Mister Exam
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-
How to use it?
- Graphing y =:
- (x+2)
- 1/(x-1)+1/(x-2)+1/(x-3)
- sqrt(x^2-4)/(x^2+4*x+4)
- sin^2√x-3+cos^2√x-3
- Limit of the function:
-
(7-2*x)^(2/(-3+x))
-
Identical expressions
- (seven - two *x)^(two /(- three +x))
- (7 minus 2 multiply by x) to the power of (2 divide by ( minus 3 plus x))
- (seven minus two multiply by x) to the power of (two divide by ( minus three plus x))
- (7-2*x)(2/(-3+x))
- 7-2*x2/-3+x
- (7-2x)^(2/(-3+x))
- (7-2x)(2/(-3+x))
- 7-2x2/-3+x
- 7-2x^2/-3+x
- (7-2*x)^(2 divide by (-3+x))
-
Similar expressions
- (7+2*x)^(2/(-3+x))
- (7-2*x)^(2/(3+x))
- (7-2*x)^(2/(-3-x))
Graphing y = (7-2*x)^(2/(-3+x))
The solution
You have entered
[src]
2
------
-3 + x
f(x) = (7 - 2*x)
$$f{\left(x \right)} = \left(7 - 2 x\right)^{\frac{2}{x - 3}}$$
f = (7 - 2*x)^(2/(x - 3))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 3$$
$$x_{1} = 3$$
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:
$$\left(7 - 2 x\right)^{\frac{2}{x - 3}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{7}{2}$$
Numerical solution
$$x_{1} = 3.49961751987821$$
so we need to solve the equation:
$$\left(7 - 2 x\right)^{\frac{2}{x - 3}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{7}{2}$$
Numerical solution
$$x_{1} = 3.49961751987821$$
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
$$\left(7 - 2 x\right)^{\frac{2}{x - 3}} \left(- \frac{2 \log{\left(7 - 2 x \right)}}{\left(x - 3\right)^{2}} - \frac{4}{\left(7 - 2 x\right) \left(x - 3\right)}\right) = 0$$
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
$$\left(7 - 2 x\right)^{\frac{2}{x - 3}} \left(- \frac{2 \log{\left(7 - 2 x \right)}}{\left(x - 3\right)^{2}} - \frac{4}{\left(7 - 2 x\right) \left(x - 3\right)}\right) = 0$$
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
$$\frac{4 \left(7 - 2 x\right)^{\frac{2}{x - 3}} \left(- \frac{2}{\left(2 x - 7\right)^{2}} + \frac{\left(\frac{2}{2 x - 7} - \frac{\log{\left(7 - 2 x \right)}}{x - 3}\right)^{2}}{x - 3} - \frac{2}{\left(x - 3\right) \left(2 x - 7\right)} + \frac{\log{\left(7 - 2 x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -50415.592773315$$
$$x_{2} = -54771.8531264758$$
$$x_{3} = -53684.2188785056$$
$$x_{4} = -31684.089326942$$
$$x_{5} = -39447.589149683$$
$$x_{6} = -56944.4262188582$$
$$x_{7} = 2.30150058684961$$
$$x_{8} = -37237.9135546137$$
$$x_{9} = -42751.1482885861$$
$$x_{10} = -38343.5230255027$$
$$x_{11} = -59113.5722496062$$
$$x_{12} = -43849.6412367176$$
$$x_{13} = -32798.5725460975$$
$$x_{14} = -33911.1051723028$$
$$x_{15} = -58029.4157142093$$
$$x_{16} = -35021.7823286231$$
$$x_{17} = -51506.1182746925$$
$$x_{18} = -41651.3439812066$$
$$x_{19} = -60196.918231627$$
$$x_{20} = -44946.8716074288$$
$$x_{21} = -40550.1761959289$$
$$x_{22} = -55858.5802535397$$
$$x_{23} = -49324.0416224826$$
$$x_{24} = -61279.4750351948$$
$$x_{25} = -52595.6501893341$$
$$x_{26} = -36130.6915423374$$
$$x_{27} = -47137.7247758638$$
$$x_{28} = -48231.4309207009$$
$$x_{29} = -46042.885137003$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 3$$
$$\lim_{x \to 3^-}\left(\frac{4 \left(7 - 2 x\right)^{\frac{2}{x - 3}} \left(- \frac{2}{\left(2 x - 7\right)^{2}} + \frac{\left(\frac{2}{2 x - 7} - \frac{\log{\left(7 - 2 x \right)}}{x - 3}\right)^{2}}{x - 3} - \frac{2}{\left(x - 3\right) \left(2 x - 7\right)} + \frac{\log{\left(7 - 2 x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3}\right) = 0.0976834074065823$$
$$\lim_{x \to 3^+}\left(\frac{4 \left(7 - 2 x\right)^{\frac{2}{x - 3}} \left(- \frac{2}{\left(2 x - 7\right)^{2}} + \frac{\left(\frac{2}{2 x - 7} - \frac{\log{\left(7 - 2 x \right)}}{x - 3}\right)^{2}}{x - 3} - \frac{2}{\left(x - 3\right) \left(2 x - 7\right)} + \frac{\log{\left(7 - 2 x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3}\right) = 0.0976834074065823$$
- limits are equal, then skip the corresponding 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
$$\left[2.30150058684961, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.30150058684961\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{4 \left(7 - 2 x\right)^{\frac{2}{x - 3}} \left(- \frac{2}{\left(2 x - 7\right)^{2}} + \frac{\left(\frac{2}{2 x - 7} - \frac{\log{\left(7 - 2 x \right)}}{x - 3}\right)^{2}}{x - 3} - \frac{2}{\left(x - 3\right) \left(2 x - 7\right)} + \frac{\log{\left(7 - 2 x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -50415.592773315$$
$$x_{2} = -54771.8531264758$$
$$x_{3} = -53684.2188785056$$
$$x_{4} = -31684.089326942$$
$$x_{5} = -39447.589149683$$
$$x_{6} = -56944.4262188582$$
$$x_{7} = 2.30150058684961$$
$$x_{8} = -37237.9135546137$$
$$x_{9} = -42751.1482885861$$
$$x_{10} = -38343.5230255027$$
$$x_{11} = -59113.5722496062$$
$$x_{12} = -43849.6412367176$$
$$x_{13} = -32798.5725460975$$
$$x_{14} = -33911.1051723028$$
$$x_{15} = -58029.4157142093$$
$$x_{16} = -35021.7823286231$$
$$x_{17} = -51506.1182746925$$
$$x_{18} = -41651.3439812066$$
$$x_{19} = -60196.918231627$$
$$x_{20} = -44946.8716074288$$
$$x_{21} = -40550.1761959289$$
$$x_{22} = -55858.5802535397$$
$$x_{23} = -49324.0416224826$$
$$x_{24} = -61279.4750351948$$
$$x_{25} = -52595.6501893341$$
$$x_{26} = -36130.6915423374$$
$$x_{27} = -47137.7247758638$$
$$x_{28} = -48231.4309207009$$
$$x_{29} = -46042.885137003$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 3$$
$$\lim_{x \to 3^-}\left(\frac{4 \left(7 - 2 x\right)^{\frac{2}{x - 3}} \left(- \frac{2}{\left(2 x - 7\right)^{2}} + \frac{\left(\frac{2}{2 x - 7} - \frac{\log{\left(7 - 2 x \right)}}{x - 3}\right)^{2}}{x - 3} - \frac{2}{\left(x - 3\right) \left(2 x - 7\right)} + \frac{\log{\left(7 - 2 x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3}\right) = 0.0976834074065823$$
$$\lim_{x \to 3^+}\left(\frac{4 \left(7 - 2 x\right)^{\frac{2}{x - 3}} \left(- \frac{2}{\left(2 x - 7\right)^{2}} + \frac{\left(\frac{2}{2 x - 7} - \frac{\log{\left(7 - 2 x \right)}}{x - 3}\right)^{2}}{x - 3} - \frac{2}{\left(x - 3\right) \left(2 x - 7\right)} + \frac{\log{\left(7 - 2 x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3}\right) = 0.0976834074065823$$
- limits are equal, then skip the corresponding 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
$$\left[2.30150058684961, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.30150058684961\right]$$
Vertical asymptotes
Have:
$$x_{1} = 3$$
$$x_{1} = 3$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left(7 - 2 x\right)^{\frac{2}{x - 3}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(7 - 2 x\right)^{\frac{2}{x - 3}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty} \left(7 - 2 x\right)^{\frac{2}{x - 3}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(7 - 2 x\right)^{\frac{2}{x - 3}} = 1$$
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 (7 - 2*x)^(2/(-3 + x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(7 - 2 x\right)^{\frac{2}{x - 3}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(7 - 2 x\right)^{\frac{2}{x - 3}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(7 - 2 x\right)^{\frac{2}{x - 3}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(7 - 2 x\right)^{\frac{2}{x - 3}}}{x}\right) = 0$$
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:
$$\left(7 - 2 x\right)^{\frac{2}{x - 3}} = \left(2 x + 7\right)^{\frac{2}{- x - 3}}$$
- No
$$\left(7 - 2 x\right)^{\frac{2}{x - 3}} = - \left(2 x + 7\right)^{\frac{2}{- x - 3}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(7 - 2 x\right)^{\frac{2}{x - 3}} = \left(2 x + 7\right)^{\frac{2}{- x - 3}}$$
- No
$$\left(7 - 2 x\right)^{\frac{2}{x - 3}} = - \left(2 x + 7\right)^{\frac{2}{- x - 3}}$$
- No
so, the function
not is
neither even, nor odd