Mister Exam
Graphing y = cot(x-pi/6)
The solution
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/ pi\
f(x) = cot|x - --|
\ 6 /
$$f{\left(x \right)} = \cot{\left(x - \frac{\pi}{6} \right)}$$
f = cot(x - pi/6)
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:
$$\cot{\left(x - \frac{\pi}{6} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{3}$$
Numerical solution
$$x_{1} = -13.6135681655558$$
$$x_{2} = -73.3038285837618$$
$$x_{3} = -98.4365698124802$$
$$x_{4} = -10.471975511966$$
$$x_{5} = 11.5191730631626$$
$$x_{6} = 83.7758040957278$$
$$x_{7} = 58.6430628670095$$
$$x_{8} = -82.7286065445312$$
$$x_{9} = -79.5870138909414$$
$$x_{10} = -89.0117918517108$$
$$x_{11} = 77.4926187885482$$
$$x_{12} = -26.1799387799149$$
$$x_{13} = 8.37758040957278$$
$$x_{14} = -85.870199198121$$
$$x_{15} = -4.18879020478639$$
$$x_{16} = -19.8967534727354$$
$$x_{17} = 52.3598775598299$$
$$x_{18} = 27.2271363311115$$
$$x_{19} = -45.0294947014537$$
$$x_{20} = -95.2949771588904$$
$$x_{21} = -67.0206432765823$$
$$x_{22} = 20.943951023932$$
$$x_{23} = -16.7551608191456$$
$$x_{24} = 5.23598775598299$$
$$x_{25} = -41.8879020478639$$
$$x_{26} = -60.7374579694027$$
$$x_{27} = 99.4837673636768$$
$$x_{28} = 74.3510261349584$$
$$x_{29} = 42.9350995990605$$
$$x_{30} = -48.1710873550435$$
$$x_{31} = 46.0766922526503$$
$$x_{32} = -23.0383461263252$$
$$x_{33} = 64.9262481741891$$
$$x_{34} = -35.6047167406843$$
$$x_{35} = -29.3215314335047$$
$$x_{36} = 36.6519142918809$$
$$x_{37} = 24.0855436775217$$
$$x_{38} = 90.0589894029074$$
$$x_{39} = 93.2005820564972$$
$$x_{40} = -1.0471975511966$$
$$x_{41} = 96.342174710087$$
$$x_{42} = 49.2182849062401$$
$$x_{43} = 2.0943951023932$$
$$x_{44} = 80.634211442138$$
$$x_{45} = 55.5014702134197$$
$$x_{46} = -76.4454212373516$$
$$x_{47} = -32.4631240870945$$
$$x_{48} = 33.5103216382911$$
$$x_{49} = 30.3687289847013$$
$$x_{50} = -63.8790506229925$$
$$x_{51} = 39.7935069454707$$
$$x_{52} = 86.9173967493176$$
$$x_{53} = -70.162235930172$$
$$x_{54} = 17.8023583703422$$
$$x_{55} = 14.6607657167524$$
$$x_{56} = -57.5958653158129$$
$$x_{57} = 61.7846555205993$$
$$x_{58} = -54.4542726622231$$
$$x_{59} = -51.3126800086333$$
$$x_{60} = -92.1533845053006$$
$$x_{61} = 71.2094334813686$$
$$x_{62} = 68.0678408277789$$
$$x_{63} = -7.33038285837618$$
$$x_{64} = -38.7463093942741$$
so we need to solve the equation:
$$\cot{\left(x - \frac{\pi}{6} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{3}$$
Numerical solution
$$x_{1} = -13.6135681655558$$
$$x_{2} = -73.3038285837618$$
$$x_{3} = -98.4365698124802$$
$$x_{4} = -10.471975511966$$
$$x_{5} = 11.5191730631626$$
$$x_{6} = 83.7758040957278$$
$$x_{7} = 58.6430628670095$$
$$x_{8} = -82.7286065445312$$
$$x_{9} = -79.5870138909414$$
$$x_{10} = -89.0117918517108$$
$$x_{11} = 77.4926187885482$$
$$x_{12} = -26.1799387799149$$
$$x_{13} = 8.37758040957278$$
$$x_{14} = -85.870199198121$$
$$x_{15} = -4.18879020478639$$
$$x_{16} = -19.8967534727354$$
$$x_{17} = 52.3598775598299$$
$$x_{18} = 27.2271363311115$$
$$x_{19} = -45.0294947014537$$
$$x_{20} = -95.2949771588904$$
$$x_{21} = -67.0206432765823$$
$$x_{22} = 20.943951023932$$
$$x_{23} = -16.7551608191456$$
$$x_{24} = 5.23598775598299$$
$$x_{25} = -41.8879020478639$$
$$x_{26} = -60.7374579694027$$
$$x_{27} = 99.4837673636768$$
$$x_{28} = 74.3510261349584$$
$$x_{29} = 42.9350995990605$$
$$x_{30} = -48.1710873550435$$
$$x_{31} = 46.0766922526503$$
$$x_{32} = -23.0383461263252$$
$$x_{33} = 64.9262481741891$$
$$x_{34} = -35.6047167406843$$
$$x_{35} = -29.3215314335047$$
$$x_{36} = 36.6519142918809$$
$$x_{37} = 24.0855436775217$$
$$x_{38} = 90.0589894029074$$
$$x_{39} = 93.2005820564972$$
$$x_{40} = -1.0471975511966$$
$$x_{41} = 96.342174710087$$
$$x_{42} = 49.2182849062401$$
$$x_{43} = 2.0943951023932$$
$$x_{44} = 80.634211442138$$
$$x_{45} = 55.5014702134197$$
$$x_{46} = -76.4454212373516$$
$$x_{47} = -32.4631240870945$$
$$x_{48} = 33.5103216382911$$
$$x_{49} = 30.3687289847013$$
$$x_{50} = -63.8790506229925$$
$$x_{51} = 39.7935069454707$$
$$x_{52} = 86.9173967493176$$
$$x_{53} = -70.162235930172$$
$$x_{54} = 17.8023583703422$$
$$x_{55} = 14.6607657167524$$
$$x_{56} = -57.5958653158129$$
$$x_{57} = 61.7846555205993$$
$$x_{58} = -54.4542726622231$$
$$x_{59} = -51.3126800086333$$
$$x_{60} = -92.1533845053006$$
$$x_{61} = 71.2094334813686$$
$$x_{62} = 68.0678408277789$$
$$x_{63} = -7.33038285837618$$
$$x_{64} = -38.7463093942741$$
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
$$- \cot^{2}{\left(x - \frac{\pi}{6} \right)} - 1 = 0$$
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
$$- \cot^{2}{\left(x - \frac{\pi}{6} \right)} - 1 = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 2 \left(\tan^{2}{\left(x + \frac{\pi}{3} \right)} + 1\right) \tan{\left(x + \frac{\pi}{3} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{3}$$
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
$$\left(-\infty, - \frac{\pi}{3}\right]$$
Convex at the intervals
$$\left[- \frac{\pi}{3}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 2 \left(\tan^{2}{\left(x + \frac{\pi}{3} \right)} + 1\right) \tan{\left(x + \frac{\pi}{3} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{3}$$
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
$$\left(-\infty, - \frac{\pi}{3}\right]$$
Convex at the intervals
$$\left[- \frac{\pi}{3}, \infty\right)$$
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:
$$y = \lim_{x \to -\infty} \cot{\left(x - \frac{\pi}{6} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \cot{\left(x - \frac{\pi}{6} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \cot{\left(x - \frac{\pi}{6} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \cot{\left(x - \frac{\pi}{6} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(x - pi/6), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(x - \frac{\pi}{6} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(x - \frac{\pi}{6} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(x - \frac{\pi}{6} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(x - \frac{\pi}{6} \right)}}{x}\right)$$
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:
$$\cot{\left(x - \frac{\pi}{6} \right)} = - \cot{\left(x + \frac{\pi}{6} \right)}$$
- No
$$\cot{\left(x - \frac{\pi}{6} \right)} = \cot{\left(x + \frac{\pi}{6} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\cot{\left(x - \frac{\pi}{6} \right)} = - \cot{\left(x + \frac{\pi}{6} \right)}$$
- No
$$\cot{\left(x - \frac{\pi}{6} \right)} = \cot{\left(x + \frac{\pi}{6} \right)}$$
- No
so, the function
not is
neither even, nor odd