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cosx*1/2cos2x

Gráfico de la función y = cosx*1/2cos2x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src]
       cos(x)         
f(x) = ------*cos(2*x)
         2            
f(x)=cos(x)2cos(2x)f{\left(x \right)} = \frac{\cos{\left(x \right)}}{2} \cos{\left(2 x \right)}
f = (cos(x)/2)*cos(2*x)
Gráfico de la función
02468-8-6-4-2-10101-1
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(x)2cos(2x)=0\frac{\cos{\left(x \right)}}{2} \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = - \frac{\pi}{2}
x2=π4x_{2} = - \frac{\pi}{4}
x3=π4x_{3} = \frac{\pi}{4}
x4=π2x_{4} = \frac{\pi}{2}
Solución numérica
x1=55.7632696012188x_{1} = -55.7632696012188
x2=73.8274273593601x_{2} = 73.8274273593601
x3=98.174770424681x_{3} = -98.174770424681
x4=70.6858347057703x_{4} = 70.6858347057703
x5=89.5353906273091x_{5} = -89.5353906273091
x6=11.7809724509617x_{6} = -11.7809724509617
x7=25.9181393921158x_{7} = 25.9181393921158
x8=40.0553063332699x_{8} = -40.0553063332699
x9=18.0641577581413x_{9} = 18.0641577581413
x10=42.4115008234622x_{10} = 42.4115008234622
x11=14.1371669411541x_{11} = 14.1371669411541
x12=55.7632696012188x_{12} = 55.7632696012188
x13=1.5707963267949x_{13} = -1.5707963267949
x14=49.4800842940392x_{14} = -49.4800842940392
x15=41.6261026600648x_{15} = -41.6261026600648
x16=67.5442420521806x_{16} = -67.5442420521806
x17=76.1836218495525x_{17} = 76.1836218495525
x18=69.9004365423729x_{18} = 69.9004365423729
x19=19.6349540849362x_{19} = -19.6349540849362
x20=89.5353906273091x_{20} = 89.5353906273091
x21=3.92699081698724x_{21} = -3.92699081698724
x22=38.484510006475x_{22} = 38.484510006475
x23=32.2013246992954x_{23} = 32.2013246992954
x24=80.1106126665397x_{24} = -80.1106126665397
x25=36.1283155162826x_{25} = 36.1283155162826
x26=84.037603483527x_{26} = 84.037603483527
x27=91.8915851175014x_{27} = -91.8915851175014
x28=98.174770424681x_{28} = 98.174770424681
x29=95.8185759344887x_{29} = 95.8185759344887
x30=45.553093477052x_{30} = 45.553093477052
x31=92.6769832808989x_{31} = 92.6769832808989
x32=85.6083998103219x_{32} = -85.6083998103219
x33=91.8915851175014x_{33} = 91.8915851175014
x34=16.4933614313464x_{34} = 16.4933614313464
x35=46.3384916404494x_{35} = 46.3384916404494
x36=1.5707963267949x_{36} = 1.5707963267949
x37=29.845130209103x_{37} = -29.845130209103
x38=23.5619449019235x_{38} = 23.5619449019235
x39=20.4203522483337x_{39} = 20.4203522483337
x40=54.1924732744239x_{40} = 54.1924732744239
x41=84.037603483527x_{41} = -84.037603483527
x42=32.9867228626928x_{42} = -32.9867228626928
x43=7.85398163397448x_{43} = 7.85398163397448
x44=93.4623814442964x_{44} = -93.4623814442964
x45=17.2787595947439x_{45} = -17.2787595947439
x46=47.9092879672443x_{46} = 47.9092879672443
x47=24.3473430653209x_{47} = 24.3473430653209
x48=5.49778714378214x_{48} = -5.49778714378214
x49=68.329640215578x_{49} = 68.329640215578
x50=62.0464549083984x_{50} = 62.0464549083984
x51=4.71238898038469x_{51} = 4.71238898038469
x52=26.7035375555132x_{52} = 26.7035375555132
x53=64.4026493985908x_{53} = 64.4026493985908
x54=47.9092879672443x_{54} = -47.9092879672443
x55=14.1371669411541x_{55} = -14.1371669411541
x56=40.0553063332699x_{56} = 40.0553063332699
x57=51.8362787842316x_{57} = -51.8362787842316
x58=58.1194640914112x_{58} = 58.1194640914112
x59=25.9181393921158x_{59} = -25.9181393921158
x60=76.1836218495525x_{60} = -76.1836218495525
x61=48.6946861306418x_{61} = 48.6946861306418
x62=99.7455667514759x_{62} = 99.7455667514759
x63=95.8185759344887x_{63} = -95.8185759344887
x64=62.0464549083984x_{64} = -62.0464549083984
x65=82.4668071567321x_{65} = -82.4668071567321
x66=71.4712328691678x_{66} = -71.4712328691678
x67=36.1283155162826x_{67} = -36.1283155162826
x68=60.4756585816035x_{68} = 60.4756585816035
x69=33.7721210260903x_{69} = -33.7721210260903
x70=77.7544181763474x_{70} = 77.7544181763474
x71=86.3937979737193x_{71} = 86.3937979737193
x72=63.6172512351933x_{72} = -63.6172512351933
x73=10.2101761241668x_{73} = -10.2101761241668
x74=23.5619449019235x_{74} = -23.5619449019235
x75=18.0641577581413x_{75} = -18.0641577581413
x76=27.4889357189107x_{76} = -27.4889357189107
x77=54.1924732744239x_{77} = -54.1924732744239
x78=7.85398163397448x_{78} = -7.85398163397448
x79=67.5442420521806x_{79} = 67.5442420521806
x80=3.92699081698724x_{80} = 3.92699081698724
x81=80.1106126665397x_{81} = 80.1106126665397
x82=29.845130209103x_{82} = 29.845130209103
x83=99.7455667514759x_{83} = -99.7455667514759
x84=82.4668071567321x_{84} = 82.4668071567321
x85=33.7721210260903x_{85} = 33.7721210260903
x86=11.7809724509617x_{86} = 11.7809724509617
x87=10.2101761241668x_{87} = 10.2101761241668
x88=73.8274273593601x_{88} = -73.8274273593601
x89=77.7544181763474x_{89} = -77.7544181763474
x90=69.9004365423729x_{90} = -69.9004365423729
x91=51.8362787842316x_{91} = 51.8362787842316
x92=45.553093477052x_{92} = -45.553093477052
x93=2.35619449019234x_{93} = 2.35619449019234
x94=60.4756585816035x_{94} = -60.4756585816035
x95=90.3207887907066x_{95} = 90.3207887907066
x96=32.2013246992954x_{96} = -32.2013246992954
x97=58.1194640914112x_{97} = -58.1194640914112
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(x)/2)*cos(2*x).
cos(0)2cos(02)\frac{\cos{\left(0 \right)}}{2} \cos{\left(0 \cdot 2 \right)}
Resultado:
f(0)=12f{\left(0 \right)} = \frac{1}{2}
Punto:
(0, 1/2)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
sin(x)cos(2x)2sin(2x)cos(x)=0- \frac{\sin{\left(x \right)} \cos{\left(2 x \right)}}{2} - \sin{\left(2 x \right)} \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=πx_{2} = \pi
x3=i(log(3)log(25i))2x_{3} = \frac{i \left(\log{\left(3 \right)} - \log{\left(-2 - \sqrt{5} i \right)}\right)}{2}
x4=i(log(3)log(2+5i))2x_{4} = \frac{i \left(\log{\left(3 \right)} - \log{\left(-2 + \sqrt{5} i \right)}\right)}{2}
Signos de extremos en los puntos:
(0, 1/2)

(pi, -1/2)

                                                                            /  /     /         ___\         \\ 
                                      /  /     /         ___\         \\    |I*\- log\-2 - I*\/ 5 / + log(3)/| 
   /     /         ___\         \  cos\I*\- log\-2 - I*\/ 5 / + log(3)//*cos|--------------------------------| 
 I*\- log\-2 - I*\/ 5 / + log(3)/                                           \               2                / 
(--------------------------------, ---------------------------------------------------------------------------)
                2                                                       2                                      

                                                                            /  /     /         ___\         \\ 
                                      /  /     /         ___\         \\    |I*\- log\-2 + I*\/ 5 / + log(3)/| 
   /     /         ___\         \  cos\I*\- log\-2 + I*\/ 5 / + log(3)//*cos|--------------------------------| 
 I*\- log\-2 + I*\/ 5 / + log(3)/                                           \               2                / 
(--------------------------------, ---------------------------------------------------------------------------)
                2                                                       2                                      


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=πx_{1} = \pi
x2=π2+atan(52)2x_{2} = - \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2}
x3=atan(52)2+π2x_{3} = - \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} + \frac{\pi}{2}
Puntos máximos de la función:
x3=0x_{3} = 0
Decrece en los intervalos
[π,)\left[\pi, \infty\right)
Crece en los intervalos
(,π2+atan(52)2][0,atan(52)2+π2]\left(-\infty, - \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2}\right] \cup \left[0, - \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} + \frac{\pi}{2}\right]
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (cos(x)/2)*cos(2*x), dividida por x con x->+oo y x ->-oo
limx(cos(x)cos(2x)2x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \cos{\left(2 x \right)}}{2 x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
limx(cos(x)cos(2x)2x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \cos{\left(2 x \right)}}{2 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(x)2cos(2x)=cos(x)2cos(2x)\frac{\cos{\left(x \right)}}{2} \cos{\left(2 x \right)} = \frac{\cos{\left(x \right)}}{2} \cos{\left(2 x \right)}
- Sí
cos(x)2cos(2x)=cos(x)2cos(2x)\frac{\cos{\left(x \right)}}{2} \cos{\left(2 x \right)} = - \frac{\cos{\left(x \right)}}{2} \cos{\left(2 x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = cosx*1/2cos2x