Mister Exam
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- Graph of implicit function
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How to use it?
- Graphing y =:
- 6x^2-x^3
-
y=|x-2|-|x+1|+x-2
- x^2-4x+4
-
4x/(x^2+1)^2
-
Identical expressions
- tan(x/ two)^(two /x)*(log(tan(x/ two))*(- two)/x^ two + two *(one / two +tan(x/ two)^ one)/((x*tan(x/ two))))
- tangent of (x divide by 2) to the power of (2 divide by x) multiply by ( logarithm of ( tangent of (x divide by 2)) multiply by ( minus 2) divide by x squared plus 2 multiply by (1 divide by 2 plus tangent of (x divide by 2) to the power of 1) divide by ((x multiply by tangent of (x divide by 2))))
- tangent of (x divide by two) to the power of (two divide by x) multiply by ( logarithm of ( tangent of (x divide by two)) multiply by ( minus two) divide by x to the power of two plus two multiply by (one divide by two plus tangent of (x divide by two) to the power of one) divide by ((x multiply by tangent of (x divide by two))))
- tan(x/2)(2/x)*(log(tan(x/2))*(-2)/x2+2*(1/2+tan(x/2)1)/((x*tan(x/2))))
- tanx/22/x*logtanx/2*-2/x2+2*1/2+tanx/21/x*tanx/2
- tan(x/2)^(2/x)*(log(tan(x/2))*(-2)/x²+2*(1/2+tan(x/2)^1)/((x*tan(x/2))))
- tan(x/2) to the power of (2/x)*(log(tan(x/2))*(-2)/x to the power of 2+2*(1/2+tan(x/2) to the power of 1)/((x*tan(x/2))))
- tan(x/2)^(2/x)(log(tan(x/2))(-2)/x^2+2(1/2+tan(x/2)^1)/((xtan(x/2))))
- tan(x/2)(2/x)(log(tan(x/2))(-2)/x2+2(1/2+tan(x/2)1)/((xtan(x/2))))
- tanx/22/xlogtanx/2-2/x2+21/2+tanx/21/xtanx/2
- tanx/2^2/xlogtanx/2-2/x^2+21/2+tanx/2^1/xtanx/2
- tan(x divide by 2)^(2 divide by x)*(log(tan(x divide by 2))*(-2) divide by x^2+2*(1 divide by 2+tan(x divide by 2)^1) divide by ((x*tan(x divide by 2))))
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Similar expressions
- tan(x/2)^(2/x)*(log(tan(x/2))*(-2)/x^2-2*(1/2+tan(x/2)^1)/((x*tan(x/2))))
- tan(x/2)^(2/x)*(log(tan(x/2))*(2)/x^2+2*(1/2+tan(x/2)^1)/((x*tan(x/2))))
- tan(x/2)^(2/x)*(log(tan(x/2))*(-2)/x^2+2*(1/2-tan(x/2)^1)/((x*tan(x/2))))
Plotting a function graph/
tan(x/2)^(2/x)*(log(tan(x/2))*(-2)/x^2+2*(1/2+tan(x/2)^1)/((x*tan(x/2))))
Graphing y = tan(x/2)^(2/x)*(log(tan(x/2))*(-2)/x^2+2*(1/2+tan(x/2)^1)/((x*tan(x/2))))
The solution
You have entered
[src]
2
- / / /x\\ /1 1/x\\\
x |log|tan|-||*(-2) 2*|- + tan |-|||
/ /x\\ | \ \2// \2 \2//|
f(x) = |tan|-|| *|---------------- + ---------------|
\ \2// | 2 /x\ |
| x x*tan|-| |
\ \2/ /
$$f{\left(x \right)} = \left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)}$$
f = ((2*(tan(x/2)^1 + 1/2))/((x*tan(x/2))) + ((-2)*log(tan(x/2)))/x^2)*tan(x/2)^(2/x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x/2)^(2/x)*((log(tan(x/2))*(-2))/x^2 + (2*(1/2 + tan(x/2)^1))/((x*tan(x/2)))), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)}}{x}\right)$$
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:
$$\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)} = \left(- \tan{\left(\frac{x}{2} \right)}\right)^{- \frac{2}{x}} \left(\frac{1 - 2 \tan{\left(\frac{x}{2} \right)}}{x \tan{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(- \tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right)$$
- No
$$\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)} = - \left(- \tan{\left(\frac{x}{2} \right)}\right)^{- \frac{2}{x}} \left(\frac{1 - 2 \tan{\left(\frac{x}{2} \right)}}{x \tan{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(- \tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right)$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)} = \left(- \tan{\left(\frac{x}{2} \right)}\right)^{- \frac{2}{x}} \left(\frac{1 - 2 \tan{\left(\frac{x}{2} \right)}}{x \tan{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(- \tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right)$$
- No
$$\left(\frac{2 \left(\tan^{1}{\left(\frac{x}{2} \right)} + \frac{1}{2}\right)}{x \tan{\left(\frac{x}{2} \right)}} + \frac{\left(-2\right) \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right) \tan^{\frac{2}{x}}{\left(\frac{x}{2} \right)} = - \left(- \tan{\left(\frac{x}{2} \right)}\right)^{- \frac{2}{x}} \left(\frac{1 - 2 \tan{\left(\frac{x}{2} \right)}}{x \tan{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(- \tan{\left(\frac{x}{2} \right)} \right)}}{x^{2}}\right)$$
- No
so, the function
not is
neither even, nor odd