Mister Exam
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-
How to use it?
- Graphing y =:
-
x^2+x-2
- log(1/2)x
- sqrt(3x)
-
arctan(e^(1/x))
-
Identical expressions
- arctan(e^(one /x))
- arc tangent of (e to the power of (1 divide by x))
- arc tangent of (e to the power of (one divide by x))
- arctan(e(1/x))
- arctane1/x
- arctane^1/x
- arctan(e^(1 divide by x))
Graphing y = arctan(e^(1/x))
The solution
You have entered
[src]
/x ___\ f(x) = atan\\/ e /
$$f{\left(x \right)} = \operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)}$$
f = atan(E^(1/x))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$- \frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{2}{x}} + 1\right)} = 0$$
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
$$- \frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{2}{x}} + 1\right)} = 0$$
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
$$\frac{\left(2 - \frac{2 e^{\frac{2}{x}}}{x \left(e^{\frac{2}{x}} + 1\right)} + \frac{1}{x}\right) e^{\frac{1}{x}}}{x^{3} \left(e^{\frac{2}{x}} + 1\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 22953.9833755605$$
$$x_{2} = -26212.7876215694$$
$$x_{3} = -21127.80572081$$
$$x_{4} = 18716.6535850519$$
$$x_{5} = 36514.5173378552$$
$$x_{6} = -40621.1114534134$$
$$x_{7} = -38078.4139226522$$
$$x_{8} = 40752.3389728928$$
$$x_{9} = 24648.9818766472$$
$$x_{10} = -19432.8866255406$$
$$x_{11} = -14348.5398262285$$
$$x_{12} = -18585.4468806785$$
$$x_{13} = 34819.4022863123$$
$$x_{14} = -12653.9861138132$$
$$x_{15} = -16890.6169386139$$
$$x_{16} = -33840.6231730589$$
$$x_{17} = 15327.0651843105$$
$$x_{18} = -34688.1768180891$$
$$x_{19} = 38209.6406795697$$
$$x_{20} = 10243.7120654759$$
$$x_{21} = 29734.1210542433$$
$$x_{22} = 12785.1624756624$$
$$x_{23} = -10112.5676113793$$
$$x_{24} = 39057.2051218745$$
$$x_{25} = -23670.2620983471$$
$$x_{26} = -32145.5238877213$$
$$x_{27} = -31297.9786810377$$
$$x_{28} = -41468.680436125$$
$$x_{29} = -36383.2911803601$$
$$x_{30} = 28039.0550600592$$
$$x_{31} = -28755.3639435987$$
$$x_{32} = -15195.8714954228$$
$$x_{33} = -27907.8337159018$$
$$x_{34} = 23801.4788865444$$
$$x_{35} = 17021.8181084218$$
$$x_{36} = 33971.8482572335$$
$$x_{37} = 11090.7927207691$$
$$x_{38} = 42447.4787845915$$
$$x_{39} = -22822.7678146879$$
$$x_{40} = -27060.308125254$$
$$x_{41} = -42316.2508311932$$
$$x_{42} = 39904.7712548156$$
$$x_{43} = 37362.0780429762$$
$$x_{44} = -35535.7328767582$$
$$x_{45} = 9396.72600086872$$
$$x_{46} = 35666.9587019242$$
$$x_{47} = 27191.5287260521$$
$$x_{48} = -13501.2424031439$$
$$x_{49} = 11937.9477960536$$
$$x_{50} = 26344.0074059076$$
$$x_{51} = -21975.2820150427$$
$$x_{52} = 16174.4297041018$$
$$x_{53} = -29602.898409969$$
$$x_{54} = 25496.4915997823$$
$$x_{55} = 30581.6599755962$$
$$x_{56} = -39773.5439733611$$
$$x_{57} = -30450.4367610008$$
$$x_{58} = 41599.9081792049$$
$$x_{59} = 20411.5510378044$$
$$x_{60} = 28886.5859664881$$
$$x_{61} = -24517.7639859043$$
$$x_{62} = -10959.6351537251$$
$$x_{63} = -38925.9780941372$$
$$x_{64} = -17738.022862144$$
$$x_{65} = -11806.779825885$$
$$x_{66} = 17869.2269967979$$
$$x_{67} = -32993.0721277129$$
$$x_{68} = 21259.0183704435$$
$$x_{69} = -25365.2727149136$$
$$x_{70} = 14479.728746347$$
$$x_{71} = 22106.4962042461$$
$$x_{72} = 19564.0955717255$$
$$x_{73} = -9265.59839966773$$
$$x_{74} = 13632.4256322851$$
$$x_{75} = -37230.8515755536$$
$$x_{76} = 33124.2967979173$$
$$x_{77} = -20280.3401242072$$
$$x_{78} = 31429.202420336$$
$$x_{79} = 32276.748110843$$
$$x_{80} = -16043.2319796366$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(2 - \frac{2 e^{\frac{2}{x}}}{x \left(e^{\frac{2}{x}} + 1\right)} + \frac{1}{x}\right) e^{\frac{1}{x}}}{x^{3} \left(e^{\frac{2}{x}} + 1\right)}\right) = 0$$
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\left(2 - \frac{2 e^{\frac{2}{x}}}{x \left(e^{\frac{2}{x}} + 1\right)} + \frac{1}{x}\right) e^{\frac{1}{x}}}{x^{3} \left(e^{\frac{2}{x}} + 1\right)}\right) = 0$$
Let's take the limit
- limits are equal, then skip the corresponding point
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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(2 - \frac{2 e^{\frac{2}{x}}}{x \left(e^{\frac{2}{x}} + 1\right)} + \frac{1}{x}\right) e^{\frac{1}{x}}}{x^{3} \left(e^{\frac{2}{x}} + 1\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 22953.9833755605$$
$$x_{2} = -26212.7876215694$$
$$x_{3} = -21127.80572081$$
$$x_{4} = 18716.6535850519$$
$$x_{5} = 36514.5173378552$$
$$x_{6} = -40621.1114534134$$
$$x_{7} = -38078.4139226522$$
$$x_{8} = 40752.3389728928$$
$$x_{9} = 24648.9818766472$$
$$x_{10} = -19432.8866255406$$
$$x_{11} = -14348.5398262285$$
$$x_{12} = -18585.4468806785$$
$$x_{13} = 34819.4022863123$$
$$x_{14} = -12653.9861138132$$
$$x_{15} = -16890.6169386139$$
$$x_{16} = -33840.6231730589$$
$$x_{17} = 15327.0651843105$$
$$x_{18} = -34688.1768180891$$
$$x_{19} = 38209.6406795697$$
$$x_{20} = 10243.7120654759$$
$$x_{21} = 29734.1210542433$$
$$x_{22} = 12785.1624756624$$
$$x_{23} = -10112.5676113793$$
$$x_{24} = 39057.2051218745$$
$$x_{25} = -23670.2620983471$$
$$x_{26} = -32145.5238877213$$
$$x_{27} = -31297.9786810377$$
$$x_{28} = -41468.680436125$$
$$x_{29} = -36383.2911803601$$
$$x_{30} = 28039.0550600592$$
$$x_{31} = -28755.3639435987$$
$$x_{32} = -15195.8714954228$$
$$x_{33} = -27907.8337159018$$
$$x_{34} = 23801.4788865444$$
$$x_{35} = 17021.8181084218$$
$$x_{36} = 33971.8482572335$$
$$x_{37} = 11090.7927207691$$
$$x_{38} = 42447.4787845915$$
$$x_{39} = -22822.7678146879$$
$$x_{40} = -27060.308125254$$
$$x_{41} = -42316.2508311932$$
$$x_{42} = 39904.7712548156$$
$$x_{43} = 37362.0780429762$$
$$x_{44} = -35535.7328767582$$
$$x_{45} = 9396.72600086872$$
$$x_{46} = 35666.9587019242$$
$$x_{47} = 27191.5287260521$$
$$x_{48} = -13501.2424031439$$
$$x_{49} = 11937.9477960536$$
$$x_{50} = 26344.0074059076$$
$$x_{51} = -21975.2820150427$$
$$x_{52} = 16174.4297041018$$
$$x_{53} = -29602.898409969$$
$$x_{54} = 25496.4915997823$$
$$x_{55} = 30581.6599755962$$
$$x_{56} = -39773.5439733611$$
$$x_{57} = -30450.4367610008$$
$$x_{58} = 41599.9081792049$$
$$x_{59} = 20411.5510378044$$
$$x_{60} = 28886.5859664881$$
$$x_{61} = -24517.7639859043$$
$$x_{62} = -10959.6351537251$$
$$x_{63} = -38925.9780941372$$
$$x_{64} = -17738.022862144$$
$$x_{65} = -11806.779825885$$
$$x_{66} = 17869.2269967979$$
$$x_{67} = -32993.0721277129$$
$$x_{68} = 21259.0183704435$$
$$x_{69} = -25365.2727149136$$
$$x_{70} = 14479.728746347$$
$$x_{71} = 22106.4962042461$$
$$x_{72} = 19564.0955717255$$
$$x_{73} = -9265.59839966773$$
$$x_{74} = 13632.4256322851$$
$$x_{75} = -37230.8515755536$$
$$x_{76} = 33124.2967979173$$
$$x_{77} = -20280.3401242072$$
$$x_{78} = 31429.202420336$$
$$x_{79} = 32276.748110843$$
$$x_{80} = -16043.2319796366$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(2 - \frac{2 e^{\frac{2}{x}}}{x \left(e^{\frac{2}{x}} + 1\right)} + \frac{1}{x}\right) e^{\frac{1}{x}}}{x^{3} \left(e^{\frac{2}{x}} + 1\right)}\right) = 0$$
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\left(2 - \frac{2 e^{\frac{2}{x}}}{x \left(e^{\frac{2}{x}} + 1\right)} + \frac{1}{x}\right) e^{\frac{1}{x}}}{x^{3} \left(e^{\frac{2}{x}} + 1\right)}\right) = 0$$
Let's take the limit
- limits are equal, then skip the corresponding point
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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = \frac{\pi}{4}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\pi}{4}$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = \frac{\pi}{4}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\pi}{4}$$
$$\lim_{x \to -\infty} \operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = \frac{\pi}{4}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\pi}{4}$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = \frac{\pi}{4}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\pi}{4}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(E^(1/x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = \operatorname{atan}{\left(e^{- \frac{1}{x}} \right)}$$
- No
$$\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = - \operatorname{atan}{\left(e^{- \frac{1}{x}} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = \operatorname{atan}{\left(e^{- \frac{1}{x}} \right)}$$
- No
$$\operatorname{atan}{\left(e^{1 \cdot \frac{1}{x}} \right)} = - \operatorname{atan}{\left(e^{- \frac{1}{x}} \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph