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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^4-x^2+2
-
(x^2-5)/(x-3)
-
(x^2-9)/(x^2-4)
-
x/(1-x^3)
-
Expresiones idénticas
- sin(tres *x)^ dos /((cinco *x^ dos))
- seno de (3 multiplicar por x) al cuadrado dividir por ((5 multiplicar por x al cuadrado ))
- seno de (tres multiplicar por x) en el grado dos dividir por ((cinco multiplicar por x en el grado dos))
- sin(3*x)2/((5*x2))
- sin3*x2/5*x2
- sin(3*x)²/((5*x²))
- sin(3*x) en el grado 2/((5*x en el grado 2))
- sin(3x)^2/((5x^2))
- sin(3x)2/((5x2))
- sin3x2/5x2
- sin3x^2/5x^2
- sin(3*x)^2 dividir por ((5*x^2))
-
Expresiones con funciones
- Seno sin
- sin(x)+5
- sin(x)-cos(x^2)+(1/4)
- sin(x)+5*cos(x)
- sin(x^(3)+x)
- sin(x)*3
Gráfico de la función y = sin(3*x)^2/((5*x^2))
Solución
Ha introducido
[src]
2
sin (3*x)
f(x) = ---------
2
5*x
$$f{\left(x \right)} = \frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}}$$
f = sin(3*x)^2/((5*x^2))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{\pi}{3}$$
Solución numérica
$$x_{1} = -48.1710872770894$$
$$x_{2} = 96.3421746573426$$
$$x_{3} = 26.1799388464653$$
$$x_{4} = -130.89969356643$$
$$x_{5} = 92.1533846087568$$
$$x_{6} = 15.707963116935$$
$$x_{7} = 76.4454211663553$$
$$x_{8} = -90.0589893539331$$
$$x_{9} = 17.8023584373541$$
$$x_{10} = -121.474913745577$$
$$x_{11} = 100.530964954858$$
$$x_{12} = 54.4542725926977$$
$$x_{13} = -61.78465549173$$
$$x_{14} = -62.8318388913378$$
$$x_{15} = -86.9173970072424$$
$$x_{16} = 28.2743338654124$$
$$x_{17} = -6.2831852907774$$
$$x_{18} = -68.0678407739678$$
$$x_{19} = 4.18879025806703$$
$$x_{20} = 156.032434810019$$
$$x_{21} = -94.2477795685686$$
$$x_{22} = -35.6047168083621$$
$$x_{23} = 32.463124019205$$
$$x_{24} = 19.8967535390473$$
$$x_{25} = 34.5575193686514$$
$$x_{26} = -55.501470245751$$
$$x_{27} = -77.4926188070681$$
$$x_{28} = -15.7079632953978$$
$$x_{29} = -59.6902604555398$$
$$x_{30} = 83.7758041540691$$
$$x_{31} = -87.9645943604236$$
$$x_{32} = 37.699111661875$$
$$x_{33} = -46.0766921943899$$
$$x_{34} = -13.6135682299916$$
$$x_{35} = 59.6902602084204$$
$$x_{36} = 43.9822971689364$$
$$x_{37} = -26.1799387014293$$
$$x_{38} = -65.9734457659649$$
$$x_{39} = 12.5663708034402$$
$$x_{40} = 6.28318867393247$$
$$x_{41} = 6.28318528436868$$
$$x_{42} = -118.333323003394$$
$$x_{43} = -82.7286059618285$$
$$x_{44} = 50.2654824464336$$
$$x_{45} = -28.2743338888228$$
$$x_{46} = -9.42477794759928$$
$$x_{47} = -67.0206261482275$$
$$x_{48} = 14.6607660440567$$
$$x_{49} = 70.1622360212769$$
$$x_{50} = 90.0589892599068$$
$$x_{51} = 61.7846555838288$$
$$x_{52} = 85.8701992718717$$
$$x_{53} = -11.5191730266485$$
$$x_{54} = 52.3598775001506$$
$$x_{55} = 78.539816419006$$
$$x_{56} = 74.3510260786411$$
$$x_{57} = -21.99114858654$$
$$x_{58} = -57.5958653852565$$
$$x_{59} = -79.5870139615191$$
$$x_{60} = -96.3421744702308$$
$$x_{61} = 39.7935070119776$$
$$x_{62} = 65.9734457518411$$
$$x_{63} = 30.3687289217073$$
$$x_{64} = -43.982297174986$$
$$x_{65} = -33.5103215679451$$
$$x_{66} = -17.8023583232207$$
$$x_{67} = 129.852495391863$$
$$x_{68} = 72.256631027734$$
$$x_{69} = -92.153384427898$$
$$x_{70} = 87.9645943339274$$
$$x_{71} = -39.7935069079438$$
$$x_{72} = 98.436569740585$$
$$x_{73} = -81.6814090351582$$
$$x_{74} = -70.162235852376$$
$$x_{75} = -24.0855436148985$$
$$x_{76} = -33.5103216799534$$
$$x_{77} = -50.2654824467186$$
$$x_{78} = -99.483767363158$$
$$x_{79} = -11.5191731054941$$
$$x_{80} = 68.0678406417633$$
$$x_{81} = 41.8879021174409$$
$$x_{82} = -72.2566310060475$$
$$x_{83} = 56.5486678933424$$
$$x_{84} = -91.1062074760563$$
$$x_{85} = 10.4719754407311$$
$$x_{86} = 48.1710874335315$$
$$x_{87} = 45.0294954114788$$
$$x_{88} = 21.9911485850598$$
$$x_{89} = -2.09439502263277$$
$$x_{90} = -83.7758040749045$$
$$x_{91} = 63.8790506949197$$
$$x_{92} = 8.37758034125385$$
$$x_{93} = -37.6991118756413$$
$$x_{94} = -4.18879011020188$$
$$x_{95} = 94.247779609354$$
$$x_{96} = -31.4159266012248$$
$$x_{97} = -90.0589893822631$$
$$x_{98} = 78.5398156383358$$
o sea hay que resolver la ecuación:
$$\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{\pi}{3}$$
Solución numérica
$$x_{1} = -48.1710872770894$$
$$x_{2} = 96.3421746573426$$
$$x_{3} = 26.1799388464653$$
$$x_{4} = -130.89969356643$$
$$x_{5} = 92.1533846087568$$
$$x_{6} = 15.707963116935$$
$$x_{7} = 76.4454211663553$$
$$x_{8} = -90.0589893539331$$
$$x_{9} = 17.8023584373541$$
$$x_{10} = -121.474913745577$$
$$x_{11} = 100.530964954858$$
$$x_{12} = 54.4542725926977$$
$$x_{13} = -61.78465549173$$
$$x_{14} = -62.8318388913378$$
$$x_{15} = -86.9173970072424$$
$$x_{16} = 28.2743338654124$$
$$x_{17} = -6.2831852907774$$
$$x_{18} = -68.0678407739678$$
$$x_{19} = 4.18879025806703$$
$$x_{20} = 156.032434810019$$
$$x_{21} = -94.2477795685686$$
$$x_{22} = -35.6047168083621$$
$$x_{23} = 32.463124019205$$
$$x_{24} = 19.8967535390473$$
$$x_{25} = 34.5575193686514$$
$$x_{26} = -55.501470245751$$
$$x_{27} = -77.4926188070681$$
$$x_{28} = -15.7079632953978$$
$$x_{29} = -59.6902604555398$$
$$x_{30} = 83.7758041540691$$
$$x_{31} = -87.9645943604236$$
$$x_{32} = 37.699111661875$$
$$x_{33} = -46.0766921943899$$
$$x_{34} = -13.6135682299916$$
$$x_{35} = 59.6902602084204$$
$$x_{36} = 43.9822971689364$$
$$x_{37} = -26.1799387014293$$
$$x_{38} = -65.9734457659649$$
$$x_{39} = 12.5663708034402$$
$$x_{40} = 6.28318867393247$$
$$x_{41} = 6.28318528436868$$
$$x_{42} = -118.333323003394$$
$$x_{43} = -82.7286059618285$$
$$x_{44} = 50.2654824464336$$
$$x_{45} = -28.2743338888228$$
$$x_{46} = -9.42477794759928$$
$$x_{47} = -67.0206261482275$$
$$x_{48} = 14.6607660440567$$
$$x_{49} = 70.1622360212769$$
$$x_{50} = 90.0589892599068$$
$$x_{51} = 61.7846555838288$$
$$x_{52} = 85.8701992718717$$
$$x_{53} = -11.5191730266485$$
$$x_{54} = 52.3598775001506$$
$$x_{55} = 78.539816419006$$
$$x_{56} = 74.3510260786411$$
$$x_{57} = -21.99114858654$$
$$x_{58} = -57.5958653852565$$
$$x_{59} = -79.5870139615191$$
$$x_{60} = -96.3421744702308$$
$$x_{61} = 39.7935070119776$$
$$x_{62} = 65.9734457518411$$
$$x_{63} = 30.3687289217073$$
$$x_{64} = -43.982297174986$$
$$x_{65} = -33.5103215679451$$
$$x_{66} = -17.8023583232207$$
$$x_{67} = 129.852495391863$$
$$x_{68} = 72.256631027734$$
$$x_{69} = -92.153384427898$$
$$x_{70} = 87.9645943339274$$
$$x_{71} = -39.7935069079438$$
$$x_{72} = 98.436569740585$$
$$x_{73} = -81.6814090351582$$
$$x_{74} = -70.162235852376$$
$$x_{75} = -24.0855436148985$$
$$x_{76} = -33.5103216799534$$
$$x_{77} = -50.2654824467186$$
$$x_{78} = -99.483767363158$$
$$x_{79} = -11.5191731054941$$
$$x_{80} = 68.0678406417633$$
$$x_{81} = 41.8879021174409$$
$$x_{82} = -72.2566310060475$$
$$x_{83} = 56.5486678933424$$
$$x_{84} = -91.1062074760563$$
$$x_{85} = 10.4719754407311$$
$$x_{86} = 48.1710874335315$$
$$x_{87} = 45.0294954114788$$
$$x_{88} = 21.9911485850598$$
$$x_{89} = -2.09439502263277$$
$$x_{90} = -83.7758040749045$$
$$x_{91} = 63.8790506949197$$
$$x_{92} = 8.37758034125385$$
$$x_{93} = -37.6991118756413$$
$$x_{94} = -4.18879011020188$$
$$x_{95} = 94.247779609354$$
$$x_{96} = -31.4159266012248$$
$$x_{97} = -90.0589893822631$$
$$x_{98} = 78.5398156383358$$
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
$$6 \frac{1}{5 x^{2}} \sin{\left(3 x \right)} \cos{\left(3 x \right)} - \frac{2 \sin^{2}{\left(3 x \right)}}{5 x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -5.74025175731026$$
$$x_{2} = -39.7935069454707$$
$$x_{3} = 48.1710873550435$$
$$x_{4} = 80.1092256793491$$
$$x_{5} = 84.2980848042281$$
$$x_{6} = -49.7396498613733$$
$$x_{7} = -100.006255101775$$
$$x_{8} = -95.8174163262761$$
$$x_{9} = -41.8879020478639$$
$$x_{10} = 90.0589894029074$$
$$x_{11} = 4.18879020478639$$
$$x_{12} = 60.212013881401$$
$$x_{13} = 26.1799387799149$$
$$x_{14} = 52.3598775598299$$
$$x_{15} = -31.9360462622872$$
$$x_{16} = -103.147881594549$$
$$x_{17} = 28.2743338823081$$
$$x_{18} = 9.42477796076938$$
$$x_{19} = 18.3198927626296$$
$$x_{20} = 50.2654824574367$$
$$x_{21} = 12.0335407481252$$
$$x_{22} = -51.8341352242202$$
$$x_{23} = -27.7467308235745$$
$$x_{24} = -78.014793341506$$
$$x_{25} = -2.0943951023932$$
$$x_{26} = -93.722995310418$$
$$x_{27} = -71.7314832814509$$
$$x_{28} = -34.0306554883025$$
$$x_{29} = 100.006255101775$$
$$x_{30} = -90.0589894029074$$
$$x_{31} = 31.9360462622872$$
$$x_{32} = 97.9118362335106$$
$$x_{33} = 68.0678408277789$$
$$x_{34} = -7.839817499563$$
$$x_{35} = 65.9734457253857$$
$$x_{36} = 82.2036561197804$$
$$x_{37} = -75.9203589492161$$
$$x_{38} = -73.8259223276238$$
$$x_{39} = -24.0855436775217$$
$$x_{40} = -68.0678408277789$$
$$x_{41} = -47.1238898038469$$
$$x_{42} = -85.870199198121$$
$$x_{43} = 21.9911485751286$$
$$x_{44} = -37.6991118430775$$
$$x_{45} = -81.6814089933346$$
$$x_{46} = -21.9911485751286$$
$$x_{47} = -46.0766922526503$$
$$x_{48} = -4.18879020478639$$
$$x_{49} = -87.9645943005142$$
$$x_{50} = 36.1252398838916$$
$$x_{51} = 19.3674182514985$$
$$x_{52} = 53.9286135759886$$
$$x_{53} = 58.1175522783976$$
$$x_{54} = 16.2247147439848$$
$$x_{55} = -12.0335407481252$$
$$x_{56} = -43.9822971502571$$
$$x_{57} = 137.182879206754$$
$$x_{58} = -83.7758040957278$$
$$x_{59} = -56.0230857030463$$
$$x_{60} = -29.8414069768057$$
$$x_{61} = 75.9203589492161$$
$$x_{62} = 142.418866962737$$
$$x_{63} = -17.8023583703422$$
$$x_{64} = 14.1293045227106$$
$$x_{65} = -91.106186954104$$
$$x_{66} = 94.2477796076938$$
$$x_{67} = 78.014793341506$$
$$x_{68} = -53.9286135759886$$
$$x_{69} = 46.0766922526503$$
$$x_{70} = -59.6902604182061$$
$$x_{71} = 87.9645943005142$$
$$x_{72} = -63.8790506229925$$
$$x_{73} = -61.7846555205993$$
$$x_{74} = -120.950398515305$$
$$x_{75} = 2.0943951023932$$
$$x_{76} = -15.707963267949$$
$$x_{77} = 40.3143496657172$$
$$x_{78} = -97.9118362335106$$
$$x_{79} = 83.7758040957278$$
$$x_{80} = 52.8813752244211$$
$$x_{81} = 96.342174710087$$
$$x_{82} = 6.28318530717959$$
$$x_{83} = 34.0306554883025$$
$$x_{84} = -50.2654824574367$$
$$x_{85} = 62.3064710135101$$
$$x_{86} = 38.2198035316744$$
$$x_{87} = -9.93719959696432$$
$$x_{88} = 9.93719959696432$$
$$x_{89} = -19.8967534727354$$
$$x_{90} = 43.9822971502571$$
$$x_{91} = -65.9734457253857$$
$$x_{92} = 92.1533845053006$$
$$x_{93} = -25.6520087701104$$
$$x_{94} = 70.162235930172$$
$$x_{95} = 72.2566310325652$$
$$x_{96} = 56.0230857030463$$
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:
$$x_{1} = -39.7935069454707$$
$$x_{2} = 48.1710873550435$$
$$x_{3} = -41.8879020478639$$
$$x_{4} = 90.0589894029074$$
$$x_{5} = 4.18879020478639$$
$$x_{6} = 26.1799387799149$$
$$x_{7} = 52.3598775598299$$
$$x_{8} = 28.2743338823081$$
$$x_{9} = 9.42477796076938$$
$$x_{10} = 50.2654824574367$$
$$x_{11} = -2.0943951023932$$
$$x_{12} = -90.0589894029074$$
$$x_{13} = 68.0678408277789$$
$$x_{14} = 65.9734457253857$$
$$x_{15} = -24.0855436775217$$
$$x_{16} = -68.0678408277789$$
$$x_{17} = -47.1238898038469$$
$$x_{18} = -85.870199198121$$
$$x_{19} = 21.9911485751286$$
$$x_{20} = -37.6991118430775$$
$$x_{21} = -81.6814089933346$$
$$x_{22} = -21.9911485751286$$
$$x_{23} = -46.0766922526503$$
$$x_{24} = -4.18879020478639$$
$$x_{25} = -87.9645943005142$$
$$x_{26} = -43.9822971502571$$
$$x_{27} = 137.182879206754$$
$$x_{28} = -83.7758040957278$$
$$x_{29} = 142.418866962737$$
$$x_{30} = -17.8023583703422$$
$$x_{31} = -91.106186954104$$
$$x_{32} = 94.2477796076938$$
$$x_{33} = 46.0766922526503$$
$$x_{34} = -59.6902604182061$$
$$x_{35} = 87.9645943005142$$
$$x_{36} = -63.8790506229925$$
$$x_{37} = -61.7846555205993$$
$$x_{38} = 2.0943951023932$$
$$x_{39} = -15.707963267949$$
$$x_{40} = 83.7758040957278$$
$$x_{41} = 96.342174710087$$
$$x_{42} = 6.28318530717959$$
$$x_{43} = -50.2654824574367$$
$$x_{44} = -19.8967534727354$$
$$x_{45} = 43.9822971502571$$
$$x_{46} = -65.9734457253857$$
$$x_{47} = 92.1533845053006$$
$$x_{48} = 70.162235930172$$
$$x_{49} = 72.2566310325652$$
Puntos máximos de la función:
$$x_{49} = -5.74025175731026$$
$$x_{49} = 80.1092256793491$$
$$x_{49} = 84.2980848042281$$
$$x_{49} = -49.7396498613733$$
$$x_{49} = -100.006255101775$$
$$x_{49} = -95.8174163262761$$
$$x_{49} = 60.212013881401$$
$$x_{49} = -31.9360462622872$$
$$x_{49} = -103.147881594549$$
$$x_{49} = 18.3198927626296$$
$$x_{49} = 12.0335407481252$$
$$x_{49} = -51.8341352242202$$
$$x_{49} = -27.7467308235745$$
$$x_{49} = -78.014793341506$$
$$x_{49} = -93.722995310418$$
$$x_{49} = -71.7314832814509$$
$$x_{49} = -34.0306554883025$$
$$x_{49} = 100.006255101775$$
$$x_{49} = 31.9360462622872$$
$$x_{49} = 97.9118362335106$$
$$x_{49} = -7.839817499563$$
$$x_{49} = 82.2036561197804$$
$$x_{49} = -75.9203589492161$$
$$x_{49} = -73.8259223276238$$
$$x_{49} = 36.1252398838916$$
$$x_{49} = 19.3674182514985$$
$$x_{49} = 53.9286135759886$$
$$x_{49} = 58.1175522783976$$
$$x_{49} = 16.2247147439848$$
$$x_{49} = -12.0335407481252$$
$$x_{49} = -56.0230857030463$$
$$x_{49} = -29.8414069768057$$
$$x_{49} = 75.9203589492161$$
$$x_{49} = 14.1293045227106$$
$$x_{49} = 78.014793341506$$
$$x_{49} = -53.9286135759886$$
$$x_{49} = -120.950398515305$$
$$x_{49} = 40.3143496657172$$
$$x_{49} = -97.9118362335106$$
$$x_{49} = 52.8813752244211$$
$$x_{49} = 34.0306554883025$$
$$x_{49} = 62.3064710135101$$
$$x_{49} = 38.2198035316744$$
$$x_{49} = -9.93719959696432$$
$$x_{49} = 9.93719959696432$$
$$x_{49} = -25.6520087701104$$
$$x_{49} = 56.0230857030463$$
Decrece en los intervalos
$$\left[142.418866962737, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -91.106186954104\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$6 \frac{1}{5 x^{2}} \sin{\left(3 x \right)} \cos{\left(3 x \right)} - \frac{2 \sin^{2}{\left(3 x \right)}}{5 x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -5.74025175731026$$
$$x_{2} = -39.7935069454707$$
$$x_{3} = 48.1710873550435$$
$$x_{4} = 80.1092256793491$$
$$x_{5} = 84.2980848042281$$
$$x_{6} = -49.7396498613733$$
$$x_{7} = -100.006255101775$$
$$x_{8} = -95.8174163262761$$
$$x_{9} = -41.8879020478639$$
$$x_{10} = 90.0589894029074$$
$$x_{11} = 4.18879020478639$$
$$x_{12} = 60.212013881401$$
$$x_{13} = 26.1799387799149$$
$$x_{14} = 52.3598775598299$$
$$x_{15} = -31.9360462622872$$
$$x_{16} = -103.147881594549$$
$$x_{17} = 28.2743338823081$$
$$x_{18} = 9.42477796076938$$
$$x_{19} = 18.3198927626296$$
$$x_{20} = 50.2654824574367$$
$$x_{21} = 12.0335407481252$$
$$x_{22} = -51.8341352242202$$
$$x_{23} = -27.7467308235745$$
$$x_{24} = -78.014793341506$$
$$x_{25} = -2.0943951023932$$
$$x_{26} = -93.722995310418$$
$$x_{27} = -71.7314832814509$$
$$x_{28} = -34.0306554883025$$
$$x_{29} = 100.006255101775$$
$$x_{30} = -90.0589894029074$$
$$x_{31} = 31.9360462622872$$
$$x_{32} = 97.9118362335106$$
$$x_{33} = 68.0678408277789$$
$$x_{34} = -7.839817499563$$
$$x_{35} = 65.9734457253857$$
$$x_{36} = 82.2036561197804$$
$$x_{37} = -75.9203589492161$$
$$x_{38} = -73.8259223276238$$
$$x_{39} = -24.0855436775217$$
$$x_{40} = -68.0678408277789$$
$$x_{41} = -47.1238898038469$$
$$x_{42} = -85.870199198121$$
$$x_{43} = 21.9911485751286$$
$$x_{44} = -37.6991118430775$$
$$x_{45} = -81.6814089933346$$
$$x_{46} = -21.9911485751286$$
$$x_{47} = -46.0766922526503$$
$$x_{48} = -4.18879020478639$$
$$x_{49} = -87.9645943005142$$
$$x_{50} = 36.1252398838916$$
$$x_{51} = 19.3674182514985$$
$$x_{52} = 53.9286135759886$$
$$x_{53} = 58.1175522783976$$
$$x_{54} = 16.2247147439848$$
$$x_{55} = -12.0335407481252$$
$$x_{56} = -43.9822971502571$$
$$x_{57} = 137.182879206754$$
$$x_{58} = -83.7758040957278$$
$$x_{59} = -56.0230857030463$$
$$x_{60} = -29.8414069768057$$
$$x_{61} = 75.9203589492161$$
$$x_{62} = 142.418866962737$$
$$x_{63} = -17.8023583703422$$
$$x_{64} = 14.1293045227106$$
$$x_{65} = -91.106186954104$$
$$x_{66} = 94.2477796076938$$
$$x_{67} = 78.014793341506$$
$$x_{68} = -53.9286135759886$$
$$x_{69} = 46.0766922526503$$
$$x_{70} = -59.6902604182061$$
$$x_{71} = 87.9645943005142$$
$$x_{72} = -63.8790506229925$$
$$x_{73} = -61.7846555205993$$
$$x_{74} = -120.950398515305$$
$$x_{75} = 2.0943951023932$$
$$x_{76} = -15.707963267949$$
$$x_{77} = 40.3143496657172$$
$$x_{78} = -97.9118362335106$$
$$x_{79} = 83.7758040957278$$
$$x_{80} = 52.8813752244211$$
$$x_{81} = 96.342174710087$$
$$x_{82} = 6.28318530717959$$
$$x_{83} = 34.0306554883025$$
$$x_{84} = -50.2654824574367$$
$$x_{85} = 62.3064710135101$$
$$x_{86} = 38.2198035316744$$
$$x_{87} = -9.93719959696432$$
$$x_{88} = 9.93719959696432$$
$$x_{89} = -19.8967534727354$$
$$x_{90} = 43.9822971502571$$
$$x_{91} = -65.9734457253857$$
$$x_{92} = 92.1533845053006$$
$$x_{93} = -25.6520087701104$$
$$x_{94} = 70.162235930172$$
$$x_{95} = 72.2566310325652$$
$$x_{96} = 56.0230857030463$$
Signos de extremos en los puntos:
(-5.740251757310256, 0.00604931376107076)
(-39.79350694547072, 3.50664956856788e-32)
(48.1710873550435, 2.1198748344427e-32)
(80.10922567934914, 3.11643023073909e-5)
(84.29808480422805, 2.81441274900475e-5)
(-49.73964986137327, 8.08360425849672e-5)
(-100.00625510177518, 1.99972760298663e-5)
(-95.81741632627612, 2.17839096702349e-5)
(-41.88790204786391, 2.73523385445254e-33)
(90.0589894029074, 7.67086757168651e-33)
(4.188790204786391, 1.88656202715141e-32)
(60.212013881400964, 5.51633179057749e-5)
(26.179938779914945, 7.0376350240366e-35)
(52.35987755982989, 7.0376350240366e-35)
(-31.93604626228723, 0.000196074171184823)
(-103.14788159454915, 1.87977051888559e-5)
(28.274333882308138, 1.17706315709556e-32)
(9.42477796076938, 2.73523385445254e-33)
(18.319892762629646, 0.000595717538029012)
(50.26548245743669, 2.73523385445254e-33)
(12.033540748125203, 0.00138009829483675)
(-51.83413522422022, 7.44355361929604e-5)
(-27.746730823574467, 0.000259742896802955)
(-78.01479334150599, 3.28600441412612e-5)
(-2.0943951023931957, 1.88656202715141e-32)
(-93.72299531041797, 2.2768382695349e-5)
(-71.73148328145086, 3.8868787642629e-5)
(-34.03065548830255, 0.000172682250874105)
(100.00625510177518, 1.99972760298663e-5)
(-90.0589894029074, 7.67086757168651e-33)
(31.93604626228723, 0.000196074171184823)
(97.91183623351056, 2.08619342068965e-5)
(68.06784082777885, 1.49633282910139e-32)
(-7.839817499563003, 0.00324813214541642)
(65.97344572538566, 3.97953835917639e-34)
(82.20365611978043, 2.95965157737277e-5)
(-75.9203589492161, 3.46980539789198e-5)
(-73.82592232762377, 3.66947031904297e-5)
(-24.08554367752175, 2.1198748344427e-32)
(-68.06784082777885, 1.49633282910139e-32)
(-47.1238898038469, 3.45350792760592e-34)
(-85.87019919812101, 1.7616330792198e-32)
(21.991148575128552, 3.97953835917639e-34)
(-37.69911184307752, 2.73523385445254e-33)
(-81.68140899333463, 4.14854992394097e-33)
(-21.991148575128552, 3.97953835917639e-34)
(-46.076692252650304, 7.33232593480711e-33)
(-4.188790204786391, 1.88656202715141e-32)
(-87.96459430051421, 3.97953835917639e-34)
(36.12523988389156, 0.000153239787803824)
(19.36741825149853, 0.000533037673910117)
(53.92861357598856, 6.87661789482822e-5)
(58.11755227839756, 5.92108205283373e-5)
(16.224714743984794, 0.000759438454034515)
(-12.033540748125203, 0.00138009829483675)
(-43.982297150257104, 3.97953835917639e-34)
(137.1828792067543, 8.27885141342781e-34)
(-83.77580409572782, 2.73523385445254e-33)
(-56.02308570304626, 6.37207046517988e-5)
(-29.84140697680573, 0.000224562492385297)
(75.9203589492161, 3.46980539789198e-5)
(142.4188669627373, 1.36518182602633e-33)
(-17.802358370342162, 1.36518182602633e-33)
(14.12930452271064, 0.00100125979806124)
(-91.106186954104, 1.94590277857176e-32)
(94.2477796076938, 3.45350792760592e-34)
(78.01479334150599, 3.28600441412612e-5)
(-53.92861357598856, 6.87661789482822e-5)
(46.076692252650304, 7.33232593480711e-33)
(-59.69026041820607, 6.52692819522687e-33)
(87.96459430051421, 3.97953835917639e-34)
(-63.879050622992466, 2.94299693078321e-32)
(-61.784655520599266, 6.08633082400166e-33)
(-120.95039851530534, 1.3671371657898e-5)
(2.0943951023931957, 1.88656202715141e-32)
(-15.707963267948966, 2.35459830964729e-32)
(40.31434966571717, 0.000123049821820816)
(-97.91183623351056, 2.08619342068965e-5)
(83.77580409572782, 2.73523385445254e-33)
(52.88137522442115, 7.151666574077e-5)
(96.342174710087, 2.1198748344427e-32)
(6.283185307179586, 2.73523385445254e-33)
(34.03065548830255, 0.000172682250874105)
(-50.26548245743669, 2.73523385445254e-33)
(62.306471013510084, 5.15170822336715e-5)
(38.21980353167436, 0.000136905237236687)
(-9.93719959696432, 0.00202308242076543)
(9.93719959696432, 0.00202308242076543)
(-19.89675347273536, 3.50664956856788e-32)
(43.982297150257104, 3.97953835917639e-34)
(-65.97344572538566, 3.97953835917639e-34)
(92.15338450530061, 7.33232593480711e-33)
(-25.652008770110395, 0.000303888250905934)
(70.16223593017205, 5.61666835098137e-33)
(72.25663103256524, 9.18748680817066e-34)
(56.02308570304626, 6.37207046517988e-5)
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:
$$x_{1} = -39.7935069454707$$
$$x_{2} = 48.1710873550435$$
$$x_{3} = -41.8879020478639$$
$$x_{4} = 90.0589894029074$$
$$x_{5} = 4.18879020478639$$
$$x_{6} = 26.1799387799149$$
$$x_{7} = 52.3598775598299$$
$$x_{8} = 28.2743338823081$$
$$x_{9} = 9.42477796076938$$
$$x_{10} = 50.2654824574367$$
$$x_{11} = -2.0943951023932$$
$$x_{12} = -90.0589894029074$$
$$x_{13} = 68.0678408277789$$
$$x_{14} = 65.9734457253857$$
$$x_{15} = -24.0855436775217$$
$$x_{16} = -68.0678408277789$$
$$x_{17} = -47.1238898038469$$
$$x_{18} = -85.870199198121$$
$$x_{19} = 21.9911485751286$$
$$x_{20} = -37.6991118430775$$
$$x_{21} = -81.6814089933346$$
$$x_{22} = -21.9911485751286$$
$$x_{23} = -46.0766922526503$$
$$x_{24} = -4.18879020478639$$
$$x_{25} = -87.9645943005142$$
$$x_{26} = -43.9822971502571$$
$$x_{27} = 137.182879206754$$
$$x_{28} = -83.7758040957278$$
$$x_{29} = 142.418866962737$$
$$x_{30} = -17.8023583703422$$
$$x_{31} = -91.106186954104$$
$$x_{32} = 94.2477796076938$$
$$x_{33} = 46.0766922526503$$
$$x_{34} = -59.6902604182061$$
$$x_{35} = 87.9645943005142$$
$$x_{36} = -63.8790506229925$$
$$x_{37} = -61.7846555205993$$
$$x_{38} = 2.0943951023932$$
$$x_{39} = -15.707963267949$$
$$x_{40} = 83.7758040957278$$
$$x_{41} = 96.342174710087$$
$$x_{42} = 6.28318530717959$$
$$x_{43} = -50.2654824574367$$
$$x_{44} = -19.8967534727354$$
$$x_{45} = 43.9822971502571$$
$$x_{46} = -65.9734457253857$$
$$x_{47} = 92.1533845053006$$
$$x_{48} = 70.162235930172$$
$$x_{49} = 72.2566310325652$$
Puntos máximos de la función:
$$x_{49} = -5.74025175731026$$
$$x_{49} = 80.1092256793491$$
$$x_{49} = 84.2980848042281$$
$$x_{49} = -49.7396498613733$$
$$x_{49} = -100.006255101775$$
$$x_{49} = -95.8174163262761$$
$$x_{49} = 60.212013881401$$
$$x_{49} = -31.9360462622872$$
$$x_{49} = -103.147881594549$$
$$x_{49} = 18.3198927626296$$
$$x_{49} = 12.0335407481252$$
$$x_{49} = -51.8341352242202$$
$$x_{49} = -27.7467308235745$$
$$x_{49} = -78.014793341506$$
$$x_{49} = -93.722995310418$$
$$x_{49} = -71.7314832814509$$
$$x_{49} = -34.0306554883025$$
$$x_{49} = 100.006255101775$$
$$x_{49} = 31.9360462622872$$
$$x_{49} = 97.9118362335106$$
$$x_{49} = -7.839817499563$$
$$x_{49} = 82.2036561197804$$
$$x_{49} = -75.9203589492161$$
$$x_{49} = -73.8259223276238$$
$$x_{49} = 36.1252398838916$$
$$x_{49} = 19.3674182514985$$
$$x_{49} = 53.9286135759886$$
$$x_{49} = 58.1175522783976$$
$$x_{49} = 16.2247147439848$$
$$x_{49} = -12.0335407481252$$
$$x_{49} = -56.0230857030463$$
$$x_{49} = -29.8414069768057$$
$$x_{49} = 75.9203589492161$$
$$x_{49} = 14.1293045227106$$
$$x_{49} = 78.014793341506$$
$$x_{49} = -53.9286135759886$$
$$x_{49} = -120.950398515305$$
$$x_{49} = 40.3143496657172$$
$$x_{49} = -97.9118362335106$$
$$x_{49} = 52.8813752244211$$
$$x_{49} = 34.0306554883025$$
$$x_{49} = 62.3064710135101$$
$$x_{49} = 38.2198035316744$$
$$x_{49} = -9.93719959696432$$
$$x_{49} = 9.93719959696432$$
$$x_{49} = -25.6520087701104$$
$$x_{49} = 56.0230857030463$$
Decrece en los intervalos
$$\left[142.418866962737, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -91.106186954104\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
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:
$$\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}} = \frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}}$$
- Sí
$$\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}} = - \frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}} = \frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}}$$
- Sí
$$\frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}} = - \frac{\sin^{2}{\left(3 x \right)}}{5 x^{2}}$$
- No
es decir, función
es
par
Gráfico