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x/cos(x)
Plotting a function graph/ x/cos(x)

Graphing y = x/cos(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
         x   
f(x) = ------
       cos(x)
f(x)=xcos⁡(x)f{\left(x \right)} = \frac{x}{\cos{\left(x \right)}} 
f = x/cos(x)
The graph of the function
02468-8-6-4-2-1010-500500
The domain of the function
The points at which the function is not precisely defined:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469
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:
xcos⁡(x)=0\frac{x}{\cos{\left(x \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x/cos(x).
0cos⁡(0)\frac{0}{\cos{\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
xsin⁡(x)cos⁡2(x)+1cos⁡(x)=0\frac{x \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{1}{\cos{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=−84.811211299318x_{1} = -84.811211299318
x2=−25.0929104121121x_{2} = -25.0929104121121
x3=72.2427897046973x_{3} = 72.2427897046973
x4=−18.7964043662102x_{4} = -18.7964043662102
x5=−47.1026627703624x_{5} = -47.1026627703624
x6=−6.12125046689807x_{6} = -6.12125046689807
x7=−59.6735041304405x_{7} = -59.6735041304405
x8=43.9595528888955x_{8} = 43.9595528888955
x9=−78.5270825679419x_{9} = -78.5270825679419
x10=−12.4864543952238x_{10} = -12.4864543952238
x11=−9.31786646179107x_{11} = -9.31786646179107
x12=−56.5309801938186x_{12} = -56.5309801938186
x13=100.521017074687x_{13} = 100.521017074687
x14=−53.3883466217256x_{14} = -53.3883466217256
x15=21.945612879981x_{15} = 21.945612879981
x16=−15.644128370333x_{16} = -15.644128370333
x17=−21.945612879981x_{17} = -21.945612879981
x18=9.31786646179107x_{18} = 9.31786646179107
x19=−34.5285657554621x_{19} = -34.5285657554621
x20=−100.521017074687x_{20} = -100.521017074687
x21=50.2455828375744x_{21} = 50.2455828375744
x22=78.5270825679419x_{22} = 78.5270825679419
x23=53.3883466217256x_{23} = 53.3883466217256
x24=56.5309801938186x_{24} = 56.5309801938186
x25=25.0929104121121x_{25} = 25.0929104121121
x26=18.7964043662102x_{26} = 18.7964043662102
x27=15.644128370333x_{27} = 15.644128370333
x28=2.79838604578389x_{28} = 2.79838604578389
x29=47.1026627703624x_{29} = 47.1026627703624
x30=91.0952098694071x_{30} = 91.0952098694071
x31=−72.2427897046973x_{31} = -72.2427897046973
x32=−28.2389365752603x_{32} = -28.2389365752603
x33=94.2371684817036x_{33} = 94.2371684817036
x34=69.100567727981x_{34} = 69.100567727981
x35=34.5285657554621x_{35} = 34.5285657554621
x36=−87.9532251106725x_{36} = -87.9532251106725
x37=−97.3791034786112x_{37} = -97.3791034786112
x38=31.3840740178899x_{38} = 31.3840740178899
x39=−62.8159348889734x_{39} = -62.8159348889734
x40=−94.2371684817036x_{40} = -94.2371684817036
x41=62.8159348889734x_{41} = 62.8159348889734
x42=−81.6691650818489x_{42} = -81.6691650818489
x43=97.3791034786112x_{43} = 97.3791034786112
x44=28.2389365752603x_{44} = 28.2389365752603
x45=75.3849592185347x_{45} = 75.3849592185347
x46=6.12125046689807x_{46} = 6.12125046689807
x47=−37.672573565113x_{47} = -37.672573565113
x48=81.6691650818489x_{48} = 81.6691650818489
x49=−31.3840740178899x_{49} = -31.3840740178899
x50=−40.8162093266346x_{50} = -40.8162093266346
x51=−43.9595528888955x_{51} = -43.9595528888955
x52=−69.100567727981x_{52} = -69.100567727981
x53=−65.9582857893902x_{53} = -65.9582857893902
x54=40.8162093266346x_{54} = 40.8162093266346
x55=−91.0952098694071x_{55} = -91.0952098694071
x56=−75.3849592185347x_{56} = -75.3849592185347
x57=87.9532251106725x_{57} = 87.9532251106725
x58=−2.79838604578389x_{58} = -2.79838604578389
x59=12.4864543952238x_{59} = 12.4864543952238
x60=59.6735041304405x_{60} = 59.6735041304405
x61=37.672573565113x_{61} = 37.672573565113
x62=84.811211299318x_{62} = 84.811211299318
x63=65.9582857893902x_{63} = 65.9582857893902
x64=−50.2455828375744x_{64} = -50.2455828375744
The values of the extrema at the points:
(-84.811211299318, 84.817106541414)

(-25.0929104121121, -25.1128284538059)

(72.2427897046973, -72.2497104791231)

(-18.7964043662102, -18.822986402218)

(-47.1026627703624, 47.1132766856486)

(-6.12125046689807, -6.20239528557313)

(-59.6735041304405, 59.6818824703587)

(43.9595528888955, 43.9709255098366)

(-78.5270825679419, 78.5334495398768)

(-12.4864543952238, -12.5264337847611)

(-9.31786646179107, 9.37137318645303)

(-56.5309801938186, -56.5398242097896)

(100.521017074687, 100.525991035798)

(-53.3883466217256, 53.3977111400996)

(21.945612879981, -21.9683846624641)

(-15.644128370333, 15.6760566619115)

(-21.945612879981, 21.9683846624641)

(9.31786646179107, -9.37137318645303)

(-34.5285657554621, 34.5430434838806)

(-100.521017074687, -100.525991035798)

(50.2455828375744, 50.255532975858)

(78.5270825679419, -78.5334495398768)

(53.3883466217256, -53.3977111400996)

(56.5309801938186, 56.5398242097896)

(25.0929104121121, 25.1128284538059)

(18.7964043662102, 18.822986402218)

(15.644128370333, -15.6760566619115)

(2.79838604578389, -2.9716938707138)

(47.1026627703624, -47.1132766856486)

(91.0952098694071, -91.1006984668687)

(-72.2427897046973, 72.2497104791231)

(-28.2389365752603, 28.256637077005)

(94.2371684817036, 94.2424740944813)

(69.100567727981, 69.1078031797371)

(34.5285657554621, -34.5430434838806)

(-87.9532251106725, -87.958909766826)

(-97.3791034786112, 97.3842379150654)

(31.3840740178899, 31.400001623573)

(-62.8159348889734, -62.8238941484508)

(-94.2371684817036, -94.2424740944813)

(62.8159348889734, 62.8238941484508)

(-81.6691650818489, -81.6752871140731)

(97.3791034786112, -97.3842379150654)

(28.2389365752603, -28.256637077005)

(75.3849592185347, 75.3915915495896)

(6.12125046689807, 6.20239528557313)

(-37.672573565113, -37.6858434829161)

(81.6691650818489, 81.6752871140731)

(-31.3840740178899, -31.400001623573)

(-40.8162093266346, 40.8284575240806)

(-43.9595528888955, -43.9709255098366)

(-69.100567727981, -69.1078031797371)

(-65.9582857893902, 65.9658659025626)

(40.8162093266346, -40.8284575240806)

(-91.0952098694071, 91.1006984668687)

(-75.3849592185347, -75.3915915495896)

(87.9532251106725, 87.958909766826)

(-2.79838604578389, 2.9716938707138)

(12.4864543952238, 12.5264337847611)

(59.6735041304405, -59.6818824703587)

(37.672573565113, 37.6858434829161)

(84.811211299318, -84.817106541414)

(65.9582857893902, -65.9658659025626)

(-50.2455828375744, -50.255532975858)


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:
Minima of the function at points:
x1=−84.811211299318x_{1} = -84.811211299318
x2=−47.1026627703624x_{2} = -47.1026627703624
x3=−59.6735041304405x_{3} = -59.6735041304405
x4=43.9595528888955x_{4} = 43.9595528888955
x5=−78.5270825679419x_{5} = -78.5270825679419
x6=−9.31786646179107x_{6} = -9.31786646179107
x7=100.521017074687x_{7} = 100.521017074687
x8=−53.3883466217256x_{8} = -53.3883466217256
x9=−15.644128370333x_{9} = -15.644128370333
x10=−21.945612879981x_{10} = -21.945612879981
x11=−34.5285657554621x_{11} = -34.5285657554621
x12=50.2455828375744x_{12} = 50.2455828375744
x13=56.5309801938186x_{13} = 56.5309801938186
x14=25.0929104121121x_{14} = 25.0929104121121
x15=18.7964043662102x_{15} = 18.7964043662102
x16=−72.2427897046973x_{16} = -72.2427897046973
x17=−28.2389365752603x_{17} = -28.2389365752603
x18=94.2371684817036x_{18} = 94.2371684817036
x19=69.100567727981x_{19} = 69.100567727981
x20=−97.3791034786112x_{20} = -97.3791034786112
x21=31.3840740178899x_{21} = 31.3840740178899
x22=62.8159348889734x_{22} = 62.8159348889734
x23=75.3849592185347x_{23} = 75.3849592185347
x24=6.12125046689807x_{24} = 6.12125046689807
x25=81.6691650818489x_{25} = 81.6691650818489
x26=−40.8162093266346x_{26} = -40.8162093266346
x27=−65.9582857893902x_{27} = -65.9582857893902
x28=−91.0952098694071x_{28} = -91.0952098694071
x29=87.9532251106725x_{29} = 87.9532251106725
x30=−2.79838604578389x_{30} = -2.79838604578389
x31=12.4864543952238x_{31} = 12.4864543952238
x32=37.672573565113x_{32} = 37.672573565113
Maxima of the function at points:
x32=−25.0929104121121x_{32} = -25.0929104121121
x32=72.2427897046973x_{32} = 72.2427897046973
x32=−18.7964043662102x_{32} = -18.7964043662102
x32=−6.12125046689807x_{32} = -6.12125046689807
x32=−12.4864543952238x_{32} = -12.4864543952238
x32=−56.5309801938186x_{32} = -56.5309801938186
x32=21.945612879981x_{32} = 21.945612879981
x32=9.31786646179107x_{32} = 9.31786646179107
x32=−100.521017074687x_{32} = -100.521017074687
x32=78.5270825679419x_{32} = 78.5270825679419
x32=53.3883466217256x_{32} = 53.3883466217256
x32=15.644128370333x_{32} = 15.644128370333
x32=2.79838604578389x_{32} = 2.79838604578389
x32=47.1026627703624x_{32} = 47.1026627703624
x32=91.0952098694071x_{32} = 91.0952098694071
x32=34.5285657554621x_{32} = 34.5285657554621
x32=−87.9532251106725x_{32} = -87.9532251106725
x32=−62.8159348889734x_{32} = -62.8159348889734
x32=−94.2371684817036x_{32} = -94.2371684817036
x32=−81.6691650818489x_{32} = -81.6691650818489
x32=97.3791034786112x_{32} = 97.3791034786112
x32=28.2389365752603x_{32} = 28.2389365752603
x32=−37.672573565113x_{32} = -37.672573565113
x32=−31.3840740178899x_{32} = -31.3840740178899
x32=−43.9595528888955x_{32} = -43.9595528888955
x32=−69.100567727981x_{32} = -69.100567727981
x32=40.8162093266346x_{32} = 40.8162093266346
x32=−75.3849592185347x_{32} = -75.3849592185347
x32=59.6735041304405x_{32} = 59.6735041304405
x32=84.811211299318x_{32} = 84.811211299318
x32=65.9582857893902x_{32} = 65.9582857893902
x32=−50.2455828375744x_{32} = -50.2455828375744
Decreasing at intervals
[100.521017074687,∞)\left[100.521017074687, \infty\right)
Increasing at intervals
(−∞,−97.3791034786112]\left(-\infty, -97.3791034786112\right]
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(2sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x)cos⁡(x)=0\frac{x \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{\cos{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469

lim⁡x→1.5707963267949−(x(2sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x)cos⁡(x))=1.36838184805714⋅1049\lim_{x \to 1.5707963267949^-}\left(\frac{x \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{\cos{\left(x \right)}}\right) = 1.36838184805714 \cdot 10^{49}
Let's take the limit
lim⁡x→1.5707963267949+(x(2sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x)cos⁡(x))=1.36838184805714⋅1049\lim_{x \to 1.5707963267949^+}\left(\frac{x \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{\cos{\left(x \right)}}\right) = 1.36838184805714 \cdot 10^{49}
Let's take the limit
- limits are equal, then skip the corresponding point
lim⁡x→4.71238898038469−(x(2sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x)cos⁡(x))=−1.52042427561904⋅1048\lim_{x \to 4.71238898038469^-}\left(\frac{x \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{\cos{\left(x \right)}}\right) = -1.52042427561904 \cdot 10^{48}
Let's take the limit
lim⁡x→4.71238898038469+(x(2sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x)cos⁡(x))=−1.52042427561904⋅1048\lim_{x \to 4.71238898038469^+}\left(\frac{x \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{\cos{\left(x \right)}}\right) = -1.52042427561904 \cdot 10^{48}
Let's take the limit
- limits are equal, then skip the corresponding point

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]
Vertical asymptotes
Have:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xcos⁡(x))=lim⁡x→−∞(xcos⁡(x))\lim_{x \to -\infty}\left(\frac{x}{\cos{\left(x \right)}}\right) = \lim_{x \to -\infty}\left(\frac{x}{\cos{\left(x \right)}}\right)
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(xcos⁡(x))y = \lim_{x \to -\infty}\left(\frac{x}{\cos{\left(x \right)}}\right)
lim⁡x→∞(xcos⁡(x))=lim⁡x→∞(xcos⁡(x))\lim_{x \to \infty}\left(\frac{x}{\cos{\left(x \right)}}\right) = \lim_{x \to \infty}\left(\frac{x}{\cos{\left(x \right)}}\right)
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(xcos⁡(x))y = \lim_{x \to \infty}\left(\frac{x}{\cos{\left(x \right)}}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞1cos⁡(x)=⟨−∞,∞⟩\lim_{x \to -\infty} \frac{1}{\cos{\left(x \right)}} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞1cos⁡(x)=⟨−∞,∞⟩\lim_{x \to \infty} \frac{1}{\cos{\left(x \right)}} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
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:
xcos⁡(x)=−xcos⁡(x)\frac{x}{\cos{\left(x \right)}} = - \frac{x}{\cos{\left(x \right)}}
- No
xcos⁡(x)=xcos⁡(x)\frac{x}{\cos{\left(x \right)}} = \frac{x}{\cos{\left(x \right)}}
- Yes
so, the function
is
odd
The graph
Graphing y = x/cos(x)
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