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- Gráfico de la función implícita
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- Gráfico de la función paramétrica
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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
2*x^2-x^3
-
2*x^2-20*x+1
-
(2*x-1)*e^(2*(1-x))
-
(2-x^2)/(9x^2-4)^1/2
-
Expresiones idénticas
- sin(diez *x)*sin(tres *x)/x^ dos
- seno de (10 multiplicar por x) multiplicar por seno de (3 multiplicar por x) dividir por x al cuadrado
- seno de (diez multiplicar por x) multiplicar por seno de (tres multiplicar por x) dividir por x en el grado dos
- sin(10*x)*sin(3*x)/x2
- sin10*x*sin3*x/x2
- sin(10*x)*sin(3*x)/x²
- sin(10*x)*sin(3*x)/x en el grado 2
- sin(10x)sin(3x)/x^2
- sin(10x)sin(3x)/x2
- sin10xsin3x/x2
- sin10xsin3x/x^2
- sin(10*x)*sin(3*x) dividir por x^2
-
Expresiones con funciones
- Seno sin
- sin^2(t/2)
- sin(2)^(2)*x*cos(x)/2
- sin(x)^2+9cos(x)^2+6sin(x)cos(x)
- sin(x)-0.2*x-0.5
- sin(x+1)/(x^2-4*x-5)
- Seno sin
- sin^2(t/2)
- sin(2)^(2)*x*cos(x)/2
- sin(x)^2+9cos(x)^2+6sin(x)cos(x)
- sin(x)-0.2*x-0.5
- sin(x+1)/(x^2-4*x-5)
Gráfico de la función y = sin(10*x)*sin(3*x)/x^2
Solución
Ha introducido
[src]
sin(10*x)*sin(3*x)
f(x) = ------------------
2
x
$$f{\left(x \right)} = \frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}}$$
f = (sin(3*x)*sin(10*x))/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:
$$\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -83.7758040957278$$
$$x_{2} = -85.7654794430014$$
$$x_{3} = -57.8053048260522$$
$$x_{4} = 8.16814089933346$$
$$x_{5} = 26.3893782901543$$
$$x_{6} = 6.2831853127796$$
$$x_{7} = 4.39822971502571$$
$$x_{8} = -27.9601746169492$$
$$x_{9} = 50.2654824488303$$
$$x_{10} = -29.845130209103$$
$$x_{11} = 76.654860747591$$
$$x_{12} = -81.681408866394$$
$$x_{13} = -35.8141562509236$$
$$x_{14} = -47.7522083345649$$
$$x_{15} = -77.4926187885482$$
$$x_{16} = -3.76991118430775$$
$$x_{17} = -95.8185759344887$$
$$x_{18} = -19.4778744522567$$
$$x_{19} = -39.7935069454707$$
$$x_{20} = -17.5929188601028$$
$$x_{21} = -61.7846555205993$$
$$x_{22} = 58.1194640914112$$
$$x_{23} = -89.8495498926681$$
$$x_{24} = 68.0678408277789$$
$$x_{25} = -53.0929158456675$$
$$x_{26} = -41.8879020478639$$
$$x_{27} = 12.2522113490002$$
$$x_{28} = 52.1504380495906$$
$$x_{29} = 106.185831691335$$
$$x_{30} = -98.0176907920015$$
$$x_{31} = 18.2212373908208$$
$$x_{32} = -93.9336203423348$$
$$x_{33} = -77.9114978090269$$
$$x_{34} = 28.2743338741964$$
$$x_{35} = -29.5309709437441$$
$$x_{36} = -37.3849525777185$$
$$x_{37} = 38.0132711084365$$
$$x_{38} = 30.159289474462$$
$$x_{39} = 36.1283155162826$$
$$x_{40} = -13.8230076757951$$
$$x_{41} = -49.9513231920777$$
$$x_{42} = 42.0973415581032$$
$$x_{43} = -59.6902605322691$$
$$x_{44} = -79.7964534011807$$
$$x_{45} = 60.0044196835651$$
$$x_{46} = -23.8761041672824$$
$$x_{47} = -9.73893722612836$$
$$x_{48} = 20.1061929829747$$
$$x_{49} = 55.5014702134197$$
$$x_{50} = 62.2035345410779$$
$$x_{51} = 35.4999969855647$$
$$x_{52} = -65.9734457811152$$
$$x_{53} = -63.7743308678728$$
$$x_{54} = 89.8495498926681$$
$$x_{55} = 34.2433599241287$$
$$x_{56} = 46.0766922526503$$
$$x_{57} = -31.7300858012569$$
$$x_{58} = 80.1106126665397$$
$$x_{59} = 64.0884901332318$$
$$x_{60} = -21.9911485884892$$
$$x_{61} = -5.96902604182061$$
$$x_{62} = -87.9645943793467$$
$$x_{63} = 56.2345084992573$$
$$x_{64} = 10.471975511966$$
$$x_{65} = 84.1946831162065$$
$$x_{66} = 2.51327412287183$$
$$x_{67} = 14.1371669411541$$
$$x_{68} = 86.0796387083603$$
$$x_{69} = 40.2123859659494$$
$$x_{70} = 94.8760981384118$$
$$x_{71} = 94.2477796093827$$
$$x_{72} = -34.5575176413363$$
$$x_{73} = -43.9822971824851$$
$$x_{74} = -71.9424717672063$$
$$x_{75} = 89.2212313619501$$
$$x_{76} = 16.0221225333079$$
$$x_{77} = 72.256631028062$$
$$x_{78} = 78.2256570743859$$
$$x_{79} = -55.9203492338983$$
$$x_{80} = -73.8274273593601$$
$$x_{81} = -14.6607657167524$$
$$x_{82} = 98.4365698124802$$
$$x_{83} = -69.7433569096934$$
$$x_{84} = -45.867252742411$$
$$x_{85} = -1.88495559215388$$
$$x_{86} = 74.1415866247191$$
$$x_{87} = -25.7610597594363$$
$$x_{88} = 25.4469004940773$$
$$x_{89} = -51.8362787842316$$
$$x_{90} = 21.9911485829191$$
$$x_{91} = -99.9026463841554$$
$$x_{92} = -13.1946891450771$$
$$x_{93} = -7.85398163397448$$
$$x_{94} = 87.9645942735851$$
$$x_{95} = 43.9822971603677$$
$$x_{96} = -86.0796387083603$$
$$x_{97} = 100.216805649514$$
$$x_{98} = 81.9955682586936$$
$$x_{99} = -67.8584013175395$$
$$x_{100} = 65.9734457312647$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -83.7758040957278$$
$$x_{2} = -85.7654794430014$$
$$x_{3} = -57.8053048260522$$
$$x_{4} = 8.16814089933346$$
$$x_{5} = 26.3893782901543$$
$$x_{6} = 6.2831853127796$$
$$x_{7} = 4.39822971502571$$
$$x_{8} = -27.9601746169492$$
$$x_{9} = 50.2654824488303$$
$$x_{10} = -29.845130209103$$
$$x_{11} = 76.654860747591$$
$$x_{12} = -81.681408866394$$
$$x_{13} = -35.8141562509236$$
$$x_{14} = -47.7522083345649$$
$$x_{15} = -77.4926187885482$$
$$x_{16} = -3.76991118430775$$
$$x_{17} = -95.8185759344887$$
$$x_{18} = -19.4778744522567$$
$$x_{19} = -39.7935069454707$$
$$x_{20} = -17.5929188601028$$
$$x_{21} = -61.7846555205993$$
$$x_{22} = 58.1194640914112$$
$$x_{23} = -89.8495498926681$$
$$x_{24} = 68.0678408277789$$
$$x_{25} = -53.0929158456675$$
$$x_{26} = -41.8879020478639$$
$$x_{27} = 12.2522113490002$$
$$x_{28} = 52.1504380495906$$
$$x_{29} = 106.185831691335$$
$$x_{30} = -98.0176907920015$$
$$x_{31} = 18.2212373908208$$
$$x_{32} = -93.9336203423348$$
$$x_{33} = -77.9114978090269$$
$$x_{34} = 28.2743338741964$$
$$x_{35} = -29.5309709437441$$
$$x_{36} = -37.3849525777185$$
$$x_{37} = 38.0132711084365$$
$$x_{38} = 30.159289474462$$
$$x_{39} = 36.1283155162826$$
$$x_{40} = -13.8230076757951$$
$$x_{41} = -49.9513231920777$$
$$x_{42} = 42.0973415581032$$
$$x_{43} = -59.6902605322691$$
$$x_{44} = -79.7964534011807$$
$$x_{45} = 60.0044196835651$$
$$x_{46} = -23.8761041672824$$
$$x_{47} = -9.73893722612836$$
$$x_{48} = 20.1061929829747$$
$$x_{49} = 55.5014702134197$$
$$x_{50} = 62.2035345410779$$
$$x_{51} = 35.4999969855647$$
$$x_{52} = -65.9734457811152$$
$$x_{53} = -63.7743308678728$$
$$x_{54} = 89.8495498926681$$
$$x_{55} = 34.2433599241287$$
$$x_{56} = 46.0766922526503$$
$$x_{57} = -31.7300858012569$$
$$x_{58} = 80.1106126665397$$
$$x_{59} = 64.0884901332318$$
$$x_{60} = -21.9911485884892$$
$$x_{61} = -5.96902604182061$$
$$x_{62} = -87.9645943793467$$
$$x_{63} = 56.2345084992573$$
$$x_{64} = 10.471975511966$$
$$x_{65} = 84.1946831162065$$
$$x_{66} = 2.51327412287183$$
$$x_{67} = 14.1371669411541$$
$$x_{68} = 86.0796387083603$$
$$x_{69} = 40.2123859659494$$
$$x_{70} = 94.8760981384118$$
$$x_{71} = 94.2477796093827$$
$$x_{72} = -34.5575176413363$$
$$x_{73} = -43.9822971824851$$
$$x_{74} = -71.9424717672063$$
$$x_{75} = 89.2212313619501$$
$$x_{76} = 16.0221225333079$$
$$x_{77} = 72.256631028062$$
$$x_{78} = 78.2256570743859$$
$$x_{79} = -55.9203492338983$$
$$x_{80} = -73.8274273593601$$
$$x_{81} = -14.6607657167524$$
$$x_{82} = 98.4365698124802$$
$$x_{83} = -69.7433569096934$$
$$x_{84} = -45.867252742411$$
$$x_{85} = -1.88495559215388$$
$$x_{86} = 74.1415866247191$$
$$x_{87} = -25.7610597594363$$
$$x_{88} = 25.4469004940773$$
$$x_{89} = -51.8362787842316$$
$$x_{90} = 21.9911485829191$$
$$x_{91} = -99.9026463841554$$
$$x_{92} = -13.1946891450771$$
$$x_{93} = -7.85398163397448$$
$$x_{94} = 87.9645942735851$$
$$x_{95} = 43.9822971603677$$
$$x_{96} = -86.0796387083603$$
$$x_{97} = 100.216805649514$$
$$x_{98} = 81.9955682586936$$
$$x_{99} = -67.8584013175395$$
$$x_{100} = 65.9734457312647$$
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
$$\frac{10 \sin{\left(3 x \right)} \cos{\left(10 x \right)} + 3 \sin{\left(10 x \right)} \cos{\left(3 x \right)}}{x^{2}} - \frac{2 \sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 99.103389444345$$
$$x_{2} = -39.8479096702856$$
$$x_{3} = -61.8390813957329$$
$$x_{4} = 18.0884749138409$$
$$x_{5} = -77.7794307044932$$
$$x_{6} = 84.0626317410054$$
$$x_{7} = -27.7980889472184$$
$$x_{8} = 32.1755939890952$$
$$x_{9} = 13.9924982173739$$
$$x_{10} = 50.2654824574367$$
$$x_{11} = 86.250192592747$$
$$x_{12} = 82.156773729881$$
$$x_{13} = -96.222327954753$$
$$x_{14} = -31.8909403767992$$
$$x_{15} = 77.1121898659549$$
$$x_{16} = -9.89851698018621$$
$$x_{17} = 46.1311038443535$$
$$x_{18} = -49.7895279421032$$
$$x_{19} = 58.5231595633446$$
$$x_{20} = 48.2906711524039$$
$$x_{21} = -97.8647727260355$$
$$x_{22} = 90.1134284388961$$
$$x_{23} = 25.1327412287183$$
$$x_{24} = -337.143135975335$$
$$x_{25} = -83.1085921640793$$
$$x_{26} = -93.7719973086379$$
$$x_{27} = 92.2730518047756$$
$$x_{28} = 98.1493809479803$$
$$x_{29} = -87.7657288307864$$
$$x_{30} = -47.8837237348124$$
$$x_{31} = -63.9986774231948$$
$$x_{32} = -99.7706256620177$$
$$x_{33} = -75.8735699888791$$
$$x_{34} = -20.0160895596232$$
$$x_{35} = 62.0714141743783$$
$$x_{36} = -79.7066667391308$$
$$x_{37} = 63.8245421747774$$
$$x_{38} = 42.2676813662533$$
$$x_{39} = 63.9986774231948$$
$$x_{40} = 96.222327954753$$
$$x_{41} = 54.1669486134151$$
$$x_{42} = -29.4410030026552$$
$$x_{43} = 88.1631527320623$$
$$x_{44} = -67.9479575416364$$
$$x_{45} = 52.2399569004501$$
$$x_{46} = -33.7968595393584$$
$$x_{47} = -55.7881991594329$$
$$x_{48} = 72.2566310325652$$
$$x_{49} = 70.2818745472965$$
$$x_{50} = -46.1311038443535$$
$$x_{51} = 12.0892717242791$$
$$x_{52} = 20.8237866800918$$
$$x_{53} = -69.874979688883$$
$$x_{54} = 28.2743338823081$$
$$x_{55} = 76.1581842903436$$
$$x_{56} = 24.1399040364683$$
$$x_{57} = -5.80445235275529$$
$$x_{58} = -91.866184251143$$
$$x_{59} = 8.94714873719536$$
$$x_{60} = 94.2477796076938$$
$$x_{61} = 21.9911485751286$$
$$x_{62} = 30.2486908490499$$
$$x_{63} = -15.707963267949$$
$$x_{64} = 4.30657505562746$$
$$x_{65} = 40.0801208399867$$
$$x_{66} = 2.14765281901655$$
$$x_{67} = 80.2537915166431$$
$$x_{68} = -24.3718940255804$$
$$x_{69} = 74.2311533708957$$
$$x_{70} = -61.4041678075017$$
$$x_{71} = 68.1222705717507$$
$$x_{72} = -57.7154777250412$$
$$x_{73} = -13.7327104235176$$
$$x_{74} = -7.99516931765348$$
$$x_{75} = 60.1655437901152$$
$$x_{76} = 38.1742200902028$$
$$x_{77} = -71.7807890035278$$
$$x_{78} = -53.8823230356843$$
$$x_{79} = 16.1824202315257$$
$$x_{80} = 6.28318530717959$$
$$x_{81} = -89.9391365139768$$
$$x_{82} = -17.8566809065234$$
$$x_{83} = -52.414295892353$$
$$x_{84} = -35.7242388879548$$
$$x_{85} = -83.8302409743508$$
$$x_{86} = -25.8922856011411$$
$$x_{87} = -3.89760750085627$$
$$x_{88} = -42.007459633629$$
$$x_{89} = -1.97040802141203$$
$$x_{90} = 61.6647595690351$$
$$x_{91} = -37.6991118430775$$
$$x_{92} = -20.5628905052217$$
$$x_{93} = 38.8658506940265$$
$$x_{94} = -74.6378291927198$$
$$x_{95} = 34.0813954679699$$
$$x_{96} = -44.7421071306256$$
$$x_{97} = 10.183352203933$$
$$x_{98} = -11.8048062508305$$
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:
$$x_{1} = 99.103389444345$$
$$x_{2} = -61.8390813957329$$
$$x_{3} = -77.7794307044932$$
$$x_{4} = 84.0626317410054$$
$$x_{5} = 13.9924982173739$$
$$x_{6} = 50.2654824574367$$
$$x_{7} = 82.156773729881$$
$$x_{8} = -96.222327954753$$
$$x_{9} = -31.8909403767992$$
$$x_{10} = -49.7895279421032$$
$$x_{11} = 58.5231595633446$$
$$x_{12} = 48.2906711524039$$
$$x_{13} = 25.1327412287183$$
$$x_{14} = -83.1085921640793$$
$$x_{15} = -93.7719973086379$$
$$x_{16} = 92.2730518047756$$
$$x_{17} = 98.1493809479803$$
$$x_{18} = -47.8837237348124$$
$$x_{19} = -75.8735699888791$$
$$x_{20} = -79.7066667391308$$
$$x_{21} = 63.8245421747774$$
$$x_{22} = 96.222327954753$$
$$x_{23} = 54.1669486134151$$
$$x_{24} = -29.4410030026552$$
$$x_{25} = 52.2399569004501$$
$$x_{26} = -33.7968595393584$$
$$x_{27} = 12.0892717242791$$
$$x_{28} = 20.8237866800918$$
$$x_{29} = 24.1399040364683$$
$$x_{30} = -5.80445235275529$$
$$x_{31} = -91.866184251143$$
$$x_{32} = 94.2477796076938$$
$$x_{33} = 4.30657505562746$$
$$x_{34} = 40.0801208399867$$
$$x_{35} = 80.2537915166431$$
$$x_{36} = -61.4041678075017$$
$$x_{37} = 68.1222705717507$$
$$x_{38} = 38.1742200902028$$
$$x_{39} = 6.28318530717959$$
$$x_{40} = -89.9391365139768$$
$$x_{41} = -17.8566809065234$$
$$x_{42} = -35.7242388879548$$
$$x_{43} = -3.89760750085627$$
$$x_{44} = -42.007459633629$$
$$x_{45} = -1.97040802141203$$
$$x_{46} = -37.6991118430775$$
$$x_{47} = 10.183352203933$$
Puntos máximos de la función:
$$x_{47} = -39.8479096702856$$
$$x_{47} = 18.0884749138409$$
$$x_{47} = -27.7980889472184$$
$$x_{47} = 32.1755939890952$$
$$x_{47} = 86.250192592747$$
$$x_{47} = 77.1121898659549$$
$$x_{47} = -9.89851698018621$$
$$x_{47} = 46.1311038443535$$
$$x_{47} = -97.8647727260355$$
$$x_{47} = 90.1134284388961$$
$$x_{47} = -337.143135975335$$
$$x_{47} = -87.7657288307864$$
$$x_{47} = -63.9986774231948$$
$$x_{47} = -99.7706256620177$$
$$x_{47} = -20.0160895596232$$
$$x_{47} = 62.0714141743783$$
$$x_{47} = 42.2676813662533$$
$$x_{47} = 63.9986774231948$$
$$x_{47} = 88.1631527320623$$
$$x_{47} = -67.9479575416364$$
$$x_{47} = -55.7881991594329$$
$$x_{47} = 72.2566310325652$$
$$x_{47} = 70.2818745472965$$
$$x_{47} = -46.1311038443535$$
$$x_{47} = -69.874979688883$$
$$x_{47} = 28.2743338823081$$
$$x_{47} = 76.1581842903436$$
$$x_{47} = 8.94714873719536$$
$$x_{47} = 21.9911485751286$$
$$x_{47} = 30.2486908490499$$
$$x_{47} = -15.707963267949$$
$$x_{47} = 2.14765281901655$$
$$x_{47} = -24.3718940255804$$
$$x_{47} = 74.2311533708957$$
$$x_{47} = -57.7154777250412$$
$$x_{47} = -13.7327104235176$$
$$x_{47} = -7.99516931765348$$
$$x_{47} = 60.1655437901152$$
$$x_{47} = -71.7807890035278$$
$$x_{47} = -53.8823230356843$$
$$x_{47} = 16.1824202315257$$
$$x_{47} = -52.414295892353$$
$$x_{47} = -83.8302409743508$$
$$x_{47} = -25.8922856011411$$
$$x_{47} = 61.6647595690351$$
$$x_{47} = -20.5628905052217$$
$$x_{47} = 38.8658506940265$$
$$x_{47} = -74.6378291927198$$
$$x_{47} = 34.0813954679699$$
$$x_{47} = -44.7421071306256$$
$$x_{47} = -11.8048062508305$$
Decrece en los intervalos
$$\left[99.103389444345, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -96.222327954753\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{10 \sin{\left(3 x \right)} \cos{\left(10 x \right)} + 3 \sin{\left(10 x \right)} \cos{\left(3 x \right)}}{x^{2}} - \frac{2 \sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 99.103389444345$$
$$x_{2} = -39.8479096702856$$
$$x_{3} = -61.8390813957329$$
$$x_{4} = 18.0884749138409$$
$$x_{5} = -77.7794307044932$$
$$x_{6} = 84.0626317410054$$
$$x_{7} = -27.7980889472184$$
$$x_{8} = 32.1755939890952$$
$$x_{9} = 13.9924982173739$$
$$x_{10} = 50.2654824574367$$
$$x_{11} = 86.250192592747$$
$$x_{12} = 82.156773729881$$
$$x_{13} = -96.222327954753$$
$$x_{14} = -31.8909403767992$$
$$x_{15} = 77.1121898659549$$
$$x_{16} = -9.89851698018621$$
$$x_{17} = 46.1311038443535$$
$$x_{18} = -49.7895279421032$$
$$x_{19} = 58.5231595633446$$
$$x_{20} = 48.2906711524039$$
$$x_{21} = -97.8647727260355$$
$$x_{22} = 90.1134284388961$$
$$x_{23} = 25.1327412287183$$
$$x_{24} = -337.143135975335$$
$$x_{25} = -83.1085921640793$$
$$x_{26} = -93.7719973086379$$
$$x_{27} = 92.2730518047756$$
$$x_{28} = 98.1493809479803$$
$$x_{29} = -87.7657288307864$$
$$x_{30} = -47.8837237348124$$
$$x_{31} = -63.9986774231948$$
$$x_{32} = -99.7706256620177$$
$$x_{33} = -75.8735699888791$$
$$x_{34} = -20.0160895596232$$
$$x_{35} = 62.0714141743783$$
$$x_{36} = -79.7066667391308$$
$$x_{37} = 63.8245421747774$$
$$x_{38} = 42.2676813662533$$
$$x_{39} = 63.9986774231948$$
$$x_{40} = 96.222327954753$$
$$x_{41} = 54.1669486134151$$
$$x_{42} = -29.4410030026552$$
$$x_{43} = 88.1631527320623$$
$$x_{44} = -67.9479575416364$$
$$x_{45} = 52.2399569004501$$
$$x_{46} = -33.7968595393584$$
$$x_{47} = -55.7881991594329$$
$$x_{48} = 72.2566310325652$$
$$x_{49} = 70.2818745472965$$
$$x_{50} = -46.1311038443535$$
$$x_{51} = 12.0892717242791$$
$$x_{52} = 20.8237866800918$$
$$x_{53} = -69.874979688883$$
$$x_{54} = 28.2743338823081$$
$$x_{55} = 76.1581842903436$$
$$x_{56} = 24.1399040364683$$
$$x_{57} = -5.80445235275529$$
$$x_{58} = -91.866184251143$$
$$x_{59} = 8.94714873719536$$
$$x_{60} = 94.2477796076938$$
$$x_{61} = 21.9911485751286$$
$$x_{62} = 30.2486908490499$$
$$x_{63} = -15.707963267949$$
$$x_{64} = 4.30657505562746$$
$$x_{65} = 40.0801208399867$$
$$x_{66} = 2.14765281901655$$
$$x_{67} = 80.2537915166431$$
$$x_{68} = -24.3718940255804$$
$$x_{69} = 74.2311533708957$$
$$x_{70} = -61.4041678075017$$
$$x_{71} = 68.1222705717507$$
$$x_{72} = -57.7154777250412$$
$$x_{73} = -13.7327104235176$$
$$x_{74} = -7.99516931765348$$
$$x_{75} = 60.1655437901152$$
$$x_{76} = 38.1742200902028$$
$$x_{77} = -71.7807890035278$$
$$x_{78} = -53.8823230356843$$
$$x_{79} = 16.1824202315257$$
$$x_{80} = 6.28318530717959$$
$$x_{81} = -89.9391365139768$$
$$x_{82} = -17.8566809065234$$
$$x_{83} = -52.414295892353$$
$$x_{84} = -35.7242388879548$$
$$x_{85} = -83.8302409743508$$
$$x_{86} = -25.8922856011411$$
$$x_{87} = -3.89760750085627$$
$$x_{88} = -42.007459633629$$
$$x_{89} = -1.97040802141203$$
$$x_{90} = 61.6647595690351$$
$$x_{91} = -37.6991118430775$$
$$x_{92} = -20.5628905052217$$
$$x_{93} = 38.8658506940265$$
$$x_{94} = -74.6378291927198$$
$$x_{95} = 34.0813954679699$$
$$x_{96} = -44.7421071306256$$
$$x_{97} = 10.183352203933$$
$$x_{98} = -11.8048062508305$$
Signos de extremos en los puntos:
(99.10338944434504, -9.16755678280232e-5)
(-39.84790967028563, 4.93438536675773e-5)
(-61.83908139573289, -2.04888936520947e-5)
(18.08847491384093, 0.00224485338972649)
(-77.7794307044932, -0.000121417965131701)
(84.06263174100542, -0.000103945735073535)
(-27.79808894721844, 0.00127946538324165)
(32.17559398909525, 0.000709502037172672)
(13.9924982173739, -0.00459834428604798)
(50.26548245743669, 4.55872309075424e-32)
(86.25019259274696, 0.000121034791431435)
(82.15677372988101, -0.000146480820727575)
(-96.222327954753, -2.96709875956392e-5)
(-31.890940376799165, -0.000972134223577643)
(77.11218986595486, 0.000151420278579595)
(-9.89851698018621, 0.0100889814081959)
(46.131103844353476, 3.68176986959297e-5)
(-49.789527942103184, -0.000398831520031318)
(58.52315956334457, -8.02098154247023e-5)
(48.29067115240387, -0.000117802921198254)
(-97.86477272603547, 0.000103232175800995)
(90.11342843889607, 9.64863098875592e-6)
(25.132741228718345, 4.55872309075424e-32)
(-337.1431359753354, 6.8931258539892e-7)
(-83.10859216407933, -0.000130358226182873)
(-93.77199730863786, -0.000112440160811573)
(92.27305180477565, -3.22651670179294e-5)
(98.14938094798033, -7.62496366586671e-5)
(-87.76572883078643, 6.66626114855716e-5)
(-47.883723734812364, -0.000320357367995719)
(-63.998677423194835, 6.70719948084586e-5)
(-99.77062566201775, 7.37917041030685e-5)
(-75.87356998887911, -0.000171745859811728)
(-20.016089559623246, 0.000685672084160346)
(62.07141417437828, 0.000190646361068794)
(-79.70666673913081, -4.32408521921406e-5)
(63.8245421747774, -1.92339804006236e-5)
(42.26768136625329, 0.00050397645323833)
(63.998677423194835, 6.70719948084586e-5)
(96.222327954753, -2.96709875956392e-5)
(54.16694861341509, -0.000250347053176728)
(-29.44100300265515, -0.000316938263866365)
(88.16315273206231, 6.60629604875086e-5)
(-67.94795754163643, 5.95018556118178e-5)
(52.23995690045012, -0.000100664688443397)
(-33.79685953935841, -0.000643064781307468)
(-55.78819915943293, 0.000236007973530126)
(72.25663103256524, -8.64051087166256e-32)
(70.28187454729652, 5.56156095614604e-5)
(-46.131103844353476, 3.68176986959297e-5)
(12.089271724279067, -0.00676416395574239)
(20.823786680091768, -0.000633514054277227)
(-69.87497968888299, 0.000150441897276682)
(28.274333882308138, -3.36030626263042e-32)
(76.15818429034358, 0.000126642436777823)
(24.13990403646829, -0.000134453418908558)
(-5.8044523527552885, -0.0293299254264352)
(-91.86618425114301, -8.70365059197808e-5)
(8.947148737195361, 0.0123481045066944)
(94.2477796076938, -2.76608127503434e-32)
(21.991148575128552, 3.18011517576988e-32)
(30.24869084904994, 0.000300238888642602)
(-15.707963267948966, 2.14546055218345e-32)
(4.306575055627461, -0.01480541654302)
(40.08012083998671, -0.000457247325920427)
(2.147652819016546, 0.0169774263518496)
(80.25379151664309, -0.000139797400079832)
(-24.37189402558045, 0.00123658252370543)
(74.2311533708957, 4.98552656932231e-5)
(-61.40416780750172, -0.000238799639789511)
(68.12227057175068, -1.68836518996505e-5)
(-57.71547772504125, 8.24704616586205e-5)
(-13.73271042351756, 0.001456639487899)
(-7.995169317653476, 0.0140817304578434)
(60.165543790115244, 0.00027313088205438)
(38.17422009020279, -0.000678457889834131)
(-71.78078900352784, 0.000191889272984975)
(-53.88232303568432, 0.000340544080661994)
(16.182420231525704, 0.00377529944960238)
(6.283185307179586, 4.55872309075424e-32)
(-89.93913651397678, -3.39614509813883e-5)
(-17.856680906523444, -0.000245719629791089)
(-52.41429589235299, 2.85196991479209e-5)
(-35.72423888795484, -0.000215256001639148)
(-83.83024097435082, 1.11491888716527e-5)
(-25.89228560114108, 0.00109562589867959)
(-3.897607500856274, -0.0483009008297943)
(-42.00745963362896, -0.000155678700747426)
(-1.970408021412031, -0.0706064843260395)
(61.664759569035084, 7.22452117010313e-5)
(-37.69911184307752, 1.33753265901809e-31)
(-20.56289050522172, 0.00212933941847605)
(38.86585069402647, 0.000181863409840976)
(-74.63782919271982, 0.000131854322012016)
(34.08139546796991, 0.000851191193328507)
(-44.742107130625634, 0.000366924941324184)
(10.183352203933012, -0.00708213023388379)
(-11.804806250830499, 0.00527041447316484)
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:
$$x_{1} = 99.103389444345$$
$$x_{2} = -61.8390813957329$$
$$x_{3} = -77.7794307044932$$
$$x_{4} = 84.0626317410054$$
$$x_{5} = 13.9924982173739$$
$$x_{6} = 50.2654824574367$$
$$x_{7} = 82.156773729881$$
$$x_{8} = -96.222327954753$$
$$x_{9} = -31.8909403767992$$
$$x_{10} = -49.7895279421032$$
$$x_{11} = 58.5231595633446$$
$$x_{12} = 48.2906711524039$$
$$x_{13} = 25.1327412287183$$
$$x_{14} = -83.1085921640793$$
$$x_{15} = -93.7719973086379$$
$$x_{16} = 92.2730518047756$$
$$x_{17} = 98.1493809479803$$
$$x_{18} = -47.8837237348124$$
$$x_{19} = -75.8735699888791$$
$$x_{20} = -79.7066667391308$$
$$x_{21} = 63.8245421747774$$
$$x_{22} = 96.222327954753$$
$$x_{23} = 54.1669486134151$$
$$x_{24} = -29.4410030026552$$
$$x_{25} = 52.2399569004501$$
$$x_{26} = -33.7968595393584$$
$$x_{27} = 12.0892717242791$$
$$x_{28} = 20.8237866800918$$
$$x_{29} = 24.1399040364683$$
$$x_{30} = -5.80445235275529$$
$$x_{31} = -91.866184251143$$
$$x_{32} = 94.2477796076938$$
$$x_{33} = 4.30657505562746$$
$$x_{34} = 40.0801208399867$$
$$x_{35} = 80.2537915166431$$
$$x_{36} = -61.4041678075017$$
$$x_{37} = 68.1222705717507$$
$$x_{38} = 38.1742200902028$$
$$x_{39} = 6.28318530717959$$
$$x_{40} = -89.9391365139768$$
$$x_{41} = -17.8566809065234$$
$$x_{42} = -35.7242388879548$$
$$x_{43} = -3.89760750085627$$
$$x_{44} = -42.007459633629$$
$$x_{45} = -1.97040802141203$$
$$x_{46} = -37.6991118430775$$
$$x_{47} = 10.183352203933$$
Puntos máximos de la función:
$$x_{47} = -39.8479096702856$$
$$x_{47} = 18.0884749138409$$
$$x_{47} = -27.7980889472184$$
$$x_{47} = 32.1755939890952$$
$$x_{47} = 86.250192592747$$
$$x_{47} = 77.1121898659549$$
$$x_{47} = -9.89851698018621$$
$$x_{47} = 46.1311038443535$$
$$x_{47} = -97.8647727260355$$
$$x_{47} = 90.1134284388961$$
$$x_{47} = -337.143135975335$$
$$x_{47} = -87.7657288307864$$
$$x_{47} = -63.9986774231948$$
$$x_{47} = -99.7706256620177$$
$$x_{47} = -20.0160895596232$$
$$x_{47} = 62.0714141743783$$
$$x_{47} = 42.2676813662533$$
$$x_{47} = 63.9986774231948$$
$$x_{47} = 88.1631527320623$$
$$x_{47} = -67.9479575416364$$
$$x_{47} = -55.7881991594329$$
$$x_{47} = 72.2566310325652$$
$$x_{47} = 70.2818745472965$$
$$x_{47} = -46.1311038443535$$
$$x_{47} = -69.874979688883$$
$$x_{47} = 28.2743338823081$$
$$x_{47} = 76.1581842903436$$
$$x_{47} = 8.94714873719536$$
$$x_{47} = 21.9911485751286$$
$$x_{47} = 30.2486908490499$$
$$x_{47} = -15.707963267949$$
$$x_{47} = 2.14765281901655$$
$$x_{47} = -24.3718940255804$$
$$x_{47} = 74.2311533708957$$
$$x_{47} = -57.7154777250412$$
$$x_{47} = -13.7327104235176$$
$$x_{47} = -7.99516931765348$$
$$x_{47} = 60.1655437901152$$
$$x_{47} = -71.7807890035278$$
$$x_{47} = -53.8823230356843$$
$$x_{47} = 16.1824202315257$$
$$x_{47} = -52.414295892353$$
$$x_{47} = -83.8302409743508$$
$$x_{47} = -25.8922856011411$$
$$x_{47} = 61.6647595690351$$
$$x_{47} = -20.5628905052217$$
$$x_{47} = 38.8658506940265$$
$$x_{47} = -74.6378291927198$$
$$x_{47} = 34.0813954679699$$
$$x_{47} = -44.7421071306256$$
$$x_{47} = -11.8048062508305$$
Decrece en los intervalos
$$\left[99.103389444345, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -96.222327954753\right]$$
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
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (sin(10*x)*sin(3*x))/x^2, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
$$\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}} = \frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}}$$
- No
$$\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}} = - \frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}} = \frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}}$$
- No
$$\frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}} = - \frac{\sin{\left(3 x \right)} \sin{\left(10 x \right)}}{x^{2}}$$
- No
es decir, función
no es
par ni impar