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Plotting a function graph/ sqrt(sin(2*x))/x

Graphing y = sqrt(sin(2*x))/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
         __________
       \/ sin(2*x) 
f(x) = ------------
            x
f(x)=sin⁡(2x)xf{\left(x \right)} = \frac{\sqrt{\sin{\left(2 x \right)}}}{x} 
f = sqrt(sin(2*x))/x
The graph of the function
02468-8-6-4-2-1010-510
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:
sin⁡(2x)x=0\frac{\sqrt{\sin{\left(2 x \right)}}}{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
Numerical solution
x1=1.5707963267949x_{1} = 1.5707963267949
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(sin(2*x))/x.
sin⁡(0⋅2)0\frac{\sqrt{\sin{\left(0 \cdot 2 \right)}}}{0}
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
cos⁡(2x)xsin⁡(2x)−sin⁡(2x)x2=0\frac{\cos{\left(2 x \right)}}{x \sqrt{\sin{\left(2 x \right)}}} - \frac{\sqrt{\sin{\left(2 x \right)}}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=−63.609391408151x_{1} = -63.609391408151
x2=66.7513539643031x_{2} = 66.7513539643031
x3=−99.7405539133885x_{3} = -99.7405539133885
x4=−62.0383960813757x_{4} = -62.0383960813757
x5=33.7573137512912x_{5} = 33.7573137512912
x6=36.9001669211527x_{6} = 36.9001669211527
x7=−46.3277006372222x_{7} = -46.3277006372222
x8=−98.1696773784321x_{8} = -98.1696773784321
x9=74.6061240613667x_{9} = 74.6061240613667
x10=11.7384800939942x_{10} = 11.7384800939942
x11=58.8963735834305x_{11} = 58.8963735834305
x12=−13.3142855057572x_{12} = -13.3142855057572
x13=22.7545872771683x_{13} = 22.7545872771683
x14=63.609391408151x_{14} = 63.609391408151
x15=−65.1803771434091x_{15} = -65.1803771434091
x16=−30.6142018882607x_{16} = -30.6142018882607
x17=−19.6094782798075x_{17} = -19.6094782798075
x18=76.1770585706428x_{18} = 76.1770585706428
x19=−18.0364644916681x_{19} = -18.0364644916681
x20=−27.4707425696429x_{20} = -27.4707425696429
x21=41.614089810742x_{21} = 41.614089810742
x22=−47.8988508196586x_{22} = -47.8988508196586
x23=−43.1853230479384x_{23} = -43.1853230479384
x24=−41.614089810742x_{24} = -41.614089810742
x25=47.8988508196586x_{25} = 47.8988508196586
x26=−57.3253446670262x_{26} = -57.3253446670262
x27=85.602559129126x_{27} = 85.602559129126
x28=18.0364644916681x_{28} = 18.0364644916681
x29=44.7565256168206x_{29} = 44.7565256168206
x30=−85.602559129126x_{30} = -85.602559129126
x31=−90.3152528534981x_{31} = -90.3152528534981
x32=−52.6121745880344x_{32} = -52.6121745880344
x33=40.0428222957763x_{33} = 40.0428222957763
x34=−5.40633666694364x_{34} = -5.40633666694364
x35=98.1696773784321x_{35} = 98.1696773784321
x36=−79.3189111701245x_{36} = -79.3189111701245
x37=99.7405539133885x_{37} = 99.7405539133885
x38=−40.0428222957763x_{38} = -40.0428222957763
x39=16.4630276170453x_{39} = 16.4630276170453
x40=−54.1832463772731x_{40} = -54.1832463772731
x41=27.4707425696429x_{41} = 27.4707425696429
x42=3.79827300987529x_{42} = 3.79827300987529
x43=14.8890337004883x_{43} = 14.8890337004883
x44=24.3268011678533x_{44} = 24.3268011678533
x45=55.7543026449315x_{45} = 55.7543026449315
x46=88.744358541403x_{46} = 88.744358541403
x47=−93.4570315962354x_{47} = -93.4570315962354
x48=60.4673904155608x_{48} = 60.4673904155608
x49=−69.8932832672187x_{49} = -69.8932832672187
x50=71.4642368192271x_{50} = 71.4642368192271
x51=2.13739113572906x_{51} = 2.13739113572906
x52=77.747987496317x_{52} = 77.747987496317
x53=−76.1770585706428x_{53} = -76.1770585706428
x54=49.4699785303278x_{54} = 49.4699785303278
x55=8.58137569421011x_{55} = 8.58137569421011
x56=19.6094782798075x_{56} = 19.6094782798075
x57=−35.3287683589234x_{57} = -35.3287683589234
x58=−55.7543026449315x_{58} = -55.7543026449315
x59=−84.0316536256143x_{59} = -84.0316536256143
x60=−32.1857948915632x_{60} = -32.1857948915632
x61=−38.4715163039797x_{61} = -38.4715163039797
x62=−60.4673904155608x_{62} = -60.4673904155608
x63=−49.4699785303278x_{63} = -49.4699785303278
x64=32.1857948915632x_{64} = 32.1857948915632
x65=46.3277006372222x_{65} = 46.3277006372222
x66=62.0383960813757x_{66} = 62.0383960813757
x67=96.5987982349427x_{67} = 96.5987982349427
x68=−71.4642368192271x_{68} = -71.4642368192271
x69=−16.4630276170453x_{69} = -16.4630276170453
x70=−96.5987982349427x_{70} = -96.5987982349427
x71=10.1611269299962x_{71} = 10.1611269299962
x72=82.4607439626894x_{72} = 82.4607439626894
x73=−2.13739113572906x_{73} = -2.13739113572906
x74=68.322322485708x_{74} = 68.322322485708
x75=−87.173460698078x_{75} = -87.173460698078
x76=30.6142018882607x_{76} = 30.6142018882607
x77=−91.8861438154664x_{77} = -91.8861438154664
x78=84.0316536256143x_{78} = 84.0316536256143
x79=−21.182163158836x_{79} = -21.182163158836
x80=−24.3268011678533x_{80} = -24.3268011678533
x81=−10.1611269299962x_{81} = -10.1611269299962
x82=−11.7384800939942x_{82} = -11.7384800939942
x83=54.1832463772731x_{83} = 54.1832463772731
x84=−25.898843096056x_{84} = -25.898843096056
x85=−74.6061240613667x_{85} = -74.6061240613667
x86=90.3152528534981x_{86} = 90.3152528534981
x87=−3.79827300987529x_{87} = -3.79827300987529
x88=−33.7573137512912x_{88} = -33.7573137512912
x89=−82.4607439626894x_{89} = -82.4607439626894
x90=80.8898298980315x_{90} = 80.8898298980315
x91=38.4715163039797x_{91} = 38.4715163039797
x92=69.8932832672187x_{92} = 69.8932832672187
x93=5.40633666694364x_{93} = 5.40633666694364
x94=−8.58137569421011x_{94} = -8.58137569421011
x95=25.898843096056x_{95} = 25.898843096056
x96=−68.322322485708x_{96} = -68.322322485708
x97=91.8861438154664x_{97} = 91.8861438154664
x98=52.6121745880344x_{98} = 52.6121745880344
x99=93.4570315962354x_{99} = 93.4570315962354
x100=−77.747987496317x_{100} = -77.747987496317
The values of the extrema at the points:
(-63.60939140815101, -0.0157199778263343*I)

(66.75135396430314, 0.0149801291078152)

(-99.74055391338845, -0.0100257601558681)

(-62.03839608137567, -0.0161180030072405)

(33.75731375129117, 0.0296167148833163*I)

(36.900166921152675, 0.0270951749854015*I)

(-46.327700637222165, -0.0215828443509178)

(-98.16967737843208, -0.0101861805750159*I)

(74.60612406136674, 0.0134031234387214*I)

(11.73848009399415, 0.0850360393335331*I)

(58.89637358343045, 0.016977750594038*I)

(-13.314285505757224, -0.0750017471309216*I)

(22.754587277168262, 0.0439259886853852)

(63.60939140815101, 0.0157199778263343)

(-65.18037714340906, -0.0153411380529724)

(-30.6142018882607, -0.0326558712102137)

(-19.60947827980746, -0.0509626466768668*I)

(76.17705857064283, 0.0131267463173753)

(-18.036464491668116, -0.0554007125582391)

(-27.470742569642862, -0.0363903155700504)

(41.614089810742044, 0.02402685466694)

(-47.89885081965864, -0.0208750533021526*I)

(-43.185323047938354, -0.0231529122413857)

(-41.614089810742044, -0.02402685466694*I)

(47.89885081965864, 0.0208750533021526)

(-57.32534466702617, -0.0174429642673855*I)

(85.60255912912602, 0.0116814952312444)

(18.036464491668116, 0.0554007125582391*I)

(44.756525616820596, 0.0223403229249365)

(-85.60255912912602, -0.0116814952312444*I)

(-90.31525285349807, -0.0110719875286708)

(-52.612174588034385, -0.0190052912967891)

(40.04282229577635, 0.0249693724726356*I)

(-5.406336666943637, -0.18341903022942)

(98.16967737843208, 0.0101861805750159)

(-79.31891117012448, -0.0126068330109199*I)

(99.74055391338845, 0.0100257601558681*I)

(-40.04282229577635, -0.0249693724726356)

(16.463027617045263, 0.0606862687517582)

(-54.18324637727305, -0.018454318078295*I)

(27.470742569642862, 0.0363903155700504*I)

(3.798273009875294, 0.258903189710353)

(14.889033700488254, 0.067087996040512*I)

(24.326801167853258, 0.0410895782745605*I)

(55.75430264493151, 0.0179343933146333*I)

(88.74435854140305, 0.0112679642422872)

(-93.45703159623535, -0.0106997982057615)

(60.46739041556085, 0.0165367089563368)

(-69.89328326721875, -0.0143067943509695*I)

(71.46423681922705, 0.0139923281824918*I)

(2.137391135729064, 0.445271188713631*I)

(77.74798749631695, 0.0128615373628376*I)

(-76.17705857064283, -0.0131267463173753*I)

(49.46997853032777, 0.0202122156003825*I)

(8.581375694210113, 0.116139144217135*I)

(19.60947827980746, 0.0509626466768668)

(-35.328768358923384, -0.028299877004799*I)

(-55.75430264493151, -0.0179343933146333)

(-84.03165362561431, -0.0118998562546094)

(-32.18579489156321, -0.0310621135424149*I)

(-38.471516303979705, -0.0259888679989479*I)

(-60.46739041556085, -0.0165367089563368*I)

(-49.46997853032777, -0.0202122156003825)

(32.18579489156321, 0.0310621135424149)

(46.327700637222165, 0.0215828443509178*I)

(62.03839608137567, 0.0161180030072405*I)

(96.59879823494268, 0.010351818331022*I)

(-71.46423681922705, -0.0139923281824918)

(-16.463027617045263, -0.0606862687517582*I)

(-96.59879823494268, -0.010351818331022)

(10.16112692999623, 0.0981774182935068)

(82.46074396268945, 0.0121265366936926)

(-2.137391135729064, -0.445271188713631)

(68.32232248570797, 0.0146357210055436*I)

(-87.17346069807797, -0.0114710038646054)

(30.6142018882607, 0.0326558712102137*I)

(-91.88614381546644, -0.0108827114792659*I)

(84.03165362561431, 0.0118998562546094*I)

(-21.18216315883595, -0.0471832636912932)

(-24.326801167853258, -0.0410895782745605)

(-10.16112692999623, -0.0981774182935068*I)

(-11.73848009399415, -0.0850360393335331)

(54.18324637727305, 0.018454318078295)

(-25.89884309605599, -0.0385973854559027*I)

(-74.60612406136674, -0.0134031234387214)

(90.31525285349807, 0.0110719875286708*I)

(-3.798273009875294, -0.258903189710353*I)

(-33.75731375129117, -0.0296167148833163)

(-82.46074396268945, -0.0121265366936926*I)

(80.88982989803151, 0.0123620212466517*I)

(38.471516303979705, 0.0259888679989479)

(69.89328326721875, 0.0143067943509695)

(5.406336666943637, 0.18341903022942*I)

(-8.581375694210113, -0.116139144217135)

(25.89884309605599, 0.0385973854559027)

(-68.32232248570797, -0.0146357210055436)

(91.88614381546644, 0.0108827114792659)

(52.612174588034385, 0.0190052912967891*I)

(93.45703159623535, 0.0106997982057615*I)

(-77.74798749631695, -0.0128615373628376)


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=−99.7405539133885x_{1} = -99.7405539133885
x2=−62.0383960813757x_{2} = -62.0383960813757
x3=−46.3277006372222x_{3} = -46.3277006372222
x4=−65.1803771434091x_{4} = -65.1803771434091
x5=−30.6142018882607x_{5} = -30.6142018882607
x6=−18.0364644916681x_{6} = -18.0364644916681
x7=−27.4707425696429x_{7} = -27.4707425696429
x8=−43.1853230479384x_{8} = -43.1853230479384
x9=−90.3152528534981x_{9} = -90.3152528534981
x10=−52.6121745880344x_{10} = -52.6121745880344
x11=−5.40633666694364x_{11} = -5.40633666694364
x12=−40.0428222957763x_{12} = -40.0428222957763
x13=−93.4570315962354x_{13} = -93.4570315962354
x14=−55.7543026449315x_{14} = -55.7543026449315
x15=−84.0316536256143x_{15} = -84.0316536256143
x16=−49.4699785303278x_{16} = -49.4699785303278
x17=−71.4642368192271x_{17} = -71.4642368192271
x18=−96.5987982349427x_{18} = -96.5987982349427
x19=−2.13739113572906x_{19} = -2.13739113572906
x20=−87.173460698078x_{20} = -87.173460698078
x21=−21.182163158836x_{21} = -21.182163158836
x22=−24.3268011678533x_{22} = -24.3268011678533
x23=−11.7384800939942x_{23} = -11.7384800939942
x24=−74.6061240613667x_{24} = -74.6061240613667
x25=−33.7573137512912x_{25} = -33.7573137512912
x26=−8.58137569421011x_{26} = -8.58137569421011
x27=−68.322322485708x_{27} = -68.322322485708
x28=−77.747987496317x_{28} = -77.747987496317
Maxima of the function at points:
x28=66.7513539643031x_{28} = 66.7513539643031
x28=22.7545872771683x_{28} = 22.7545872771683
x28=63.609391408151x_{28} = 63.609391408151
x28=76.1770585706428x_{28} = 76.1770585706428
x28=41.614089810742x_{28} = 41.614089810742
x28=47.8988508196586x_{28} = 47.8988508196586
x28=85.602559129126x_{28} = 85.602559129126
x28=44.7565256168206x_{28} = 44.7565256168206
x28=98.1696773784321x_{28} = 98.1696773784321
x28=16.4630276170453x_{28} = 16.4630276170453
x28=3.79827300987529x_{28} = 3.79827300987529
x28=88.744358541403x_{28} = 88.744358541403
x28=60.4673904155608x_{28} = 60.4673904155608
x28=19.6094782798075x_{28} = 19.6094782798075
x28=32.1857948915632x_{28} = 32.1857948915632
x28=10.1611269299962x_{28} = 10.1611269299962
x28=82.4607439626894x_{28} = 82.4607439626894
x28=54.1832463772731x_{28} = 54.1832463772731
x28=38.4715163039797x_{28} = 38.4715163039797
x28=69.8932832672187x_{28} = 69.8932832672187
x28=25.898843096056x_{28} = 25.898843096056
x28=91.8861438154664x_{28} = 91.8861438154664
Decreasing at intervals
[−2.13739113572906,3.79827300987529]\left[-2.13739113572906, 3.79827300987529\right]
Increasing at intervals
(−∞,−99.7405539133885]\left(-\infty, -99.7405539133885\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⁡(2x)−cos⁡2(2x)sin⁡32(2x)−2cos⁡(2x)xsin⁡(2x)+2sin⁡(2x)x2x=0\frac{- 2 \sqrt{\sin{\left(2 x \right)}} - \frac{\cos^{2}{\left(2 x \right)}}{\sin^{\frac{3}{2}}{\left(2 x \right)}} - \frac{2 \cos{\left(2 x \right)}}{x \sqrt{\sin{\left(2 x \right)}}} + \frac{2 \sqrt{\sin{\left(2 x \right)}}}{x^{2}}}{x} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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→−∞(sin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(sin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\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 sqrt(sin(2*x))/x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(2x)x2)=0\lim_{x \to -\infty}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(2x)x2)=0\lim_{x \to \infty}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{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:
sin⁡(2x)x=−−sin⁡(2x)x\frac{\sqrt{\sin{\left(2 x \right)}}}{x} = - \frac{\sqrt{- \sin{\left(2 x \right)}}}{x}
- No
sin⁡(2x)x=−sin⁡(2x)x\frac{\sqrt{\sin{\left(2 x \right)}}}{x} = \frac{\sqrt{- \sin{\left(2 x \right)}}}{x}
- No
so, the function
not is
neither even, nor odd
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