Mister Exam
Graphing y = log(cot(x))^tan(x)
The solution
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tan(x) f(x) = log (cot(x))
$$f{\left(x \right)} = \log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}$$
f = log(cot(x))^tan(x)
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:
$$\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{4}$$
Numerical solution
$$x_{1} = -18.0641577581413$$
$$x_{2} = -77.7544181763474$$
$$x_{3} = -90.3207887907066$$
$$x_{4} = 69.9004365423729$$
$$x_{5} = -99.7455667514759$$
$$x_{6} = -33.7721210260903$$
$$x_{7} = -55.7632696012188$$
$$x_{8} = 91.8915851175014$$
$$x_{9} = -24.3473430653209$$
$$x_{10} = 10.2101761241668$$
$$x_{11} = 54.1924732744239$$
$$x_{12} = -71.4712328691678$$
$$x_{13} = 60.4756585816035$$
$$x_{14} = 25.9181393921158$$
$$x_{15} = -84.037603483527$$
$$x_{16} = 76.1836218495525$$
$$x_{17} = -40.0553063332699$$
$$x_{18} = 98.174770424681$$
$$x_{19} = 3.92699081698724$$
$$x_{20} = -62.0464549083984$$
$$x_{21} = 32.2013246992954$$
$$x_{22} = 38.484510006475$$
$$x_{23} = 16.4933614313464$$
$$x_{24} = -49.4800842940392$$
$$x_{25} = -11.7809724509617$$
$$x_{26} = -27.4889357189107$$
$$x_{27} = 82.4668071567321$$
$$x_{28} = 47.9092879672443$$
$$x_{29} = -5.49778714378214$$
so we need to solve the equation:
$$\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{4}$$
Numerical solution
$$x_{1} = -18.0641577581413$$
$$x_{2} = -77.7544181763474$$
$$x_{3} = -90.3207887907066$$
$$x_{4} = 69.9004365423729$$
$$x_{5} = -99.7455667514759$$
$$x_{6} = -33.7721210260903$$
$$x_{7} = -55.7632696012188$$
$$x_{8} = 91.8915851175014$$
$$x_{9} = -24.3473430653209$$
$$x_{10} = 10.2101761241668$$
$$x_{11} = 54.1924732744239$$
$$x_{12} = -71.4712328691678$$
$$x_{13} = 60.4756585816035$$
$$x_{14} = 25.9181393921158$$
$$x_{15} = -84.037603483527$$
$$x_{16} = 76.1836218495525$$
$$x_{17} = -40.0553063332699$$
$$x_{18} = 98.174770424681$$
$$x_{19} = 3.92699081698724$$
$$x_{20} = -62.0464549083984$$
$$x_{21} = 32.2013246992954$$
$$x_{22} = 38.484510006475$$
$$x_{23} = 16.4933614313464$$
$$x_{24} = -49.4800842940392$$
$$x_{25} = -11.7809724509617$$
$$x_{26} = -27.4889357189107$$
$$x_{27} = 82.4668071567321$$
$$x_{28} = 47.9092879672443$$
$$x_{29} = -5.49778714378214$$
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:
$$y = \lim_{x \to -\infty} \log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(cot(x))^tan(x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}}{x}\right)$$
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:
$$\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}} = \log{\left(- \cot{\left(x \right)} \right)}^{- \tan{\left(x \right)}}$$
- No
$$\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}} = - \log{\left(- \cot{\left(x \right)} \right)}^{- \tan{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}} = \log{\left(- \cot{\left(x \right)} \right)}^{- \tan{\left(x \right)}}$$
- No
$$\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}} = - \log{\left(- \cot{\left(x \right)} \right)}^{- \tan{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd