Mister Exam
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-
How to use it?
- Graphing y =:
- 2x^2+8x+6
-
sqrt(arctan(sqrt(2x+3)))
- y=x^3-6x+1
- sinx-1/2
- Canonical form:
- sqrt(arctan(sqrt(2x+3)))
-
Identical expressions
- sqrt(arctan(sqrt(2x+ three)))
- square root of (arc tangent of ( square root of (2x plus 3)))
- square root of (arc tangent of ( square root of (2x plus three)))
- √(arctan(√(2x+3)))
- sqrtarctansqrt2x+3
-
Similar expressions
- sqrt(arctan(sqrt(2x-3)))
Graphing y = sqrt(arctan(sqrt(2x+3)))
The solution
You have entered
[src]
___________________
/ / _________\
f(x) = \/ atan\\/ 2*x + 3 /
$$f{\left(x \right)} = \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}$$
f = sqrt(atan(sqrt(2*x + 3)))
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:
$$\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{3}{2}$$
Numerical solution
$$x_{1} = -1.5$$
so we need to solve the equation:
$$\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{3}{2}$$
Numerical solution
$$x_{1} = -1.5$$
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
$$\frac{1}{2 \sqrt{2 x + 3} \cdot \left(2 x + 4\right) \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}} = 0$$
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
$$\frac{1}{2 \sqrt{2 x + 3} \cdot \left(2 x + 4\right) \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}} = 0$$
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
$$- \frac{\frac{4}{\left(x + 2\right) \sqrt{2 x + 3}} + \frac{4}{\left(2 x + 3\right)^{\frac{3}{2}}} + \frac{1}{\left(x + 2\right) \left(2 x + 3\right) \operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}}{16 \left(x + 2\right) \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 107577.56403671$$
$$x_{2} = 62903.8628749537$$
$$x_{3} = 60555.049520562$$
$$x_{4} = 79353.594352919$$
$$x_{5} = 67602.4245400132$$
$$x_{6} = 116990.63189526$$
$$x_{7} = 86407.0907612323$$
$$x_{8} = 39433.7739186636$$
$$x_{9} = 105224.644640647$$
$$x_{10} = 109930.626877423$$
$$x_{11} = 74652.3743804977$$
$$x_{12} = 77002.8666831961$$
$$x_{13} = 69952.1408924897$$
$$x_{14} = 32403.2572766674$$
$$x_{15} = 51163.3629419719$$
$$x_{16} = 37089.5579402752$$
$$x_{17} = 95814.5013977822$$
$$x_{18} = 58206.5716651308$$
$$x_{19} = 102871.873473385$$
$$x_{20} = 55858.4494176989$$
$$x_{21} = 46469.998895868$$
$$x_{22} = 48816.4507038095$$
$$x_{23} = 98166.7963429332$$
$$x_{24} = 88758.6644284932$$
$$x_{25} = 72302.1284767751$$
$$x_{26} = 114637.164996904$$
$$x_{27} = 44124.041963804$$
$$x_{28} = 84055.7155798312$$
$$x_{29} = 41778.6188611969$$
$$x_{30} = 93462.376769134$$
$$x_{31} = 112283.828630407$$
$$x_{32} = 91110.4288464257$$
$$x_{33} = 34746.0296041462$$
$$x_{34} = 81704.5471562711$$
$$x_{35} = 100519.255590868$$
$$x_{36} = 65252.9934430143$$
$$x_{37} = 53510.7049758468$$
$$x_{38} = 119344.225445893$$
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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\frac{4}{\left(x + 2\right) \sqrt{2 x + 3}} + \frac{4}{\left(2 x + 3\right)^{\frac{3}{2}}} + \frac{1}{\left(x + 2\right) \left(2 x + 3\right) \operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}}{16 \left(x + 2\right) \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 107577.56403671$$
$$x_{2} = 62903.8628749537$$
$$x_{3} = 60555.049520562$$
$$x_{4} = 79353.594352919$$
$$x_{5} = 67602.4245400132$$
$$x_{6} = 116990.63189526$$
$$x_{7} = 86407.0907612323$$
$$x_{8} = 39433.7739186636$$
$$x_{9} = 105224.644640647$$
$$x_{10} = 109930.626877423$$
$$x_{11} = 74652.3743804977$$
$$x_{12} = 77002.8666831961$$
$$x_{13} = 69952.1408924897$$
$$x_{14} = 32403.2572766674$$
$$x_{15} = 51163.3629419719$$
$$x_{16} = 37089.5579402752$$
$$x_{17} = 95814.5013977822$$
$$x_{18} = 58206.5716651308$$
$$x_{19} = 102871.873473385$$
$$x_{20} = 55858.4494176989$$
$$x_{21} = 46469.998895868$$
$$x_{22} = 48816.4507038095$$
$$x_{23} = 98166.7963429332$$
$$x_{24} = 88758.6644284932$$
$$x_{25} = 72302.1284767751$$
$$x_{26} = 114637.164996904$$
$$x_{27} = 44124.041963804$$
$$x_{28} = 84055.7155798312$$
$$x_{29} = 41778.6188611969$$
$$x_{30} = 93462.376769134$$
$$x_{31} = 112283.828630407$$
$$x_{32} = 91110.4288464257$$
$$x_{33} = 34746.0296041462$$
$$x_{34} = 81704.5471562711$$
$$x_{35} = 100519.255590868$$
$$x_{36} = 65252.9934430143$$
$$x_{37} = 53510.7049758468$$
$$x_{38} = 119344.225445893$$
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:
Have no bends at the whole real axis
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = \frac{\sqrt{2} \sqrt{\pi}}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\sqrt{2} \sqrt{\pi}}{2}$$
$$\lim_{x \to \infty} \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = \frac{\sqrt{2} \sqrt{\pi}}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\sqrt{2} \sqrt{\pi}}{2}$$
$$\lim_{x \to -\infty} \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = \frac{\sqrt{2} \sqrt{\pi}}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\sqrt{2} \sqrt{\pi}}{2}$$
$$\lim_{x \to \infty} \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = \frac{\sqrt{2} \sqrt{\pi}}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\sqrt{2} \sqrt{\pi}}{2}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(atan(sqrt(2*x + 3))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}}{x}\right) = 0$$
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:
$$\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = \sqrt{\operatorname{atan}{\left(\sqrt{- 2 x + 3} \right)}}$$
- No
$$\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = - \sqrt{\operatorname{atan}{\left(\sqrt{- 2 x + 3} \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = \sqrt{\operatorname{atan}{\left(\sqrt{- 2 x + 3} \right)}}$$
- No
$$\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = - \sqrt{\operatorname{atan}{\left(\sqrt{- 2 x + 3} \right)}}$$
- No
so, the function
not is
neither even, nor odd
The graph