Mister Exam
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-
How to use it?
- Graphing y =:
- x^3/(x-2)^2
- x^3+x^2-2
- x^3-6x^2+5
- x^3-6x^2+2x-6
-
Identical expressions
- one /(y^ three)
- 1 divide by (y cubed )
- one divide by (y to the power of three)
- 1/(y3)
- 1/y3
- 1/(y³)
- 1/(y to the power of 3)
- 1/y^3
- 1 divide by (y^3)
Graphing y = 1/(y^3)
The solution
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$y_{1} = 0$$
$$y_{1} = 0$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis Y at f = 0
so we need to solve the equation:
$$1 \cdot \frac{1}{y^{3}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis Y
so we need to solve the equation:
$$1 \cdot \frac{1}{y^{3}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis Y
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d y} f{\left(y \right)} = $$
the first derivative
$$- \frac{3}{y y^{3}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d y} f{\left(y \right)} = $$
the first derivative
$$- \frac{3}{y y^{3}} = 0$$
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
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
the second derivative
$$\frac{12}{y^{5}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
the second derivative
$$\frac{12}{y^{5}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$y_{1} = 0$$
$$y_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at y->+oo and y->-oo
$$\lim_{y \to -\infty}\left(1 \cdot \frac{1}{y^{3}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{y \to \infty}\left(1 \cdot \frac{1}{y^{3}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{y \to -\infty}\left(1 \cdot \frac{1}{y^{3}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{y \to \infty}\left(1 \cdot \frac{1}{y^{3}}\right) = 0$$
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 1/y^3, divided by y at y->+oo and y ->-oo
$$\lim_{y \to -\infty}\left(\frac{1}{y y^{3}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{y \to \infty}\left(\frac{1}{y y^{3}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{y \to -\infty}\left(\frac{1}{y y^{3}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{y \to \infty}\left(\frac{1}{y y^{3}}\right) = 0$$
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(-y) и f = -f(-y).
So, check:
$$1 \cdot \frac{1}{y^{3}} = - \frac{1}{y^{3}}$$
- No
$$1 \cdot \frac{1}{y^{3}} = \frac{1}{y^{3}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$1 \cdot \frac{1}{y^{3}} = - \frac{1}{y^{3}}$$
- No
$$1 \cdot \frac{1}{y^{3}} = \frac{1}{y^{3}}$$
- No
so, the function
not is
neither even, nor odd
The graph