Mister Exam
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- The intersection points of functions
-
How to use it?
- Graphing y =:
-
(x^2+3)/(x-1)
- 3x-x^2
- x*sin(1/x)
- xsin1/x
- Derivative of:
-
3*y
- 3*y
-
Identical expressions
- three *y
- 3 multiply by y
- three multiply by y
- 3y
-
Similar expressions
- 15-3y+4y
- 3ylog(y)-(1/36)exp((-(36y-(36/e))^4))
- 3y-3
- (6-3y)/2
- (-3(y-3)^2+36)/9
Graphing y = 3*y
The solution
The graph of the function
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:
$$3 y = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = 0$$
Numerical solution
$$y_{1} = 0$$
so we need to solve the equation:
$$3 y = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = 0$$
Numerical solution
$$y_{1} = 0$$
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
$$3 = 0$$
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
$$3 = 0$$
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
$$0 = 0$$
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
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at y->+oo and y->-oo
$$\lim_{y \to -\infty}\left(3 y\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{y \to \infty}\left(3 y\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{y \to -\infty}\left(3 y\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{y \to \infty}\left(3 y\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*y, divided by y at y->+oo and y ->-oo
$$\lim_{y \to -\infty} 3 = 3$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 3 y$$
$$\lim_{y \to \infty} 3 = 3$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 3 y$$
$$\lim_{y \to -\infty} 3 = 3$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 3 y$$
$$\lim_{y \to \infty} 3 = 3$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 3 y$$
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:
$$3 y = - 3 y$$
- No
$$3 y = 3 y$$
- Yes
so, the function
is
odd
So, check:
$$3 y = - 3 y$$
- No
$$3 y = 3 y$$
- Yes
so, the function
is
odd