Mister Exam
Graphing y = x*exp((-1)/x^2)
The solution
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-1
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2
x
f(x) = x*e
$$f{\left(x \right)} = x e^{- \frac{1}{x^{2}}}$$
f = x*exp(-1/x^2)
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:
$$x e^{- \frac{1}{x^{2}}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -0.0244202520149352$$
$$x_{2} = 0.0293452508727927$$
$$x_{3} = -0.133632892750401$$
$$x_{4} = -0.0227537545194084$$
$$x_{5} = -0.183894714065453$$
$$x_{6} = 0.0321766152374528$$
$$x_{7} = 0.0232158501114876$$
$$x_{8} = 0.0249532855619005$$
$$x_{9} = 0.036926488011599$$
$$x_{10} = -0.0417393197169768$$
$$x_{11} = -0.021763591131046$$
$$x_{12} = -0.0232833999730666$$
$$x_{13} = -0.0250313381193013$$
$$x_{14} = 0.12309154644048$$
$$x_{15} = -0.0208559930312259$$
$$x_{16} = -0.038525807483289$$
$$x_{17} = -0.0256737822249551$$
$$x_{18} = -0.030346654331782$$
$$x_{19} = -0.0477072054397396$$
$$x_{20} = 0.0221859806652269$$
$$x_{21} = 0.181652878435138$$
$$x_{22} = 0.0662451871546496$$
$$x_{23} = 0.146795111116457$$
$$x_{24} = -0.0770662641835069$$
$$x_{25} = -0.165868912312062$$
$$x_{26} = 0.0217045588314378$$
$$x_{27} = 0.0415228642966208$$
$$x_{28} = -0.0455371039234589$$
$$x_{29} = 0.0585083569901389$$
$$x_{30} = 0.0203779658308351$$
$$x_{31} = 0.0826198554087581$$
$$x_{32} = -0.120519864876116$$
$$x_{33} = 0.0383413017870185$$
$$x_{34} = 0.0302320264679165$$
$$x_{35} = 0.0269717092416228$$
$$x_{36} = -0.0312959333072649$$
$$x_{37} = -0.0204299952964932$$
$$x_{38} = 0.0452796302067717$$
$$x_{39} = 0.0433200278767269$$
$$x_{40} = 0.131882716433614$$
$$x_{41} = 0.0277190622277982$$
$$x_{42} = 0.0212435817779227$$
$$x_{43} = -0.0500942204868285$$
$$x_{44} = -0.020021048405059$$
$$x_{45} = 0.0243459571462513$$
$$x_{46} = -0.124999938811044$$
$$x_{47} = -0.0333844437437797$$
$$x_{48} = -0.0370976070237541$$
$$x_{49} = 0.0199710780829502$$
$$x_{50} = 0.0311740398062829$$
$$x_{51} = 0.119018992007155$$
$$x_{52} = 0.0552792644801734$$
$$x_{53} = -0.0238382833443369$$
$$x_{54} = 0.109682267911704$$
$$x_{55} = 0.1636531623797$$
$$x_{56} = -0.0294532436103678$$
$$x_{57} = 0.0398687609878657$$
$$x_{58} = -0.0213001304582747$$
$$x_{59} = -0.0357714467483564$$
$$x_{60} = 0.0208017745000695$$
$$x_{61} = -0.0400682866023276$$
$$x_{62} = -0.111197477764175$$
$$x_{63} = 0.0621373289334657$$
$$x_{64} = -0.0910509761609461$$
$$x_{65} = 0.0356123095325598$$
$$x_{66} = 0.0988943301147501$$
$$x_{67} = -0.0278154024442179$$
$$x_{68} = -0.0323064873348789$$
$$x_{69} = -0.0345367744823859$$
$$x_{70} = -0.100125924597671$$
$$x_{71} = -0.0556633038384204$$
$$x_{72} = 0.0262635845882966$$
$$x_{73} = -0.0667970907316034$$
$$x_{74} = 0.0523874949317494$$
$$x_{75} = -0.043555660742004$$
$$x_{76} = -0.0834790129684383$$
$$x_{77} = 0.047424711041097$$
$$x_{78} = -0.0263500607496007$$
$$x_{79} = -0.0527323347377053$$
$$x_{80} = -0.0589386675893727$$
$$x_{81} = -0.0626227878905622$$
$$x_{82} = -0.0270629178212322$$
$$x_{83} = -0.0715661066999077$$
$$x_{84} = 0.0497828778122316$$
$$x_{85} = 0.0343884039881248$$
$$x_{86} = -0.0286109120312936$$
$$x_{87} = -0.0222476638299303$$
$$x_{88} = 0.0763330847355461$$
$$x_{89} = -0.148803455740981$$
$$x_{90} = 0.0285089947797144$$
$$x_{91} = 0.0237674812678401$$
$$x_{92} = 0.0255916796116544$$
$$x_{93} = 0.0226892378873187$$
$$x_{94} = 0.033245783719257$$
$$x_{95} = 0.0900305071100009$$
$$x_{96} = 0.0709331616586192$$
so we need to solve the equation:
$$x e^{- \frac{1}{x^{2}}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -0.0244202520149352$$
$$x_{2} = 0.0293452508727927$$
$$x_{3} = -0.133632892750401$$
$$x_{4} = -0.0227537545194084$$
$$x_{5} = -0.183894714065453$$
$$x_{6} = 0.0321766152374528$$
$$x_{7} = 0.0232158501114876$$
$$x_{8} = 0.0249532855619005$$
$$x_{9} = 0.036926488011599$$
$$x_{10} = -0.0417393197169768$$
$$x_{11} = -0.021763591131046$$
$$x_{12} = -0.0232833999730666$$
$$x_{13} = -0.0250313381193013$$
$$x_{14} = 0.12309154644048$$
$$x_{15} = -0.0208559930312259$$
$$x_{16} = -0.038525807483289$$
$$x_{17} = -0.0256737822249551$$
$$x_{18} = -0.030346654331782$$
$$x_{19} = -0.0477072054397396$$
$$x_{20} = 0.0221859806652269$$
$$x_{21} = 0.181652878435138$$
$$x_{22} = 0.0662451871546496$$
$$x_{23} = 0.146795111116457$$
$$x_{24} = -0.0770662641835069$$
$$x_{25} = -0.165868912312062$$
$$x_{26} = 0.0217045588314378$$
$$x_{27} = 0.0415228642966208$$
$$x_{28} = -0.0455371039234589$$
$$x_{29} = 0.0585083569901389$$
$$x_{30} = 0.0203779658308351$$
$$x_{31} = 0.0826198554087581$$
$$x_{32} = -0.120519864876116$$
$$x_{33} = 0.0383413017870185$$
$$x_{34} = 0.0302320264679165$$
$$x_{35} = 0.0269717092416228$$
$$x_{36} = -0.0312959333072649$$
$$x_{37} = -0.0204299952964932$$
$$x_{38} = 0.0452796302067717$$
$$x_{39} = 0.0433200278767269$$
$$x_{40} = 0.131882716433614$$
$$x_{41} = 0.0277190622277982$$
$$x_{42} = 0.0212435817779227$$
$$x_{43} = -0.0500942204868285$$
$$x_{44} = -0.020021048405059$$
$$x_{45} = 0.0243459571462513$$
$$x_{46} = -0.124999938811044$$
$$x_{47} = -0.0333844437437797$$
$$x_{48} = -0.0370976070237541$$
$$x_{49} = 0.0199710780829502$$
$$x_{50} = 0.0311740398062829$$
$$x_{51} = 0.119018992007155$$
$$x_{52} = 0.0552792644801734$$
$$x_{53} = -0.0238382833443369$$
$$x_{54} = 0.109682267911704$$
$$x_{55} = 0.1636531623797$$
$$x_{56} = -0.0294532436103678$$
$$x_{57} = 0.0398687609878657$$
$$x_{58} = -0.0213001304582747$$
$$x_{59} = -0.0357714467483564$$
$$x_{60} = 0.0208017745000695$$
$$x_{61} = -0.0400682866023276$$
$$x_{62} = -0.111197477764175$$
$$x_{63} = 0.0621373289334657$$
$$x_{64} = -0.0910509761609461$$
$$x_{65} = 0.0356123095325598$$
$$x_{66} = 0.0988943301147501$$
$$x_{67} = -0.0278154024442179$$
$$x_{68} = -0.0323064873348789$$
$$x_{69} = -0.0345367744823859$$
$$x_{70} = -0.100125924597671$$
$$x_{71} = -0.0556633038384204$$
$$x_{72} = 0.0262635845882966$$
$$x_{73} = -0.0667970907316034$$
$$x_{74} = 0.0523874949317494$$
$$x_{75} = -0.043555660742004$$
$$x_{76} = -0.0834790129684383$$
$$x_{77} = 0.047424711041097$$
$$x_{78} = -0.0263500607496007$$
$$x_{79} = -0.0527323347377053$$
$$x_{80} = -0.0589386675893727$$
$$x_{81} = -0.0626227878905622$$
$$x_{82} = -0.0270629178212322$$
$$x_{83} = -0.0715661066999077$$
$$x_{84} = 0.0497828778122316$$
$$x_{85} = 0.0343884039881248$$
$$x_{86} = -0.0286109120312936$$
$$x_{87} = -0.0222476638299303$$
$$x_{88} = 0.0763330847355461$$
$$x_{89} = -0.148803455740981$$
$$x_{90} = 0.0285089947797144$$
$$x_{91} = 0.0237674812678401$$
$$x_{92} = 0.0255916796116544$$
$$x_{93} = 0.0226892378873187$$
$$x_{94} = 0.033245783719257$$
$$x_{95} = 0.0900305071100009$$
$$x_{96} = 0.0709331616586192$$
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
$$e^{- \frac{1}{x^{2}}} + \frac{2 e^{- \frac{1}{x^{2}}}}{x^{2}} = 0$$
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
$$e^{- \frac{1}{x^{2}}} + \frac{2 e^{- \frac{1}{x^{2}}}}{x^{2}} = 0$$
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
$$\frac{2 \left(-1 + \frac{2}{x^{2}}\right) e^{- \frac{1}{x^{2}}}}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \sqrt{2}$$
$$x_{2} = \sqrt{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(-1 + \frac{2}{x^{2}}\right) e^{- \frac{1}{x^{2}}}}{x^{3}}\right) = 0$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(-1 + \frac{2}{x^{2}}\right) e^{- \frac{1}{x^{2}}}}{x^{3}}\right) = 0$$
- 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:
Concave at the intervals
$$\left(-\infty, - \sqrt{2}\right]$$
Convex at the intervals
$$\left[\sqrt{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(-1 + \frac{2}{x^{2}}\right) e^{- \frac{1}{x^{2}}}}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \sqrt{2}$$
$$x_{2} = \sqrt{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(-1 + \frac{2}{x^{2}}\right) e^{- \frac{1}{x^{2}}}}{x^{3}}\right) = 0$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(-1 + \frac{2}{x^{2}}\right) e^{- \frac{1}{x^{2}}}}{x^{3}}\right) = 0$$
- 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:
Concave at the intervals
$$\left(-\infty, - \sqrt{2}\right]$$
Convex at the intervals
$$\left[\sqrt{2}, \infty\right)$$
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
$$\lim_{x \to -\infty}\left(x e^{- \frac{1}{x^{2}}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x e^{- \frac{1}{x^{2}}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x e^{- \frac{1}{x^{2}}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x e^{- \frac{1}{x^{2}}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*exp(-1/x^2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} e^{- \frac{1}{x^{2}}} = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty} e^{- \frac{1}{x^{2}}} = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty} e^{- \frac{1}{x^{2}}} = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty} e^{- \frac{1}{x^{2}}} = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
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:
$$x e^{- \frac{1}{x^{2}}} = - x e^{- \frac{1}{x^{2}}}$$
- No
$$x e^{- \frac{1}{x^{2}}} = x e^{- \frac{1}{x^{2}}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x e^{- \frac{1}{x^{2}}} = - x e^{- \frac{1}{x^{2}}}$$
- No
$$x e^{- \frac{1}{x^{2}}} = x e^{- \frac{1}{x^{2}}}$$
- No
so, the function
not is
neither even, nor odd