Mister Exam
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-
How to use it?
- Graphing y =:
- x^3-((x^4)/4)
- x^3-2*x^2+x
- x^2+x-1
- (x^2-1)/(x-1)
-
Identical expressions
- ln(tg(pi*x/ two))
- ln(tg( Pi multiply by x divide by 2))
- ln(tg( Pi multiply by x divide by two))
- ln(tg(pix/2))
- lntgpix/2
- ln(tg(pi*x divide by 2))
Graphing y = ln(tg(pi*x/2))
The solution
You have entered
[src]
/ /pi*x\\
f(x) = log|tan|----||
\ \ 2 //
$$f{\left(x \right)} = \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}$$
f = log(tan(pi*x/2))
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:
$$\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
Numerical solution
$$x_{1} = 82.5$$
$$x_{2} = 28.5$$
$$x_{3} = -99.5$$
$$x_{4} = -49.5$$
$$x_{5} = -31.5$$
$$x_{6} = 84.5$$
$$x_{7} = 22.5$$
$$x_{8} = 66.5$$
$$x_{9} = -85.5$$
$$x_{10} = 78.5$$
$$x_{11} = -51.5$$
$$x_{12} = -59.5$$
$$x_{13} = 70.5$$
$$x_{14} = -37.5$$
$$x_{15} = -47.5$$
$$x_{16} = 98.5$$
$$x_{17} = -87.5$$
$$x_{18} = 10.5$$
$$x_{19} = -55.5$$
$$x_{20} = -1.5$$
$$x_{21} = -89.5$$
$$x_{22} = 34.5$$
$$x_{23} = -65.5$$
$$x_{24} = 62.5$$
$$x_{25} = 58.5$$
$$x_{26} = -93.5$$
$$x_{27} = -83.5$$
$$x_{28} = 16.5$$
$$x_{29} = -39.5$$
$$x_{30} = 96.5$$
$$x_{31} = -79.5$$
$$x_{32} = -11.5$$
$$x_{33} = -43.5$$
$$x_{34} = 4.5$$
$$x_{35} = -19.5$$
$$x_{36} = 100.5$$
$$x_{37} = 12.5$$
$$x_{38} = 52.5$$
$$x_{39} = 80.5$$
$$x_{40} = -97.5$$
$$x_{41} = -73.5$$
$$x_{42} = -67.5$$
$$x_{43} = 2.5$$
$$x_{44} = 8.5$$
$$x_{45} = 92.5$$
$$x_{46} = -21.5$$
$$x_{47} = 64.5$$
$$x_{48} = 32.5$$
$$x_{49} = 56.5$$
$$x_{50} = 48.5$$
$$x_{51} = -13.5$$
$$x_{52} = -41.5$$
$$x_{53} = 42.5$$
$$x_{54} = 6.5$$
$$x_{55} = 90.5$$
$$x_{56} = -17.5$$
$$x_{57} = -7.5$$
$$x_{58} = -29.5$$
$$x_{59} = 24.5$$
$$x_{60} = 44.5$$
$$x_{61} = 68.5$$
$$x_{62} = -61.5$$
$$x_{63} = 38.5$$
$$x_{64} = 86.5$$
$$x_{65} = -91.5$$
$$x_{66} = 50.5$$
$$x_{67} = -15.5$$
$$x_{68} = -81.5$$
$$x_{69} = 74.5$$
$$x_{70} = -63.5$$
$$x_{71} = -33.5$$
$$x_{72} = -35.5$$
$$x_{73} = -23.5$$
$$x_{74} = -69.5$$
$$x_{75} = 40.5$$
$$x_{76} = -27.5$$
$$x_{77} = -9.5$$
$$x_{78} = 72.5$$
$$x_{79} = 36.5$$
$$x_{80} = -25.5$$
$$x_{81} = -77.5$$
$$x_{82} = 88.5$$
$$x_{83} = -5.5$$
$$x_{84} = -57.5$$
$$x_{85} = 60.5$$
$$x_{86} = 30.5$$
$$x_{87} = -95.5$$
$$x_{88} = 26.5$$
$$x_{89} = 18.5$$
$$x_{90} = 46.5$$
$$x_{91} = 54.5$$
$$x_{92} = -3.5$$
$$x_{93} = -75.5$$
$$x_{94} = 94.5$$
$$x_{95} = -53.5$$
$$x_{96} = 76.5$$
$$x_{97} = 20.5$$
$$x_{98} = -71.5$$
$$x_{99} = -45.5$$
$$x_{100} = 14.5$$
so we need to solve the equation:
$$\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
Numerical solution
$$x_{1} = 82.5$$
$$x_{2} = 28.5$$
$$x_{3} = -99.5$$
$$x_{4} = -49.5$$
$$x_{5} = -31.5$$
$$x_{6} = 84.5$$
$$x_{7} = 22.5$$
$$x_{8} = 66.5$$
$$x_{9} = -85.5$$
$$x_{10} = 78.5$$
$$x_{11} = -51.5$$
$$x_{12} = -59.5$$
$$x_{13} = 70.5$$
$$x_{14} = -37.5$$
$$x_{15} = -47.5$$
$$x_{16} = 98.5$$
$$x_{17} = -87.5$$
$$x_{18} = 10.5$$
$$x_{19} = -55.5$$
$$x_{20} = -1.5$$
$$x_{21} = -89.5$$
$$x_{22} = 34.5$$
$$x_{23} = -65.5$$
$$x_{24} = 62.5$$
$$x_{25} = 58.5$$
$$x_{26} = -93.5$$
$$x_{27} = -83.5$$
$$x_{28} = 16.5$$
$$x_{29} = -39.5$$
$$x_{30} = 96.5$$
$$x_{31} = -79.5$$
$$x_{32} = -11.5$$
$$x_{33} = -43.5$$
$$x_{34} = 4.5$$
$$x_{35} = -19.5$$
$$x_{36} = 100.5$$
$$x_{37} = 12.5$$
$$x_{38} = 52.5$$
$$x_{39} = 80.5$$
$$x_{40} = -97.5$$
$$x_{41} = -73.5$$
$$x_{42} = -67.5$$
$$x_{43} = 2.5$$
$$x_{44} = 8.5$$
$$x_{45} = 92.5$$
$$x_{46} = -21.5$$
$$x_{47} = 64.5$$
$$x_{48} = 32.5$$
$$x_{49} = 56.5$$
$$x_{50} = 48.5$$
$$x_{51} = -13.5$$
$$x_{52} = -41.5$$
$$x_{53} = 42.5$$
$$x_{54} = 6.5$$
$$x_{55} = 90.5$$
$$x_{56} = -17.5$$
$$x_{57} = -7.5$$
$$x_{58} = -29.5$$
$$x_{59} = 24.5$$
$$x_{60} = 44.5$$
$$x_{61} = 68.5$$
$$x_{62} = -61.5$$
$$x_{63} = 38.5$$
$$x_{64} = 86.5$$
$$x_{65} = -91.5$$
$$x_{66} = 50.5$$
$$x_{67} = -15.5$$
$$x_{68} = -81.5$$
$$x_{69} = 74.5$$
$$x_{70} = -63.5$$
$$x_{71} = -33.5$$
$$x_{72} = -35.5$$
$$x_{73} = -23.5$$
$$x_{74} = -69.5$$
$$x_{75} = 40.5$$
$$x_{76} = -27.5$$
$$x_{77} = -9.5$$
$$x_{78} = 72.5$$
$$x_{79} = 36.5$$
$$x_{80} = -25.5$$
$$x_{81} = -77.5$$
$$x_{82} = 88.5$$
$$x_{83} = -5.5$$
$$x_{84} = -57.5$$
$$x_{85} = 60.5$$
$$x_{86} = 30.5$$
$$x_{87} = -95.5$$
$$x_{88} = 26.5$$
$$x_{89} = 18.5$$
$$x_{90} = 46.5$$
$$x_{91} = 54.5$$
$$x_{92} = -3.5$$
$$x_{93} = -75.5$$
$$x_{94} = 94.5$$
$$x_{95} = -53.5$$
$$x_{96} = 76.5$$
$$x_{97} = 20.5$$
$$x_{98} = -71.5$$
$$x_{99} = -45.5$$
$$x_{100} = 14.5$$
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
$$\frac{\pi \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{2 \tan{\left(\frac{\pi x}{2} \right)}} = 0$$
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
$$\frac{\pi \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{2 \tan{\left(\frac{\pi x}{2} \right)}} = 0$$
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
$$\frac{\pi^{2} \left(2 - \frac{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}{\tan^{2}{\left(\frac{\pi x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{1}{2}$$
$$x_{2} = \frac{1}{2}$$
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
$$\left(-\infty, - \frac{1}{2}\right] \cup \left[\frac{1}{2}, \infty\right)$$
Convex at the intervals
$$\left[- \frac{1}{2}, \frac{1}{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\pi^{2} \left(2 - \frac{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}{\tan^{2}{\left(\frac{\pi x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{1}{2}$$
$$x_{2} = \frac{1}{2}$$
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
$$\left(-\infty, - \frac{1}{2}\right] \cup \left[\frac{1}{2}, \infty\right)$$
Convex at the intervals
$$\left[- \frac{1}{2}, \frac{1}{2}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = \lim_{x \to -\infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}$$
$$\lim_{x \to \infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = \lim_{x \to \infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}$$
$$\lim_{x \to -\infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = \lim_{x \to -\infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}$$
$$\lim_{x \to \infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = \lim_{x \to \infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(tan(pi*x/2)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)$$
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:
$$\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = \log{\left(- \tan{\left(\frac{\pi x}{2} \right)} \right)}$$
- No
$$\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = - \log{\left(- \tan{\left(\frac{\pi x}{2} \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = \log{\left(- \tan{\left(\frac{\pi x}{2} \right)} \right)}$$
- No
$$\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = - \log{\left(- \tan{\left(\frac{\pi x}{2} \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph