Mister Exam
Other calculators
- Graph of implicit function
- Derivative Step by Step
- Integral Step by Step
- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- x^3-((x^4)/4)
- x^3-2*x^2+x
- x^2+x-1
- (x^2-1)/(x-1)
-
Identical expressions
- s
Graphing y = s
The solution
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis S at f = 0
so we need to solve the equation:
$$s = 0$$
Solve this equation
The points of intersection with the axis S:
Analytical solution
$$s_{1} = 0$$
Numerical solution
$$s_{1} = 0$$
so we need to solve the equation:
$$s = 0$$
Solve this equation
The points of intersection with the axis S:
Analytical solution
$$s_{1} = 0$$
Numerical solution
$$s_{1} = 0$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d s} f{\left(s \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d s} f{\left(s \right)} = $$
the first derivative
$$1 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d s} f{\left(s \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d s} f{\left(s \right)} = $$
the first derivative
$$1 = 0$$
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
$$\frac{d^{2}}{d s^{2}} f{\left(s \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d s^{2}} f{\left(s \right)} = $$
the second derivative
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d s^{2}} f{\left(s \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d s^{2}} f{\left(s \right)} = $$
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 s->+oo and s->-oo
$$\lim_{s \to -\infty} s = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{s \to \infty} s = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{s \to -\infty} s = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{s \to \infty} s = \infty$$
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 s, divided by s at s->+oo and s ->-oo
$$\lim_{s \to -\infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = s$$
$$\lim_{s \to \infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = s$$
$$\lim_{s \to -\infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = s$$
$$\lim_{s \to \infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = s$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-s) и f = -f(-s).
So, check:
$$s = - s$$
- No
$$s = s$$
- Yes
so, the function
is
odd
So, check:
$$s = - s$$
- No
$$s = s$$
- Yes
so, the function
is
odd
The graph