Mister Exam
Graphing y = tan(x-pi/6)
The solution
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/ pi\
f(x) = tan|x - --|
\ 6 /
$$f{\left(x \right)} = \tan{\left(x - \frac{\pi}{6} \right)}$$
f = tan(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:
$$\tan{\left(x - \frac{\pi}{6} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{6}$$
Numerical solution
$$x_{1} = -52.8834763354282$$
$$x_{2} = -90.5825881785057$$
$$x_{3} = -21.4675497995303$$
$$x_{4} = -81.1578102177363$$
$$x_{5} = 69.6386371545737$$
$$x_{6} = 25.6563400043166$$
$$x_{7} = -12.0427718387609$$
$$x_{8} = 22.5147473507269$$
$$x_{9} = 82.2050077689329$$
$$x_{10} = -96.8657734856853$$
$$x_{11} = 101.054563690472$$
$$x_{12} = -24.60914245312$$
$$x_{13} = 41.3643032722656$$
$$x_{14} = 28.7979326579064$$
$$x_{15} = -18.3259571459405$$
$$x_{16} = -43.4586983746588$$
$$x_{17} = -27.7507351067098$$
$$x_{18} = 6.80678408277789$$
$$x_{19} = 60.2138591938044$$
$$x_{20} = -2.61799387799149$$
$$x_{21} = 63.3554518473942$$
$$x_{22} = 0.523598775598299$$
$$x_{23} = -56.025068989018$$
$$x_{24} = 3.66519142918809$$
$$x_{25} = -84.2994028713261$$
$$x_{26} = -71.733032256967$$
$$x_{27} = 38.2227106186758$$
$$x_{28} = 31.9395253114962$$
$$x_{29} = -37.1755130674792$$
$$x_{30} = -30.8923277602996$$
$$x_{31} = 53.9306738866248$$
$$x_{32} = 85.3466004225227$$
$$x_{33} = 50.789081233035$$
$$x_{34} = 88.4881930761125$$
$$x_{35} = 13.0899693899575$$
$$x_{36} = -62.3082542961976$$
$$x_{37} = 66.497044500984$$
$$x_{38} = -74.8746249105567$$
$$x_{39} = 47.6474885794452$$
$$x_{40} = -34.0339204138894$$
$$x_{41} = 44.5058959258554$$
$$x_{42} = 91.6297857297023$$
$$x_{43} = -78.0162175641465$$
$$x_{44} = 94.7713783832921$$
$$x_{45} = -59.1666616426078$$
$$x_{46} = 79.0634151153431$$
$$x_{47} = 16.2315620435473$$
$$x_{48} = -8.90117918517108$$
$$x_{49} = 75.9218224617533$$
$$x_{50} = -5.75958653158129$$
$$x_{51} = 97.9129710368819$$
$$x_{52} = -46.6002910282486$$
$$x_{53} = -100.007366139275$$
$$x_{54} = -15.1843644923507$$
$$x_{55} = 9.94837673636768$$
$$x_{56} = -49.7418836818384$$
$$x_{57} = 72.7802298081635$$
$$x_{58} = 19.3731546971371$$
$$x_{59} = -87.4409955249159$$
$$x_{60} = 57.0722665402146$$
$$x_{61} = -93.7241808320955$$
$$x_{62} = -65.4498469497874$$
$$x_{63} = 35.081117965086$$
$$x_{64} = -40.317105721069$$
$$x_{65} = -68.5914396033772$$
so we need to solve the equation:
$$\tan{\left(x - \frac{\pi}{6} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{6}$$
Numerical solution
$$x_{1} = -52.8834763354282$$
$$x_{2} = -90.5825881785057$$
$$x_{3} = -21.4675497995303$$
$$x_{4} = -81.1578102177363$$
$$x_{5} = 69.6386371545737$$
$$x_{6} = 25.6563400043166$$
$$x_{7} = -12.0427718387609$$
$$x_{8} = 22.5147473507269$$
$$x_{9} = 82.2050077689329$$
$$x_{10} = -96.8657734856853$$
$$x_{11} = 101.054563690472$$
$$x_{12} = -24.60914245312$$
$$x_{13} = 41.3643032722656$$
$$x_{14} = 28.7979326579064$$
$$x_{15} = -18.3259571459405$$
$$x_{16} = -43.4586983746588$$
$$x_{17} = -27.7507351067098$$
$$x_{18} = 6.80678408277789$$
$$x_{19} = 60.2138591938044$$
$$x_{20} = -2.61799387799149$$
$$x_{21} = 63.3554518473942$$
$$x_{22} = 0.523598775598299$$
$$x_{23} = -56.025068989018$$
$$x_{24} = 3.66519142918809$$
$$x_{25} = -84.2994028713261$$
$$x_{26} = -71.733032256967$$
$$x_{27} = 38.2227106186758$$
$$x_{28} = 31.9395253114962$$
$$x_{29} = -37.1755130674792$$
$$x_{30} = -30.8923277602996$$
$$x_{31} = 53.9306738866248$$
$$x_{32} = 85.3466004225227$$
$$x_{33} = 50.789081233035$$
$$x_{34} = 88.4881930761125$$
$$x_{35} = 13.0899693899575$$
$$x_{36} = -62.3082542961976$$
$$x_{37} = 66.497044500984$$
$$x_{38} = -74.8746249105567$$
$$x_{39} = 47.6474885794452$$
$$x_{40} = -34.0339204138894$$
$$x_{41} = 44.5058959258554$$
$$x_{42} = 91.6297857297023$$
$$x_{43} = -78.0162175641465$$
$$x_{44} = 94.7713783832921$$
$$x_{45} = -59.1666616426078$$
$$x_{46} = 79.0634151153431$$
$$x_{47} = 16.2315620435473$$
$$x_{48} = -8.90117918517108$$
$$x_{49} = 75.9218224617533$$
$$x_{50} = -5.75958653158129$$
$$x_{51} = 97.9129710368819$$
$$x_{52} = -46.6002910282486$$
$$x_{53} = -100.007366139275$$
$$x_{54} = -15.1843644923507$$
$$x_{55} = 9.94837673636768$$
$$x_{56} = -49.7418836818384$$
$$x_{57} = 72.7802298081635$$
$$x_{58} = 19.3731546971371$$
$$x_{59} = -87.4409955249159$$
$$x_{60} = 57.0722665402146$$
$$x_{61} = -93.7241808320955$$
$$x_{62} = -65.4498469497874$$
$$x_{63} = 35.081117965086$$
$$x_{64} = -40.317105721069$$
$$x_{65} = -68.5914396033772$$
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
$$\tan^{2}{\left(x - \frac{\pi}{6} \right)} + 1 = 0$$
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
$$\tan^{2}{\left(x - \frac{\pi}{6} \right)} + 1 = 0$$
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
$$- 2 \left(\cot^{2}{\left(x + \frac{\pi}{3} \right)} + 1\right) \cot{\left(x + \frac{\pi}{3} \right)} = 0$$
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
$$\left[\frac{\pi}{6}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{\pi}{6}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 2 \left(\cot^{2}{\left(x + \frac{\pi}{3} \right)} + 1\right) \cot{\left(x + \frac{\pi}{3} \right)} = 0$$
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
$$\left[\frac{\pi}{6}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{\pi}{6}\right]$$
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:
$$y = \lim_{x \to -\infty} \tan{\left(x - \frac{\pi}{6} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \tan{\left(x - \frac{\pi}{6} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \tan{\left(x - \frac{\pi}{6} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \tan{\left(x - \frac{\pi}{6} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x - pi/6), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x - \frac{\pi}{6} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x - \frac{\pi}{6} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x - \frac{\pi}{6} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x - \frac{\pi}{6} \right)}}{x}\right)$$
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:
$$\tan{\left(x - \frac{\pi}{6} \right)} = - \tan{\left(x + \frac{\pi}{6} \right)}$$
- No
$$\tan{\left(x - \frac{\pi}{6} \right)} = \tan{\left(x + \frac{\pi}{6} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(x - \frac{\pi}{6} \right)} = - \tan{\left(x + \frac{\pi}{6} \right)}$$
- No
$$\tan{\left(x - \frac{\pi}{6} \right)} = \tan{\left(x + \frac{\pi}{6} \right)}$$
- No
so, the function
not is
neither even, nor odd