Mister Exam
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-
How to use it?
- Graphing y =:
- x*cbrt((x-1)^2)
- -x^4+8*x^2-16
- x^3-x^2-2x
- x^3-8x
- Derivative of:
-
(1+x)^(1/x)
- Limit of the function:
-
(1+x)^(1/x)
- Canonical form:
- (1+x)^(1/x)
-
Identical expressions
- (one +x)^(one /x)
- (1 plus x) to the power of (1 divide by x)
- (one plus x) to the power of (one divide by x)
- (1+x)(1/x)
- 1+x1/x
- 1+x^1/x
- (1+x)^(1 divide by x)
-
Similar expressions
- (1-x)^(1/x)
Graphing y = (1+x)^(1/x)
The solution
You have entered
[src]
x _______ f(x) = \/ 1 + x
$$f{\left(x \right)} = \left(x + 1\right)^{\frac{1}{x}}$$
f = (x + 1)^(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:
$$\left(x + 1\right)^{\frac{1}{x}} = 0$$
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:
$$\left(x + 1\right)^{\frac{1}{x}} = 0$$
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
$$\left(x + 1\right)^{\frac{1}{x}} \left(\frac{1}{x \left(x + 1\right)} - \frac{\log{\left(x + 1 \right)}}{x^{2}}\right) = 0$$
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
$$\left(x + 1\right)^{\frac{1}{x}} \left(\frac{1}{x \left(x + 1\right)} - \frac{\log{\left(x + 1 \right)}}{x^{2}}\right) = 0$$
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
$$\frac{\left(x + 1\right)^{\frac{1}{x}} \left(- \frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x}\right)^{2}}{x} - \frac{2}{x \left(x + 1\right)} + \frac{2 \log{\left(x + 1 \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 29980.4129184309$$
$$x_{2} = 46714.933356596$$
$$x_{3} = 25461.2257552751$$
$$x_{4} = 47821.3520867522$$
$$x_{5} = 24326.4600582814$$
$$x_{6} = 54440.8206019301$$
$$x_{7} = 35591.3063506813$$
$$x_{8} = 34472.1244637095$$
$$x_{9} = 28853.3975120921$$
$$x_{10} = 48926.8209166782$$
$$x_{11} = 37825.5858265746$$
$$x_{12} = 43389.6836739697$$
$$x_{13} = 27724.5899421305$$
$$x_{14} = 32229.39409249$$
$$x_{15} = 50031.3669817923$$
$$x_{16} = 44499.1301965023$$
$$x_{17} = 38940.7849760761$$
$$x_{18} = 31105.7195519117$$
$$x_{19} = 42279.1627921657$$
$$x_{20} = 45607.5361227612$$
$$x_{21} = 53339.7200843891$$
$$x_{22} = 33351.507260858$$
$$x_{23} = 52237.7926464356$$
$$x_{24} = 40054.7526229321$$
$$x_{25} = 41167.5317686861$$
$$x_{26} = 51135.0160556731$$
$$x_{27} = 36709.1092974302$$
$$x_{28} = 26593.8992266672$$
$$x_{29} = 55541.1154065021$$
$$x_{30} = 56640.6247473501$$
$$x_{31} = 57739.3679767074$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(x + 1\right)^{\frac{1}{x}} \left(- \frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x}\right)^{2}}{x} - \frac{2}{x \left(x + 1\right)} + \frac{2 \log{\left(x + 1 \right)}}{x^{2}}\right)}{x}\right) = \frac{11 e}{12}$$
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{\frac{1}{x}} \left(- \frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x}\right)^{2}}{x} - \frac{2}{x \left(x + 1\right)} + \frac{2 \log{\left(x + 1 \right)}}{x^{2}}\right)}{x}\right) = \frac{11 e}{12}$$
- 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
$$\frac{\left(x + 1\right)^{\frac{1}{x}} \left(- \frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x}\right)^{2}}{x} - \frac{2}{x \left(x + 1\right)} + \frac{2 \log{\left(x + 1 \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 29980.4129184309$$
$$x_{2} = 46714.933356596$$
$$x_{3} = 25461.2257552751$$
$$x_{4} = 47821.3520867522$$
$$x_{5} = 24326.4600582814$$
$$x_{6} = 54440.8206019301$$
$$x_{7} = 35591.3063506813$$
$$x_{8} = 34472.1244637095$$
$$x_{9} = 28853.3975120921$$
$$x_{10} = 48926.8209166782$$
$$x_{11} = 37825.5858265746$$
$$x_{12} = 43389.6836739697$$
$$x_{13} = 27724.5899421305$$
$$x_{14} = 32229.39409249$$
$$x_{15} = 50031.3669817923$$
$$x_{16} = 44499.1301965023$$
$$x_{17} = 38940.7849760761$$
$$x_{18} = 31105.7195519117$$
$$x_{19} = 42279.1627921657$$
$$x_{20} = 45607.5361227612$$
$$x_{21} = 53339.7200843891$$
$$x_{22} = 33351.507260858$$
$$x_{23} = 52237.7926464356$$
$$x_{24} = 40054.7526229321$$
$$x_{25} = 41167.5317686861$$
$$x_{26} = 51135.0160556731$$
$$x_{27} = 36709.1092974302$$
$$x_{28} = 26593.8992266672$$
$$x_{29} = 55541.1154065021$$
$$x_{30} = 56640.6247473501$$
$$x_{31} = 57739.3679767074$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\left(x + 1\right)^{\frac{1}{x}} \left(- \frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x}\right)^{2}}{x} - \frac{2}{x \left(x + 1\right)} + \frac{2 \log{\left(x + 1 \right)}}{x^{2}}\right)}{x}\right) = \frac{11 e}{12}$$
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{\frac{1}{x}} \left(- \frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x}\right)^{2}}{x} - \frac{2}{x \left(x + 1\right)} + \frac{2 \log{\left(x + 1 \right)}}{x^{2}}\right)}{x}\right) = \frac{11 e}{12}$$
- 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
$$\lim_{x \to -\infty} \left(x + 1\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(x + 1\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty} \left(x + 1\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(x + 1\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1 + x)^(1/x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{\frac{1}{x}}}{x}\right) = 0$$
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:
$$\left(x + 1\right)^{\frac{1}{x}} = \left(1 - x\right)^{- \frac{1}{x}}$$
- No
$$\left(x + 1\right)^{\frac{1}{x}} = - \left(1 - x\right)^{- \frac{1}{x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(x + 1\right)^{\frac{1}{x}} = \left(1 - x\right)^{- \frac{1}{x}}$$
- No
$$\left(x + 1\right)^{\frac{1}{x}} = - \left(1 - x\right)^{- \frac{1}{x}}$$
- No
so, the function
not is
neither even, nor odd