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Trazado el gráfico de la función/ 10*|3*tg(x)*cos(x)|

Gráfico de la función y = 10*|3*tg(x)*cos(x)|

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = 10*|3*tan(x)*cos(x)|
f(x)=10∣cos⁡(x)3tan⁡(x)∣f{\left(x \right)} = 10 \left|{\cos{\left(x \right)} 3 \tan{\left(x \right)}}\right| 
f = 10*Abs(cos(x)*(3*tan(x)))
Gráfico de la función
-1.0-0.8-0.6-0.4-0.21.00.00.20.40.60.8050
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:
10∣cos⁡(x)3tan⁡(x)∣=010 \left|{\cos{\left(x \right)} 3 \tan{\left(x \right)}}\right| = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=43.9822971502571x_{1} = 43.9822971502571
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=−43.9822971502571x_{3} = -43.9822971502571
x4=−72.2566310325652x_{4} = -72.2566310325652
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=81.6814089933346x_{6} = 81.6814089933346
x7=−31.4159265358979x_{7} = -31.4159265358979
x8=−78.5398163397448x_{8} = -78.5398163397448
x9=97.3893722612836x_{9} = 97.3893722612836
x10=−25.1327412287183x_{10} = -25.1327412287183
x11=9.42477796076938x_{11} = 9.42477796076938
x12=84.8230016469244x_{12} = 84.8230016469244
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=6.28318530717959x_{15} = 6.28318530717959
x16=−50.2654824574367x_{16} = -50.2654824574367
x17=28.2743338823081x_{17} = 28.2743338823081
x18=−75.398223686155x_{18} = -75.398223686155
x19=−28.2743338823081x_{19} = -28.2743338823081
x20=−56.5486677646163x_{20} = -56.5486677646163
x21=−65.9734457253857x_{21} = -65.9734457253857
x22=−91.106186954104x_{22} = -91.106186954104
x23=−285.884931476671x_{23} = -285.884931476671
x24=50.2654824574367x_{24} = 50.2654824574367
x25=−69.1150383789755x_{25} = -69.1150383789755
x26=−100.530964914873x_{26} = -100.530964914873
x27=56.5486677646163x_{27} = 56.5486677646163
x28=−87.9645943005142x_{28} = -87.9645943005142
x29=40.8407044966673x_{29} = 40.8407044966673
x30=18.8495559215388x_{30} = 18.8495559215388
x31=650.309679293087x_{31} = 650.309679293087
x32=100.530964914873x_{32} = 100.530964914873
x33=62.8318530717959x_{33} = 62.8318530717959
x34=−53.4070751110265x_{34} = -53.4070751110265
x35=94.2477796076938x_{35} = 94.2477796076938
x36=−3.14159265358979x_{36} = -3.14159265358979
x37=21.9911485751286x_{37} = 21.9911485751286
x38=12.5663706143592x_{38} = 12.5663706143592
x39=−427.256600888212x_{39} = -427.256600888212
x40=34.5575191894877x_{40} = 34.5575191894877
x41=−15.707963267949x_{41} = -15.707963267949
x42=53.4070751110265x_{42} = 53.4070751110265
x43=65.9734457253857x_{43} = 65.9734457253857
x44=87.9645943005142x_{44} = 87.9645943005142
x45=59.6902604182061x_{45} = 59.6902604182061
x46=−6.28318530717959x_{46} = -6.28318530717959
x47=75.398223686155x_{47} = 75.398223686155
x48=−37.6991118430775x_{48} = -37.6991118430775
x49=−12.5663706143592x_{49} = -12.5663706143592
x50=31.4159265358979x_{50} = 31.4159265358979
x51=−81.6814089933346x_{51} = -81.6814089933346
x52=78.5398163397448x_{52} = 78.5398163397448
x53=15.707963267949x_{53} = 15.707963267949
x54=72.2566310325652x_{54} = 72.2566310325652
x55=37.6991118430775x_{55} = 37.6991118430775
x56=−47.1238898038469x_{56} = -47.1238898038469
x57=0x_{57} = 0
x58=−9.42477796076938x_{58} = -9.42477796076938
x59=−34.5575191894877x_{59} = -34.5575191894877
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 10*Abs((3*tan(x))*cos(x)).
10∣cos⁡(0)3tan⁡(0)∣10 \left|{\cos{\left(0 \right)} 3 \tan{\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
10((3tan⁡2(x)+3)cos⁡(x)−3sin⁡(x)tan⁡(x))sign⁡(cos⁡(x)tan⁡(x))=010 \left(\left(3 \tan^{2}{\left(x \right)} + 3\right) \cos{\left(x \right)} - 3 \sin{\left(x \right)} \tan{\left(x \right)}\right) \operatorname{sign}{\left(\cos{\left(x \right)} \tan{\left(x \right)} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1229.9335238804x_{1} = 1229.9335238804
x2=−54.9778714378214x_{2} = -54.9778714378214
x3=−26.7035375555132x_{3} = -26.7035375555132
x4=7.85398163397448x_{4} = 7.85398163397448
x5=−262.322986574748x_{5} = -262.322986574748
x6=73.8274273593601x_{6} = 73.8274273593601
x7=61.261056745001x_{7} = 61.261056745001
x8=0x_{8} = 0
x9=−26.7035375555132x_{9} = -26.7035375555132
x10=−306.305283725005x_{10} = -306.305283725005
x11=−1.5707963267949x_{11} = -1.5707963267949
x12=−95.8185759344887x_{12} = -95.8185759344887
x13=−39.2699081698724x_{13} = -39.2699081698724
x14=14.1371669411541x_{14} = 14.1371669411541
x15=10.9955742875643x_{15} = 10.9955742875643
x16=−42.4115008234622x_{16} = -42.4115008234622
x17=58.1194640914112x_{17} = 58.1194640914112
x18=−29.845130209103x_{18} = -29.845130209103
x19=70.6858347057703x_{19} = 70.6858347057703
x20=54.9778714378214x_{20} = 54.9778714378214
x21=54.9778714378214x_{21} = 54.9778714378214
x22=−36.1283155162826x_{22} = -36.1283155162826
x23=23.5619449019235x_{23} = 23.5619449019235
x24=237.190245346029x_{24} = 237.190245346029
x25=17.2787595947438x_{25} = 17.2787595947438
x26=80.1106126665397x_{26} = 80.1106126665397
x27=−7.85398163397449x_{27} = -7.85398163397449
x28=−92.6769832808989x_{28} = -92.6769832808989
x29=−10.9955742875643x_{29} = -10.9955742875643
x30=−86.3937979737193x_{30} = -86.3937979737193
x31=−54.9778714378214x_{31} = -54.9778714378214
x32=39.2699081698724x_{32} = 39.2699081698724
x33=−32.9867228626928x_{33} = -32.9867228626928
x34=36.1283155162826x_{34} = 36.1283155162826
x35=98.9601685880785x_{35} = 98.9601685880785
x36=−61.261056745001x_{36} = -61.261056745001
x37=−67.5442420521806x_{37} = -67.5442420521806
x38=92.6769832808989x_{38} = 92.6769832808989
x39=−58.1194640914112x_{39} = -58.1194640914112
x40=26.7035375555132x_{40} = 26.7035375555132
x41=−4.71238898038469x_{41} = -4.71238898038469
x42=−48.6946861306418x_{42} = -48.6946861306418
x43=51.8362787842316x_{43} = 51.8362787842316
x44=−73.8274273593601x_{44} = -73.8274273593601
x45=86.3937979737193x_{45} = 86.3937979737193
x46=−98.9601685880785x_{46} = -98.9601685880785
x47=−89.5353906273091x_{47} = -89.5353906273091
x48=−14.1371669411541x_{48} = -14.1371669411541
x49=−64.4026493985908x_{49} = -64.4026493985908
x50=95.8185759344887x_{50} = 95.8185759344887
x51=1.5707963267949x_{51} = 1.5707963267949
x52=45.553093477052x_{52} = 45.553093477052
x53=−17.2787595947439x_{53} = -17.2787595947439
x54=4.71238898038469x_{54} = 4.71238898038469
x55=76.9690200129499x_{55} = 76.9690200129499
x56=−45.553093477052x_{56} = -45.553093477052
x57=20.4203522483337x_{57} = 20.4203522483337
x58=−86.3937979737193x_{58} = -86.3937979737193
x59=−83.2522053201295x_{59} = -83.2522053201295
x60=−20.4203522483337x_{60} = -20.4203522483337
x61=−80.1106126665397x_{61} = -80.1106126665397
x62=61.261056745001x_{62} = 61.261056745001
x63=32.9867228626928x_{63} = 32.9867228626928
x64=64.4026493985908x_{64} = 64.4026493985908
x65=−23.5619449019235x_{65} = -23.5619449019235
x66=32.9867228626928x_{66} = 32.9867228626928
x67=29.845130209103x_{67} = 29.845130209103
x68=42.4115008234622x_{68} = 42.4115008234622
x69=89.5353906273091x_{69} = 89.5353906273091
x70=−51.8362787842316x_{70} = -51.8362787842316
x71=−70.6858347057703x_{71} = -70.6858347057703
x72=83.2522053201295x_{72} = 83.2522053201295
x73=67.5442420521806x_{73} = 67.5442420521806
x74=−76.9690200129499x_{74} = -76.9690200129499
x75=−2279.22547017939x_{75} = -2279.22547017939
Signos de extremos en los puntos:
(1229.933523880404, 30)

(-54.977871437821385, 30)

(-26.70353755551324, 30)

(7.853981633974483, 30)

(-262.32298657474774, 30)

(73.82742735936014, 30)

(61.261056745000964, 30)

(0, 0)

(-26.703537555513243, 30)

(-306.3052837250048, 30)

(-1.5707963267948966, 30)

(-95.81857593448869, 30)

(-39.269908169872416, 30)

(14.137166941154069, 30)

(10.995574287564276, 30)

(-42.41150082346223, 30)

(58.119464091411174, 30)

(-29.845130209103033, 30)

(70.68583470577035, 30)

(54.97787143782138, 30)

(54.977871437821385, 30)

(-36.12831551628262, 30)

(23.56194490192345, 30)

(237.1902453460294, 30)

(17.278759594743807, 30)

(80.11061266653974, 30)

(-7.853981633974486, 30)

(-92.6769832808989, 30)

(-10.995574287564276, 30)

(-86.39379797371932, 30)

(-54.97787143782138, 30)

(39.269908169872416, 30)

(-32.98672286269283, 30)

(36.12831551628262, 30)

(98.96016858807849, 30)

(-61.26105674500097, 30)

(-67.54424205218055, 30)

(92.67698328089891, 30)

(-58.119464091411174, 30)

(26.703537555513243, 30)

(-4.712388980384691, 30)

(-48.6946861306418, 30)

(51.83627878423159, 30)

(-73.82742735936013, 30)

(86.39379797371932, 30)

(-98.96016858807849, 30)

(-89.53539062730911, 30)

(-14.137166941154069, 30)

(-64.40264939859077, 30)

(95.81857593448869, 30)

(1.5707963267948966, 30)

(45.553093477052, 30)

(-17.278759594743864, 30)

(4.71238898038469, 30)

(76.96902001294994, 30)

(-45.553093477052, 30)

(20.420352248333657, 30)

(-86.3937979737193, 30)

(-83.25220532012952, 30)

(-20.420352248333657, 30)

(-80.11061266653972, 30)

(61.26105674500097, 30)

(32.98672286269282, 30)

(64.40264939859077, 30)

(-23.56194490192345, 30)

(32.98672286269283, 30)

(29.845130209103036, 30)

(42.411500823462205, 30)

(89.53539062730911, 30)

(-51.83627878423159, 30)

(-70.68583470577035, 30)

(83.25220532012952, 30)

(67.54424205218055, 30)

(-76.96902001294994, 30)

(-2279.225470179395, 30)


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=1229.9335238804x_{1} = 1229.9335238804
x1=−54.9778714378214x_{1} = -54.9778714378214
x1=−26.7035375555132x_{1} = -26.7035375555132
x1=7.85398163397448x_{1} = 7.85398163397448
x1=−262.322986574748x_{1} = -262.322986574748
x1=73.8274273593601x_{1} = 73.8274273593601
x1=61.261056745001x_{1} = 61.261056745001
x1=−26.7035375555132x_{1} = -26.7035375555132
x1=−306.305283725005x_{1} = -306.305283725005
x1=−1.5707963267949x_{1} = -1.5707963267949
x1=−95.8185759344887x_{1} = -95.8185759344887
x1=−39.2699081698724x_{1} = -39.2699081698724
x1=14.1371669411541x_{1} = 14.1371669411541
x1=10.9955742875643x_{1} = 10.9955742875643
x1=−42.4115008234622x_{1} = -42.4115008234622
x1=58.1194640914112x_{1} = 58.1194640914112
x1=−29.845130209103x_{1} = -29.845130209103
x1=70.6858347057703x_{1} = 70.6858347057703
x1=54.9778714378214x_{1} = 54.9778714378214
x1=54.9778714378214x_{1} = 54.9778714378214
x1=−36.1283155162826x_{1} = -36.1283155162826
x1=23.5619449019235x_{1} = 23.5619449019235
x1=237.190245346029x_{1} = 237.190245346029
x1=17.2787595947438x_{1} = 17.2787595947438
x1=80.1106126665397x_{1} = 80.1106126665397
x1=−7.85398163397449x_{1} = -7.85398163397449
x1=−92.6769832808989x_{1} = -92.6769832808989
x1=−10.9955742875643x_{1} = -10.9955742875643
x1=−86.3937979737193x_{1} = -86.3937979737193
x1=−54.9778714378214x_{1} = -54.9778714378214
x1=39.2699081698724x_{1} = 39.2699081698724
x1=−32.9867228626928x_{1} = -32.9867228626928
x1=36.1283155162826x_{1} = 36.1283155162826
x1=98.9601685880785x_{1} = 98.9601685880785
x1=−61.261056745001x_{1} = -61.261056745001
x1=−67.5442420521806x_{1} = -67.5442420521806
x1=92.6769832808989x_{1} = 92.6769832808989
x1=−58.1194640914112x_{1} = -58.1194640914112
x1=26.7035375555132x_{1} = 26.7035375555132
x1=−4.71238898038469x_{1} = -4.71238898038469
x1=−48.6946861306418x_{1} = -48.6946861306418
x1=51.8362787842316x_{1} = 51.8362787842316
x1=−73.8274273593601x_{1} = -73.8274273593601
x1=86.3937979737193x_{1} = 86.3937979737193
x1=−98.9601685880785x_{1} = -98.9601685880785
x1=−89.5353906273091x_{1} = -89.5353906273091
x1=−14.1371669411541x_{1} = -14.1371669411541
x1=−64.4026493985908x_{1} = -64.4026493985908
x1=95.8185759344887x_{1} = 95.8185759344887
x1=1.5707963267949x_{1} = 1.5707963267949
x1=45.553093477052x_{1} = 45.553093477052
x1=−17.2787595947439x_{1} = -17.2787595947439
x1=4.71238898038469x_{1} = 4.71238898038469
x1=76.9690200129499x_{1} = 76.9690200129499
x1=−45.553093477052x_{1} = -45.553093477052
x1=20.4203522483337x_{1} = 20.4203522483337
x1=−86.3937979737193x_{1} = -86.3937979737193
x1=−83.2522053201295x_{1} = -83.2522053201295
x1=−20.4203522483337x_{1} = -20.4203522483337
x1=−80.1106126665397x_{1} = -80.1106126665397
x1=61.261056745001x_{1} = 61.261056745001
x1=32.9867228626928x_{1} = 32.9867228626928
x1=64.4026493985908x_{1} = 64.4026493985908
x1=−23.5619449019235x_{1} = -23.5619449019235
x1=32.9867228626928x_{1} = 32.9867228626928
x1=29.845130209103x_{1} = 29.845130209103
x1=42.4115008234622x_{1} = 42.4115008234622
x1=89.5353906273091x_{1} = 89.5353906273091
x1=−51.8362787842316x_{1} = -51.8362787842316
x1=−70.6858347057703x_{1} = -70.6858347057703
x1=83.2522053201295x_{1} = 83.2522053201295
x1=67.5442420521806x_{1} = 67.5442420521806
x1=−76.9690200129499x_{1} = -76.9690200129499
x1=−2279.22547017939x_{1} = -2279.22547017939
Decrece en los intervalos
(−∞,−2279.22547017939]∪[0,∞)\left(-\infty, -2279.22547017939\right] \cup \left[0, \infty\right)
Crece en los intervalos
(−∞,0]∪[1229.9335238804,∞)\left(-\infty, 0\right] \cup \left[1229.9335238804, \infty\right)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(10∣cos⁡(x)3tan⁡(x)∣)y = \lim_{x \to -\infty}\left(10 \left|{\cos{\left(x \right)} 3 \tan{\left(x \right)}}\right|\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(10∣cos⁡(x)3tan⁡(x)∣)y = \lim_{x \to \infty}\left(10 \left|{\cos{\left(x \right)} 3 \tan{\left(x \right)}}\right|\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 10*Abs((3*tan(x))*cos(x)), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(10∣cos⁡(x)3tan⁡(x)∣x)y = x \lim_{x \to -\infty}\left(\frac{10 \left|{\cos{\left(x \right)} 3 \tan{\left(x \right)}}\right|}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(10∣cos⁡(x)3tan⁡(x)∣x)y = x \lim_{x \to \infty}\left(\frac{10 \left|{\cos{\left(x \right)} 3 \tan{\left(x \right)}}\right|}{x}\right)
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:
10∣cos⁡(x)3tan⁡(x)∣=30∣cos⁡(x)tan⁡(x)∣10 \left|{\cos{\left(x \right)} 3 \tan{\left(x \right)}}\right| = 30 \left|{\cos{\left(x \right)} \tan{\left(x \right)}}\right|
- No
10∣cos⁡(x)3tan⁡(x)∣=−30∣cos⁡(x)tan⁡(x)∣10 \left|{\cos{\left(x \right)} 3 \tan{\left(x \right)}}\right| = - 30 \left|{\cos{\left(x \right)} \tan{\left(x \right)}}\right|
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор