Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • -x^2+5x-4
  • |x^2+8x+12|
  • (x^2-4)/(2x+5)
  • -x^2-2x
  • Limit of the function:
  • -cos(1/x)/x^2 -cos(1/x)/x^2

  • Identical expressions

  • -cos(one /x)/x^ two
  • minus co sinus of e of (1 divide by x) divide by x squared
  • minus co sinus of e of (one divide by x) divide by x to the power of two
  • -cos(1/x)/x2
  • -cos1/x/x2
  • -cos(1/x)/x²
  • -cos(1/x)/x to the power of 2
  • -cos1/x/x^2
  • -cos(1 divide by x) divide by x^2

  • Similar expressions

  • cos(1/x)/x^2
Plotting a function graph/ -cos(1/x)/x^2

Graphing y = -cos(1/x)/x^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           /1\ 
       -cos|-| 
           \x/ 
f(x) = --------
           2   
          x
f(x)=(−1)cos⁡(1x)x2f{\left(x \right)} = \frac{\left(-1\right) \cos{\left(\frac{1}{x} \right)}}{x^{2}} 
f = (-cos(1/x))/x^2
The graph of the function
02468-8-6-4-2-1010-250250
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:
(−1)cos⁡(1x)x2=0\frac{\left(-1\right) \cos{\left(\frac{1}{x} \right)}}{x^{2}} = 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))/x^2.
(−1)cos⁡(10)02\frac{\left(-1\right) \cos{\left(\frac{1}{0} \right)}}{0^{2}}
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)x2x2+2cos⁡(1x)x3=0- \frac{\sin{\left(\frac{1}{x} \right)}}{x^{2} x^{2}} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=−21127.8065465298x_{1} = -21127.8065465298
x2=26344.0078318137x_{2} = 26344.0078318137
x3=−39773.5440970563x_{3} = -39773.5440970563
x4=42447.4788863631x_{4} = 42447.4788863631
x5=39057.2052525216x_{5} = 39057.2052525216
x6=14479.7313150492x_{6} = 14479.7313150492
x7=33971.8484557919x_{7} = 33971.8484557919
x8=−14348.5424650725x_{8} = -14348.5424650725
x9=11937.9523841008x_{9} = 11937.9523841008
x10=−19432.887686874x_{10} = -19432.887686874
x11=39904.771377312x_{11} = 39904.771377312
x12=40752.3390879025x_{12} = 40752.3390879025
x13=−16043.2338667424x_{13} = -16043.2338667424
x14=22953.9840195429x_{14} = 22953.9840195429
x15=−15195.8737165349x_{15} = -15195.8737165349
x16=−21975.2827488203x_{16} = -21975.2827488203
x17=−35535.7330502043x_{17} = -35535.7330502043
x18=30581.6602478055x_{18} = 30581.6602478055
x19=25496.4920696083x_{19} = 25496.4920696083
x20=31429.2026711056x_{20} = 31429.2026711056
x21=27191.5291133465x_{21} = 27191.5291133465
x22=−37230.8517263672x_{22} = -37230.8517263672
x23=−18585.4480940284x_{23} = -18585.4480940284
x24=19564.09661214x_{24} = 19564.09661214
x25=33124.297012115x_{25} = 33124.297012115
x26=12785.166209279x_{26} = 12785.166209279
x27=17021.8196886792x_{27} = 17021.8196886792
x28=−29602.8987100326x_{28} = -29602.8987100326
x29=−24517.764514177x_{29} = -24517.764514177
x30=20411.5519538498x_{30} = 20411.5519538498
x31=41599.9082873259x_{31} = 41599.9082873259
x32=−13501.2455714192x_{32} = -13501.2455714192
x33=23801.4794641224x_{33} = 23801.4794641224
x34=13632.4287111505x_{34} = 13632.4287111505
x35=−40621.1115695252x_{35} = -40621.1115695252
x36=38209.640819107x_{36} = 38209.640819107
x37=28886.5862895034x_{37} = 28886.5862895034
x38=−12653.9899632038x_{38} = -12653.9899632038
x39=−28755.3642709925x_{39} = -28755.3642709925
x40=32276.7483423665x_{40} = 32276.7483423665
x41=−38078.4140636161x_{41} = -38078.4140636161
x42=24648.982396646x_{42} = 24648.982396646
x43=−34688.1770045647x_{43} = -34688.1770045647
x44=−41468.6805452608x_{44} = -41468.6805452608
x45=36514.5174977468x_{45} = 36514.5174977468
x46=−22822.7684696829x_{46} = -22822.7684696829
x47=−16890.6185554691x_{47} = -16890.6185554691
x48=−38925.9782260908x_{48} = -38925.9782260908
x49=−31297.9789349291x_{49} = -31297.9789349291
x50=29734.1213504079x_{50} = 29734.1213504079
x51=−30450.4370366936x_{51} = -30450.4370366936
x52=−17738.0242579883x_{52} = -17738.0242579883
x53=21259.0191811792x_{53} = 21259.0191811792
x54=17869.2283625365x_{54} = 17869.2283625365
x55=37362.078192229x_{55} = 37362.078192229
x56=−25365.273191964x_{56} = -25365.273191964
x57=15327.0673496263x_{57} = 15327.0673496263
x58=−33840.6233739026x_{58} = -33840.6233739026
x59=−27907.8340740483x_{59} = -27907.8340740483
x60=−27060.3085181287x_{60} = -27060.3085181287
x61=34819.4024707177x_{61} = 34819.4024707177
x62=−32993.0723444395x_{62} = -32993.0723444395
x63=18716.6547734102x_{63} = 18716.6547734102
x64=22106.496925216x_{64} = 22106.496925216
x65=−23670.2626854463x_{65} = -23670.2626854463
x66=−36383.2913419629x_{66} = -36383.2913419629
x67=−11806.7845665482x_{67} = -11806.7845665482
x68=28039.0554132717x_{68} = 28039.0554132717
x69=−26212.7880538116x_{69} = -26212.7880538116
x70=−42316.2509339008x_{70} = -42316.2509339008
x71=16174.4315462694x_{71} = 16174.4315462694
x72=−32145.5241220507x_{72} = -32145.5241220507
x73=−20280.3410578956x_{73} = -20280.3410578956
x74=35666.9588734903x_{74} = 35666.9588734903
The values of the extrema at the points:
(-21127.806546529755, -2.24022261018734e-9)

(26344.00783181366, -1.44090817738494e-9)

(-39773.54409705632, -6.32137300056214e-10)

(42447.47888636313, -5.55004108311864e-10)

(39057.20525252162, -6.55537698157871e-10)

(14479.731315049154, -4.76956742501154e-9)

(33971.84845579192, -8.66486186602981e-10)

(-14348.542465072484, -4.85718252877262e-9)

(11937.952384100769, -7.0168196438427e-9)

(-19432.88768687398, -2.64804473797296e-9)

(39904.77137731199, -6.27986555233478e-10)

(40752.339087902525, -6.0213644937939e-10)

(-16043.233866742383, -3.88522496240994e-9)

(22953.984019542928, -1.89794599074655e-9)

(-15195.873716534898, -4.33060574372337e-9)

(-21975.282748820264, -2.07076614527114e-9)

(-35535.733050204326, -7.91898357833571e-10)

(30581.660247805507, -1.06924660873998e-9)

(25496.492069608328, -1.5382932530713e-9)

(31429.202671105595, -1.01235602697324e-9)

(27191.529113346467, -1.35248587494898e-9)

(-37230.85172636723, -7.21429764890631e-10)

(-18585.448094028365, -2.89503572032555e-9)

(19564.096612140027, -2.61264499394054e-9)

(33124.29701211498, -9.11395037502451e-10)

(12785.16620927896, -6.11768684110306e-9)

(17021.819688679218, -3.45134224857833e-9)

(-29602.89871003259, -1.14112053667174e-9)

(-24517.764514177044, -1.66355920109288e-9)

(20411.551953849834, -2.40020284403386e-9)

(41599.908287325896, -5.77850180859856e-10)

(-13501.245571419151, -5.4859560696488e-9)

(23801.479464122403, -1.76519258172597e-9)

(13632.428711150475, -5.3808827430285e-9)

(-40621.111569525165, -6.06033167213377e-10)

(38209.64081910699, -6.84942460287209e-10)

(28886.586289503364, -1.19841588408651e-9)

(-12653.98996320384, -6.24518096509154e-9)

(-28755.364270992468, -1.20937852507445e-9)

(32276.748342366496, -9.59887746221052e-10)

(-38078.414063616125, -6.89671525667098e-10)

(24648.982396646003, -1.64589456193039e-9)

(-34688.17700456475, -8.31068930621954e-10)

(-41468.680545260824, -5.81513184145046e-10)

(36514.5174977468, -7.50013131646209e-10)

(-22822.768469682887, -1.91983254960971e-9)

(-16890.61855546906, -3.50516842376101e-9)

(-38925.97822609083, -6.59965037892555e-10)

(-31297.97893492907, -1.0208628786012e-9)

(29734.12135040786, -1.13107077399244e-9)

(-30450.43703669357, -1.07848209520273e-9)

(-17738.024257988258, -3.17826064554935e-9)

(21259.019181179232, -2.21265422816479e-9)

(17869.228362536483, -3.13175948424326e-9)

(37362.07819222895, -7.16370923011444e-10)

(-25365.273191964017, -1.55425012629189e-9)

(15327.067349626257, -4.25678648968596e-9)

(-33840.62337390262, -8.73219229198603e-10)

(-27907.834074048253, -1.28394888738824e-9)

(-27060.308518128684, -1.36563460190668e-9)

(34819.40247071765, -8.24816560090003e-10)

(-32993.07234443946, -9.18659311019976e-10)

(18716.6547734102, -2.85458868205407e-9)

(22106.496925216008, -2.04625683567672e-9)

(-23670.262685446316, -1.78481761852769e-9)

(-36383.29134196288, -7.55433137804917e-10)

(-11806.784566548204, -7.17359279569276e-9)

(28039.055413271715, -1.27195937864428e-9)

(-26212.7880538116, -1.45537050026306e-9)

(-42316.25093390082, -5.58451719181666e-10)

(16174.431546269394, -3.82245117419289e-9)

(-32145.524122050676, -9.6774063535741e-10)

(-20280.34105789558, -2.43136125002401e-9)

(35666.95887349029, -7.8608197479349e-10)


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
Decreasing 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
−6cos⁡(1x)+2sin⁡(1x)+cos⁡(1x)xx+4sin⁡(1x)xx4=0\frac{- 6 \cos{\left(\frac{1}{x} \right)} + \frac{2 \sin{\left(\frac{1}{x} \right)} + \frac{\cos{\left(\frac{1}{x} \right)}}{x}}{x} + \frac{4 \sin{\left(\frac{1}{x} \right)}}{x}}{x^{4}} = 0
Solve this equation
The roots of this equation
x1=7868.13411654533x_{1} = 7868.13411654533
x2=9830.62215810209x_{2} = 9830.62215810209
x3=10484.7968264945x_{3} = 10484.7968264945
x4=−3691.84789237652x_{4} = -3691.84789237652
x5=−10232.9706326181x_{5} = -10232.9706326181
x6=2200.17500708681x_{6} = 2200.17500708681
x7=−7180.22328373159x_{7} = -7180.22328373159
x8=9176.45262843974x_{8} = 9176.45262843974
x9=7432.0360008205x_{9} = 7432.0360008205
x10=−7398.27020502735x_{10} = -7398.27020502735
x11=4379.58573892125x_{11} = 4379.58573892125
x12=10048.6798595208x_{12} = 10048.6798595208
x13=6995.94316924641x_{13} = 6995.94316924641
x14=4597.59438348294x_{14} = 4597.59438348294
x15=−7616.31846211567x_{15} = -7616.31846211567
x16=−2820.0445671961x_{16} = -2820.0445671961
x17=−10451.0292903727x_{17} = -10451.0292903727
x18=9612.56502540175x_{18} = 9612.56502540175
x19=6341.81621748738x_{19} = 6341.81621748738
x20=8304.2366831176x_{20} = 8304.2366831176
x21=−9142.68562865806x_{21} = -9142.68562865806
x22=3071.71819362602x_{22} = 3071.71819362602
x23=−10887.1479971977x_{23} = -10887.1479971977
x24=−2602.14405445455x_{24} = -2602.14405445455
x25=−8052.4185491709x_{25} = -8052.4185491709
x26=6559.85667705587x_{26} = 6559.85667705587
x27=−9578.79782229995x_{27} = -9578.79782229995
x28=5687.70855299094x_{28} = 5687.70855299094
x29=−10014.9124790511x_{29} = -10014.9124790511
x30=6123.77788058949x_{30} = 6123.77788058949
x31=−8924.63056842638x_{31} = -8924.63056842638
x32=3289.66540661109x_{32} = 3289.66540661109
x33=−5435.91532349258x_{33} = -5435.91532349258
x34=8740.34303372537x_{34} = 8740.34303372537
x35=−4345.82655525773x_{35} = -4345.82655525773
x36=10266.7380933985x_{36} = 10266.7380933985
x37=−4781.84767656339x_{37} = -4781.84767656339
x38=2418.01054350765x_{38} = 2418.01054350765
x39=8522.28942175551x_{39} = 8522.28942175551
x40=8086.18488852658x_{40} = 8086.18488852658
x41=5905.74190186146x_{41} = 5905.74190186146
x42=−6526.09187529655x_{42} = -6526.09187529655
x43=1764.68006787969x_{43} = 1764.68006787969
x44=5251.65105771383x_{44} = 5251.65105771383
x45=7213.98886455141x_{45} = 7213.98886455141
x46=3725.60318476762x_{46} = 3725.60318476762
x47=4815.60861662379x_{47} = 4815.60861662379
x48=−1948.6726316607x_{48} = -1948.6726316607
x49=9394.50850104819x_{49} = 9394.50850104819
x50=6777.89905415137x_{50} = 6777.89905415137
x51=−3037.96956849881x_{51} = -3037.96956849881
x52=−8706.57626853843x_{52} = -8706.57626853843
x53=−6962.17782388771x_{53} = -6962.17782388771
x54=−6744.13396743094x_{54} = -6744.13396743094
x55=−7834.36794334472x_{55} = -7834.36794334472
x56=1982.39163387668x_{56} = 1982.39163387668
x57=−2384.27480939291x_{57} = -2384.27480939291
x58=10702.856028279x_{58} = 10702.856028279
x59=−9796.85486335916x_{59} = -9796.85486335916
x60=−8488.52278763815x_{60} = -8488.52278763815
x61=−1513.38566285405x_{61} = -1513.38566285405
x62=−6308.05173071026x_{62} = -6308.05173071026
x63=3943.58893603249x_{63} = 3943.58893603249
x64=−6090.01374318718x_{64} = -6090.01374318718
x65=−1730.97458860345x_{65} = -1730.97458860345
x66=−4563.83425879039x_{66} = -4563.83425879039
x67=8958.39745520388x_{67} = 8958.39745520388
x68=2853.78987580662x_{68} = 2853.78987580662
x69=10920.9156706618x_{69} = 10920.9156706618
x70=5469.6781490859x_{70} = 5469.6781490859
x71=−9360.74139605879x_{71} = -9360.74139605879
x72=−4999.8660591042x_{72} = -4999.8660591042
x73=−3473.87449018124x_{73} = -3473.87449018124
x74=−3909.83212815293x_{74} = -3909.83212815293
x75=−5217.88878257435x_{75} = -5217.88878257435
x76=2635.88517720966x_{76} = 2635.88517720966
x77=5033.62771029841x_{77} = 5033.62771029841
x78=4161.58356413872x_{78} = 4161.58356413872
x79=−2166.44637500511x_{79} = -2166.44637500511
x80=−8270.47019056052x_{80} = -8270.47019056052
x81=−5871.97815342411x_{81} = -5871.97815342411
x82=−10669.0884213832x_{82} = -10669.0884213832
x83=3507.62797296392x_{83} = 3507.62797296392
x84=−4127.8254743997x_{84} = -4127.8254743997
x85=−3255.91410899798x_{85} = -3255.91410899798
x86=1547.0712912054x_{86} = 1547.0712912054
x87=−5653.94523932684x_{87} = -5653.94523932684
x88=7650.08445470459x_{88} = 7650.08445470459
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→−∞((−1)cos⁡(1x)x2)=0\lim_{x \to -\infty}\left(\frac{\left(-1\right) \cos{\left(\frac{1}{x} \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞((−1)cos⁡(1x)x2)=0\lim_{x \to \infty}\left(\frac{\left(-1\right) \cos{\left(\frac{1}{x} \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (-cos(1/x))/x^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−cos⁡(1x)xx2)=0\lim_{x \to -\infty}\left(- \frac{\cos{\left(\frac{1}{x} \right)}}{x x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−cos⁡(1x)xx2)=0\lim_{x \to \infty}\left(- \frac{\cos{\left(\frac{1}{x} \right)}}{x x^{2}}\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:
(−1)cos⁡(1x)x2=(−1)cos⁡(1x)x2\frac{\left(-1\right) \cos{\left(\frac{1}{x} \right)}}{x^{2}} = \frac{\left(-1\right) \cos{\left(\frac{1}{x} \right)}}{x^{2}}
- Yes
(−1)cos⁡(1x)x2=−(−1)cos⁡(1x)x2\frac{\left(-1\right) \cos{\left(\frac{1}{x} \right)}}{x^{2}} = - \frac{\left(-1\right) \cos{\left(\frac{1}{x} \right)}}{x^{2}}
- No
so, the function
is
even
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