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