Mister Exam
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-
How to use it?
- Graphing y =:
- x^3+6x^2-1
- (x^3-3*x)/(x^2-1)
- -x^3+3x^2-7
- x^3-3x^2+3x-1
- Derivative of:
-
1/3*x
- Canonical form:
- 1/3*x
-
Identical expressions
- one / three *x
- 1 divide by 3 multiply by x
- one divide by three multiply by x
- 1/3x
- 1 divide by 3*x
-
Similar expressions
- (1/3)*x^3-x^2-3*x+9
- 1/3*(x-3)*sqrt(x)
- (1/3)*(x^3-16*x^2+69*x-54)
- abs(3*x-1)*(1-1/(3*x))^(1/3)
Graphing y = 1/3*x
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:
$$\frac{x}{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\frac{x}{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
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
$$\frac{1}{3} = 0$$
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
$$\frac{1}{3} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\lim_{x \to -\infty}\left(\frac{x}{3}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x}{3}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{x}{3}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x}{3}\right) = \infty$$
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 x/3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \frac{1}{3} = \frac{1}{3}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{x}{3}$$
$$\lim_{x \to \infty} \frac{1}{3} = \frac{1}{3}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{x}{3}$$
$$\lim_{x \to -\infty} \frac{1}{3} = \frac{1}{3}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{x}{3}$$
$$\lim_{x \to \infty} \frac{1}{3} = \frac{1}{3}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{x}{3}$$
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:
$$\frac{x}{3} = - \frac{x}{3}$$
- No
$$\frac{x}{3} = \frac{x}{3}$$
- Yes
so, the function
is
odd
So, check:
$$\frac{x}{3} = - \frac{x}{3}$$
- No
$$\frac{x}{3} = \frac{x}{3}$$
- Yes
so, the function
is
odd