Mister Exam
Graphing y = exp-exp^(-1/x)
The solution
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-1
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x x
f(x) = e - E
$$f{\left(x \right)} = e^{x} - e^{- \frac{1}{x}}$$
f = exp(x) - 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:
$$e^{x} - e^{- \frac{1}{x}} = 0$$
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:
$$e^{x} - e^{- \frac{1}{x}} = 0$$
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
$$e^{x} - \frac{e^{- \frac{1}{x}}}{x^{2}} = 0$$
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
$$e^{x} - \frac{e^{- \frac{1}{x}}}{x^{2}} = 0$$
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
$$e^{x} + \frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{e^{- \frac{1}{x}}}{x^{4}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -27695.201429547$$
$$x_{2} = -28542.7681732437$$
$$x_{3} = -37866.1120404105$$
$$x_{4} = -20914.7749461278$$
$$x_{5} = -19219.7123808585$$
$$x_{6} = -38713.6959163438$$
$$x_{7} = -22609.8590326263$$
$$x_{8} = -13287.278555532$$
$$x_{9} = -33628.2078852771$$
$$x_{10} = -40408.8662068897$$
$$x_{11} = -35323.3662132531$$
$$x_{12} = 0.128513790124308$$
$$x_{13} = -26847.6370204238$$
$$x_{14} = -15829.6788313566$$
$$x_{15} = -31085.48052493$$
$$x_{16} = -14134.729248906$$
$$x_{17} = -24304.9602057369$$
$$x_{18} = -41256.4525181345$$
$$x_{19} = -16677.1729258949$$
$$x_{20} = -32780.6306440145$$
$$x_{21} = -34475.7864395299$$
$$x_{22} = -36170.9471216022$$
$$x_{23} = -18372.1909931076$$
$$x_{24} = -30237.9078887233$$
$$x_{25} = -17524.6774514389$$
$$x_{26} = -12439.8481955955$$
$$x_{27} = 0.329545076041446$$
$$x_{28} = -37018.5290874254$$
$$x_{29} = -21762.3146082237$$
$$x_{30} = -14982.1968951896$$
$$x_{31} = -11592.4425720895$$
$$x_{32} = -29390.3370520836$$
$$x_{33} = -39561.2806564628$$
$$x_{34} = -20067.2406387144$$
$$x_{35} = -26000.0751709394$$
$$x_{36} = -23457.4077112318$$
$$x_{37} = -25152.5161360503$$
$$x_{38} = -42104.0395446566$$
$$x_{39} = -31933.0548190758$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(e^{x} + \frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{e^{- \frac{1}{x}}}{x^{4}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(e^{x} + \frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{e^{- \frac{1}{x}}}{x^{4}}\right) = 1$$
- 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
$$\left(-\infty, 0.128513790124308\right] \cup \left[0.329545076041446, \infty\right)$$
Convex at the intervals
$$\left[0.128513790124308, 0.329545076041446\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$e^{x} + \frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{e^{- \frac{1}{x}}}{x^{4}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -27695.201429547$$
$$x_{2} = -28542.7681732437$$
$$x_{3} = -37866.1120404105$$
$$x_{4} = -20914.7749461278$$
$$x_{5} = -19219.7123808585$$
$$x_{6} = -38713.6959163438$$
$$x_{7} = -22609.8590326263$$
$$x_{8} = -13287.278555532$$
$$x_{9} = -33628.2078852771$$
$$x_{10} = -40408.8662068897$$
$$x_{11} = -35323.3662132531$$
$$x_{12} = 0.128513790124308$$
$$x_{13} = -26847.6370204238$$
$$x_{14} = -15829.6788313566$$
$$x_{15} = -31085.48052493$$
$$x_{16} = -14134.729248906$$
$$x_{17} = -24304.9602057369$$
$$x_{18} = -41256.4525181345$$
$$x_{19} = -16677.1729258949$$
$$x_{20} = -32780.6306440145$$
$$x_{21} = -34475.7864395299$$
$$x_{22} = -36170.9471216022$$
$$x_{23} = -18372.1909931076$$
$$x_{24} = -30237.9078887233$$
$$x_{25} = -17524.6774514389$$
$$x_{26} = -12439.8481955955$$
$$x_{27} = 0.329545076041446$$
$$x_{28} = -37018.5290874254$$
$$x_{29} = -21762.3146082237$$
$$x_{30} = -14982.1968951896$$
$$x_{31} = -11592.4425720895$$
$$x_{32} = -29390.3370520836$$
$$x_{33} = -39561.2806564628$$
$$x_{34} = -20067.2406387144$$
$$x_{35} = -26000.0751709394$$
$$x_{36} = -23457.4077112318$$
$$x_{37} = -25152.5161360503$$
$$x_{38} = -42104.0395446566$$
$$x_{39} = -31933.0548190758$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(e^{x} + \frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{e^{- \frac{1}{x}}}{x^{4}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(e^{x} + \frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{e^{- \frac{1}{x}}}{x^{4}}\right) = 1$$
- 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
$$\left(-\infty, 0.128513790124308\right] \cup \left[0.329545076041446, \infty\right)$$
Convex at the intervals
$$\left[0.128513790124308, 0.329545076041446\right]$$
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
$$\lim_{x \to -\infty}\left(e^{x} - e^{- \frac{1}{x}}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(e^{x} - e^{- \frac{1}{x}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(e^{x} - e^{- \frac{1}{x}}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(e^{x} - e^{- \frac{1}{x}}\right) = \infty$$
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 exp(x) - E^(-1/x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e^{- \frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{x} - e^{- \frac{1}{x}}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e^{- \frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{x} - e^{- \frac{1}{x}}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$e^{x} - e^{- \frac{1}{x}} = - e^{\frac{1}{x}} + e^{- x}$$
- No
$$e^{x} - e^{- \frac{1}{x}} = e^{\frac{1}{x}} - e^{- x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$e^{x} - e^{- \frac{1}{x}} = - e^{\frac{1}{x}} + e^{- x}$$
- No
$$e^{x} - e^{- \frac{1}{x}} = e^{\frac{1}{x}} - e^{- x}$$
- No
so, the function
not is
neither even, nor odd