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Plotting a function graph/ ctg(x)

Graphing y = ctg(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cot(x)
f(x)=cot⁡(x)f{\left(x \right)} = \cot{\left(x \right)} 
f = cot(x)
The graph of the function
2.03.04.05.06.07.08.09.010.0-100000100000
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:
cot⁡(x)=0\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=14.1371669411541x_{1} = 14.1371669411541
x2=73.8274273593601x_{2} = 73.8274273593601
x3=−92.6769832808989x_{3} = -92.6769832808989
x4=−48.6946861306418x_{4} = -48.6946861306418
x5=89.5353906273091x_{5} = 89.5353906273091
x6=−23.5619449019235x_{6} = -23.5619449019235
x7=−86.3937979737193x_{7} = -86.3937979737193
x8=39.2699081698724x_{8} = 39.2699081698724
x9=−17.2787595947439x_{9} = -17.2787595947439
x10=20.4203522483337x_{10} = 20.4203522483337
x11=−26.7035375555132x_{11} = -26.7035375555132
x12=61.261056745001x_{12} = 61.261056745001
x13=42.4115008234622x_{13} = 42.4115008234622
x14=−64.4026493985908x_{14} = -64.4026493985908
x15=−83.2522053201295x_{15} = -83.2522053201295
x16=−4.71238898038469x_{16} = -4.71238898038469
x17=−42.4115008234622x_{17} = -42.4115008234622
x18=−29.845130209103x_{18} = -29.845130209103
x19=17.2787595947439x_{19} = 17.2787595947439
x20=51.8362787842316x_{20} = 51.8362787842316
x21=1.5707963267949x_{21} = 1.5707963267949
x22=−67.5442420521806x_{22} = -67.5442420521806
x23=−36.1283155162826x_{23} = -36.1283155162826
x24=45.553093477052x_{24} = 45.553093477052
x25=−80.1106126665397x_{25} = -80.1106126665397
x26=86.3937979737193x_{26} = 86.3937979737193
x27=−73.8274273593601x_{27} = -73.8274273593601
x28=32.9867228626928x_{28} = 32.9867228626928
x29=64.4026493985908x_{29} = 64.4026493985908
x30=−1.5707963267949x_{30} = -1.5707963267949
x31=95.8185759344887x_{31} = 95.8185759344887
x32=−20.4203522483337x_{32} = -20.4203522483337
x33=−10.9955742875643x_{33} = -10.9955742875643
x34=−98.9601685880785x_{34} = -98.9601685880785
x35=92.6769832808989x_{35} = 92.6769832808989
x36=36.1283155162826x_{36} = 36.1283155162826
x37=−32.9867228626928x_{37} = -32.9867228626928
x38=−39.2699081698724x_{38} = -39.2699081698724
x39=−58.1194640914112x_{39} = -58.1194640914112
x40=−61.261056745001x_{40} = -61.261056745001
x41=4.71238898038469x_{41} = 4.71238898038469
x42=−76.9690200129499x_{42} = -76.9690200129499
x43=−95.8185759344887x_{43} = -95.8185759344887
x44=48.6946861306418x_{44} = 48.6946861306418
x45=−51.8362787842316x_{45} = -51.8362787842316
x46=23.5619449019235x_{46} = 23.5619449019235
x47=67.5442420521806x_{47} = 67.5442420521806
x48=−14.1371669411541x_{48} = -14.1371669411541
x49=76.9690200129499x_{49} = 76.9690200129499
x50=98.9601685880785x_{50} = 98.9601685880785
x51=80.1106126665397x_{51} = 80.1106126665397
x52=−7.85398163397448x_{52} = -7.85398163397448
x53=7.85398163397448x_{53} = 7.85398163397448
x54=83.2522053201295x_{54} = 83.2522053201295
x55=58.1194640914112x_{55} = 58.1194640914112
x56=−45.553093477052x_{56} = -45.553093477052
x57=70.6858347057703x_{57} = 70.6858347057703
x58=54.9778714378214x_{58} = 54.9778714378214
x59=26.7035375555132x_{59} = 26.7035375555132
x60=−89.5353906273091x_{60} = -89.5353906273091
x61=10.9955742875643x_{61} = 10.9955742875643
x62=−70.6858347057703x_{62} = -70.6858347057703
x63=−54.9778714378214x_{63} = -54.9778714378214
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 cot(x).
cot⁡(0)\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
−cot⁡2(x)−1=0- \cot^{2}{\left(x \right)} - 1 = 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
2(cot⁡2(x)+1)cot⁡(x)=02 \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
lim⁡x→−∞cot⁡(x)=−cot⁡(∞)\lim_{x \to -\infty} \cot{\left(x \right)} = - \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−cot⁡(∞)y = - \cot{\left(\infty \right)}
lim⁡x→∞cot⁡(x)=cot⁡(∞)\lim_{x \to \infty} \cot{\left(x \right)} = \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=cot⁡(∞)y = \cot{\left(\infty \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 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→−∞(cot⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\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:
cot⁡(x)=−cot⁡(x)\cot{\left(x \right)} = - \cot{\left(x \right)}
- No
cot⁡(x)=cot⁡(x)\cot{\left(x \right)} = \cot{\left(x \right)}
- Yes
so, the function
is
odd
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