721133195, 1934.66466356844)
(50.2699027802066, 2526.84093261722)
(-19.907911810810152, -396.102917172654)
(-22.001245923609222, -483.832752880189)
(72.25970627226545, -5221.2429425173)
(-37.70500493524829, 1421.44522703497)
(-15.722089200925566, -246.962165843019)
(-26.18842246153318, -685.611356749139)
(-76.44832799274067, -5844.12464333712)
(-28.282189747769714, -799.660127269992)
(100.53317531919265, 10106.6971248661)
(92.1557958386718, 8492.46849315845)
(0, 0)
(-5.27787047164924, -27.6363188099095)
(48.17569980571475, 2320.67586145916)
(-79.58980591967777, 6334.31499580275)
(39.79909002390769, 1583.74539126285)
(8.403967688070194, 70.4054940215946)
(87.96712044854304, 7737.9920673583)
(-24.094764167894162, -580.335565594115)
(-54.45835304860955, 2965.49001951849)
(32.46996705689075, -1054.07660868777)
(-46.08151428864634, 2123.28377178932)
(-92.1557958386718, 8492.46849315845)
(-39.79909002390769, 1583.74539126285)
(90.0614568087362, 8110.84378942169)
(63.88252910386553, -4080.75532063342)
(-65.97681379739119, -4352.71775364898)
(-41.89320609325366, 1754.81853674717)
(2.1927779109074463, 4.60036024565405)
(19.907911810810152, -396.102917172654)
(-13.629859255346949, -185.551239039262)
(26.18842246153318, -685.611356749139)
(24.094764167894162, -580.335565594115)
(46.08151428864634, 2123.28377178932)
(-99.48600103367775, -9897.24218693458)
(-94.25013736039777, 8882.86617857006)
(12.58401321153674, 158.135632959759)
(15.722089200925566, -246.962165843019)
(56.55259706124914, 3197.9740353083)
(-85.87278695340147, 7373.91332696667)
(98.4388272431212, 9689.98049442277)
(4.240766257255545, 17.7659120606034)
(3.20985344776581, -10.0878773357935)
(41.89320609325366, 1754.81853674717)
(-30.376043517046433, -922.481877774411)
(-50.2699027802066, 2526.84093261722)
(-83.7784565381466, 7018.60756824496)
(76.44832799274067, -5844.12464333712)
(-48.17569980571475, 2320.67586145916)
(-63.88252910386553, -4080.75532063342)
(68.07110528362693, -4633.45316829715)
(-55.50547363006835, -3080.63540471643)
(28.282189747769714, -799.660127269992)
(10.493124973438015, 109.884119985328)
(70.16540295446222, -4922.96156458467)
(30.376043517046433, -922.481877774411)
(96.34448121143284, 9282.03684565777)
(22.001245923609222, -483.832752880189)
(-34.56394769912969, -1194.44432031071)
(83.7784565381466, 7018.60756824496)
(43.987348721133195, 1934.66466356844)
(85.87278695340147, 7373.91332696667)
(-33.51695090847467, 1123.16384189537)
(-68.07110528362693, -4633.45316829715)
(74.3540147599617, -5528.29730210002)
(-7.360491922426915, -53.9559771019712)
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:
$$x_{1} = -57.5997231867839$$
$$x_{2} = -11.5384110184102$$
$$x_{3} = 65.9768137973912$$
$$x_{4} = 61.7882518935787$$
$$x_{5} = 17.8148265565878$$
$$x_{6} = -72.2597062722654$$
$$x_{7} = 59.6939829539286$$
$$x_{8} = -17.8148265565878$$
$$x_{9} = -51.3170101463671$$
$$x_{10} = -70.1654029544622$$
$$x_{11} = -9.44825895659546$$
$$x_{12} = 78.5426455910908$$
$$x_{13} = -59.6939829539286$$
$$x_{14} = -61.7882518935787$$
$$x_{15} = -19.9079118108102$$
$$x_{16} = -22.0012459236092$$
$$x_{17} = 72.2597062722654$$
$$x_{18} = -15.7220892009256$$
$$x_{19} = -26.1884224615332$$
$$x_{20} = -76.4483279927407$$
$$x_{21} = -28.2821897477697$$
$$x_{22} = 0$$
$$x_{23} = -5.27787047164924$$
$$x_{24} = -24.0947641678942$$
$$x_{25} = 32.4699670568908$$
$$x_{26} = 63.8825291038655$$
$$x_{27} = -65.9768137973912$$
$$x_{28} = 19.9079118108102$$
$$x_{29} = -13.6298592553469$$
$$x_{30} = 26.1884224615332$$
$$x_{31} = 24.0947641678942$$
$$x_{32} = -99.4860010336778$$
$$x_{33} = 15.7220892009256$$
$$x_{34} = 3.20985344776581$$
$$x_{35} = -30.3760435170464$$
$$x_{36} = 76.4483279927407$$
$$x_{37} = -63.8825291038655$$
$$x_{38} = 68.0711052836269$$
$$x_{39} = -55.5054736300684$$
$$x_{40} = 28.2821897477697$$
$$x_{41} = 70.1654029544622$$
$$x_{42} = 30.3760435170464$$
$$x_{43} = 22.0012459236092$$
$$x_{44} = -34.5639476991297$$
$$x_{45} = -68.0711052836269$$
$$x_{46} = 74.3540147599617$$
$$x_{47} = -7.36049192242691$$
Puntos máximos de la función:
$$x_{47} = 94.2501373603978$$
$$x_{47} = 6.31822725550968$$
$$x_{47} = -35.6109562885689$$
$$x_{47} = -4.24076625725554$$
$$x_{47} = -87.967120448543$$
$$x_{47} = -81.6841294396421$$
$$x_{47} = -2.19277791090745$$
$$x_{47} = -90.0614568087362$$
$$x_{47} = -77.4954862683366$$
$$x_{47} = 37.7050049352483$$
$$x_{47} = 52.3641211184374$$
$$x_{47} = 54.4583530486096$$
$$x_{47} = 35.6109562885689$$
$$x_{47} = -43.9873487211332$$
$$x_{47} = 50.2699027802066$$
$$x_{47} = -37.7050049352483$$
$$x_{47} = 100.533175319193$$
$$x_{47} = 92.1557958386718$$
$$x_{47} = 48.1756998057147$$
$$x_{47} = -79.5898059196778$$
$$x_{47} = 39.7990900239077$$
$$x_{47} = 8.40396768807019$$
$$x_{47} = 87.967120448543$$
$$x_{47} = -54.4583530486096$$
$$x_{47} = -46.0815142886463$$
$$x_{47} = -92.1557958386718$$
$$x_{47} = -39.7990900239077$$
$$x_{47} = 90.0614568087362$$
$$x_{47} = -41.8932060932537$$
$$x_{47} = 2.19277791090745$$
$$x_{47} = 46.0815142886463$$
$$x_{47} = -94.2501373603978$$
$$x_{47} = 12.5840132115367$$
$$x_{47} = 56.5525970612491$$
$$x_{47} = -85.8727869534015$$
$$x_{47} = 98.4388272431212$$
$$x_{47} = 4.24076625725554$$
$$x_{47} = 41.8932060932537$$
$$x_{47} = -50.2699027802066$$
$$x_{47} = -83.7784565381466$$
$$x_{47} = -48.1756998057147$$
$$x_{47} = 10.493124973438$$
$$x_{47} = 96.3444812114328$$
$$x_{47} = 83.7784565381466$$
$$x_{47} = 43.9873487211332$$
$$x_{47} = 85.8727869534015$$
$$x_{47} = -33.5169509084747$$
Decrece en los intervalos
$$\left[78.5426455910908, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.4860010336778\right]$$
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
$$- 9 x^{2} \cos{\left(3 x \right)} - 12 x \sin{\left(3 x \right)} + 2 \cos{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 17.3044117895173$$
$$x_{2} = -84.3046744803925$$
$$x_{3} = -16.2588593807667$$
$$x_{4} = 97.9175098295664$$
$$x_{5} = 31.95342940113$$
$$x_{6} = -82.2104136545356$$
$$x_{7} = 47.6568129963247$$
$$x_{8} = -29.8600083023031$$
$$x_{9} = 3.77988002313343$$
$$x_{10} = -78.021913622938$$
$$x_{11} = 14.1684778276527$$
$$x_{12} = -49.7508157556872$$
$$x_{13} = 90.5874942021338$$
$$x_{14} = -8.95060704504339$$
$$x_{15} = -25.6736416013905$$
$$x_{16} = 82.2104136545356$$
$$x_{17} = -93.7289224390323$$
$$x_{18} = -51.8448501894803$$
$$x_{19} = 0.199913807009275$$
$$x_{20} = 56.0329998874158$$
$$x_{21} = 88.493215194763$$
$$x_{22} = -65.4566362695214$$
$$x_{23} = -5.83494568158038$$
$$x_{24} = 42.4219754185456$$
$$x_{25} = 73.8334465048004$$
$$x_{26} = -56.0329998874158$$
$$x_{27} = 39.2812198811439$$
$$x_{28} = 78.021913622938$$
$$x_{29} = -63.3624655178403$$
$$x_{30} = -21.4882164075572$$
$$x_{31} = 36.1406096777871$$
$$x_{32} = 12.0794723044888$$
$$x_{33} = 16.2588593807667$$
$$x_{34} = 51.8448501894803$$
$$x_{35} = -9.99268998886934$$
$$x_{36} = -69.6450182357207$$
$$x_{37} = 66.5037269419935$$
$$x_{38} = -97.9175098295664$$
$$x_{39} = -27.7667337891131$$
$$x_{40} = -89.5403540197371$$
$$x_{41} = -71.7392270890376$$
$$x_{42} = -91.6346356954313$$
$$x_{43} = 29.8600083023031$$
$$x_{44} = 1.79524321800934$$
$$x_{45} = -40.3281239196866$$
$$x_{46} = -58.1271093320215$$
$$x_{47} = 100.011809894359$$
$$x_{48} = 60.2212386345931$$
$$x_{49} = -3.77988002313343$$
$$x_{50} = 7.90984185417305$$
$$x_{51} = 22.5344558708549$$
$$x_{52} = -2.77132823698164$$
$$x_{53} = 95.8232139183403$$
$$x_{54} = 84.3046744803925$$
$$x_{55} = -38.234331899335$$
$$x_{56} = -53.938912611874$$
$$x_{57} = -1.79524321800934$$
$$x_{58} = -12.0794723044888$$
$$x_{59} = 64.4095490707657$$
$$x_{60} = 38.234331899335$$
$$x_{61} = 58.1271093320215$$
$$x_{62} = -43.4689207900395$$
$$x_{63} = 71.7392270890376$$
$$x_{64} = 62.3153857944182$$
$$x_{65} = 27.7667337891131$$
$$x_{66} = -47.6568129963247$$
$$x_{67} = 93.7289224390323$$
$$x_{68} = 40.3281239196866$$
$$x_{69} = -67.5508209267318$$
$$x_{70} = -60.2212386345931$$
$$x_{71} = -45.5628462738552$$
$$x_{72} = -14.1684778276527$$
$$x_{73} = 75.9276756093914$$
$$x_{74} = 9.99268998886934$$
$$x_{75} = -23.5807801067948$$
$$x_{76} = -75.9276756093914$$
$$x_{77} = -4.80347720658609$$
$$x_{78} = -34.0469701077371$$
$$x_{79} = 20.4420746646229$$
$$x_{80} = -31.95342940113$$
$$x_{81} = 53.938912611874$$
$$x_{82} = -100.011809894359$$
$$x_{83} = 86.3989418144419$$
$$x_{84} = 49.7508157556872$$
$$x_{85} = -87.4460777759607$$
$$x_{86} = 5.83494568158038$$
$$x_{87} = -80.1161598470679$$
$$x_{88} = -7.90984185417305$$
$$x_{89} = 44.5158780138332$$
$$x_{90} = 80.1161598470679$$
$$x_{91} = -73.8334465048004$$
$$x_{92} = -36.1406096777871$$
$$x_{93} = 24.6271783553447$$
$$x_{94} = 34.0469701077371$$
$$x_{95} = 18.3501507736041$$
$$x_{96} = -95.8232139183403$$
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
$$\left[90.5874942021338, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.011809894359\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 9 x^{2} \cos{\left(3 x \right)} - 12 x \sin{\left(3 x \right)} + 2 \cos{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 17.3044117895173$$
$$x_{2} = -84.3046744803925$$
$$x_{3} = -16.2588593807667$$
$$x_{4} = 97.9175098295664$$
$$x_{5} = 31.95342940113$$
$$x_{6} = -82.2104136545356$$
$$x_{7} = 47.6568129963247$$
$$x_{8} = -29.8600083023031$$
$$x_{9} = 3.77988002313343$$
$$x_{10} = -78.021913622938$$
$$x_{11} = 14.1684778276527$$
$$x_{12} = -49.7508157556872$$
$$x_{13} = 90.5874942021338$$
$$x_{14} = -8.95060704504339$$
$$x_{15} = -25.6736416013905$$
$$x_{16} = 82.2104136545356$$
$$x_{17} = -93.7289224390323$$
$$x_{18} = -51.8448501894803$$
$$x_{19} = 0.199913807009275$$
$$x_{20} = 56.0329998874158$$
$$x_{21} = 88.493215194763$$
$$x_{22} = -65.4566362695214$$
$$x_{23} = -5.83494568158038$$
$$x_{24} = 42.4219754185456$$
$$x_{25} = 73.8334465048004$$
$$x_{26} = -56.0329998874158$$
$$x_{27} = 39.2812198811439$$
$$x_{28} = 78.021913622938$$
$$x_{29} = -63.3624655178403$$
$$x_{30} = -21.4882164075572$$
$$x_{31} = 36.1406096777871$$
$$x_{32} = 12.0794723044888$$
$$x_{33} = 16.2588593807667$$
$$x_{34} = 51.8448501894803$$
$$x_{35} = -9.99268998886934$$
$$x_{36} = -69.6450182357207$$
$$x_{37} = 66.5037269419935$$
$$x_{38} = -97.9175098295664$$
$$x_{39} = -27.7667337891131$$
$$x_{40} = -89.5403540197371$$
$$x_{41} = -71.7392270890376$$
$$x_{42} = -91.6346356954313$$
$$x_{43} = 29.8600083023031$$
$$x_{44} = 1.79524321800934$$
$$x_{45} = -40.3281239196866$$
$$x_{46} = -58.1271093320215$$
$$x_{47} = 100.011809894359$$
$$x_{48} = 60.2212386345931$$
$$x_{49} = -3.77988002313343$$
$$x_{50} = 7.90984185417305$$
$$x_{51} = 22.5344558708549$$
$$x_{52} = -2.77132823698164$$
$$x_{53} = 95.8232139183403$$
$$x_{54} = 84.3046744803925$$
$$x_{55} = -38.234331899335$$
$$x_{56} = -53.938912611874$$
$$x_{57} = -1.79524321800934$$
$$x_{58} = -12.0794723044888$$
$$x_{59} = 64.4095490707657$$
$$x_{60} = 38.234331899335$$
$$x_{61} = 58.1271093320215$$
$$x_{62} = -43.4689207900395$$
$$x_{63} = 71.7392270890376$$
$$x_{64} = 62.3153857944182$$
$$x_{65} = 27.7667337891131$$
$$x_{66} = -47.6568129963247$$
$$x_{67} = 93.7289224390323$$
$$x_{68} = 40.3281239196866$$
$$x_{69} = -67.5508209267318$$
$$x_{70} = -60.2212386345931$$
$$x_{71} = -45.5628462738552$$
$$x_{72} = -14.1684778276527$$
$$x_{73} = 75.9276756093914$$
$$x_{74} = 9.99268998886934$$
$$x_{75} = -23.5807801067948$$
$$x_{76} = -75.9276756093914$$
$$x_{77} = -4.80347720658609$$
$$x_{78} = -34.0469701077371$$
$$x_{79} = 20.4420746646229$$
$$x_{80} = -31.95342940113$$
$$x_{81} = 53.938912611874$$
$$x_{82} = -100.011809894359$$
$$x_{83} = 86.3989418144419$$
$$x_{84} = 49.7508157556872$$
$$x_{85} = -87.4460777759607$$
$$x_{86} = 5.83494568158038$$
$$x_{87} = -80.1161598470679$$
$$x_{88} = -7.90984185417305$$
$$x_{89} = 44.5158780138332$$
$$x_{90} = 80.1161598470679$$
$$x_{91} = -73.8334465048004$$
$$x_{92} = -36.1406096777871$$
$$x_{93} = 24.6271783553447$$
$$x_{94} = 34.0469701077371$$
$$x_{95} = 18.3501507736041$$
$$x_{96} = -95.8232139183403$$
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
$$\left[90.5874942021338, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.011809894359\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(x^{2} \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(x^{2} \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(x^{2} \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(x^{2} \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
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 x^2*cos(3*x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(x \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle x$$
$$\lim_{x \to \infty}\left(x \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle x$$
$$\lim_{x \to -\infty}\left(x \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle x$$
$$\lim_{x \to \infty}\left(x \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle x$$
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:
$$x^{2} \cos{\left(3 x \right)} = x^{2} \cos{\left(3 x \right)}$$
- Sí
$$x^{2} \cos{\left(3 x \right)} = - x^{2} \cos{\left(3 x \right)}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$x^{2} \cos{\left(3 x \right)} = x^{2} \cos{\left(3 x \right)}$$
- Sí
$$x^{2} \cos{\left(3 x \right)} = - x^{2} \cos{\left(3 x \right)}$$
- No
es decir, función
es
par
Gráfico