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Plotting a function graph/ x*sin(x)

Graphing y = x*sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = x*sin(x)
f(x)=xsin⁡(x)f{\left(x \right)} = x \sin{\left(x \right)} 
f = x*sin(x)
The graph of the function
0.02.55.07.510.012.515.017.520.022.525.027.530.0-5050
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:
xsin⁡(x)=0x \sin{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=59.6902604182061x_{1} = 59.6902604182061
x2=−100.530964914873x_{2} = -100.530964914873
x3=−15.707963267949x_{3} = -15.707963267949
x4=−25.1327412287183x_{4} = -25.1327412287183
x5=84.8230016469244x_{5} = 84.8230016469244
x6=0x_{6} = 0
x7=−75.398223686155x_{7} = -75.398223686155
x8=97.3893722612836x_{8} = 97.3893722612836
x9=−50.2654824574367x_{9} = -50.2654824574367
x10=81.6814089933346x_{10} = 81.6814089933346
x11=−72.2566310325652x_{11} = -72.2566310325652
x12=91.106186954104x_{12} = 91.106186954104
x13=50.2654824574367x_{13} = 50.2654824574367
x14=−43.9822971502571x_{14} = -43.9822971502571
x15=−37.6991118430775x_{15} = -37.6991118430775
x16=25.1327412287183x_{16} = 25.1327412287183
x17=−65.9734457253857x_{17} = -65.9734457253857
x18=−53.4070751110265x_{18} = -53.4070751110265
x19=−18.8495559215388x_{19} = -18.8495559215388
x20=−59.6902604182061x_{20} = -59.6902604182061
x21=15.707963267949x_{21} = 15.707963267949
x22=9.42477796076938x_{22} = 9.42477796076938
x23=18.8495559215388x_{23} = 18.8495559215388
x24=−56.5486677646163x_{24} = -56.5486677646163
x25=−6.28318530717959x_{25} = -6.28318530717959
x26=−62.8318530717959x_{26} = -62.8318530717959
x27=12.5663706143592x_{27} = 12.5663706143592
x28=56.5486677646163x_{28} = 56.5486677646163
x29=40.8407044966673x_{29} = 40.8407044966673
x30=3.14159265358979x_{30} = 3.14159265358979
x31=−21.9911485751286x_{31} = -21.9911485751286
x32=−84.8230016469244x_{32} = -84.8230016469244
x33=6.28318530717959x_{33} = 6.28318530717959
x34=69.1150383789755x_{34} = 69.1150383789755
x35=72.2566310325652x_{35} = 72.2566310325652
x36=−78.5398163397448x_{36} = -78.5398163397448
x37=37.6991118430775x_{37} = 37.6991118430775
x38=21.9911485751286x_{38} = 21.9911485751286
x39=47.1238898038469x_{39} = 47.1238898038469
x40=34.5575191894877x_{40} = 34.5575191894877
x41=−97.3893722612836x_{41} = -97.3893722612836
x42=−31.4159265358979x_{42} = -31.4159265358979
x43=100.530964914873x_{43} = 100.530964914873
x44=−47.1238898038469x_{44} = -47.1238898038469
x45=28.2743338823081x_{45} = 28.2743338823081
x46=94.2477796076938x_{46} = 94.2477796076938
x47=−40.8407044966673x_{47} = -40.8407044966673
x48=−12.5663706143592x_{48} = -12.5663706143592
x49=−34.5575191894877x_{49} = -34.5575191894877
x50=−28.2743338823081x_{50} = -28.2743338823081
x51=78.5398163397448x_{51} = 78.5398163397448
x52=−94.2477796076938x_{52} = -94.2477796076938
x53=−91.106186954104x_{53} = -91.106186954104
x54=697.433569096934x_{54} = 697.433569096934
x55=43.9822971502571x_{55} = 43.9822971502571
x56=75.398223686155x_{56} = 75.398223686155
x57=62.8318530717959x_{57} = 62.8318530717959
x58=−3.14159265358979x_{58} = -3.14159265358979
x59=87.9645943005142x_{59} = 87.9645943005142
x60=53.4070751110265x_{60} = 53.4070751110265
x61=−81.6814089933346x_{61} = -81.6814089933346
x62=−87.9645943005142x_{62} = -87.9645943005142
x63=65.9734457253857x_{63} = 65.9734457253857
x64=−69.1150383789755x_{64} = -69.1150383789755
x65=31.4159265358979x_{65} = 31.4159265358979
x66=−9.42477796076938x_{66} = -9.42477796076938
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*sin(x).
0sin⁡(0)0 \sin{\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
xcos⁡(x)+sin⁡(x)=0x \cos{\left(x \right)} + \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−64.4181717218392x_{1} = -64.4181717218392
x2=51.855560729152x_{2} = 51.855560729152
x3=−58.1366632448992x_{3} = -58.1366632448992
x4=26.7409160147873x_{4} = 26.7409160147873
x5=−17.3363779239834x_{5} = -17.3363779239834
x6=−61.2773745335697x_{6} = -61.2773745335697
x7=−42.4350618814099x_{7} = -42.4350618814099
x8=102.111554139654x_{8} = 102.111554139654
x9=29.8785865061074x_{9} = 29.8785865061074
x10=64.4181717218392x_{10} = 64.4181717218392
x11=−2.02875783811043x_{11} = -2.02875783811043
x12=−33.0170010333572x_{12} = -33.0170010333572
x13=73.8409691490209x_{13} = 73.8409691490209
x14=−80.1230928148503x_{14} = -80.1230928148503
x15=89.5465575382492x_{15} = 89.5465575382492
x16=−20.469167402741x_{16} = -20.469167402741
x17=−26.7409160147873x_{17} = -26.7409160147873
x18=−36.1559664195367x_{18} = -36.1559664195367
x19=33.0170010333572x_{19} = 33.0170010333572
x20=20.469167402741x_{20} = 20.469167402741
x21=54.9960525574964x_{21} = 54.9960525574964
x22=7.97866571241324x_{22} = 7.97866571241324
x23=−14.2074367251912x_{23} = -14.2074367251912
x24=39.295350981473x_{24} = 39.295350981473
x25=36.1559664195367x_{25} = 36.1559664195367
x26=83.2642147040886x_{26} = 83.2642147040886
x27=86.4053708116885x_{27} = 86.4053708116885
x28=−92.687771772017x_{28} = -92.687771772017
x29=−29.8785865061074x_{29} = -29.8785865061074
x30=−67.5590428388084x_{30} = -67.5590428388084
x31=76.9820093304187x_{31} = 76.9820093304187
x32=−11.085538406497x_{32} = -11.085538406497
x33=70.69997803861x_{33} = 70.69997803861
x34=−51.855560729152x_{34} = -51.855560729152
x35=48.7152107175577x_{35} = 48.7152107175577
x36=17.3363779239834x_{36} = 17.3363779239834
x37=−4.91318043943488x_{37} = -4.91318043943488
x38=−86.4053708116885x_{38} = -86.4053708116885
x39=92.687771772017x_{39} = 92.687771772017
x40=−39.295350981473x_{40} = -39.295350981473
x41=−73.8409691490209x_{41} = -73.8409691490209
x42=80.1230928148503x_{42} = 80.1230928148503
x43=58.1366632448992x_{43} = 58.1366632448992
x44=−45.57503179559x_{44} = -45.57503179559
x45=67.5590428388084x_{45} = 67.5590428388084
x46=−89.5465575382492x_{46} = -89.5465575382492
x47=−70.69997803861x_{47} = -70.69997803861
x48=95.8290108090195x_{48} = 95.8290108090195
x49=11.085538406497x_{49} = 11.085538406497
x50=−95.8290108090195x_{50} = -95.8290108090195
x51=0x_{51} = 0
x52=98.9702722883957x_{52} = 98.9702722883957
x53=2.02875783811043x_{53} = 2.02875783811043
x54=−83.2642147040886x_{54} = -83.2642147040886
x55=4.91318043943488x_{55} = 4.91318043943488
x56=−23.6042847729804x_{56} = -23.6042847729804
x57=−48.7152107175577x_{57} = -48.7152107175577
x58=−76.9820093304187x_{58} = -76.9820093304187
x59=61.2773745335697x_{59} = 61.2773745335697
x60=42.4350618814099x_{60} = 42.4350618814099
x61=−54.9960525574964x_{61} = -54.9960525574964
x62=−7.97866571241324x_{62} = -7.97866571241324
x63=−98.9702722883957x_{63} = -98.9702722883957
x64=23.6042847729804x_{64} = 23.6042847729804
x65=45.57503179559x_{65} = 45.57503179559
x66=14.2074367251912x_{66} = 14.2074367251912
The values of the extrema at the points:
(-64.41817172183916, 64.4104113393753)

(51.85556072915197, 51.8459212502015)

(-58.13666324489916, 58.1280647280857)

(26.74091601478731, 26.7222376646974)

(-17.33637792398336, -17.3076086078585)

(-61.277374533569656, -61.2692165444766)

(-42.43506188140989, -42.4232840772591)

(102.11155413965392, 102.106657886316)

(29.878586506107393, -29.8618661591868)

(64.41817172183916, 64.4104113393753)

(-2.028757838110434, 1.81970574115965)

(-33.017001033357246, 33.0018677308454)

(73.8409691490209, -73.8341987715416)

(-80.12309281485025, -80.1168531456592)

(89.54655753824919, 89.5409743728852)

(-20.46916740274095, 20.4447840582523)

(-26.74091601478731, 26.7222376646974)

(-36.15596641953672, -36.1421453722421)

(33.017001033357246, 33.0018677308454)

(20.46916740274095, 20.4447840582523)

(54.99605255749639, -54.9869632496976)

(7.978665712413241, 7.91672737158778)

(-14.207436725191188, 14.1723741137743)

(39.295350981472986, 39.2826330068918)

(36.15596641953672, -36.1421453722421)

(83.26421470408864, 83.2582103729533)

(86.40537081168854, -86.3995847156108)

(-92.687771772017, -92.6823777880592)

(-29.878586506107393, -29.8618661591868)

(-67.5590428388084, -67.5516431209725)

(76.98200933041872, 76.9755151282637)

(-11.085538406497022, -11.04070801593)

(70.69997803861, 70.6929069615931)

(-51.85556072915197, 51.8459212502015)

(48.715210717557724, -48.7049502253679)

(17.33637792398336, -17.3076086078585)

(-4.913180439434884, -4.81446988971227)

(-86.40537081168854, -86.3995847156108)

(92.687771772017, -92.6823777880592)

(-39.295350981472986, 39.2826330068918)

(-73.8409691490209, -73.8341987715416)

(80.12309281485025, -80.1168531456592)

(58.13666324489916, 58.1280647280857)

(-45.57503179559002, 45.5640648360268)

(67.5590428388084, -67.5516431209725)

(-89.54655753824919, 89.5409743728852)

(-70.69997803861, 70.6929069615931)

(95.82901080901948, 95.8237936084657)

(11.085538406497022, -11.04070801593)

(-95.82901080901948, 95.8237936084657)

(0, 0)

(98.9702722883957, -98.9652206531187)

(2.028757838110434, 1.81970574115965)

(-83.26421470408864, 83.2582103729533)

(4.913180439434884, -4.81446988971227)

(-23.604284772980407, -23.5831306496334)

(-48.715210717557724, -48.7049502253679)

(-76.98200933041872, 76.9755151282637)

(61.277374533569656, -61.2692165444766)

(42.43506188140989, -42.4232840772591)

(-54.99605255749639, -54.9869632496976)

(-7.978665712413241, 7.91672737158778)

(-98.9702722883957, -98.9652206531187)

(23.604284772980407, -23.5831306496334)

(45.57503179559002, 45.5640648360268)

(14.207436725191188, 14.1723741137743)


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=−17.3363779239834x_{1} = -17.3363779239834
x2=−61.2773745335697x_{2} = -61.2773745335697
x3=−42.4350618814099x_{3} = -42.4350618814099
x4=29.8785865061074x_{4} = 29.8785865061074
x5=73.8409691490209x_{5} = 73.8409691490209
x6=−80.1230928148503x_{6} = -80.1230928148503
x7=−36.1559664195367x_{7} = -36.1559664195367
x8=54.9960525574964x_{8} = 54.9960525574964
x9=36.1559664195367x_{9} = 36.1559664195367
x10=86.4053708116885x_{10} = 86.4053708116885
x11=−92.687771772017x_{11} = -92.687771772017
x12=−29.8785865061074x_{12} = -29.8785865061074
x13=−67.5590428388084x_{13} = -67.5590428388084
x14=−11.085538406497x_{14} = -11.085538406497
x15=48.7152107175577x_{15} = 48.7152107175577
x16=17.3363779239834x_{16} = 17.3363779239834
x17=−4.91318043943488x_{17} = -4.91318043943488
x18=−86.4053708116885x_{18} = -86.4053708116885
x19=92.687771772017x_{19} = 92.687771772017
x20=−73.8409691490209x_{20} = -73.8409691490209
x21=80.1230928148503x_{21} = 80.1230928148503
x22=67.5590428388084x_{22} = 67.5590428388084
x23=11.085538406497x_{23} = 11.085538406497
x24=0x_{24} = 0
x25=98.9702722883957x_{25} = 98.9702722883957
x26=4.91318043943488x_{26} = 4.91318043943488
x27=−23.6042847729804x_{27} = -23.6042847729804
x28=−48.7152107175577x_{28} = -48.7152107175577
x29=61.2773745335697x_{29} = 61.2773745335697
x30=42.4350618814099x_{30} = 42.4350618814099
x31=−54.9960525574964x_{31} = -54.9960525574964
x32=−98.9702722883957x_{32} = -98.9702722883957
x33=23.6042847729804x_{33} = 23.6042847729804
Maxima of the function at points:
x33=−64.4181717218392x_{33} = -64.4181717218392
x33=51.855560729152x_{33} = 51.855560729152
x33=−58.1366632448992x_{33} = -58.1366632448992
x33=26.7409160147873x_{33} = 26.7409160147873
x33=102.111554139654x_{33} = 102.111554139654
x33=64.4181717218392x_{33} = 64.4181717218392
x33=−2.02875783811043x_{33} = -2.02875783811043
x33=−33.0170010333572x_{33} = -33.0170010333572
x33=89.5465575382492x_{33} = 89.5465575382492
x33=−20.469167402741x_{33} = -20.469167402741
x33=−26.7409160147873x_{33} = -26.7409160147873
x33=33.0170010333572x_{33} = 33.0170010333572
x33=20.469167402741x_{33} = 20.469167402741
x33=7.97866571241324x_{33} = 7.97866571241324
x33=−14.2074367251912x_{33} = -14.2074367251912
x33=39.295350981473x_{33} = 39.295350981473
x33=83.2642147040886x_{33} = 83.2642147040886
x33=76.9820093304187x_{33} = 76.9820093304187
x33=70.69997803861x_{33} = 70.69997803861
x33=−51.855560729152x_{33} = -51.855560729152
x33=−39.295350981473x_{33} = -39.295350981473
x33=58.1366632448992x_{33} = 58.1366632448992
x33=−45.57503179559x_{33} = -45.57503179559
x33=−89.5465575382492x_{33} = -89.5465575382492
x33=−70.69997803861x_{33} = -70.69997803861
x33=95.8290108090195x_{33} = 95.8290108090195
x33=−95.8290108090195x_{33} = -95.8290108090195
x33=2.02875783811043x_{33} = 2.02875783811043
x33=−83.2642147040886x_{33} = -83.2642147040886
x33=−76.9820093304187x_{33} = -76.9820093304187
x33=−7.97866571241324x_{33} = -7.97866571241324
x33=45.57503179559x_{33} = 45.57503179559
x33=14.2074367251912x_{33} = 14.2074367251912
Decreasing at intervals
[98.9702722883957,∞)\left[98.9702722883957, \infty\right)
Increasing at intervals
(−∞,−98.9702722883957]\left(-\infty, -98.9702722883957\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
−xsin⁡(x)+2cos⁡(x)=0- x \sin{\left(x \right)} + 2 \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=15.8336114149477x_{1} = 15.8336114149477
x2=100.550852725424x_{2} = 100.550852725424
x3=−9.62956034329743x_{3} = -9.62956034329743
x4=25.2119030642106x_{4} = 25.2119030642106
x5=−72.2842925036825x_{5} = -72.2842925036825
x6=66.0037377708277x_{6} = 66.0037377708277
x7=47.1662676027767x_{7} = 47.1662676027767
x8=−75.4247339745236x_{8} = -75.4247339745236
x9=−94.2689923093066x_{9} = -94.2689923093066
x10=−91.1281305511393x_{10} = -91.1281305511393
x11=−56.5839987378634x_{11} = -56.5839987378634
x12=−34.6152330552306x_{12} = -34.6152330552306
x13=−78.5652673845995x_{13} = -78.5652673845995
x14=34.6152330552306x_{14} = 34.6152330552306
x15=−69.1439554764926x_{15} = -69.1439554764926
x16=−97.4099011706723x_{16} = -97.4099011706723
x17=50.3052188363296x_{17} = 50.3052188363296
x18=72.2842925036825x_{18} = 72.2842925036825
x19=−12.7222987717666x_{19} = -12.7222987717666
x20=3.6435971674254x_{20} = 3.6435971674254
x21=−66.0037377708277x_{21} = -66.0037377708277
x22=−59.7237354324305x_{22} = -59.7237354324305
x23=−22.0814757672807x_{23} = -22.0814757672807
x24=59.7237354324305x_{24} = 59.7237354324305
x25=84.8465692433091x_{25} = 84.8465692433091
x26=87.9873209346887x_{26} = 87.9873209346887
x27=75.4247339745236x_{27} = 75.4247339745236
x28=−3.6435971674254x_{28} = -3.6435971674254
x29=−47.1662676027767x_{29} = -47.1662676027767
x30=−128.820822990274x_{30} = -128.820822990274
x31=28.3447768697864x_{31} = 28.3447768697864
x32=−15.8336114149477x_{32} = -15.8336114149477
x33=12.7222987717666x_{33} = 12.7222987717666
x34=−84.8465692433091x_{34} = -84.8465692433091
x35=−28.3447768697864x_{35} = -28.3447768697864
x36=−100.550852725424x_{36} = -100.550852725424
x37=−40.8895777660408x_{37} = -40.8895777660408
x38=−50.3052188363296x_{38} = -50.3052188363296
x39=−81.7058821480364x_{39} = -81.7058821480364
x40=62.863657228703x_{40} = 62.863657228703
x41=22.0814757672807x_{41} = 22.0814757672807
x42=91.1281305511393x_{42} = 91.1281305511393
x43=81.7058821480364x_{43} = 81.7058821480364
x44=31.479374920314x_{44} = 31.479374920314
x45=−31.479374920314x_{45} = -31.479374920314
x46=37.7520396346102x_{46} = 37.7520396346102
x47=−87.9873209346887x_{47} = -87.9873209346887
x48=−37.7520396346102x_{48} = -37.7520396346102
x49=56.5839987378634x_{49} = 56.5839987378634
x50=−1.0768739863118x_{50} = -1.0768739863118
x51=−6.57833373272234x_{51} = -6.57833373272234
x52=18.954681766529x_{52} = 18.954681766529
x53=−44.0276918992479x_{53} = -44.0276918992479
x54=78.5652673845995x_{54} = 78.5652673845995
x55=69.1439554764926x_{55} = 69.1439554764926
x56=9.62956034329743x_{56} = 9.62956034329743
x57=40.8895777660408x_{57} = 40.8895777660408
x58=44.0276918992479x_{58} = 44.0276918992479
x59=−18.954681766529x_{59} = -18.954681766529
x60=−25.2119030642106x_{60} = -25.2119030642106
x61=53.4444796697636x_{61} = 53.4444796697636
x62=−62.863657228703x_{62} = -62.863657228703
x63=6.57833373272234x_{63} = 6.57833373272234
x64=94.2689923093066x_{64} = 94.2689923093066
x65=97.4099011706723x_{65} = 97.4099011706723
x66=1.0768739863118x_{66} = 1.0768739863118
x67=−53.4444796697636x_{67} = -53.4444796697636

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
[97.4099011706723,∞)\left[97.4099011706723, \infty\right)
Convex at the intervals
(−∞,−100.550852725424]\left(-\infty, -100.550852725424\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xsin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(xsin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞sin⁡(x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
lim⁡x→∞sin⁡(x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
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:
xsin⁡(x)=xsin⁡(x)x \sin{\left(x \right)} = x \sin{\left(x \right)}
- Yes
xsin⁡(x)=−xsin⁡(x)x \sin{\left(x \right)} = - x \sin{\left(x \right)}
- No
so, the function
is
even
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