Mister Exam
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-
How to use it?
- Graphing y =:
- x^3-6x^2-15x+7
- (x^2-6x+13)/(x-3)
- sin(x)
- y(0)
- Limit of the function:
-
(1-cos(x))*cot(x)
-
Identical expressions
- (one -cos(x))*cot(x)
- (1 minus co sinus of e of (x)) multiply by cotangent of (x)
- (one minus co sinus of e of (x)) multiply by cotangent of (x)
- (1-cos(x))cot(x)
- 1-cosxcotx
-
Similar expressions
- (1+cos(x))*cot(x)
- (1-cosx)*cot(x)
Graphing y = (1-cos(x))*cot(x)
The solution
You have entered
[src]
f(x) = (1 - cos(x))*cot(x)
$$f{\left(x \right)} = \left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}$$
f = (1 - cos(x))*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:
$$\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 62.8318530717959$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = -50.2654824574367$$
$$x_{4} = 14.1371669411541$$
$$x_{5} = -70.6858347057703$$
$$x_{6} = 32.9867228626928$$
$$x_{7} = 61.261056745001$$
$$x_{8} = -36.1283155162826$$
$$x_{9} = 7.85398163397448$$
$$x_{10} = 42.4115008234622$$
$$x_{11} = -43.9822971502571$$
$$x_{12} = 98.9601685880785$$
$$x_{13} = 83.2522053201295$$
$$x_{14} = -37.6991118430775$$
$$x_{15} = -89.5353906273091$$
$$x_{16} = -12.5663706143592$$
$$x_{17} = -54.9778714378214$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = -67.5442420521806$$
$$x_{20} = -51.8362787842316$$
$$x_{21} = -83.2522053201295$$
$$x_{22} = -14.1371669411541$$
$$x_{23} = -56.5486677646163$$
$$x_{24} = 50.2654824574367$$
$$x_{25} = 92.6769832808989$$
$$x_{26} = 64.4026493985908$$
$$x_{27} = 48.6946861306418$$
$$x_{28} = 67.5442420521806$$
$$x_{29} = -87.9645943005142$$
$$x_{30} = 76.9690200129499$$
$$x_{31} = -32.9867228626928$$
$$x_{32} = 23.5619449019235$$
$$x_{33} = 10.9955742875643$$
$$x_{34} = 50.2654824574367$$
$$x_{35} = 6.28318530717959$$
$$x_{36} = -80.1106126665397$$
$$x_{37} = 31.4159265358979$$
$$x_{38} = -64.4026493985908$$
$$x_{39} = -29.845130209103$$
$$x_{40} = -86.3937979737193$$
$$x_{41} = 4.71238898038469$$
$$x_{42} = 17.2787595947439$$
$$x_{43} = 20.4203522483337$$
$$x_{44} = -76.9690200129499$$
$$x_{45} = 94.2477796076938$$
$$x_{46} = 69.1150383789755$$
$$x_{47} = 37.6991118430775$$
$$x_{48} = -1.5707963267949$$
$$x_{49} = 86.3937979737193$$
$$x_{50} = 39.2699081698724$$
$$x_{51} = -39.2699081698724$$
$$x_{52} = -6.28318530717959$$
$$x_{53} = 36.1283155162826$$
$$x_{54} = -75.398223686155$$
$$x_{55} = 51.8362787842316$$
$$x_{56} = 80.1106126665397$$
$$x_{57} = -92.6769832808989$$
$$x_{58} = 89.5353906273091$$
$$x_{59} = 58.1194640914112$$
$$x_{60} = 73.8274273593601$$
$$x_{61} = -58.1194640914112$$
$$x_{62} = -20.4203522483337$$
$$x_{63} = 54.9778714378214$$
$$x_{64} = -25.1327412287183$$
$$x_{65} = -95.8185759344887$$
$$x_{66} = 29.845130209103$$
$$x_{67} = 81.6814089933346$$
$$x_{68} = 26.7035375555132$$
$$x_{69} = 100.530964914873$$
$$x_{70} = -23.5619449019235$$
$$x_{71} = -42.4115008234622$$
$$x_{72} = 25.1327412287183$$
$$x_{73} = -61.261056745001$$
$$x_{74} = -48.6946861306418$$
$$x_{75} = 70.6858347057703$$
$$x_{76} = -98.9601685880785$$
$$x_{77} = 12.5663706143592$$
$$x_{78} = 87.9645943005142$$
$$x_{79} = -100.530964914873$$
$$x_{80} = -26.7035375555132$$
$$x_{81} = -10.9955742875643$$
$$x_{82} = 18.8495559215388$$
$$x_{83} = 43.9822971502571$$
$$x_{84} = -31.4159265358979$$
$$x_{85} = 45.553093477052$$
$$x_{86} = 56.5486677646163$$
$$x_{87} = -81.6814089933346$$
$$x_{88} = -73.8274273593601$$
$$x_{89} = -4.71238898038469$$
$$x_{90} = -45.553093477052$$
$$x_{91} = -7.85398163397448$$
$$x_{92} = -17.2787595947439$$
$$x_{93} = 95.8185759344887$$
$$x_{94} = 75.398223686155$$
so we need to solve the equation:
$$\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 62.8318530717959$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = -50.2654824574367$$
$$x_{4} = 14.1371669411541$$
$$x_{5} = -70.6858347057703$$
$$x_{6} = 32.9867228626928$$
$$x_{7} = 61.261056745001$$
$$x_{8} = -36.1283155162826$$
$$x_{9} = 7.85398163397448$$
$$x_{10} = 42.4115008234622$$
$$x_{11} = -43.9822971502571$$
$$x_{12} = 98.9601685880785$$
$$x_{13} = 83.2522053201295$$
$$x_{14} = -37.6991118430775$$
$$x_{15} = -89.5353906273091$$
$$x_{16} = -12.5663706143592$$
$$x_{17} = -54.9778714378214$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = -67.5442420521806$$
$$x_{20} = -51.8362787842316$$
$$x_{21} = -83.2522053201295$$
$$x_{22} = -14.1371669411541$$
$$x_{23} = -56.5486677646163$$
$$x_{24} = 50.2654824574367$$
$$x_{25} = 92.6769832808989$$
$$x_{26} = 64.4026493985908$$
$$x_{27} = 48.6946861306418$$
$$x_{28} = 67.5442420521806$$
$$x_{29} = -87.9645943005142$$
$$x_{30} = 76.9690200129499$$
$$x_{31} = -32.9867228626928$$
$$x_{32} = 23.5619449019235$$
$$x_{33} = 10.9955742875643$$
$$x_{34} = 50.2654824574367$$
$$x_{35} = 6.28318530717959$$
$$x_{36} = -80.1106126665397$$
$$x_{37} = 31.4159265358979$$
$$x_{38} = -64.4026493985908$$
$$x_{39} = -29.845130209103$$
$$x_{40} = -86.3937979737193$$
$$x_{41} = 4.71238898038469$$
$$x_{42} = 17.2787595947439$$
$$x_{43} = 20.4203522483337$$
$$x_{44} = -76.9690200129499$$
$$x_{45} = 94.2477796076938$$
$$x_{46} = 69.1150383789755$$
$$x_{47} = 37.6991118430775$$
$$x_{48} = -1.5707963267949$$
$$x_{49} = 86.3937979737193$$
$$x_{50} = 39.2699081698724$$
$$x_{51} = -39.2699081698724$$
$$x_{52} = -6.28318530717959$$
$$x_{53} = 36.1283155162826$$
$$x_{54} = -75.398223686155$$
$$x_{55} = 51.8362787842316$$
$$x_{56} = 80.1106126665397$$
$$x_{57} = -92.6769832808989$$
$$x_{58} = 89.5353906273091$$
$$x_{59} = 58.1194640914112$$
$$x_{60} = 73.8274273593601$$
$$x_{61} = -58.1194640914112$$
$$x_{62} = -20.4203522483337$$
$$x_{63} = 54.9778714378214$$
$$x_{64} = -25.1327412287183$$
$$x_{65} = -95.8185759344887$$
$$x_{66} = 29.845130209103$$
$$x_{67} = 81.6814089933346$$
$$x_{68} = 26.7035375555132$$
$$x_{69} = 100.530964914873$$
$$x_{70} = -23.5619449019235$$
$$x_{71} = -42.4115008234622$$
$$x_{72} = 25.1327412287183$$
$$x_{73} = -61.261056745001$$
$$x_{74} = -48.6946861306418$$
$$x_{75} = 70.6858347057703$$
$$x_{76} = -98.9601685880785$$
$$x_{77} = 12.5663706143592$$
$$x_{78} = 87.9645943005142$$
$$x_{79} = -100.530964914873$$
$$x_{80} = -26.7035375555132$$
$$x_{81} = -10.9955742875643$$
$$x_{82} = 18.8495559215388$$
$$x_{83} = 43.9822971502571$$
$$x_{84} = -31.4159265358979$$
$$x_{85} = 45.553093477052$$
$$x_{86} = 56.5486677646163$$
$$x_{87} = -81.6814089933346$$
$$x_{88} = -73.8274273593601$$
$$x_{89} = -4.71238898038469$$
$$x_{90} = -45.553093477052$$
$$x_{91} = -7.85398163397448$$
$$x_{92} = -17.2787595947439$$
$$x_{93} = 95.8185759344887$$
$$x_{94} = 75.398223686155$$
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
$$\left(1 - \cos{\left(x \right)}\right) \left(- \cot^{2}{\left(x \right)} - 1\right) + \sin{\left(x \right)} \cot{\left(x \right)} = 0$$
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
$$\left(1 - \cos{\left(x \right)}\right) \left(- \cot^{2}{\left(x \right)} - 1\right) + \sin{\left(x \right)} \cot{\left(x \right)} = 0$$
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
$$- 2 \left(\cos{\left(x \right)} - 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \cos{\left(x \right)} \cot{\left(x \right)} = 0$$
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
$$- 2 \left(\cos{\left(x \right)} - 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \cos{\left(x \right)} \cot{\left(x \right)} = 0$$
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:
$$y = \lim_{x \to -\infty}\left(\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1 - cos(x))*cot(x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}}{x}\right)$$
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:
$$\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)} = - \left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}$$
- No
$$\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)} = \left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)} = - \left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}$$
- No
$$\left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)} = \left(1 - \cos{\left(x \right)}\right) \cot{\left(x \right)}$$
- Yes
so, the function
is
odd