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Trazado el gráfico de la función/ sqrt(x)/tan(x)

Gráfico de la función y = sqrt(x)/tan(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         ___ 
       \/ x  
f(x) = ------
       tan(x)
f(x)=xtan⁡(x)f{\left(x \right)} = \frac{\sqrt{x}}{\tan{\left(x \right)}} 
f = sqrt(x)/tan(x)
Gráfico de la función
02468-8-6-4-2-1010-20002000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
xtan⁡(x)=0\frac{\sqrt{x}}{\tan{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=48.6946861306418x_{1} = 48.6946861306418
x2=54.9778714378214x_{2} = 54.9778714378214
x3=−98.9601685880785x_{3} = -98.9601685880785
x4=67.5442420521806x_{4} = 67.5442420521806
x5=76.9690200129499x_{5} = 76.9690200129499
x6=36.1283155162826x_{6} = 36.1283155162826
x7=58.1194640914112x_{7} = 58.1194640914112
x8=14.1371669411541x_{8} = 14.1371669411541
x9=−29.845130209103x_{9} = -29.845130209103
x10=61.261056745001x_{10} = 61.261056745001
x11=−36.1283155162826x_{11} = -36.1283155162826
x12=−4.71238898038469x_{12} = -4.71238898038469
x13=−39.2699081698724x_{13} = -39.2699081698724
x14=1.5707963267949x_{14} = 1.5707963267949
x15=−14.1371669411541x_{15} = -14.1371669411541
x16=−64.4026493985908x_{16} = -64.4026493985908
x17=−67.5442420521806x_{17} = -67.5442420521806
x18=92.6769832808989x_{18} = 92.6769832808989
x19=−51.8362787842316x_{19} = -51.8362787842316
x20=−86.3937979737193x_{20} = -86.3937979737193
x21=42.4115008234622x_{21} = 42.4115008234622
x22=−17.2787595947439x_{22} = -17.2787595947439
x23=−45.553093477052x_{23} = -45.553093477052
x24=−89.5353906273091x_{24} = -89.5353906273091
x25=−1.5707963267949x_{25} = -1.5707963267949
x26=39.2699081698724x_{26} = 39.2699081698724
x27=23.5619449019235x_{27} = 23.5619449019235
x28=7.85398163397448x_{28} = 7.85398163397448
x29=−58.1194640914112x_{29} = -58.1194640914112
x30=−61.261056745001x_{30} = -61.261056745001
x31=−73.8274273593601x_{31} = -73.8274273593601
x32=73.8274273593601x_{32} = 73.8274273593601
x33=29.845130209103x_{33} = 29.845130209103
x34=4.71238898038469x_{34} = 4.71238898038469
x35=86.3937979737193x_{35} = 86.3937979737193
x36=64.4026493985908x_{36} = 64.4026493985908
x37=89.5353906273091x_{37} = 89.5353906273091
x38=−20.4203522483337x_{38} = -20.4203522483337
x39=−26.7035375555132x_{39} = -26.7035375555132
x40=98.9601685880785x_{40} = 98.9601685880785
x41=51.8362787842316x_{41} = 51.8362787842316
x42=83.2522053201295x_{42} = 83.2522053201295
x43=−48.6946861306418x_{43} = -48.6946861306418
x44=−54.9778714378214x_{44} = -54.9778714378214
x45=70.6858347057703x_{45} = 70.6858347057703
x46=−95.8185759344887x_{46} = -95.8185759344887
x47=26.7035375555132x_{47} = 26.7035375555132
x48=80.1106126665397x_{48} = 80.1106126665397
x49=−23.5619449019235x_{49} = -23.5619449019235
x50=−7.85398163397448x_{50} = -7.85398163397448
x51=−83.2522053201295x_{51} = -83.2522053201295
x52=−76.9690200129499x_{52} = -76.9690200129499
x53=−42.4115008234622x_{53} = -42.4115008234622
x54=−32.9867228626928x_{54} = -32.9867228626928
x55=17.2787595947439x_{55} = 17.2787595947439
x56=32.9867228626928x_{56} = 32.9867228626928
x57=20.4203522483337x_{57} = 20.4203522483337
x58=−70.6858347057703x_{58} = -70.6858347057703
x59=−10.9955742875643x_{59} = -10.9955742875643
x60=−92.6769832808989x_{60} = -92.6769832808989
x61=45.553093477052x_{61} = 45.553093477052
x62=10.9955742875643x_{62} = 10.9955742875643
x63=−80.1106126665397x_{63} = -80.1106126665397
x64=95.8185759344887x_{64} = 95.8185759344887
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 sqrt(x)/tan(x).
0tan⁡(0)\frac{\sqrt{0}}{\tan{\left(0 \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
x(−tan⁡2(x)−1)tan⁡2(x)+12xtan⁡(x)=0\frac{\sqrt{x} \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{1}{2 \sqrt{x} \tan{\left(x \right)}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
2x(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)−tan⁡2(x)+1xtan⁡(x)−14x32tan⁡(x)=0\frac{2 \sqrt{x} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x} \tan{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}}}{\tan{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−70.6787603857481x_{1} = -70.6787603857481
x2=−54.9687752392829x_{2} = -54.9687752392829
x3=−98.9551157707129x_{3} = -98.9551157707129
x4=17.2497696527847x_{4} = 17.2497696527847
x5=73.8206541354127x_{5} = 73.8206541354127
x6=−32.9715576965888x_{6} = -32.9715576965888
x7=−45.5421144075225x_{7} = -45.5421144075225
x8=45.5421144075225x_{8} = 45.5421144075225
x9=−4.60357245433596x_{9} = -4.60357245433596
x10=−76.9625232987661x_{10} = -76.9625232987661
x11=39.2571712992709x_{11} = 39.2571712992709
x12=32.9715576965888x_{12} = 32.9715576965888
x13=−42.39970801623x_{13} = -42.39970801623
x14=−7.7897512567741x_{14} = -7.7897512567741
x15=36.1144702082836x_{15} = 36.1144702082836
x16=7.7897512567741x_{16} = 7.7897512567741
x17=−10.949895994345x_{17} = -10.949895994345
x18=26.6847992011637x_{18} = 26.6847992011637
x19=14.101702831878x_{19} = 14.101702831878
x20=−83.2461990037822x_{20} = -83.2461990037822
x21=80.104370769356x_{21} = 80.104370769356
x22=48.6844157231755x_{22} = 48.6844157231755
x23=83.2461990037822x_{23} = 83.2461990037822
x24=−86.3880101011813x_{24} = -86.3880101011813
x25=76.9625232987661x_{25} = 76.9625232987661
x26=−89.5298058659662x_{26} = -89.5298058659662
x27=70.6787603857481x_{27} = 70.6787603857481
x28=58.1108597415507x_{28} = 58.1108597415507
x29=4.60357245433596x_{29} = 4.60357245433596
x30=−20.3958349886934x_{30} = -20.3958349886934
x31=−95.8133574317394x_{31} = -95.8133574317394
x32=−14.101702831878x_{32} = -14.101702831878
x33=1.11791403207435x_{33} = 1.11791403207435
x34=54.9687752392829x_{34} = 54.9687752392829
x35=20.3958349886934x_{35} = 20.3958349886934
x36=92.6715878578012x_{36} = 92.6715878578012
x37=−58.1108597415507x_{37} = -58.1108597415507
x38=−36.1144702082836x_{38} = -36.1144702082836
x39=−1.11791403207435x_{39} = -1.11791403207435
x40=−61.2528937747202x_{40} = -61.2528937747202
x41=−73.8206541354127x_{41} = -73.8206541354127
x42=89.5298058659662x_{42} = 89.5298058659662
x43=−92.6715878578012x_{43} = -92.6715878578012
x44=67.5368386175974x_{44} = 67.5368386175974
x45=95.8133574317394x_{45} = 95.8133574317394
x46=−51.8266310849037x_{46} = -51.8266310849037
x47=−17.2497696527847x_{47} = -17.2497696527847
x48=−39.2571712992709x_{48} = -39.2571712992709
x49=51.8266310849037x_{49} = 51.8266310849037
x50=64.3948847286922x_{50} = 64.3948847286922
x51=−29.8283668577611x_{51} = -29.8283668577611
x52=−26.6847992011637x_{52} = -26.6847992011637
x53=−48.6844157231755x_{53} = -48.6844157231755
x54=10.949895994345x_{54} = 10.949895994345
x55=−67.5368386175974x_{55} = -67.5368386175974
x56=−80.104370769356x_{56} = -80.104370769356
x57=−23.5407035002745x_{57} = -23.5407035002745
x58=−64.3948847286922x_{58} = -64.3948847286922
x59=86.3880101011813x_{59} = 86.3880101011813
x60=29.8283668577611x_{60} = 29.8283668577611
x61=42.39970801623x_{61} = 42.39970801623
x62=61.2528937747202x_{62} = 61.2528937747202
x63=98.9551157707129x_{63} = 98.9551157707129
x64=23.5407035002745x_{64} = 23.5407035002745
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(2x(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)−tan⁡2(x)+1xtan⁡(x)−14x32tan⁡(x))=−∞i\lim_{x \to 0^-}\left(\frac{2 \sqrt{x} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x} \tan{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}}}{\tan{\left(x \right)}}\right) = - \infty i
lim⁡x→0+(2x(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)−tan⁡2(x)+1xtan⁡(x)−14x32tan⁡(x))=∞\lim_{x \to 0^+}\left(\frac{2 \sqrt{x} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x} \tan{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}}}{\tan{\left(x \right)}}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
(−∞,1.11791403207435]\left(-\infty, 1.11791403207435\right]
Convexa en los intervalos
[98.9551157707129,∞)\left[98.9551157707129, \infty\right)
Asíntotas verticales
Hay:
x1=0x_{1} = 0
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→−∞(xtan⁡(x))y = \lim_{x \to -\infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(xtan⁡(x))y = \lim_{x \to \infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sqrt(x)/tan(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→−∞(1xtan⁡(x))y = x \lim_{x \to -\infty}\left(\frac{1}{\sqrt{x} \tan{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(1xtan⁡(x))y = x \lim_{x \to \infty}\left(\frac{1}{\sqrt{x} \tan{\left(x \right)}}\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:
xtan⁡(x)=−−xtan⁡(x)\frac{\sqrt{x}}{\tan{\left(x \right)}} = - \frac{\sqrt{- x}}{\tan{\left(x \right)}}
- No
xtan⁡(x)=−xtan⁡(x)\frac{\sqrt{x}}{\tan{\left(x \right)}} = \frac{\sqrt{- x}}{\tan{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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