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Plotting a function graph/ 2ctgx

Graphing y = 2ctgx

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*cot(x)
f(x)=2cot⁡(x)f{\left(x \right)} = 2 \cot{\left(x \right)} 
f = 2*cot(x)
The graph of the function
05-20-15-10-5201015-2000020000
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
2cot⁡(x)=02 \cot{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
Numerical solution
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
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*cot(x).
2cot⁡(0)2 \cot{\left(0 \right)}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−2cot⁡2(x)−2=0- 2 \cot^{2}{\left(x \right)} - 2 = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
4(cot⁡2(x)+1)cot⁡(x)=04 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,π2]\left(-\infty, \frac{\pi}{2}\right]
Convex at the intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(2cot⁡(x))y = \lim_{x \to -\infty}\left(2 \cot{\left(x \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(2cot⁡(x))y = \lim_{x \to \infty}\left(2 \cot{\left(x \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*cot(x), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(2cot⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{2 \cot{\left(x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(2cot⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{2 \cot{\left(x \right)}}{x}\right)
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
2cot⁡(x)=−2cot⁡(x)2 \cot{\left(x \right)} = - 2 \cot{\left(x \right)}
- No
2cot⁡(x)=2cot⁡(x)2 \cot{\left(x \right)} = 2 \cot{\left(x \right)}
- Yes
so, the function
is
odd
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