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Plotting a function graph/ x*tan(x)/2*x-1

Graphing y = x*tan(x)/2*x-1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       x*tan(x)      
f(x) = --------*x - 1
          2
f(x)=xxtan⁡(x)2−1f{\left(x \right)} = x \frac{x \tan{\left(x \right)}}{2} - 1 
f = x*((x*tan(x))/2) - 1
The graph of the function
02468-8-6-4-2-1010-20002000
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:
xxtan⁡(x)2−1=0x \frac{x \tan{\left(x \right)}}{2} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=40.8419034929538x_{1} = 40.8419034929538
x2=28.2768351921315x_{2} = 28.2768351921315
x3=−56.5480423114571x_{3} = -56.5480423114571
x4=59.6908217437871x_{4} = 59.6908217437871
x5=87.9648527713967x_{5} = 87.9648527713967
x6=−87.9643358265937x_{6} = -87.9643358265937
x7=100.531162807029x_{7} = 100.531162807029
x8=−40.8395053595645x_{8} = -40.8395053595645
x9=−65.9729862125151x_{9} = -65.9729862125151
x10=−72.2562479616654x_{10} = -72.2562479616654
x11=31.4179526954364x_{11} = 31.4179526954364
x12=−15.6998493708897x_{12} = -15.6998493708897
x13=12.5790096508774x_{13} = 12.5790096508774
x14=3.32098433785097x_{14} = 3.32098433785097
x15=9.44718335686126x_{15} = 9.44718335686126
x16=−37.697704500272x_{16} = -37.697704500272
x17=−75.397871873221x_{17} = -75.397871873221
x18=−12.5536805291747x_{18} = -12.5536805291747
x19=53.4077762771625x_{19} = 53.4077762771625
x20=34.5591937559364x_{20} = 34.5591937559364
x21=91.1064279068882x_{21} = 91.1064279068882
x22=−34.5558442983964x_{22} = -34.5558442983964
x23=−9.40215753025406x_{23} = -9.40215753025406
x24=43.9833309909146x_{24} = 43.9833309909146
x25=72.2570140953418x_{25} = 72.2570140953418
x26=−94.2475544484359x_{26} = -94.2475544484359
x27=−50.2646908609224x_{27} = -50.2646908609224
x28=−43.9812632123854x_{28} = -43.9812632123854
x29=56.5492931901055x_{29} = 56.5492931901055
x30=65.9739052254543x_{30} = 65.9739052254543
x31=69.1154570564556x_{31} = 69.1154570564556
x32=−21.98701148377x_{32} = -21.98701148377
x33=−53.4063739080669x_{33} = -53.4063739080669
x34=97.3895831265144x_{34} = 97.3895831265144
x35=−25.1295741541754x_{35} = -25.1295741541754
x36=−28.2718316870543x_{36} = -28.2718316870543
x37=−59.6896990715094x_{37} = -59.6896990715094
x38=18.8551814584896x_{38} = 18.8551814584896
x39=−69.1146196913502x_{39} = -69.1146196913502
x40=75.3985754925229x_{40} = 75.3985754925229
x41=−18.8439236610695x_{41} = -18.8439236610695
x42=−91.1059459987708x_{42} = -91.1059459987708
x43=78.5401405648442x_{43} = 78.5401405648442
x44=21.9952825556968x_{44} = 21.9952825556968
x45=−100.53076702116x_{45} = -100.53076702116
x46=50.2662740040889x_{46} = 50.2662740040889
x47=94.2480047648001x_{47} = 94.2480047648001
x48=−84.8227236720407x_{48} = -84.8227236720407
x49=84.8232796181644x_{49} = 84.8232796181644
x50=−78.5394921092915x_{50} = -78.5394921092915
x51=−97.3891613942266x_{51} = -97.3891613942266
x52=−62.8313464577514x_{52} = -62.8313464577514
x53=47.1247904019234x_{53} = 47.1247904019234
x54=37.700518975765x_{54} = 37.700518975765
x55=−31.4138998535201x_{55} = -31.4138998535201
x56=62.8323596695016x_{56} = 62.8323596695016
x57=15.7160604354572x_{57} = 15.7160604354572
x58=−2.90960090827191x_{58} = -2.90960090827191
x59=−6.23173011544337x_{59} = -6.23173011544337
x60=25.1359067076945x_{60} = 25.1359067076945
x61=6.33301061245153x_{61} = 6.33301061245153
x62=81.6817087579409x_{62} = 81.6817087579409
x63=−81.6811092243278x_{63} = -81.6811092243278
x64=−47.1229891369188x_{64} = -47.1229891369188
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to ((x*tan(x))/2)*x - 1.
−1+00tan⁡(0)2-1 + 0 \frac{0 \tan{\left(0 \right)}}{2}
The result:
f(0)=−1f{\left(0 \right)} = -1
The point:
(0, -1)
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
x(x(tan⁡2(x)+1)2+tan⁡(x)2)+xtan⁡(x)2=0x \left(\frac{x \left(\tan^{2}{\left(x \right)} + 1\right)}{2} + \frac{\tan{\left(x \right)}}{2}\right) + \frac{x \tan{\left(x \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
The values of the extrema at the points:
(0, -1)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
The function has no maxima
Doesn't change the value at the entire real axis
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
x(tan⁡2(x)+1)+x(x(tan⁡2(x)+1)tan⁡(x)+tan⁡2(x)+1)+tan⁡(x)=0x \left(\tan^{2}{\left(x \right)} + 1\right) + x \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=100.511071202847x_{1} = 100.511071202847
x2=−28.2036276766186x_{2} = -28.2036276766186
x3=15.5808081405648x_{3} = 15.5808081405648
x4=72.2289536776301x_{4} = 72.2289536776301
x5=−31.3522859596756x_{5} = -31.3522859596756
x6=−43.93683212641x_{6} = -43.93683212641
x7=40.7917435045268x_{7} = 40.7917435045268
x8=12.4075419598293x_{8} = 12.4075419598293
x9=−97.3688368609732x_{9} = -97.3688368609732
x10=25.0532054465023x_{10} = 25.0532054465023
x11=−72.2289536776301x_{11} = -72.2289536776301
x12=−21.9002649847656x_{12} = -21.9002649847656
x13=−50.2256989613876x_{13} = -50.2256989613876
x14=−62.8000247676753x_{14} = -62.8000247676753
x15=43.93683212641x_{15} = 43.93683212641
x16=−75.3716994163885x_{16} = -75.3716994163885
x17=−5.96726435810305x_{17} = -5.96726435810305
x18=−40.7917435045268x_{18} = -40.7917435045268
x19=91.0842354292587x_{19} = 91.0842354292587
x20=37.6460725978858x_{20} = 37.6460725978858
x21=5.96726435810305x_{21} = 5.96726435810305
x22=−47.0814548431779x_{22} = -47.0814548431779
x23=2.51787226577809x_{23} = 2.51787226577809
x24=−2.51787226577809x_{24} = -2.51787226577809
x25=28.2036276766186x_{25} = 28.2036276766186
x26=56.513303680917x_{26} = 56.513303680917
x27=−9.21332735720748x_{27} = -9.21332735720748
x28=−56.513303680917x_{28} = -56.513303680917
x29=75.3716994163885x_{29} = 75.3716994163885
x30=87.9418588589466x_{30} = 87.9418588589466
x31=−12.4075419598293x_{31} = -12.4075419598293
x32=−59.6567572450692x_{32} = -59.6567572450692
x33=−94.2265597445368x_{33} = -94.2265597445368
x34=69.086103133906x_{34} = 69.086103133906
x35=81.6569248399483x_{35} = 81.6569248399483
x36=−25.0532054465023x_{36} = -25.0532054465023
x37=18.7435508863884x_{37} = 18.7435508863884
x38=−37.6460725978858x_{38} = -37.6460725978858
x39=34.4996607566446x_{39} = 34.4996607566446
x40=−69.086103133906x_{40} = -69.086103133906
x41=59.6567572450692x_{41} = 59.6567572450692
x42=−34.4996607566446x_{42} = -34.4996607566446
x43=47.0814548431779x_{43} = 47.0814548431779
x44=0x_{44} = 0
x45=−65.9431328173515x_{45} = -65.9431328173515
x46=−87.9418588589466x_{46} = -87.9418588589466
x47=−81.6569248399483x_{47} = -81.6569248399483
x48=78.5143529238898x_{48} = 78.5143529238898
x49=−78.5143529238898x_{49} = -78.5143529238898
x50=94.2265597445368x_{50} = 94.2265597445368
x51=31.3522859596756x_{51} = 31.3522859596756
x52=−15.5808081405648x_{52} = -15.5808081405648
x53=50.2256989613876x_{53} = 50.2256989613876
x54=−53.3696312584227x_{54} = -53.3696312584227
x55=21.9002649847656x_{55} = 21.9002649847656
x56=−91.0842354292587x_{56} = -91.0842354292587
x57=−84.7994242285037x_{57} = -84.7994242285037
x58=9.21332735720748x_{58} = 9.21332735720748
x59=−100.511071202847x_{59} = -100.511071202847
x60=84.7994242285037x_{60} = 84.7994242285037
x61=65.9431328173515x_{61} = 65.9431328173515
x62=53.3696312584227x_{62} = 53.3696312584227
x63=−18.7435508863884x_{63} = -18.7435508863884
x64=62.8000247676753x_{64} = 62.8000247676753
x65=97.3688368609732x_{65} = 97.3688368609732

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
[100.511071202847,∞)\left[100.511071202847, \infty\right)
Convex at the intervals
(−∞,−100.511071202847]\left(-\infty, -100.511071202847\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→−∞(xxtan⁡(x)2−1)y = \lim_{x \to -\infty}\left(x \frac{x \tan{\left(x \right)}}{2} - 1\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(xxtan⁡(x)2−1)y = \lim_{x \to \infty}\left(x \frac{x \tan{\left(x \right)}}{2} - 1\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of ((x*tan(x))/2)*x - 1, 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→−∞(xxtan⁡(x)2−1x)y = x \lim_{x \to -\infty}\left(\frac{x \frac{x \tan{\left(x \right)}}{2} - 1}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(xxtan⁡(x)2−1x)y = x \lim_{x \to \infty}\left(\frac{x \frac{x \tan{\left(x \right)}}{2} - 1}{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:
xxtan⁡(x)2−1=−x2tan⁡(x)2−1x \frac{x \tan{\left(x \right)}}{2} - 1 = - \frac{x^{2} \tan{\left(x \right)}}{2} - 1
- No
xxtan⁡(x)2−1=x2tan⁡(x)2+1x \frac{x \tan{\left(x \right)}}{2} - 1 = \frac{x^{2} \tan{\left(x \right)}}{2} + 1
- No
so, the function
not is
neither even, nor odd
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