Mister Exam
Graphing y = cot(x/2)*sin(x)
The solution
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/x\
f(x) = cot|-|*sin(x)
\2/
$$f{\left(x \right)} = \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}$$
f = sin(x)*cot(x/2)
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:
$$\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 91.1061873718352$$
$$x_{2} = -91.106187265474$$
$$x_{3} = -21.9911490521325$$
$$x_{4} = 65.9734452390837$$
$$x_{5} = -9.42477744529557$$
$$x_{6} = -15.7079635641079$$
$$x_{7} = -34.5575188899093$$
$$x_{8} = -53.4070745786761$$
$$x_{9} = -65.9734449870253$$
$$x_{10} = 9.42477748794163$$
$$x_{11} = 34.5575197055812$$
$$x_{12} = 53.4070766553897$$
$$x_{13} = 21.9911489072506$$
$$x_{14} = 9.42477826738203$$
$$x_{15} = 65.9734460390947$$
$$x_{16} = -21.9911485864417$$
$$x_{17} = 34.5575195449229$$
$$x_{18} = 72.2566310277176$$
$$x_{19} = -1127.83176318906$$
$$x_{20} = -84.8230012511693$$
$$x_{21} = -72.2566308657983$$
$$x_{22} = -47.1238893275319$$
$$x_{23} = 78.5398152766482$$
$$x_{24} = 3.14159306054457$$
$$x_{25} = -9.4247781365785$$
$$x_{26} = -40.8407049290801$$
$$x_{27} = -15.7079632965989$$
$$x_{28} = -53.4070745963886$$
$$x_{29} = 34.5575190219169$$
$$x_{30} = 28.2743343711514$$
$$x_{31} = 40.8407045848602$$
$$x_{32} = 65.9734457529812$$
$$x_{33} = -84.8230020565447$$
$$x_{34} = 78.5398161804942$$
$$x_{35} = 72.2566315166773$$
$$x_{36} = -59.6902606928653$$
$$x_{37} = 3.1415922548952$$
$$x_{38} = 84.8230021335997$$
$$x_{39} = 59.6902606104322$$
$$x_{40} = 78.5398166181283$$
$$x_{41} = -91.1061864815274$$
$$x_{42} = 15.7079627593774$$
$$x_{43} = -47.1238901083229$$
$$x_{44} = 15.7079629803241$$
$$x_{45} = 40.8407045792514$$
$$x_{46} = 47.1238902162437$$
$$x_{47} = 97.389372581711$$
$$x_{48} = -28.2743337069329$$
$$x_{49} = -65.9734457649277$$
$$x_{50} = 28.2743335663982$$
$$x_{51} = -97.3893724533348$$
$$x_{52} = -97.3893716284562$$
$$x_{53} = -15.707962774825$$
$$x_{54} = -59.6902604578012$$
$$x_{55} = -72.2566311847166$$
$$x_{56} = -3.14159217367683$$
$$x_{57} = 78.5398168562347$$
$$x_{58} = 15.7079634518075$$
$$x_{59} = -28.2743340989896$$
$$x_{60} = -34.5575196658297$$
$$x_{61} = 40.8407049800347$$
$$x_{62} = 21.9911480932338$$
$$x_{63} = 84.8230013636028$$
$$x_{64} = -40.8407049008781$$
$$x_{65} = 59.6902599104079$$
$$x_{66} = -97.3893717476911$$
$$x_{67} = 53.4070746418597$$
$$x_{68} = -40.8407040952604$$
$$x_{69} = -78.5398160472843$$
$$x_{70} = 97.3893717959212$$
$$x_{71} = 28.2743338651796$$
$$x_{72} = 91.1061865667532$$
$$x_{73} = -65.9734453607004$$
$$x_{74} = -28.2743343914215$$
$$x_{75} = -3.14159295109225$$
$$x_{76} = 40.8407042062167$$
$$x_{77} = -53.407075294995$$
$$x_{78} = -21.991148226056$$
$$x_{79} = 59.6902600526626$$
$$x_{80} = -72.2566315419804$$
$$x_{81} = 21.9911485852059$$
$$x_{82} = 72.2566306985$$
$$x_{83} = -78.5398168194507$$
$$x_{84} = -65.9734461969855$$
$$x_{85} = 78.5398149750205$$
$$x_{86} = 47.123889410773$$
$$x_{87} = 53.407075424589$$
$$x_{88} = -59.6902599212271$$
$$x_{89} = -9.42477752082051$$
so we need to solve the equation:
$$\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 91.1061873718352$$
$$x_{2} = -91.106187265474$$
$$x_{3} = -21.9911490521325$$
$$x_{4} = 65.9734452390837$$
$$x_{5} = -9.42477744529557$$
$$x_{6} = -15.7079635641079$$
$$x_{7} = -34.5575188899093$$
$$x_{8} = -53.4070745786761$$
$$x_{9} = -65.9734449870253$$
$$x_{10} = 9.42477748794163$$
$$x_{11} = 34.5575197055812$$
$$x_{12} = 53.4070766553897$$
$$x_{13} = 21.9911489072506$$
$$x_{14} = 9.42477826738203$$
$$x_{15} = 65.9734460390947$$
$$x_{16} = -21.9911485864417$$
$$x_{17} = 34.5575195449229$$
$$x_{18} = 72.2566310277176$$
$$x_{19} = -1127.83176318906$$
$$x_{20} = -84.8230012511693$$
$$x_{21} = -72.2566308657983$$
$$x_{22} = -47.1238893275319$$
$$x_{23} = 78.5398152766482$$
$$x_{24} = 3.14159306054457$$
$$x_{25} = -9.4247781365785$$
$$x_{26} = -40.8407049290801$$
$$x_{27} = -15.7079632965989$$
$$x_{28} = -53.4070745963886$$
$$x_{29} = 34.5575190219169$$
$$x_{30} = 28.2743343711514$$
$$x_{31} = 40.8407045848602$$
$$x_{32} = 65.9734457529812$$
$$x_{33} = -84.8230020565447$$
$$x_{34} = 78.5398161804942$$
$$x_{35} = 72.2566315166773$$
$$x_{36} = -59.6902606928653$$
$$x_{37} = 3.1415922548952$$
$$x_{38} = 84.8230021335997$$
$$x_{39} = 59.6902606104322$$
$$x_{40} = 78.5398166181283$$
$$x_{41} = -91.1061864815274$$
$$x_{42} = 15.7079627593774$$
$$x_{43} = -47.1238901083229$$
$$x_{44} = 15.7079629803241$$
$$x_{45} = 40.8407045792514$$
$$x_{46} = 47.1238902162437$$
$$x_{47} = 97.389372581711$$
$$x_{48} = -28.2743337069329$$
$$x_{49} = -65.9734457649277$$
$$x_{50} = 28.2743335663982$$
$$x_{51} = -97.3893724533348$$
$$x_{52} = -97.3893716284562$$
$$x_{53} = -15.707962774825$$
$$x_{54} = -59.6902604578012$$
$$x_{55} = -72.2566311847166$$
$$x_{56} = -3.14159217367683$$
$$x_{57} = 78.5398168562347$$
$$x_{58} = 15.7079634518075$$
$$x_{59} = -28.2743340989896$$
$$x_{60} = -34.5575196658297$$
$$x_{61} = 40.8407049800347$$
$$x_{62} = 21.9911480932338$$
$$x_{63} = 84.8230013636028$$
$$x_{64} = -40.8407049008781$$
$$x_{65} = 59.6902599104079$$
$$x_{66} = -97.3893717476911$$
$$x_{67} = 53.4070746418597$$
$$x_{68} = -40.8407040952604$$
$$x_{69} = -78.5398160472843$$
$$x_{70} = 97.3893717959212$$
$$x_{71} = 28.2743338651796$$
$$x_{72} = 91.1061865667532$$
$$x_{73} = -65.9734453607004$$
$$x_{74} = -28.2743343914215$$
$$x_{75} = -3.14159295109225$$
$$x_{76} = 40.8407042062167$$
$$x_{77} = -53.407075294995$$
$$x_{78} = -21.991148226056$$
$$x_{79} = 59.6902600526626$$
$$x_{80} = -72.2566315419804$$
$$x_{81} = 21.9911485852059$$
$$x_{82} = 72.2566306985$$
$$x_{83} = -78.5398168194507$$
$$x_{84} = -65.9734461969855$$
$$x_{85} = 78.5398149750205$$
$$x_{86} = 47.123889410773$$
$$x_{87} = 53.407075424589$$
$$x_{88} = -59.6902599212271$$
$$x_{89} = -9.42477752082051$$
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
$$\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}\right) \sin{\left(x \right)} + \cos{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = 0$$
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
$$\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}\right) \sin{\left(x \right)} + \cos{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = 0$$
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
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}}{2} - \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
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
$$\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\pi}{2}, \infty\right)$$
Convex at the intervals
$$\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}}{2} - \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
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
$$\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\pi}{2}, \infty\right)$$
Convex at the intervals
$$\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]$$
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:
$$y = \lim_{x \to -\infty}\left(\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(x/2)*sin(x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}}{x}\right)$$
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:
$$\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}$$
- No
$$\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = - \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}$$
- No
$$\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)} = - \sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd