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Trazado el gráfico de la función/ ctg(x)/x

Gráfico de la función y = ctg(x)/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cot(x)
f(x) = ------
         x
f(x)=cot⁡(x)xf{\left(x \right)} = \frac{\cot{\left(x \right)}}{x} 
f = cot(x)/x
Gráfico de la función
02468-8-6-4-2-1010-500500
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:
cot⁡(x)x=0\frac{\cot{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
Solución numérica
x1=4.71238898038469x_{1} = 4.71238898038469
x2=17.2787595947439x_{2} = 17.2787595947439
x3=−89.5353906273091x_{3} = -89.5353906273091
x4=64.4026493985908x_{4} = 64.4026493985908
x5=70.6858347057703x_{5} = 70.6858347057703
x6=36.1283155162826x_{6} = 36.1283155162826
x7=−98.9601685880785x_{7} = -98.9601685880785
x8=48.6946861306418x_{8} = 48.6946861306418
x9=−58.1194640914112x_{9} = -58.1194640914112
x10=7.85398163397448x_{10} = 7.85398163397448
x11=39.2699081698724x_{11} = 39.2699081698724
x12=−95.8185759344887x_{12} = -95.8185759344887
x13=−1.5707963267949x_{13} = -1.5707963267949
x14=−92.6769832808989x_{14} = -92.6769832808989
x15=−23.5619449019235x_{15} = -23.5619449019235
x16=23.5619449019235x_{16} = 23.5619449019235
x17=61.261056745001x_{17} = 61.261056745001
x18=29.845130209103x_{18} = 29.845130209103
x19=−32.9867228626928x_{19} = -32.9867228626928
x20=−51.8362787842316x_{20} = -51.8362787842316
x21=−80.1106126665397x_{21} = -80.1106126665397
x22=−83.2522053201295x_{22} = -83.2522053201295
x23=67.5442420521806x_{23} = 67.5442420521806
x24=98.9601685880785x_{24} = 98.9601685880785
x25=92.6769832808989x_{25} = 92.6769832808989
x26=−39.2699081698724x_{26} = -39.2699081698724
x27=86.3937979737193x_{27} = 86.3937979737193
x28=45.553093477052x_{28} = 45.553093477052
x29=−67.5442420521806x_{29} = -67.5442420521806
x30=51.8362787842316x_{30} = 51.8362787842316
x31=76.9690200129499x_{31} = 76.9690200129499
x32=−26.7035375555132x_{32} = -26.7035375555132
x33=−4.71238898038469x_{33} = -4.71238898038469
x34=95.8185759344887x_{34} = 95.8185759344887
x35=−86.3937979737193x_{35} = -86.3937979737193
x36=−10.9955742875643x_{36} = -10.9955742875643
x37=83.2522053201295x_{37} = 83.2522053201295
x38=−7.85398163397448x_{38} = -7.85398163397448
x39=−36.1283155162826x_{39} = -36.1283155162826
x40=−17.2787595947439x_{40} = -17.2787595947439
x41=−14.1371669411541x_{41} = -14.1371669411541
x42=20.4203522483337x_{42} = 20.4203522483337
x43=54.9778714378214x_{43} = 54.9778714378214
x44=−70.6858347057703x_{44} = -70.6858347057703
x45=−48.6946861306418x_{45} = -48.6946861306418
x46=−54.9778714378214x_{46} = -54.9778714378214
x47=−45.553093477052x_{47} = -45.553093477052
x48=14.1371669411541x_{48} = 14.1371669411541
x49=−73.8274273593601x_{49} = -73.8274273593601
x50=26.7035375555132x_{50} = 26.7035375555132
x51=89.5353906273091x_{51} = 89.5353906273091
x52=10.9955742875643x_{52} = 10.9955742875643
x53=80.1106126665397x_{53} = 80.1106126665397
x54=73.8274273593601x_{54} = 73.8274273593601
x55=58.1194640914112x_{55} = 58.1194640914112
x56=−61.261056745001x_{56} = -61.261056745001
x57=1.5707963267949x_{57} = 1.5707963267949
x58=−20.4203522483337x_{58} = -20.4203522483337
x59=−42.4115008234622x_{59} = -42.4115008234622
x60=32.9867228626928x_{60} = 32.9867228626928
x61=42.4115008234622x_{61} = 42.4115008234622
x62=−76.9690200129499x_{62} = -76.9690200129499
x63=−64.4026493985908x_{63} = -64.4026493985908
x64=−29.845130209103x_{64} = -29.845130209103
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 cot(x)/x.
cot⁡(0)0\frac{\cot{\left(0 \right)}}{0}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
−cot⁡2(x)−1x−cot⁡(x)x2=0\frac{- \cot^{2}{\left(x \right)} - 1}{x} - \frac{\cot{\left(x \right)}}{x^{2}} = 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
2((cot⁡2(x)+1)cot⁡(x)+cot⁡2(x)+1x+cot⁡(x)x2)x=0\frac{2 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{\cot^{2}{\left(x \right)} + 1}{x} + \frac{\cot{\left(x \right)}}{x^{2}}\right)}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−36.155945326761x_{1} = -36.155945326761
x2=−17.3361885185616x_{2} = -17.3361885185616
x3=61.2773701920955x_{3} = 61.2773701920955
x4=98.9702712572776x_{4} = 98.9702712572776
x5=−70.6999752111609x_{5} = -70.6999752111609
x6=−58.136658161712x_{6} = -58.136658161712
x7=−73.8409666670918x_{7} = -73.8409666670918
x8=−92.6877705167662x_{8} = -92.6877705167662
x9=92.6877705167662x_{9} = 92.6877705167662
x10=42.4350488238474x_{10} = 42.4350488238474
x11=−14.2070947841125x_{11} = -14.2070947841125
x12=45.5750212521166x_{12} = 45.5750212521166
x13=−4.90592869090344x_{13} = -4.90592869090344
x14=67.5590395986284x_{14} = 67.5590395986284
x15=−33.0169733513735x_{15} = -33.0169733513735
x16=−95.8290096731724x_{16} = -95.8290096731724
x17=58.136658161712x_{17} = 58.136658161712
x18=64.4181679845522x_{18} = 64.4181679845522
x19=48.7152020823059x_{19} = 48.7152020823059
x20=7.97681246070588x_{20} = 7.97681246070588
x21=83.2642129727806x_{21} = 83.2642129727806
x22=17.3361885185616x_{22} = 17.3361885185616
x23=−26.7408640091706x_{23} = -26.7408640091706
x24=−45.5750212521166x_{24} = -45.5750212521166
x25=23.6042092770644x_{25} = 23.6042092770644
x26=1.96571540226556x_{26} = 1.96571540226556
x27=−42.4350488238474x_{27} = -42.4350488238474
x28=4.90592869090344x_{28} = 4.90592869090344
x29=76.9820071399389x_{29} = 76.9820071399389
x30=36.155945326761x_{30} = 36.155945326761
x31=−48.7152020823059x_{31} = -48.7152020823059
x32=70.6999752111609x_{32} = 70.6999752111609
x33=−83.2642129727806x_{33} = -83.2642129727806
x34=−39.2953345433102x_{34} = -39.2953345433102
x35=−54.9960465536225x_{35} = -54.9960465536225
x36=51.8555535682362x_{36} = 51.8555535682362
x37=80.1230908719237x_{37} = 80.1230908719237
x38=−76.9820071399389x_{38} = -76.9820071399389
x39=−11.0848274012762x_{39} = -11.0848274012762
x40=−98.9702712572776x_{40} = -98.9702712572776
x41=95.8290096731724x_{41} = 95.8290096731724
x42=39.2953345433102x_{42} = 39.2953345433102
x43=−51.8555535682362x_{43} = -51.8555535682362
x44=−29.8785491828649x_{44} = -29.8785491828649
x45=26.7408640091706x_{45} = 26.7408640091706
x46=33.0169733513735x_{46} = 33.0169733513735
x47=86.4053692623541x_{47} = 86.4053692623541
x48=−7.97681246070588x_{48} = -7.97681246070588
x49=−20.46905190384x_{49} = -20.46905190384
x50=−1.96571540226556x_{50} = -1.96571540226556
x51=20.46905190384x_{51} = 20.46905190384
x52=89.5465561462572x_{52} = 89.5465561462572
x53=−89.5465561462572x_{53} = -89.5465561462572
x54=73.8409666670918x_{54} = 73.8409666670918
x55=−23.6042092770644x_{55} = -23.6042092770644
x56=14.2070947841125x_{56} = 14.2070947841125
x57=−64.4181679845522x_{57} = -64.4181679845522
x58=54.9960465536225x_{58} = 54.9960465536225
x59=−80.1230908719237x_{59} = -80.1230908719237
x60=11.0848274012762x_{60} = 11.0848274012762
x61=29.8785491828649x_{61} = 29.8785491828649
x62=−61.2773701920955x_{62} = -61.2773701920955
x63=−86.4053692623541x_{63} = -86.4053692623541
x64=−67.5590395986284x_{64} = -67.5590395986284
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−(2((cot⁡2(x)+1)cot⁡(x)+cot⁡2(x)+1x+cot⁡(x)x2)x)=∞\lim_{x \to 0^-}\left(\frac{2 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{\cot^{2}{\left(x \right)} + 1}{x} + \frac{\cot{\left(x \right)}}{x^{2}}\right)}{x}\right) = \infty
lim⁡x→0+(2((cot⁡2(x)+1)cot⁡(x)+cot⁡2(x)+1x+cot⁡(x)x2)x)=∞\lim_{x \to 0^+}\left(\frac{2 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{\cot^{2}{\left(x \right)} + 1}{x} + \frac{\cot{\left(x \right)}}{x^{2}}\right)}{x}\right) = \infty
- los límites son iguales, es decir omitimos el punto correspondiente

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
[−1.96571540226556,1.96571540226556]\left[-1.96571540226556, 1.96571540226556\right]
Convexa en los intervalos
(−∞,−98.9702712572776]\left(-\infty, -98.9702712572776\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→−∞(cot⁡(x)x)y = \lim_{x \to -\infty}\left(\frac{\cot{\left(x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(cot⁡(x)x)y = \lim_{x \to \infty}\left(\frac{\cot{\left(x \right)}}{x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cot(x)/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→−∞(cot⁡(x)x2)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(x \right)}}{x^{2}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(cot⁡(x)x2)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(x \right)}}{x^{2}}\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:
cot⁡(x)x=cot⁡(x)x\frac{\cot{\left(x \right)}}{x} = \frac{\cot{\left(x \right)}}{x}
- No
cot⁡(x)x=−cot⁡(x)x\frac{\cot{\left(x \right)}}{x} = - \frac{\cot{\left(x \right)}}{x}
- No
es decir, función
no es
par ni impar
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