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Plotting a function graph/ sec(x)*tan(x)

Graphing y = sec(x)*tan(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = sec(x)*tan(x)
f(x)=tan⁡(x)sec⁡(x)f{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)} 
f = tan(x)*sec(x)
The graph of the function
02468-8-6-4-2-1010-1000010000
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)sec⁡(x)=0\tan{\left(x \right)} \sec{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

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

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

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x)sec⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \sec{\left(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⁡(x)sec⁡(x)=−tan⁡(x)sec⁡(x)\tan{\left(x \right)} \sec{\left(x \right)} = - \tan{\left(x \right)} \sec{\left(x \right)}
- No
tan⁡(x)sec⁡(x)=tan⁡(x)sec⁡(x)\tan{\left(x \right)} \sec{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}
- Yes
so, the function
is
odd
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