Mister Exam
Graphing y = abs(tan(x-pi/3)+1)
The solution
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| / pi\ |
f(x) = |tan|x - --| + 1|
| \ 3 / |
$$f{\left(x \right)} = \left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right|$$
f = Abs(tan(x - pi/3) + 1)
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:
$$\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{12}$$
Numerical solution
$$x_{1} = 100.792764302673$$
$$x_{2} = 25.3945406165175$$
$$x_{3} = 53.6688744988256$$
$$x_{4} = -50.0036830696375$$
$$x_{5} = -21.7293491873294$$
$$x_{6} = -84.5612022591253$$
$$x_{7} = -37.4373124552784$$
$$x_{8} = 97.6511716490827$$
$$x_{9} = 9.68657734856853$$
$$x_{10} = 41.1025038844665$$
$$x_{11} = -100.269165527074$$
$$x_{12} = 88.2263936883134$$
$$x_{13} = 3.40339204138894$$
$$x_{14} = 44.2440965380563$$
$$x_{15} = 94.5095789954929$$
$$x_{16} = -59.4284610304069$$
$$x_{17} = -46.8620904160477$$
$$x_{18} = -75.1364242983559$$
$$x_{19} = -12.30457122656$$
$$x_{20} = 12.8281700021583$$
$$x_{21} = 0.261799387799149$$
$$x_{22} = -2.87979326579064$$
$$x_{23} = -31.1541271480988$$
$$x_{24} = -87.7027949127151$$
$$x_{25} = -81.4196096055355$$
$$x_{26} = -97.1275728734844$$
$$x_{27} = -18.5877565337396$$
$$x_{28} = -78.2780169519457$$
$$x_{29} = -68.8532389911763$$
$$x_{30} = -6.02138591938044$$
$$x_{31} = -90.8443875663049$$
$$x_{32} = 50.5272818452358$$
$$x_{33} = 22.2529479629277$$
$$x_{34} = 31.6777259236971$$
$$x_{35} = 63.093652459595$$
$$x_{36} = 19.1113553093379$$
$$x_{37} = 28.5361332701073$$
$$x_{38} = -65.7116463375865$$
$$x_{39} = 34.8193185772869$$
$$x_{40} = -93.9859802198946$$
$$x_{41} = 72.5184304203644$$
$$x_{42} = 91.3679863419031$$
$$x_{43} = 81.9432083811338$$
$$x_{44} = -43.720497762458$$
$$x_{45} = -56.2868683768171$$
$$x_{46} = 69.3768377667746$$
$$x_{47} = -71.9948316447661$$
$$x_{48} = 47.3856891916461$$
$$x_{49} = 6.54498469497874$$
$$x_{50} = 37.9609112308767$$
$$x_{51} = 85.0848010347236$$
$$x_{52} = 15.9697626557481$$
$$x_{53} = -28.012534494509$$
$$x_{54} = 59.9520598060052$$
$$x_{55} = 56.8104671524154$$
$$x_{56} = -40.5789051088682$$
$$x_{57} = 75.6600230739542$$
$$x_{58} = -24.8709418409192$$
$$x_{59} = -34.2957198016886$$
$$x_{60} = -53.1452757232273$$
$$x_{61} = 78.801615727544$$
$$x_{62} = -9.16297857297023$$
$$x_{63} = 66.2352451131848$$
$$x_{64} = -62.5700536839967$$
$$x_{65} = -15.4461638801498$$
so we need to solve the equation:
$$\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{12}$$
Numerical solution
$$x_{1} = 100.792764302673$$
$$x_{2} = 25.3945406165175$$
$$x_{3} = 53.6688744988256$$
$$x_{4} = -50.0036830696375$$
$$x_{5} = -21.7293491873294$$
$$x_{6} = -84.5612022591253$$
$$x_{7} = -37.4373124552784$$
$$x_{8} = 97.6511716490827$$
$$x_{9} = 9.68657734856853$$
$$x_{10} = 41.1025038844665$$
$$x_{11} = -100.269165527074$$
$$x_{12} = 88.2263936883134$$
$$x_{13} = 3.40339204138894$$
$$x_{14} = 44.2440965380563$$
$$x_{15} = 94.5095789954929$$
$$x_{16} = -59.4284610304069$$
$$x_{17} = -46.8620904160477$$
$$x_{18} = -75.1364242983559$$
$$x_{19} = -12.30457122656$$
$$x_{20} = 12.8281700021583$$
$$x_{21} = 0.261799387799149$$
$$x_{22} = -2.87979326579064$$
$$x_{23} = -31.1541271480988$$
$$x_{24} = -87.7027949127151$$
$$x_{25} = -81.4196096055355$$
$$x_{26} = -97.1275728734844$$
$$x_{27} = -18.5877565337396$$
$$x_{28} = -78.2780169519457$$
$$x_{29} = -68.8532389911763$$
$$x_{30} = -6.02138591938044$$
$$x_{31} = -90.8443875663049$$
$$x_{32} = 50.5272818452358$$
$$x_{33} = 22.2529479629277$$
$$x_{34} = 31.6777259236971$$
$$x_{35} = 63.093652459595$$
$$x_{36} = 19.1113553093379$$
$$x_{37} = 28.5361332701073$$
$$x_{38} = -65.7116463375865$$
$$x_{39} = 34.8193185772869$$
$$x_{40} = -93.9859802198946$$
$$x_{41} = 72.5184304203644$$
$$x_{42} = 91.3679863419031$$
$$x_{43} = 81.9432083811338$$
$$x_{44} = -43.720497762458$$
$$x_{45} = -56.2868683768171$$
$$x_{46} = 69.3768377667746$$
$$x_{47} = -71.9948316447661$$
$$x_{48} = 47.3856891916461$$
$$x_{49} = 6.54498469497874$$
$$x_{50} = 37.9609112308767$$
$$x_{51} = 85.0848010347236$$
$$x_{52} = 15.9697626557481$$
$$x_{53} = -28.012534494509$$
$$x_{54} = 59.9520598060052$$
$$x_{55} = 56.8104671524154$$
$$x_{56} = -40.5789051088682$$
$$x_{57} = 75.6600230739542$$
$$x_{58} = -24.8709418409192$$
$$x_{59} = -34.2957198016886$$
$$x_{60} = -53.1452757232273$$
$$x_{61} = 78.801615727544$$
$$x_{62} = -9.16297857297023$$
$$x_{63} = 66.2352451131848$$
$$x_{64} = -62.5700536839967$$
$$x_{65} = -15.4461638801498$$
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
$$\left(\tan^{2}{\left(x - \frac{\pi}{3} \right)} + 1\right) \operatorname{sign}{\left(1 - \cot{\left(x + \frac{\pi}{6} \right)} \right)} = 0$$
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
$$\left(\tan^{2}{\left(x - \frac{\pi}{3} \right)} + 1\right) \operatorname{sign}{\left(1 - \cot{\left(x + \frac{\pi}{6} \right)} \right)} = 0$$
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
$$2 \left(\left(\cot^{2}{\left(x + \frac{\pi}{6} \right)} + 1\right) \delta\left(\cot{\left(x + \frac{\pi}{6} \right)} - 1\right) + \cot{\left(x + \frac{\pi}{6} \right)} \operatorname{sign}{\left(\cot{\left(x + \frac{\pi}{6} \right)} - 1 \right)}\right) \left(\cot^{2}{\left(x + \frac{\pi}{6} \right)} + 1\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(\left(\cot^{2}{\left(x + \frac{\pi}{6} \right)} + 1\right) \delta\left(\cot{\left(x + \frac{\pi}{6} \right)} - 1\right) + \cot{\left(x + \frac{\pi}{6} \right)} \operatorname{sign}{\left(\cot{\left(x + \frac{\pi}{6} \right)} - 1 \right)}\right) \left(\cot^{2}{\left(x + \frac{\pi}{6} \right)} + 1\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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:
$$y = \lim_{x \to -\infty} \left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right|$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right|$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right|$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right|$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(tan(x - pi/3) + 1), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right|}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right|}{x}\right)$$
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:
$$\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right| = \sqrt{\tan{\left(x + \frac{\pi}{3} \right)} \tan{\left(x + \frac{\pi}{3} \right)} - \tan{\left(x + \frac{\pi}{3} \right)} - \tan{\left(x + \frac{\pi}{3} \right)} + 1}$$
- No
$$\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right| = - \sqrt{\tan{\left(x + \frac{\pi}{3} \right)} \tan{\left(x + \frac{\pi}{3} \right)} - \tan{\left(x + \frac{\pi}{3} \right)} - \tan{\left(x + \frac{\pi}{3} \right)} + 1}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right| = \sqrt{\tan{\left(x + \frac{\pi}{3} \right)} \tan{\left(x + \frac{\pi}{3} \right)} - \tan{\left(x + \frac{\pi}{3} \right)} - \tan{\left(x + \frac{\pi}{3} \right)} + 1}$$
- No
$$\left|{\tan{\left(x - \frac{\pi}{3} \right)} + 1}\right| = - \sqrt{\tan{\left(x + \frac{\pi}{3} \right)} \tan{\left(x + \frac{\pi}{3} \right)} - \tan{\left(x + \frac{\pi}{3} \right)} - \tan{\left(x + \frac{\pi}{3} \right)} + 1}$$
- No
so, the function
not is
neither even, nor odd