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Plotting a function graph/ cos(1/x)

Graphing y = cos(1/x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /1\
f(x) = cos|-|
          \x/
f(x)=cos⁡(1x)f{\left(x \right)} = \cos{\left(\frac{1}{x} \right)} 
f = cos(1/x)
The graph of the function
0.001.000.100.200.300.400.500.600.700.800.902-2
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
cos⁡(1x)=0\cos{\left(\frac{1}{x} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=23πx_{1} = \frac{2}{3 \pi}
x2=2πx_{2} = \frac{2}{\pi}
Numerical solution
x1=0.212206590789194x_{1} = 0.212206590789194
x2=0.636619772367581x_{2} = 0.636619772367581
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(1/x).
cos⁡(10)\cos{\left(\frac{1}{0} \right)}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
sin⁡(1x)x2=0\frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=1πx_{1} = \frac{1}{\pi}
The values of the extrema at the points:
 1      
(--, -1)
 pi


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=1πx_{1} = \frac{1}{\pi}
The function has no maxima
Decreasing at intervals
[1π,∞)\left[\frac{1}{\pi}, \infty\right)
Increasing at intervals
(−∞,1π]\left(-\infty, \frac{1}{\pi}\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
−2sin⁡(1x)+cos⁡(1x)xx3=0- \frac{2 \sin{\left(\frac{1}{x} \right)} + \frac{\cos{\left(\frac{1}{x} \right)}}{x}}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=2852.18048814786x_{1} = 2852.18048814786
x2=10920.4961326302x_{2} = 10920.4961326302
x3=−8487.98297415597x_{3} = -8487.98297415597
x4=−3690.60517565222x_{4} = -3690.60517565222
x5=−7397.65078485226x_{5} = -7397.65078485226
x6=10702.4279394484x_{6} = 10702.4279394484
x7=9830.15607053174x_{7} = 9830.15607053174
x8=4596.59700962761x_{8} = 4596.59700962761
x9=−4562.82951485885x_{9} = -4562.82951485885
x10=10484.3598310079x_{10} = 10484.3598310079
x11=8085.61819208994x_{11} = 8085.61819208994
x12=7649.4854303998x_{12} = 7649.4854303998
x13=−9142.18445899447x_{13} = -9142.18445899447
x14=1980.06831995953x_{14} = 1980.06831995953
x15=5032.71688419619x_{15} = 5032.71688419619
x16=7213.35359972748x_{16} = 7213.35359972748
x17=6559.15800566404x_{17} = 6559.15800566404
x18=−2164.32241907986x_{18} = -2164.32241907986
x19=5904.96575692001x_{19} = 5904.96575692001
x20=1762.06641001527x_{20} = 1762.06641001527
x21=−4998.94909118433x_{21} = -4998.94909118433
x22=−1510.33069137687x_{22} = -1510.33069137687
x23=−5871.19755389099x_{23} = -5871.19755389099
x24=7867.55170454519x_{24} = 7867.55170454519
x25=−8706.04998202671x_{25} = -8706.04998202671
x26=−6307.32515246242x_{26} = -6307.32515246242
x27=−8269.91613624535x_{27} = -8269.91613624535
x28=−3908.6588920217x_{28} = -3908.6588920217
x29=6123.0294019367x_{29} = 6123.0294019367
x30=−8924.11714739678x_{30} = -8924.11714739678
x31=−4344.77129946241x_{31} = -4344.77129946241
x32=−3254.50438518975x_{32} = -3254.50438518975
x33=−2382.34641253713x_{33} = -2382.34641253713
x34=9612.08836001311x_{34} = 9612.08836001311
x35=−9360.25190659859x_{35} = -9360.25190659859
x36=10048.2238904202x_{36} = 10048.2238904202
x37=−3036.45826575513x_{37} = -3036.45826575513
x38=−7615.71678829324x_{38} = -7615.71678829324
x39=4814.65648065893x_{39} = 4814.65648065893
x40=−8051.84948210571x_{40} = -8051.84948210571
x41=−10232.5228831544x_{41} = -10232.5228831544
x42=4160.48145172456x_{42} = 4160.48145172456
x43=−1728.30948622997x_{43} = -1728.30948622997
x44=−3472.55353426121x_{44} = -3472.55353426121
x45=5468.84004252579x_{45} = 5468.84004252579
x46=6341.09349991192x_{46} = 6341.09349991192
x47=10266.2918127088x_{47} = 10266.2918127088
x48=−6525.38959659195x_{48} = -6525.38959659195
x49=2634.14196282981x_{49} = 2634.14196282981
x50=−10014.4549765932x_{50} = -10014.4549765932
x51=−5217.01018907647x_{51} = -5217.01018907647
x52=8739.81877537654x_{52} = 8739.81877537654
x53=2416.10916381092x_{53} = 2416.10916381092
x54=−7179.58503821156x_{54} = -7179.58503821156
x55=6777.22287982727x_{55} = 6777.22287982727
x56=3506.31973323777x_{56} = 3506.31973323777
x57=−5653.13449897004x_{57} = -5653.13449897004
x58=−10886.7271614234x_{58} = -10886.7271614234
x59=−1292.38283163016x_{59} = -1292.38283163016
x60=−4780.88882660632x_{60} = -4780.88882660632
x61=7431.41938844086x_{61} = 7431.41938844086
x62=−2600.37816709516x_{62} = -2600.37816709516
x63=3724.37172736816x_{63} = 3724.37172736816
x64=1544.08382506109x_{64} = 1544.08382506109
x65=1326.13012084789x_{65} = 1326.13012084789
x66=−2818.4158797084x_{66} = -2818.4158797084
x67=6995.28808794494x_{67} = 6995.28808794494
x68=−10450.5908866988x_{68} = -10450.5908866988
x69=−9578.31948091886x_{69} = -9578.31948091886
x70=−5435.0720199532x_{70} = -5435.0720199532
x71=−6961.51957250769x_{71} = -6961.51957250769
x72=8521.75174184478x_{72} = 8521.75174184478
x73=4378.53861045658x_{73} = 4378.53861045658
x74=−10668.6589812788x_{74} = -10668.6589812788
x75=5250.77810403946x_{75} = 5250.77810403946
x76=9394.02076648111x_{76} = 9394.02076648111
x77=−6089.26112283023x_{77} = -6089.26112283023
x78=9175.95329827666x_{78} = 9175.95329827666
x79=3942.42573932x_{79} = 3942.42573932
x80=3070.22351896922x_{80} = 3070.22351896922
x81=−1946.30879913818x_{81} = -1946.30879913818
x82=−6743.45441499989x_{82} = -6743.45441499989
x83=8303.68487622058x_{83} = 8303.68487622058
x84=−9796.38717349232x_{84} = -9796.38717349232
x85=−7833.78302708594x_{85} = -7833.78302708594
x86=−4126.71435428679x_{86} = -4126.71435428679
x87=3288.2701585224x_{87} = 3288.2701585224
x88=2198.08379682573x_{88} = 2198.08379682573
x89=8957.88596455289x_{89} = 8957.88596455289
x90=5686.90261697445x_{90} = 5686.90261697445
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=0x_{1} = 0

True

True

- the limits are not equal, so
x1=0x_{1} = 0
- is an inflection 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞cos⁡(1x)=1\lim_{x \to -\infty} \cos{\left(\frac{1}{x} \right)} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞cos⁡(1x)=1\lim_{x \to \infty} \cos{\left(\frac{1}{x} \right)} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(1/x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(1x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(1x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
cos⁡(1x)=cos⁡(1x)\cos{\left(\frac{1}{x} \right)} = \cos{\left(\frac{1}{x} \right)}
- Yes
cos⁡(1x)=−cos⁡(1x)\cos{\left(\frac{1}{x} \right)} = - \cos{\left(\frac{1}{x} \right)}
- No
so, the function
is
even
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