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Plotting a function graph/ 1/sqrtxsinx

Graphing y = 1/sqrtxsinx

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(x)
f(x) = ------
         ___ 
       \/ x
f(x)=sin⁡(x)xf{\left(x \right)} = \frac{\sin{\left(x \right)}}{\sqrt{x}} 
f = sin(x)/sqrt(x)
The graph of the function
3.000.250.500.751.001.251.501.752.002.252.502.750.01.0
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⁡(x)x=0\frac{\sin{\left(x \right)}}{\sqrt{x}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
Numerical solution
x1=43.9822971502571x_{1} = 43.9822971502571
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=−43.9822971502571x_{3} = -43.9822971502571
x4=−72.2566310325652x_{4} = -72.2566310325652
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=81.6814089933346x_{6} = 81.6814089933346
x7=−31.4159265358979x_{7} = -31.4159265358979
x8=−78.5398163397448x_{8} = -78.5398163397448
x9=97.3893722612836x_{9} = 97.3893722612836
x10=9.42477796076938x_{10} = 9.42477796076938
x11=−25.1327412287183x_{11} = -25.1327412287183
x12=84.8230016469244x_{12} = 84.8230016469244
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=6.28318530717959x_{15} = 6.28318530717959
x16=119.380520836412x_{16} = 119.380520836412
x17=3.14159265358979x_{17} = 3.14159265358979
x18=131.946891450771x_{18} = 131.946891450771
x19=−50.2654824574367x_{19} = -50.2654824574367
x20=28.2743338823081x_{20} = 28.2743338823081
x21=−75.398223686155x_{21} = -75.398223686155
x22=−28.2743338823081x_{22} = -28.2743338823081
x23=−56.5486677646163x_{23} = -56.5486677646163
x24=−65.9734457253857x_{24} = -65.9734457253857
x25=−40.8407044966673x_{25} = -40.8407044966673
x26=−91.106186954104x_{26} = -91.106186954104
x27=50.2654824574367x_{27} = 50.2654824574367
x28=−69.1150383789755x_{28} = -69.1150383789755
x29=−100.530964914873x_{29} = -100.530964914873
x30=56.5486677646163x_{30} = 56.5486677646163
x31=−62.8318530717959x_{31} = -62.8318530717959
x32=−87.9645943005142x_{32} = -87.9645943005142
x33=40.8407044966673x_{33} = 40.8407044966673
x34=100.530964914873x_{34} = 100.530964914873
x35=18.8495559215388x_{35} = 18.8495559215388
x36=62.8318530717959x_{36} = 62.8318530717959
x37=−53.4070751110265x_{37} = -53.4070751110265
x38=94.2477796076938x_{38} = 94.2477796076938
x39=−3.14159265358979x_{39} = -3.14159265358979
x40=21.9911485751286x_{40} = 21.9911485751286
x41=12.5663706143592x_{41} = 12.5663706143592
x42=−84.8230016469244x_{42} = -84.8230016469244
x43=34.5575191894877x_{43} = 34.5575191894877
x44=47.1238898038469x_{44} = 47.1238898038469
x45=−15.707963267949x_{45} = -15.707963267949
x46=53.4070751110265x_{46} = 53.4070751110265
x47=65.9734457253857x_{47} = 65.9734457253857
x48=87.9645943005142x_{48} = 87.9645943005142
x49=91.106186954104x_{49} = 91.106186954104
x50=59.6902604182061x_{50} = 59.6902604182061
x51=69.1150383789755x_{51} = 69.1150383789755
x52=−6.28318530717959x_{52} = -6.28318530717959
x53=75.398223686155x_{53} = 75.398223686155
x54=−37.6991118430775x_{54} = -37.6991118430775
x55=−12.5663706143592x_{55} = -12.5663706143592
x56=−18.8495559215388x_{56} = -18.8495559215388
x57=31.4159265358979x_{57} = 31.4159265358979
x58=−81.6814089933346x_{58} = -81.6814089933346
x59=78.5398163397448x_{59} = 78.5398163397448
x60=15.707963267949x_{60} = 15.707963267949
x61=72.2566310325652x_{61} = 72.2566310325652
x62=37.6991118430775x_{62} = 37.6991118430775
x63=25.1327412287183x_{63} = 25.1327412287183
x64=−47.1238898038469x_{64} = -47.1238898038469
x65=521.504380495906x_{65} = 521.504380495906
x66=−9.42477796076938x_{66} = -9.42477796076938
x67=−34.5575191894877x_{67} = -34.5575191894877
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)/sqrt(x).
sin⁡(0)0\frac{\sin{\left(0 \right)}}{\sqrt{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⁡(x)x−sin⁡(x)2x32=0\frac{\cos{\left(x \right)}}{\sqrt{x}} - \frac{\sin{\left(x \right)}}{2 x^{\frac{3}{2}}} = 0
Solve this equation
The roots of this equation
x1=−39.2571723324086x_{1} = -39.2571723324086
x2=−26.6848024909251x_{2} = -26.6848024909251
x3=−58.1108600600615x_{3} = -58.1108600600615
x4=−1.16556118520721x_{4} = -1.16556118520721
x5=−7.78988375114457x_{5} = -7.78988375114457
x6=7.78988375114457x_{6} = 7.78988375114457
x7=−51.8266315338985x_{7} = -51.8266315338985
x8=67.5368388204916x_{8} = 67.5368388204916
x9=−80.1043708909521x_{9} = -80.1043708909521
x10=−67.5368388204916x_{10} = -67.5368388204916
x11=20.3958423573092x_{11} = 20.3958423573092
x12=64.3948849627586x_{12} = 64.3948849627586
x13=117.80548025038x_{13} = 117.80548025038
x14=70.6787605627689x_{14} = 70.6787605627689
x15=−45.5421150692309x_{15} = -45.5421150692309
x16=32.9715594404485x_{16} = 32.9715594404485
x17=4.60421677720058x_{17} = 4.60421677720058
x18=−23.5407082923052x_{18} = -23.5407082923052
x19=−64.3948849627586x_{19} = -64.3948849627586
x20=−70.6787605627689x_{20} = -70.6787605627689
x21=−76.9625234358705x_{21} = -76.9625234358705
x22=−73.8206542907788x_{22} = -73.8206542907788
x23=73.8206542907788x_{23} = 73.8206542907788
x24=48.6844162648433x_{24} = 48.6844162648433
x25=80.1043708909521x_{25} = 80.1043708909521
x26=−20.3958423573092x_{26} = -20.3958423573092
x27=−92.6715879363332x_{27} = -92.6715879363332
x28=26.6848024909251x_{28} = 26.6848024909251
x29=−36.1144715353049x_{29} = -36.1144715353049
x30=158.64727737108x_{30} = 158.64727737108
x31=92.6715879363332x_{31} = 92.6715879363332
x32=10.9499436485412x_{32} = 10.9499436485412
x33=83.2461991121237x_{33} = 83.2461991121237
x34=−98.9551158352145x_{34} = -98.9551158352145
x35=54.9687756155963x_{35} = 54.9687756155963
x36=−29.8283692130955x_{36} = -29.8283692130955
x37=14.1017251335659x_{37} = 14.1017251335659
x38=−4.60421677720058x_{38} = -4.60421677720058
x39=215.19677332017x_{39} = 215.19677332017
x40=−61.2528940466862x_{40} = -61.2528940466862
x41=−83.2461991121237x_{41} = -83.2461991121237
x42=−246.612995841404x_{42} = -246.612995841404
x43=−54.9687756155963x_{43} = -54.9687756155963
x44=76.9625234358705x_{44} = 76.9625234358705
x45=36.1144715353049x_{45} = 36.1144715353049
x46=−48.6844162648433x_{46} = -48.6844162648433
x47=29.8283692130955x_{47} = 29.8283692130955
x48=58.1108600600615x_{48} = 58.1108600600615
x49=−42.3997088362447x_{49} = -42.3997088362447
x50=17.2497818346079x_{50} = 17.2497818346079
x51=−32.9715594404485x_{51} = -32.9715594404485
x52=23.5407082923052x_{52} = 23.5407082923052
x53=1.16556118520721x_{53} = 1.16556118520721
x54=−17.2497818346079x_{54} = -17.2497818346079
x55=−10.9499436485412x_{55} = -10.9499436485412
x56=51.8266315338985x_{56} = 51.8266315338985
x57=45.5421150692309x_{57} = 45.5421150692309
x58=−86.3880101981266x_{58} = -86.3880101981266
x59=89.5298059530594x_{59} = 89.5298059530594
x60=−95.8133575027966x_{60} = -95.8133575027966
x61=39.2571723324086x_{61} = 39.2571723324086
x62=−89.5298059530594x_{62} = -89.5298059530594
x63=61.2528940466862x_{63} = 61.2528940466862
x64=95.8133575027966x_{64} = 95.8133575027966
x65=−14.1017251335659x_{65} = -14.1017251335659
x66=98.9551158352145x_{66} = 98.9551158352145
x67=86.3880101981266x_{67} = 86.3880101981266
x68=42.3997088362447x_{68} = 42.3997088362447
The values of the extrema at the points:
(-39.25717233240859, 0.159589851348603*I)

(-26.68480249092507, 0.19354937797769*I)

(-58.110860060061505, 0.131176268600912*I)

(-1.1655611852072114, 0.851241066782324*I)

(-7.789883751144573, 0.357554083426262*I)

(7.789883751144573, 0.357554083426262)

(-51.82663153389846, 0.138900336703391*I)

(67.53683882049161, -0.121679588990783)

(-80.1043708909521, -0.111728362291416*I)

(-67.53683882049161, -0.121679588990783*I)

(20.395842357309167, 0.221359780635401)

(64.39488496275855, 0.124612389237314)

(117.80548025038037, -0.0921326029924126)

(70.67876056276886, 0.118944583684481)

(-45.5421150692309, 0.148172370731446*I)

(32.97155944044848, 0.17413269656851)

(4.604216777200577, -0.463314891176637)

(-23.54070829230515, -0.206059336815155*I)

(-64.39488496275855, 0.124612389237314*I)

(-70.67876056276886, 0.118944583684481*I)

(-76.96252343587051, 0.113985913925499*I)

(-73.82065429077876, -0.116386094038002*I)

(73.82065429077876, -0.116386094038002)

(48.68441626484328, -0.143311853691665)

(80.1043708909521, -0.111728362291416)

(-20.395842357309167, 0.221359780635401*I)

(-92.67158793633321, -0.103877233902111*I)

(26.68480249092507, 0.19354937797769)

(-36.11447153530485, -0.166386370791913*I)

(158.6472773710796, 0.0793928754394215)

(92.67158793633321, -0.103877233902111)

(10.94994364854116, -0.301885161430297)

(83.24619911212368, 0.109599849994829)

(-98.95511583521451, -0.100525289012326*I)

(54.96877561559635, -0.134872684738376)

(-29.828369213095506, -0.183072974858657*I)

(14.101725133565873, 0.266128298234218)

(-4.604216777200577, -0.463314891176637*I)

(215.1967733201699, 0.0681680624478802)

(-61.252894046686194, -0.127768037744087*I)

(-83.24619911212368, 0.109599849994829*I)

(-246.61299584140428, 0.0636782512070729*I)

(-54.96877561559635, -0.134872684738376*I)

(76.96252343587051, 0.113985913925499)

(36.11447153530485, -0.166386370791913)

(-48.68441626484328, -0.143311853691665*I)

(29.828369213095506, -0.183072974858657)

(58.110860060061505, 0.131176268600912)

(-42.39970883624466, -0.15356362930828*I)

(17.249781834607894, -0.240672145897842)

(-32.97155944044848, 0.17413269656851*I)

(23.54070829230515, -0.206059336815155)

(1.1655611852072114, 0.851241066782324)

(-17.249781834607894, -0.240672145897842*I)

(-10.94994364854116, -0.301885161430297*I)

(51.82663153389846, 0.138900336703391)

(45.5421150692309, 0.148172370731446)

(-86.38801019812658, -0.107588534144322*I)

(89.52980595305935, 0.105684039776562)

(-95.81335750279658, 0.102160040658152*I)

(39.25717233240859, 0.159589851348603)

(-89.52980595305935, 0.105684039776562*I)

(61.252894046686194, -0.127768037744087)

(95.81335750279658, 0.102160040658152)

(-14.101725133565873, 0.266128298234218*I)

(98.95511583521451, -0.100525289012326)

(86.38801019812658, -0.107588534144322)

(42.39970883624466, -0.15356362930828)


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=67.5368388204916x_{1} = 67.5368388204916
x2=117.80548025038x_{2} = 117.80548025038
x3=4.60421677720058x_{3} = 4.60421677720058
x4=73.8206542907788x_{4} = 73.8206542907788
x5=48.6844162648433x_{5} = 48.6844162648433
x6=80.1043708909521x_{6} = 80.1043708909521
x7=92.6715879363332x_{7} = 92.6715879363332
x8=10.9499436485412x_{8} = 10.9499436485412
x9=54.9687756155963x_{9} = 54.9687756155963
x10=36.1144715353049x_{10} = 36.1144715353049
x11=29.8283692130955x_{11} = 29.8283692130955
x12=17.2497818346079x_{12} = 17.2497818346079
x13=23.5407082923052x_{13} = 23.5407082923052
x14=61.2528940466862x_{14} = 61.2528940466862
x15=98.9551158352145x_{15} = 98.9551158352145
x16=86.3880101981266x_{16} = 86.3880101981266
x17=42.3997088362447x_{17} = 42.3997088362447
Maxima of the function at points:
x17=7.78988375114457x_{17} = 7.78988375114457
x17=20.3958423573092x_{17} = 20.3958423573092
x17=64.3948849627586x_{17} = 64.3948849627586
x17=70.6787605627689x_{17} = 70.6787605627689
x17=32.9715594404485x_{17} = 32.9715594404485
x17=26.6848024909251x_{17} = 26.6848024909251
x17=158.64727737108x_{17} = 158.64727737108
x17=83.2461991121237x_{17} = 83.2461991121237
x17=14.1017251335659x_{17} = 14.1017251335659
x17=215.19677332017x_{17} = 215.19677332017
x17=76.9625234358705x_{17} = 76.9625234358705
x17=58.1108600600615x_{17} = 58.1108600600615
x17=1.16556118520721x_{17} = 1.16556118520721
x17=51.8266315338985x_{17} = 51.8266315338985
x17=45.5421150692309x_{17} = 45.5421150692309
x17=89.5298059530594x_{17} = 89.5298059530594
x17=39.2571723324086x_{17} = 39.2571723324086
x17=95.8133575027966x_{17} = 95.8133575027966
Decreasing at intervals
[117.80548025038,∞)\left[117.80548025038, \infty\right)
Increasing at intervals
(−∞,4.60421677720058]\left(-\infty, 4.60421677720058\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
−sin⁡(x)−cos⁡(x)x+3sin⁡(x)4x2x=0\frac{- \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{4 x^{2}}}{\sqrt{x}} = 0
Solve this equation
The roots of this equation
x1=12.4860672578708x_{1} = 12.4860672578708
x2=−47.1026555912318x_{2} = -47.1026555912318
x3=−37.6725595300203x_{3} = -37.6725595300203
x4=−28.2389032383054x_{4} = -28.2389032383054
x5=169.640108376141x_{5} = 169.640108376141
x6=37.6725595300203x_{6} = 37.6725595300203
x7=−43.9595440566684x_{7} = -43.9595440566684
x8=53.3883416918471x_{8} = 53.3883416918471
x9=−94.2371675854493x_{9} = -94.2371675854493
x10=25.0928628865337x_{10} = 25.0928628865337
x11=−84.8112100697664x_{11} = -84.8112100697664
x12=62.8159318625173x_{12} = 62.8159318625173
x13=−147.648081727825x_{13} = -147.648081727825
x14=−87.95322400825x_{14} = -87.95322400825
x15=−40.8161982919721x_{15} = -40.8161982919721
x16=43.9595440566684x_{16} = 43.9595440566684
x17=65.9582831752547x_{17} = 65.9582831752547
x18=91.0952088771736x_{18} = 91.0952088771736
x19=81.6691637048431x_{19} = 81.6691637048431
x20=−31.3840497369889x_{20} = -31.3840497369889
x21=15.6439318755503x_{21} = 15.6439318755503
x22=−59.6735006001685x_{22} = -59.6735006001685
x23=−56.5309760413753x_{23} = -56.5309760413753
x24=−9.31693112610028x_{24} = -9.31693112610028
x25=28.2389032383054x_{25} = 28.2389032383054
x26=−62.8159318625173x_{26} = -62.8159318625173
x27=−75.384957467622x_{27} = -75.384957467622
x28=31.3840497369889x_{28} = 31.3840497369889
x29=75.384957467622x_{29} = 75.384957467622
x30=−78.5270810189266x_{30} = -78.5270810189266
x31=−113.088492608463x_{31} = -113.088492608463
x32=−69.1005654545348x_{32} = -69.1005654545348
x33=−91.0952088771736x_{33} = -91.0952088771736
x34=−81.6691637048431x_{34} = -81.6691637048431
x35=72.2427877152145x_{35} = 72.2427877152145
x36=2.75936321522763x_{36} = 2.75936321522763
x37=−65.9582831752547x_{37} = -65.9582831752547
x38=−6.11791002392407x_{38} = -6.11791002392407
x39=18.796291187414x_{39} = 18.796291187414
x40=−25.0928628865337x_{40} = -25.0928628865337
x41=40.8161982919721x_{41} = 40.8161982919721
x42=9.31693112610028x_{42} = 9.31693112610028
x43=78.5270810189266x_{43} = 78.5270810189266
x44=59.6735006001685x_{44} = 59.6735006001685
x45=−18.796291187414x_{45} = -18.796291187414
x46=97.3791026663451x_{46} = 97.3791026663451
x47=56.5309760413753x_{47} = 56.5309760413753
x48=87.95322400825x_{48} = 87.95322400825
x49=−97.3791026663451x_{49} = -97.3791026663451
x50=−100.521016336234x_{50} = -100.521016336234
x51=21.9455418081046x_{51} = 21.9455418081046
x52=69.1005654545348x_{52} = 69.1005654545348
x53=34.5285475249278x_{53} = 34.5285475249278
x54=−2.75936321522763x_{54} = -2.75936321522763
x55=−50.2455769233645x_{55} = -50.2455769233645
x56=100.521016336234x_{56} = 100.521016336234
x57=84.8112100697664x_{57} = 84.8112100697664
x58=−53.3883416918471x_{58} = -53.3883416918471
x59=6.11791002392407x_{59} = 6.11791002392407
x60=47.1026555912318x_{60} = 47.1026555912318
x61=−12.4860672578708x_{61} = -12.4860672578708
x62=94.2371675854493x_{62} = 94.2371675854493
x63=−34.5285475249278x_{63} = -34.5285475249278
x64=−15.6439318755503x_{64} = -15.6439318755503
x65=−72.2427877152145x_{65} = -72.2427877152145
x66=−21.9455418081046x_{66} = -21.9455418081046
x67=50.2455769233645x_{67} = 50.2455769233645
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

lim⁡x→0−(−sin⁡(x)−cos⁡(x)x+3sin⁡(x)4x2x)=−∞i\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{4 x^{2}}}{\sqrt{x}}\right) = - \infty i
lim⁡x→0+(−sin⁡(x)−cos⁡(x)x+3sin⁡(x)4x2x)=−∞\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{4 x^{2}}}{\sqrt{x}}\right) = -\infty
- 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:
Concave at the intervals
[97.3791026663451,∞)\left[97.3791026663451, \infty\right)
Convex at the intervals
(−∞,2.75936321522763]\left(-\infty, 2.75936321522763\right]
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⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{x}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{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 sin(x)/sqrt(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)xx)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{x} x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)xx)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{x} 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:
sin⁡(x)x=−sin⁡(x)−x\frac{\sin{\left(x \right)}}{\sqrt{x}} = - \frac{\sin{\left(x \right)}}{\sqrt{- x}}
- No
sin⁡(x)x=sin⁡(x)−x\frac{\sin{\left(x \right)}}{\sqrt{x}} = \frac{\sin{\left(x \right)}}{\sqrt{- x}}
- No
so, the function
not is
neither even, nor odd
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