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Plotting a function graph/ cot(5*x)

Graphing y = cot(5*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cot(5*x)
f(x)=cot⁡(5x)f{\left(x \right)} = \cot{\left(5 x \right)} 
f = cot(5*x)
The graph of the function
02468-8-6-4-2-1010-10001000
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⁡(5x)=0\cot{\left(5 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π10x_{1} = \frac{\pi}{10}
Numerical solution
x1=88.2787535658732x_{1} = 88.2787535658732
x2=4.08407044966673x_{2} = 4.08407044966673
x3=−85.7654794430014x_{3} = -85.7654794430014
x4=−48.0663675999238x_{4} = -48.0663675999238
x5=42.4115008234622x_{5} = 42.4115008234622
x6=−27.9601746169492x_{6} = -27.9601746169492
x7=−29.845130209103x_{7} = -29.845130209103
x8=−26.0752190247953x_{8} = -26.0752190247953
x9=−75.712382951514x_{9} = -75.712382951514
x10=39.8982267005904x_{10} = 39.8982267005904
x11=−36.1283155162826x_{11} = -36.1283155162826
x12=86.3937979737193x_{12} = 86.3937979737193
x13=76.340701482232x_{13} = 76.340701482232
x14=−19.7920337176157x_{14} = -19.7920337176157
x15=−16.0221225333079x_{15} = -16.0221225333079
x16=27.9601746169492x_{16} = 27.9601746169492
x17=54.3495529071034x_{17} = 54.3495529071034
x18=−95.8185759344887x_{18} = -95.8185759344887
x19=−38.0132711084365x_{19} = -38.0132711084365
x20=−14.1371669411541x_{20} = -14.1371669411541
x21=46.18141200777x_{21} = 46.18141200777
x22=96.4468944652067x_{22} = 96.4468944652067
x23=58.1194640914112x_{23} = 58.1194640914112
x24=−39.8982267005904x_{24} = -39.8982267005904
x25=−70.0575161750524x_{25} = -70.0575161750524
x26=12.2522113490002x_{26} = 12.2522113490002
x27=2.19911485751286x_{27} = 2.19911485751286
x28=−23.5619449019235x_{28} = -23.5619449019235
x29=−93.9336203423348x_{29} = -93.9336203423348
x30=92.0486647501809x_{30} = 92.0486647501809
x31=−77.5973385436679x_{31} = -77.5973385436679
x32=98.3318500573605x_{32} = 98.3318500573605
x33=5.96902604182061x_{33} = 5.96902604182061
x34=22.3053078404875x_{34} = 22.3053078404875
x35=49.9513231920777x_{35} = 49.9513231920777
x36=38.0132711084365x_{36} = 38.0132711084365
x37=−67.5442420521806x_{37} = -67.5442420521806
x38=−11.6238928182822x_{38} = -11.6238928182822
x39=−80.1106126665397x_{39} = -80.1106126665397
x40=−41.7831822927443x_{40} = -41.7831822927443
x41=−1.5707963267949x_{41} = -1.5707963267949
x42=36.1283155162826x_{42} = 36.1283155162826
x43=−99.5884871187965x_{43} = -99.5884871187965
x44=−49.9513231920777x_{44} = -49.9513231920777
x45=44.2964564156161x_{45} = 44.2964564156161
x46=93.9336203423348x_{46} = 93.9336203423348
x47=10.3672557568463x_{47} = 10.3672557568463
x48=83.8805238508475x_{48} = 83.8805238508475
x49=60.0044196835651x_{49} = 60.0044196835651
x50=66.2876049907446x_{50} = 66.2876049907446
x51=−60.0044196835651x_{51} = -60.0044196835651
x52=−45.553093477052x_{52} = -45.553093477052
x53=−4.08407044966673x_{53} = -4.08407044966673
x54=−89.5353906273091x_{54} = -89.5353906273091
x55=−9.73893722612836x_{55} = -9.73893722612836
x56=−97.7035315266426x_{56} = -97.7035315266426
x57=26.0752190247953x_{57} = 26.0752190247953
x58=71.9424717672063x_{58} = 71.9424717672063
x59=48.0663675999238x_{59} = 48.0663675999238
x60=−33.6150413934108x_{60} = -33.6150413934108
x61=20.4203522483337x_{61} = 20.4203522483337
x62=−61.8893752757189x_{62} = -61.8893752757189
x63=17.9070781254618x_{63} = 17.9070781254618
x64=−21.6769893097696x_{64} = -21.6769893097696
x65=74.4557458900781x_{65} = 74.4557458900781
x66=52.4645973149496x_{66} = 52.4645973149496
x67=32.3584043319749x_{67} = 32.3584043319749
x68=−81.9955682586936x_{68} = -81.9955682586936
x69=−87.6504350351552x_{69} = -87.6504350351552
x70=−63.7743308678728x_{70} = -63.7743308678728
x71=−58.1194640914112x_{71} = -58.1194640914112
x72=34.2433599241287x_{72} = 34.2433599241287
x73=−31.7300858012569x_{73} = -31.7300858012569
x74=−92.0486647501809x_{74} = -92.0486647501809
x75=24.1902634326414x_{75} = 24.1902634326414
x76=80.1106126665397x_{76} = 80.1106126665397
x77=7.85398163397448x_{77} = 7.85398163397448
x78=−5.96902604182061x_{78} = -5.96902604182061
x79=68.1725605828985x_{79} = 68.1725605828985
x80=29.845130209103x_{80} = 29.845130209103
x81=56.2345084992573x_{81} = 56.2345084992573
x82=14.1371669411541x_{82} = 14.1371669411541
x83=−17.9070781254618x_{83} = -17.9070781254618
x84=−53.7212343763855x_{84} = -53.7212343763855
x85=−65.6592864600267x_{85} = -65.6592864600267
x86=−71.9424717672063x_{86} = -71.9424717672063
x87=61.8893752757189x_{87} = 61.8893752757189
x88=16.0221225333079x_{88} = 16.0221225333079
x89=78.2256570743859x_{89} = 78.2256570743859
x90=90.1637091580271x_{90} = 90.1637091580271
x91=−55.6061899685393x_{91} = -55.6061899685393
x92=64.4026493985908x_{92} = 64.4026493985908
x93=−73.8274273593601x_{93} = -73.8274273593601
x94=−43.6681378848981x_{94} = -43.6681378848981
x95=−51.8362787842316x_{95} = -51.8362787842316
x96=−7.85398163397448x_{96} = -7.85398163397448
x97=100.216805649514x_{97} = 100.216805649514
x98=81.9955682586936x_{98} = 81.9955682586936
x99=70.0575161750524x_{99} = 70.0575161750524
x100=−83.8805238508475x_{100} = -83.8805238508475
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(5*x).
cot⁡(0⋅5)\cot{\left(0 \cdot 5 \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
−5cot⁡2(5x)−5=0- 5 \cot^{2}{\left(5 x \right)} - 5 = 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
50(cot⁡2(5x)+1)cot⁡(5x)=050 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} = 0
Solve this equation
The roots of this equation
x1=π10x_{1} = \frac{\pi}{10}

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
(−∞,π10]\left(-\infty, \frac{\pi}{10}\right]
Convex at the intervals
[π10,∞)\left[\frac{\pi}{10}, \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⁡(5x)=−cot⁡(∞)\lim_{x \to -\infty} \cot{\left(5 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⁡(5x)=cot⁡(∞)\lim_{x \to \infty} \cot{\left(5 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(5*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⁡(5x)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(5 x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(5x)x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(5 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⁡(5x)=−cot⁡(5x)\cot{\left(5 x \right)} = - \cot{\left(5 x \right)}
- No
cot⁡(5x)=cot⁡(5x)\cot{\left(5 x \right)} = \cot{\left(5 x \right)}
- Yes
so, the function
is
odd
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