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 =:
-
1/(x^2+4)
-
x^2*exp(-x)
-
-cos(x)-sin(x)
-
-exp(x)-exp(-x)-exp(2*x)
- Derivada de:
-
x*sin(3*x)
- Integral de d{x}:
- x*sin(3*x)
- Límite de la función:
-
x*sin(3*x)
-
Expresiones idénticas
- x*sin(tres *x)
- x multiplicar por seno de (3 multiplicar por x)
- x multiplicar por seno de (tres multiplicar por x)
- xsin(3x)
- xsin3x
-
Expresiones semejantes
- sinx*sin3x
- -cos(x)-sin(x)-3*cos(3*x)*exp(2*x)/160-exp(2*x)*sin(3*x)/160
- (sinx)(sin3x)
- sin(x)*sin(3*x)
- y=sinx*sin3x
-
Expresiones con funciones
- Seno sin
- sin(1/x)
- sin(x)*cos(x)
- sin(x)/cos(x)
- sin(2*x)/x
- sin(x)-cos(x)
Gráfico de la función y = x*sin(3*x)
Solución
Ha introducido
[src]
f(x) = x*sin(3*x)
$$f{\left(x \right)} = x \sin{\left(3 x \right)}$$
f = x*sin(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:
$$x \sin{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{2 \pi}{3}$$
$$x_{3} = - \frac{\pi}{3}$$
$$x_{4} = \frac{\pi}{3}$$
$$x_{5} = \frac{2 \pi}{3}$$
$$x_{6} = \pi$$
Solución numérica
$$x_{1} = -39.7935069454707$$
$$x_{2} = -26.1799387799149$$
$$x_{3} = 48.1710873550435$$
$$x_{4} = 32.4631240870945$$
$$x_{5} = -41.8879020478639$$
$$x_{6} = 90.0589894029074$$
$$x_{7} = 4.18879020478639$$
$$x_{8} = -53.4070751110265$$
$$x_{9} = 26.1799387799149$$
$$x_{10} = 52.3598775598299$$
$$x_{11} = -1.0471975511966$$
$$x_{12} = 28.2743338823081$$
$$x_{13} = 19.8967534727354$$
$$x_{14} = 50.2654824574367$$
$$x_{15} = 8.37758040957278$$
$$x_{16} = -2.0943951023932$$
$$x_{17} = -86.9173967493176$$
$$x_{18} = 24.0855436775217$$
$$x_{19} = -11.5191730631626$$
$$x_{20} = -90.0589894029074$$
$$x_{21} = -33.5103216382911$$
$$x_{22} = 68.0678408277789$$
$$x_{23} = 65.9734457253857$$
$$x_{24} = 0$$
$$x_{25} = 3.14159265358979$$
$$x_{26} = -24.0855436775217$$
$$x_{27} = -30.3687289847013$$
$$x_{28} = -68.0678408277789$$
$$x_{29} = 78.5398163397448$$
$$x_{30} = 98.4365698124802$$
$$x_{31} = -70.162235930172$$
$$x_{32} = 59.6902604182061$$
$$x_{33} = 17.8023583703422$$
$$x_{34} = 100.530964914873$$
$$x_{35} = -85.870199198121$$
$$x_{36} = -72.2566310325652$$
$$x_{37} = 21.9911485751286$$
$$x_{38} = -79.5870138909414$$
$$x_{39} = -37.6991118430775$$
$$x_{40} = -81.6814089933346$$
$$x_{41} = -21.9911485751286$$
$$x_{42} = -46.0766922526503$$
$$x_{43} = -13.6135681655558$$
$$x_{44} = -4.18879020478639$$
$$x_{45} = -87.9645943005142$$
$$x_{46} = -77.4926187885482$$
$$x_{47} = 41.8879020478639$$
$$x_{48} = -99.4837673636768$$
$$x_{49} = 1.0471975511966$$
$$x_{50} = -94.2477796076938$$
$$x_{51} = 15.707963267949$$
$$x_{52} = 30.3687289847013$$
$$x_{53} = -43.9822971502571$$
$$x_{54} = 39.7935069454707$$
$$x_{55} = -6.28318530717959$$
$$x_{56} = -8.37758040957278$$
$$x_{57} = -83.7758040957278$$
$$x_{58} = -28.2743338823081$$
$$x_{59} = -95.2949771588904$$
$$x_{60} = -17.8023583703422$$
$$x_{61} = 94.2477796076938$$
$$x_{62} = -48.1710873550435$$
$$x_{63} = 46.0766922526503$$
$$x_{64} = -59.6902604182061$$
$$x_{65} = 87.9645943005142$$
$$x_{66} = -63.8790506229925$$
$$x_{67} = -61.7846555205993$$
$$x_{68} = 63.8790506229925$$
$$x_{69} = -92.1533845053006$$
$$x_{70} = 2.0943951023932$$
$$x_{71} = 74.3510261349584$$
$$x_{72} = -15.707963267949$$
$$x_{73} = 83.7758040957278$$
$$x_{74} = 96.342174710087$$
$$x_{75} = 6.28318530717959$$
$$x_{76} = 13.6135681655558$$
$$x_{77} = -50.2654824574367$$
$$x_{78} = 54.4542726622231$$
$$x_{79} = -57.5958653158129$$
$$x_{80} = 37.6991118430775$$
$$x_{81} = -19.8967534727354$$
$$x_{82} = 43.9822971502571$$
$$x_{83} = 56.5486677646163$$
$$x_{84} = -65.9734457253857$$
$$x_{85} = 58.6430628670095$$
$$x_{86} = -55.5014702134197$$
$$x_{87} = 76.4454212373516$$
$$x_{88} = 75.398223686155$$
$$x_{89} = 92.1533845053006$$
$$x_{90} = 70.162235930172$$
$$x_{91} = -35.6047167406843$$
$$x_{92} = 72.2566310325652$$
$$x_{93} = 10.471975511966$$
o sea hay que resolver la ecuación:
$$x \sin{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{2 \pi}{3}$$
$$x_{3} = - \frac{\pi}{3}$$
$$x_{4} = \frac{\pi}{3}$$
$$x_{5} = \frac{2 \pi}{3}$$
$$x_{6} = \pi$$
Solución numérica
$$x_{1} = -39.7935069454707$$
$$x_{2} = -26.1799387799149$$
$$x_{3} = 48.1710873550435$$
$$x_{4} = 32.4631240870945$$
$$x_{5} = -41.8879020478639$$
$$x_{6} = 90.0589894029074$$
$$x_{7} = 4.18879020478639$$
$$x_{8} = -53.4070751110265$$
$$x_{9} = 26.1799387799149$$
$$x_{10} = 52.3598775598299$$
$$x_{11} = -1.0471975511966$$
$$x_{12} = 28.2743338823081$$
$$x_{13} = 19.8967534727354$$
$$x_{14} = 50.2654824574367$$
$$x_{15} = 8.37758040957278$$
$$x_{16} = -2.0943951023932$$
$$x_{17} = -86.9173967493176$$
$$x_{18} = 24.0855436775217$$
$$x_{19} = -11.5191730631626$$
$$x_{20} = -90.0589894029074$$
$$x_{21} = -33.5103216382911$$
$$x_{22} = 68.0678408277789$$
$$x_{23} = 65.9734457253857$$
$$x_{24} = 0$$
$$x_{25} = 3.14159265358979$$
$$x_{26} = -24.0855436775217$$
$$x_{27} = -30.3687289847013$$
$$x_{28} = -68.0678408277789$$
$$x_{29} = 78.5398163397448$$
$$x_{30} = 98.4365698124802$$
$$x_{31} = -70.162235930172$$
$$x_{32} = 59.6902604182061$$
$$x_{33} = 17.8023583703422$$
$$x_{34} = 100.530964914873$$
$$x_{35} = -85.870199198121$$
$$x_{36} = -72.2566310325652$$
$$x_{37} = 21.9911485751286$$
$$x_{38} = -79.5870138909414$$
$$x_{39} = -37.6991118430775$$
$$x_{40} = -81.6814089933346$$
$$x_{41} = -21.9911485751286$$
$$x_{42} = -46.0766922526503$$
$$x_{43} = -13.6135681655558$$
$$x_{44} = -4.18879020478639$$
$$x_{45} = -87.9645943005142$$
$$x_{46} = -77.4926187885482$$
$$x_{47} = 41.8879020478639$$
$$x_{48} = -99.4837673636768$$
$$x_{49} = 1.0471975511966$$
$$x_{50} = -94.2477796076938$$
$$x_{51} = 15.707963267949$$
$$x_{52} = 30.3687289847013$$
$$x_{53} = -43.9822971502571$$
$$x_{54} = 39.7935069454707$$
$$x_{55} = -6.28318530717959$$
$$x_{56} = -8.37758040957278$$
$$x_{57} = -83.7758040957278$$
$$x_{58} = -28.2743338823081$$
$$x_{59} = -95.2949771588904$$
$$x_{60} = -17.8023583703422$$
$$x_{61} = 94.2477796076938$$
$$x_{62} = -48.1710873550435$$
$$x_{63} = 46.0766922526503$$
$$x_{64} = -59.6902604182061$$
$$x_{65} = 87.9645943005142$$
$$x_{66} = -63.8790506229925$$
$$x_{67} = -61.7846555205993$$
$$x_{68} = 63.8790506229925$$
$$x_{69} = -92.1533845053006$$
$$x_{70} = 2.0943951023932$$
$$x_{71} = 74.3510261349584$$
$$x_{72} = -15.707963267949$$
$$x_{73} = 83.7758040957278$$
$$x_{74} = 96.342174710087$$
$$x_{75} = 6.28318530717959$$
$$x_{76} = 13.6135681655558$$
$$x_{77} = -50.2654824574367$$
$$x_{78} = 54.4542726622231$$
$$x_{79} = -57.5958653158129$$
$$x_{80} = 37.6991118430775$$
$$x_{81} = -19.8967534727354$$
$$x_{82} = 43.9822971502571$$
$$x_{83} = 56.5486677646163$$
$$x_{84} = -65.9734457253857$$
$$x_{85} = 58.6430628670095$$
$$x_{86} = -55.5014702134197$$
$$x_{87} = 76.4454212373516$$
$$x_{88} = 75.398223686155$$
$$x_{89} = 92.1533845053006$$
$$x_{90} = 70.162235930172$$
$$x_{91} = -35.6047167406843$$
$$x_{92} = 72.2566310325652$$
$$x_{93} = 10.471975511966$$
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
$$3 x \cos{\left(3 x \right)} + \sin{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -14.1450206271366$$
$$x_{2} = -47.6498203678514$$
$$x_{3} = -100.008477152089$$
$$x_{4} = 23.5666593462033$$
$$x_{5} = -78.0176417347899$$
$$x_{6} = 93.7253663237826$$
$$x_{7} = 100.008477152089$$
$$x_{8} = 95.8197355146347$$
$$x_{9} = 16.2384035725192$$
$$x_{10} = -87.4422661984441$$
$$x_{11} = 49.7441173016936$$
$$x_{12} = 89.5366315785916$$
$$x_{13} = -75.9232859178705$$
$$x_{14} = 84.3007208972085$$
$$x_{15} = -25.6606697768062$$
$$x_{16} = -12.0519888065122$$
$$x_{17} = 22.5196809462695$$
$$x_{18} = -3.69517946883234$$
$$x_{19} = -84.3007208972085$$
$$x_{20} = -45.5555324591521$$
$$x_{21} = 18.3320175191655$$
$$x_{22} = 66.4987153630436$$
$$x_{23} = 3.69517946883234$$
$$x_{24} = 64.4043745938079$$
$$x_{25} = 82.2063593736386$$
$$x_{26} = 40.3198613996847$$
$$x_{27} = 92.6781821675128$$
$$x_{28} = -65.4515445438056$$
$$x_{29} = 12.0519888065122$$
$$x_{30} = -60.2157043931142$$
$$x_{31} = -53.9327340398572$$
$$x_{32} = -51.8384221669793$$
$$x_{33} = -58.1213757786594$$
$$x_{34} = 51.8384221669793$$
$$x_{35} = 60.2157043931142$$
$$x_{36} = 36.1313906251304$$
$$x_{37} = -29.8488525127497$$
$$x_{38} = 0$$
$$x_{39} = 58.1213757786594$$
$$x_{40} = 29.8488525127497$$
$$x_{41} = -9.95952883536913$$
$$x_{42} = 73.8289323297373$$
$$x_{43} = -56.0270521345864$$
$$x_{44} = -95.8197355146347$$
$$x_{45} = -91.6309983173967$$
$$x_{46} = -69.6402326441114$$
$$x_{47} = 54.9798923539233$$
$$x_{48} = -4.7358122417304$$
$$x_{49} = -27.7547382346962$$
$$x_{50} = -38.225617263561$$
$$x_{51} = -89.5366315785916$$
$$x_{52} = -34.037184713218$$
$$x_{53} = 7.8680949243268$$
$$x_{54} = 78.0176417347899$$
$$x_{55} = 0.676252612703478$$
$$x_{56} = -106.291596785411$$
$$x_{57} = 53.9327340398572$$
$$x_{58} = 62.3100374768166$$
$$x_{59} = 34.037184713218$$
$$x_{60} = -67.5458870110976$$
$$x_{61} = -93.7253663237826$$
$$x_{62} = -36.1313906251304$$
$$x_{63} = -23.5666593462033$$
$$x_{64} = 75.9232859178705$$
$$x_{65} = -97.9141058139495$$
$$x_{66} = -7.8680949243268$$
$$x_{67} = 20.4257915111899$$
$$x_{68} = 27.7547382346962$$
$$x_{69} = 42.4141204421759$$
$$x_{70} = 88.4894487126566$$
$$x_{71} = 67.5458870110976$$
$$x_{72} = -5.77879264132779$$
$$x_{73} = -80.1119996057056$$
$$x_{74} = 38.225617263561$$
$$x_{75} = -71.7345811655909$$
$$x_{76} = -73.8289323297373$$
$$x_{77} = -61.2628704049539$$
$$x_{78} = 97.9141058139495$$
$$x_{79} = 86.3950840487432$$
$$x_{80} = 31.9430036030065$$
$$x_{81} = -16.2384035725192$$
$$x_{82} = 56.0270521345864$$
$$x_{83} = 14.1450206271366$$
$$x_{84} = 9.95952883536913$$
$$x_{85} = -21.4727239072797$$
$$x_{86} = 71.7345811655909$$
$$x_{87} = 26.7076976049501$$
$$x_{88} = 80.1119996057056$$
$$x_{89} = -41.3669891991005$$
$$x_{90} = -49.7441173016936$$
$$x_{91} = 5.77879264132779$$
$$x_{92} = -31.9430036030065$$
$$x_{93} = -1.63772681314496$$
$$x_{94} = -82.2063593736386$$
$$x_{95} = -43.4612548800528$$
$$x_{96} = 44.5083922872857$$
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:
$$x_{1} = -14.1450206271366$$
$$x_{2} = -47.6498203678514$$
$$x_{3} = -100.008477152089$$
$$x_{4} = 93.7253663237826$$
$$x_{5} = 100.008477152089$$
$$x_{6} = 95.8197355146347$$
$$x_{7} = 16.2384035725192$$
$$x_{8} = -87.4422661984441$$
$$x_{9} = 49.7441173016936$$
$$x_{10} = 89.5366315785916$$
$$x_{11} = -12.0519888065122$$
$$x_{12} = 22.5196809462695$$
$$x_{13} = -3.69517946883234$$
$$x_{14} = -45.5555324591521$$
$$x_{15} = 18.3320175191655$$
$$x_{16} = 66.4987153630436$$
$$x_{17} = 3.69517946883234$$
$$x_{18} = 64.4043745938079$$
$$x_{19} = 12.0519888065122$$
$$x_{20} = -60.2157043931142$$
$$x_{21} = -53.9327340398572$$
$$x_{22} = -51.8384221669793$$
$$x_{23} = -58.1213757786594$$
$$x_{24} = 51.8384221669793$$
$$x_{25} = 60.2157043931142$$
$$x_{26} = 0$$
$$x_{27} = 58.1213757786594$$
$$x_{28} = -9.95952883536913$$
$$x_{29} = -56.0270521345864$$
$$x_{30} = -95.8197355146347$$
$$x_{31} = -91.6309983173967$$
$$x_{32} = -89.5366315785916$$
$$x_{33} = 7.8680949243268$$
$$x_{34} = -106.291596785411$$
$$x_{35} = 53.9327340398572$$
$$x_{36} = 62.3100374768166$$
$$x_{37} = -93.7253663237826$$
$$x_{38} = -97.9141058139495$$
$$x_{39} = -7.8680949243268$$
$$x_{40} = 20.4257915111899$$
$$x_{41} = -5.77879264132779$$
$$x_{42} = 97.9141058139495$$
$$x_{43} = -16.2384035725192$$
$$x_{44} = 56.0270521345864$$
$$x_{45} = 14.1450206271366$$
$$x_{46} = 9.95952883536913$$
$$x_{47} = 26.7076976049501$$
$$x_{48} = -41.3669891991005$$
$$x_{49} = -49.7441173016936$$
$$x_{50} = 5.77879264132779$$
$$x_{51} = -1.63772681314496$$
$$x_{52} = -43.4612548800528$$
Puntos máximos de la función:
$$x_{52} = 23.5666593462033$$
$$x_{52} = -78.0176417347899$$
$$x_{52} = -75.9232859178705$$
$$x_{52} = 84.3007208972085$$
$$x_{52} = -25.6606697768062$$
$$x_{52} = -84.3007208972085$$
$$x_{52} = 82.2063593736386$$
$$x_{52} = 40.3198613996847$$
$$x_{52} = 92.6781821675128$$
$$x_{52} = -65.4515445438056$$
$$x_{52} = 36.1313906251304$$
$$x_{52} = -29.8488525127497$$
$$x_{52} = 29.8488525127497$$
$$x_{52} = 73.8289323297373$$
$$x_{52} = -69.6402326441114$$
$$x_{52} = 54.9798923539233$$
$$x_{52} = -4.7358122417304$$
$$x_{52} = -27.7547382346962$$
$$x_{52} = -38.225617263561$$
$$x_{52} = -34.037184713218$$
$$x_{52} = 78.0176417347899$$
$$x_{52} = 0.676252612703478$$
$$x_{52} = 34.037184713218$$
$$x_{52} = -67.5458870110976$$
$$x_{52} = -36.1313906251304$$
$$x_{52} = -23.5666593462033$$
$$x_{52} = 75.9232859178705$$
$$x_{52} = 27.7547382346962$$
$$x_{52} = 42.4141204421759$$
$$x_{52} = 88.4894487126566$$
$$x_{52} = 67.5458870110976$$
$$x_{52} = -80.1119996057056$$
$$x_{52} = 38.225617263561$$
$$x_{52} = -71.7345811655909$$
$$x_{52} = -73.8289323297373$$
$$x_{52} = -61.2628704049539$$
$$x_{52} = 86.3950840487432$$
$$x_{52} = 31.9430036030065$$
$$x_{52} = -21.4727239072797$$
$$x_{52} = 71.7345811655909$$
$$x_{52} = 80.1119996057056$$
$$x_{52} = -31.9430036030065$$
$$x_{52} = -82.2063593736386$$
$$x_{52} = 44.5083922872857$$
Decrece en los intervalos
$$\left[100.008477152089, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -106.291596785411\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$3 x \cos{\left(3 x \right)} + \sin{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -14.1450206271366$$
$$x_{2} = -47.6498203678514$$
$$x_{3} = -100.008477152089$$
$$x_{4} = 23.5666593462033$$
$$x_{5} = -78.0176417347899$$
$$x_{6} = 93.7253663237826$$
$$x_{7} = 100.008477152089$$
$$x_{8} = 95.8197355146347$$
$$x_{9} = 16.2384035725192$$
$$x_{10} = -87.4422661984441$$
$$x_{11} = 49.7441173016936$$
$$x_{12} = 89.5366315785916$$
$$x_{13} = -75.9232859178705$$
$$x_{14} = 84.3007208972085$$
$$x_{15} = -25.6606697768062$$
$$x_{16} = -12.0519888065122$$
$$x_{17} = 22.5196809462695$$
$$x_{18} = -3.69517946883234$$
$$x_{19} = -84.3007208972085$$
$$x_{20} = -45.5555324591521$$
$$x_{21} = 18.3320175191655$$
$$x_{22} = 66.4987153630436$$
$$x_{23} = 3.69517946883234$$
$$x_{24} = 64.4043745938079$$
$$x_{25} = 82.2063593736386$$
$$x_{26} = 40.3198613996847$$
$$x_{27} = 92.6781821675128$$
$$x_{28} = -65.4515445438056$$
$$x_{29} = 12.0519888065122$$
$$x_{30} = -60.2157043931142$$
$$x_{31} = -53.9327340398572$$
$$x_{32} = -51.8384221669793$$
$$x_{33} = -58.1213757786594$$
$$x_{34} = 51.8384221669793$$
$$x_{35} = 60.2157043931142$$
$$x_{36} = 36.1313906251304$$
$$x_{37} = -29.8488525127497$$
$$x_{38} = 0$$
$$x_{39} = 58.1213757786594$$
$$x_{40} = 29.8488525127497$$
$$x_{41} = -9.95952883536913$$
$$x_{42} = 73.8289323297373$$
$$x_{43} = -56.0270521345864$$
$$x_{44} = -95.8197355146347$$
$$x_{45} = -91.6309983173967$$
$$x_{46} = -69.6402326441114$$
$$x_{47} = 54.9798923539233$$
$$x_{48} = -4.7358122417304$$
$$x_{49} = -27.7547382346962$$
$$x_{50} = -38.225617263561$$
$$x_{51} = -89.5366315785916$$
$$x_{52} = -34.037184713218$$
$$x_{53} = 7.8680949243268$$
$$x_{54} = 78.0176417347899$$
$$x_{55} = 0.676252612703478$$
$$x_{56} = -106.291596785411$$
$$x_{57} = 53.9327340398572$$
$$x_{58} = 62.3100374768166$$
$$x_{59} = 34.037184713218$$
$$x_{60} = -67.5458870110976$$
$$x_{61} = -93.7253663237826$$
$$x_{62} = -36.1313906251304$$
$$x_{63} = -23.5666593462033$$
$$x_{64} = 75.9232859178705$$
$$x_{65} = -97.9141058139495$$
$$x_{66} = -7.8680949243268$$
$$x_{67} = 20.4257915111899$$
$$x_{68} = 27.7547382346962$$
$$x_{69} = 42.4141204421759$$
$$x_{70} = 88.4894487126566$$
$$x_{71} = 67.5458870110976$$
$$x_{72} = -5.77879264132779$$
$$x_{73} = -80.1119996057056$$
$$x_{74} = 38.225617263561$$
$$x_{75} = -71.7345811655909$$
$$x_{76} = -73.8289323297373$$
$$x_{77} = -61.2628704049539$$
$$x_{78} = 97.9141058139495$$
$$x_{79} = 86.3950840487432$$
$$x_{80} = 31.9430036030065$$
$$x_{81} = -16.2384035725192$$
$$x_{82} = 56.0270521345864$$
$$x_{83} = 14.1450206271366$$
$$x_{84} = 9.95952883536913$$
$$x_{85} = -21.4727239072797$$
$$x_{86} = 71.7345811655909$$
$$x_{87} = 26.7076976049501$$
$$x_{88} = 80.1119996057056$$
$$x_{89} = -41.3669891991005$$
$$x_{90} = -49.7441173016936$$
$$x_{91} = 5.77879264132779$$
$$x_{92} = -31.9430036030065$$
$$x_{93} = -1.63772681314496$$
$$x_{94} = -82.2063593736386$$
$$x_{95} = -43.4612548800528$$
$$x_{96} = 44.5083922872857$$
Signos de extremos en los puntos:
(-14.145020627136628, -14.1410946924197)
(-47.64982036785143, -47.6486544974204)
(-100.00847715208945, -100.007921648254)
(23.566659346203334, 23.564302320531)
(-78.01764173478989, 78.0169296548843)
(93.7253663237826, -93.724773581063)
(100.00847715208945, -100.007921648254)
(95.81973551463474, -95.8191557274852)
(16.238403572519243, -16.234983408456)
(-87.44226619844414, -87.4416308655269)
(49.74411730169364, -49.7430005126604)
(89.53663157859164, -89.5360111065335)
(-75.92328591787046, 75.9225541956887)
(84.3007208972085, 84.3000618885604)
(-25.66066977680624, 25.6585050427546)
(-12.05198880651224, -12.0473817907474)
(22.519680946269467, -22.5172143736575)
(-3.695179468832341, -3.68023600531)
(-84.3007208972085, 84.3000618885604)
(-45.555532459152055, -45.5543129953057)
(18.332017519165465, -18.3289877498992)
(66.49871536304362, -66.4978799407601)
(3.695179468832341, -3.68023600531)
(64.40437459380786, -64.4035120058269)
(82.20635937363859, 82.2056835759154)
(40.319861399684655, 40.318483599613)
(92.6781821675128, 92.6775827274368)
(-65.45154454380558, 65.4506957559693)
(12.05198880651224, -12.0473817907474)
(-60.21570439311422, -60.2147818052389)
(-53.93273403985717, -53.9317039796355)
(-51.83842216697935, -51.8373504940684)
(-58.121375778659406, -58.1204199481347)
(51.83842216697935, -51.8373504940684)
(60.21570439311422, -60.2147818052389)
(36.131390625130436, 36.1298531252298)
(-29.848852512749733, 29.8469914576284)
(0, 0)
(58.121375778659406, -58.1204199481347)
(29.848852512749733, 29.8469914576284)
(-9.959528835369131, -9.9539553863956)
(73.82893232973727, 73.8281798509399)
(-56.027052134586405, -56.0260605764261)
(-95.81973551463474, -95.8191557274852)
(-91.63099831739673, -91.6303920268926)
(-69.64023264411136, 69.6394349069578)
(54.97989235392331, 54.9788819113479)
(-4.735812241730396, 4.72412470459143)
(-27.754738234696216, 27.7527367909844)
(-38.225617263561, 38.2241639871628)
(-89.53663157859164, -89.5360111065335)
(-34.037184713217975, 34.0355526287721)
(7.868094924326803, -7.86104354987779)
(78.01764173478989, 78.0169296548843)
(0.676252612703478, 0.606568580386551)
(-106.29159678541116, -106.291074118075)
(53.93273403985717, -53.9317039796355)
(62.31003747681657, -62.3091458971384)
(34.037184713217975, 34.0355526287721)
(-67.54588701109756, 67.5450645399848)
(-93.7253663237826, -93.724773581063)
(-36.131390625130436, 36.1298531252298)
(-23.566659346203334, 23.564302320531)
(75.92328591787046, 75.9225541956887)
(-97.91410581394948, -97.9135384281556)
(-7.868094924326803, -7.86104354987779)
(20.425791511189885, -20.4230721814922)
(27.754738234696216, 27.7527367909844)
(42.41412044217588, 42.4128106665257)
(88.4894487126566, 88.4888208980964)
(67.54588701109756, 67.5450645399848)
(-5.778792641327787, -5.76920286928617)
(-80.1119996057056, 80.111306141125)
(38.225617263561, 38.2241639871628)
(-71.73458116559091, 71.7338067182464)
(-73.82893232973727, 73.8281798509399)
(-61.262870404953894, 61.2619635861633)
(97.91410581394948, -97.9135384281556)
(86.39508404874319, 86.3944410152197)
(31.943003603006492, 31.9412645361552)
(-16.238403572519243, -16.234983408456)
(56.027052134586405, -56.0260605764261)
(14.145020627136628, -14.1410946924197)
(9.959528835369131, -9.9539553863956)
(-21.47272390727972, 21.4701371131251)
(71.73458116559091, 71.7338067182464)
(26.707697604950084, -26.7056177152197)
(80.1119996057056, 80.111306141125)
(-41.36698919910045, -41.3656462720145)
(-49.74411730169364, -49.7430005126604)
(5.778792641327787, -5.76920286928617)
(-31.943003603006492, 31.9412645361552)
(-1.6377268131449612, -1.60482329657076)
(-82.20635937363859, 82.2056835759154)
(-43.46125488005277, -43.4599766586844)
(44.50839228728569, 44.5071441357397)
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:
$$x_{1} = -14.1450206271366$$
$$x_{2} = -47.6498203678514$$
$$x_{3} = -100.008477152089$$
$$x_{4} = 93.7253663237826$$
$$x_{5} = 100.008477152089$$
$$x_{6} = 95.8197355146347$$
$$x_{7} = 16.2384035725192$$
$$x_{8} = -87.4422661984441$$
$$x_{9} = 49.7441173016936$$
$$x_{10} = 89.5366315785916$$
$$x_{11} = -12.0519888065122$$
$$x_{12} = 22.5196809462695$$
$$x_{13} = -3.69517946883234$$
$$x_{14} = -45.5555324591521$$
$$x_{15} = 18.3320175191655$$
$$x_{16} = 66.4987153630436$$
$$x_{17} = 3.69517946883234$$
$$x_{18} = 64.4043745938079$$
$$x_{19} = 12.0519888065122$$
$$x_{20} = -60.2157043931142$$
$$x_{21} = -53.9327340398572$$
$$x_{22} = -51.8384221669793$$
$$x_{23} = -58.1213757786594$$
$$x_{24} = 51.8384221669793$$
$$x_{25} = 60.2157043931142$$
$$x_{26} = 0$$
$$x_{27} = 58.1213757786594$$
$$x_{28} = -9.95952883536913$$
$$x_{29} = -56.0270521345864$$
$$x_{30} = -95.8197355146347$$
$$x_{31} = -91.6309983173967$$
$$x_{32} = -89.5366315785916$$
$$x_{33} = 7.8680949243268$$
$$x_{34} = -106.291596785411$$
$$x_{35} = 53.9327340398572$$
$$x_{36} = 62.3100374768166$$
$$x_{37} = -93.7253663237826$$
$$x_{38} = -97.9141058139495$$
$$x_{39} = -7.8680949243268$$
$$x_{40} = 20.4257915111899$$
$$x_{41} = -5.77879264132779$$
$$x_{42} = 97.9141058139495$$
$$x_{43} = -16.2384035725192$$
$$x_{44} = 56.0270521345864$$
$$x_{45} = 14.1450206271366$$
$$x_{46} = 9.95952883536913$$
$$x_{47} = 26.7076976049501$$
$$x_{48} = -41.3669891991005$$
$$x_{49} = -49.7441173016936$$
$$x_{50} = 5.77879264132779$$
$$x_{51} = -1.63772681314496$$
$$x_{52} = -43.4612548800528$$
Puntos máximos de la función:
$$x_{52} = 23.5666593462033$$
$$x_{52} = -78.0176417347899$$
$$x_{52} = -75.9232859178705$$
$$x_{52} = 84.3007208972085$$
$$x_{52} = -25.6606697768062$$
$$x_{52} = -84.3007208972085$$
$$x_{52} = 82.2063593736386$$
$$x_{52} = 40.3198613996847$$
$$x_{52} = 92.6781821675128$$
$$x_{52} = -65.4515445438056$$
$$x_{52} = 36.1313906251304$$
$$x_{52} = -29.8488525127497$$
$$x_{52} = 29.8488525127497$$
$$x_{52} = 73.8289323297373$$
$$x_{52} = -69.6402326441114$$
$$x_{52} = 54.9798923539233$$
$$x_{52} = -4.7358122417304$$
$$x_{52} = -27.7547382346962$$
$$x_{52} = -38.225617263561$$
$$x_{52} = -34.037184713218$$
$$x_{52} = 78.0176417347899$$
$$x_{52} = 0.676252612703478$$
$$x_{52} = 34.037184713218$$
$$x_{52} = -67.5458870110976$$
$$x_{52} = -36.1313906251304$$
$$x_{52} = -23.5666593462033$$
$$x_{52} = 75.9232859178705$$
$$x_{52} = 27.7547382346962$$
$$x_{52} = 42.4141204421759$$
$$x_{52} = 88.4894487126566$$
$$x_{52} = 67.5458870110976$$
$$x_{52} = -80.1119996057056$$
$$x_{52} = 38.225617263561$$
$$x_{52} = -71.7345811655909$$
$$x_{52} = -73.8289323297373$$
$$x_{52} = -61.2628704049539$$
$$x_{52} = 86.3950840487432$$
$$x_{52} = 31.9430036030065$$
$$x_{52} = -21.4727239072797$$
$$x_{52} = 71.7345811655909$$
$$x_{52} = 80.1119996057056$$
$$x_{52} = -31.9430036030065$$
$$x_{52} = -82.2063593736386$$
$$x_{52} = 44.5083922872857$$
Decrece en los intervalos
$$\left[100.008477152089, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -106.291596785411\right]$$
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
$$3 \left(- 3 x \sin{\left(3 x \right)} + 2 \cos{\left(3 x \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -57.5997231867839$$
$$x_{2} = -11.5384110184102$$
$$x_{3} = 94.2501373603978$$
$$x_{4} = 105.769053656653$$
$$x_{5} = 65.9768137973912$$
$$x_{6} = 6.31822725550968$$
$$x_{7} = -35.6109562885689$$
$$x_{8} = -4.24076625725554$$
$$x_{9} = 61.7882518935787$$
$$x_{10} = 17.8148265565878$$
$$x_{11} = -72.2597062722654$$
$$x_{12} = 59.6939829539286$$
$$x_{13} = -87.967120448543$$
$$x_{14} = -81.6841294396421$$
$$x_{15} = -2.19277791090745$$
$$x_{16} = -17.8148265565878$$
$$x_{17} = -90.0614568087362$$
$$x_{18} = -77.4954862683366$$
$$x_{19} = -70.1654029544622$$
$$x_{20} = 37.7050049352483$$
$$x_{21} = -9.44825895659546$$
$$x_{22} = 78.5426455910908$$
$$x_{23} = -0.358957995437268$$
$$x_{24} = 52.3641211184374$$
$$x_{25} = -59.6939829539286$$
$$x_{26} = 54.4583530486096$$
$$x_{27} = -61.7882518935787$$
$$x_{28} = -43.9873487211332$$
$$x_{29} = 50.2699027802066$$
$$x_{30} = -19.9079118108102$$
$$x_{31} = -22.0012459236092$$
$$x_{32} = 72.2597062722654$$
$$x_{33} = -37.7050049352483$$
$$x_{34} = -15.7220892009256$$
$$x_{35} = -26.1884224615332$$
$$x_{36} = 0.358957995437268$$
$$x_{37} = -28.2821897477697$$
$$x_{38} = 100.533175319193$$
$$x_{39} = 92.1557958386718$$
$$x_{40} = 48.1756998057147$$
$$x_{41} = -79.5898059196778$$
$$x_{42} = 39.7990900239077$$
$$x_{43} = 8.40396768807019$$
$$x_{44} = 87.967120448543$$
$$x_{45} = -24.0947641678942$$
$$x_{46} = -3.20985344776581$$
$$x_{47} = 32.4699670568908$$
$$x_{48} = -46.0815142886463$$
$$x_{49} = -92.1557958386718$$
$$x_{50} = -53.4112354844189$$
$$x_{51} = -39.7990900239077$$
$$x_{52} = 90.0614568087362$$
$$x_{53} = 63.8825291038655$$
$$x_{54} = -65.9768137973912$$
$$x_{55} = -41.8932060932537$$
$$x_{56} = 2.19277791090745$$
$$x_{57} = 19.9079118108102$$
$$x_{58} = -13.6298592553469$$
$$x_{59} = 26.1884224615332$$
$$x_{60} = 24.0947641678942$$
$$x_{61} = 46.0815142886463$$
$$x_{62} = -99.4860010336778$$
$$x_{63} = -94.2501373603978$$
$$x_{64} = 12.5840132115367$$
$$x_{65} = 15.7220892009256$$
$$x_{66} = 56.5525970612491$$
$$x_{67} = -85.8727869534015$$
$$x_{68} = 98.4388272431212$$
$$x_{69} = 4.24076625725554$$
$$x_{70} = 34.5639476991297$$
$$x_{71} = 41.8932060932537$$
$$x_{72} = -30.3760435170464$$
$$x_{73} = -50.2699027802066$$
$$x_{74} = -83.7784565381466$$
$$x_{75} = 76.4483279927407$$
$$x_{76} = -95.2973090043489$$
$$x_{77} = -48.1756998057147$$
$$x_{78} = -63.8825291038655$$
$$x_{79} = 68.0711052836269$$
$$x_{80} = -55.5054736300684$$
$$x_{81} = 28.2821897477697$$
$$x_{82} = 10.493124973438$$
$$x_{83} = 70.1654029544622$$
$$x_{84} = 30.3760435170464$$
$$x_{85} = -1.2145323891418$$
$$x_{86} = 96.3444812114328$$
$$x_{87} = 22.0012459236092$$
$$x_{88} = 83.7784565381466$$
$$x_{89} = 43.9873487211332$$
$$x_{90} = 85.8727869534015$$
$$x_{91} = 1.2145323891418$$
$$x_{92} = -33.5169509084747$$
$$x_{93} = -68.0711052836269$$
$$x_{94} = 74.3540147599617$$
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
$$\left[105.769053656653, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.2501373603978\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$3 \left(- 3 x \sin{\left(3 x \right)} + 2 \cos{\left(3 x \right)}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -57.5997231867839$$
$$x_{2} = -11.5384110184102$$
$$x_{3} = 94.2501373603978$$
$$x_{4} = 105.769053656653$$
$$x_{5} = 65.9768137973912$$
$$x_{6} = 6.31822725550968$$
$$x_{7} = -35.6109562885689$$
$$x_{8} = -4.24076625725554$$
$$x_{9} = 61.7882518935787$$
$$x_{10} = 17.8148265565878$$
$$x_{11} = -72.2597062722654$$
$$x_{12} = 59.6939829539286$$
$$x_{13} = -87.967120448543$$
$$x_{14} = -81.6841294396421$$
$$x_{15} = -2.19277791090745$$
$$x_{16} = -17.8148265565878$$
$$x_{17} = -90.0614568087362$$
$$x_{18} = -77.4954862683366$$
$$x_{19} = -70.1654029544622$$
$$x_{20} = 37.7050049352483$$
$$x_{21} = -9.44825895659546$$
$$x_{22} = 78.5426455910908$$
$$x_{23} = -0.358957995437268$$
$$x_{24} = 52.3641211184374$$
$$x_{25} = -59.6939829539286$$
$$x_{26} = 54.4583530486096$$
$$x_{27} = -61.7882518935787$$
$$x_{28} = -43.9873487211332$$
$$x_{29} = 50.2699027802066$$
$$x_{30} = -19.9079118108102$$
$$x_{31} = -22.0012459236092$$
$$x_{32} = 72.2597062722654$$
$$x_{33} = -37.7050049352483$$
$$x_{34} = -15.7220892009256$$
$$x_{35} = -26.1884224615332$$
$$x_{36} = 0.358957995437268$$
$$x_{37} = -28.2821897477697$$
$$x_{38} = 100.533175319193$$
$$x_{39} = 92.1557958386718$$
$$x_{40} = 48.1756998057147$$
$$x_{41} = -79.5898059196778$$
$$x_{42} = 39.7990900239077$$
$$x_{43} = 8.40396768807019$$
$$x_{44} = 87.967120448543$$
$$x_{45} = -24.0947641678942$$
$$x_{46} = -3.20985344776581$$
$$x_{47} = 32.4699670568908$$
$$x_{48} = -46.0815142886463$$
$$x_{49} = -92.1557958386718$$
$$x_{50} = -53.4112354844189$$
$$x_{51} = -39.7990900239077$$
$$x_{52} = 90.0614568087362$$
$$x_{53} = 63.8825291038655$$
$$x_{54} = -65.9768137973912$$
$$x_{55} = -41.8932060932537$$
$$x_{56} = 2.19277791090745$$
$$x_{57} = 19.9079118108102$$
$$x_{58} = -13.6298592553469$$
$$x_{59} = 26.1884224615332$$
$$x_{60} = 24.0947641678942$$
$$x_{61} = 46.0815142886463$$
$$x_{62} = -99.4860010336778$$
$$x_{63} = -94.2501373603978$$
$$x_{64} = 12.5840132115367$$
$$x_{65} = 15.7220892009256$$
$$x_{66} = 56.5525970612491$$
$$x_{67} = -85.8727869534015$$
$$x_{68} = 98.4388272431212$$
$$x_{69} = 4.24076625725554$$
$$x_{70} = 34.5639476991297$$
$$x_{71} = 41.8932060932537$$
$$x_{72} = -30.3760435170464$$
$$x_{73} = -50.2699027802066$$
$$x_{74} = -83.7784565381466$$
$$x_{75} = 76.4483279927407$$
$$x_{76} = -95.2973090043489$$
$$x_{77} = -48.1756998057147$$
$$x_{78} = -63.8825291038655$$
$$x_{79} = 68.0711052836269$$
$$x_{80} = -55.5054736300684$$
$$x_{81} = 28.2821897477697$$
$$x_{82} = 10.493124973438$$
$$x_{83} = 70.1654029544622$$
$$x_{84} = 30.3760435170464$$
$$x_{85} = -1.2145323891418$$
$$x_{86} = 96.3444812114328$$
$$x_{87} = 22.0012459236092$$
$$x_{88} = 83.7784565381466$$
$$x_{89} = 43.9873487211332$$
$$x_{90} = 85.8727869534015$$
$$x_{91} = 1.2145323891418$$
$$x_{92} = -33.5169509084747$$
$$x_{93} = -68.0711052836269$$
$$x_{94} = 74.3540147599617$$
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
$$\left[105.769053656653, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -94.2501373603978\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(x \sin{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(x \sin{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(x \sin{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(x \sin{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
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*sin(3*x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty} \sin{\left(3 x \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -1, 1\right\rangle x$$
$$\lim_{x \to \infty} \sin{\left(3 x \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -1, 1\right\rangle x$$
$$\lim_{x \to -\infty} \sin{\left(3 x \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -1, 1\right\rangle x$$
$$\lim_{x \to \infty} \sin{\left(3 x \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -1, 1\right\rangle x$$
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:
$$x \sin{\left(3 x \right)} = x \sin{\left(3 x \right)}$$
- Sí
$$x \sin{\left(3 x \right)} = - x \sin{\left(3 x \right)}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$x \sin{\left(3 x \right)} = x \sin{\left(3 x \right)}$$
- Sí
$$x \sin{\left(3 x \right)} = - x \sin{\left(3 x \right)}$$
- No
es decir, función
es
par
Gráfico