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Trazado el gráfico de la función/ cos(7x)+cos(3x)

Gráfico de la función y = cos(7x)+cos(3x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = cos(7*x) + cos(3*x)
f(x)=cos⁡(3x)+cos⁡(7x)f{\left(x \right)} = \cos{\left(3 x \right)} + \cos{\left(7 x \right)} 
f = cos(3*x) + cos(7*x)
Gráfico de la función
02468-8-6-4-2-10105-5
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⁡(3x)+cos⁡(7x)=0\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−9π10x_{1} = - \frac{9 \pi}{10}
x2=−3π4x_{2} = - \frac{3 \pi}{4}
x3=−π2x_{3} = - \frac{\pi}{2}
x4=−π4x_{4} = - \frac{\pi}{4}
x5=π10x_{5} = \frac{\pi}{10}
x6=π4x_{6} = \frac{\pi}{4}
x7=π2x_{7} = \frac{\pi}{2}
x8=3π4x_{8} = \frac{3 \pi}{4}
x9=−ilog⁡(−105+58+25+58+i4+5i4)x_{9} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x10=−ilog⁡(−25+58+105+58−5i4−i4)x_{10} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
x11=−ilog⁡(−105−516+25−516+25+516+105+516+i4+5i4)x_{11} = - i \log{\left(- \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x12=−ilog⁡(−105+516−105−516−25+516+25−516−i4+5i4)x_{12} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x13=−ilog⁡(−105+516−25+516−25−516+105−516−5i4−i4)x_{13} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
x14=−ilog⁡(−25−516+25+516+105−516+105+516−5i4+i4)x_{14} = - i \log{\left(- \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{5} i}{4} + \frac{i}{4} \right)}
Solución numérica
x1=93.9336203423348x_{1} = 93.9336203423348
x2=−61.8893752757189x_{2} = -61.8893752757189
x3=95.1902574037707x_{3} = 95.1902574037707
x4=32.2013246992954x_{4} = 32.2013246992954
x5=10.2101761241668x_{5} = 10.2101761241668
x6=14.1371669411541x_{6} = 14.1371669411541
x7=−63.7743308678728x_{7} = -63.7743308678728
x8=60.0044196835651x_{8} = 60.0044196835651
x9=24.1902634326414x_{9} = 24.1902634326414
x10=−21.6769893097696x_{10} = -21.6769893097696
x11=16.0221225333079x_{11} = 16.0221225333079
x12=−27.9601746169492x_{12} = -27.9601746169492
x13=−31.7300858012569x_{13} = -31.7300858012569
x14=−49.9513231920777x_{14} = -49.9513231920777
x15=68.1725605828985x_{15} = 68.1725605828985
x16=−51.8362787842316x_{16} = -51.8362787842316
x17=−3.92699081698724x_{17} = -3.92699081698724
x18=66.2876049907446x_{18} = 66.2876049907446
x19=−69.9004365423729x_{19} = -69.9004365423729
x20=−58.9048622548086x_{20} = -58.9048622548086
x21=−9.73893722612836x_{21} = -9.73893722612836
x22=78.2256570743859x_{22} = 78.2256570743859
x23=27.9601746169492x_{23} = 27.9601746169492
x24=−77.7544181763474x_{24} = -77.7544181763474
x25=−91.8915851175014x_{25} = -91.8915851175014
x26=−0.785398163397448x_{26} = -0.785398163397448
x27=22.3053078404875x_{27} = 22.3053078404875
x28=90.1637091580271x_{28} = 90.1637091580271
x29=12.2522113490002x_{29} = 12.2522113490002
x30=70.0575161750524x_{30} = 70.0575161750524
x31=84.037603483527x_{31} = 84.037603483527
x32=91.8915851175014x_{32} = 91.8915851175014
x33=−55.7632696012188x_{33} = -55.7632696012188
x34=−16.6504410640259x_{34} = -16.6504410640259
x35=−36.9137136796801x_{35} = -36.9137136796801
x36=2.19911485751286x_{36} = 2.19911485751286
x37=−5.96902604182061x_{37} = -5.96902604182061
x38=48.0663675999238x_{38} = 48.0663675999238
x39=26.0752190247953x_{39} = 26.0752190247953
x40=38.484510006475x_{40} = 38.484510006475
x41=−29.845130209103x_{41} = -29.845130209103
x42=−60.0044196835651x_{42} = -60.0044196835651
x43=−90.3207887907066x_{43} = -90.3207887907066
x44=−81.9955682586936x_{44} = -81.9955682586936
x45=4.08407044966673x_{45} = 4.08407044966673
x46=98.174770424681x_{46} = 98.174770424681
x47=56.2345084992573x_{47} = 56.2345084992573
x48=76.1836218495525x_{48} = 76.1836218495525
x49=44.2964564156161x_{49} = 44.2964564156161
x50=−71.9424717672063x_{50} = -71.9424717672063
x51=−80.8960108299372x_{51} = -80.8960108299372
x52=88.2787535658732x_{52} = 88.2787535658732
x53=−19.7920337176157x_{53} = -19.7920337176157
x54=−68.329640215578x_{54} = -68.329640215578
x55=−87.6504350351552x_{55} = -87.6504350351552
x56=−39.8982267005904x_{56} = -39.8982267005904
x57=40.0553063332699x_{57} = 40.0553063332699
x58=36.1283155162826x_{58} = 36.1283155162826
x59=−25.9181393921158x_{59} = -25.9181393921158
x60=−33.7721210260903x_{60} = -33.7721210260903
x61=−75.712382951514x_{61} = -75.712382951514
x62=58.7477826221291x_{62} = 58.7477826221291
x63=−93.9336203423348x_{63} = -93.9336203423348
x64=80.1106126665397x_{64} = 80.1106126665397
x65=58.1194640914112x_{65} = 58.1194640914112
x66=−49.3230046613598x_{66} = -49.3230046613598
x67=−38.0132711084365x_{67} = -38.0132711084365
x68=162.577419823272x_{68} = 162.577419823272
x69=100.216805649514x_{69} = 100.216805649514
x70=−16.0221225333079x_{70} = -16.0221225333079
x71=92.0486647501809x_{71} = 92.0486647501809
x72=71.9424717672063x_{72} = 71.9424717672063
x73=−53.7212343763855x_{73} = -53.7212343763855
x74=−95.8185759344887x_{74} = -95.8185759344887
x75=−99.7455667514759x_{75} = -99.7455667514759
x76=34.2433599241287x_{76} = 34.2433599241287
x77=−43.6681378848981x_{77} = -43.6681378848981
x78=74.6128255227576x_{78} = 74.6128255227576
x79=−41.7831822927443x_{79} = -41.7831822927443
x80=62.0464549083984x_{80} = 62.0464549083984
x81=46.18141200777x_{81} = 46.18141200777
x82=54.1924732744239x_{82} = 54.1924732744239
x83=−17.9070781254618x_{83} = -17.9070781254618
x84=33.6150413934108x_{84} = 33.6150413934108
x85=19.6349540849362x_{85} = 19.6349540849362
x86=0.314159265358979x_{86} = 0.314159265358979
x87=18.0641577581413x_{87} = 18.0641577581413
x88=5.96902604182061x_{88} = 5.96902604182061
x89=38.0132711084365x_{89} = 38.0132711084365
x90=−97.7035315266426x_{90} = -97.7035315266426
x91=−11.7809724509617x_{91} = -11.7809724509617
x92=−73.8274273593601x_{92} = -73.8274273593601
x93=−65.6592864600267x_{93} = -65.6592864600267
x94=81.9955682586936x_{94} = 81.9955682586936
x95=−85.7654794430014x_{95} = -85.7654794430014
x96=−7.85398163397448x_{96} = -7.85398163397448
x97=−47.9092879672443x_{97} = -47.9092879672443
x98=49.9513231920777x_{98} = 49.9513231920777
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(7*x) + cos(3*x).
cos⁡(0⋅7)+cos⁡(0⋅3)\cos{\left(0 \cdot 7 \right)} + \cos{\left(0 \cdot 3 \right)}
Resultado:
f(0)=2f{\left(0 \right)} = 2
Punto:
(0, 2)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(3x)+cos⁡(7x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\cos{\left(3 x \right)} + \cos{\left(7 x \right)}\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(cos⁡(3x)+cos⁡(7x))=⟨−2,2⟩\lim_{x \to \infty}\left(\cos{\left(3 x \right)} + \cos{\left(7 x \right)}\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(7*x) + cos(3*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(3x)+cos⁡(7x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(3 x \right)} + \cos{\left(7 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(3x)+cos⁡(7x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(3 x \right)} + \cos{\left(7 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:
cos⁡(3x)+cos⁡(7x)=cos⁡(3x)+cos⁡(7x)\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = \cos{\left(3 x \right)} + \cos{\left(7 x \right)}
- Sí
cos⁡(3x)+cos⁡(7x)=−cos⁡(3x)−cos⁡(7x)\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = - \cos{\left(3 x \right)} - \cos{\left(7 x \right)}
- No
es decir, función
es
par
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