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cot(5*x)
cot(5*x)
Graphing y = cot(5*x)
The solution
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f(x) = cot(5*x)
$$f{\left(x \right)} = \cot{\left(5 x \right)}$$
f = cot(5*x)
The graph of the function
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:
$$\cot{\left(5 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{10}$$
Numerical solution
$$x_{1} = 88.2787535658732$$
$$x_{2} = 4.08407044966673$$
$$x_{3} = -85.7654794430014$$
$$x_{4} = -48.0663675999238$$
$$x_{5} = 42.4115008234622$$
$$x_{6} = -27.9601746169492$$
$$x_{7} = -29.845130209103$$
$$x_{8} = -26.0752190247953$$
$$x_{9} = -75.712382951514$$
$$x_{10} = 39.8982267005904$$
$$x_{11} = -36.1283155162826$$
$$x_{12} = 86.3937979737193$$
$$x_{13} = 76.340701482232$$
$$x_{14} = -19.7920337176157$$
$$x_{15} = -16.0221225333079$$
$$x_{16} = 27.9601746169492$$
$$x_{17} = 54.3495529071034$$
$$x_{18} = -95.8185759344887$$
$$x_{19} = -38.0132711084365$$
$$x_{20} = -14.1371669411541$$
$$x_{21} = 46.18141200777$$
$$x_{22} = 96.4468944652067$$
$$x_{23} = 58.1194640914112$$
$$x_{24} = -39.8982267005904$$
$$x_{25} = -70.0575161750524$$
$$x_{26} = 12.2522113490002$$
$$x_{27} = 2.19911485751286$$
$$x_{28} = -23.5619449019235$$
$$x_{29} = -93.9336203423348$$
$$x_{30} = 92.0486647501809$$
$$x_{31} = -77.5973385436679$$
$$x_{32} = 98.3318500573605$$
$$x_{33} = 5.96902604182061$$
$$x_{34} = 22.3053078404875$$
$$x_{35} = 49.9513231920777$$
$$x_{36} = 38.0132711084365$$
$$x_{37} = -67.5442420521806$$
$$x_{38} = -11.6238928182822$$
$$x_{39} = -80.1106126665397$$
$$x_{40} = -41.7831822927443$$
$$x_{41} = -1.5707963267949$$
$$x_{42} = 36.1283155162826$$
$$x_{43} = -99.5884871187965$$
$$x_{44} = -49.9513231920777$$
$$x_{45} = 44.2964564156161$$
$$x_{46} = 93.9336203423348$$
$$x_{47} = 10.3672557568463$$
$$x_{48} = 83.8805238508475$$
$$x_{49} = 60.0044196835651$$
$$x_{50} = 66.2876049907446$$
$$x_{51} = -60.0044196835651$$
$$x_{52} = -45.553093477052$$
$$x_{53} = -4.08407044966673$$
$$x_{54} = -89.5353906273091$$
$$x_{55} = -9.73893722612836$$
$$x_{56} = -97.7035315266426$$
$$x_{57} = 26.0752190247953$$
$$x_{58} = 71.9424717672063$$
$$x_{59} = 48.0663675999238$$
$$x_{60} = -33.6150413934108$$
$$x_{61} = 20.4203522483337$$
$$x_{62} = -61.8893752757189$$
$$x_{63} = 17.9070781254618$$
$$x_{64} = -21.6769893097696$$
$$x_{65} = 74.4557458900781$$
$$x_{66} = 52.4645973149496$$
$$x_{67} = 32.3584043319749$$
$$x_{68} = -81.9955682586936$$
$$x_{69} = -87.6504350351552$$
$$x_{70} = -63.7743308678728$$
$$x_{71} = -58.1194640914112$$
$$x_{72} = 34.2433599241287$$
$$x_{73} = -31.7300858012569$$
$$x_{74} = -92.0486647501809$$
$$x_{75} = 24.1902634326414$$
$$x_{76} = 80.1106126665397$$
$$x_{77} = 7.85398163397448$$
$$x_{78} = -5.96902604182061$$
$$x_{79} = 68.1725605828985$$
$$x_{80} = 29.845130209103$$
$$x_{81} = 56.2345084992573$$
$$x_{82} = 14.1371669411541$$
$$x_{83} = -17.9070781254618$$
$$x_{84} = -53.7212343763855$$
$$x_{85} = -65.6592864600267$$
$$x_{86} = -71.9424717672063$$
$$x_{87} = 61.8893752757189$$
$$x_{88} = 16.0221225333079$$
$$x_{89} = 78.2256570743859$$
$$x_{90} = 90.1637091580271$$
$$x_{91} = -55.6061899685393$$
$$x_{92} = 64.4026493985908$$
$$x_{93} = -73.8274273593601$$
$$x_{94} = -43.6681378848981$$
$$x_{95} = -51.8362787842316$$
$$x_{96} = -7.85398163397448$$
$$x_{97} = 100.216805649514$$
$$x_{98} = 81.9955682586936$$
$$x_{99} = 70.0575161750524$$
$$x_{100} = -83.8805238508475$$
so we need to solve the equation:
$$\cot{\left(5 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{10}$$
Numerical solution
$$x_{1} = 88.2787535658732$$
$$x_{2} = 4.08407044966673$$
$$x_{3} = -85.7654794430014$$
$$x_{4} = -48.0663675999238$$
$$x_{5} = 42.4115008234622$$
$$x_{6} = -27.9601746169492$$
$$x_{7} = -29.845130209103$$
$$x_{8} = -26.0752190247953$$
$$x_{9} = -75.712382951514$$
$$x_{10} = 39.8982267005904$$
$$x_{11} = -36.1283155162826$$
$$x_{12} = 86.3937979737193$$
$$x_{13} = 76.340701482232$$
$$x_{14} = -19.7920337176157$$
$$x_{15} = -16.0221225333079$$
$$x_{16} = 27.9601746169492$$
$$x_{17} = 54.3495529071034$$
$$x_{18} = -95.8185759344887$$
$$x_{19} = -38.0132711084365$$
$$x_{20} = -14.1371669411541$$
$$x_{21} = 46.18141200777$$
$$x_{22} = 96.4468944652067$$
$$x_{23} = 58.1194640914112$$
$$x_{24} = -39.8982267005904$$
$$x_{25} = -70.0575161750524$$
$$x_{26} = 12.2522113490002$$
$$x_{27} = 2.19911485751286$$
$$x_{28} = -23.5619449019235$$
$$x_{29} = -93.9336203423348$$
$$x_{30} = 92.0486647501809$$
$$x_{31} = -77.5973385436679$$
$$x_{32} = 98.3318500573605$$
$$x_{33} = 5.96902604182061$$
$$x_{34} = 22.3053078404875$$
$$x_{35} = 49.9513231920777$$
$$x_{36} = 38.0132711084365$$
$$x_{37} = -67.5442420521806$$
$$x_{38} = -11.6238928182822$$
$$x_{39} = -80.1106126665397$$
$$x_{40} = -41.7831822927443$$
$$x_{41} = -1.5707963267949$$
$$x_{42} = 36.1283155162826$$
$$x_{43} = -99.5884871187965$$
$$x_{44} = -49.9513231920777$$
$$x_{45} = 44.2964564156161$$
$$x_{46} = 93.9336203423348$$
$$x_{47} = 10.3672557568463$$
$$x_{48} = 83.8805238508475$$
$$x_{49} = 60.0044196835651$$
$$x_{50} = 66.2876049907446$$
$$x_{51} = -60.0044196835651$$
$$x_{52} = -45.553093477052$$
$$x_{53} = -4.08407044966673$$
$$x_{54} = -89.5353906273091$$
$$x_{55} = -9.73893722612836$$
$$x_{56} = -97.7035315266426$$
$$x_{57} = 26.0752190247953$$
$$x_{58} = 71.9424717672063$$
$$x_{59} = 48.0663675999238$$
$$x_{60} = -33.6150413934108$$
$$x_{61} = 20.4203522483337$$
$$x_{62} = -61.8893752757189$$
$$x_{63} = 17.9070781254618$$
$$x_{64} = -21.6769893097696$$
$$x_{65} = 74.4557458900781$$
$$x_{66} = 52.4645973149496$$
$$x_{67} = 32.3584043319749$$
$$x_{68} = -81.9955682586936$$
$$x_{69} = -87.6504350351552$$
$$x_{70} = -63.7743308678728$$
$$x_{71} = -58.1194640914112$$
$$x_{72} = 34.2433599241287$$
$$x_{73} = -31.7300858012569$$
$$x_{74} = -92.0486647501809$$
$$x_{75} = 24.1902634326414$$
$$x_{76} = 80.1106126665397$$
$$x_{77} = 7.85398163397448$$
$$x_{78} = -5.96902604182061$$
$$x_{79} = 68.1725605828985$$
$$x_{80} = 29.845130209103$$
$$x_{81} = 56.2345084992573$$
$$x_{82} = 14.1371669411541$$
$$x_{83} = -17.9070781254618$$
$$x_{84} = -53.7212343763855$$
$$x_{85} = -65.6592864600267$$
$$x_{86} = -71.9424717672063$$
$$x_{87} = 61.8893752757189$$
$$x_{88} = 16.0221225333079$$
$$x_{89} = 78.2256570743859$$
$$x_{90} = 90.1637091580271$$
$$x_{91} = -55.6061899685393$$
$$x_{92} = 64.4026493985908$$
$$x_{93} = -73.8274273593601$$
$$x_{94} = -43.6681378848981$$
$$x_{95} = -51.8362787842316$$
$$x_{96} = -7.85398163397448$$
$$x_{97} = 100.216805649514$$
$$x_{98} = 81.9955682586936$$
$$x_{99} = 70.0575161750524$$
$$x_{100} = -83.8805238508475$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- 5 \cot^{2}{\left(5 x \right)} - 5 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- 5 \cot^{2}{\left(5 x \right)} - 5 = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$50 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{10}$$
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
$$\left(-\infty, \frac{\pi}{10}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{10}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$50 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{10}$$
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
$$\left(-\infty, \frac{\pi}{10}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{10}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \cot{\left(5 x \right)} = - \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \cot{\left(\infty \right)}$$
$$\lim_{x \to \infty} \cot{\left(5 x \right)} = \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \cot{\left(\infty \right)}$$
$$\lim_{x \to -\infty} \cot{\left(5 x \right)} = - \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \cot{\left(\infty \right)}$$
$$\lim_{x \to \infty} \cot{\left(5 x \right)} = \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \cot{\left(\infty \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(5*x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(5 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(5 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(5 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(5 x \right)}}{x}\right)$$
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:
$$\cot{\left(5 x \right)} = - \cot{\left(5 x \right)}$$
- No
$$\cot{\left(5 x \right)} = \cot{\left(5 x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\cot{\left(5 x \right)} = - \cot{\left(5 x \right)}$$
- No
$$\cot{\left(5 x \right)} = \cot{\left(5 x \right)}$$
- Yes
so, the function
is
odd