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- Puntos de cruce de las funciones
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- Gráfico de la función y =:
-
x-8/x^4
-
y=x²-2x-8
-
-x^4+x^2
-
x*e^(-x^1)
-
Expresiones idénticas
- abs(cos(x*(treinta y tres *pi))/(uno -(sesenta y seis *x)^ dos))
- abs( coseno de (x multiplicar por (33 multiplicar por número pi )) dividir por (1 menos (66 multiplicar por x) al cuadrado ))
- abs( coseno de (x multiplicar por (treinta y tres multiplicar por número pi )) dividir por (uno menos (sesenta y seis multiplicar por x) en el grado dos))
- abs(cos(x*(33*pi))/(1-(66*x)2))
- abscosx*33*pi/1-66*x2
- abs(cos(x*(33*pi))/(1-(66*x)²))
- abs(cos(x*(33*pi))/(1-(66*x) en el grado 2))
- abs(cos(x(33pi))/(1-(66x)^2))
- abs(cos(x(33pi))/(1-(66x)2))
- abscosx33pi/1-66x2
- abscosx33pi/1-66x^2
- abs(cos(x*(33*pi)) dividir por (1-(66*x)^2))
-
Expresiones semejantes
- abs(cos(x*(33*pi))/(1+(66*x)^2))
-
Expresiones con funciones
- abs
- abs(y*0.27)
- abs(6x-12)-abs(9x+36)
- abs(x+1)+2*abs(x+5)-2*x-3
- abs((1/(x))-1)
- abs(cos(x*2)*sin(x))
- Coseno cos
- cos(3*z)
- cos(x)*((2*x^3-5)^3)^4
- cos2x-sinx
- cos(x-1)-2
- cos(x)-sin(2)*5*x+e/x
Gráfico de la función y = abs(cos(x*(33*pi))/(1-(66*x)^2))
Solución
Ha introducido
[src]
|cos(x*33*pi)|
f(x) = |------------|
| 2 |
|1 - (66*x) |
$$f{\left(x \right)} = \left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right|$$
f = Abs(cos((33*pi)*x)/(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:
$$\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 26055.2744583996$$
$$x_{2} = -13241.8943406374$$
$$x_{3} = -18353.3068232892$$
$$x_{4} = 44.7121212121212$$
$$x_{5} = 0.257575757575758$$
$$x_{6} = -12232.7209047432$$
$$x_{7} = 210.621212121212$$
$$x_{8} = 513.833333333333$$
$$x_{9} = 8.37878787878788$$
$$x_{10} = -30.9848484848485$$
$$x_{11} = 18263.0002524105$$
$$x_{12} = -683.257575757576$$
$$x_{13} = -7.1969696969697$$
$$x_{14} = 3.10606060606061$$
$$x_{15} = 72077.6222823594$$
$$x_{16} = 4769.22837565766$$
$$x_{17} = 144764.773623337$$
$$x_{18} = 96407.2927910052$$
$$x_{19} = -4.68181818181818$$
$$x_{20} = -57709.3036741969$$
$$x_{21} = 61.5606060606061$$
$$x_{22} = -99.1363636363636$$
$$x_{23} = -115.106060606061$$
$$x_{24} = 78.9545454545455$$
$$x_{25} = 30.0757575757576$$
$$x_{26} = 50.8333333333333$$
$$x_{27} = 715.19696969697$$
$$x_{28} = -11431.0822574421$$
$$x_{29} = 16577.9982980195$$
$$x_{30} = -10709.8354272508$$
$$x_{31} = 3302.49995855877$$
$$x_{32} = -180.257575757576$$
$$x_{33} = -109914.804741886$$
$$x_{34} = 74.469696969697$$
$$x_{35} = -23.7121212121212$$
$$x_{36} = -14.8333333333333$$
$$x_{37} = 52425.819018951$$
$$x_{38} = -48337.0303099831$$
$$x_{39} = -426147.968049528$$
$$x_{40} = -7402.40909090909$$
$$x_{41} = 37.9848484848485$$
$$x_{42} = -11151.0145603888$$
o sea hay que resolver la ecuación:
$$\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 26055.2744583996$$
$$x_{2} = -13241.8943406374$$
$$x_{3} = -18353.3068232892$$
$$x_{4} = 44.7121212121212$$
$$x_{5} = 0.257575757575758$$
$$x_{6} = -12232.7209047432$$
$$x_{7} = 210.621212121212$$
$$x_{8} = 513.833333333333$$
$$x_{9} = 8.37878787878788$$
$$x_{10} = -30.9848484848485$$
$$x_{11} = 18263.0002524105$$
$$x_{12} = -683.257575757576$$
$$x_{13} = -7.1969696969697$$
$$x_{14} = 3.10606060606061$$
$$x_{15} = 72077.6222823594$$
$$x_{16} = 4769.22837565766$$
$$x_{17} = 144764.773623337$$
$$x_{18} = 96407.2927910052$$
$$x_{19} = -4.68181818181818$$
$$x_{20} = -57709.3036741969$$
$$x_{21} = 61.5606060606061$$
$$x_{22} = -99.1363636363636$$
$$x_{23} = -115.106060606061$$
$$x_{24} = 78.9545454545455$$
$$x_{25} = 30.0757575757576$$
$$x_{26} = 50.8333333333333$$
$$x_{27} = 715.19696969697$$
$$x_{28} = -11431.0822574421$$
$$x_{29} = 16577.9982980195$$
$$x_{30} = -10709.8354272508$$
$$x_{31} = 3302.49995855877$$
$$x_{32} = -180.257575757576$$
$$x_{33} = -109914.804741886$$
$$x_{34} = 74.469696969697$$
$$x_{35} = -23.7121212121212$$
$$x_{36} = -14.8333333333333$$
$$x_{37} = 52425.819018951$$
$$x_{38} = -48337.0303099831$$
$$x_{39} = -426147.968049528$$
$$x_{40} = -7402.40909090909$$
$$x_{41} = 37.9848484848485$$
$$x_{42} = -11151.0145603888$$
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
$$\frac{\left(\frac{8712 x \cos{\left(33 \pi x \right)}}{\left(1 - 4356 x^{2}\right)^{2}} - \frac{33 \pi \sin{\left(33 \pi x \right)}}{1 - 4356 x^{2}}\right) \sqrt{18974736 x^{4} - 8712 x^{2} + 1} \cos{\left(33 \pi x \right)}}{\left(1 - 4356 x^{2}\right) \left|{\cos{\left(33 \pi x \right)}}\right|} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -47.999996123309$$
$$x_{2} = 75.9999975515638$$
$$x_{3} = -15.9999883699152$$
$$x_{4} = -79.9999976739856$$
$$x_{5} = -33.9999945270238$$
$$x_{6} = 41.9999955694958$$
$$x_{7} = 77.9999976143442$$
$$x_{8} = 47.999996123309$$
$$x_{9} = 83.9999977847482$$
$$x_{10} = 85.9999978362657$$
$$x_{11} = -63.9999970924819$$
$$x_{12} = -23.999992246615$$
$$x_{13} = 27.9999933542423$$
$$x_{14} = 61.999996998691$$
$$x_{15} = 39.9999953479705$$
$$x_{16} = 65.9999971805885$$
$$x_{17} = 31.9999941849626$$
$$x_{18} = -59.9999968986473$$
$$x_{19} = 2.03021137191148$$
$$x_{20} = 69.9999973416978$$
$$x_{21} = -45.9999959547572$$
$$x_{22} = -67.9999972635124$$
$$x_{23} = 29.9999937972931$$
$$x_{24} = -5.9393626091462$$
$$x_{25} = -93.9999980204133$$
$$x_{26} = -25.9999928430298$$
$$x_{27} = -13.9999867084698$$
$$x_{28} = -37.9999951031268$$
$$x_{29} = -97.9999981012127$$
$$x_{30} = 73.9999974853898$$
$$x_{31} = -75.9999975515638$$
$$x_{32} = -89.9999979324316$$
$$x_{33} = 63.9999970924819$$
$$x_{34} = -35.9999948310781$$
$$x_{35} = -95.9999980616547$$
$$x_{36} = -51.9999964215161$$
$$x_{37} = 25.9999928430298$$
$$x_{38} = 87.9999978854414$$
$$x_{39} = 15.9999883699152$$
$$x_{40} = 17.9999896621493$$
$$x_{41} = -9.96967830496806$$
$$x_{42} = 95.9999980616547$$
$$x_{43} = -73.9999974853898$$
$$x_{44} = -65.9999971805885$$
$$x_{45} = 21.999991541761$$
$$x_{46} = 81.9999977307176$$
$$x_{47} = 55.9999966771221$$
$$x_{48} = -85.9999978362657$$
$$x_{49} = 79.9999976739856$$
$$x_{50} = 37.9999951031268$$
$$x_{51} = 59.9999968986473$$
$$x_{52} = 17.8181713748426$$
$$x_{53} = 93.9999980204133$$
$$x_{54} = -17.9999896621493$$
$$x_{55} = -4.39389704379771$$
$$x_{56} = -43.9999957708824$$
$$x_{57} = 23.999992246615$$
$$x_{58} = 57.9999967917041$$
$$x_{59} = 0$$
$$x_{60} = -3.96965009342653$$
$$x_{61} = -53.9999965540525$$
$$x_{62} = -21.999991541761$$
$$x_{63} = -29.9999937972931$$
$$x_{64} = -27.9999933542423$$
$$x_{65} = 53.9999965540525$$
$$x_{66} = 91.9999979773788$$
$$x_{67} = 35.9999948310781$$
$$x_{68} = -7.969673621005$$
$$x_{69} = -87.9999978854414$$
$$x_{70} = -81.9999977307176$$
$$x_{71} = -49.9999962783767$$
$$x_{72} = 49.9999962783767$$
$$x_{73} = 45.9999959547572$$
$$x_{74} = -83.9999977847482$$
$$x_{75} = 99.9999981391885$$
$$x_{76} = -19.9999906959359$$
$$x_{77} = -99.9999981391885$$
$$x_{78} = -31.9999941849626$$
$$x_{79} = 67.9999972635124$$
$$x_{80} = 11.9999844932065$$
$$x_{81} = 6.18178808022675$$
$$x_{82} = -39.9999953479705$$
$$x_{83} = -69.9999973416978$$
$$x_{84} = -57.9999967917041$$
$$x_{85} = -71.9999974155395$$
$$x_{86} = 19.9999906959359$$
$$x_{87} = 51.9999964215161$$
$$x_{88} = -11.9999844932065$$
$$x_{89} = -61.999996998691$$
$$x_{90} = -91.9999979773788$$
$$x_{91} = 13.9999867084698$$
$$x_{92} = 97.9999981012127$$
$$x_{93} = 71.9999974155395$$
$$x_{94} = 33.9999945270238$$
$$x_{95} = -77.9999976143442$$
$$x_{96} = 89.9999979324316$$
$$x_{97} = -41.9999955694958$$
$$x_{98} = -55.9999966771221$$
$$x_{99} = 43.9999957708824$$
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:
La función no tiene puntos mínimos
Puntos máximos de la función:
$$x_{99} = -47.999996123309$$
$$x_{99} = 75.9999975515638$$
$$x_{99} = -15.9999883699152$$
$$x_{99} = -79.9999976739856$$
$$x_{99} = -33.9999945270238$$
$$x_{99} = 41.9999955694958$$
$$x_{99} = 77.9999976143442$$
$$x_{99} = 47.999996123309$$
$$x_{99} = 83.9999977847482$$
$$x_{99} = 85.9999978362657$$
$$x_{99} = -63.9999970924819$$
$$x_{99} = -23.999992246615$$
$$x_{99} = 27.9999933542423$$
$$x_{99} = 61.999996998691$$
$$x_{99} = 39.9999953479705$$
$$x_{99} = 65.9999971805885$$
$$x_{99} = 31.9999941849626$$
$$x_{99} = -59.9999968986473$$
$$x_{99} = 2.03021137191148$$
$$x_{99} = 69.9999973416978$$
$$x_{99} = -45.9999959547572$$
$$x_{99} = -67.9999972635124$$
$$x_{99} = 29.9999937972931$$
$$x_{99} = -5.9393626091462$$
$$x_{99} = -93.9999980204133$$
$$x_{99} = -25.9999928430298$$
$$x_{99} = -13.9999867084698$$
$$x_{99} = -37.9999951031268$$
$$x_{99} = -97.9999981012127$$
$$x_{99} = 73.9999974853898$$
$$x_{99} = -75.9999975515638$$
$$x_{99} = -89.9999979324316$$
$$x_{99} = 63.9999970924819$$
$$x_{99} = -35.9999948310781$$
$$x_{99} = -95.9999980616547$$
$$x_{99} = -51.9999964215161$$
$$x_{99} = 25.9999928430298$$
$$x_{99} = 87.9999978854414$$
$$x_{99} = 15.9999883699152$$
$$x_{99} = 17.9999896621493$$
$$x_{99} = -9.96967830496806$$
$$x_{99} = 95.9999980616547$$
$$x_{99} = -73.9999974853898$$
$$x_{99} = -65.9999971805885$$
$$x_{99} = 21.999991541761$$
$$x_{99} = 81.9999977307176$$
$$x_{99} = 55.9999966771221$$
$$x_{99} = -85.9999978362657$$
$$x_{99} = 79.9999976739856$$
$$x_{99} = 37.9999951031268$$
$$x_{99} = 59.9999968986473$$
$$x_{99} = 17.8181713748426$$
$$x_{99} = 93.9999980204133$$
$$x_{99} = -17.9999896621493$$
$$x_{99} = -4.39389704379771$$
$$x_{99} = -43.9999957708824$$
$$x_{99} = 23.999992246615$$
$$x_{99} = 57.9999967917041$$
$$x_{99} = 0$$
$$x_{99} = -3.96965009342653$$
$$x_{99} = -53.9999965540525$$
$$x_{99} = -21.999991541761$$
$$x_{99} = -29.9999937972931$$
$$x_{99} = -27.9999933542423$$
$$x_{99} = 53.9999965540525$$
$$x_{99} = 91.9999979773788$$
$$x_{99} = 35.9999948310781$$
$$x_{99} = -7.969673621005$$
$$x_{99} = -87.9999978854414$$
$$x_{99} = -81.9999977307176$$
$$x_{99} = -49.9999962783767$$
$$x_{99} = 49.9999962783767$$
$$x_{99} = 45.9999959547572$$
$$x_{99} = -83.9999977847482$$
$$x_{99} = 99.9999981391885$$
$$x_{99} = -19.9999906959359$$
$$x_{99} = -99.9999981391885$$
$$x_{99} = -31.9999941849626$$
$$x_{99} = 67.9999972635124$$
$$x_{99} = 11.9999844932065$$
$$x_{99} = 6.18178808022675$$
$$x_{99} = -39.9999953479705$$
$$x_{99} = -69.9999973416978$$
$$x_{99} = -57.9999967917041$$
$$x_{99} = -71.9999974155395$$
$$x_{99} = 19.9999906959359$$
$$x_{99} = 51.9999964215161$$
$$x_{99} = -11.9999844932065$$
$$x_{99} = -61.999996998691$$
$$x_{99} = -91.9999979773788$$
$$x_{99} = 13.9999867084698$$
$$x_{99} = 97.9999981012127$$
$$x_{99} = 71.9999974155395$$
$$x_{99} = 33.9999945270238$$
$$x_{99} = -77.9999976143442$$
$$x_{99} = 89.9999979324316$$
$$x_{99} = -41.9999955694958$$
$$x_{99} = -55.9999966771221$$
$$x_{99} = 43.9999957708824$$
Decrece en los intervalos
$$\left(-\infty, -99.9999981391885\right]$$
Crece en los intervalos
$$\left[99.9999981391885, \infty\right)$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\left(\frac{8712 x \cos{\left(33 \pi x \right)}}{\left(1 - 4356 x^{2}\right)^{2}} - \frac{33 \pi \sin{\left(33 \pi x \right)}}{1 - 4356 x^{2}}\right) \sqrt{18974736 x^{4} - 8712 x^{2} + 1} \cos{\left(33 \pi x \right)}}{\left(1 - 4356 x^{2}\right) \left|{\cos{\left(33 \pi x \right)}}\right|} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -47.999996123309$$
$$x_{2} = 75.9999975515638$$
$$x_{3} = -15.9999883699152$$
$$x_{4} = -79.9999976739856$$
$$x_{5} = -33.9999945270238$$
$$x_{6} = 41.9999955694958$$
$$x_{7} = 77.9999976143442$$
$$x_{8} = 47.999996123309$$
$$x_{9} = 83.9999977847482$$
$$x_{10} = 85.9999978362657$$
$$x_{11} = -63.9999970924819$$
$$x_{12} = -23.999992246615$$
$$x_{13} = 27.9999933542423$$
$$x_{14} = 61.999996998691$$
$$x_{15} = 39.9999953479705$$
$$x_{16} = 65.9999971805885$$
$$x_{17} = 31.9999941849626$$
$$x_{18} = -59.9999968986473$$
$$x_{19} = 2.03021137191148$$
$$x_{20} = 69.9999973416978$$
$$x_{21} = -45.9999959547572$$
$$x_{22} = -67.9999972635124$$
$$x_{23} = 29.9999937972931$$
$$x_{24} = -5.9393626091462$$
$$x_{25} = -93.9999980204133$$
$$x_{26} = -25.9999928430298$$
$$x_{27} = -13.9999867084698$$
$$x_{28} = -37.9999951031268$$
$$x_{29} = -97.9999981012127$$
$$x_{30} = 73.9999974853898$$
$$x_{31} = -75.9999975515638$$
$$x_{32} = -89.9999979324316$$
$$x_{33} = 63.9999970924819$$
$$x_{34} = -35.9999948310781$$
$$x_{35} = -95.9999980616547$$
$$x_{36} = -51.9999964215161$$
$$x_{37} = 25.9999928430298$$
$$x_{38} = 87.9999978854414$$
$$x_{39} = 15.9999883699152$$
$$x_{40} = 17.9999896621493$$
$$x_{41} = -9.96967830496806$$
$$x_{42} = 95.9999980616547$$
$$x_{43} = -73.9999974853898$$
$$x_{44} = -65.9999971805885$$
$$x_{45} = 21.999991541761$$
$$x_{46} = 81.9999977307176$$
$$x_{47} = 55.9999966771221$$
$$x_{48} = -85.9999978362657$$
$$x_{49} = 79.9999976739856$$
$$x_{50} = 37.9999951031268$$
$$x_{51} = 59.9999968986473$$
$$x_{52} = 17.8181713748426$$
$$x_{53} = 93.9999980204133$$
$$x_{54} = -17.9999896621493$$
$$x_{55} = -4.39389704379771$$
$$x_{56} = -43.9999957708824$$
$$x_{57} = 23.999992246615$$
$$x_{58} = 57.9999967917041$$
$$x_{59} = 0$$
$$x_{60} = -3.96965009342653$$
$$x_{61} = -53.9999965540525$$
$$x_{62} = -21.999991541761$$
$$x_{63} = -29.9999937972931$$
$$x_{64} = -27.9999933542423$$
$$x_{65} = 53.9999965540525$$
$$x_{66} = 91.9999979773788$$
$$x_{67} = 35.9999948310781$$
$$x_{68} = -7.969673621005$$
$$x_{69} = -87.9999978854414$$
$$x_{70} = -81.9999977307176$$
$$x_{71} = -49.9999962783767$$
$$x_{72} = 49.9999962783767$$
$$x_{73} = 45.9999959547572$$
$$x_{74} = -83.9999977847482$$
$$x_{75} = 99.9999981391885$$
$$x_{76} = -19.9999906959359$$
$$x_{77} = -99.9999981391885$$
$$x_{78} = -31.9999941849626$$
$$x_{79} = 67.9999972635124$$
$$x_{80} = 11.9999844932065$$
$$x_{81} = 6.18178808022675$$
$$x_{82} = -39.9999953479705$$
$$x_{83} = -69.9999973416978$$
$$x_{84} = -57.9999967917041$$
$$x_{85} = -71.9999974155395$$
$$x_{86} = 19.9999906959359$$
$$x_{87} = 51.9999964215161$$
$$x_{88} = -11.9999844932065$$
$$x_{89} = -61.999996998691$$
$$x_{90} = -91.9999979773788$$
$$x_{91} = 13.9999867084698$$
$$x_{92} = 97.9999981012127$$
$$x_{93} = 71.9999974155395$$
$$x_{94} = 33.9999945270238$$
$$x_{95} = -77.9999976143442$$
$$x_{96} = 89.9999979324316$$
$$x_{97} = -41.9999955694958$$
$$x_{98} = -55.9999966771221$$
$$x_{99} = 43.9999957708824$$
Signos de extremos en los puntos:
(-47.99999612330898, 9.96390934646259e-8*cos(1.9998720691965*pi))
(75.99999755156375, 3.97452277185799e-8*cos(1.99991920160392*pi))
(-15.999988369915169, 8.96753714808963e-7*cos(1.99961620720057*pi))
(-79.99999767398558, 3.58700676516742e-8*cos(1.9999232415239*pi))
(-33.99999452702376, 1.98588694535917e-7*cos(1.99981939178406*pi))
(41.99999556949581, 1.30140867174804e-7*cos(1.99985379336158*pi))
(77.99999761434418, 3.77331416981893e-8*cos(1.99992127335781*pi))
(47.99999612330898, 9.96390934646259e-8*cos(1.9998720691965*pi))
(83.99999778474817, 3.25352084699585e-8*cos(1.99992689668989*pi))
(85.99999783626566, 3.10395389486259e-8*cos(1.99992859676695*pi))
(-63.9999970924819, 5.60469836698694e-8*cos(1.99990405190283*pi))
(-23.99999224661502, 3.98556686129234e-7*cos(1.99974413829557*pi))
(27.99999335424234, 2.92817075999548e-7*cos(1.99978068999712*pi))
(61.999996998690975, 5.97212401984126e-8*cos(1.99990095680209*pi))
(39.99999534797052, 1.43480311076943e-7*cos(1.99984648302711*pi))
(65.99999718058852, 5.27016627866856e-8*cos(1.99990695942142*pi))
(31.999994184962553, 2.2418803348382e-7*cos(1.99980810376428*pi))
(-59.999996898647325, 6.37690138217689e-8*cos(1.99989765536179*pi))
(2.030211371911484, -5.56998218602049e-5*cos(0.996975273078959*pi))
(69.99999734169776, 4.68507019547142e-8*cos(1.99991227602641*pi))
(-45.99999595475715, 1.08491718652833e-7*cos(1.99986650698611*pi))
(-67.9999972635124, 4.96471542519055e-8*cos(1.99990969590908*pi))
(29.999993797293147, 2.55076183193094e-7*cos(1.9997953106739*pi))
(-5.939362609146204, 6.50781613097177e-6*cos(1.99896610182472*pi))
(-93.99999802041329, 2.59810351992055e-8*cos(1.99993467363856*pi))
(-25.999992843029784, 3.39598543984468e-7*cos(1.99976381998283*pi))
(-13.999986708469844, 1.17127100090182e-6*cos(1.99956137950488*pi))
(-37.999995103126764, 1.58980960561168e-7*cos(1.99983840318328*pi))
(-97.99999810121274, 2.39034179269069e-8*cos(1.99993734002055*pi))
(73.9999974853898, 4.19226509519247e-8*cos(1.99991701786348*pi))
(-75.99999755156375, 3.97452277185799e-8*cos(1.99991920160392*pi))
(-89.99999793243164, 2.8341781288977e-8*cos(1.99993177024407*pi))
(63.9999970924819, 5.60469836698694e-8*cos(1.99990405190283*pi))
(-35.99999483107813, 1.77136202141122e-7*cos(1.99982942557835*pi))
(-95.99999806165468, 2.49097684869321e-8*cos(1.99993603460462*pi))
(-51.99999642151605, 8.48995793170411e-8*cos(1.99988191002967*pi))
(25.999992843029784, 3.39598543984468e-7*cos(1.99976381998283*pi))
(87.99999788544145, 2.96446835259181e-8*cos(1.99993021956789*pi))
(15.999988369915169, 8.96753714808963e-7*cos(1.99961620720057*pi))
(17.99998966214929, 7.08545795495413e-7*cos(1.99965885092661*pi))
(-9.96967830496806, -2.30967483205559e-6*cos(0.999384063945968*pi))
(95.99999806165468, 2.49097684869321e-8*cos(1.99993603460462*pi))
(-73.9999974853898, 4.19226509519247e-8*cos(1.99991701786348*pi))
(-65.99999718058852, 5.27016627866856e-8*cos(1.99990695942142*pi))
(21.999991541761023, 4.74315489249707e-7*cos(1.99972087811375*pi))
(81.99999773071764, 3.41416466285947e-8*cos(1.99992511368191*pi))
(55.9999966771221, 7.32042268609138e-8*cos(1.99989034502914*pi))
(-85.99999783626566, 3.10395389486259e-8*cos(1.99992859676695*pi))
(79.99999767398558, 3.58700676516742e-8*cos(1.9999232415239*pi))
(37.999995103126764, 1.58980960561168e-7*cos(1.99983840318328*pi))
(59.999996898647325, 6.37690138217689e-8*cos(1.99989765536179*pi))
(17.818171374842642, 7.2307971743048e-7*cos(1.99965536980721*pi))
(93.99999802041329, 2.59810351992055e-8*cos(1.99993467363856*pi))
(-17.99998966214929, 7.08545795495413e-7*cos(1.99965885092661*pi))
(-4.393897043797715, -1.18909770285512e-5*cos(0.998602445324593*pi))
(-43.99999577088242, 1.18578761745392e-7*cos(1.99986043911986*pi))
(23.99999224661502, 3.98556686129234e-7*cos(1.99974413829557*pi))
(57.99999679170411, 6.82427028687052e-8*cos(1.99989412623563*pi))
(0, 1)
(-3.969650093426533, -1.45684719172338e-5*cos(0.998453083075589*pi))
(-53.99999655405252, 7.87271806444296e-8*cos(1.9998862837333*pi))
(-21.999991541761023, 4.74315489249707e-7*cos(1.99972087811375*pi))
(-29.999993797293147, 2.55076183193094e-7*cos(1.9997953106739*pi))
(-27.99999335424234, 2.92817075999548e-7*cos(1.99978068999712*pi))
(53.99999655405252, 7.87271806444296e-8*cos(1.9998862837333*pi))
(91.99999797737878, 2.71229238784605e-8*cos(1.99993325349988*pi))
(35.99999483107813, 1.77136202141122e-7*cos(1.99982942557835*pi))
(-7.9696736210050005, -3.61437014336132e-6*cos(0.999229493164989*pi))
(-87.99999788544145, 2.96446835259181e-8*cos(1.99993021956789*pi))
(-81.99999773071764, 3.41416466285947e-8*cos(1.99992511368191*pi))
(-49.999996278376656, 9.18273866567815e-8*cos(1.99987718642956*pi))
(49.999996278376656, 9.18273866567815e-8*cos(1.99987718642956*pi))
(45.99999595475715, 1.08491718652833e-7*cos(1.99986650698611*pi))
(-83.99999778474817, 3.25352084699585e-8*cos(1.99992689668989*pi))
(99.99999813918849, 2.2956842520043e-8*cos(1.99993859322012*pi))
(-19.99999069593595, 5.73921891832805e-7*cos(1.99969296588631*pi))
(-99.99999813918849, 2.2956842520043e-8*cos(1.99993859322012*pi))
(-31.999994184962553, 2.2418803348382e-7*cos(1.99980810376428*pi))
(67.9999972635124, 4.96471542519055e-8*cos(1.99990969590908*pi))
(11.999984493206487, 1.59423174087219e-6*cos(1.99948827581409*pi))
(6.181788080226745, 6.00739947543784e-6*cos(1.99900664748259*pi))
(-39.99999534797052, 1.43480311076943e-7*cos(1.99984648302711*pi))
(-69.99999734169776, 4.68507019547142e-8*cos(1.99991227602641*pi))
(-57.99999679170411, 6.82427028687052e-8*cos(1.99989412623563*pi))
(-71.9999974155395, 4.42840351145172e-8*cos(1.99991471280327*pi))
(19.99999069593595, 5.73921891832805e-7*cos(1.99969296588631*pi))
(51.99999642151605, 8.48995793170411e-8*cos(1.99988191002967*pi))
(-11.999984493206487, 1.59423174087219e-6*cos(1.99948827581409*pi))
(-61.999996998690975, 5.97212401984126e-8*cos(1.99990095680209*pi))
(-91.99999797737878, 2.71229238784605e-8*cos(1.99993325349988*pi))
(13.999986708469844, 1.17127100090182e-6*cos(1.99956137950488*pi))
(97.99999810121274, 2.39034179269069e-8*cos(1.99993734002055*pi))
(71.9999974155395, 4.42840351145172e-8*cos(1.99991471280327*pi))
(33.99999452702376, 1.98588694535917e-7*cos(1.99981939178406*pi))
(-77.99999761434418, 3.77331416981893e-8*cos(1.99992127335781*pi))
(89.99999793243164, 2.8341781288977e-8*cos(1.99993177024407*pi))
(-41.99999556949581, 1.30140867174804e-7*cos(1.99985379336158*pi))
(-55.9999966771221, 7.32042268609138e-8*cos(1.99989034502914*pi))
(43.99999577088242, 1.18578761745392e-7*cos(1.99986043911986*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:
La función no tiene puntos mínimos
Puntos máximos de la función:
$$x_{99} = -47.999996123309$$
$$x_{99} = 75.9999975515638$$
$$x_{99} = -15.9999883699152$$
$$x_{99} = -79.9999976739856$$
$$x_{99} = -33.9999945270238$$
$$x_{99} = 41.9999955694958$$
$$x_{99} = 77.9999976143442$$
$$x_{99} = 47.999996123309$$
$$x_{99} = 83.9999977847482$$
$$x_{99} = 85.9999978362657$$
$$x_{99} = -63.9999970924819$$
$$x_{99} = -23.999992246615$$
$$x_{99} = 27.9999933542423$$
$$x_{99} = 61.999996998691$$
$$x_{99} = 39.9999953479705$$
$$x_{99} = 65.9999971805885$$
$$x_{99} = 31.9999941849626$$
$$x_{99} = -59.9999968986473$$
$$x_{99} = 2.03021137191148$$
$$x_{99} = 69.9999973416978$$
$$x_{99} = -45.9999959547572$$
$$x_{99} = -67.9999972635124$$
$$x_{99} = 29.9999937972931$$
$$x_{99} = -5.9393626091462$$
$$x_{99} = -93.9999980204133$$
$$x_{99} = -25.9999928430298$$
$$x_{99} = -13.9999867084698$$
$$x_{99} = -37.9999951031268$$
$$x_{99} = -97.9999981012127$$
$$x_{99} = 73.9999974853898$$
$$x_{99} = -75.9999975515638$$
$$x_{99} = -89.9999979324316$$
$$x_{99} = 63.9999970924819$$
$$x_{99} = -35.9999948310781$$
$$x_{99} = -95.9999980616547$$
$$x_{99} = -51.9999964215161$$
$$x_{99} = 25.9999928430298$$
$$x_{99} = 87.9999978854414$$
$$x_{99} = 15.9999883699152$$
$$x_{99} = 17.9999896621493$$
$$x_{99} = -9.96967830496806$$
$$x_{99} = 95.9999980616547$$
$$x_{99} = -73.9999974853898$$
$$x_{99} = -65.9999971805885$$
$$x_{99} = 21.999991541761$$
$$x_{99} = 81.9999977307176$$
$$x_{99} = 55.9999966771221$$
$$x_{99} = -85.9999978362657$$
$$x_{99} = 79.9999976739856$$
$$x_{99} = 37.9999951031268$$
$$x_{99} = 59.9999968986473$$
$$x_{99} = 17.8181713748426$$
$$x_{99} = 93.9999980204133$$
$$x_{99} = -17.9999896621493$$
$$x_{99} = -4.39389704379771$$
$$x_{99} = -43.9999957708824$$
$$x_{99} = 23.999992246615$$
$$x_{99} = 57.9999967917041$$
$$x_{99} = 0$$
$$x_{99} = -3.96965009342653$$
$$x_{99} = -53.9999965540525$$
$$x_{99} = -21.999991541761$$
$$x_{99} = -29.9999937972931$$
$$x_{99} = -27.9999933542423$$
$$x_{99} = 53.9999965540525$$
$$x_{99} = 91.9999979773788$$
$$x_{99} = 35.9999948310781$$
$$x_{99} = -7.969673621005$$
$$x_{99} = -87.9999978854414$$
$$x_{99} = -81.9999977307176$$
$$x_{99} = -49.9999962783767$$
$$x_{99} = 49.9999962783767$$
$$x_{99} = 45.9999959547572$$
$$x_{99} = -83.9999977847482$$
$$x_{99} = 99.9999981391885$$
$$x_{99} = -19.9999906959359$$
$$x_{99} = -99.9999981391885$$
$$x_{99} = -31.9999941849626$$
$$x_{99} = 67.9999972635124$$
$$x_{99} = 11.9999844932065$$
$$x_{99} = 6.18178808022675$$
$$x_{99} = -39.9999953479705$$
$$x_{99} = -69.9999973416978$$
$$x_{99} = -57.9999967917041$$
$$x_{99} = -71.9999974155395$$
$$x_{99} = 19.9999906959359$$
$$x_{99} = 51.9999964215161$$
$$x_{99} = -11.9999844932065$$
$$x_{99} = -61.999996998691$$
$$x_{99} = -91.9999979773788$$
$$x_{99} = 13.9999867084698$$
$$x_{99} = 97.9999981012127$$
$$x_{99} = 71.9999974155395$$
$$x_{99} = 33.9999945270238$$
$$x_{99} = -77.9999976143442$$
$$x_{99} = 89.9999979324316$$
$$x_{99} = -41.9999955694958$$
$$x_{99} = -55.9999966771221$$
$$x_{99} = 43.9999957708824$$
Decrece en los intervalos
$$\left(-\infty, -99.9999981391885\right]$$
Crece en los intervalos
$$\left[99.9999981391885, \infty\right)$$
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
$$\lim_{x \to -\infty} \left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty} \left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty} \left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty} \left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = 0$$
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 Abs(cos(x*(33*pi))/(1 - (66*x)^2)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right|}{x}\right) = 0$$
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:
$$\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = \left|{\frac{\cos{\left(33 \pi x \right)}}{4356 x^{2} - 1}}\right|$$
- No
$$\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = - \left|{\frac{\cos{\left(33 \pi x \right)}}{4356 x^{2} - 1}}\right|$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = \left|{\frac{\cos{\left(33 \pi x \right)}}{4356 x^{2} - 1}}\right|$$
- No
$$\left|{\frac{\cos{\left(33 \pi x \right)}}{1 - \left(66 x\right)^{2}}}\right| = - \left|{\frac{\cos{\left(33 \pi x \right)}}{4356 x^{2} - 1}}\right|$$
- No
es decir, función
no es
par ni impar