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Otras calculadoras
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- 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 =:
-
-sqrt(-1+x^2)
-
(e^x)*sin(x)
-
-exp(-2*x)-exp(3*x)
-
-exp(-3*x)-exp(2*x)
-
Expresiones idénticas
- sin^ dos (cero .6x)cos(cero .3x^ dos)- cero . dos
- seno de al cuadrado (0.6x) coseno de (0.3x al cuadrado ) menos 0.2
- seno de en el grado dos (cero .6x) coseno de (cero .3x en el grado dos) menos cero . dos
- sin2(0.6x)cos(0.3x2)-0.2
- sin20.6xcos0.3x2-0.2
- sin²(0.6x)cos(0.3x²)-0.2
- sin en el grado 2(0.6x)cos(0.3x en el grado 2)-0.2
- sin^20.6xcos0.3x^2-0.2
-
Expresiones semejantes
- sin^2(0.6x)cos(0.3x^2)+0.2
-
Expresiones con funciones
- Seno sin
- sin((x+2)^(1/2)-2^(1/2))
- sin(sqrt(x+2)-sqrt2)
- sin(x)+|sin(x)|
- sin(x)^2+cos(x)^2-10*x
- sin(3*x)^(5)
- Coseno cos
- cos(x)+4/5
- cos(2x^2+3)+10x
- cos(200000*pi*t)+cos(20000*pi*t)*cos(200000*pi*t)+cos(40000*pi*t)*cos(200000*pi*t)/5
- cos(a)*sin(a)
- cos^2(20*x)/sin(40x)
Gráfico de la función y = sin^2(0.6x)cos(0.3x^2)-0.2
Solución
Ha introducido
[src]
/ 2\
2/3*x\ |3*x | 1
f(x) = sin |---|*cos|----| - -
\ 5 / \ 10 / 5
$$f{\left(x \right)} = \sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}$$
f = sin(3*x/5)^2*cos(3*x^2/10) - 1/5
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:
$$\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -58.6340361622731$$
$$x_{2} = 46.2438572735719$$
$$x_{3} = 4.24265637838059$$
$$x_{4} = -76.0597830203121$$
$$x_{5} = -95.1091510965776$$
$$x_{6} = 98.241666763985$$
$$x_{7} = -49.7589282877285$$
$$x_{8} = 75.9997840718253$$
$$x_{9} = 20.0132028439202$$
$$x_{10} = 92.188731379259$$
$$x_{11} = -13.8879759641433$$
$$x_{12} = 88.1597668529561$$
$$x_{13} = 40.8972533755532$$
$$x_{14} = -61.7660700144777$$
$$x_{15} = -72.2377545691392$$
$$x_{16} = 77.5489882018227$$
$$x_{17} = 35.7739321982008$$
$$x_{18} = 96.1285273652439$$
$$x_{19} = 22.0228510294214$$
$$x_{20} = 30.4064665112622$$
$$x_{21} = 6.23742577801944$$
$$x_{22} = -20.0132028439202$$
$$x_{23} = -17.8512076451557$$
$$x_{24} = 56.2710224440924$$
$$x_{25} = -4.42777434041541$$
$$x_{26} = 27.9151547970739$$
$$x_{27} = 18.1808582243302$$
$$x_{28} = -4.24265637838059$$
$$x_{29} = -66.2862823235775$$
$$x_{30} = -27.7617574730486$$
$$x_{31} = 61.7660700144777$$
$$x_{32} = 80.9939185478441$$
$$x_{33} = -8.2076208329856$$
$$x_{34} = -82.1474497208596$$
$$x_{35} = 64.202758572524$$
$$x_{36} = -21.8869886572274$$
$$x_{37} = -71.7497525134781$$
$$x_{38} = 86.9783867431852$$
$$x_{39} = -100.700313554161$$
$$x_{40} = -2.12143949765553$$
$$x_{41} = -87.0474717154031$$
$$x_{42} = -89.9003931445741$$
$$x_{43} = -23.8752347374123$$
$$x_{44} = -37.4459217319169$$
$$x_{45} = 56.2008685038142$$
$$x_{46} = 49.3357984688241$$
$$x_{47} = -70.6344426679678$$
$$x_{48} = -307.570952740127$$
$$x_{49} = 82.7377187373806$$
$$x_{50} = -7.63407149021458$$
$$x_{51} = 72.3778263457328$$
$$x_{52} = -39.8404584632007$$
$$x_{53} = -63.7630176520401$$
$$x_{54} = -6.23742577801944$$
$$x_{55} = -56.6322740616215$$
$$x_{56} = -45.5809738487768$$
$$x_{57} = 48.2517425077724$$
$$x_{58} = 53.9950555510981$$
$$x_{59} = -11.9380482663461$$
$$x_{60} = 64.9170564165521$$
$$x_{61} = -97.7485108470373$$
$$x_{62} = -35.7739321982008$$
$$x_{63} = -33.8726869406973$$
$$x_{64} = 17.0106484361766$$
$$x_{65} = -42.6758991338471$$
$$x_{66} = 8.2076208329856$$
$$x_{67} = 2.12143949765553$$
$$x_{68} = 39.310151080048$$
$$x_{69} = 14.5898084613639$$
$$x_{70} = -92.9129469075419$$
$$x_{71} = -29.7311428377902$$
$$x_{72} = 43.4644891019756$$
$$x_{73} = -69.8254725579034$$
o sea hay que resolver la ecuación:
$$\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = -58.6340361622731$$
$$x_{2} = 46.2438572735719$$
$$x_{3} = 4.24265637838059$$
$$x_{4} = -76.0597830203121$$
$$x_{5} = -95.1091510965776$$
$$x_{6} = 98.241666763985$$
$$x_{7} = -49.7589282877285$$
$$x_{8} = 75.9997840718253$$
$$x_{9} = 20.0132028439202$$
$$x_{10} = 92.188731379259$$
$$x_{11} = -13.8879759641433$$
$$x_{12} = 88.1597668529561$$
$$x_{13} = 40.8972533755532$$
$$x_{14} = -61.7660700144777$$
$$x_{15} = -72.2377545691392$$
$$x_{16} = 77.5489882018227$$
$$x_{17} = 35.7739321982008$$
$$x_{18} = 96.1285273652439$$
$$x_{19} = 22.0228510294214$$
$$x_{20} = 30.4064665112622$$
$$x_{21} = 6.23742577801944$$
$$x_{22} = -20.0132028439202$$
$$x_{23} = -17.8512076451557$$
$$x_{24} = 56.2710224440924$$
$$x_{25} = -4.42777434041541$$
$$x_{26} = 27.9151547970739$$
$$x_{27} = 18.1808582243302$$
$$x_{28} = -4.24265637838059$$
$$x_{29} = -66.2862823235775$$
$$x_{30} = -27.7617574730486$$
$$x_{31} = 61.7660700144777$$
$$x_{32} = 80.9939185478441$$
$$x_{33} = -8.2076208329856$$
$$x_{34} = -82.1474497208596$$
$$x_{35} = 64.202758572524$$
$$x_{36} = -21.8869886572274$$
$$x_{37} = -71.7497525134781$$
$$x_{38} = 86.9783867431852$$
$$x_{39} = -100.700313554161$$
$$x_{40} = -2.12143949765553$$
$$x_{41} = -87.0474717154031$$
$$x_{42} = -89.9003931445741$$
$$x_{43} = -23.8752347374123$$
$$x_{44} = -37.4459217319169$$
$$x_{45} = 56.2008685038142$$
$$x_{46} = 49.3357984688241$$
$$x_{47} = -70.6344426679678$$
$$x_{48} = -307.570952740127$$
$$x_{49} = 82.7377187373806$$
$$x_{50} = -7.63407149021458$$
$$x_{51} = 72.3778263457328$$
$$x_{52} = -39.8404584632007$$
$$x_{53} = -63.7630176520401$$
$$x_{54} = -6.23742577801944$$
$$x_{55} = -56.6322740616215$$
$$x_{56} = -45.5809738487768$$
$$x_{57} = 48.2517425077724$$
$$x_{58} = 53.9950555510981$$
$$x_{59} = -11.9380482663461$$
$$x_{60} = 64.9170564165521$$
$$x_{61} = -97.7485108470373$$
$$x_{62} = -35.7739321982008$$
$$x_{63} = -33.8726869406973$$
$$x_{64} = 17.0106484361766$$
$$x_{65} = -42.6758991338471$$
$$x_{66} = 8.2076208329856$$
$$x_{67} = 2.12143949765553$$
$$x_{68} = 39.310151080048$$
$$x_{69} = 14.5898084613639$$
$$x_{70} = -92.9129469075419$$
$$x_{71} = -29.7311428377902$$
$$x_{72} = 43.4644891019756$$
$$x_{73} = -69.8254725579034$$
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
$$- \frac{3 x \sin^{2}{\left(\frac{3 x}{5} \right)} \sin{\left(\frac{3 x^{2}}{10} \right)}}{5} + \frac{6 \sin{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 12.1229824381761$$
$$x_{2} = -65.6843933727047$$
$$x_{3} = 3.13113402113334$$
$$x_{4} = 36.7819344329021$$
$$x_{5} = -31.7237228244603$$
$$x_{6} = 60.1068937789405$$
$$x_{7} = 88.8543310730566$$
$$x_{8} = 52.2615470317133$$
$$x_{9} = 94.4598843953214$$
$$x_{10} = -89.3310053485053$$
$$x_{11} = -73.5135729831908$$
$$x_{12} = -15.707963267949$$
$$x_{13} = -11.6944767794746$$
$$x_{14} = -83.7448097110512$$
$$x_{15} = -91.7576303504263$$
$$x_{16} = -19.936259585824$$
$$x_{17} = 95.9422194402264$$
$$x_{18} = 14.09408526094$$
$$x_{19} = -96.4320105798712$$
$$x_{20} = 37.879473611331$$
$$x_{21} = 99.4837673636768$$
$$x_{22} = 48.0007642143196$$
$$x_{23} = -33.6306745660586$$
$$x_{24} = 50.2364958899188$$
$$x_{25} = -21.721908390206$$
$$x_{26} = -80.6419446072301$$
$$x_{27} = -30.0065529310323$$
$$x_{28} = -10.471975511966$$
$$x_{29} = 99.7436693920312$$
$$x_{30} = 4.33188285755461$$
$$x_{31} = -55.8621401192102$$
$$x_{32} = -52.5880166938635$$
$$x_{33} = -37.7414741229502$$
$$x_{34} = 44.2524858552252$$
$$x_{35} = -68.0708694375468$$
$$x_{36} = -85.3116773421505$$
$$x_{37} = -5.82391963347914$$
$$x_{38} = -7.92441056952944$$
$$x_{39} = 31.4159265358979$$
$$x_{40} = 94.2477796076938$$
$$x_{41} = -50.2364958899188$$
$$x_{42} = -26.0420466134315$$
$$x_{43} = 38.1541974640602$$
$$x_{44} = 73.8657984508439$$
$$x_{45} = -35.7389993261695$$
$$x_{46} = -63.0067797559502$$
$$x_{47} = 78.4595441521704$$
$$x_{48} = 82.248610897273$$
$$x_{49} = 23.9975905109106$$
$$x_{50} = 1.57411458733581$$
$$x_{51} = 5.82391963347914$$
$$x_{52} = -77.9997678736386$$
$$x_{53} = -57.8052881194235$$
$$x_{54} = 68.0154824215815$$
$$x_{55} = 31.7237228244603$$
$$x_{56} = 58.2511405363585$$
$$x_{57} = 62.2442013680375$$
$$x_{58} = -13.7223199085531$$
$$x_{59} = 69.6318519740652$$
$$x_{60} = -73.0756715190628$$
$$x_{61} = -53.7622175617027$$
$$x_{62} = -94.0098795829772$$
$$x_{63} = 72.9339457850776$$
$$x_{64} = 80.2515741614971$$
$$x_{65} = 76.5101444891375$$
$$x_{66} = 89.0078563237483$$
$$x_{67} = -89.2149356986415$$
$$x_{68} = 17.4331276401965$$
$$x_{69} = -63.7440034171542$$
$$x_{70} = -41.8835996793582$$
$$x_{71} = 22.1924510958835$$
$$x_{72} = 16.2184676662057$$
$$x_{73} = -26.1799387799149$$
$$x_{74} = 54.2465002870835$$
$$x_{75} = -47.3498033640974$$
$$x_{76} = -39.6328034914763$$
$$x_{77} = -4.33188285755461$$
$$x_{78} = 7.92441056952944$$
$$x_{79} = -97.9401776591651$$
$$x_{80} = 20.1924350450536$$
$$x_{81} = 83.8942472369898$$
$$x_{82} = 86.1055874271444$$
$$x_{83} = 0$$
$$x_{84} = 45.7630668857082$$
$$x_{85} = 41.9735858539592$$
$$x_{86} = 10.1128795090719$$
$$x_{87} = -47.8919861327274$$
$$x_{88} = -59.7576352717305$$
$$x_{89} = 66.0023260880075$$
$$x_{90} = 20.943951023932$$
$$x_{91} = 34.2466972080798$$
$$x_{92} = 56.2354443005911$$
$$x_{93} = -69.9318160427244$$
$$x_{94} = -99.6400212424982$$
$$x_{95} = -81.9298215839724$$
$$x_{96} = -75.7538875442276$$
$$x_{97} = 26.186626750821$$
$$x_{98} = -27.8402787235296$$
$$x_{99} = 62.8609426250491$$
$$x_{100} = -1.57411458733581$$
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:
$$x_{1} = 3.13113402113334$$
$$x_{2} = 36.7819344329021$$
$$x_{3} = 60.1068937789405$$
$$x_{4} = 52.2615470317133$$
$$x_{5} = -15.707963267949$$
$$x_{6} = -11.6944767794746$$
$$x_{7} = 95.9422194402264$$
$$x_{8} = 14.09408526094$$
$$x_{9} = 37.879473611331$$
$$x_{10} = 50.2364958899188$$
$$x_{11} = -21.721908390206$$
$$x_{12} = -80.6419446072301$$
$$x_{13} = -10.471975511966$$
$$x_{14} = 44.2524858552252$$
$$x_{15} = -85.3116773421505$$
$$x_{16} = -5.82391963347914$$
$$x_{17} = 31.4159265358979$$
$$x_{18} = 94.2477796076938$$
$$x_{19} = -50.2364958899188$$
$$x_{20} = -26.0420466134315$$
$$x_{21} = 38.1541974640602$$
$$x_{22} = 73.8657984508439$$
$$x_{23} = -63.0067797559502$$
$$x_{24} = 23.9975905109106$$
$$x_{25} = 5.82391963347914$$
$$x_{26} = -77.9997678736386$$
$$x_{27} = -57.8052881194235$$
$$x_{28} = 69.6318519740652$$
$$x_{29} = 80.2515741614971$$
$$x_{30} = 76.5101444891375$$
$$x_{31} = 89.0078563237483$$
$$x_{32} = 17.4331276401965$$
$$x_{33} = 22.1924510958835$$
$$x_{34} = 16.2184676662057$$
$$x_{35} = 54.2465002870835$$
$$x_{36} = 20.1924350450536$$
$$x_{37} = 0$$
$$x_{38} = -47.8919861327274$$
$$x_{39} = -59.7576352717305$$
$$x_{40} = 20.943951023932$$
$$x_{41} = -69.9318160427244$$
$$x_{42} = -81.9298215839724$$
$$x_{43} = 26.186626750821$$
$$x_{44} = 62.8609426250491$$
Puntos máximos de la función:
$$x_{44} = 12.1229824381761$$
$$x_{44} = -65.6843933727047$$
$$x_{44} = -31.7237228244603$$
$$x_{44} = 88.8543310730566$$
$$x_{44} = 94.4598843953214$$
$$x_{44} = -89.3310053485053$$
$$x_{44} = -73.5135729831908$$
$$x_{44} = -83.7448097110512$$
$$x_{44} = -91.7576303504263$$
$$x_{44} = -19.936259585824$$
$$x_{44} = -96.4320105798712$$
$$x_{44} = 99.4837673636768$$
$$x_{44} = 48.0007642143196$$
$$x_{44} = -33.6306745660586$$
$$x_{44} = -30.0065529310323$$
$$x_{44} = 99.7436693920312$$
$$x_{44} = 4.33188285755461$$
$$x_{44} = -55.8621401192102$$
$$x_{44} = -52.5880166938635$$
$$x_{44} = -37.7414741229502$$
$$x_{44} = -68.0708694375468$$
$$x_{44} = -7.92441056952944$$
$$x_{44} = -35.7389993261695$$
$$x_{44} = 78.4595441521704$$
$$x_{44} = 82.248610897273$$
$$x_{44} = 1.57411458733581$$
$$x_{44} = 68.0154824215815$$
$$x_{44} = 31.7237228244603$$
$$x_{44} = 58.2511405363585$$
$$x_{44} = 62.2442013680375$$
$$x_{44} = -13.7223199085531$$
$$x_{44} = -73.0756715190628$$
$$x_{44} = -53.7622175617027$$
$$x_{44} = -94.0098795829772$$
$$x_{44} = 72.9339457850776$$
$$x_{44} = -89.2149356986415$$
$$x_{44} = -63.7440034171542$$
$$x_{44} = -41.8835996793582$$
$$x_{44} = -26.1799387799149$$
$$x_{44} = -47.3498033640974$$
$$x_{44} = -39.6328034914763$$
$$x_{44} = -4.33188285755461$$
$$x_{44} = 7.92441056952944$$
$$x_{44} = -97.9401776591651$$
$$x_{44} = 83.8942472369898$$
$$x_{44} = 86.1055874271444$$
$$x_{44} = 45.7630668857082$$
$$x_{44} = 41.9735858539592$$
$$x_{44} = 10.1128795090719$$
$$x_{44} = 66.0023260880075$$
$$x_{44} = 34.2466972080798$$
$$x_{44} = 56.2354443005911$$
$$x_{44} = -99.6400212424982$$
$$x_{44} = -75.7538875442276$$
$$x_{44} = -27.8402787235296$$
$$x_{44} = -1.57411458733581$$
Decrece en los intervalos
$$\left[95.9422194402264, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -85.3116773421505\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{3 x \sin^{2}{\left(\frac{3 x}{5} \right)} \sin{\left(\frac{3 x^{2}}{10} \right)}}{5} + \frac{6 \sin{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)}}{5} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 12.1229824381761$$
$$x_{2} = -65.6843933727047$$
$$x_{3} = 3.13113402113334$$
$$x_{4} = 36.7819344329021$$
$$x_{5} = -31.7237228244603$$
$$x_{6} = 60.1068937789405$$
$$x_{7} = 88.8543310730566$$
$$x_{8} = 52.2615470317133$$
$$x_{9} = 94.4598843953214$$
$$x_{10} = -89.3310053485053$$
$$x_{11} = -73.5135729831908$$
$$x_{12} = -15.707963267949$$
$$x_{13} = -11.6944767794746$$
$$x_{14} = -83.7448097110512$$
$$x_{15} = -91.7576303504263$$
$$x_{16} = -19.936259585824$$
$$x_{17} = 95.9422194402264$$
$$x_{18} = 14.09408526094$$
$$x_{19} = -96.4320105798712$$
$$x_{20} = 37.879473611331$$
$$x_{21} = 99.4837673636768$$
$$x_{22} = 48.0007642143196$$
$$x_{23} = -33.6306745660586$$
$$x_{24} = 50.2364958899188$$
$$x_{25} = -21.721908390206$$
$$x_{26} = -80.6419446072301$$
$$x_{27} = -30.0065529310323$$
$$x_{28} = -10.471975511966$$
$$x_{29} = 99.7436693920312$$
$$x_{30} = 4.33188285755461$$
$$x_{31} = -55.8621401192102$$
$$x_{32} = -52.5880166938635$$
$$x_{33} = -37.7414741229502$$
$$x_{34} = 44.2524858552252$$
$$x_{35} = -68.0708694375468$$
$$x_{36} = -85.3116773421505$$
$$x_{37} = -5.82391963347914$$
$$x_{38} = -7.92441056952944$$
$$x_{39} = 31.4159265358979$$
$$x_{40} = 94.2477796076938$$
$$x_{41} = -50.2364958899188$$
$$x_{42} = -26.0420466134315$$
$$x_{43} = 38.1541974640602$$
$$x_{44} = 73.8657984508439$$
$$x_{45} = -35.7389993261695$$
$$x_{46} = -63.0067797559502$$
$$x_{47} = 78.4595441521704$$
$$x_{48} = 82.248610897273$$
$$x_{49} = 23.9975905109106$$
$$x_{50} = 1.57411458733581$$
$$x_{51} = 5.82391963347914$$
$$x_{52} = -77.9997678736386$$
$$x_{53} = -57.8052881194235$$
$$x_{54} = 68.0154824215815$$
$$x_{55} = 31.7237228244603$$
$$x_{56} = 58.2511405363585$$
$$x_{57} = 62.2442013680375$$
$$x_{58} = -13.7223199085531$$
$$x_{59} = 69.6318519740652$$
$$x_{60} = -73.0756715190628$$
$$x_{61} = -53.7622175617027$$
$$x_{62} = -94.0098795829772$$
$$x_{63} = 72.9339457850776$$
$$x_{64} = 80.2515741614971$$
$$x_{65} = 76.5101444891375$$
$$x_{66} = 89.0078563237483$$
$$x_{67} = -89.2149356986415$$
$$x_{68} = 17.4331276401965$$
$$x_{69} = -63.7440034171542$$
$$x_{70} = -41.8835996793582$$
$$x_{71} = 22.1924510958835$$
$$x_{72} = 16.2184676662057$$
$$x_{73} = -26.1799387799149$$
$$x_{74} = 54.2465002870835$$
$$x_{75} = -47.3498033640974$$
$$x_{76} = -39.6328034914763$$
$$x_{77} = -4.33188285755461$$
$$x_{78} = 7.92441056952944$$
$$x_{79} = -97.9401776591651$$
$$x_{80} = 20.1924350450536$$
$$x_{81} = 83.8942472369898$$
$$x_{82} = 86.1055874271444$$
$$x_{83} = 0$$
$$x_{84} = 45.7630668857082$$
$$x_{85} = 41.9735858539592$$
$$x_{86} = 10.1128795090719$$
$$x_{87} = -47.8919861327274$$
$$x_{88} = -59.7576352717305$$
$$x_{89} = 66.0023260880075$$
$$x_{90} = 20.943951023932$$
$$x_{91} = 34.2466972080798$$
$$x_{92} = 56.2354443005911$$
$$x_{93} = -69.9318160427244$$
$$x_{94} = -99.6400212424982$$
$$x_{95} = -81.9298215839724$$
$$x_{96} = -75.7538875442276$$
$$x_{97} = 26.186626750821$$
$$x_{98} = -27.8402787235296$$
$$x_{99} = 62.8609426250491$$
$$x_{100} = -1.57411458733581$$
Signos de extremos en los puntos:
(12.122982438176072, 0.495439634053406)
(-65.68439337270475, 0.780316943676428)
(3.1311340211333363, -1.08999126604949)
(36.78193443290206, -0.204985961287315)
(-31.7237228244603, -0.168050506243539)
(60.10689377894048, -1.19588439917299)
(88.85433107305663, -0.191341641142858)
(52.26154703171333, -0.202917876124195)
(94.45988439532135, -0.18410751714817)
(-89.3310053485053, -0.164002502563617)
(-73.5135729831908, -0.184598138754976)
(-15.707963267948966, -0.2)
(-11.694476779474574, -0.640337701401968)
(-83.7448097110512, -0.199787532971672)
(-91.75763035042625, 0.794126207264612)
(-19.936259585823986, 0.119776909853067)
(95.94221944022644, -0.923047352632901)
(14.094085260939979, -0.875664717267834)
(-96.43201057987125, 0.733767223026068)
(37.879473611331, -0.650452170951313)
(99.48376736367679, -0.2)
(48.00076421431957, 0.0515440249608699)
(-33.630674566058595, 0.742493286590499)
(50.236495889918785, -1.11441671326918)
(-21.72190839020604, -0.399209306910576)
(-80.64194460723007, -1.10718977219525)
(-30.006552931032346, 0.35910584967736)
(-10.471975511965978, -0.2)
(99.74366939203122, -0.176072633940323)
(4.3318828575546116, 0.0115793187313457)
(-55.86214011921016, 0.543782261197991)
(-52.58801669386346, -0.182051216332967)
(-37.741474122950194, 0.168972427432094)
(44.25248585522518, -1.17703617181545)
(-68.07086943754685, -0.199999796161617)
(-85.31167734215052, -0.834349949338148)
(-5.823919633479142, -0.287284389954009)
(-7.9244105695294405, 0.798158544445862)
(31.41592653589793, -0.2)
(94.2477796076938, -0.2)
(-50.236495889918785, -1.11441671326918)
(-26.04204661343155, -0.205010729742474)
(38.15419746406023, -0.814406348788211)
(73.86579845084388, -0.309122850092553)
(-35.73899932616946, 0.0700640897824696)
(-63.00677975595021, -0.210508659248956)
(78.4595441521704, -0.1979509813381)
(82.24861089727298, 0.429317438751723)
(23.997590510910637, -1.1329867259313)
(1.5741145873358113, 0.283245446606709)
(5.823919633479142, -0.287284389954009)
(-77.99976787363863, -0.301077139843918)
(-57.8052881194235, -0.215147988824541)
(68.01548242158147, -0.199279610133012)
(31.7237228244603, -0.168050506243539)
(58.251140536358484, -0.0537240222852314)
(62.24420136803749, -0.0811988567497575)
(-13.722319908553107, 0.661370704411558)
(69.6318519740652, -0.850487500941559)
(-73.0756715190628, -0.181733756148313)
(-53.76221756170269, 0.355581748402606)
(-94.00987958297723, -0.179981430174821)
(72.93394578507758, -0.151904392214644)
(80.25157416149706, -0.93227455078432)
(76.51014448913746, -1.08044513961129)
(89.00785632374833, -0.200000582741816)
(-89.21493569864154, -0.185458665369386)
(17.4331276401965, -0.937741164942956)
(-63.74400341715418, 0.0704320815719325)
(-41.883599679358234, -0.199999640286369)
(22.192451095883523, -0.661571265986302)
(16.218467666205665, -0.284711532019328)
(-26.179938779914945, -0.2)
(54.24650028708346, -1.01935942458116)
(-47.34980336409741, -0.18255636863491)
(-39.63280349147627, 0.753275602992737)
(-4.3318828575546116, 0.0115793187313457)
(7.9244105695294405, 0.798158544445862)
(-97.94017765916512, 0.438828621083014)
(20.19243504505362, -0.386054928630667)
(83.8942472369898, -0.195219108292072)
(86.10558742714439, 0.770385386183498)
(0, -1/5)
(45.7630668857082, 0.330628292915112)
(41.97358585395921, -0.198062458158328)
(10.112879509071856, -0.166087039538411)
(-47.89198613272742, -0.397076999169344)
(-59.757635271730514, -1.12688150616752)
(66.00232608800746, 0.694033745010953)
(20.943951023931955, -0.2)
(34.24669720807977, 0.783762087428377)
(56.23544430059111, 0.330538403541457)
(-69.93181604272439, -1.00883448566664)
(-99.64002124249825, -0.191429329624297)
(-81.92982158397238, -1.00029303609286)
(-75.75388754422764, 0.789878017542022)
(26.18662675082099, -0.200000844865974)
(-27.840278723529604, 0.50385476335627)
(62.86094262504909, -0.200146513330916)
(-1.5741145873358113, 0.283245446606709)
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:
$$x_{1} = 3.13113402113334$$
$$x_{2} = 36.7819344329021$$
$$x_{3} = 60.1068937789405$$
$$x_{4} = 52.2615470317133$$
$$x_{5} = -15.707963267949$$
$$x_{6} = -11.6944767794746$$
$$x_{7} = 95.9422194402264$$
$$x_{8} = 14.09408526094$$
$$x_{9} = 37.879473611331$$
$$x_{10} = 50.2364958899188$$
$$x_{11} = -21.721908390206$$
$$x_{12} = -80.6419446072301$$
$$x_{13} = -10.471975511966$$
$$x_{14} = 44.2524858552252$$
$$x_{15} = -85.3116773421505$$
$$x_{16} = -5.82391963347914$$
$$x_{17} = 31.4159265358979$$
$$x_{18} = 94.2477796076938$$
$$x_{19} = -50.2364958899188$$
$$x_{20} = -26.0420466134315$$
$$x_{21} = 38.1541974640602$$
$$x_{22} = 73.8657984508439$$
$$x_{23} = -63.0067797559502$$
$$x_{24} = 23.9975905109106$$
$$x_{25} = 5.82391963347914$$
$$x_{26} = -77.9997678736386$$
$$x_{27} = -57.8052881194235$$
$$x_{28} = 69.6318519740652$$
$$x_{29} = 80.2515741614971$$
$$x_{30} = 76.5101444891375$$
$$x_{31} = 89.0078563237483$$
$$x_{32} = 17.4331276401965$$
$$x_{33} = 22.1924510958835$$
$$x_{34} = 16.2184676662057$$
$$x_{35} = 54.2465002870835$$
$$x_{36} = 20.1924350450536$$
$$x_{37} = 0$$
$$x_{38} = -47.8919861327274$$
$$x_{39} = -59.7576352717305$$
$$x_{40} = 20.943951023932$$
$$x_{41} = -69.9318160427244$$
$$x_{42} = -81.9298215839724$$
$$x_{43} = 26.186626750821$$
$$x_{44} = 62.8609426250491$$
Puntos máximos de la función:
$$x_{44} = 12.1229824381761$$
$$x_{44} = -65.6843933727047$$
$$x_{44} = -31.7237228244603$$
$$x_{44} = 88.8543310730566$$
$$x_{44} = 94.4598843953214$$
$$x_{44} = -89.3310053485053$$
$$x_{44} = -73.5135729831908$$
$$x_{44} = -83.7448097110512$$
$$x_{44} = -91.7576303504263$$
$$x_{44} = -19.936259585824$$
$$x_{44} = -96.4320105798712$$
$$x_{44} = 99.4837673636768$$
$$x_{44} = 48.0007642143196$$
$$x_{44} = -33.6306745660586$$
$$x_{44} = -30.0065529310323$$
$$x_{44} = 99.7436693920312$$
$$x_{44} = 4.33188285755461$$
$$x_{44} = -55.8621401192102$$
$$x_{44} = -52.5880166938635$$
$$x_{44} = -37.7414741229502$$
$$x_{44} = -68.0708694375468$$
$$x_{44} = -7.92441056952944$$
$$x_{44} = -35.7389993261695$$
$$x_{44} = 78.4595441521704$$
$$x_{44} = 82.248610897273$$
$$x_{44} = 1.57411458733581$$
$$x_{44} = 68.0154824215815$$
$$x_{44} = 31.7237228244603$$
$$x_{44} = 58.2511405363585$$
$$x_{44} = 62.2442013680375$$
$$x_{44} = -13.7223199085531$$
$$x_{44} = -73.0756715190628$$
$$x_{44} = -53.7622175617027$$
$$x_{44} = -94.0098795829772$$
$$x_{44} = 72.9339457850776$$
$$x_{44} = -89.2149356986415$$
$$x_{44} = -63.7440034171542$$
$$x_{44} = -41.8835996793582$$
$$x_{44} = -26.1799387799149$$
$$x_{44} = -47.3498033640974$$
$$x_{44} = -39.6328034914763$$
$$x_{44} = -4.33188285755461$$
$$x_{44} = 7.92441056952944$$
$$x_{44} = -97.9401776591651$$
$$x_{44} = 83.8942472369898$$
$$x_{44} = 86.1055874271444$$
$$x_{44} = 45.7630668857082$$
$$x_{44} = 41.9735858539592$$
$$x_{44} = 10.1128795090719$$
$$x_{44} = 66.0023260880075$$
$$x_{44} = 34.2466972080798$$
$$x_{44} = 56.2354443005911$$
$$x_{44} = -99.6400212424982$$
$$x_{44} = -75.7538875442276$$
$$x_{44} = -27.8402787235296$$
$$x_{44} = -1.57411458733581$$
Decrece en los intervalos
$$\left[95.9422194402264, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -85.3116773421505\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}\right) = \left\langle - \frac{6}{5}, \frac{4}{5}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{6}{5}, \frac{4}{5}\right\rangle$$
$$\lim_{x \to \infty}\left(\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}\right) = \left\langle - \frac{6}{5}, \frac{4}{5}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{6}{5}, \frac{4}{5}\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}\right) = \left\langle - \frac{6}{5}, \frac{4}{5}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{6}{5}, \frac{4}{5}\right\rangle$$
$$\lim_{x \to \infty}\left(\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}\right) = \left\langle - \frac{6}{5}, \frac{4}{5}\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{6}{5}, \frac{4}{5}\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(3*x/5)^2*cos(3*x^2/10) - 1/5, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}}{x}\right) = 0$$
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:
$$\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5} = \sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}$$
- Sí
$$\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5} = - \sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} + \frac{1}{5}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5} = \sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5}$$
- Sí
$$\sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} - \frac{1}{5} = - \sin^{2}{\left(\frac{3 x}{5} \right)} \cos{\left(\frac{3 x^{2}}{10} \right)} + \frac{1}{5}$$
- No
es decir, función
es
par