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Trazado el gráfico de la función/ ((1-sqrt(cos(x)))*sin(x))/x

Gráfico de la función y = ((1-sqrt(cos(x)))*sin(x))/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /      ________\       
       \1 - \/ cos(x) /*sin(x)
f(x) = -----------------------
                  x
f(x)=(1−cos⁡(x))sin⁡(x)xf{\left(x \right)} = \frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x} 
f = ((1 - sqrt(cos(x)))*sin(x))/x
Gráfico de la función
02468-8-6-4-2-10101.0-1.0
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{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:
(1−cos⁡(x))sin⁡(x)x=0\frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=28.2743338823081x_{1} = 28.2743338823081
x2=91.106186954104x_{2} = 91.106186954104
x3=−31.4160025281077x_{3} = -31.4160025281077
x4=−62.8316722125381x_{4} = -62.8316722125381
x5=−50.2654105793664x_{5} = -50.2654105793664
x6=65.9734457253857x_{6} = 65.9734457253857
x7=−56.548902423916x_{7} = -56.548902423916
x8=−78.5398163397448x_{8} = -78.5398163397448
x9=−21.9911485751286x_{9} = -21.9911485751286
x10=−81.6814265304604x_{10} = -81.6814265304604
x11=34.5575191894877x_{11} = 34.5575191894877
x12=−59.6902604182061x_{12} = -59.6902604182061
x13=25.13292309214x_{13} = 25.13292309214
x14=62.8317275324877x_{14} = 62.8317275324877
x15=−18.8497328770223x_{15} = -18.8497328770223
x16=−12.5665930631181x_{16} = -12.5665930631181
x17=−47.1238898038469x_{17} = -47.1238898038469
x18=94.2477801894766x_{18} = 94.2477801894766
x19=−69.1151724017491x_{19} = -69.1151724017491
x20=−18.8493683789461x_{20} = -18.8493683789461
x21=−87.9646059771989x_{21} = -87.9646059771989
x22=3.14159265358979x_{22} = 3.14159265358979
x23=9.42477796076938x_{23} = 9.42477796076938
x24=43.981925840808x_{24} = 43.981925840808
x25=−100.530839085825x_{25} = -100.530839085825
x26=18.8494253857312x_{26} = 18.8494253857312
x27=37.699190730448x_{27} = 37.699190730448
x28=−56.5485382276347x_{28} = -56.5485382276347
x29=31.4160604014781x_{29} = 31.4160604014781
x30=−3.14159265358979x_{30} = -3.14159265358979
x31=43.9823032528597x_{31} = 43.9823032528597
x32=56.5485984144687x_{32} = 56.5485984144687
x33=−12.566234718141x_{33} = -12.566234718141
x34=−40.8407044966673x_{34} = -40.8407044966673
x35=81.6814918473805x_{35} = 81.6814918473805
x36=−84.8230016469244x_{36} = -84.8230016469244
x37=−62.8320362012315x_{37} = -62.8320362012315
x38=50.2654784080386x_{38} = 50.2654784080386
x39=62.8320917832409x_{39} = 62.8320917832409
x40=18.8497853935164x_{40} = 18.8497853935164
x41=12.5662967174173x_{41} = 12.5662967174173
x42=6.28317665322009x_{42} = 6.28317665322009
x43=25.132559128045x_{43} = 25.132559128045
x44=−43.9823032321115x_{44} = -43.9823032321115
x45=−6.28310744612189x_{45} = -6.28310744612189
x46=47.1238898038469x_{46} = 47.1238898038469
x47=59.6902604182061x_{47} = 59.6902604182061
x48=−25.1325015716212x_{48} = -25.1325015716212
x49=40.8407044966673x_{49} = 40.8407044966673
x50=−94.2477117863401x_{50} = -94.2477117863401
x51=75.3983616656452x_{51} = 75.3983616656452
x52=97.3893722612836x_{52} = 97.3893722612836
x53=−37.6991249632092x_{53} = -37.6991249632092
x54=53.4070751110265x_{54} = 53.4070751110265
x55=100.530899375963x_{55} = 100.530899375963
x56=78.5398163397448x_{56} = 78.5398163397448
x57=72.2566310325652x_{57} = 72.2566310325652
x58=69.1152255253406x_{58} = 69.1152255253406
x59=−75.398303545531x_{59} = -75.398303545531
x60=87.9646063182847x_{60} = 87.9646063182847
x61=−15.707963267949x_{61} = -15.707963267949
x62=31.4156922588288x_{62} = 31.4156922588288
x63=−9.42477796076938x_{63} = -9.42477796076938
x64=−69.1148064262417x_{64} = -69.1148064262417
x65=−25.1328709111794x_{65} = -25.1328709111794
x66=15.707963267949x_{66} = 15.707963267949
x67=−72.2566310325652x_{67} = -72.2566310325652
x68=−91.106186954104x_{68} = -91.106186954104
x69=21.9911485751286x_{69} = 21.9911485751286
x70=75.3979958816101x_{70} = 75.3979958816101
x71=−65.9734457253857x_{71} = -65.9734457253857
x72=−28.2743338823081x_{72} = -28.2743338823081
x73=−34.5575191894877x_{73} = -34.5575191894877
x74=69.114861676761x_{74} = 69.114861676761
x75=−53.4070751110265x_{75} = -53.4070751110265
x76=84.8230016469244x_{76} = 84.8230016469244
x77=−97.3893722612836x_{77} = -97.3893722612836
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en ((1 - sqrt(cos(x)))*sin(x))/x.
(1−cos⁡(0))sin⁡(0)0\frac{\left(1 - \sqrt{\cos{\left(0 \right)}}\right) \sin{\left(0 \right)}}{0}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((1−cos⁡(x))sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((1−cos⁡(x))sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((1 - sqrt(cos(x)))*sin(x))/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((1−cos⁡(x))sin⁡(x)x2)=0\lim_{x \to -\infty}\left(\frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((1−cos⁡(x))sin⁡(x)x2)=0\lim_{x \to \infty}\left(\frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{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:
(1−cos⁡(x))sin⁡(x)x=(1−cos⁡(x))sin⁡(x)x\frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x} = \frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x}
- No
(1−cos⁡(x))sin⁡(x)x=−(1−cos⁡(x))sin⁡(x)x\frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x} = - \frac{\left(1 - \sqrt{\cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x}
- No
es decir, función
no es
par ni impar
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