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Trazado el gráfico de la función/ cos^2(20*x)/sin(40x)

Gráfico de la función y = cos^2(20*x)/sin(40x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2      
       cos (20*x)
f(x) = ----------
       sin(40*x)
f(x)=cos⁡2(20x)sin⁡(40x)f{\left(x \right)} = \frac{\cos^{2}{\left(20 x \right)}}{\sin{\left(40 x \right)}} 
f = cos(20*x)^2/sin(40*x)
Gráfico de la función
02468-8-6-4-2-1010-200200
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=0.0785398163397448x_{2} = 0.0785398163397448
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:
cos⁡2(20x)sin⁡(40x)=0\frac{\cos^{2}{\left(20 x \right)}}{\sin{\left(40 x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−64.009950316892x_{1} = -64.009950316892
x2=68.2511003992383x_{2} = 68.2511003992383
x3=80.0320728502x_{3} = 80.0320728502
x4=−51.7577389678918x_{4} = -51.7577389678918
x5=100.295345465854x_{5} = 100.295345465854
x6=98.2533102410208x_{6} = 98.2533102410208
x7=−41.7046424764045x_{7} = -41.7046424764045
x8=−91.9701249338412x_{8} = -91.9701249338412
x9=−75.7909227678538x_{9} = -75.7909227678538
x10=46.2599518241097x_{10} = 46.2599518241097
x11=50.0298630084175x_{11} = 50.0298630084175
x12=−31.9657052502761x_{12} = -31.9657052502761
x13=14.2157067574938x_{13} = 14.2157067574938
x14=54.2710130907637x_{14} = 54.2710130907637
x15=−17.9856179418016x_{15} = -17.9856179418016
x16=8.24668071567321x_{16} = 8.24668071567321
x17=27.8816348006094x_{17} = 27.8816348006094
x18=44.2179165992763x_{18} = 44.2179165992763
x19=−95.740036118149x_{19} = -95.740036118149
x20=24.2688032489812x_{20} = 24.2688032489812
x21=−59.7688002345458x_{21} = -59.7688002345458
x22=38.2488905574557x_{22} = 38.2488905574557
x23=25.9966792084555x_{23} = 25.9966792084555
x24=16.2577419823272x_{24} = 16.2577419823272
x25=−35.7356164345839x_{25} = -35.7356164345839
x26=94.0121601586746x_{26} = 94.0121601586746
x27=−7.77544181763474x_{27} = -7.77544181763474
x28=42.0188017417635x_{28} = 42.0188017417635
x29=77.9900376253666x_{29} = 77.9900376253666
x30=−27.8816348006094x_{30} = -27.8816348006094
x31=48.1449074162636x_{31} = 48.1449074162636
x32=11.2311937365835x_{32} = 11.2311937365835
x33=40.4480054149686x_{33} = 40.4480054149686
x34=64.009950316892x_{34} = 64.009950316892
x35=−11.8595122673015x_{35} = -11.8595122673015
x36=88.3572933822129x_{36} = 88.3572933822129
x37=62.1249947247382x_{37} = 62.1249947247382
x38=−39.9767665169301x_{38} = -39.9767665169301
x39=84.2732229325462x_{39} = 84.2732229325462
x40=90.2422489743668x_{40} = 90.2422489743668
x41=−57.7267650097125x_{41} = -57.7267650097125
x42=36.2068553326224x_{42} = 36.2068553326224
x43=−34.0077404751095x_{43} = -34.0077404751095
x44=−37.7776516594173x_{44} = -37.7776516594173
x45=96.2112750161874x_{45} = 96.2112750161874
x46=−43.7466777012379x_{46} = -43.7466777012379
x47=−29.7665903927633x_{47} = -29.7665903927633
x48=−61.9679150920587x_{48} = -61.9679150920587
x49=−49.872783375738x_{49} = -49.872783375738
x50=2.12057504117311x_{50} = 2.12057504117311
x51=−23.6404847182632x_{51} = -23.6404847182632
x52=−87.728974851495x_{52} = -87.728974851495
x53=34.0077404751095x_{53} = 34.0077404751095
x54=−4.00553063332699x_{54} = -4.00553063332699
x55=70.2931356240716x_{55} = 70.2931356240716
x56=74.2201264410589x_{56} = 74.2201264410589
x57=−2.43473430653209x_{57} = -2.43473430653209
x58=−79.8749932175205x_{58} = -79.8749932175205
x59=58.8263224384689x_{59} = 58.8263224384689
x60=86.0010988920206x_{60} = 86.0010988920206
x61=−71.706852318187x_{61} = -71.706852318187
x62=−69.9789763587126x_{62} = -69.9789763587126
x63=−55.9988890502381x_{63} = -55.9988890502381
x64=60.2400391325843x_{64} = 60.2400391325843
x65=−97.7820713429823x_{65} = -97.7820713429823
x66=−53.6426945600457x_{66} = -53.6426945600457
x67=30.2378292908018x_{67} = 30.2378292908018
x68=−77.9900376253666x_{68} = -77.9900376253666
x69=−65.7378262763664x_{69} = -65.7378262763664
x70=20.0276531666349x_{70} = 20.0276531666349
x71=−45.7887129260712x_{71} = -45.7887129260712
x72=55.9988890502381x_{72} = 55.9988890502381
x73=−21.7555291261093x_{73} = -21.7555291261093
x74=−25.9966792084555x_{74} = -25.9966792084555
x75=22.2267680241478x_{75} = 22.2267680241478
x76=6.20464549083984x_{76} = 6.20464549083984
x77=52.2289778659303x_{77} = 52.2289778659303
x78=−5.73340659280137x_{78} = -5.73340659280137
x79=−81.7599488096744x_{79} = -81.7599488096744
x80=−73.7488875430204x_{80} = -73.7488875430204
x81=82.2311877077128x_{81} = 82.2311877077128
x82=−47.9878277835841x_{82} = -47.9878277835841
x83=−67.7798615011998x_{83} = -67.7798615011998
x84=−9.97455667514759x_{84} = -9.97455667514759
x85=32.1227848829556x_{85} = 32.1227848829556
x86=−89.7710100763283x_{86} = -89.7710100763283
x87=−15.7865030842887x_{87} = -15.7865030842887
x88=92.1272045665207x_{88} = 92.1272045665207
x89=4.00553063332699x_{89} = 4.00553063332699
x90=72.021011583546x_{90} = 72.021011583546
x91=18.1426975744811x_{91} = 18.1426975744811
x92=−86.0010988920206x_{92} = -86.0010988920206
x93=12.016591899981x_{93} = 12.016591899981
x94=−99.9811862004952x_{94} = -99.9811862004952
x95=−83.4878247691488x_{95} = -83.4878247691488
x96=−93.8550805259951x_{96} = -93.8550805259951
x97=−13.7444678594553x_{97} = -13.7444678594553
x98=76.2621616658922x_{98} = 76.2621616658922
x99=66.2090651744049x_{99} = 66.2090651744049
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 cos(20*x)^2/sin(40*x).
cos⁡2(0⋅20)sin⁡(0⋅40)\frac{\cos^{2}{\left(0 \cdot 20 \right)}}{\sin{\left(0 \cdot 40 \right)}}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=0.0785398163397448x_{2} = 0.0785398163397448
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→−∞(cos⁡2(20x)sin⁡(40x))y = \lim_{x \to -\infty}\left(\frac{\cos^{2}{\left(20 x \right)}}{\sin{\left(40 x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(cos⁡2(20x)sin⁡(40x))y = \lim_{x \to \infty}\left(\frac{\cos^{2}{\left(20 x \right)}}{\sin{\left(40 x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(20*x)^2/sin(40*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→−∞(cos⁡2(20x)xsin⁡(40x))y = x \lim_{x \to -\infty}\left(\frac{\cos^{2}{\left(20 x \right)}}{x \sin{\left(40 x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(cos⁡2(20x)xsin⁡(40x))y = x \lim_{x \to \infty}\left(\frac{\cos^{2}{\left(20 x \right)}}{x \sin{\left(40 x \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:
cos⁡2(20x)sin⁡(40x)=−cos⁡2(20x)sin⁡(40x)\frac{\cos^{2}{\left(20 x \right)}}{\sin{\left(40 x \right)}} = - \frac{\cos^{2}{\left(20 x \right)}}{\sin{\left(40 x \right)}}
- No
cos⁡2(20x)sin⁡(40x)=cos⁡2(20x)sin⁡(40x)\frac{\cos^{2}{\left(20 x \right)}}{\sin{\left(40 x \right)}} = \frac{\cos^{2}{\left(20 x \right)}}{\sin{\left(40 x \right)}}
- Sí
es decir, función
es
impar
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