Mister Exam
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-
How to use it?
- Graphing y =:
- x/(2-x)
- x^2-5x-3
- x^2-4x-2
-
x^1/3
-
Identical expressions
- x^ two +ln(x^ two)
- x squared plus ln(x squared )
- x to the power of two plus ln(x to the power of two)
- x2+ln(x2)
- x2+lnx2
- x²+ln(x²)
- x to the power of 2+ln(x to the power of 2)
- x^2+lnx^2
-
Similar expressions
- x^2-ln(x^2)
Graphing y = x^2+ln(x^2)
The solution
You have entered
[src]
2 / 2\ f(x) = x + log\x /
$$f{\left(x \right)} = x^{2} + \log{\left(x^{2} \right)}$$
f = x^2 + log(x^2)
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:
$$x^{2} + \log{\left(x^{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{1}{e^{\frac{W\left(1\right)}{2}}}$$
$$x_{2} = e^{- \frac{W\left(1\right)}{2}}$$
Numerical solution
$$x_{1} = 0.753089164979675$$
$$x_{2} = -0.753089164979675$$
so we need to solve the equation:
$$x^{2} + \log{\left(x^{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{1}{e^{\frac{W\left(1\right)}{2}}}$$
$$x_{2} = e^{- \frac{W\left(1\right)}{2}}$$
Numerical solution
$$x_{1} = 0.753089164979675$$
$$x_{2} = -0.753089164979675$$
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
$$2 x + \frac{2}{x} = 0$$
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
$$2 x + \frac{2}{x} = 0$$
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
$$2 \left(1 - \frac{1}{x^{2}}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1$$
$$x_{2} = 1$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left(-\infty, -1\right] \cup \left[1, \infty\right)$$
Convex at the intervals
$$\left[-1, 1\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(1 - \frac{1}{x^{2}}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1$$
$$x_{2} = 1$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left(-\infty, -1\right] \cup \left[1, \infty\right)$$
Convex at the intervals
$$\left[-1, 1\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(x^{2} + \log{\left(x^{2} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{2} + \log{\left(x^{2} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x^{2} + \log{\left(x^{2} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{2} + \log{\left(x^{2} \right)}\right) = \infty$$
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^2 + log(x^2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x^{2} + \log{\left(x^{2} \right)}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x^{2} + \log{\left(x^{2} \right)}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{x^{2} + \log{\left(x^{2} \right)}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x^{2} + \log{\left(x^{2} \right)}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$x^{2} + \log{\left(x^{2} \right)} = x^{2} + \log{\left(x^{2} \right)}$$
- Yes
$$x^{2} + \log{\left(x^{2} \right)} = - x^{2} - \log{\left(x^{2} \right)}$$
- No
so, the function
is
even
So, check:
$$x^{2} + \log{\left(x^{2} \right)} = x^{2} + \log{\left(x^{2} \right)}$$
- Yes
$$x^{2} + \log{\left(x^{2} \right)} = - x^{2} - \log{\left(x^{2} \right)}$$
- No
so, the function
is
even