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Trazado el gráfico de la función/ cos(|x|+3,14)

Gráfico de la función y = cos(|x|+3,14)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /      157\
f(x) = cos||x| + ---|
          \       50/
f(x)=cos⁡(∣x∣+15750)f{\left(x \right)} = \cos{\left(\left|{x}\right| + \frac{157}{50} \right)} 
f = cos(|x| + 157/50)
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:
cos⁡(∣x∣+15750)=0\cos{\left(\left|{x}\right| + \frac{157}{50} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−15750+3π2x_{1} = - \frac{157}{50} + \frac{3 \pi}{2}
Solución numérica
x1=64.4042420521805x_{1} = 64.4042420521805
x2=51.8378714378214x_{2} = 51.8378714378214
x3=58.121056745001x_{3} = 58.121056745001
x4=−54.9794640914112x_{4} = -54.9794640914112
x5=7.85557428756428x_{5} = 7.85557428756428
x6=−76.9706126665397x_{6} = -76.9706126665397
x7=73.8290200129499x_{7} = 73.8290200129499
x8=−67.5458347057703x_{8} = -67.5458347057703
x9=752.413033188345x_{9} = 752.413033188345
x10=83.2537979737193x_{10} = 83.2537979737193
x11=−98.9617612416683x_{11} = -98.9617612416683
x12=26.705130209103x_{12} = 26.705130209103
x13=−10.9971669411541x_{13} = -10.9971669411541
x14=48.6962787842316x_{14} = 48.6962787842316
x15=20.4219449019234x_{15} = 20.4219449019234
x16=76.9706126665397x_{16} = 76.9706126665397
x17=−32.9883155162826x_{17} = -32.9883155162826
x18=86.3953906273091x_{18} = 86.3953906273091
x19=−58.121056745001x_{19} = -58.121056745001
x20=−4.71398163397448x_{20} = -4.71398163397448
x21=39.2715008234622x_{21} = 39.2715008234622
x22=−36.1299081698724x_{22} = -36.1299081698724
x23=67.5458347057703x_{23} = 67.5458347057703
x24=36.1299081698724x_{24} = 36.1299081698724
x25=−7.85557428756428x_{25} = -7.85557428756428
x26=−83.2537979737193x_{26} = -83.2537979737193
x27=−42.413093477052x_{27} = -42.413093477052
x28=32.9883155162826x_{28} = 32.9883155162826
x29=45.5546861306418x_{29} = 45.5546861306418
x30=−70.6874273593601x_{30} = -70.6874273593601
x31=80.1122053201295x_{31} = 80.1122053201295
x32=−95.8201685880785x_{32} = -95.8201685880785
x33=42.413093477052x_{33} = 42.413093477052
x34=17.2803522483337x_{34} = 17.2803522483337
x35=−80.1122053201295x_{35} = -80.1122053201295
x36=−29.8467228626928x_{36} = -29.8467228626928
x37=61.2626493985908x_{37} = 61.2626493985908
x38=−48.6962787842316x_{38} = -48.6962787842316
x39=54.9794640914112x_{39} = 54.9794640914112
x40=−61.2626493985908x_{40} = -61.2626493985908
x41=23.5635375555132x_{41} = 23.5635375555132
x42=−89.5369832808989x_{42} = -89.5369832808989
x43=89.5369832808989x_{43} = 89.5369832808989
x44=29.8467228626928x_{44} = 29.8467228626928
x45=−20.4219449019234x_{45} = -20.4219449019234
x46=10.9971669411541x_{46} = 10.9971669411541
x47=70.6874273593601x_{47} = 70.6874273593601
x48=1.57238898038469x_{48} = 1.57238898038469
x49=−86.3953906273091x_{49} = -86.3953906273091
x50=−14.1387595947439x_{50} = -14.1387595947439
x51=−92.6785759344887x_{51} = -92.6785759344887
x52=937.766999750143x_{52} = 937.766999750143
x53=95.8201685880785x_{53} = 95.8201685880785
x54=−23.5635375555132x_{54} = -23.5635375555132
x55=4.71398163397448x_{55} = 4.71398163397448
x56=−39.2715008234622x_{56} = -39.2715008234622
x57=−64.4042420521805x_{57} = -64.4042420521805
x58=−45.5546861306418x_{58} = -45.5546861306418
x59=−1.57238898038469x_{59} = -1.57238898038469
x60=14.1387595947439x_{60} = 14.1387595947439
x61=92.6785759344887x_{61} = 92.6785759344887
x62=−51.8378714378214x_{62} = -51.8378714378214
x63=−26.705130209103x_{63} = -26.705130209103
x64=98.9617612416683x_{64} = 98.9617612416683
x65=−17.2803522483337x_{65} = -17.2803522483337
x66=−73.8290200129499x_{66} = -73.8290200129499
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| + 157/50).
cos⁡(∣0∣+15750)\cos{\left(\left|{0}\right| + \frac{157}{50} \right)}
Resultado:
f(0)=cos⁡(15750)f{\left(0 \right)} = \cos{\left(\frac{157}{50} \right)}
Punto:
(0, cos(157/50))
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∣+15750)sign⁡(x)=0- \sin{\left(\left|{x}\right| + \frac{157}{50} \right)} \operatorname{sign}{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=9.42637061435917x_{1} = 9.42637061435917
x2=34.5591118430775x_{2} = 34.5591118430775
x3=15.7095559215388x_{3} = 15.7095559215388
x4=−47.1254824574367x_{4} = -47.1254824574367
x5=0x_{5} = 0
x6=47.1254824574367x_{6} = 47.1254824574367
x7=72.258223686155x_{7} = 72.258223686155
x8=−9.42637061435917x_{8} = -9.42637061435917
x9=−37.7007044966673x_{9} = -37.7007044966673
x10=12.567963267949x_{10} = 12.567963267949
x11=21.9927412287183x_{11} = 21.9927412287183
x12=−59.6918530717959x_{12} = -59.6918530717959
x13=97.3909649148734x_{13} = 97.3909649148734
x14=69.1166310325652x_{14} = 69.1166310325652
x15=−78.5414089933346x_{15} = -78.5414089933346
x16=−69.1166310325652x_{16} = -69.1166310325652
x17=−292.169709437441x_{17} = -292.169709437441
x18=−3.14318530717959x_{18} = -3.14318530717959
x19=37.7007044966673x_{19} = 37.7007044966673
x20=−97.3909649148734x_{20} = -97.3909649148734
x21=−100.532557568463x_{21} = -100.532557568463
x22=−53.4086677646163x_{22} = -53.4086677646163
x23=−28.2759265358979x_{23} = -28.2759265358979
x24=−62.8334457253857x_{24} = -62.8334457253857
x25=−34.5591118430775x_{25} = -34.5591118430775
x26=56.5502604182061x_{26} = 56.5502604182061
x27=31.4175191894877x_{27} = 31.4175191894877
x28=−31.4175191894877x_{28} = -31.4175191894877
x29=65.9750383789755x_{29} = 65.9750383789755
x30=91.1077796076938x_{30} = 91.1077796076938
x31=75.3998163397448x_{31} = 75.3998163397448
x32=−103.674150222053x_{32} = -103.674150222053
x33=94.2493722612836x_{33} = 94.2493722612836
x34=84.8245943005142x_{34} = 84.8245943005142
x35=−56.5502604182061x_{35} = -56.5502604182061
x36=59.6918530717959x_{36} = 59.6918530717959
x37=50.2670751110265x_{37} = 50.2670751110265
x38=−94.2493722612836x_{38} = -94.2493722612836
x39=−150.7980400259x_{39} = -150.7980400259
x40=28.2759265358979x_{40} = 28.2759265358979
x41=−40.8422971502571x_{41} = -40.8422971502571
x42=−135.090076757951x_{42} = -135.090076757951
x43=−12.567963267949x_{43} = -12.567963267949
x44=−15.7095559215388x_{44} = -15.7095559215388
x45=53.4086677646163x_{45} = 53.4086677646163
x46=−87.966186954104x_{46} = -87.966186954104
x47=−43.9838898038469x_{47} = -43.9838898038469
x48=−75.3998163397448x_{48} = -75.3998163397448
x49=6.28477796076938x_{49} = 6.28477796076938
x50=−21.9927412287183x_{50} = -21.9927412287183
x51=81.6830016469244x_{51} = 81.6830016469244
x52=78.5414089933346x_{52} = 78.5414089933346
x53=−25.1343338823081x_{53} = -25.1343338823081
x54=153.93963267949x_{54} = 153.93963267949
x55=62.8334457253857x_{55} = 62.8334457253857
x56=43.9838898038469x_{56} = 43.9838898038469
x57=−81.6830016469244x_{57} = -81.6830016469244
x58=−72.258223686155x_{58} = -72.258223686155
x59=−65.9750383789755x_{59} = -65.9750383789755
x60=3.14318530717959x_{60} = 3.14318530717959
x61=87.966186954104x_{61} = 87.966186954104
x62=−84.8245943005142x_{62} = -84.8245943005142
x63=18.8511485751286x_{63} = 18.8511485751286
x64=100.532557568463x_{64} = 100.532557568463
x65=−50.2670751110265x_{65} = -50.2670751110265
x66=−6.28477796076938x_{66} = -6.28477796076938
x67=−91.1077796076938x_{67} = -91.1077796076938
x68=−18.8511485751286x_{68} = -18.8511485751286
x69=25.1343338823081x_{69} = 25.1343338823081
x70=40.8422971502571x_{70} = 40.8422971502571
Signos de extremos en los puntos:
(9.426370614359174, 1)

(34.55911184307752, 1)

(15.70955592153876, 1)

(-47.12548245743669, 1)

         /157\ 
(0, cos|---|)
         \ 50/ 

(47.12548245743669, 1)

(72.25822368615503, 1)

(-9.426370614359174, 1)

(-37.70070449666731, -1)

(12.567963267948967, -1)

(21.992741228718344, 1)

(-59.69185307179586, 1)

(97.39096491487338, 1)

(69.11663103256524, -1)

(-78.54140899333463, 1)

(-69.11663103256524, -1)

(-292.1697094374406, 1)

(-3.1431853071795866, 1)

(37.70070449666731, -1)

(-97.39096491487338, 1)

(-100.53255756846318, -1)

(-53.408667764616276, 1)

(-28.275926535897934, 1)

(-62.83344572538566, -1)

(-34.55911184307752, 1)

(56.55026041820607, -1)

(31.417519189487727, -1)

(-31.417519189487727, -1)

(65.97503837897546, 1)

(91.1077796076938, 1)

(75.39981633974483, -1)

(-103.67415022205297, 1)

(94.2493722612836, -1)

(84.8245943005142, 1)

(-56.55026041820607, -1)

(59.69185307179586, 1)

(50.267075111026486, -1)

(-94.2493722612836, -1)

(-150.79804002589987, -1)

(28.275926535897934, 1)

(-40.8422971502571, 1)

(-135.0900767579509, 1)

(-12.567963267948967, -1)

(-15.70955592153876, 1)

(53.408667764616276, 1)

(-87.966186954104, -1)

(-43.9838898038469, -1)

(-75.39981633974483, -1)

(6.28477796076938, -1)

(-21.992741228718344, 1)

(81.68300164692442, -1)

(78.54140899333463, 1)

(-25.134333882308137, -1)

(153.93963267948965, 1)

(62.83344572538566, -1)

(43.9838898038469, -1)

(-81.68300164692442, -1)

(-72.25822368615503, 1)

(-65.97503837897546, 1)

(3.1431853071795866, 1)

(87.966186954104, -1)

(-84.8245943005142, 1)

(18.85114857512855, -1)

(100.53255756846318, -1)

(-50.267075111026486, -1)

(-6.28477796076938, -1)

(-91.1077796076938, 1)

(-18.85114857512855, -1)

(25.134333882308137, -1)

(40.8422971502571, 1)


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=−37.7007044966673x_{1} = -37.7007044966673
x2=12.567963267949x_{2} = 12.567963267949
x3=69.1166310325652x_{3} = 69.1166310325652
x4=−69.1166310325652x_{4} = -69.1166310325652
x5=37.7007044966673x_{5} = 37.7007044966673
x6=−100.532557568463x_{6} = -100.532557568463
x7=−62.8334457253857x_{7} = -62.8334457253857
x8=56.5502604182061x_{8} = 56.5502604182061
x9=31.4175191894877x_{9} = 31.4175191894877
x10=−31.4175191894877x_{10} = -31.4175191894877
x11=75.3998163397448x_{11} = 75.3998163397448
x12=94.2493722612836x_{12} = 94.2493722612836
x13=−56.5502604182061x_{13} = -56.5502604182061
x14=50.2670751110265x_{14} = 50.2670751110265
x15=−94.2493722612836x_{15} = -94.2493722612836
x16=−150.7980400259x_{16} = -150.7980400259
x17=−12.567963267949x_{17} = -12.567963267949
x18=−87.966186954104x_{18} = -87.966186954104
x19=−43.9838898038469x_{19} = -43.9838898038469
x20=−75.3998163397448x_{20} = -75.3998163397448
x21=6.28477796076938x_{21} = 6.28477796076938
x22=81.6830016469244x_{22} = 81.6830016469244
x23=−25.1343338823081x_{23} = -25.1343338823081
x24=62.8334457253857x_{24} = 62.8334457253857
x25=43.9838898038469x_{25} = 43.9838898038469
x26=−81.6830016469244x_{26} = -81.6830016469244
x27=87.966186954104x_{27} = 87.966186954104
x28=18.8511485751286x_{28} = 18.8511485751286
x29=100.532557568463x_{29} = 100.532557568463
x30=−50.2670751110265x_{30} = -50.2670751110265
x31=−6.28477796076938x_{31} = -6.28477796076938
x32=−18.8511485751286x_{32} = -18.8511485751286
x33=25.1343338823081x_{33} = 25.1343338823081
Puntos máximos de la función:
x33=9.42637061435917x_{33} = 9.42637061435917
x33=34.5591118430775x_{33} = 34.5591118430775
x33=15.7095559215388x_{33} = 15.7095559215388
x33=−47.1254824574367x_{33} = -47.1254824574367
x33=0x_{33} = 0
x33=47.1254824574367x_{33} = 47.1254824574367
x33=72.258223686155x_{33} = 72.258223686155
x33=−9.42637061435917x_{33} = -9.42637061435917
x33=21.9927412287183x_{33} = 21.9927412287183
x33=−59.6918530717959x_{33} = -59.6918530717959
x33=97.3909649148734x_{33} = 97.3909649148734
x33=−78.5414089933346x_{33} = -78.5414089933346
x33=−292.169709437441x_{33} = -292.169709437441
x33=−3.14318530717959x_{33} = -3.14318530717959
x33=−97.3909649148734x_{33} = -97.3909649148734
x33=−53.4086677646163x_{33} = -53.4086677646163
x33=−28.2759265358979x_{33} = -28.2759265358979
x33=−34.5591118430775x_{33} = -34.5591118430775
x33=65.9750383789755x_{33} = 65.9750383789755
x33=91.1077796076938x_{33} = 91.1077796076938
x33=−103.674150222053x_{33} = -103.674150222053
x33=84.8245943005142x_{33} = 84.8245943005142
x33=59.6918530717959x_{33} = 59.6918530717959
x33=28.2759265358979x_{33} = 28.2759265358979
x33=−40.8422971502571x_{33} = -40.8422971502571
x33=−135.090076757951x_{33} = -135.090076757951
x33=−15.7095559215388x_{33} = -15.7095559215388
x33=53.4086677646163x_{33} = 53.4086677646163
x33=−21.9927412287183x_{33} = -21.9927412287183
x33=78.5414089933346x_{33} = 78.5414089933346
x33=153.93963267949x_{33} = 153.93963267949
x33=−72.258223686155x_{33} = -72.258223686155
x33=−65.9750383789755x_{33} = -65.9750383789755
x33=3.14318530717959x_{33} = 3.14318530717959
x33=−84.8245943005142x_{33} = -84.8245943005142
x33=−91.1077796076938x_{33} = -91.1077796076938
x33=40.8422971502571x_{33} = 40.8422971502571
Decrece en los intervalos
[100.532557568463,∞)\left[100.532557568463, \infty\right)
Crece en los intervalos
(−∞,−150.7980400259]\left(-\infty, -150.7980400259\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
−(2sin⁡(∣x∣+15750)δ(x)+cos⁡(∣x∣+15750)sign⁡2(x))=0- (2 \sin{\left(\left|{x}\right| + \frac{157}{50} \right)} \delta\left(x\right) + \cos{\left(\left|{x}\right| + \frac{157}{50} \right)} \operatorname{sign}^{2}{\left(x \right)}) = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞cos⁡(∣x∣+15750)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(\left|{x}\right| + \frac{157}{50} \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→∞cos⁡(∣x∣+15750)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(\left|{x}\right| + \frac{157}{50} \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 cos(|x| + 157/50), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(∣x∣+15750)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\left|{x}\right| + \frac{157}{50} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(∣x∣+15750)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\left|{x}\right| + \frac{157}{50} \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⁡(∣x∣+15750)=cos⁡(∣x∣+15750)\cos{\left(\left|{x}\right| + \frac{157}{50} \right)} = \cos{\left(\left|{x}\right| + \frac{157}{50} \right)}
- Sí
cos⁡(∣x∣+15750)=−cos⁡(∣x∣+15750)\cos{\left(\left|{x}\right| + \frac{157}{50} \right)} = - \cos{\left(\left|{x}\right| + \frac{157}{50} \right)}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор