Mister Exam
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-
How to use it?
- Graphing y =:
- 2x^3-3x^2-16
- 1/2x^2-1/5x^5
- x/(x-x^2)
- x/(x^3-1)
- Derivative of:
- tan(sqrt(x))
- Integral of d{x}:
- tan(sqrt(x))
- Limit of the function:
-
tan(sqrt(x))
-
Identical expressions
- tan(sqrt(x))
- tangent of ( square root of (x))
- tan(√(x))
- tansqrtx
-
Similar expressions
- -2-4*tan(sqrt(x)/2)/(-1+tan(sqrt(x)/2)^2)
- atan(sqrt(x)/2)/2-sqrt(x)/(4+x)
- 5/atan(sqrt(x^2+3))
- tan(sqrt(x^3))
- x^2*atan(sqrt(x^2-1))-(sqrt(x^2-1))
Graphing y = tan(sqrt(x))
The solution
You have entered
[src]
/ ___\ f(x) = tan\\/ x /
$$f{\left(x \right)} = \tan{\left(\sqrt{x} \right)}$$
f = tan(sqrt(x))
The graph of the function
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
$$\frac{\tan^{2}{\left(\sqrt{x} \right)} + 1}{2 \sqrt{x}} = 0$$
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
$$\frac{\tan^{2}{\left(\sqrt{x} \right)} + 1}{2 \sqrt{x}} = 0$$
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
$$\frac{\left(\frac{2 \tan{\left(\sqrt{x} \right)}}{x} - \frac{1}{x^{\frac{3}{2}}}\right) \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)}{4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -215.646441112928$$
$$x_{2} = -159.300211154543$$
$$x_{3} = -319.944464400342$$
$$x_{4} = -281.534964865174$$
$$x_{5} = -255.653948442012$$
$$x_{6} = -239.164396466154$$
$$x_{7} = -266.21937178092$$
$$x_{8} = 40.4702140127457$$
$$x_{9} = 89.8227352964555$$
$$x_{10} = -291.438556885432$$
$$x_{11} = -196.422278015437$$
$$x_{12} = -324.544296616901$$
$$x_{13} = -351.39586749775$$
$$x_{14} = -360.093833886562$$
$$x_{15} = 0.426763243887731$$
$$x_{16} = -167.448812652849$$
$$x_{17} = -381.367939721434$$
$$x_{18} = -189.606715430135$$
$$x_{19} = -244.755301839975$$
$$x_{20} = -227.657722313128$$
$$x_{21} = -182.536654996717$$
$$x_{22} = -203.016957146004$$
$$x_{23} = -338.121531240487$$
$$x_{24} = -276.494842038143$$
$$x_{25} = -221.721376352388$$
$$x_{26} = -310.623473077084$$
$$x_{27} = -150.618992203896$$
$$x_{28} = -377.163339190757$$
$$x_{29} = -209.417691117761$$
$$x_{30} = -175.169296525612$$
$$x_{31} = -271.391010027857$$
$$x_{32} = -333.631278587251$$
$$x_{33} = -305.898446693039$$
$$x_{34} = -141.252489071196$$
$$x_{35} = -364.400591794112$$
$$x_{36} = -250.249395058521$$
$$x_{37} = -385.549032813221$$
$$x_{38} = -260.975376832951$$
$$x_{39} = -347.00255883551$$
$$x_{40} = -368.680499222418$$
$$x_{41} = -296.30844941084$$
$$x_{42} = -355.759265272666$$
$$x_{43} = 10.8393052762345$$
$$x_{44} = -329.106032699268$$
$$x_{45} = -372.934462707114$$
$$x_{46} = -301.12755788971$$
$$x_{47} = -233.468381463196$$
$$x_{48} = -342.57818867327$$
$$x_{49} = -315.304809752847$$
$$x_{50} = -286.51508260015$$
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
$$\left[89.8227352964555, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0.426763243887731\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(\frac{2 \tan{\left(\sqrt{x} \right)}}{x} - \frac{1}{x^{\frac{3}{2}}}\right) \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)}{4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -215.646441112928$$
$$x_{2} = -159.300211154543$$
$$x_{3} = -319.944464400342$$
$$x_{4} = -281.534964865174$$
$$x_{5} = -255.653948442012$$
$$x_{6} = -239.164396466154$$
$$x_{7} = -266.21937178092$$
$$x_{8} = 40.4702140127457$$
$$x_{9} = 89.8227352964555$$
$$x_{10} = -291.438556885432$$
$$x_{11} = -196.422278015437$$
$$x_{12} = -324.544296616901$$
$$x_{13} = -351.39586749775$$
$$x_{14} = -360.093833886562$$
$$x_{15} = 0.426763243887731$$
$$x_{16} = -167.448812652849$$
$$x_{17} = -381.367939721434$$
$$x_{18} = -189.606715430135$$
$$x_{19} = -244.755301839975$$
$$x_{20} = -227.657722313128$$
$$x_{21} = -182.536654996717$$
$$x_{22} = -203.016957146004$$
$$x_{23} = -338.121531240487$$
$$x_{24} = -276.494842038143$$
$$x_{25} = -221.721376352388$$
$$x_{26} = -310.623473077084$$
$$x_{27} = -150.618992203896$$
$$x_{28} = -377.163339190757$$
$$x_{29} = -209.417691117761$$
$$x_{30} = -175.169296525612$$
$$x_{31} = -271.391010027857$$
$$x_{32} = -333.631278587251$$
$$x_{33} = -305.898446693039$$
$$x_{34} = -141.252489071196$$
$$x_{35} = -364.400591794112$$
$$x_{36} = -250.249395058521$$
$$x_{37} = -385.549032813221$$
$$x_{38} = -260.975376832951$$
$$x_{39} = -347.00255883551$$
$$x_{40} = -368.680499222418$$
$$x_{41} = -296.30844941084$$
$$x_{42} = -355.759265272666$$
$$x_{43} = 10.8393052762345$$
$$x_{44} = -329.106032699268$$
$$x_{45} = -372.934462707114$$
$$x_{46} = -301.12755788971$$
$$x_{47} = -233.468381463196$$
$$x_{48} = -342.57818867327$$
$$x_{49} = -315.304809752847$$
$$x_{50} = -286.51508260015$$
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
$$\left[89.8227352964555, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0.426763243887731\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \tan{\left(\sqrt{x} \right)} = i$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = i$$
$$\lim_{x \to \infty} \tan{\left(\sqrt{x} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty} \tan{\left(\sqrt{x} \right)} = i$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = i$$
$$\lim_{x \to \infty} \tan{\left(\sqrt{x} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(sqrt(x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(\sqrt{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\sqrt{x} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(\sqrt{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\sqrt{x} \right)}}{x}\right)$$
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:
$$\tan{\left(\sqrt{x} \right)} = \tan{\left(\sqrt{- x} \right)}$$
- No
$$\tan{\left(\sqrt{x} \right)} = - \tan{\left(\sqrt{- x} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(\sqrt{x} \right)} = \tan{\left(\sqrt{- x} \right)}$$
- No
$$\tan{\left(\sqrt{x} \right)} = - \tan{\left(\sqrt{- x} \right)}$$
- No
so, the function
not is
neither even, nor odd