Mister Exam
Graphing y = ctg(4x)*cos(4x)
The solution
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f(x) = cot(4*x)*cos(4*x)
$$f{\left(x \right)} = \cos{\left(4 x \right)} \cot{\left(4 x \right)}$$
f = cos(4*x)*cot(4*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:
$$\cos{\left(4 x \right)} \cot{\left(4 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{8}$$
$$x_{2} = \frac{\pi}{8}$$
Numerical solution
$$x_{1} = 12.1736716122318$$
$$x_{2} = -21.5984496198436$$
$$x_{3} = -45.9457926721836$$
$$x_{4} = -3.5342919993209$$
$$x_{5} = -75.7909227335983$$
$$x_{6} = -23.9546440649227$$
$$x_{7} = 92.2842841747879$$
$$x_{8} = -82.0741079971979$$
$$x_{9} = -60.0829594119627$$
$$x_{10} = -61.653755896837$$
$$x_{11} = -79.7179136077217$$
$$x_{12} = 20.0276531264367$$
$$x_{13} = -1.96349546669253$$
$$x_{14} = 66.3661447194599$$
$$x_{15} = -38.0918108260377$$
$$x_{16} = -39.6626073145636$$
$$x_{17} = 8.24668071815482$$
$$x_{18} = 5.89048634868838$$
$$x_{19} = -13.7444678635245$$
$$x_{20} = -78.1471170599108$$
$$x_{21} = 44.3749961357343$$
$$x_{22} = -97.7820713170403$$
$$x_{23} = 48.3019870121468$$
$$x_{24} = 67.9369411998163$$
$$x_{25} = -20.0276531365334$$
$$x_{26} = 62.4391538011772$$
$$x_{27} = -16.1006622392305$$
$$x_{28} = -87.57189538582$$
$$x_{29} = 30.2378293000822$$
$$x_{30} = 78.1471174281992$$
$$x_{31} = -100.138265656808$$
$$x_{32} = -65.5807467955783$$
$$x_{33} = -57.7267650264066$$
$$x_{34} = 34.1648202052439$$
$$x_{35} = 18.4568566055691$$
$$x_{36} = 89.928089784421$$
$$x_{37} = -17.6714587325842$$
$$x_{38} = -93.8550803321541$$
$$x_{39} = 88.3572933026447$$
$$x_{40} = 70.2931355934875$$
$$x_{41} = -5.89048609822403$$
$$x_{42} = -34.1648198530086$$
$$x_{43} = -31.808625563961$$
$$x_{44} = 71.8639321223223$$
$$x_{45} = 22.3838475513636$$
$$x_{46} = 84.430302394159$$
$$x_{47} = -42.0188017207944$$
$$x_{48} = 49.8727835290591$$
$$x_{49} = -86.0010988907069$$
$$x_{50} = -56.1559684590926$$
$$x_{51} = 96.2112750483112$$
$$x_{52} = 56.1559688055632$$
$$x_{53} = -64.009950305446$$
$$x_{54} = -43.5895982070279$$
$$x_{55} = -35.7356164450251$$
$$x_{56} = 74.2201264650387$$
$$x_{57} = 1.96349544751322$$
$$x_{58} = 93.8550807184008$$
$$x_{59} = 40.4480052052875$$
$$x_{60} = 52.2289778823416$$
$$x_{61} = 16.8860606369495$$
$$x_{62} = -83.6449044794693$$
$$x_{63} = -27.8816347272427$$
$$x_{64} = 27.8816349380003$$
$$x_{65} = 45.9457926155531$$
$$x_{66} = -67.9369412987172$$
$$x_{67} = -53.7997741493363$$
$$x_{68} = 42.0188017163919$$
$$x_{69} = 26.3108384307197$$
$$x_{70} = 23.9546440314963$$
$$x_{71} = 4.31968984915916$$
$$x_{72} = -9.81747697702242$$
$$x_{73} = 86.0010988906926$$
$$x_{74} = 64.0099503042551$$
so we need to solve the equation:
$$\cos{\left(4 x \right)} \cot{\left(4 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{8}$$
$$x_{2} = \frac{\pi}{8}$$
Numerical solution
$$x_{1} = 12.1736716122318$$
$$x_{2} = -21.5984496198436$$
$$x_{3} = -45.9457926721836$$
$$x_{4} = -3.5342919993209$$
$$x_{5} = -75.7909227335983$$
$$x_{6} = -23.9546440649227$$
$$x_{7} = 92.2842841747879$$
$$x_{8} = -82.0741079971979$$
$$x_{9} = -60.0829594119627$$
$$x_{10} = -61.653755896837$$
$$x_{11} = -79.7179136077217$$
$$x_{12} = 20.0276531264367$$
$$x_{13} = -1.96349546669253$$
$$x_{14} = 66.3661447194599$$
$$x_{15} = -38.0918108260377$$
$$x_{16} = -39.6626073145636$$
$$x_{17} = 8.24668071815482$$
$$x_{18} = 5.89048634868838$$
$$x_{19} = -13.7444678635245$$
$$x_{20} = -78.1471170599108$$
$$x_{21} = 44.3749961357343$$
$$x_{22} = -97.7820713170403$$
$$x_{23} = 48.3019870121468$$
$$x_{24} = 67.9369411998163$$
$$x_{25} = -20.0276531365334$$
$$x_{26} = 62.4391538011772$$
$$x_{27} = -16.1006622392305$$
$$x_{28} = -87.57189538582$$
$$x_{29} = 30.2378293000822$$
$$x_{30} = 78.1471174281992$$
$$x_{31} = -100.138265656808$$
$$x_{32} = -65.5807467955783$$
$$x_{33} = -57.7267650264066$$
$$x_{34} = 34.1648202052439$$
$$x_{35} = 18.4568566055691$$
$$x_{36} = 89.928089784421$$
$$x_{37} = -17.6714587325842$$
$$x_{38} = -93.8550803321541$$
$$x_{39} = 88.3572933026447$$
$$x_{40} = 70.2931355934875$$
$$x_{41} = -5.89048609822403$$
$$x_{42} = -34.1648198530086$$
$$x_{43} = -31.808625563961$$
$$x_{44} = 71.8639321223223$$
$$x_{45} = 22.3838475513636$$
$$x_{46} = 84.430302394159$$
$$x_{47} = -42.0188017207944$$
$$x_{48} = 49.8727835290591$$
$$x_{49} = -86.0010988907069$$
$$x_{50} = -56.1559684590926$$
$$x_{51} = 96.2112750483112$$
$$x_{52} = 56.1559688055632$$
$$x_{53} = -64.009950305446$$
$$x_{54} = -43.5895982070279$$
$$x_{55} = -35.7356164450251$$
$$x_{56} = 74.2201264650387$$
$$x_{57} = 1.96349544751322$$
$$x_{58} = 93.8550807184008$$
$$x_{59} = 40.4480052052875$$
$$x_{60} = 52.2289778823416$$
$$x_{61} = 16.8860606369495$$
$$x_{62} = -83.6449044794693$$
$$x_{63} = -27.8816347272427$$
$$x_{64} = 27.8816349380003$$
$$x_{65} = 45.9457926155531$$
$$x_{66} = -67.9369412987172$$
$$x_{67} = -53.7997741493363$$
$$x_{68} = 42.0188017163919$$
$$x_{69} = 26.3108384307197$$
$$x_{70} = 23.9546440314963$$
$$x_{71} = 4.31968984915916$$
$$x_{72} = -9.81747697702242$$
$$x_{73} = 86.0010988906926$$
$$x_{74} = 64.0099503042551$$
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:
$$y = \lim_{x \to -\infty}\left(\cos{\left(4 x \right)} \cot{\left(4 x \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\cos{\left(4 x \right)} \cot{\left(4 x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\cos{\left(4 x \right)} \cot{\left(4 x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\cos{\left(4 x \right)} \cot{\left(4 x \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(4*x)*cos(4*x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(4 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(4 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(4 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(4 x \right)}}{x}\right)$$
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:
$$\cos{\left(4 x \right)} \cot{\left(4 x \right)} = - \cos{\left(4 x \right)} \cot{\left(4 x \right)}$$
- No
$$\cos{\left(4 x \right)} \cot{\left(4 x \right)} = \cos{\left(4 x \right)} \cot{\left(4 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\cos{\left(4 x \right)} \cot{\left(4 x \right)} = - \cos{\left(4 x \right)} \cot{\left(4 x \right)}$$
- No
$$\cos{\left(4 x \right)} \cot{\left(4 x \right)} = \cos{\left(4 x \right)} \cot{\left(4 x \right)}$$
- No
so, the function
not is
neither even, nor odd