Mister Exam
Graphing y = ctg3x/5
The solution
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cot(3*x)
f(x) = --------
5
$$f{\left(x \right)} = \frac{\cot{\left(3 x \right)}}{5}$$
f = cot(3*x)/5
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:
$$\frac{\cot{\left(3 x \right)}}{5} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{6}$$
Numerical solution
$$x_{1} = -12.0427718387609$$
$$x_{2} = -23.5619449019235$$
$$x_{3} = 44.5058959258554$$
$$x_{4} = -29.845130209103$$
$$x_{5} = -21.4675497995303$$
$$x_{6} = 42.4115008234622$$
$$x_{7} = 46.6002910282486$$
$$x_{8} = 36.1283155162826$$
$$x_{9} = -34.0339204138894$$
$$x_{10} = 34.0339204138894$$
$$x_{11} = 2.61799387799149$$
$$x_{12} = -5.75958653158129$$
$$x_{13} = 66.497044500984$$
$$x_{14} = -41.3643032722656$$
$$x_{15} = 5.75958653158129$$
$$x_{16} = 9.94837673636768$$
$$x_{17} = -80.1106126665397$$
$$x_{18} = 86.3937979737193$$
$$x_{19} = 64.4026493985908$$
$$x_{20} = -95.8185759344887$$
$$x_{21} = 47.6474885794452$$
$$x_{22} = 90.5825881785057$$
$$x_{23} = 60.2138591938044$$
$$x_{24} = -1.5707963267949$$
$$x_{25} = -18.3259571459405$$
$$x_{26} = 71.733032256967$$
$$x_{27} = -60.2138591938044$$
$$x_{28} = -78.0162175641465$$
$$x_{29} = 53.9306738866248$$
$$x_{30} = -19.3731546971371$$
$$x_{31} = 73.8274273593601$$
$$x_{32} = 31.9395253114962$$
$$x_{33} = -40.317105721069$$
$$x_{34} = 12.0427718387609$$
$$x_{35} = 84.2994028713261$$
$$x_{36} = 78.0162175641465$$
$$x_{37} = -93.7241808320955$$
$$x_{38} = 22.5147473507269$$
$$x_{39} = -3.66519142918809$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = -65.4498469497874$$
$$x_{42} = -38.2227106186758$$
$$x_{43} = 100.007366139275$$
$$x_{44} = 97.9129710368819$$
$$x_{45} = 49.7418836818384$$
$$x_{46} = -56.025068989018$$
$$x_{47} = -62.3082542961976$$
$$x_{48} = -51.8362787842316$$
$$x_{49} = 58.1194640914112$$
$$x_{50} = -87.4409955249159$$
$$x_{51} = -73.8274273593601$$
$$x_{52} = 51.8362787842316$$
$$x_{53} = -85.3466004225227$$
$$x_{54} = 27.7507351067098$$
$$x_{55} = -45.553093477052$$
$$x_{56} = 29.845130209103$$
$$x_{57} = 75.9218224617533$$
$$x_{58} = 80.1106126665397$$
$$x_{59} = -58.1194640914112$$
$$x_{60} = -49.7418836818384$$
$$x_{61} = -36.1283155162826$$
$$x_{62} = -71.733032256967$$
$$x_{63} = 16.2315620435473$$
$$x_{64} = 24.60914245312$$
$$x_{65} = -100.007366139275$$
$$x_{66} = -31.9395253114962$$
$$x_{67} = -9.94837673636768$$
$$x_{68} = -97.9129710368819$$
$$x_{69} = 38.2227106186758$$
$$x_{70} = 69.6386371545737$$
$$x_{71} = 3.66519142918809$$
$$x_{72} = -67.5442420521806$$
$$x_{73} = -43.4586983746588$$
$$x_{74} = 62.3082542961976$$
$$x_{75} = -7.85398163397448$$
$$x_{76} = 82.2050077689329$$
$$x_{77} = 7.85398163397448$$
$$x_{78} = -14.1371669411541$$
$$x_{79} = 25.6563400043166$$
$$x_{80} = 18.3259571459405$$
$$x_{81} = -53.9306738866248$$
$$x_{82} = -75.9218224617533$$
$$x_{83} = 93.7241808320955$$
$$x_{84} = -91.6297857297023$$
$$x_{85} = 14.1371669411541$$
$$x_{86} = -16.2315620435473$$
$$x_{87} = -82.2050077689329$$
$$x_{88} = 40.317105721069$$
$$x_{89} = 20.4203522483337$$
$$x_{90} = -69.6386371545737$$
$$x_{91} = 95.8185759344887$$
$$x_{92} = -89.5353906273091$$
$$x_{93} = -47.6474885794452$$
$$x_{94} = 92.6769832808989$$
$$x_{95} = 68.5914396033772$$
$$x_{96} = 88.4881930761125$$
$$x_{97} = -25.6563400043166$$
$$x_{98} = -63.3554518473942$$
$$x_{99} = 56.025068989018$$
$$x_{100} = -84.2994028713261$$
so we need to solve the equation:
$$\frac{\cot{\left(3 x \right)}}{5} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{6}$$
Numerical solution
$$x_{1} = -12.0427718387609$$
$$x_{2} = -23.5619449019235$$
$$x_{3} = 44.5058959258554$$
$$x_{4} = -29.845130209103$$
$$x_{5} = -21.4675497995303$$
$$x_{6} = 42.4115008234622$$
$$x_{7} = 46.6002910282486$$
$$x_{8} = 36.1283155162826$$
$$x_{9} = -34.0339204138894$$
$$x_{10} = 34.0339204138894$$
$$x_{11} = 2.61799387799149$$
$$x_{12} = -5.75958653158129$$
$$x_{13} = 66.497044500984$$
$$x_{14} = -41.3643032722656$$
$$x_{15} = 5.75958653158129$$
$$x_{16} = 9.94837673636768$$
$$x_{17} = -80.1106126665397$$
$$x_{18} = 86.3937979737193$$
$$x_{19} = 64.4026493985908$$
$$x_{20} = -95.8185759344887$$
$$x_{21} = 47.6474885794452$$
$$x_{22} = 90.5825881785057$$
$$x_{23} = 60.2138591938044$$
$$x_{24} = -1.5707963267949$$
$$x_{25} = -18.3259571459405$$
$$x_{26} = 71.733032256967$$
$$x_{27} = -60.2138591938044$$
$$x_{28} = -78.0162175641465$$
$$x_{29} = 53.9306738866248$$
$$x_{30} = -19.3731546971371$$
$$x_{31} = 73.8274273593601$$
$$x_{32} = 31.9395253114962$$
$$x_{33} = -40.317105721069$$
$$x_{34} = 12.0427718387609$$
$$x_{35} = 84.2994028713261$$
$$x_{36} = 78.0162175641465$$
$$x_{37} = -93.7241808320955$$
$$x_{38} = 22.5147473507269$$
$$x_{39} = -3.66519142918809$$
$$x_{40} = -27.7507351067098$$
$$x_{41} = -65.4498469497874$$
$$x_{42} = -38.2227106186758$$
$$x_{43} = 100.007366139275$$
$$x_{44} = 97.9129710368819$$
$$x_{45} = 49.7418836818384$$
$$x_{46} = -56.025068989018$$
$$x_{47} = -62.3082542961976$$
$$x_{48} = -51.8362787842316$$
$$x_{49} = 58.1194640914112$$
$$x_{50} = -87.4409955249159$$
$$x_{51} = -73.8274273593601$$
$$x_{52} = 51.8362787842316$$
$$x_{53} = -85.3466004225227$$
$$x_{54} = 27.7507351067098$$
$$x_{55} = -45.553093477052$$
$$x_{56} = 29.845130209103$$
$$x_{57} = 75.9218224617533$$
$$x_{58} = 80.1106126665397$$
$$x_{59} = -58.1194640914112$$
$$x_{60} = -49.7418836818384$$
$$x_{61} = -36.1283155162826$$
$$x_{62} = -71.733032256967$$
$$x_{63} = 16.2315620435473$$
$$x_{64} = 24.60914245312$$
$$x_{65} = -100.007366139275$$
$$x_{66} = -31.9395253114962$$
$$x_{67} = -9.94837673636768$$
$$x_{68} = -97.9129710368819$$
$$x_{69} = 38.2227106186758$$
$$x_{70} = 69.6386371545737$$
$$x_{71} = 3.66519142918809$$
$$x_{72} = -67.5442420521806$$
$$x_{73} = -43.4586983746588$$
$$x_{74} = 62.3082542961976$$
$$x_{75} = -7.85398163397448$$
$$x_{76} = 82.2050077689329$$
$$x_{77} = 7.85398163397448$$
$$x_{78} = -14.1371669411541$$
$$x_{79} = 25.6563400043166$$
$$x_{80} = 18.3259571459405$$
$$x_{81} = -53.9306738866248$$
$$x_{82} = -75.9218224617533$$
$$x_{83} = 93.7241808320955$$
$$x_{84} = -91.6297857297023$$
$$x_{85} = 14.1371669411541$$
$$x_{86} = -16.2315620435473$$
$$x_{87} = -82.2050077689329$$
$$x_{88} = 40.317105721069$$
$$x_{89} = 20.4203522483337$$
$$x_{90} = -69.6386371545737$$
$$x_{91} = 95.8185759344887$$
$$x_{92} = -89.5353906273091$$
$$x_{93} = -47.6474885794452$$
$$x_{94} = 92.6769832808989$$
$$x_{95} = 68.5914396033772$$
$$x_{96} = 88.4881930761125$$
$$x_{97} = -25.6563400043166$$
$$x_{98} = -63.3554518473942$$
$$x_{99} = 56.025068989018$$
$$x_{100} = -84.2994028713261$$
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
$$- \frac{3 \cot^{2}{\left(3 x \right)}}{5} - \frac{3}{5} = 0$$
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
$$- \frac{3 \cot^{2}{\left(3 x \right)}}{5} - \frac{3}{5} = 0$$
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
$$\frac{18 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cot{\left(3 x \right)}}{5} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{6}$$
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
$$\left(-\infty, \frac{\pi}{6}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{6}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{18 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cot{\left(3 x \right)}}{5} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{6}$$
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
$$\left(-\infty, \frac{\pi}{6}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{6}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\cot{\left(3 x \right)}}{5}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\cot{\left(3 x \right)}}{5}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\cot{\left(3 x \right)}}{5}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\cot{\left(3 x \right)}}{5}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(3*x)/5, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(3 x \right)}}{5 x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(3 x \right)}}{5 x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(3 x \right)}}{5 x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(3 x \right)}}{5 x}\right)$$
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:
$$\frac{\cot{\left(3 x \right)}}{5} = - \frac{\cot{\left(3 x \right)}}{5}$$
- No
$$\frac{\cot{\left(3 x \right)}}{5} = \frac{\cot{\left(3 x \right)}}{5}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\cot{\left(3 x \right)}}{5} = - \frac{\cot{\left(3 x \right)}}{5}$$
- No
$$\frac{\cot{\left(3 x \right)}}{5} = \frac{\cot{\left(3 x \right)}}{5}$$
- No
so, the function
not is
neither even, nor odd