Mister Exam
Graphing y = f
The solution
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis F at f = 0
so we need to solve the equation:
$$f = 0$$
Solve this equation
The points of intersection with the axis F:
Analytical solution
$$f_{1} = 0$$
Numerical solution
$$f_{1} = 0$$
so we need to solve the equation:
$$f = 0$$
Solve this equation
The points of intersection with the axis F:
Analytical solution
$$f_{1} = 0$$
Numerical solution
$$f_{1} = 0$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d f} f{\left(f \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d f} f{\left(f \right)} = $$
the first derivative
$$1 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d f} f{\left(f \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d f} f{\left(f \right)} = $$
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
$$\frac{d^{2}}{d f^{2}} f{\left(f \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d f^{2}} f{\left(f \right)} = $$
the second derivative
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d f^{2}} f{\left(f \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d f^{2}} f{\left(f \right)} = $$
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 f->+oo and f->-oo
$$\lim_{f \to -\infty} f = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{f \to \infty} f = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{f \to -\infty} f = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{f \to \infty} f = \infty$$
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 f, divided by f at f->+oo and f ->-oo
$$\lim_{f \to -\infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = f$$
$$\lim_{f \to \infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = f$$
$$\lim_{f \to -\infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = f$$
$$\lim_{f \to \infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = f$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-f) и f = -f(-f).
So, check:
$$f = - f$$
- No
$$f = f$$
- Yes
so, the function
is
odd
So, check:
$$f = - f$$
- No
$$f = f$$
- Yes
so, the function
is
odd