Mister Exam
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-
How to use it?
- Graphing y =:
- x^3-6x^2-15x+7
- x^3-12x^2+145
- -(x-2)^2-7
- 1/(x^2+2x-3)
-
Identical expressions
- exp^(two -x)/(two -x)*x
- exponent of to the power of (2 minus x) divide by (2 minus x) multiply by x
- exponent of to the power of (two minus x) divide by (two minus x) multiply by x
- exp(2-x)/(2-x)*x
- exp2-x/2-x*x
- exp^(2-x)/(2-x)x
- exp(2-x)/(2-x)x
- exp2-x/2-xx
- exp^2-x/2-xx
- exp^(2-x) divide by (2-x)*x
-
Similar expressions
- exp^(2+x)/(2-x)*x
- exp^(2-x)/(2+x)*x
Graphing y = exp^(2-x)/(2-x)*x
The solution
You have entered
[src]
2 - x
e *x
f(x) = --------
2 - x
$$f{\left(x \right)} = \frac{x e^{2 - x}}{2 - x}$$
f = x*E^(2 - x)/(2 - x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 2$$
$$x_{1} = 2$$
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:
$$\frac{x e^{2 - x}}{2 - x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 33.0571776157327$$
$$x_{2} = 121.178102581297$$
$$x_{3} = 87.1740763981455$$
$$x_{4} = 51.1521535131471$$
$$x_{5} = 91.1748368518887$$
$$x_{6} = 69.1683555957443$$
$$x_{7} = 49.1486096800712$$
$$x_{8} = 55.15769628377$$
$$x_{9} = 89.1744717733504$$
$$x_{10} = 57.1598899821613$$
$$x_{11} = 115.177672108263$$
$$x_{12} = 111.177340056896$$
$$x_{13} = 41.124918517446$$
$$x_{14} = 93.175174668055$$
$$x_{15} = 107.176964578548$$
$$x_{16} = 53.1551454444043$$
$$x_{17} = 99.1760496128704$$
$$x_{18} = 101.176302029271$$
$$x_{19} = 71.1692685841321$$
$$x_{20} = 119.177967199791$$
$$x_{21} = 43.132871687516$$
$$x_{22} = 63.1649073358433$$
$$x_{23} = 95.1754878837313$$
$$x_{24} = 79.1721141752132$$
$$x_{25} = 83.1731803564069$$
$$x_{26} = 77.1715033698701$$
$$x_{27} = 65.1661941485493$$
$$x_{28} = 117.177823918444$$
$$x_{29} = 85.1736472548213$$
$$x_{30} = 73.1700898713663$$
$$x_{31} = 61.1634503178614$$
$$x_{32} = 45.1392149059937$$
$$x_{33} = 81.1726711050428$$
$$x_{34} = 103.176537720436$$
$$x_{35} = 109.177158203895$$
$$x_{36} = 35.0830976103373$$
$$x_{37} = 75.1708314419527$$
$$x_{38} = 67.1673365541753$$
$$x_{39} = 39.1147302712608$$
$$x_{40} = 113.177511076155$$
$$x_{41} = 105.176758137916$$
$$x_{42} = 31.0180079017645$$
$$x_{43} = 0$$
$$x_{44} = 37.1013285290104$$
$$x_{45} = 97.1757788421662$$
$$x_{46} = 47.1443649826676$$
$$x_{47} = 59.1617911706003$$
so we need to solve the equation:
$$\frac{x e^{2 - x}}{2 - x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 33.0571776157327$$
$$x_{2} = 121.178102581297$$
$$x_{3} = 87.1740763981455$$
$$x_{4} = 51.1521535131471$$
$$x_{5} = 91.1748368518887$$
$$x_{6} = 69.1683555957443$$
$$x_{7} = 49.1486096800712$$
$$x_{8} = 55.15769628377$$
$$x_{9} = 89.1744717733504$$
$$x_{10} = 57.1598899821613$$
$$x_{11} = 115.177672108263$$
$$x_{12} = 111.177340056896$$
$$x_{13} = 41.124918517446$$
$$x_{14} = 93.175174668055$$
$$x_{15} = 107.176964578548$$
$$x_{16} = 53.1551454444043$$
$$x_{17} = 99.1760496128704$$
$$x_{18} = 101.176302029271$$
$$x_{19} = 71.1692685841321$$
$$x_{20} = 119.177967199791$$
$$x_{21} = 43.132871687516$$
$$x_{22} = 63.1649073358433$$
$$x_{23} = 95.1754878837313$$
$$x_{24} = 79.1721141752132$$
$$x_{25} = 83.1731803564069$$
$$x_{26} = 77.1715033698701$$
$$x_{27} = 65.1661941485493$$
$$x_{28} = 117.177823918444$$
$$x_{29} = 85.1736472548213$$
$$x_{30} = 73.1700898713663$$
$$x_{31} = 61.1634503178614$$
$$x_{32} = 45.1392149059937$$
$$x_{33} = 81.1726711050428$$
$$x_{34} = 103.176537720436$$
$$x_{35} = 109.177158203895$$
$$x_{36} = 35.0830976103373$$
$$x_{37} = 75.1708314419527$$
$$x_{38} = 67.1673365541753$$
$$x_{39} = 39.1147302712608$$
$$x_{40} = 113.177511076155$$
$$x_{41} = 105.176758137916$$
$$x_{42} = 31.0180079017645$$
$$x_{43} = 0$$
$$x_{44} = 37.1013285290104$$
$$x_{45} = 97.1757788421662$$
$$x_{46} = 47.1443649826676$$
$$x_{47} = 59.1617911706003$$
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
$$- \frac{x e^{2 - x}}{2 - x} + \frac{x e^{2 - x}}{\left(2 - x\right)^{2}} + \frac{e^{2 - x}}{2 - x} = 0$$
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
$$- \frac{x e^{2 - x}}{2 - x} + \frac{x e^{2 - x}}{\left(2 - x\right)^{2}} + \frac{e^{2 - x}}{2 - x} = 0$$
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
$$\frac{\left(- x - \frac{2 x}{x - 2} - \frac{2 x}{\left(x - 2\right)^{2}} + 2 + \frac{2}{x - 2}\right) e^{2 - x}}{x - 2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt[3]{26 + 6 \sqrt{33}}}{3} + \frac{8}{3 \sqrt[3]{26 + 6 \sqrt{33}}} + \frac{4}{3}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 2$$
$$\lim_{x \to 2^-}\left(\frac{\left(- x - \frac{2 x}{x - 2} - \frac{2 x}{\left(x - 2\right)^{2}} + 2 + \frac{2}{x - 2}\right) e^{2 - x}}{x - 2}\right) = \infty$$
Let's take the limit
$$\lim_{x \to 2^+}\left(\frac{\left(- x - \frac{2 x}{x - 2} - \frac{2 x}{\left(x - 2\right)^{2}} + 2 + \frac{2}{x - 2}\right) e^{2 - x}}{x - 2}\right) = -\infty$$
Let's take the limit
- the limits are not equal, so
$$x_{1} = 2$$
- 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
$$\left[- \frac{\sqrt[3]{26 + 6 \sqrt{33}}}{3} + \frac{8}{3 \sqrt[3]{26 + 6 \sqrt{33}}} + \frac{4}{3}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{\sqrt[3]{26 + 6 \sqrt{33}}}{3} + \frac{8}{3 \sqrt[3]{26 + 6 \sqrt{33}}} + \frac{4}{3}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(- x - \frac{2 x}{x - 2} - \frac{2 x}{\left(x - 2\right)^{2}} + 2 + \frac{2}{x - 2}\right) e^{2 - x}}{x - 2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt[3]{26 + 6 \sqrt{33}}}{3} + \frac{8}{3 \sqrt[3]{26 + 6 \sqrt{33}}} + \frac{4}{3}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 2$$
$$\lim_{x \to 2^-}\left(\frac{\left(- x - \frac{2 x}{x - 2} - \frac{2 x}{\left(x - 2\right)^{2}} + 2 + \frac{2}{x - 2}\right) e^{2 - x}}{x - 2}\right) = \infty$$
Let's take the limit
$$\lim_{x \to 2^+}\left(\frac{\left(- x - \frac{2 x}{x - 2} - \frac{2 x}{\left(x - 2\right)^{2}} + 2 + \frac{2}{x - 2}\right) e^{2 - x}}{x - 2}\right) = -\infty$$
Let's take the limit
- the limits are not equal, so
$$x_{1} = 2$$
- 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
$$\left[- \frac{\sqrt[3]{26 + 6 \sqrt{33}}}{3} + \frac{8}{3 \sqrt[3]{26 + 6 \sqrt{33}}} + \frac{4}{3}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{\sqrt[3]{26 + 6 \sqrt{33}}}{3} + \frac{8}{3 \sqrt[3]{26 + 6 \sqrt{33}}} + \frac{4}{3}\right]$$
Vertical asymptotes
Have:
$$x_{1} = 2$$
$$x_{1} = 2$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x e^{2 - x}}{2 - x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x e^{2 - x}}{2 - x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{x e^{2 - x}}{2 - x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x e^{2 - x}}{2 - x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of E^(2 - x)*x/(2 - x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{2 - x}}{2 - x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{e^{2 - x}}{2 - x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{e^{2 - x}}{2 - x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{e^{2 - x}}{2 - x}\right) = 0$$
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:
$$\frac{x e^{2 - x}}{2 - x} = - \frac{x e^{x + 2}}{x + 2}$$
- No
$$\frac{x e^{2 - x}}{2 - x} = \frac{x e^{x + 2}}{x + 2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x e^{2 - x}}{2 - x} = - \frac{x e^{x + 2}}{x + 2}$$
- No
$$\frac{x e^{2 - x}}{2 - x} = \frac{x e^{x + 2}}{x + 2}$$
- No
so, the function
not is
neither even, nor odd
The graph