Mister Exam
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-
How to use it?
- Graphing y =:
- arctg(ctg(x))
-
sin^4x-cos^4x
-
logx^2
- log1/2x
-
Identical expressions
- arctg(ctg(x))
- arctgctgx
Graphing y = arctg(ctg(x))
The solution
You have entered
[src]
f(x) = atan(cot(x))
$$f{\left(x \right)} = \operatorname{atan}{\left(\cot{\left(x \right)} \right)}$$
f = atan(cot(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:
$$\operatorname{atan}{\left(\cot{\left(x \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 20.4203522483337$$
$$x_{2} = -73.8274273593601$$
$$x_{3} = 73.8274273593601$$
$$x_{4} = -23.5619449019235$$
$$x_{5} = 83.2522053201295$$
$$x_{6} = 95.8185759344887$$
$$x_{7} = 48.6946861306418$$
$$x_{8} = 36.1283155162826$$
$$x_{9} = 70.6858347057703$$
$$x_{10} = -67.5442420521806$$
$$x_{11} = -83.2522053201295$$
$$x_{12} = 17.2787595947439$$
$$x_{13} = -80.1106126665397$$
$$x_{14} = 76.9690200129499$$
$$x_{15} = 86.3937979737193$$
$$x_{16} = -20.4203522483337$$
$$x_{17} = -48.6946861306418$$
$$x_{18} = -51.8362787842316$$
$$x_{19} = 61.261056745001$$
$$x_{20} = -61.261056745001$$
$$x_{21} = 64.4026493985908$$
$$x_{22} = -70.6858347057703$$
$$x_{23} = -26.7035375555132$$
$$x_{24} = 98.9601685880785$$
$$x_{25} = 80.1106126665397$$
$$x_{26} = -14.1371669411541$$
$$x_{27} = -32.9867228626928$$
$$x_{28} = -95.8185759344887$$
$$x_{29} = 14.1371669411541$$
$$x_{30} = 10.9955742875643$$
$$x_{31} = 4.71238898038469$$
$$x_{32} = -42.4115008234622$$
$$x_{33} = 39.2699081698724$$
$$x_{34} = 42.4115008234622$$
$$x_{35} = -39.2699081698724$$
$$x_{36} = 26.7035375555132$$
$$x_{37} = -86.3937979737193$$
$$x_{38} = 7.85398163397448$$
$$x_{39} = 54.9778714378214$$
$$x_{40} = 32.9867228626928$$
$$x_{41} = -7.85398163397448$$
$$x_{42} = 1.5707963267949$$
$$x_{43} = 92.6769832808989$$
$$x_{44} = -54.9778714378214$$
$$x_{45} = 51.8362787842316$$
$$x_{46} = -4.71238898038469$$
$$x_{47} = -98.9601685880785$$
$$x_{48} = -92.6769832808989$$
$$x_{49} = -64.4026493985908$$
$$x_{50} = -17.2787595947439$$
$$x_{51} = -45.553093477052$$
$$x_{52} = 67.5442420521806$$
$$x_{53} = -10.9955742875643$$
$$x_{54} = -1.5707963267949$$
$$x_{55} = 89.5353906273091$$
$$x_{56} = -29.845130209103$$
$$x_{57} = 45.553093477052$$
$$x_{58} = 58.1194640914112$$
$$x_{59} = 23.5619449019235$$
$$x_{60} = -89.5353906273091$$
$$x_{61} = 29.845130209103$$
$$x_{62} = -76.9690200129499$$
$$x_{63} = -58.1194640914112$$
$$x_{64} = -36.1283155162826$$
so we need to solve the equation:
$$\operatorname{atan}{\left(\cot{\left(x \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 20.4203522483337$$
$$x_{2} = -73.8274273593601$$
$$x_{3} = 73.8274273593601$$
$$x_{4} = -23.5619449019235$$
$$x_{5} = 83.2522053201295$$
$$x_{6} = 95.8185759344887$$
$$x_{7} = 48.6946861306418$$
$$x_{8} = 36.1283155162826$$
$$x_{9} = 70.6858347057703$$
$$x_{10} = -67.5442420521806$$
$$x_{11} = -83.2522053201295$$
$$x_{12} = 17.2787595947439$$
$$x_{13} = -80.1106126665397$$
$$x_{14} = 76.9690200129499$$
$$x_{15} = 86.3937979737193$$
$$x_{16} = -20.4203522483337$$
$$x_{17} = -48.6946861306418$$
$$x_{18} = -51.8362787842316$$
$$x_{19} = 61.261056745001$$
$$x_{20} = -61.261056745001$$
$$x_{21} = 64.4026493985908$$
$$x_{22} = -70.6858347057703$$
$$x_{23} = -26.7035375555132$$
$$x_{24} = 98.9601685880785$$
$$x_{25} = 80.1106126665397$$
$$x_{26} = -14.1371669411541$$
$$x_{27} = -32.9867228626928$$
$$x_{28} = -95.8185759344887$$
$$x_{29} = 14.1371669411541$$
$$x_{30} = 10.9955742875643$$
$$x_{31} = 4.71238898038469$$
$$x_{32} = -42.4115008234622$$
$$x_{33} = 39.2699081698724$$
$$x_{34} = 42.4115008234622$$
$$x_{35} = -39.2699081698724$$
$$x_{36} = 26.7035375555132$$
$$x_{37} = -86.3937979737193$$
$$x_{38} = 7.85398163397448$$
$$x_{39} = 54.9778714378214$$
$$x_{40} = 32.9867228626928$$
$$x_{41} = -7.85398163397448$$
$$x_{42} = 1.5707963267949$$
$$x_{43} = 92.6769832808989$$
$$x_{44} = -54.9778714378214$$
$$x_{45} = 51.8362787842316$$
$$x_{46} = -4.71238898038469$$
$$x_{47} = -98.9601685880785$$
$$x_{48} = -92.6769832808989$$
$$x_{49} = -64.4026493985908$$
$$x_{50} = -17.2787595947439$$
$$x_{51} = -45.553093477052$$
$$x_{52} = 67.5442420521806$$
$$x_{53} = -10.9955742875643$$
$$x_{54} = -1.5707963267949$$
$$x_{55} = 89.5353906273091$$
$$x_{56} = -29.845130209103$$
$$x_{57} = 45.553093477052$$
$$x_{58} = 58.1194640914112$$
$$x_{59} = 23.5619449019235$$
$$x_{60} = -89.5353906273091$$
$$x_{61} = 29.845130209103$$
$$x_{62} = -76.9690200129499$$
$$x_{63} = -58.1194640914112$$
$$x_{64} = -36.1283155162826$$
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
$$\frac{- \cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} = 0$$
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
$$\frac{- \cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} = 0$$
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
$$0 = 0$$
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
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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:
$$\operatorname{atan}{\left(\cot{\left(x \right)} \right)} = - \operatorname{atan}{\left(\cot{\left(x \right)} \right)}$$
- No
$$\operatorname{atan}{\left(\cot{\left(x \right)} \right)} = \operatorname{atan}{\left(\cot{\left(x \right)} \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\operatorname{atan}{\left(\cot{\left(x \right)} \right)} = - \operatorname{atan}{\left(\cot{\left(x \right)} \right)}$$
- No
$$\operatorname{atan}{\left(\cot{\left(x \right)} \right)} = \operatorname{atan}{\left(\cot{\left(x \right)} \right)}$$
- Yes
so, the function
is
odd