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¿Cómo usar?
- Gráfico de la función y =:
-
(x+5)^2-9
-
x^(7/2)-3
-
x^3/
-
x^4-3*x^2+4
-
Expresiones idénticas
- cos(pi* treinta y tres x)/(uno -(dos *33*x)^ dos)
- coseno de ( número pi multiplicar por 33x) dividir por (1 menos (2 multiplicar por 33 multiplicar por x) al cuadrado )
- coseno de ( número pi multiplicar por treinta y tres x) dividir por (uno menos (dos multiplicar por 33 multiplicar por x) en el grado dos)
- cos(pi*33x)/(1-(2*33*x)2)
- cospi*33x/1-2*33*x2
- cos(pi*33x)/(1-(2*33*x)²)
- cos(pi*33x)/(1-(2*33*x) en el grado 2)
- cos(pi33x)/(1-(233x)^2)
- cos(pi33x)/(1-(233x)2)
- cospi33x/1-233x2
- cospi33x/1-233x^2
- cos(pi*33x) dividir por (1-(2*33*x)^2)
-
Expresiones semejantes
- cos(pi*33x)/(1+(2*33*x)^2)
-
Expresiones con funciones
- Coseno cos
- cos(x*sqrt(5))+sin(x*sqrt(5))
- cos(x)^2/(2*(1+cos(x)^2))
- cos(x)-exp(x)+(0,5)
- cos(x)/2-2
- cos(x)*(1-cos(x)-3*x/5)
Gráfico de la función y = cos(pi*33x)/(1-(2*33*x)^2)
Solución
Ha introducido
[src]
cos(pi*33*x)
f(x) = ------------
2
1 - (66*x)
$$f{\left(x \right)} = \frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}}$$
f = cos(x*(33*pi))/(1 - (66*x)^2)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -0.0151515151515152$$
$$x_{2} = 0.0151515151515152$$
$$x_{1} = -0.0151515151515152$$
$$x_{2} = 0.0151515151515152$$
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:
$$\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{1}{22}$$
Solución numérica
$$x_{1} = -53.4393939393939$$
$$x_{2} = -6.77272727272727$$
$$x_{3} = 144.621212121212$$
$$x_{4} = 0.257575757575758$$
$$x_{5} = -6.98484848484848$$
$$x_{6} = -95.7424242424242$$
$$x_{7} = 168.530303030303$$
$$x_{8} = 8.59090909090909$$
$$x_{9} = -59.3787878787879$$
$$x_{10} = 36.7121212121212$$
$$x_{11} = -99.1969696969697$$
$$x_{12} = 46.7727272727273$$
$$x_{13} = 332.045454545455$$
$$x_{14} = -65.7424242424242$$
$$x_{15} = 8.37878787878788$$
$$x_{16} = 91.7424242424242$$
$$x_{17} = -97.1060606060606$$
$$x_{18} = -63.3787878787879$$
$$x_{19} = -59.0454545454545$$
$$x_{20} = 86.8333333333333$$
$$x_{21} = 52.9242424242424$$
$$x_{22} = -90.6515151515152$$
$$x_{23} = -93.7121212121212$$
$$x_{24} = 59.2575757575758$$
$$x_{25} = -29.9545454545455$$
$$x_{26} = -52.8636363636364$$
$$x_{27} = -10.4393939393939$$
$$x_{28} = 48.8333333333333$$
$$x_{29} = -71.3787878787879$$
$$x_{30} = 38.1060606060606$$
$$x_{31} = -4.68181818181818$$
$$x_{32} = 10.3787878787879$$
$$x_{33} = 67.6212121212121$$
$$x_{34} = 76.469696969697$$
$$x_{35} = -25.1666666666667$$
$$x_{36} = 56.9242424242424$$
$$x_{37} = 97.0454545454545$$
$$x_{38} = 13.2272727272727$$
$$x_{39} = -67.2878787878788$$
$$x_{40} = -43.1060606060606$$
$$x_{41} = -69.2272727272727$$
$$x_{42} = 92.8333333333333$$
$$x_{43} = -79.3787878787879$$
$$x_{44} = -87.2575757575758$$
$$x_{45} = 26.6212121212121$$
$$x_{46} = -88.4090909090909$$
$$x_{47} = -70.3787878787879$$
$$x_{48} = 50.8333333333333$$
$$x_{49} = -43.9848484848485$$
$$x_{50} = -15.0757575757576$$
$$x_{51} = -37.3787878787879$$
$$x_{52} = 54.9848484848485$$
$$x_{53} = 20.6515151515152$$
$$x_{54} = 86.7424242460235$$
$$x_{55} = 190.136363645277$$
$$x_{56} = 91.1363636363636$$
$$x_{57} = -85.1969696969697$$
$$x_{58} = 78.3181818181818$$
$$x_{59} = 65.5606060606061$$
$$x_{60} = -89.1363636363636$$
$$x_{61} = -55.3181818181818$$
$$x_{62} = 8.1969696969697$$
$$x_{63} = 71.9545454545455$$
$$x_{64} = 7.10606060606061$$
$$x_{65} = -142.227272727273$$
$$x_{66} = -38.3787878787881$$
$$x_{67} = 28.6515151515152$$
$$x_{68} = -95.4090909090909$$
$$x_{69} = -61.2878787878788$$
$$x_{70} = 55.530303030303$$
$$x_{71} = 40.6818181818182$$
$$x_{72} = -28.530303030303$$
$$x_{73} = 68.0757575757576$$
$$x_{74} = -43566.2272796674$$
$$x_{75} = -11.2575757575758$$
$$x_{76} = -25.7727272727273$$
o sea hay que resolver la ecuación:
$$\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{1}{22}$$
Solución numérica
$$x_{1} = -53.4393939393939$$
$$x_{2} = -6.77272727272727$$
$$x_{3} = 144.621212121212$$
$$x_{4} = 0.257575757575758$$
$$x_{5} = -6.98484848484848$$
$$x_{6} = -95.7424242424242$$
$$x_{7} = 168.530303030303$$
$$x_{8} = 8.59090909090909$$
$$x_{9} = -59.3787878787879$$
$$x_{10} = 36.7121212121212$$
$$x_{11} = -99.1969696969697$$
$$x_{12} = 46.7727272727273$$
$$x_{13} = 332.045454545455$$
$$x_{14} = -65.7424242424242$$
$$x_{15} = 8.37878787878788$$
$$x_{16} = 91.7424242424242$$
$$x_{17} = -97.1060606060606$$
$$x_{18} = -63.3787878787879$$
$$x_{19} = -59.0454545454545$$
$$x_{20} = 86.8333333333333$$
$$x_{21} = 52.9242424242424$$
$$x_{22} = -90.6515151515152$$
$$x_{23} = -93.7121212121212$$
$$x_{24} = 59.2575757575758$$
$$x_{25} = -29.9545454545455$$
$$x_{26} = -52.8636363636364$$
$$x_{27} = -10.4393939393939$$
$$x_{28} = 48.8333333333333$$
$$x_{29} = -71.3787878787879$$
$$x_{30} = 38.1060606060606$$
$$x_{31} = -4.68181818181818$$
$$x_{32} = 10.3787878787879$$
$$x_{33} = 67.6212121212121$$
$$x_{34} = 76.469696969697$$
$$x_{35} = -25.1666666666667$$
$$x_{36} = 56.9242424242424$$
$$x_{37} = 97.0454545454545$$
$$x_{38} = 13.2272727272727$$
$$x_{39} = -67.2878787878788$$
$$x_{40} = -43.1060606060606$$
$$x_{41} = -69.2272727272727$$
$$x_{42} = 92.8333333333333$$
$$x_{43} = -79.3787878787879$$
$$x_{44} = -87.2575757575758$$
$$x_{45} = 26.6212121212121$$
$$x_{46} = -88.4090909090909$$
$$x_{47} = -70.3787878787879$$
$$x_{48} = 50.8333333333333$$
$$x_{49} = -43.9848484848485$$
$$x_{50} = -15.0757575757576$$
$$x_{51} = -37.3787878787879$$
$$x_{52} = 54.9848484848485$$
$$x_{53} = 20.6515151515152$$
$$x_{54} = 86.7424242460235$$
$$x_{55} = 190.136363645277$$
$$x_{56} = 91.1363636363636$$
$$x_{57} = -85.1969696969697$$
$$x_{58} = 78.3181818181818$$
$$x_{59} = 65.5606060606061$$
$$x_{60} = -89.1363636363636$$
$$x_{61} = -55.3181818181818$$
$$x_{62} = 8.1969696969697$$
$$x_{63} = 71.9545454545455$$
$$x_{64} = 7.10606060606061$$
$$x_{65} = -142.227272727273$$
$$x_{66} = -38.3787878787881$$
$$x_{67} = 28.6515151515152$$
$$x_{68} = -95.4090909090909$$
$$x_{69} = -61.2878787878788$$
$$x_{70} = 55.530303030303$$
$$x_{71} = 40.6818181818182$$
$$x_{72} = -28.530303030303$$
$$x_{73} = 68.0757575757576$$
$$x_{74} = -43566.2272796674$$
$$x_{75} = -11.2575757575758$$
$$x_{76} = -25.7727272727273$$
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
$$\frac{8712 x \cos{\left(x 33 \pi \right)}}{\left(1 - \left(66 x\right)^{2}\right)^{2}} - \frac{33 \pi \sin{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -7.63633926847275$$
$$x_{2} = 19.9999906959359$$
$$x_{3} = -77.9999976143442$$
$$x_{4} = 2.36355763288854$$
$$x_{5} = 71.9999974155395$$
$$x_{6} = -95.9999980616547$$
$$x_{7} = -81.9999977307176$$
$$x_{8} = 87.9999978854414$$
$$x_{9} = 45.9999959547572$$
$$x_{10} = 23.999992246615$$
$$x_{11} = -2.30294950019578$$
$$x_{12} = 95.9999980616547$$
$$x_{13} = -55.9999966771221$$
$$x_{14} = -25.9999928430298$$
$$x_{15} = -13.9999867084698$$
$$x_{16} = -61.999996998691$$
$$x_{17} = -87.9999978854414$$
$$x_{18} = 17.9999896621493$$
$$x_{19} = -9.66664741683294$$
$$x_{20} = -45.9999959547572$$
$$x_{21} = -67.9999972635124$$
$$x_{22} = -17.9999896621493$$
$$x_{23} = 31.9999941849626$$
$$x_{24} = -35.9999948310781$$
$$x_{25} = 63.9999970924819$$
$$x_{26} = 9.9999813918312$$
$$x_{27} = -59.9999968986473$$
$$x_{28} = -73.9999974853898$$
$$x_{29} = -23.999992246615$$
$$x_{30} = 57.9999967917041$$
$$x_{31} = 43.9999957708824$$
$$x_{32} = 77.9999976143442$$
$$x_{33} = 73.9999974853898$$
$$x_{34} = -19.9999906959359$$
$$x_{35} = 61.999996998691$$
$$x_{36} = 8.0302798578292$$
$$x_{37} = 17.3333225978722$$
$$x_{38} = 81.9999977307176$$
$$x_{39} = 65.9999971805885$$
$$x_{40} = 53.9999965540525$$
$$x_{41} = 55.9999966771221$$
$$x_{42} = 67.9999972635124$$
$$x_{43} = 10.9999830834915$$
$$x_{44} = -99.9999981391885$$
$$x_{45} = 91.9999979773788$$
$$x_{46} = 59.9999968986473$$
$$x_{47} = -63.9999970924819$$
$$x_{48} = -39.9999953479705$$
$$x_{49} = 37.9999951031268$$
$$x_{50} = 39.9999953479705$$
$$x_{51} = 83.9999977847482$$
$$x_{52} = 49.9999962783767$$
$$x_{53} = -9.9999813918312$$
$$x_{54} = 69.9999973416978$$
$$x_{55} = 33.9999945270238$$
$$x_{56} = -41.9999955694958$$
$$x_{57} = 29.9999937972931$$
$$x_{58} = 27.9999933542423$$
$$x_{59} = -43.9999957708824$$
$$x_{60} = 35.9999948310781$$
$$x_{61} = 13.9999867084698$$
$$x_{62} = 99.9999981391885$$
$$x_{63} = -11.9999844932065$$
$$x_{64} = 47.999996123309$$
$$x_{65} = 21.999991541761$$
$$x_{66} = -15.9999883699152$$
$$x_{67} = 0$$
$$x_{68} = -85.9999978362657$$
$$x_{69} = -31.9999941849626$$
$$x_{70} = 11.9999844932065$$
$$x_{71} = -69.9999973416978$$
$$x_{72} = -93.9999980204133$$
$$x_{73} = 89.9999979324316$$
$$x_{74} = -49.9999962783767$$
$$x_{75} = -21.999991541761$$
$$x_{76} = -75.9999975515638$$
$$x_{77} = 41.9999955694958$$
$$x_{78} = -79.9999976739856$$
$$x_{79} = -8.0302798578292$$
$$x_{80} = -57.9999967917041$$
$$x_{81} = 97.9999981012127$$
$$x_{82} = -53.9999965540525$$
$$x_{83} = 79.9999976739856$$
$$x_{84} = -97.9999981012127$$
$$x_{85} = -83.9999977847482$$
$$x_{86} = -91.9999979773788$$
$$x_{87} = -65.9999971805885$$
$$x_{88} = -51.9999964215161$$
$$x_{89} = 85.9999978362657$$
$$x_{90} = 15.9999883699152$$
$$x_{91} = 25.9999928430298$$
$$x_{92} = -37.9999951031268$$
$$x_{93} = -29.9999937972931$$
$$x_{94} = 93.9999980204133$$
$$x_{95} = 75.9999975515638$$
$$x_{96} = 51.9999964215161$$
$$x_{97} = -89.9999979324316$$
$$x_{98} = -27.9999933542423$$
$$x_{99} = -47.999996123309$$
$$x_{100} = -33.9999945270238$$
$$x_{101} = -71.9999974155395$$
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:
$$x_{1} = -7.63633926847275$$
$$x_{2} = 19.9999906959359$$
$$x_{3} = -77.9999976143442$$
$$x_{4} = 2.36355763288854$$
$$x_{5} = 71.9999974155395$$
$$x_{6} = -95.9999980616547$$
$$x_{7} = -81.9999977307176$$
$$x_{8} = 87.9999978854414$$
$$x_{9} = 45.9999959547572$$
$$x_{10} = 23.999992246615$$
$$x_{11} = -2.30294950019578$$
$$x_{12} = 95.9999980616547$$
$$x_{13} = -55.9999966771221$$
$$x_{14} = -25.9999928430298$$
$$x_{15} = -13.9999867084698$$
$$x_{16} = -61.999996998691$$
$$x_{17} = -87.9999978854414$$
$$x_{18} = 17.9999896621493$$
$$x_{19} = -45.9999959547572$$
$$x_{20} = -67.9999972635124$$
$$x_{21} = -17.9999896621493$$
$$x_{22} = 31.9999941849626$$
$$x_{23} = -35.9999948310781$$
$$x_{24} = 63.9999970924819$$
$$x_{25} = 9.9999813918312$$
$$x_{26} = -59.9999968986473$$
$$x_{27} = -73.9999974853898$$
$$x_{28} = -23.999992246615$$
$$x_{29} = 57.9999967917041$$
$$x_{30} = 43.9999957708824$$
$$x_{31} = 77.9999976143442$$
$$x_{32} = 73.9999974853898$$
$$x_{33} = -19.9999906959359$$
$$x_{34} = 61.999996998691$$
$$x_{35} = 17.3333225978722$$
$$x_{36} = 81.9999977307176$$
$$x_{37} = 65.9999971805885$$
$$x_{38} = 53.9999965540525$$
$$x_{39} = 55.9999966771221$$
$$x_{40} = 67.9999972635124$$
$$x_{41} = -99.9999981391885$$
$$x_{42} = 91.9999979773788$$
$$x_{43} = 59.9999968986473$$
$$x_{44} = -63.9999970924819$$
$$x_{45} = -39.9999953479705$$
$$x_{46} = 37.9999951031268$$
$$x_{47} = 39.9999953479705$$
$$x_{48} = 83.9999977847482$$
$$x_{49} = 49.9999962783767$$
$$x_{50} = -9.9999813918312$$
$$x_{51} = 69.9999973416978$$
$$x_{52} = 33.9999945270238$$
$$x_{53} = -41.9999955694958$$
$$x_{54} = 29.9999937972931$$
$$x_{55} = 27.9999933542423$$
$$x_{56} = -43.9999957708824$$
$$x_{57} = 35.9999948310781$$
$$x_{58} = 13.9999867084698$$
$$x_{59} = 99.9999981391885$$
$$x_{60} = -11.9999844932065$$
$$x_{61} = 47.999996123309$$
$$x_{62} = 21.999991541761$$
$$x_{63} = -15.9999883699152$$
$$x_{64} = -85.9999978362657$$
$$x_{65} = -31.9999941849626$$
$$x_{66} = 11.9999844932065$$
$$x_{67} = -69.9999973416978$$
$$x_{68} = -93.9999980204133$$
$$x_{69} = 89.9999979324316$$
$$x_{70} = -49.9999962783767$$
$$x_{71} = -21.999991541761$$
$$x_{72} = -75.9999975515638$$
$$x_{73} = 41.9999955694958$$
$$x_{74} = -79.9999976739856$$
$$x_{75} = -57.9999967917041$$
$$x_{76} = 97.9999981012127$$
$$x_{77} = -53.9999965540525$$
$$x_{78} = 79.9999976739856$$
$$x_{79} = -97.9999981012127$$
$$x_{80} = -83.9999977847482$$
$$x_{81} = -91.9999979773788$$
$$x_{82} = -65.9999971805885$$
$$x_{83} = -51.9999964215161$$
$$x_{84} = 85.9999978362657$$
$$x_{85} = 15.9999883699152$$
$$x_{86} = 25.9999928430298$$
$$x_{87} = -37.9999951031268$$
$$x_{88} = -29.9999937972931$$
$$x_{89} = 93.9999980204133$$
$$x_{90} = 75.9999975515638$$
$$x_{91} = 51.9999964215161$$
$$x_{92} = -89.9999979324316$$
$$x_{93} = -27.9999933542423$$
$$x_{94} = -47.999996123309$$
$$x_{95} = -33.9999945270238$$
$$x_{96} = -71.9999974155395$$
Puntos máximos de la función:
$$x_{96} = -9.66664741683294$$
$$x_{96} = 8.0302798578292$$
$$x_{96} = 10.9999830834915$$
$$x_{96} = 0$$
$$x_{96} = -8.0302798578292$$
Decrece en los intervalos
$$\left[99.9999981391885, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.9999981391885\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{8712 x \cos{\left(x 33 \pi \right)}}{\left(1 - \left(66 x\right)^{2}\right)^{2}} - \frac{33 \pi \sin{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -7.63633926847275$$
$$x_{2} = 19.9999906959359$$
$$x_{3} = -77.9999976143442$$
$$x_{4} = 2.36355763288854$$
$$x_{5} = 71.9999974155395$$
$$x_{6} = -95.9999980616547$$
$$x_{7} = -81.9999977307176$$
$$x_{8} = 87.9999978854414$$
$$x_{9} = 45.9999959547572$$
$$x_{10} = 23.999992246615$$
$$x_{11} = -2.30294950019578$$
$$x_{12} = 95.9999980616547$$
$$x_{13} = -55.9999966771221$$
$$x_{14} = -25.9999928430298$$
$$x_{15} = -13.9999867084698$$
$$x_{16} = -61.999996998691$$
$$x_{17} = -87.9999978854414$$
$$x_{18} = 17.9999896621493$$
$$x_{19} = -9.66664741683294$$
$$x_{20} = -45.9999959547572$$
$$x_{21} = -67.9999972635124$$
$$x_{22} = -17.9999896621493$$
$$x_{23} = 31.9999941849626$$
$$x_{24} = -35.9999948310781$$
$$x_{25} = 63.9999970924819$$
$$x_{26} = 9.9999813918312$$
$$x_{27} = -59.9999968986473$$
$$x_{28} = -73.9999974853898$$
$$x_{29} = -23.999992246615$$
$$x_{30} = 57.9999967917041$$
$$x_{31} = 43.9999957708824$$
$$x_{32} = 77.9999976143442$$
$$x_{33} = 73.9999974853898$$
$$x_{34} = -19.9999906959359$$
$$x_{35} = 61.999996998691$$
$$x_{36} = 8.0302798578292$$
$$x_{37} = 17.3333225978722$$
$$x_{38} = 81.9999977307176$$
$$x_{39} = 65.9999971805885$$
$$x_{40} = 53.9999965540525$$
$$x_{41} = 55.9999966771221$$
$$x_{42} = 67.9999972635124$$
$$x_{43} = 10.9999830834915$$
$$x_{44} = -99.9999981391885$$
$$x_{45} = 91.9999979773788$$
$$x_{46} = 59.9999968986473$$
$$x_{47} = -63.9999970924819$$
$$x_{48} = -39.9999953479705$$
$$x_{49} = 37.9999951031268$$
$$x_{50} = 39.9999953479705$$
$$x_{51} = 83.9999977847482$$
$$x_{52} = 49.9999962783767$$
$$x_{53} = -9.9999813918312$$
$$x_{54} = 69.9999973416978$$
$$x_{55} = 33.9999945270238$$
$$x_{56} = -41.9999955694958$$
$$x_{57} = 29.9999937972931$$
$$x_{58} = 27.9999933542423$$
$$x_{59} = -43.9999957708824$$
$$x_{60} = 35.9999948310781$$
$$x_{61} = 13.9999867084698$$
$$x_{62} = 99.9999981391885$$
$$x_{63} = -11.9999844932065$$
$$x_{64} = 47.999996123309$$
$$x_{65} = 21.999991541761$$
$$x_{66} = -15.9999883699152$$
$$x_{67} = 0$$
$$x_{68} = -85.9999978362657$$
$$x_{69} = -31.9999941849626$$
$$x_{70} = 11.9999844932065$$
$$x_{71} = -69.9999973416978$$
$$x_{72} = -93.9999980204133$$
$$x_{73} = 89.9999979324316$$
$$x_{74} = -49.9999962783767$$
$$x_{75} = -21.999991541761$$
$$x_{76} = -75.9999975515638$$
$$x_{77} = 41.9999955694958$$
$$x_{78} = -79.9999976739856$$
$$x_{79} = -8.0302798578292$$
$$x_{80} = -57.9999967917041$$
$$x_{81} = 97.9999981012127$$
$$x_{82} = -53.9999965540525$$
$$x_{83} = 79.9999976739856$$
$$x_{84} = -97.9999981012127$$
$$x_{85} = -83.9999977847482$$
$$x_{86} = -91.9999979773788$$
$$x_{87} = -65.9999971805885$$
$$x_{88} = -51.9999964215161$$
$$x_{89} = 85.9999978362657$$
$$x_{90} = 15.9999883699152$$
$$x_{91} = 25.9999928430298$$
$$x_{92} = -37.9999951031268$$
$$x_{93} = -29.9999937972931$$
$$x_{94} = 93.9999980204133$$
$$x_{95} = 75.9999975515638$$
$$x_{96} = 51.9999964215161$$
$$x_{97} = -89.9999979324316$$
$$x_{98} = -27.9999933542423$$
$$x_{99} = -47.999996123309$$
$$x_{100} = -33.9999945270238$$
$$x_{101} = -71.9999974155395$$
Signos de extremos en los puntos:
(-7.636339268472752, -3.9368005122624e-6*cos(1.9991958596008*pi))
(19.99999069593595, -5.73921891832805e-7*cos(1.99969296588631*pi))
(-77.99999761434418, -3.77331416981893e-8*cos(1.99992127335781*pi))
(2.3635576328885395, -4.10958136179002e-5*cos(1.99740188532181*pi))
(71.9999974155395, -4.42840351145172e-8*cos(1.99991471280327*pi))
(-95.99999806165468, -2.49097684869321e-8*cos(1.99993603460462*pi))
(-81.99999773071764, -3.41416466285947e-8*cos(1.99992511368191*pi))
(87.99999788544145, -2.96446835259181e-8*cos(1.99993021956789*pi))
(45.99999595475715, -1.08491718652833e-7*cos(1.99986650698611*pi))
(23.99999224661502, -3.98556686129234e-7*cos(1.99974413829557*pi))
(-2.3029495001957763, -4.32874595339753e-5*cos(1.99733350646062*pi))
(95.99999806165468, -2.49097684869321e-8*cos(1.99993603460462*pi))
(-55.9999966771221, -7.32042268609138e-8*cos(1.99989034502914*pi))
(-25.999992843029784, -3.39598543984468e-7*cos(1.99976381998283*pi))
(-13.999986708469844, -1.17127100090182e-6*cos(1.99956137950488*pi))
(-61.999996998690975, -5.97212401984126e-8*cos(1.99990095680209*pi))
(-87.99999788544145, -2.96446835259181e-8*cos(1.99993021956789*pi))
(17.99998966214929, -7.08545795495413e-7*cos(1.99965885092661*pi))
(-9.66664741683294, -2.45675268398155e-6*cos(0.999364755487022*pi))
(-45.99999595475715, -1.08491718652833e-7*cos(1.99986650698611*pi))
(-67.9999972635124, -4.96471542519055e-8*cos(1.99990969590908*pi))
(-17.99998966214929, -7.08545795495413e-7*cos(1.99965885092661*pi))
(31.999994184962553, -2.2418803348382e-7*cos(1.99980810376428*pi))
(-35.99999483107813, -1.77136202141122e-7*cos(1.99982942557835*pi))
(63.9999970924819, -5.60469836698694e-8*cos(1.99990405190283*pi))
(9.999981391831204, -2.29569792780215e-6*cos(1.99938593042975*pi))
(-59.999996898647325, -6.37690138217689e-8*cos(1.99989765536179*pi))
(-73.9999974853898, -4.19226509519247e-8*cos(1.99991701786348*pi))
(-23.99999224661502, -3.98556686129234e-7*cos(1.99974413829557*pi))
(57.99999679170411, -6.82427028687052e-8*cos(1.99989412623563*pi))
(43.99999577088242, -1.18578761745392e-7*cos(1.99986043911986*pi))
(77.99999761434418, -3.77331416981893e-8*cos(1.99992127335781*pi))
(73.9999974853898, -4.19226509519247e-8*cos(1.99991701786348*pi))
(-19.99999069593595, -5.73921891832805e-7*cos(1.99969296588631*pi))
(61.999996998690975, -5.97212401984126e-8*cos(1.99990095680209*pi))
(8.030279857829203, -3.56001897943089e-6*cos(0.999235308363723*pi))
(17.333322597872222, -7.64097574153017e-7*cos(1.99964572978331*pi))
(81.99999773071764, -3.41416466285947e-8*cos(1.99992511368191*pi))
(65.99999718058852, -5.27016627866856e-8*cos(1.99990695942142*pi))
(53.99999655405252, -7.87271806444296e-8*cos(1.9998862837333*pi))
(55.9999966771221, -7.32042268609138e-8*cos(1.99989034502914*pi))
(67.9999972635124, -4.96471542519055e-8*cos(1.99990969590908*pi))
(10.999983083491474, -1.89726903332876e-6*cos(0.999441755218641*pi))
(-99.99999813918849, -2.2956842520043e-8*cos(1.99993859322012*pi))
(91.99999797737878, -2.71229238784605e-8*cos(1.99993325349988*pi))
(59.999996898647325, -6.37690138217689e-8*cos(1.99989765536179*pi))
(-63.9999970924819, -5.60469836698694e-8*cos(1.99990405190283*pi))
(-39.99999534797052, -1.43480311076943e-7*cos(1.99984648302711*pi))
(37.999995103126764, -1.58980960561168e-7*cos(1.99983840318328*pi))
(39.99999534797052, -1.43480311076943e-7*cos(1.99984648302711*pi))
(83.99999778474817, -3.25352084699585e-8*cos(1.99992689668989*pi))
(49.999996278376656, -9.18273866567815e-8*cos(1.99987718642956*pi))
(-9.999981391831204, -2.29569792780215e-6*cos(1.99938593042975*pi))
(69.99999734169776, -4.68507019547142e-8*cos(1.99991227602641*pi))
(33.99999452702376, -1.98588694535917e-7*cos(1.99981939178406*pi))
(-41.99999556949581, -1.30140867174804e-7*cos(1.99985379336158*pi))
(29.999993797293147, -2.55076183193094e-7*cos(1.9997953106739*pi))
(27.99999335424234, -2.92817075999548e-7*cos(1.99978068999712*pi))
(-43.99999577088242, -1.18578761745392e-7*cos(1.99986043911986*pi))
(35.99999483107813, -1.77136202141122e-7*cos(1.99982942557835*pi))
(13.999986708469844, -1.17127100090182e-6*cos(1.99956137950488*pi))
(99.99999813918849, -2.2956842520043e-8*cos(1.99993859322012*pi))
(-11.999984493206487, -1.59423174087219e-6*cos(1.99948827581409*pi))
(47.99999612330898, -9.96390934646259e-8*cos(1.9998720691965*pi))
(21.999991541761023, -4.74315489249707e-7*cos(1.99972087811375*pi))
(-15.999988369915169, -8.96753714808963e-7*cos(1.99961620720057*pi))
(0, 1)
(-85.99999783626566, -3.10395389486259e-8*cos(1.99992859676695*pi))
(-31.999994184962553, -2.2418803348382e-7*cos(1.99980810376428*pi))
(11.999984493206487, -1.59423174087219e-6*cos(1.99948827581409*pi))
(-69.99999734169776, -4.68507019547142e-8*cos(1.99991227602641*pi))
(-93.99999802041329, -2.59810351992055e-8*cos(1.99993467363856*pi))
(89.99999793243164, -2.8341781288977e-8*cos(1.99993177024407*pi))
(-49.999996278376656, -9.18273866567815e-8*cos(1.99987718642956*pi))
(-21.999991541761023, -4.74315489249707e-7*cos(1.99972087811375*pi))
(-75.99999755156375, -3.97452277185799e-8*cos(1.99991920160392*pi))
(41.99999556949581, -1.30140867174804e-7*cos(1.99985379336158*pi))
(-79.99999767398558, -3.58700676516742e-8*cos(1.9999232415239*pi))
(-8.030279857829203, -3.56001897943089e-6*cos(0.999235308363723*pi))
(-57.99999679170411, -6.82427028687052e-8*cos(1.99989412623563*pi))
(97.99999810121274, -2.39034179269069e-8*cos(1.99993734002055*pi))
(-53.99999655405252, -7.87271806444296e-8*cos(1.9998862837333*pi))
(79.99999767398558, -3.58700676516742e-8*cos(1.9999232415239*pi))
(-97.99999810121274, -2.39034179269069e-8*cos(1.99993734002055*pi))
(-83.99999778474817, -3.25352084699585e-8*cos(1.99992689668989*pi))
(-91.99999797737878, -2.71229238784605e-8*cos(1.99993325349988*pi))
(-65.99999718058852, -5.27016627866856e-8*cos(1.99990695942142*pi))
(-51.99999642151605, -8.48995793170411e-8*cos(1.99988191002967*pi))
(85.99999783626566, -3.10395389486259e-8*cos(1.99992859676695*pi))
(15.999988369915169, -8.96753714808963e-7*cos(1.99961620720057*pi))
(25.999992843029784, -3.39598543984468e-7*cos(1.99976381998283*pi))
(-37.999995103126764, -1.58980960561168e-7*cos(1.99983840318328*pi))
(-29.999993797293147, -2.55076183193094e-7*cos(1.9997953106739*pi))
(93.99999802041329, -2.59810351992055e-8*cos(1.99993467363856*pi))
(75.99999755156375, -3.97452277185799e-8*cos(1.99991920160392*pi))
(51.99999642151605, -8.48995793170411e-8*cos(1.99988191002967*pi))
(-89.99999793243164, -2.8341781288977e-8*cos(1.99993177024407*pi))
(-27.99999335424234, -2.92817075999548e-7*cos(1.99978068999712*pi))
(-47.99999612330898, -9.96390934646259e-8*cos(1.9998720691965*pi))
(-33.99999452702376, -1.98588694535917e-7*cos(1.99981939178406*pi))
(-71.9999974155395, -4.42840351145172e-8*cos(1.99991471280327*pi))
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:
$$x_{1} = -7.63633926847275$$
$$x_{2} = 19.9999906959359$$
$$x_{3} = -77.9999976143442$$
$$x_{4} = 2.36355763288854$$
$$x_{5} = 71.9999974155395$$
$$x_{6} = -95.9999980616547$$
$$x_{7} = -81.9999977307176$$
$$x_{8} = 87.9999978854414$$
$$x_{9} = 45.9999959547572$$
$$x_{10} = 23.999992246615$$
$$x_{11} = -2.30294950019578$$
$$x_{12} = 95.9999980616547$$
$$x_{13} = -55.9999966771221$$
$$x_{14} = -25.9999928430298$$
$$x_{15} = -13.9999867084698$$
$$x_{16} = -61.999996998691$$
$$x_{17} = -87.9999978854414$$
$$x_{18} = 17.9999896621493$$
$$x_{19} = -45.9999959547572$$
$$x_{20} = -67.9999972635124$$
$$x_{21} = -17.9999896621493$$
$$x_{22} = 31.9999941849626$$
$$x_{23} = -35.9999948310781$$
$$x_{24} = 63.9999970924819$$
$$x_{25} = 9.9999813918312$$
$$x_{26} = -59.9999968986473$$
$$x_{27} = -73.9999974853898$$
$$x_{28} = -23.999992246615$$
$$x_{29} = 57.9999967917041$$
$$x_{30} = 43.9999957708824$$
$$x_{31} = 77.9999976143442$$
$$x_{32} = 73.9999974853898$$
$$x_{33} = -19.9999906959359$$
$$x_{34} = 61.999996998691$$
$$x_{35} = 17.3333225978722$$
$$x_{36} = 81.9999977307176$$
$$x_{37} = 65.9999971805885$$
$$x_{38} = 53.9999965540525$$
$$x_{39} = 55.9999966771221$$
$$x_{40} = 67.9999972635124$$
$$x_{41} = -99.9999981391885$$
$$x_{42} = 91.9999979773788$$
$$x_{43} = 59.9999968986473$$
$$x_{44} = -63.9999970924819$$
$$x_{45} = -39.9999953479705$$
$$x_{46} = 37.9999951031268$$
$$x_{47} = 39.9999953479705$$
$$x_{48} = 83.9999977847482$$
$$x_{49} = 49.9999962783767$$
$$x_{50} = -9.9999813918312$$
$$x_{51} = 69.9999973416978$$
$$x_{52} = 33.9999945270238$$
$$x_{53} = -41.9999955694958$$
$$x_{54} = 29.9999937972931$$
$$x_{55} = 27.9999933542423$$
$$x_{56} = -43.9999957708824$$
$$x_{57} = 35.9999948310781$$
$$x_{58} = 13.9999867084698$$
$$x_{59} = 99.9999981391885$$
$$x_{60} = -11.9999844932065$$
$$x_{61} = 47.999996123309$$
$$x_{62} = 21.999991541761$$
$$x_{63} = -15.9999883699152$$
$$x_{64} = -85.9999978362657$$
$$x_{65} = -31.9999941849626$$
$$x_{66} = 11.9999844932065$$
$$x_{67} = -69.9999973416978$$
$$x_{68} = -93.9999980204133$$
$$x_{69} = 89.9999979324316$$
$$x_{70} = -49.9999962783767$$
$$x_{71} = -21.999991541761$$
$$x_{72} = -75.9999975515638$$
$$x_{73} = 41.9999955694958$$
$$x_{74} = -79.9999976739856$$
$$x_{75} = -57.9999967917041$$
$$x_{76} = 97.9999981012127$$
$$x_{77} = -53.9999965540525$$
$$x_{78} = 79.9999976739856$$
$$x_{79} = -97.9999981012127$$
$$x_{80} = -83.9999977847482$$
$$x_{81} = -91.9999979773788$$
$$x_{82} = -65.9999971805885$$
$$x_{83} = -51.9999964215161$$
$$x_{84} = 85.9999978362657$$
$$x_{85} = 15.9999883699152$$
$$x_{86} = 25.9999928430298$$
$$x_{87} = -37.9999951031268$$
$$x_{88} = -29.9999937972931$$
$$x_{89} = 93.9999980204133$$
$$x_{90} = 75.9999975515638$$
$$x_{91} = 51.9999964215161$$
$$x_{92} = -89.9999979324316$$
$$x_{93} = -27.9999933542423$$
$$x_{94} = -47.999996123309$$
$$x_{95} = -33.9999945270238$$
$$x_{96} = -71.9999974155395$$
Puntos máximos de la función:
$$x_{96} = -9.66664741683294$$
$$x_{96} = 8.0302798578292$$
$$x_{96} = 10.9999830834915$$
$$x_{96} = 0$$
$$x_{96} = -8.0302798578292$$
Decrece en los intervalos
$$\left[99.9999981391885, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.9999981391885\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -0.0151515151515152$$
$$x_{2} = 0.0151515151515152$$
$$x_{1} = -0.0151515151515152$$
$$x_{2} = 0.0151515151515152$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}}\right) = 0$$
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 cos((pi*33)*x)/(1 - (66*x)^2), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x 33 \pi \right)}}{x \left(1 - \left(66 x\right)^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x 33 \pi \right)}}{x \left(1 - \left(66 x\right)^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x 33 \pi \right)}}{x \left(1 - \left(66 x\right)^{2}\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x 33 \pi \right)}}{x \left(1 - \left(66 x\right)^{2}\right)}\right) = 0$$
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:
$$\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}} = \frac{\cos{\left(33 \pi x \right)}}{1 - 4356 x^{2}}$$
- No
$$\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}} = - \frac{\cos{\left(33 \pi x \right)}}{1 - 4356 x^{2}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}} = \frac{\cos{\left(33 \pi x \right)}}{1 - 4356 x^{2}}$$
- No
$$\frac{\cos{\left(x 33 \pi \right)}}{1 - \left(66 x\right)^{2}} = - \frac{\cos{\left(33 \pi x \right)}}{1 - 4356 x^{2}}$$
- No
es decir, función
no es
par ni impar