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Trazado el gráfico de la función/ sin(5*x)^2/(-cos(2*x)+cos(3*x))

Gráfico de la función y = sin(5*x)^2/(-cos(2*x)+cos(3*x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
               2           
            sin (5*x)      
f(x) = --------------------
       -cos(2*x) + cos(3*x)
f(x)=sin⁡2(5x)−cos⁡(2x)+cos⁡(3x)f{\left(x \right)} = \frac{\sin^{2}{\left(5 x \right)}}{- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}} 
f = sin(5*x)^2/(-cos(2*x) + cos(3*x))
Gráfico de la función
02468-8-6-4-2-1010-2010
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−5.02654824574367x_{1} = -5.02654824574367
x2=−3.76991118430775x_{2} = -3.76991118430775
x3=−2.51327412287183x_{3} = -2.51327412287183
x4=−1.25663706143592x_{4} = -1.25663706143592
x5=0x_{5} = 0
x6=1.25663706143592x_{6} = 1.25663706143592
x7=2.51327412287183x_{7} = 2.51327412287183
x8=3.76991118430775x_{8} = 3.76991118430775
x9=5.02654824574367x_{9} = 5.02654824574367
x10=6.28318530717959x_{10} = 6.28318530717959
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(5x)−cos⁡(2x)+cos⁡(3x)=0\frac{\sin^{2}{\left(5 x \right)}}{- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π5x_{1} = \frac{\pi}{5}
Solución numérica
x1=62.2035345883974x_{1} = 62.2035345883974
x2=−76.0265421878971x_{2} = -76.0265421878971
x3=−13.8230076757951x_{3} = -13.8230076757951
x4=28.2743338657239x_{4} = 28.2743338657239
x5=99.9026464709377x_{5} = 99.9026464709377
x6=23.2477856724586x_{6} = 23.2477856724586
x7=89.8495499097163x_{7} = 89.8495499097163
x8=−87.3362757820503x_{8} = -87.3362757820503
x9=−21.9911485866827x_{9} = -21.9911485866827
x10=−5.6548668424001x_{10} = -5.6548668424001
x11=33.9292006587698x_{11} = 33.9292006587698
x12=16.3362817986669x_{12} = 16.3362817986669
x13=72.2566310277517x_{13} = 72.2566310277517
x14=58.4336233583946x_{14} = 58.4336233583946
x15=−64.0884901332318x_{15} = -64.0884901332318
x16=−73.5132681058356x_{16} = -73.5132681058356
x17=88.5929127614729x_{17} = 88.5929127614729
x18=−20.1061929829747x_{18} = -20.1061929829747
x19=−61.5752160103599x_{19} = -61.5752160103599
x20=−15.7079632947321x_{20} = -15.7079632947321
x21=74.1415866247191x_{21} = 74.1415866247191
x22=4.39822969465763x_{22} = 4.39822969465763
x23=−25.7610597506209x_{23} = -25.7610597506209
x24=44.6106156512842x_{24} = 44.6106156512842
x25=−77.9114978090269x_{25} = -77.9114978090269
x26=−43.3539785165215x_{26} = -43.3539785165215
x27=18.2212374166027x_{27} = 18.2212374166027
x28=−33.9292006587698x_{28} = -33.9292006587698
x29=14.4513262943339x_{29} = 14.4513262943339
x30=−49.6371640035054x_{30} = -49.6371640035054
x31=8.16814097431833x_{31} = 8.16814097431833
x32=−79.7964533620005x_{32} = -79.7964533620005
x33=85.4513201776424x_{33} = 85.4513201776424
x34=92.3628239729803x_{34} = 92.3628239729803
x35=−10.0530964914873x_{35} = -10.0530964914873
x36=−83.5663644524131x_{36} = -83.5663644524131
x37=−11.9380522433082x_{37} = -11.9380522433082
x38=−57.8053048260522x_{38} = -57.8053048260522
x39=5.65486684217021x_{39} = 5.65486684217021
x40=65.9734457509078x_{40} = 65.9734457509078
x41=−39.5840674140102x_{41} = -39.5840674140102
x42=55.9203493083275x_{42} = 55.9203493083275
x43=−52.1504380012018x_{43} = -52.1504380012018
x44=−32.0442450209278x_{44} = -32.0442450209278
x45=49.6371639501047x_{45} = 49.6371639501047
x46=−28.9026524130261x_{46} = -28.9026524130261
x47=−38.3274302802565x_{47} = -38.3274302802565
x48=−96.1327351435332x_{48} = -96.1327351435332
x49=48.3805268285267x_{49} = 48.3805268285267
x50=40.2123859659494x_{50} = 40.2123859659494
x51=98.0176907920015x_{51} = 98.0176907920015
x52=79.7964532982376x_{52} = 79.7964532982376
x53=−47.7522083345649x_{53} = -47.7522083345649
x54=−84.8230014117419x_{54} = -84.8230014117419
x55=11.9380521444101x_{55} = 11.9380521444101
x56=−23.8761041672824x_{56} = -23.8761041672824
x57=20.1061929829747x_{57} = 20.1061929829747
x58=−93.6194611639668x_{58} = -93.6194611639668
x59=−65.9734457671014x_{59} = -65.9734457671014
x60=−82.3097274122509x_{60} = -82.3097274122509
x61=70.999993997043x_{61} = 70.999993997043
x62=32.0442449414914x_{62} = 32.0442449414914
x63=64.0884901332318x_{63} = 64.0884901332318
x64=38.3274303192548x_{64} = 38.3274303192548
x65=−3.76991118430775x_{65} = -3.76991118430775
x66=54.0353936417444x_{66} = 54.0353936417444
x67=93.619461038341x_{67} = 93.619461038341
x68=−89.8495497995961x_{68} = -89.8495497995961
x69=−98.0176907920015x_{69} = -98.0176907920015
x70=45.8672527783101x_{70} = 45.8672527783101
x71=60.318578948924x_{71} = 60.318578948924
x72=82.3097274813775x_{72} = 82.3097274813775
x73=−8.16814086834085x_{73} = -8.16814086834085
x74=−86.079638662179x_{74} = -86.079638662179
x75=−54.0353936417444x_{75} = -54.0353936417444
x76=84.1946831162065x_{76} = 84.1946831162065
x77=35.8141562422107x_{77} = 35.8141562422107
x78=−67.8584013175395x_{78} = -67.8584013175395
x79=1.88495563619798x_{79} = 1.88495563619798
x80=76.0265421674286x_{80} = 76.0265421674286
x81=−69.7433569144253x_{81} = -69.7433569144253
x82=28.9026524130261x_{82} = 28.9026524130261
x83=−42.0973415112003x_{83} = -42.0973415112003
x84=10.0530964914873x_{84} = 10.0530964914873
x85=66.6017642561036x_{85} = 66.6017642561036
x86=30.159289474462x_{86} = 30.159289474462
x87=−59.6902604535354x_{87} = -59.6902604535354
x88=97.3893723474895x_{88} = 97.3893723474895
x89=77.9114978090269x_{89} = 77.9114978090269
x90=−99.9026463767368x_{90} = -99.9026463767368
x91=21.9911485849706x_{91} = 21.9911485849706
x92=−35.8141561941907x_{92} = -35.8141561941907
x93=52.150438134161x_{93} = 52.150438134161
x94=−91.734505484822x_{94} = -91.734505484822
x95=−2.51327412287183x_{95} = -2.51327412287183
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 sin(5*x)^2/(-cos(2*x) + cos(3*x)).
sin⁡2(0⋅5)−cos⁡(0⋅2)+cos⁡(0⋅3)\frac{\sin^{2}{\left(0 \cdot 5 \right)}}{- \cos{\left(0 \cdot 2 \right)} + \cos{\left(0 \cdot 3 \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Asíntotas verticales
Hay:
x1=−5.02654824574367x_{1} = -5.02654824574367
x2=−3.76991118430775x_{2} = -3.76991118430775
x3=−2.51327412287183x_{3} = -2.51327412287183
x4=−1.25663706143592x_{4} = -1.25663706143592
x5=0x_{5} = 0
x6=1.25663706143592x_{6} = 1.25663706143592
x7=2.51327412287183x_{7} = 2.51327412287183
x8=3.76991118430775x_{8} = 3.76991118430775
x9=5.02654824574367x_{9} = 5.02654824574367
x10=6.28318530717959x_{10} = 6.28318530717959
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(sin⁡2(5x)−cos⁡(2x)+cos⁡(3x))y = \lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(5 x \right)}}{- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(sin⁡2(5x)−cos⁡(2x)+cos⁡(3x))y = \lim_{x \to \infty}\left(\frac{\sin^{2}{\left(5 x \right)}}{- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(5*x)^2/(-cos(2*x) + cos(3*x)), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(sin⁡2(5x)x(−cos⁡(2x)+cos⁡(3x)))y = x \lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(5 x \right)}}{x \left(- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}\right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(sin⁡2(5x)x(−cos⁡(2x)+cos⁡(3x)))y = x \lim_{x \to \infty}\left(\frac{\sin^{2}{\left(5 x \right)}}{x \left(- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}\right)}\right)
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(5x)−cos⁡(2x)+cos⁡(3x)=sin⁡2(5x)−cos⁡(2x)+cos⁡(3x)\frac{\sin^{2}{\left(5 x \right)}}{- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}} = \frac{\sin^{2}{\left(5 x \right)}}{- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}}
- Sí
sin⁡2(5x)−cos⁡(2x)+cos⁡(3x)=−sin⁡2(5x)−cos⁡(2x)+cos⁡(3x)\frac{\sin^{2}{\left(5 x \right)}}{- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}} = - \frac{\sin^{2}{\left(5 x \right)}}{- \cos{\left(2 x \right)} + \cos{\left(3 x \right)}}
- No
es decir, función
es
par
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