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Plotting a function graph/ tgx/2

Graphing y = tgx/2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       tan(x)
f(x) = ------
         2
f(x)=tan⁡(x)2f{\left(x \right)} = \frac{\tan{\left(x \right)}}{2} 
f = tan(x)/2
The graph of the function
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.0-1000010000
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:
tan⁡(x)2=0\frac{\tan{\left(x \right)}}{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=62.8318530717959x_{1} = 62.8318530717959
x2=−50.2654824574367x_{2} = -50.2654824574367
x3=47.1238898038469x_{3} = 47.1238898038469
x4=84.8230016469244x_{4} = 84.8230016469244
x5=−53.4070751110265x_{5} = -53.4070751110265
x6=91.106186954104x_{6} = 91.106186954104
x7=−84.8230016469244x_{7} = -84.8230016469244
x8=25.1327412287183x_{8} = 25.1327412287183
x9=−3.14159265358979x_{9} = -3.14159265358979
x10=−6.28318530717959x_{10} = -6.28318530717959
x11=−40.8407044966673x_{11} = -40.8407044966673
x12=−18.8495559215388x_{12} = -18.8495559215388
x13=78.5398163397448x_{13} = 78.5398163397448
x14=−75.398223686155x_{14} = -75.398223686155
x15=−9.42477796076938x_{15} = -9.42477796076938
x16=72.2566310325652x_{16} = 72.2566310325652
x17=−43.9822971502571x_{17} = -43.9822971502571
x18=31.4159265358979x_{18} = 31.4159265358979
x19=9.42477796076938x_{19} = 9.42477796076938
x20=40.8407044966673x_{20} = 40.8407044966673
x21=−69.1150383789755x_{21} = -69.1150383789755
x22=12.5663706143592x_{22} = 12.5663706143592
x23=87.9645943005142x_{23} = 87.9645943005142
x24=59.6902604182061x_{24} = 59.6902604182061
x25=−37.6991118430775x_{25} = -37.6991118430775
x26=−100.530964914873x_{26} = -100.530964914873
x27=−91.106186954104x_{27} = -91.106186954104
x28=97.3893722612836x_{28} = 97.3893722612836
x29=0x_{29} = 0
x30=−12.5663706143592x_{30} = -12.5663706143592
x31=−78.5398163397448x_{31} = -78.5398163397448
x32=18.8495559215388x_{32} = 18.8495559215388
x33=34.5575191894877x_{33} = 34.5575191894877
x34=−94.2477796076938x_{34} = -94.2477796076938
x35=43.9822971502571x_{35} = 43.9822971502571
x36=−31.4159265358979x_{36} = -31.4159265358979
x37=−81.6814089933346x_{37} = -81.6814089933346
x38=−65.9734457253857x_{38} = -65.9734457253857
x39=75.398223686155x_{39} = 75.398223686155
x40=56.5486677646163x_{40} = 56.5486677646163
x41=3.14159265358979x_{41} = 3.14159265358979
x42=15.707963267949x_{42} = 15.707963267949
x43=−56.5486677646163x_{43} = -56.5486677646163
x44=−21.9911485751286x_{44} = -21.9911485751286
x45=50.2654824574367x_{45} = 50.2654824574367
x46=−15.707963267949x_{46} = -15.707963267949
x47=28.2743338823081x_{47} = 28.2743338823081
x48=94.2477796076938x_{48} = 94.2477796076938
x49=−59.6902604182061x_{49} = -59.6902604182061
x50=−62.8318530717959x_{50} = -62.8318530717959
x51=69.1150383789755x_{51} = 69.1150383789755
x52=−34.5575191894877x_{52} = -34.5575191894877
x53=−97.3893722612836x_{53} = -97.3893722612836
x54=21.9911485751286x_{54} = 21.9911485751286
x55=65.9734457253857x_{55} = 65.9734457253857
x56=37.6991118430775x_{56} = 37.6991118430775
x57=−87.9645943005142x_{57} = -87.9645943005142
x58=−72.2566310325652x_{58} = -72.2566310325652
x59=−25.1327412287183x_{59} = -25.1327412287183
x60=−28.2743338823081x_{60} = -28.2743338823081
x61=81.6814089933346x_{61} = 81.6814089933346
x62=6.28318530717959x_{62} = 6.28318530717959
x63=100.530964914873x_{63} = 100.530964914873
x64=53.4070751110265x_{64} = 53.4070751110265
x65=−47.1238898038469x_{65} = -47.1238898038469
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x)/2.
tan⁡(0)2\frac{\tan{\left(0 \right)}}{2}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
tan⁡2(x)2+12=0\frac{\tan^{2}{\left(x \right)}}{2} + \frac{1}{2} = 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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
(tan⁡2(x)+1)tan⁡(x)=0\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0

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
[0,∞)\left[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(tan⁡(x)2)y = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{2}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(tan⁡(x)2)y = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{2}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x)/2, divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(tan⁡(x)2x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{2 x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x)2x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{2 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⁡(x)2=−tan⁡(x)2\frac{\tan{\left(x \right)}}{2} = - \frac{\tan{\left(x \right)}}{2}
- No
tan⁡(x)2=tan⁡(x)2\frac{\tan{\left(x \right)}}{2} = \frac{\tan{\left(x \right)}}{2}
- No
so, the function
not is
neither even, nor odd
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