Mister Exam
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-
How to use it?
- Graphing y =:
- -x^2+4x
- x^2-2x-8
- x+3x^2
- x^3+x^2
- Limit of the function:
-
1-x/2
-
Identical expressions
- one -x/ two
- 1 minus x divide by 2
- one minus x divide by two
- 1-x divide by 2
-
Similar expressions
- exp(-1/2-x/2-lambertw(exp(-1-x)/(2*x))/2)/sqrt(x)
- 1+x/2
- -sqrt(2)*sqrt(-1/(-1-x))/2
- 4x/sqrt(x^2-1)-x/2
- (1-x)/(2+3x)
- asin((1-x)/2)
Graphing y = 1-x/2
The solution
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$- \frac{x}{2} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 2$$
Numerical solution
$$x_{1} = 2$$
so we need to solve the equation:
$$- \frac{x}{2} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 2$$
Numerical solution
$$x_{1} = 2$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{1}{2} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{1}{2} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
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 x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- \frac{x}{2} + 1\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \frac{x}{2} + 1\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- \frac{x}{2} + 1\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \frac{x}{2} + 1\right) = -\infty$$
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 1 - x/2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- \frac{x}{2} + 1}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - \frac{x}{2}$$
$$\lim_{x \to \infty}\left(\frac{- \frac{x}{2} + 1}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{x}{2}$$
$$\lim_{x \to -\infty}\left(\frac{- \frac{x}{2} + 1}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - \frac{x}{2}$$
$$\lim_{x \to \infty}\left(\frac{- \frac{x}{2} + 1}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{x}{2}$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$- \frac{x}{2} + 1 = \frac{x}{2} + 1$$
- No
$$- \frac{x}{2} + 1 = - \frac{x}{2} - 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \frac{x}{2} + 1 = \frac{x}{2} + 1$$
- No
$$- \frac{x}{2} + 1 = - \frac{x}{2} - 1$$
- No
so, the function
not is
neither even, nor odd