Mister Exam
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-
How to use it?
- Graphing y =:
- x^2+3x+3
- -x^2-2x+1
- x^2+14x+15
- x^2+3
- Integral of d{x}:
-
x/sqrt(1+x^2)
- Limit of the function:
-
x/sqrt(1+x^2)
-
Identical expressions
- x/sqrt(one +x^ two)
- x divide by square root of (1 plus x squared )
- x divide by square root of (one plus x to the power of two)
- x/√(1+x^2)
- x/sqrt(1+x2)
- x/sqrt1+x2
- x/sqrt(1+x²)
- x/sqrt(1+x to the power of 2)
- x/sqrt1+x^2
- x divide by sqrt(1+x^2)
-
Similar expressions
- x/sqrt(1-x^2)
Graphing y = x/sqrt(1+x^2)
The solution
You have entered
[src]
x
f(x) = -----------
________
/ 2
\/ 1 + x
$$f{\left(x \right)} = \frac{x}{\sqrt{x^{2} + 1}}$$
f = x/sqrt(x^2 + 1)
The graph of the function
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:
$$\frac{x}{\sqrt{x^{2} + 1}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\frac{x}{\sqrt{x^{2} + 1}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
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
$$- \frac{x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{x^{2} + 1}} = 0$$
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
$$- \frac{x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{x^{2} + 1}} = 0$$
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
$$\frac{x \left(\frac{3 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{x \left(\frac{3 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x}{\sqrt{x^{2} + 1}}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(\frac{x}{\sqrt{x^{2} + 1}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{x}{\sqrt{x^{2} + 1}}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(\frac{x}{\sqrt{x^{2} + 1}}\right) = 1$$
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 x/sqrt(1 + x^2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \frac{1}{\sqrt{x^{2} + 1}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{\sqrt{x^{2} + 1}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty} \frac{1}{\sqrt{x^{2} + 1}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{\sqrt{x^{2} + 1}} = 0$$
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:
$$\frac{x}{\sqrt{x^{2} + 1}} = - \frac{x}{\sqrt{x^{2} + 1}}$$
- No
$$\frac{x}{\sqrt{x^{2} + 1}} = \frac{x}{\sqrt{x^{2} + 1}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x}{\sqrt{x^{2} + 1}} = - \frac{x}{\sqrt{x^{2} + 1}}$$
- No
$$\frac{x}{\sqrt{x^{2} + 1}} = \frac{x}{\sqrt{x^{2} + 1}}$$
- No
so, the function
not is
neither even, nor odd