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Trazado el gráfico de la función/ sin(4*x)*cos(5*x)

Gráfico de la función y = sin(4*x)*cos(5*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = sin(4*x)*cos(5*x)
f(x)=sin⁡(4x)cos⁡(5x)f{\left(x \right)} = \sin{\left(4 x \right)} \cos{\left(5 x \right)} 
f = sin(4*x)*cos(5*x)
Gráfico de la función
02468-8-6-4-2-10102-2
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⁡(4x)cos⁡(5x)=0\sin{\left(4 x \right)} \cos{\left(5 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−π10x_{2} = - \frac{\pi}{10}
x3=π10x_{3} = \frac{\pi}{10}
Solución numérica
x1=−9.73893722612836x_{1} = -9.73893722612836
x2=−58.1194640633422x_{2} = -58.1194640633422
x3=−66.2876049907446x_{3} = -66.2876049907446
x4=−65.9734457253857x_{4} = -65.9734457253857
x5=38.6415896391545x_{5} = 38.6415896391545
x6=46.18141200777x_{6} = 46.18141200777
x7=50.2654824574367x_{7} = 50.2654824574367
x8=73.1991088286422x_{8} = 73.1991088286422
x9=26.0752190247953x_{9} = 26.0752190247953
x10=84.037603483527x_{10} = 84.037603483527
x11=54.1924732744239x_{11} = 54.1924732744239
x12=18.8495559215388x_{12} = 18.8495559215388
x13=29.8451301331324x_{13} = 29.8451301331324
x14=56.5486677646163x_{14} = 56.5486677646163
x15=92.0486647501809x_{15} = 92.0486647501809
x16=−5.96902604182061x_{16} = -5.96902604182061
x17=51.8362786498421x_{17} = 51.8362786498421
x18=−41.7831822927443x_{18} = -41.7831822927443
x19=−60.0044196835651x_{19} = -60.0044196835651
x20=−47.9092879672443x_{20} = -47.9092879672443
x21=−49.9513231920777x_{21} = -49.9513231920777
x22=44.924774946334x_{22} = 44.924774946334
x23=60.0044196835651x_{23} = 60.0044196835651
x24=−14.1371669407303x_{24} = -14.1371669407303
x25=21.9911485751286x_{25} = 21.9911485751286
x26=38.0132711084365x_{26} = 38.0132711084365
x27=−69.9004365423729x_{27} = -69.9004365423729
x28=−87.9645943005142x_{28} = -87.9645943005142
x29=4.08407044966673x_{29} = 4.08407044966673
x30=−3.92699081698724x_{30} = -3.92699081698724
x31=−55.7632696012188x_{31} = -55.7632696012188
x32=−17.9070781254618x_{32} = -17.9070781254618
x33=62.0464549083984x_{33} = 62.0464549083984
x34=−93.9336203423348x_{34} = -93.9336203423348
x35=−51.8362787041329x_{35} = -51.8362787041329
x36=94.2477796076938x_{36} = 94.2477796076938
x37=16.0221225333079x_{37} = 16.0221225333079
x38=−31.7300858012569x_{38} = -31.7300858012569
x39=12.2522113490002x_{39} = 12.2522113490002
x40=−77.7544181763474x_{40} = -77.7544181763474
x41=−29.8451301291032x_{41} = -29.8451301291032
x42=81.9955682586936x_{42} = 81.9955682586936
x43=42.4115009489444x_{43} = 42.4115009489444
x44=78.8539756051038x_{44} = 78.8539756051038
x45=−95.8185758707203x_{45} = -95.8185758707203
x46=70.0575161750524x_{46} = 70.0575161750524
x47=−61.8893752757189x_{47} = -61.8893752757189
x48=−53.7212343763855x_{48} = -53.7212343763855
x49=−81.9955682586936x_{49} = -81.9955682586936
x50=−39.8982267005904x_{50} = -39.8982267005904
x51=48.0663675999238x_{51} = 48.0663675999238
x52=2.19911485751286x_{52} = 2.19911485751286
x53=40.0553063332699x_{53} = 40.0553063332699
x54=66.7588438887831x_{54} = 66.7588438887831
x55=−85.7654794430014x_{55} = -85.7654794430014
x56=−71.9424717672063x_{56} = -71.9424717672063
x57=98.174770424681x_{57} = 98.174770424681
x58=10.2101761241668x_{58} = 10.2101761241668
x59=100.216805649514x_{59} = 100.216805649514
x60=64.4026494609915x_{60} = 64.4026494609915
x61=−16.0221225333079x_{61} = -16.0221225333079
x62=−73.8274272865168x_{62} = -73.8274272865168
x63=90.1637091580271x_{63} = 90.1637091580271
x64=−36.1283154987097x_{64} = -36.1283154987097
x65=78.2256570743859x_{65} = 78.2256570743859
x66=43.9822971502571x_{66} = 43.9822971502571
x67=86.3937979933609x_{67} = 86.3937979933609
x68=−99.7455667514759x_{68} = -99.7455667514759
x69=−33.7721210260903x_{69} = -33.7721210260903
x70=76.1836218495525x_{70} = 76.1836218495525
x71=18.0641577581413x_{71} = 18.0641577581413
x72=0x_{72} = 0
x73=−2.82743338823081x_{73} = -2.82743338823081
x74=−21.9911485751286x_{74} = -21.9911485751286
x75=24.1902634326414x_{75} = 24.1902634326414
x76=7.85398160566834x_{76} = 7.85398160566834
x77=−19.7920337176157x_{77} = -19.7920337176157
x78=−80.1106126319001x_{78} = -80.1106126319001
x79=68.1725605828985x_{79} = 68.1725605828985
x80=−75.712382951514x_{80} = -75.712382951514
x81=−44.924774946334x_{81} = -44.924774946334
x82=−63.7743308678728x_{82} = -63.7743308678728
x83=−83.8805238508475x_{83} = -83.8805238508475
x84=−11.7809724509617x_{84} = -11.7809724509617
x85=14.765485471872x_{85} = 14.765485471872
x86=−38.0132711084365x_{86} = -38.0132711084365
x87=−91.8915851175014x_{87} = -91.8915851175014
x88=32.2013246992954x_{88} = 32.2013246992954
x89=−97.7035315266426x_{89} = -97.7035315266426
x90=−43.9822971502571x_{90} = -43.9822971502571
x91=34.2433599241287x_{91} = 34.2433599241287
x92=28.5884931476671x_{92} = 28.5884931476671
x93=−7.85398160320836x_{93} = -7.85398160320836
x94=−27.9601746169492x_{94} = -27.9601746169492
x95=−25.9181393921158x_{95} = -25.9181393921158
x96=91.420346219463x_{96} = 91.420346219463
x97=72.2566310325652x_{97} = 72.2566310325652
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(4*x)*cos(5*x).
sin⁡(0⋅4)cos⁡(0⋅5)\sin{\left(0 \cdot 4 \right)} \cos{\left(0 \cdot 5 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(4x)cos⁡(5x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin{\left(4 x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞(sin⁡(4x)cos⁡(5x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin{\left(4 x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(4*x)*cos(5*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(4x)cos⁡(5x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(4 x \right)} \cos{\left(5 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(4x)cos⁡(5x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(4 x \right)} \cos{\left(5 x \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:
sin⁡(4x)cos⁡(5x)=−sin⁡(4x)cos⁡(5x)\sin{\left(4 x \right)} \cos{\left(5 x \right)} = - \sin{\left(4 x \right)} \cos{\left(5 x \right)}
- No
sin⁡(4x)cos⁡(5x)=sin⁡(4x)cos⁡(5x)\sin{\left(4 x \right)} \cos{\left(5 x \right)} = \sin{\left(4 x \right)} \cos{\left(5 x \right)}
- No
es decir, función
no es
par ni impar
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