Mister Exam
Graphing y = x^2*(exp(1/x)-1)
The solution
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/ 1 \
| - |
2 | x |
f(x) = x *\e - 1/
$$f{\left(x \right)} = x^{2} \left(e^{\frac{1}{x}} - 1\right)$$
f = x^2*(exp(1/x) - 1)
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:
$$x^{2} \left(e^{\frac{1}{x}} - 1\right) = 0$$
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:
$$x^{2} \left(e^{\frac{1}{x}} - 1\right) = 0$$
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
$$2 x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}} = 0$$
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
$$2 x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}} = 0$$
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
$$2 e^{\frac{1}{x}} - 2 + \frac{\left(2 + \frac{1}{x}\right) e^{\frac{1}{x}}}{x} - \frac{4 e^{\frac{1}{x}}}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -41572.8577126531$$
$$x_{2} = -27163.6741592809$$
$$x_{3} = 33016.689278842$$
$$x_{4} = -22078.1076567522$$
$$x_{5} = 24540.7184974027$$
$$x_{6} = -21230.5159966738$$
$$x_{7} = -26316.0781549163$$
$$x_{8} = 26235.9089676166$$
$$x_{9} = -39877.6563212567$$
$$x_{10} = 33864.2883219652$$
$$x_{11} = 34711.8876179679$$
$$x_{12} = -32249.2593003649$$
$$x_{13} = -28011.2706635918$$
$$x_{14} = 19455.1658143876$$
$$x_{15} = 14369.6642623984$$
$$x_{16} = 7589.25045421953$$
$$x_{17} = -34792.0562988759$$
$$x_{18} = 18607.5775454055$$
$$x_{19} = 10979.3930753798$$
$$x_{20} = -14449.8361989337$$
$$x_{21} = -10212.0162696652$$
$$x_{22} = -22925.7002508004$$
$$x_{23} = 22845.5306872408$$
$$x_{24} = -25468.4827006091$$
$$x_{25} = -19535.3359672021$$
$$x_{26} = 8436.76567769069$$
$$x_{27} = -12754.6920516308$$
$$x_{28} = -38182.4555602223$$
$$x_{29} = 10131.8403739245$$
$$x_{30} = 36407.086898158$$
$$x_{31} = 39797.4878019001$$
$$x_{32} = -16992.576626392$$
$$x_{33} = -29706.4650002999$$
$$x_{34} = 38949.8873148361$$
$$x_{35} = 28778.6986357815$$
$$x_{36} = -20382.9253876527$$
$$x_{37} = -18687.7478979172$$
$$x_{38} = 32169.0905085256$$
$$x_{39} = -35639.6557979241$$
$$x_{40} = -30554.0627587705$$
$$x_{41} = -13602.2621062988$$
$$x_{42} = 16064.8227859878$$
$$x_{43} = 25388.3134331762$$
$$x_{44} = 35559.4871488201$$
$$x_{45} = -9364.47437918346$$
$$x_{46} = -11907.126904562$$
$$x_{47} = 17759.9907920548$$
$$x_{48} = -17840.1613734673$$
$$x_{49} = -24620.8878534009$$
$$x_{50} = 27083.5050447204$$
$$x_{51} = -28858.8676236457$$
$$x_{52} = 16912.4057808903$$
$$x_{53} = 11826.9531024289$$
$$x_{54} = 21150.3461737677$$
$$x_{55} = -33096.8580308478$$
$$x_{56} = 9284.29698337792$$
$$x_{57} = 42340.290161585$$
$$x_{58} = 20302.7554100964$$
$$x_{59} = -40725.2569430912$$
$$x_{60} = 31321.4920330961$$
$$x_{61} = -37334.8554429973$$
$$x_{62} = 13522.0896629389$$
$$x_{63} = -39030.0558567963$$
$$x_{64} = -11059.5678037563$$
$$x_{65} = 21997.9379709968$$
$$x_{66} = -36487.255517655$$
$$x_{67} = 23693.1242242781$$
$$x_{68} = -6821.94305309729$$
$$x_{69} = -33944.4570370887$$
$$x_{70} = 30473.8938770785$$
$$x_{71} = 41492.6892344268$$
$$x_{72} = -23773.2936784993$$
$$x_{73} = 38102.2869941325$$
$$x_{74} = -16144.9939383678$$
$$x_{75} = -42420.4586210683$$
$$x_{76} = -8516.9450458411$$
$$x_{77} = -31401.6608680414$$
$$x_{78} = -15297.4136539103$$
$$x_{79} = 15217.2421420941$$
$$x_{80} = 37254.6868511149$$
$$x_{81} = -7669.43248826304$$
$$x_{82} = 27931.1016152579$$
$$x_{83} = 40645.0884449418$$
$$x_{84} = 29626.296067797$$
$$x_{85} = 6741.75729176678$$
$$x_{86} = 12674.518996728$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(2 e^{\frac{1}{x}} - 2 + \frac{\left(2 + \frac{1}{x}\right) e^{\frac{1}{x}}}{x} - \frac{4 e^{\frac{1}{x}}}{x}\right) = -2$$
$$\lim_{x \to 0^+}\left(2 e^{\frac{1}{x}} - 2 + \frac{\left(2 + \frac{1}{x}\right) e^{\frac{1}{x}}}{x} - \frac{4 e^{\frac{1}{x}}}{x}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection 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
$$\left[42340.290161585, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -33096.8580308478\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 e^{\frac{1}{x}} - 2 + \frac{\left(2 + \frac{1}{x}\right) e^{\frac{1}{x}}}{x} - \frac{4 e^{\frac{1}{x}}}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -41572.8577126531$$
$$x_{2} = -27163.6741592809$$
$$x_{3} = 33016.689278842$$
$$x_{4} = -22078.1076567522$$
$$x_{5} = 24540.7184974027$$
$$x_{6} = -21230.5159966738$$
$$x_{7} = -26316.0781549163$$
$$x_{8} = 26235.9089676166$$
$$x_{9} = -39877.6563212567$$
$$x_{10} = 33864.2883219652$$
$$x_{11} = 34711.8876179679$$
$$x_{12} = -32249.2593003649$$
$$x_{13} = -28011.2706635918$$
$$x_{14} = 19455.1658143876$$
$$x_{15} = 14369.6642623984$$
$$x_{16} = 7589.25045421953$$
$$x_{17} = -34792.0562988759$$
$$x_{18} = 18607.5775454055$$
$$x_{19} = 10979.3930753798$$
$$x_{20} = -14449.8361989337$$
$$x_{21} = -10212.0162696652$$
$$x_{22} = -22925.7002508004$$
$$x_{23} = 22845.5306872408$$
$$x_{24} = -25468.4827006091$$
$$x_{25} = -19535.3359672021$$
$$x_{26} = 8436.76567769069$$
$$x_{27} = -12754.6920516308$$
$$x_{28} = -38182.4555602223$$
$$x_{29} = 10131.8403739245$$
$$x_{30} = 36407.086898158$$
$$x_{31} = 39797.4878019001$$
$$x_{32} = -16992.576626392$$
$$x_{33} = -29706.4650002999$$
$$x_{34} = 38949.8873148361$$
$$x_{35} = 28778.6986357815$$
$$x_{36} = -20382.9253876527$$
$$x_{37} = -18687.7478979172$$
$$x_{38} = 32169.0905085256$$
$$x_{39} = -35639.6557979241$$
$$x_{40} = -30554.0627587705$$
$$x_{41} = -13602.2621062988$$
$$x_{42} = 16064.8227859878$$
$$x_{43} = 25388.3134331762$$
$$x_{44} = 35559.4871488201$$
$$x_{45} = -9364.47437918346$$
$$x_{46} = -11907.126904562$$
$$x_{47} = 17759.9907920548$$
$$x_{48} = -17840.1613734673$$
$$x_{49} = -24620.8878534009$$
$$x_{50} = 27083.5050447204$$
$$x_{51} = -28858.8676236457$$
$$x_{52} = 16912.4057808903$$
$$x_{53} = 11826.9531024289$$
$$x_{54} = 21150.3461737677$$
$$x_{55} = -33096.8580308478$$
$$x_{56} = 9284.29698337792$$
$$x_{57} = 42340.290161585$$
$$x_{58} = 20302.7554100964$$
$$x_{59} = -40725.2569430912$$
$$x_{60} = 31321.4920330961$$
$$x_{61} = -37334.8554429973$$
$$x_{62} = 13522.0896629389$$
$$x_{63} = -39030.0558567963$$
$$x_{64} = -11059.5678037563$$
$$x_{65} = 21997.9379709968$$
$$x_{66} = -36487.255517655$$
$$x_{67} = 23693.1242242781$$
$$x_{68} = -6821.94305309729$$
$$x_{69} = -33944.4570370887$$
$$x_{70} = 30473.8938770785$$
$$x_{71} = 41492.6892344268$$
$$x_{72} = -23773.2936784993$$
$$x_{73} = 38102.2869941325$$
$$x_{74} = -16144.9939383678$$
$$x_{75} = -42420.4586210683$$
$$x_{76} = -8516.9450458411$$
$$x_{77} = -31401.6608680414$$
$$x_{78} = -15297.4136539103$$
$$x_{79} = 15217.2421420941$$
$$x_{80} = 37254.6868511149$$
$$x_{81} = -7669.43248826304$$
$$x_{82} = 27931.1016152579$$
$$x_{83} = 40645.0884449418$$
$$x_{84} = 29626.296067797$$
$$x_{85} = 6741.75729176678$$
$$x_{86} = 12674.518996728$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(2 e^{\frac{1}{x}} - 2 + \frac{\left(2 + \frac{1}{x}\right) e^{\frac{1}{x}}}{x} - \frac{4 e^{\frac{1}{x}}}{x}\right) = -2$$
$$\lim_{x \to 0^+}\left(2 e^{\frac{1}{x}} - 2 + \frac{\left(2 + \frac{1}{x}\right) e^{\frac{1}{x}}}{x} - \frac{4 e^{\frac{1}{x}}}{x}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection 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
$$\left[42340.290161585, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -33096.8580308478\right]$$
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
$$\lim_{x \to -\infty}\left(x^{2} \left(e^{\frac{1}{x}} - 1\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{2} \left(e^{\frac{1}{x}} - 1\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x^{2} \left(e^{\frac{1}{x}} - 1\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{2} \left(e^{\frac{1}{x}} - 1\right)\right) = \infty$$
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^2*(exp(1/x) - 1), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(x \left(e^{\frac{1}{x}} - 1\right)\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(x \left(e^{\frac{1}{x}} - 1\right)\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(x \left(e^{\frac{1}{x}} - 1\right)\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(x \left(e^{\frac{1}{x}} - 1\right)\right) = 1$$
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:
$$x^{2} \left(e^{\frac{1}{x}} - 1\right) = x^{2} \left(-1 + e^{- \frac{1}{x}}\right)$$
- No
$$x^{2} \left(e^{\frac{1}{x}} - 1\right) = - x^{2} \left(-1 + e^{- \frac{1}{x}}\right)$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x^{2} \left(e^{\frac{1}{x}} - 1\right) = x^{2} \left(-1 + e^{- \frac{1}{x}}\right)$$
- No
$$x^{2} \left(e^{\frac{1}{x}} - 1\right) = - x^{2} \left(-1 + e^{- \frac{1}{x}}\right)$$
- No
so, the function
not is
neither even, nor odd