Mister Exam
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-
How to use it?
- Graphing y =:
- 2x^2+6x
- 18x-x^3
-
-2(x+1)/(x^2+2x)
- 12x-x^3
-
Identical expressions
- (x+ln(x))/(x- four)
- (x plus ln(x)) divide by (x minus 4)
- (x plus ln(x)) divide by (x minus four)
- x+lnx/x-4
- (x+ln(x)) divide by (x-4)
-
Similar expressions
- (x+ln(x))/(x+4)
- (x-ln(x))/(x-4)
Graphing y = (x+ln(x))/(x-4)
The solution
You have entered
[src]
x + log(x)
f(x) = ----------
x - 4
$$f{\left(x \right)} = \frac{x + \log{\left(x \right)}}{x - 4}$$
f = (x + log(x))/(x - 4)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 4$$
$$x_{1} = 4$$
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:
$$\frac{x + \log{\left(x \right)}}{x - 4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = W\left(1\right)$$
Numerical solution
$$x_{1} = 0.567143290409784$$
so we need to solve the equation:
$$\frac{x + \log{\left(x \right)}}{x - 4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = W\left(1\right)$$
Numerical solution
$$x_{1} = 0.567143290409784$$
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
$$\frac{1 + \frac{1}{x}}{x - 4} - \frac{x + \log{\left(x \right)}}{\left(x - 4\right)^{2}} = 0$$
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
$$\frac{1 + \frac{1}{x}}{x - 4} - \frac{x + \log{\left(x \right)}}{\left(x - 4\right)^{2}} = 0$$
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
$$\frac{- \frac{2 \left(1 + \frac{1}{x}\right)}{x - 4} + \frac{2 \left(x + \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{1}{x^{2}}}{x - 4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 38783.8985529685$$
$$x_{2} = 45808.020325951$$
$$x_{3} = 29719.8418978506$$
$$x_{4} = 37778.7731403642$$
$$x_{5} = 31737.8127530236$$
$$x_{6} = 40792.827428034$$
$$x_{7} = 30729.1207207176$$
$$x_{8} = 0.8114375561742$$
$$x_{9} = 47811.4762989485$$
$$x_{10} = 46809.925612916$$
$$x_{11} = 34760.5793720106$$
$$x_{12} = 28709.9529706705$$
$$x_{13} = 35767.1290742184$$
$$x_{14} = 48812.6813980915$$
$$x_{15} = 44805.7509925326$$
$$x_{16} = 42800.0800645983$$
$$x_{17} = 43803.1077057361$$
$$x_{18} = 41796.6571384727$$
$$x_{19} = 32745.9397987124$$
$$x_{20} = 36773.1883391132$$
$$x_{21} = 27699.4289684316$$
$$x_{22} = 33753.5222833995$$
$$x_{23} = 39788.5788225481$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 4$$
$$\lim_{x \to 4^-}\left(\frac{- \frac{2 \left(1 + \frac{1}{x}\right)}{x - 4} + \frac{2 \left(x + \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{1}{x^{2}}}{x - 4}\right) = -\infty$$
$$\lim_{x \to 4^+}\left(\frac{- \frac{2 \left(1 + \frac{1}{x}\right)}{x - 4} + \frac{2 \left(x + \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{1}{x^{2}}}{x - 4}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 4$$
- is an inflection point
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
$$\left(-\infty, 0.8114375561742\right]$$
Convex at the intervals
$$\left[0.8114375561742, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \frac{2 \left(1 + \frac{1}{x}\right)}{x - 4} + \frac{2 \left(x + \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{1}{x^{2}}}{x - 4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 38783.8985529685$$
$$x_{2} = 45808.020325951$$
$$x_{3} = 29719.8418978506$$
$$x_{4} = 37778.7731403642$$
$$x_{5} = 31737.8127530236$$
$$x_{6} = 40792.827428034$$
$$x_{7} = 30729.1207207176$$
$$x_{8} = 0.8114375561742$$
$$x_{9} = 47811.4762989485$$
$$x_{10} = 46809.925612916$$
$$x_{11} = 34760.5793720106$$
$$x_{12} = 28709.9529706705$$
$$x_{13} = 35767.1290742184$$
$$x_{14} = 48812.6813980915$$
$$x_{15} = 44805.7509925326$$
$$x_{16} = 42800.0800645983$$
$$x_{17} = 43803.1077057361$$
$$x_{18} = 41796.6571384727$$
$$x_{19} = 32745.9397987124$$
$$x_{20} = 36773.1883391132$$
$$x_{21} = 27699.4289684316$$
$$x_{22} = 33753.5222833995$$
$$x_{23} = 39788.5788225481$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 4$$
$$\lim_{x \to 4^-}\left(\frac{- \frac{2 \left(1 + \frac{1}{x}\right)}{x - 4} + \frac{2 \left(x + \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{1}{x^{2}}}{x - 4}\right) = -\infty$$
$$\lim_{x \to 4^+}\left(\frac{- \frac{2 \left(1 + \frac{1}{x}\right)}{x - 4} + \frac{2 \left(x + \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{1}{x^{2}}}{x - 4}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 4$$
- is an inflection point
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
$$\left(-\infty, 0.8114375561742\right]$$
Convex at the intervals
$$\left[0.8114375561742, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = 4$$
$$x_{1} = 4$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x + \log{\left(x \right)}}{x - 4}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x + \log{\left(x \right)}}{x - 4}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{x + \log{\left(x \right)}}{x - 4}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x + \log{\left(x \right)}}{x - 4}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x + log(x))/(x - 4), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x + \log{\left(x \right)}}{x \left(x - 4\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x + \log{\left(x \right)}}{x \left(x - 4\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x + \log{\left(x \right)}}{x \left(x - 4\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x + \log{\left(x \right)}}{x \left(x - 4\right)}\right) = 0$$
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:
$$\frac{x + \log{\left(x \right)}}{x - 4} = \frac{- x + \log{\left(- x \right)}}{- x - 4}$$
- No
$$\frac{x + \log{\left(x \right)}}{x - 4} = - \frac{- x + \log{\left(- x \right)}}{- x - 4}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x + \log{\left(x \right)}}{x - 4} = \frac{- x + \log{\left(- x \right)}}{- x - 4}$$
- No
$$\frac{x + \log{\left(x \right)}}{x - 4} = - \frac{- x + \log{\left(- x \right)}}{- x - 4}$$
- No
so, the function
not is
neither even, nor odd