Mister Exam
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- The intersection points of functions
-
How to use it?
- Graphing y =:
- (4x^2+1)/x
- (6-x^3)/(x^2)
- -x^4+2x^2+1
- x^4+2x^2+1
- Derivative of:
-
x^(2/3)
- Integral of d{x}:
-
x^(2/3)
- Limit of the function:
-
x^(2/3)
-
Identical expressions
- x^(two / three)
- x to the power of (2 divide by 3)
- x to the power of (two divide by three)
- x(2/3)
- x2/3
- x^2/3
- x^(2 divide by 3)
-
Similar expressions
- x^2/(3+x)^2
- x^2/3
- (2-(4*x^2))/(3*x)-sin3x
Graphing y = x^(2/3)
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:
$$x^{\frac{2}{3}} = 0$$
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:
$$x^{\frac{2}{3}} = 0$$
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
$$\frac{2}{3 \sqrt[3]{x}} = 0$$
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
$$\frac{2}{3 \sqrt[3]{x}} = 0$$
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
$$- \frac{2}{9 x^{\frac{4}{3}}} = 0$$
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
$$- \frac{2}{9 x^{\frac{4}{3}}} = 0$$
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
$$\lim_{x \to -\infty} x^{\frac{2}{3}} = \infty \left(-1\right)^{\frac{2}{3}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \infty \left(-1\right)^{\frac{2}{3}}$$
$$\lim_{x \to \infty} x^{\frac{2}{3}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} x^{\frac{2}{3}} = \infty \left(-1\right)^{\frac{2}{3}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \infty \left(-1\right)^{\frac{2}{3}}$$
$$\lim_{x \to \infty} x^{\frac{2}{3}} = \infty$$
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^(2/3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \frac{1}{\sqrt[3]{x}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{\sqrt[3]{x}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty} \frac{1}{\sqrt[3]{x}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{\sqrt[3]{x}} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$x^{\frac{2}{3}} = \left(- x\right)^{\frac{2}{3}}$$
- No
$$x^{\frac{2}{3}} = - \left(- x\right)^{\frac{2}{3}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x^{\frac{2}{3}} = \left(- x\right)^{\frac{2}{3}}$$
- No
$$x^{\frac{2}{3}} = - \left(- x\right)^{\frac{2}{3}}$$
- No
so, the function
not is
neither even, nor odd