Mister Exam
Graphing y = |tgx-1|
The solution
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f(x) = |tan(x) - 1|
$$f{\left(x \right)} = \left|{\tan{\left(x \right)} - 1}\right|$$
f = Abs(tan(x) - 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:
$$\left|{\tan{\left(x \right)} - 1}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{4}$$
Numerical solution
$$x_{1} = 10.2101761241668$$
$$x_{2} = 101.316363078271$$
$$x_{3} = -55.7632696012188$$
$$x_{4} = -93.4623814442964$$
$$x_{5} = -43.1968989868597$$
$$x_{6} = -46.3384916404494$$
$$x_{7} = -96.6039740978861$$
$$x_{8} = -74.6128255227576$$
$$x_{9} = -5.49778714378214$$
$$x_{10} = 51.0508806208341$$
$$x_{11} = 95.0331777710912$$
$$x_{12} = 3.92699081698724$$
$$x_{13} = -18.0641577581413$$
$$x_{14} = -65.1880475619882$$
$$x_{15} = 32.2013246992954$$
$$x_{16} = 7.06858347057703$$
$$x_{17} = -30.6305283725005$$
$$x_{18} = -36.9137136796801$$
$$x_{19} = -62.0464549083984$$
$$x_{20} = 22.776546738526$$
$$x_{21} = -52.621676947629$$
$$x_{22} = 19.6349540849362$$
$$x_{23} = -11.7809724509617$$
$$x_{24} = 44.7676953136546$$
$$x_{25} = -24.3473430653209$$
$$x_{26} = -14.9225651045515$$
$$x_{27} = -49.4800842940392$$
$$x_{28} = -84.037603483527$$
$$x_{29} = 54.1924732744239$$
$$x_{30} = 88.7499924639117$$
$$x_{31} = 38.484510006475$$
$$x_{32} = -99.7455667514759$$
$$x_{33} = 69.9004365423729$$
$$x_{34} = 98.174770424681$$
$$x_{35} = 63.6172512351933$$
$$x_{36} = -8.63937979737193$$
$$x_{37} = 82.4668071567321$$
$$x_{38} = -71.4712328691678$$
$$x_{39} = -21.2057504117311$$
$$x_{40} = -68.329640215578$$
$$x_{41} = 41.6261026600648$$
$$x_{42} = 76.1836218495525$$
$$x_{43} = 60.4756585816035$$
$$x_{44} = -58.9048622548086$$
$$x_{45} = 35.3429173528852$$
$$x_{46} = 0.785398163397448$$
$$x_{47} = -80.8960108299372$$
$$x_{48} = -77.7544181763474$$
$$x_{49} = -90.3207887907066$$
$$x_{50} = 79.3252145031423$$
$$x_{51} = -87.1791961371168$$
$$x_{52} = -2.35619449019234$$
$$x_{53} = 57.3340659280137$$
$$x_{54} = -27.4889357189107$$
$$x_{55} = 85.6083998103219$$
$$x_{56} = 25.9181393921158$$
$$x_{57} = -40.0553063332699$$
$$x_{58} = 13.3517687777566$$
$$x_{59} = 16.4933614313464$$
$$x_{60} = 66.7588438887831$$
$$x_{61} = -33.7721210260903$$
$$x_{62} = 73.0420291959627$$
$$x_{63} = 29.0597320457056$$
$$x_{64} = 47.9092879672443$$
$$x_{65} = 91.8915851175014$$
so we need to solve the equation:
$$\left|{\tan{\left(x \right)} - 1}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{4}$$
Numerical solution
$$x_{1} = 10.2101761241668$$
$$x_{2} = 101.316363078271$$
$$x_{3} = -55.7632696012188$$
$$x_{4} = -93.4623814442964$$
$$x_{5} = -43.1968989868597$$
$$x_{6} = -46.3384916404494$$
$$x_{7} = -96.6039740978861$$
$$x_{8} = -74.6128255227576$$
$$x_{9} = -5.49778714378214$$
$$x_{10} = 51.0508806208341$$
$$x_{11} = 95.0331777710912$$
$$x_{12} = 3.92699081698724$$
$$x_{13} = -18.0641577581413$$
$$x_{14} = -65.1880475619882$$
$$x_{15} = 32.2013246992954$$
$$x_{16} = 7.06858347057703$$
$$x_{17} = -30.6305283725005$$
$$x_{18} = -36.9137136796801$$
$$x_{19} = -62.0464549083984$$
$$x_{20} = 22.776546738526$$
$$x_{21} = -52.621676947629$$
$$x_{22} = 19.6349540849362$$
$$x_{23} = -11.7809724509617$$
$$x_{24} = 44.7676953136546$$
$$x_{25} = -24.3473430653209$$
$$x_{26} = -14.9225651045515$$
$$x_{27} = -49.4800842940392$$
$$x_{28} = -84.037603483527$$
$$x_{29} = 54.1924732744239$$
$$x_{30} = 88.7499924639117$$
$$x_{31} = 38.484510006475$$
$$x_{32} = -99.7455667514759$$
$$x_{33} = 69.9004365423729$$
$$x_{34} = 98.174770424681$$
$$x_{35} = 63.6172512351933$$
$$x_{36} = -8.63937979737193$$
$$x_{37} = 82.4668071567321$$
$$x_{38} = -71.4712328691678$$
$$x_{39} = -21.2057504117311$$
$$x_{40} = -68.329640215578$$
$$x_{41} = 41.6261026600648$$
$$x_{42} = 76.1836218495525$$
$$x_{43} = 60.4756585816035$$
$$x_{44} = -58.9048622548086$$
$$x_{45} = 35.3429173528852$$
$$x_{46} = 0.785398163397448$$
$$x_{47} = -80.8960108299372$$
$$x_{48} = -77.7544181763474$$
$$x_{49} = -90.3207887907066$$
$$x_{50} = 79.3252145031423$$
$$x_{51} = -87.1791961371168$$
$$x_{52} = -2.35619449019234$$
$$x_{53} = 57.3340659280137$$
$$x_{54} = -27.4889357189107$$
$$x_{55} = 85.6083998103219$$
$$x_{56} = 25.9181393921158$$
$$x_{57} = -40.0553063332699$$
$$x_{58} = 13.3517687777566$$
$$x_{59} = 16.4933614313464$$
$$x_{60} = 66.7588438887831$$
$$x_{61} = -33.7721210260903$$
$$x_{62} = 73.0420291959627$$
$$x_{63} = 29.0597320457056$$
$$x_{64} = 47.9092879672443$$
$$x_{65} = 91.8915851175014$$
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
$$\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{sign}{\left(\tan{\left(x \right)} - 1 \right)} = 0$$
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
$$\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{sign}{\left(\tan{\left(x \right)} - 1 \right)} = 0$$
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
$$2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \delta\left(\tan{\left(x \right)} - 1\right) + \tan{\left(x \right)} \operatorname{sign}{\left(\tan{\left(x \right)} - 1 \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) = 0$$
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
$$2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \delta\left(\tan{\left(x \right)} - 1\right) + \tan{\left(x \right)} \operatorname{sign}{\left(\tan{\left(x \right)} - 1 \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) = 0$$
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:
$$y = \lim_{x \to -\infty} \left|{\tan{\left(x \right)} - 1}\right|$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \left|{\tan{\left(x \right)} - 1}\right|$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \left|{\tan{\left(x \right)} - 1}\right|$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \left|{\tan{\left(x \right)} - 1}\right|$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(tan(x) - 1), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left|{\tan{\left(x \right)} - 1}\right|}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left|{\tan{\left(x \right)} - 1}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left|{\tan{\left(x \right)} - 1}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left|{\tan{\left(x \right)} - 1}\right|}{x}\right)$$
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:
$$\left|{\tan{\left(x \right)} - 1}\right| = \left|{\tan{\left(x \right)} + 1}\right|$$
- No
$$\left|{\tan{\left(x \right)} - 1}\right| = - \left|{\tan{\left(x \right)} + 1}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{\tan{\left(x \right)} - 1}\right| = \left|{\tan{\left(x \right)} + 1}\right|$$
- No
$$\left|{\tan{\left(x \right)} - 1}\right| = - \left|{\tan{\left(x \right)} + 1}\right|$$
- No
so, the function
not is
neither even, nor odd