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  • How to use it?

  • Graphing y =:
  • x(2-x)^2
  • -x^2+5x+4
  • 3x^2-4x
  • y=2sin3x+1

  • Identical expressions

  • three *sec(x)*tg(x)
  • 3 multiply by sec(x) multiply by tg(x)
  • three multiply by sec(x) multiply by tg(x)
  • 3sec(x)tg(x)
  • 3secxtgx
Plotting a function graph/ 3*sec(x)*tg(x)

Graphing y = 3*sec(x)*tg(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = 3*sec(x)*tan(x)
f(x)=tan⁡(x)3sec⁡(x)f{\left(x \right)} = \tan{\left(x \right)} 3 \sec{\left(x \right)} 
f = tan(x)*(3*sec(x))
The graph of the function
0.001.000.100.200.300.400.500.600.700.800.90010
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)3sec⁡(x)=0\tan{\left(x \right)} 3 \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=−31.4159265358979x_{2} = -31.4159265358979
x3=84.8230016469244x_{3} = 84.8230016469244
x4=−91.106186954104x_{4} = -91.106186954104
x5=−97.3893722612836x_{5} = -97.3893722612836
x6=91.106186954104x_{6} = 91.106186954104
x7=6.28318530717959x_{7} = 6.28318530717959
x8=−72.2566310325652x_{8} = -72.2566310325652
x9=−47.1238898038469x_{9} = -47.1238898038469
x10=94.2477796076938x_{10} = 94.2477796076938
x11=50.2654824574367x_{11} = 50.2654824574367
x12=56.5486677646163x_{12} = 56.5486677646163
x13=43.9822971502571x_{13} = 43.9822971502571
x14=47.1238898038469x_{14} = 47.1238898038469
x15=−50.2654824574367x_{15} = -50.2654824574367
x16=37.6991118430775x_{16} = 37.6991118430775
x17=−28.2743338823081x_{17} = -28.2743338823081
x18=65.9734457253857x_{18} = 65.9734457253857
x19=15.707963267949x_{19} = 15.707963267949
x20=28.2743338823081x_{20} = 28.2743338823081
x21=−62.8318530717959x_{21} = -62.8318530717959
x22=40.8407044966673x_{22} = 40.8407044966673
x23=−40.8407044966673x_{23} = -40.8407044966673
x24=−6.28318530717959x_{24} = -6.28318530717959
x25=−81.6814089933346x_{25} = -81.6814089933346
x26=−15.707963267949x_{26} = -15.707963267949
x27=−59.6902604182061x_{27} = -59.6902604182061
x28=72.2566310325652x_{28} = 72.2566310325652
x29=3.14159265358979x_{29} = 3.14159265358979
x30=−25.1327412287183x_{30} = -25.1327412287183
x31=21.9911485751286x_{31} = 21.9911485751286
x32=−75.398223686155x_{32} = -75.398223686155
x33=−56.5486677646163x_{33} = -56.5486677646163
x34=−69.1150383789755x_{34} = -69.1150383789755
x35=−84.8230016469244x_{35} = -84.8230016469244
x36=78.5398163397448x_{36} = 78.5398163397448
x37=9.42477796076938x_{37} = 9.42477796076938
x38=−53.4070751110265x_{38} = -53.4070751110265
x39=62.8318530717959x_{39} = 62.8318530717959
x40=−18.8495559215388x_{40} = -18.8495559215388
x41=25.1327412287183x_{41} = 25.1327412287183
x42=100.530964914873x_{42} = 100.530964914873
x43=−87.9645943005142x_{43} = -87.9645943005142
x44=−9.42477796076938x_{44} = -9.42477796076938
x45=75.398223686155x_{45} = 75.398223686155
x46=81.6814089933346x_{46} = 81.6814089933346
x47=87.9645943005142x_{47} = 87.9645943005142
x48=12.5663706143592x_{48} = 12.5663706143592
x49=−34.5575191894877x_{49} = -34.5575191894877
x50=69.1150383789755x_{50} = 69.1150383789755
x51=−3.14159265358979x_{51} = -3.14159265358979
x52=0x_{52} = 0
x53=−21.9911485751286x_{53} = -21.9911485751286
x54=−37.6991118430775x_{54} = -37.6991118430775
x55=31.4159265358979x_{55} = 31.4159265358979
x56=−78.5398163397448x_{56} = -78.5398163397448
x57=−12.5663706143592x_{57} = -12.5663706143592
x58=−94.2477796076938x_{58} = -94.2477796076938
x59=97.3893722612836x_{59} = 97.3893722612836
x60=−100.530964914873x_{60} = -100.530964914873
x61=59.6902604182061x_{61} = 59.6902604182061
x62=53.4070751110265x_{62} = 53.4070751110265
x63=34.5575191894877x_{63} = 34.5575191894877
x64=−65.9734457253857x_{64} = -65.9734457253857
x65=18.8495559215388x_{65} = 18.8495559215388
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (3*sec(x))*tan(x).
tan⁡(0)3sec⁡(0)\tan{\left(0 \right)} 3 \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
3(tan⁡2(x)+1)sec⁡(x)+3tan⁡2(x)sec⁡(x)=03 \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + 3 \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
3(6tan⁡2(x)+5)tan⁡(x)sec⁡(x)=03 \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)3sec⁡(x))y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} 3 \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)3sec⁡(x))y = \lim_{x \to \infty}\left(\tan{\left(x \right)} 3 \sec{\left(x \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (3*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→−∞(3tan⁡(x)sec⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{3 \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→∞(3tan⁡(x)sec⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{3 \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)3sec⁡(x)=−3tan⁡(x)sec⁡(x)\tan{\left(x \right)} 3 \sec{\left(x \right)} = - 3 \tan{\left(x \right)} \sec{\left(x \right)}
- No
tan⁡(x)3sec⁡(x)=3tan⁡(x)sec⁡(x)\tan{\left(x \right)} 3 \sec{\left(x \right)} = 3 \tan{\left(x \right)} \sec{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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