Mister Exam
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-
How to use it?
- Graphing y =:
- y=x^2+3x-1
- |x|+x
- x+x
- -x*x
- Derivative of:
-
x-sqrt(x^2-x)
- Limit of the function:
-
x-sqrt(x^2-x)
-
Identical expressions
- x-sqrt(x^ two -x)
- x minus square root of (x squared minus x)
- x minus square root of (x to the power of two minus x)
- x-√(x^2-x)
- x-sqrt(x2-x)
- x-sqrtx2-x
- x-sqrt(x²-x)
- x-sqrt(x to the power of 2-x)
- x-sqrtx^2-x
-
Similar expressions
- x+sqrt(x^2-x)
- x-sqrt(x^2+x)
Graphing y = x-sqrt(x^2-x)
The solution
You have entered
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f(x) = x - \/ x - x
$$f{\left(x \right)} = x - \sqrt{x^{2} - x}$$
f = x - sqrt(x^2 - x)
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:
$$x - \sqrt{x^{2} - x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 1.62504262805869 \cdot 10^{28}$$
$$x_{2} = 1.37955649810789 \cdot 10^{30}$$
$$x_{3} = 1.59623125568211 \cdot 10^{31}$$
$$x_{4} = 9.06980847762811 \cdot 10^{31}$$
$$x_{5} = 1.7164221864142 \cdot 10^{32}$$
$$x_{6} = 5.58897734788599 \cdot 10^{32}$$
$$x_{7} = 1.09439022896545 \cdot 10^{31}$$
$$x_{8} = 2.304591345438 \cdot 10^{31}$$
$$x_{9} = 7.6321877357881 \cdot 10^{30}$$
$$x_{10} = 9.7077991997921 \cdot 10^{32}$$
$$x_{11} = 3.77554352063195 \cdot 10^{28}$$
$$x_{12} = 2.31825812568162 \cdot 10^{32}$$
$$x_{13} = 2.05928325659106 \cdot 10^{28}$$
$$x_{14} = 2.76146021831672 \cdot 10^{27}$$
$$x_{15} = 1.25369341817483 \cdot 10^{32}$$
$$x_{16} = 1.99641097694614 \cdot 10^{29}$$
$$x_{17} = 3.32735312594339 \cdot 10^{29}$$
$$x_{18} = 2.14529294218284 \cdot 10^{30}$$
$$x_{19} = 7.53662826956762 \cdot 10^{32}$$
$$x_{20} = 1.09037430017554 \cdot 10^{28}$$
$$x_{21} = 8.73637380888125 \cdot 10^{29}$$
$$x_{22} = 6.55211009888955 \cdot 10^{31}$$
$$x_{23} = 1.31066551931379 \cdot 10^{27}$$
$$x_{24} = 7.41847063311719 \cdot 10^{30}$$
$$x_{25} = 1.2977405244039 \cdot 10^{33}$$
$$x_{26} = 4.97172436641002 \cdot 10^{30}$$
$$x_{27} = 1.17301676415335 \cdot 10^{29}$$
$$x_{28} = 1.63459608816256 \cdot 10^{33}$$
$$x_{29} = 4.18832868866255 \cdot 10^{32}$$
$$x_{30} = 5.58765904997679 \cdot 10^{27}$$
$$x_{31} = 8.7492894917681 \cdot 10^{32}$$
$$x_{32} = 5.44000241662644 \cdot 10^{29}$$
$$x_{33} = 0$$
$$x_{34} = 4.66226651004785 \cdot 10^{31}$$
$$x_{35} = 6.73762407499375 \cdot 10^{28}$$
$$x_{36} = 5.63878719726942 \cdot 10^{34}$$
$$x_{37} = 3.29670571199155 \cdot 10^{31}$$
so we need to solve the equation:
$$x - \sqrt{x^{2} - x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 1.62504262805869 \cdot 10^{28}$$
$$x_{2} = 1.37955649810789 \cdot 10^{30}$$
$$x_{3} = 1.59623125568211 \cdot 10^{31}$$
$$x_{4} = 9.06980847762811 \cdot 10^{31}$$
$$x_{5} = 1.7164221864142 \cdot 10^{32}$$
$$x_{6} = 5.58897734788599 \cdot 10^{32}$$
$$x_{7} = 1.09439022896545 \cdot 10^{31}$$
$$x_{8} = 2.304591345438 \cdot 10^{31}$$
$$x_{9} = 7.6321877357881 \cdot 10^{30}$$
$$x_{10} = 9.7077991997921 \cdot 10^{32}$$
$$x_{11} = 3.77554352063195 \cdot 10^{28}$$
$$x_{12} = 2.31825812568162 \cdot 10^{32}$$
$$x_{13} = 2.05928325659106 \cdot 10^{28}$$
$$x_{14} = 2.76146021831672 \cdot 10^{27}$$
$$x_{15} = 1.25369341817483 \cdot 10^{32}$$
$$x_{16} = 1.99641097694614 \cdot 10^{29}$$
$$x_{17} = 3.32735312594339 \cdot 10^{29}$$
$$x_{18} = 2.14529294218284 \cdot 10^{30}$$
$$x_{19} = 7.53662826956762 \cdot 10^{32}$$
$$x_{20} = 1.09037430017554 \cdot 10^{28}$$
$$x_{21} = 8.73637380888125 \cdot 10^{29}$$
$$x_{22} = 6.55211009888955 \cdot 10^{31}$$
$$x_{23} = 1.31066551931379 \cdot 10^{27}$$
$$x_{24} = 7.41847063311719 \cdot 10^{30}$$
$$x_{25} = 1.2977405244039 \cdot 10^{33}$$
$$x_{26} = 4.97172436641002 \cdot 10^{30}$$
$$x_{27} = 1.17301676415335 \cdot 10^{29}$$
$$x_{28} = 1.63459608816256 \cdot 10^{33}$$
$$x_{29} = 4.18832868866255 \cdot 10^{32}$$
$$x_{30} = 5.58765904997679 \cdot 10^{27}$$
$$x_{31} = 8.7492894917681 \cdot 10^{32}$$
$$x_{32} = 5.44000241662644 \cdot 10^{29}$$
$$x_{33} = 0$$
$$x_{34} = 4.66226651004785 \cdot 10^{31}$$
$$x_{35} = 6.73762407499375 \cdot 10^{28}$$
$$x_{36} = 5.63878719726942 \cdot 10^{34}$$
$$x_{37} = 3.29670571199155 \cdot 10^{31}$$
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
$$- \frac{x - \frac{1}{2}}{\sqrt{x^{2} - x}} + 1 = 0$$
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
$$- \frac{x - \frac{1}{2}}{\sqrt{x^{2} - x}} + 1 = 0$$
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
$$\frac{-1 + \frac{\left(2 x - 1\right)^{2}}{4 x \left(x - 1\right)}}{\sqrt{x \left(x - 1\right)}} = 0$$
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
$$\frac{-1 + \frac{\left(2 x - 1\right)^{2}}{4 x \left(x - 1\right)}}{\sqrt{x \left(x - 1\right)}} = 0$$
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
$$\lim_{x \to -\infty}\left(x - \sqrt{x^{2} - x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{1}{2}$$
$$\lim_{x \to -\infty}\left(x - \sqrt{x^{2} - x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{1}{2}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x - sqrt(x^2 - x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x - \sqrt{x^{2} - x}}{x}\right) = 2$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 2 x$$
$$\lim_{x \to \infty}\left(\frac{x - \sqrt{x^{2} - x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x - \sqrt{x^{2} - x}}{x}\right) = 2$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 2 x$$
$$\lim_{x \to \infty}\left(\frac{x - \sqrt{x^{2} - x}}{x}\right) = 0$$
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:
$$x - \sqrt{x^{2} - x} = - x - \sqrt{x^{2} + x}$$
- No
$$x - \sqrt{x^{2} - x} = x + \sqrt{x^{2} + x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x - \sqrt{x^{2} - x} = - x - \sqrt{x^{2} + x}$$
- No
$$x - \sqrt{x^{2} - x} = x + \sqrt{x^{2} + x}$$
- No
so, the function
not is
neither even, nor odd