Mister Exam

Graphing y = 2*t

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(t) = 2*t
$$f{\left(t \right)} = 2 t$$
f = 2*t
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis T at f = 0
so we need to solve the equation:
$$2 t = 0$$
Solve this equation
The points of intersection with the axis T:

Analytical solution
$$t_{1} = 0$$
Numerical solution
$$t_{1} = 0$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when t equals 0:
substitute t = 0 to 2*t.
$$0 \cdot 2$$
The result:
$$f{\left(0 \right)} = 0$$
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d t} f{\left(t \right)} = $$
the first derivative
$$2 = 0$$
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
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = $$
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 t->+oo and t->-oo
$$\lim_{t \to -\infty}\left(2 t\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{t \to \infty}\left(2 t\right) = \infty$$
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 2*t, divided by t at t->+oo and t ->-oo
$$\lim_{t \to -\infty} 2 = 2$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 2 t$$
$$\lim_{t \to \infty} 2 = 2$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 2 t$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-t) и f = -f(-t).
So, check:
$$2 t = - 2 t$$
- No
$$2 t = 2 t$$
- Yes
so, the function
is
odd