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Trazado el gráfico de la función/ -1,09*cos((pi*x)/(1,5))+0,197*cos((pi*x)/2*1,5)^2

Gráfico de la función y = -1,09*cos((pi*x)/(1,5))+0,197*cos((pi*x)/2*1,5)^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                 /pi*x  \
                                 |----*3|
                /pi*x\          2| 2    |
         109*cos|----|   197*cos |------|
                \3/2 /           \  2   /
f(x) = - ------------- + ----------------
              100              1000
f(x)=197cos⁡2(3πx22)1000−109cos⁡(πx32)100f{\left(x \right)} = \frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} 
f = 197*cos(3*((pi*x)/2)/2)^2/1000 - 109*cos((pi*x)/(3/2))/100
Gráfico de la función
02468-8-6-4-2-10102.5-2.5
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:
197cos⁡2(3πx22)1000−109cos⁡(πx32)100=0\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−45.7169563127939x_{1} = -45.7169563127939
x2=45.7169563127939x_{2} = 45.7169563127939
x3=9.71695631279395x_{3} = 9.71695631279395
x4=−63.7000099786517x_{4} = -63.7000099786517
x5=2.28304368720605x_{5} = 2.28304368720605
x6=−86.2830436872061x_{6} = -86.2830436872061
x7=−74.2830436872061x_{7} = -74.2830436872061
x8=98.2830436872061x_{8} = 98.2830436872061
x9=74.2830436872061x_{9} = 74.2830436872061
x10=−32.2999900213483x_{10} = -32.2999900213483
x11=62.2830436872061x_{11} = 62.2830436872061
x12=27.7000099786517x_{12} = 27.7000099786517
x13=−89.3367652043401x_{13} = -89.3367652043401
x14=−21.7169563127939x_{14} = -21.7169563127939
x15=92.2999900213483x_{15} = 92.2999900213483
x16=38.2830436872061x_{16} = 38.2830436872061
x17=51.7000099786517x_{17} = 51.7000099786517
x18=−68.2999900213483x_{18} = -68.2999900213483
x19=69.7169563127939x_{19} = 69.7169563127939
x20=54.6632347956599x_{20} = 54.6632347956599
x21=90.6632347956599x_{21} = 90.6632347956599
x22=3.70000997865169x_{22} = 3.70000997865169
x23=−75.7000099786517x_{23} = -75.7000099786517
x24=−44.2999900213483x_{24} = -44.2999900213483
x25=−33.7169563127939x_{25} = -33.7169563127939
x26=−81.7169563127939x_{26} = -81.7169563127939
x27=−39.7000099786517x_{27} = -39.7000099786517
x28=42.6632347956599x_{28} = 42.6632347956599
x29=−53.3367652043401x_{29} = -53.3367652043401
x30=30.6632347956599x_{30} = 30.6632347956599
x31=68.2999900213483x_{31} = 68.2999900213483
x32=−77.3367652043401x_{32} = -77.3367652043401
x33=−3.70000997865169x_{33} = -3.70000997865169
x34=81.7169563127939x_{34} = 81.7169563127939
x35=−14.2830436872061x_{35} = -14.2830436872061
x36=−8.29999002134831x_{36} = -8.29999002134831
x37=−41.3367652043401x_{37} = -41.3367652043401
x38=−93.7169563127939x_{38} = -93.7169563127939
x39=−9.71695631279395x_{39} = -9.71695631279395
x40=57.7169563127939x_{40} = 57.7169563127939
x41=50.2830436872061x_{41} = 50.2830436872061
x42=63.7000099786517x_{42} = 63.7000099786517
x43=−51.7000099786517x_{43} = -51.7000099786517
x44=26.2830436872061x_{44} = 26.2830436872061
x45=−27.7000099786517x_{45} = -27.7000099786517
x46=−99.7000099786517x_{46} = -99.7000099786517
x47=6.66323479565987x_{47} = 6.66323479565987
x48=80.2999900213483x_{48} = 80.2999900213483
x49=−69.7169563127939x_{49} = -69.7169563127939
x50=−87.7000099786517x_{50} = -87.7000099786517
x51=102.66323479566x_{51} = 102.66323479566
x52=−2.28304368720605x_{52} = -2.28304368720605
x53=33.7169563127939x_{53} = 33.7169563127939
x54=−15.7000099786517x_{54} = -15.7000099786517
x55=−56.2999900213483x_{55} = -56.2999900213483
x56=−62.2830436872061x_{56} = -62.2830436872061
x57=−5.33676520434013x_{57} = -5.33676520434013
x58=−38.2830436872061x_{58} = -38.2830436872061
x59=39.7000099786517x_{59} = 39.7000099786517
x60=75.7000099786517x_{60} = 75.7000099786517
x61=44.2999900213483x_{61} = 44.2999900213483
x62=15.7000099786517x_{62} = 15.7000099786517
x63=−50.2830436872061x_{63} = -50.2830436872061
x64=−26.2830436872061x_{64} = -26.2830436872061
x65=−80.2999900213483x_{65} = -80.2999900213483
x66=86.2830436872061x_{66} = 86.2830436872061
x67=20.2999900213483x_{67} = 20.2999900213483
x68=8.29999002134831x_{68} = 8.29999002134831
x69=−20.2999900213483x_{69} = -20.2999900213483
x70=14.2830436872061x_{70} = 14.2830436872061
x71=21.7169563127939x_{71} = 21.7169563127939
x72=18.6632347956599x_{72} = 18.6632347956599
x73=66.6632347956599x_{73} = 66.6632347956599
x74=−17.3367652043401x_{74} = -17.3367652043401
x75=−92.2999900213483x_{75} = -92.2999900213483
x76=−29.3367652043401x_{76} = -29.3367652043401
x77=−65.3367652043401x_{77} = -65.3367652043401
x78=−98.2830436872061x_{78} = -98.2830436872061
x79=−57.7169563127939x_{79} = -57.7169563127939
x80=87.7000099786517x_{80} = 87.7000099786517
x81=56.2999900213483x_{81} = 56.2999900213483
x82=78.6632347956599x_{82} = 78.6632347956599
x83=99.7000099786517x_{83} = 99.7000099786517
x84=93.7169563127939x_{84} = 93.7169563127939
x85=32.2999900213483x_{85} = 32.2999900213483
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 -109*cos((pi*x)/(3/2))/100 + 197*cos(((pi*x)/2)*3/2)^2/1000.
−109cos⁡(0π32)100+197cos⁡2(30π22)1000- \frac{109 \cos{\left(\frac{0 \pi}{\frac{3}{2}} \right)}}{100} + \frac{197 \cos^{2}{\left(\frac{3 \frac{0 \pi}{2}}{2} \right)}}{1000}
Resultado:
f(0)=−8931000f{\left(0 \right)} = - \frac{893}{1000}
Punto:
(0, -893/1000)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(197cos⁡2(3πx22)1000−109cos⁡(πx32)100)=⟨−109100,12871000⟩\lim_{x \to -\infty}\left(\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}\right) = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−109100,12871000⟩y = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle
lim⁡x→∞(197cos⁡2(3πx22)1000−109cos⁡(πx32)100)=⟨−109100,12871000⟩\lim_{x \to \infty}\left(\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}\right) = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−109100,12871000⟩y = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función -109*cos((pi*x)/(3/2))/100 + 197*cos(((pi*x)/2)*3/2)^2/1000, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(197cos⁡2(3πx22)1000−109cos⁡(πx32)100x)=0\lim_{x \to -\infty}\left(\frac{\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(197cos⁡2(3πx22)1000−109cos⁡(πx32)100x)=0\lim_{x \to \infty}\left(\frac{\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}}{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:
197cos⁡2(3πx22)1000−109cos⁡(πx32)100=−109cos⁡(2πx3)100+197cos⁡2(3πx4)1000\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} = - \frac{109 \cos{\left(\frac{2 \pi x}{3} \right)}}{100} + \frac{197 \cos^{2}{\left(\frac{3 \pi x}{4} \right)}}{1000}
- No
197cos⁡2(3πx22)1000−109cos⁡(πx32)100=109cos⁡(2πx3)100−197cos⁡2(3πx4)1000\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} = \frac{109 \cos{\left(\frac{2 \pi x}{3} \right)}}{100} - \frac{197 \cos^{2}{\left(\frac{3 \pi x}{4} \right)}}{1000}
- No
es decir, función
no es
par ni impar
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