Mister Exam
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-
How to use it?
- Graphing y =:
- y=-x^4+8x^2+16
-
y=x^3-3x^2-9x-3
-
sin(0.5x)
- ctg(4x)
- Derivative of:
-
ctg(4x)
-
Identical expressions
- ctg(4x)
- ctg4x
-
Similar expressions
- 2x-arctg4x
- arctg(4x-1)
- arctg(4*x-1)
Graphing y = ctg(4x)
The solution
You have entered
[src]
f(x) = cot(4*x)
$$f{\left(x \right)} = \cot{\left(4 x \right)}$$
f = 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:
$$\cot{\left(4 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{8}$$
Numerical solution
$$x_{1} = 26.3108384738145$$
$$x_{2} = 4.31968989868597$$
$$x_{3} = -82.0741080750334$$
$$x_{4} = 14.5298660228528$$
$$x_{5} = -47.5165888855456$$
$$x_{6} = 20.0276531666349$$
$$x_{7} = -86.0010988920206$$
$$x_{8} = 75.7909227678538$$
$$x_{9} = -49.872783375738$$
$$x_{10} = -21.5984494934298$$
$$x_{11} = -3.53429173528852$$
$$x_{12} = 48.3019870489431$$
$$x_{13} = 71.8639319508665$$
$$x_{14} = -9.8174770424681$$
$$x_{15} = -87.5718952188155$$
$$x_{16} = -89.9280897090078$$
$$x_{17} = 78.1471172580461$$
$$x_{18} = 100.138265833175$$
$$x_{19} = 9.8174770424681$$
$$x_{20} = 64.009950316892$$
$$x_{21} = 93.8550805259951$$
$$x_{22} = 22.3838476568273$$
$$x_{23} = 89.9280897090078$$
$$x_{24} = -1.96349540849362$$
$$x_{25} = 56.1559686829176$$
$$x_{26} = -39.6626072515711$$
$$x_{27} = -69.5077374606742$$
$$x_{28} = -42.0188017417635$$
$$x_{29} = -91.4988860358027$$
$$x_{30} = -57.7267650097125$$
$$x_{31} = -27.8816348006094$$
$$x_{32} = 44.3749962319558$$
$$x_{33} = 40.4480054149686$$
$$x_{34} = -45.9457925587507$$
$$x_{35} = 92.2842841992002$$
$$x_{36} = -96.2112750161874$$
$$x_{37} = -35.7356164345839$$
$$x_{38} = 60.0829594999048$$
$$x_{39} = 74.2201264410589$$
$$x_{40} = -100.138265833175$$
$$x_{41} = -67.9369411338793$$
$$x_{42} = 97.7820713429823$$
$$x_{43} = 84.4303025652257$$
$$x_{44} = -13.7444678594553$$
$$x_{45} = -20.0276531666349$$
$$x_{46} = -56.1559686829176$$
$$x_{47} = 42.0188017417635$$
$$x_{48} = -78.1471172580461$$
$$x_{49} = 52.2289778659303$$
$$x_{50} = -64.009950316892$$
$$x_{51} = 23.9546439836222$$
$$x_{52} = -65.5807466436869$$
$$x_{53} = 12.1736715326604$$
$$x_{54} = -61.6537558266997$$
$$x_{55} = 49.872783375738$$
$$x_{56} = -74.2201264410589$$
$$x_{57} = 16.1006623496477$$
$$x_{58} = 62.4391539900971$$
$$x_{59} = -97.7820713429823$$
$$x_{60} = 96.2112750161874$$
$$x_{61} = -71.8639319508665$$
$$x_{62} = 34.164820107789$$
$$x_{63} = 70.2931356240716$$
$$x_{64} = 30.2378292908018$$
$$x_{65} = -75.7909227678538$$
$$x_{66} = -31.8086256175967$$
$$x_{67} = 1.96349540849362$$
$$x_{68} = -60.0829594999048$$
$$x_{69} = 18.45685683984$$
$$x_{70} = -43.5895980685584$$
$$x_{71} = 36.5210145979813$$
$$x_{72} = 82.0741080750334$$
$$x_{73} = 67.9369411338793$$
$$x_{74} = -7.46128255227576$$
$$x_{75} = -16.1006623496477$$
$$x_{76} = 45.9457925587507$$
$$x_{77} = 8.24668071567321$$
$$x_{78} = 5.89048622548086$$
$$x_{79} = -23.9546439836222$$
$$x_{80} = -53.7997741927252$$
$$x_{81} = -38.0918109247762$$
$$x_{82} = -52.2289778659303$$
$$x_{83} = 86.0010988920206$$
$$x_{84} = -5.89048622548086$$
$$x_{85} = -30.2378292908018$$
$$x_{86} = 58.5121631731099$$
$$x_{87} = 38.0918109247762$$
$$x_{88} = -17.6714586764426$$
$$x_{89} = -34.164820107789$$
$$x_{90} = 66.3661448070844$$
$$x_{91} = 31.8086256175967$$
$$x_{92} = -83.6449044018282$$
$$x_{93} = 88.3572933822129$$
$$x_{94} = 27.8816348006094$$
$$x_{95} = -93.8550805259951$$
$$x_{96} = -25.5254403104171$$
$$x_{97} = 80.5033117482384$$
$$x_{98} = -12.1736715326604$$
$$x_{99} = -79.717913584841$$
$$x_{100} = 53.7997741927252$$
so we need to solve the equation:
$$\cot{\left(4 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{8}$$
Numerical solution
$$x_{1} = 26.3108384738145$$
$$x_{2} = 4.31968989868597$$
$$x_{3} = -82.0741080750334$$
$$x_{4} = 14.5298660228528$$
$$x_{5} = -47.5165888855456$$
$$x_{6} = 20.0276531666349$$
$$x_{7} = -86.0010988920206$$
$$x_{8} = 75.7909227678538$$
$$x_{9} = -49.872783375738$$
$$x_{10} = -21.5984494934298$$
$$x_{11} = -3.53429173528852$$
$$x_{12} = 48.3019870489431$$
$$x_{13} = 71.8639319508665$$
$$x_{14} = -9.8174770424681$$
$$x_{15} = -87.5718952188155$$
$$x_{16} = -89.9280897090078$$
$$x_{17} = 78.1471172580461$$
$$x_{18} = 100.138265833175$$
$$x_{19} = 9.8174770424681$$
$$x_{20} = 64.009950316892$$
$$x_{21} = 93.8550805259951$$
$$x_{22} = 22.3838476568273$$
$$x_{23} = 89.9280897090078$$
$$x_{24} = -1.96349540849362$$
$$x_{25} = 56.1559686829176$$
$$x_{26} = -39.6626072515711$$
$$x_{27} = -69.5077374606742$$
$$x_{28} = -42.0188017417635$$
$$x_{29} = -91.4988860358027$$
$$x_{30} = -57.7267650097125$$
$$x_{31} = -27.8816348006094$$
$$x_{32} = 44.3749962319558$$
$$x_{33} = 40.4480054149686$$
$$x_{34} = -45.9457925587507$$
$$x_{35} = 92.2842841992002$$
$$x_{36} = -96.2112750161874$$
$$x_{37} = -35.7356164345839$$
$$x_{38} = 60.0829594999048$$
$$x_{39} = 74.2201264410589$$
$$x_{40} = -100.138265833175$$
$$x_{41} = -67.9369411338793$$
$$x_{42} = 97.7820713429823$$
$$x_{43} = 84.4303025652257$$
$$x_{44} = -13.7444678594553$$
$$x_{45} = -20.0276531666349$$
$$x_{46} = -56.1559686829176$$
$$x_{47} = 42.0188017417635$$
$$x_{48} = -78.1471172580461$$
$$x_{49} = 52.2289778659303$$
$$x_{50} = -64.009950316892$$
$$x_{51} = 23.9546439836222$$
$$x_{52} = -65.5807466436869$$
$$x_{53} = 12.1736715326604$$
$$x_{54} = -61.6537558266997$$
$$x_{55} = 49.872783375738$$
$$x_{56} = -74.2201264410589$$
$$x_{57} = 16.1006623496477$$
$$x_{58} = 62.4391539900971$$
$$x_{59} = -97.7820713429823$$
$$x_{60} = 96.2112750161874$$
$$x_{61} = -71.8639319508665$$
$$x_{62} = 34.164820107789$$
$$x_{63} = 70.2931356240716$$
$$x_{64} = 30.2378292908018$$
$$x_{65} = -75.7909227678538$$
$$x_{66} = -31.8086256175967$$
$$x_{67} = 1.96349540849362$$
$$x_{68} = -60.0829594999048$$
$$x_{69} = 18.45685683984$$
$$x_{70} = -43.5895980685584$$
$$x_{71} = 36.5210145979813$$
$$x_{72} = 82.0741080750334$$
$$x_{73} = 67.9369411338793$$
$$x_{74} = -7.46128255227576$$
$$x_{75} = -16.1006623496477$$
$$x_{76} = 45.9457925587507$$
$$x_{77} = 8.24668071567321$$
$$x_{78} = 5.89048622548086$$
$$x_{79} = -23.9546439836222$$
$$x_{80} = -53.7997741927252$$
$$x_{81} = -38.0918109247762$$
$$x_{82} = -52.2289778659303$$
$$x_{83} = 86.0010988920206$$
$$x_{84} = -5.89048622548086$$
$$x_{85} = -30.2378292908018$$
$$x_{86} = 58.5121631731099$$
$$x_{87} = 38.0918109247762$$
$$x_{88} = -17.6714586764426$$
$$x_{89} = -34.164820107789$$
$$x_{90} = 66.3661448070844$$
$$x_{91} = 31.8086256175967$$
$$x_{92} = -83.6449044018282$$
$$x_{93} = 88.3572933822129$$
$$x_{94} = 27.8816348006094$$
$$x_{95} = -93.8550805259951$$
$$x_{96} = -25.5254403104171$$
$$x_{97} = 80.5033117482384$$
$$x_{98} = -12.1736715326604$$
$$x_{99} = -79.717913584841$$
$$x_{100} = 53.7997741927252$$
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
$$- 4 \cot^{2}{\left(4 x \right)} - 4 = 0$$
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
$$- 4 \cot^{2}{\left(4 x \right)} - 4 = 0$$
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
$$32 \left(\cot^{2}{\left(4 x \right)} + 1\right) \cot{\left(4 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{8}$$
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
$$\left(-\infty, \frac{\pi}{8}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{8}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$32 \left(\cot^{2}{\left(4 x \right)} + 1\right) \cot{\left(4 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{8}$$
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
$$\left(-\infty, \frac{\pi}{8}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{8}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \cot{\left(4 x \right)} = - \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \cot{\left(\infty \right)}$$
$$\lim_{x \to \infty} \cot{\left(4 x \right)} = \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \cot{\left(\infty \right)}$$
$$\lim_{x \to -\infty} \cot{\left(4 x \right)} = - \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \cot{\left(\infty \right)}$$
$$\lim_{x \to \infty} \cot{\left(4 x \right)} = \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \cot{\left(\infty \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(4*x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(4 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(4 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(4 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(4 x \right)}}{x}\right)$$
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:
$$\cot{\left(4 x \right)} = - \cot{\left(4 x \right)}$$
- No
$$\cot{\left(4 x \right)} = \cot{\left(4 x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\cot{\left(4 x \right)} = - \cot{\left(4 x \right)}$$
- No
$$\cot{\left(4 x \right)} = \cot{\left(4 x \right)}$$
- Yes
so, the function
is
odd