Mister Exam
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-
How to use it?
- Graphing y =:
- x^3-3x^2-x+3
- x^x
- 5x^3-3x^5
- x-3
- Integral of d{x}:
- 1/y
- Derivative of:
-
1/y
- Limit of the function:
-
1/y
-
Identical expressions
- one /y
- 1 divide by y
- one divide by y
-
Similar expressions
- z=1/(y^2)+2y
- sqrt(y^2+1)/y
- 1/(y-1)-5*1/(y*(y-1))
- 1/(y^3)
Graphing y = 1/y
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:
$$\frac{1}{y} = 0$$
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:
$$\frac{1}{y} = 0$$
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
$$- \frac{1}{y^{2}} = 0$$
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
$$- \frac{1}{y^{2}} = 0$$
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
$$\frac{2}{y^{3}} = 0$$
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
$$\frac{2}{y^{3}} = 0$$
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
$$\lim_{y \to -\infty} \frac{1}{y} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{y \to \infty} \frac{1}{y} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{y \to -\infty} \frac{1}{y} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{y \to \infty} \frac{1}{y} = 0$$
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, divided by y at y->+oo and y ->-oo
$$\lim_{y \to -\infty} \frac{1}{y^{2}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{y \to \infty} \frac{1}{y^{2}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{y \to -\infty} \frac{1}{y^{2}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{y \to \infty} \frac{1}{y^{2}} = 0$$
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:
$$\frac{1}{y} = - \frac{1}{y}$$
- No
$$\frac{1}{y} = \frac{1}{y}$$
- Yes
so, the function
is
odd
So, check:
$$\frac{1}{y} = - \frac{1}{y}$$
- No
$$\frac{1}{y} = \frac{1}{y}$$
- Yes
so, the function
is
odd