Mister Exam
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-
How to use it?
- Graphing y =:
- x/((x^2-1)^(1/3))
- x*sqrt(9-x)
- x/(9-x)
- x+cos2x
- Derivative of:
-
x/(9-x)
- Limit of the function:
-
x/(9-x)
-
Identical expressions
- x/(nine -x)
- x divide by (9 minus x)
- x divide by (nine minus x)
- x/9-x
- x divide by (9-x)
-
Similar expressions
- 1/(x/9)-x/(2*x)
- 2^(x/(9-x^2))
- x/(9+x)
- (-9+3*sqrt(x))/(9-x)
- -2*exp(x)/9-x*exp(x)/3
Graphing y = x/(9-x)
The solution
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 9$$
$$x_{1} = 9$$
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:
$$\frac{x}{9 - x} = 0$$
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:
$$\frac{x}{9 - x} = 0$$
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
$$\frac{x}{\left(9 - x\right)^{2}} + \frac{1}{9 - x} = 0$$
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
$$\frac{x}{\left(9 - x\right)^{2}} + \frac{1}{9 - x} = 0$$
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
$$\frac{2 \left(- \frac{x}{x - 9} + 1\right)}{\left(x - 9\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(- \frac{x}{x - 9} + 1\right)}{\left(x - 9\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 9$$
$$x_{1} = 9$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x}{9 - x}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(\frac{x}{9 - x}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
$$\lim_{x \to -\infty}\left(\frac{x}{9 - x}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(\frac{x}{9 - x}\right) = -1$$
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/(9 - x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \frac{1}{9 - x} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{9 - x} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty} \frac{1}{9 - x} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{9 - x} = 0$$
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:
$$\frac{x}{9 - x} = - \frac{x}{x + 9}$$
- No
$$\frac{x}{9 - x} = \frac{x}{x + 9}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x}{9 - x} = - \frac{x}{x + 9}$$
- No
$$\frac{x}{9 - x} = \frac{x}{x + 9}$$
- No
so, the function
not is
neither even, nor odd