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|sinx|
Trazado el gráfico de la función/ |sinx|

Gráfico de la función y = |sinx|

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = |sin(x)|
f(x)=∣sin⁡(x)∣f{\left(x \right)} = \left|{\sin{\left(x \right)}}\right| 
f = Abs(sin(x))
Gráfico de la función
02468-8-6-4-2-101002
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⁡(x)∣=0\left|{\sin{\left(x \right)}}\right| = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=−59.6902604182061x_{1} = -59.6902604182061
x2=87.9645943005142x_{2} = 87.9645943005142
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=31.4159265358979x_{4} = 31.4159265358979
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=−37.6991118430775x_{6} = -37.6991118430775
x7=−81.6814089933346x_{7} = -81.6814089933346
x8=650.309679293087x_{8} = 650.309679293087
x9=−21.9911485751286x_{9} = -21.9911485751286
x10=−15.707963267949x_{10} = -15.707963267949
x11=−427.256600888212x_{11} = -427.256600888212
x12=−12.5663706143592x_{12} = -12.5663706143592
x13=12.5663706143592x_{13} = 12.5663706143592
x14=−87.9645943005142x_{14} = -87.9645943005142
x15=53.4070751110265x_{15} = 53.4070751110265
x16=−100.530964914873x_{16} = -100.530964914873
x17=−3.14159265358979x_{17} = -3.14159265358979
x18=34.5575191894877x_{18} = 34.5575191894877
x19=−94.2477796076938x_{19} = -94.2477796076938
x20=6.28318530717959x_{20} = 6.28318530717959
x21=−69.1150383789755x_{21} = -69.1150383789755
x22=97.3893722612836x_{22} = 97.3893722612836
x23=65.9734457253857x_{23} = 65.9734457253857
x24=0x_{24} = 0
x25=−50.2654824574367x_{25} = -50.2654824574367
x26=15.707963267949x_{26} = 15.707963267949
x27=−25.1327412287183x_{27} = -25.1327412287183
x28=40.8407044966673x_{28} = 40.8407044966673
x29=18.8495559215388x_{29} = 18.8495559215388
x30=−78.5398163397448x_{30} = -78.5398163397448
x31=−53.4070751110265x_{31} = -53.4070751110265
x32=37.6991118430775x_{32} = 37.6991118430775
x33=−43.9822971502571x_{33} = -43.9822971502571
x34=−6.28318530717959x_{34} = -6.28318530717959
x35=43.9822971502571x_{35} = 43.9822971502571
x36=56.5486677646163x_{36} = 56.5486677646163
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=−28.2743338823081x_{38} = -28.2743338823081
x39=78.5398163397448x_{39} = 78.5398163397448
x40=−3760.48640634698x_{40} = -3760.48640634698
x41=75.398223686155x_{41} = 75.398223686155
x42=59.6902604182061x_{42} = 59.6902604182061
x43=−34.5575191894877x_{43} = -34.5575191894877
x44=81.6814089933346x_{44} = 81.6814089933346
x45=−47.1238898038469x_{45} = -47.1238898038469
x46=100.530964914873x_{46} = 100.530964914873
x47=−9.42477796076938x_{47} = -9.42477796076938
x48=−75.398223686155x_{48} = -75.398223686155
x49=−72.2566310325652x_{49} = -72.2566310325652
x50=−31.4159265358979x_{50} = -31.4159265358979
x51=28.2743338823081x_{51} = 28.2743338823081
x52=−285.884931476671x_{52} = -285.884931476671
x53=−91.106186954104x_{53} = -91.106186954104
x54=21.9911485751286x_{54} = 21.9911485751286
x55=62.8318530717959x_{55} = 62.8318530717959
x56=9.42477796076938x_{56} = 9.42477796076938
x57=50.2654824574367x_{57} = 50.2654824574367
x58=94.2477796076938x_{58} = 94.2477796076938
x59=72.2566310325652x_{59} = 72.2566310325652
x60=84.8230016469244x_{60} = 84.8230016469244
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 Abs(sin(x)).
∣sin⁡(0)∣\left|{\sin{\left(0 \right)}}\right|
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
cos⁡(x)sign⁡(sin⁡(x))=0\cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−2279.22547017939x_{1} = -2279.22547017939
x2=32.9867228626928x_{2} = 32.9867228626928
x3=73.8274273593601x_{3} = 73.8274273593601
x4=4.71238898038469x_{4} = 4.71238898038469
x5=39.2699081698724x_{5} = 39.2699081698724
x6=95.8185759344887x_{6} = 95.8185759344887
x7=45.553093477052x_{7} = 45.553093477052
x8=70.6858347057703x_{8} = 70.6858347057703
x9=−10.9955742875643x_{9} = -10.9955742875643
x10=−58.1194640914112x_{10} = -58.1194640914112
x11=−23.5619449019235x_{11} = -23.5619449019235
x12=26.7035375555132x_{12} = 26.7035375555132
x13=−26.7035375555132x_{13} = -26.7035375555132
x14=−89.5353906273091x_{14} = -89.5353906273091
x15=−17.2787595947439x_{15} = -17.2787595947439
x16=−42.4115008234622x_{16} = -42.4115008234622
x17=−61.261056745001x_{17} = -61.261056745001
x18=92.6769832808989x_{18} = 92.6769832808989
x19=−76.9690200129499x_{19} = -76.9690200129499
x20=−92.6769832808989x_{20} = -92.6769832808989
x21=−98.9601685880785x_{21} = -98.9601685880785
x22=61.261056745001x_{22} = 61.261056745001
x23=−54.9778714378214x_{23} = -54.9778714378214
x24=42.4115008234622x_{24} = 42.4115008234622
x25=−64.4026493985908x_{25} = -64.4026493985908
x26=67.5442420521806x_{26} = 67.5442420521806
x27=−7.85398163397448x_{27} = -7.85398163397448
x28=80.1106126665397x_{28} = 80.1106126665397
x29=−14.1371669411541x_{29} = -14.1371669411541
x30=14.1371669411541x_{30} = 14.1371669411541
x31=−1.5707963267949x_{31} = -1.5707963267949
x32=1.5707963267949x_{32} = 1.5707963267949
x33=29.845130209103x_{33} = 29.845130209103
x34=10.9955742875643x_{34} = 10.9955742875643
x35=17.2787595947439x_{35} = 17.2787595947439
x36=−51.8362787842316x_{36} = -51.8362787842316
x37=−29.845130209103x_{37} = -29.845130209103
x38=0x_{38} = 0
x39=−183.783170235003x_{39} = -183.783170235003
x40=−48.6946861306418x_{40} = -48.6946861306418
x41=−73.8274273593601x_{41} = -73.8274273593601
x42=23.5619449019235x_{42} = 23.5619449019235
x43=20.4203522483337x_{43} = 20.4203522483337
x44=−86.3937979737193x_{44} = -86.3937979737193
x45=54.9778714378214x_{45} = 54.9778714378214
x46=58.1194640914112x_{46} = 58.1194640914112
x47=51.8362787842316x_{47} = 51.8362787842316
x48=−67.5442420521806x_{48} = -67.5442420521806
x49=237.190245346029x_{49} = 237.190245346029
x50=−4.71238898038469x_{50} = -4.71238898038469
x51=−70.6858347057703x_{51} = -70.6858347057703
x52=−45.553093477052x_{52} = -45.553093477052
x53=48.6946861306418x_{53} = 48.6946861306418
x54=−83.2522053201295x_{54} = -83.2522053201295
x55=−95.8185759344887x_{55} = -95.8185759344887
x56=89.5353906273091x_{56} = 89.5353906273091
x57=−39.2699081698724x_{57} = -39.2699081698724
x58=−306.305283725005x_{58} = -306.305283725005
x59=76.9690200129499x_{59} = 76.9690200129499
x60=−32.9867228626928x_{60} = -32.9867228626928
x61=−20.4203522483337x_{61} = -20.4203522483337
x62=−36.1283155162826x_{62} = -36.1283155162826
x63=7.85398163397448x_{63} = 7.85398163397448
x64=−80.1106126665397x_{64} = -80.1106126665397
x65=86.3937979737193x_{65} = 86.3937979737193
x66=98.9601685880785x_{66} = 98.9601685880785
x67=36.1283155162826x_{67} = 36.1283155162826
x68=64.4026493985908x_{68} = 64.4026493985908
x69=83.2522053201295x_{69} = 83.2522053201295
Signos de extremos en los puntos:
(-2279.225470179395, 1)

(32.98672286269283, 1)

(73.82742735936014, 1)

(4.71238898038469, 1)

(39.269908169872416, 1)

(95.81857593448869, 1)

(45.553093477052, 1)

(70.68583470577035, 1)

(-10.995574287564276, 1)

(-58.119464091411174, 1)

(-23.56194490192345, 1)

(26.703537555513243, 1)

(-26.703537555513243, 1)

(-89.53539062730911, 1)

(-17.278759594743864, 1)

(-42.411500823462205, 1)

(-61.26105674500097, 1)

(92.6769832808989, 1)

(-76.96902001294994, 1)

(-92.6769832808989, 1)

(-98.96016858807849, 1)

(61.26105674500097, 1)

(-54.977871437821385, 1)

(42.411500823462205, 1)

(-64.40264939859077, 1)

(67.54424205218055, 1)

(-7.853981633974483, 1)

(80.11061266653972, 1)

(-14.137166941154069, 1)

(14.137166941154069, 1)

(-1.5707963267948966, 1)

(1.5707963267948966, 1)

(29.845130209103036, 1)

(10.995574287564276, 1)

(17.278759594743864, 1)

(-51.83627878423159, 1)

(-29.845130209103036, 1)

(0, 0)

(-183.7831702350029, 1)

(-48.6946861306418, 1)

(-73.82742735936014, 1)

(23.56194490192345, 1)

(20.420352248333657, 1)

(-86.39379797371932, 1)

(54.977871437821385, 1)

(58.119464091411174, 1)

(51.83627878423159, 1)

(-67.54424205218055, 1)

(237.1902453460294, 1)

(-4.71238898038469, 1)

(-70.68583470577035, 1)

(-45.553093477052, 1)

(48.6946861306418, 1)

(-83.25220532012952, 1)

(-95.81857593448869, 1)

(89.53539062730911, 1)

(-39.269908169872416, 1)

(-306.3052837250048, 1)

(76.96902001294994, 1)

(-32.98672286269283, 1)

(-20.420352248333657, 1)

(-36.12831551628262, 1)

(7.853981633974483, 1)

(-80.11061266653972, 1)

(86.39379797371932, 1)

(98.96016858807849, 1)

(36.12831551628262, 1)

(64.40264939859077, 1)

(83.25220532012952, 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=0x_{1} = 0
Puntos máximos de la función:
x1=−2279.22547017939x_{1} = -2279.22547017939
x1=32.9867228626928x_{1} = 32.9867228626928
x1=73.8274273593601x_{1} = 73.8274273593601
x1=4.71238898038469x_{1} = 4.71238898038469
x1=39.2699081698724x_{1} = 39.2699081698724
x1=95.8185759344887x_{1} = 95.8185759344887
x1=45.553093477052x_{1} = 45.553093477052
x1=70.6858347057703x_{1} = 70.6858347057703
x1=−10.9955742875643x_{1} = -10.9955742875643
x1=−58.1194640914112x_{1} = -58.1194640914112
x1=−23.5619449019235x_{1} = -23.5619449019235
x1=26.7035375555132x_{1} = 26.7035375555132
x1=−26.7035375555132x_{1} = -26.7035375555132
x1=−89.5353906273091x_{1} = -89.5353906273091
x1=−17.2787595947439x_{1} = -17.2787595947439
x1=−42.4115008234622x_{1} = -42.4115008234622
x1=−61.261056745001x_{1} = -61.261056745001
x1=92.6769832808989x_{1} = 92.6769832808989
x1=−76.9690200129499x_{1} = -76.9690200129499
x1=−92.6769832808989x_{1} = -92.6769832808989
x1=−98.9601685880785x_{1} = -98.9601685880785
x1=61.261056745001x_{1} = 61.261056745001
x1=−54.9778714378214x_{1} = -54.9778714378214
x1=42.4115008234622x_{1} = 42.4115008234622
x1=−64.4026493985908x_{1} = -64.4026493985908
x1=67.5442420521806x_{1} = 67.5442420521806
x1=−7.85398163397448x_{1} = -7.85398163397448
x1=80.1106126665397x_{1} = 80.1106126665397
x1=−14.1371669411541x_{1} = -14.1371669411541
x1=14.1371669411541x_{1} = 14.1371669411541
x1=−1.5707963267949x_{1} = -1.5707963267949
x1=1.5707963267949x_{1} = 1.5707963267949
x1=29.845130209103x_{1} = 29.845130209103
x1=10.9955742875643x_{1} = 10.9955742875643
x1=17.2787595947439x_{1} = 17.2787595947439
x1=−51.8362787842316x_{1} = -51.8362787842316
x1=−29.845130209103x_{1} = -29.845130209103
x1=−183.783170235003x_{1} = -183.783170235003
x1=−48.6946861306418x_{1} = -48.6946861306418
x1=−73.8274273593601x_{1} = -73.8274273593601
x1=23.5619449019235x_{1} = 23.5619449019235
x1=20.4203522483337x_{1} = 20.4203522483337
x1=−86.3937979737193x_{1} = -86.3937979737193
x1=54.9778714378214x_{1} = 54.9778714378214
x1=58.1194640914112x_{1} = 58.1194640914112
x1=51.8362787842316x_{1} = 51.8362787842316
x1=−67.5442420521806x_{1} = -67.5442420521806
x1=237.190245346029x_{1} = 237.190245346029
x1=−4.71238898038469x_{1} = -4.71238898038469
x1=−70.6858347057703x_{1} = -70.6858347057703
x1=−45.553093477052x_{1} = -45.553093477052
x1=48.6946861306418x_{1} = 48.6946861306418
x1=−83.2522053201295x_{1} = -83.2522053201295
x1=−95.8185759344887x_{1} = -95.8185759344887
x1=89.5353906273091x_{1} = 89.5353906273091
x1=−39.2699081698724x_{1} = -39.2699081698724
x1=−306.305283725005x_{1} = -306.305283725005
x1=76.9690200129499x_{1} = 76.9690200129499
x1=−32.9867228626928x_{1} = -32.9867228626928
x1=−20.4203522483337x_{1} = -20.4203522483337
x1=−36.1283155162826x_{1} = -36.1283155162826
x1=7.85398163397448x_{1} = 7.85398163397448
x1=−80.1106126665397x_{1} = -80.1106126665397
x1=86.3937979737193x_{1} = 86.3937979737193
x1=98.9601685880785x_{1} = 98.9601685880785
x1=36.1283155162826x_{1} = 36.1283155162826
x1=64.4026493985908x_{1} = 64.4026493985908
x1=83.2522053201295x_{1} = 83.2522053201295
Decrece en los intervalos
(−∞,−2279.22547017939]∪[0,∞)\left(-\infty, -2279.22547017939\right] \cup \left[0, \infty\right)
Crece en los intervalos
(−∞,0]∪[237.190245346029,∞)\left(-\infty, 0\right] \cup \left[237.190245346029, \infty\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
−sin⁡(x)sign⁡(sin⁡(x))+2cos⁡2(x)δ(sin⁡(x))=0- \sin{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} + 2 \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\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→−∞∣sin⁡(x)∣=∣⟨−1,1⟩∣\lim_{x \to -\infty} \left|{\sin{\left(x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=∣⟨−1,1⟩∣y = \left|{\left\langle -1, 1\right\rangle}\right|
lim⁡x→∞∣sin⁡(x)∣=∣⟨−1,1⟩∣\lim_{x \to \infty} \left|{\sin{\left(x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=∣⟨−1,1⟩∣y = \left|{\left\langle -1, 1\right\rangle}\right|
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función Abs(sin(x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(∣sin⁡(x)∣x)=0\lim_{x \to -\infty}\left(\frac{\left|{\sin{\left(x \right)}}\right|}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(∣sin⁡(x)∣x)=0\lim_{x \to \infty}\left(\frac{\left|{\sin{\left(x \right)}}\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⁡(x)∣=∣sin⁡(x)∣\left|{\sin{\left(x \right)}}\right| = \left|{\sin{\left(x \right)}}\right|
- Sí
∣sin⁡(x)∣=−∣sin⁡(x)∣\left|{\sin{\left(x \right)}}\right| = - \left|{\sin{\left(x \right)}}\right|
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = |sinx|
    © Sr Examen – Калькулятор