Mister Exam

Other calculators

Plotting a function graph/ x*cos(x)

Graphing y = x*cos(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = x*cos(x)
f(x)=xcos⁡(x)f{\left(x \right)} = x \cos{\left(x \right)} 
f = x*cos(x)
The graph of the function
0.01.02.03.04.05.06.07.08.09.010.0-2020
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:
xcos⁡(x)=0x \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Numerical solution
x1=48.6946861306418x_{1} = 48.6946861306418
x2=54.9778714378214x_{2} = 54.9778714378214
x3=−98.9601685880785x_{3} = -98.9601685880785
x4=67.5442420521806x_{4} = 67.5442420521806
x5=76.9690200129499x_{5} = 76.9690200129499
x6=36.1283155162826x_{6} = 36.1283155162826
x7=58.1194640914112x_{7} = 58.1194640914112
x8=14.1371669411541x_{8} = 14.1371669411541
x9=−29.845130209103x_{9} = -29.845130209103
x10=61.261056745001x_{10} = 61.261056745001
x11=−36.1283155162826x_{11} = -36.1283155162826
x12=−4.71238898038469x_{12} = -4.71238898038469
x13=−39.2699081698724x_{13} = -39.2699081698724
x14=1.5707963267949x_{14} = 1.5707963267949
x15=−14.1371669411541x_{15} = -14.1371669411541
x16=−64.4026493985908x_{16} = -64.4026493985908
x17=−67.5442420521806x_{17} = -67.5442420521806
x18=92.6769832808989x_{18} = 92.6769832808989
x19=−51.8362787842316x_{19} = -51.8362787842316
x20=−86.3937979737193x_{20} = -86.3937979737193
x21=42.4115008234622x_{21} = 42.4115008234622
x22=−17.2787595947439x_{22} = -17.2787595947439
x23=−45.553093477052x_{23} = -45.553093477052
x24=−89.5353906273091x_{24} = -89.5353906273091
x25=−1.5707963267949x_{25} = -1.5707963267949
x26=39.2699081698724x_{26} = 39.2699081698724
x27=23.5619449019235x_{27} = 23.5619449019235
x28=7.85398163397448x_{28} = 7.85398163397448
x29=−114.668131856027x_{29} = -114.668131856027
x30=−58.1194640914112x_{30} = -58.1194640914112
x31=−61.261056745001x_{31} = -61.261056745001
x32=−73.8274273593601x_{32} = -73.8274273593601
x33=73.8274273593601x_{33} = 73.8274273593601
x34=29.845130209103x_{34} = 29.845130209103
x35=4.71238898038469x_{35} = 4.71238898038469
x36=0x_{36} = 0
x37=86.3937979737193x_{37} = 86.3937979737193
x38=64.4026493985908x_{38} = 64.4026493985908
x39=89.5353906273091x_{39} = 89.5353906273091
x40=−20.4203522483337x_{40} = -20.4203522483337
x41=−26.7035375555132x_{41} = -26.7035375555132
x42=98.9601685880785x_{42} = 98.9601685880785
x43=51.8362787842316x_{43} = 51.8362787842316
x44=83.2522053201295x_{44} = 83.2522053201295
x45=−48.6946861306418x_{45} = -48.6946861306418
x46=−54.9778714378214x_{46} = -54.9778714378214
x47=70.6858347057703x_{47} = 70.6858347057703
x48=−95.8185759344887x_{48} = -95.8185759344887
x49=26.7035375555132x_{49} = 26.7035375555132
x50=80.1106126665397x_{50} = 80.1106126665397
x51=114.668131856027x_{51} = 114.668131856027
x52=−23.5619449019235x_{52} = -23.5619449019235
x53=−7.85398163397448x_{53} = -7.85398163397448
x54=−83.2522053201295x_{54} = -83.2522053201295
x55=−76.9690200129499x_{55} = -76.9690200129499
x56=−42.4115008234622x_{56} = -42.4115008234622
x57=−32.9867228626928x_{57} = -32.9867228626928
x58=17.2787595947439x_{58} = 17.2787595947439
x59=32.9867228626928x_{59} = 32.9867228626928
x60=20.4203522483337x_{60} = 20.4203522483337
x61=−70.6858347057703x_{61} = -70.6858347057703
x62=−10.9955742875643x_{62} = -10.9955742875643
x63=−92.6769832808989x_{63} = -92.6769832808989
x64=45.553093477052x_{64} = 45.553093477052
x65=10.9955742875643x_{65} = 10.9955742875643
x66=−80.1106126665397x_{66} = -80.1106126665397
x67=95.8185759344887x_{67} = 95.8185759344887
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*cos(x).
0cos⁡(0)0 \cos{\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
−xsin⁡(x)+cos⁡(x)=0- x \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=56.5663442798215x_{1} = 56.5663442798215
x2=9.52933440536196x_{2} = 9.52933440536196
x3=−81.6936492356017x_{3} = -81.6936492356017
x4=97.3996388790738x_{4} = 97.3996388790738
x5=−59.7070073053355x_{5} = -59.7070073053355
x6=−18.90240995686x_{6} = -18.90240995686
x7=−65.9885986984904x_{7} = -65.9885986984904
x8=59.7070073053355x_{8} = 59.7070073053355
x9=−147.661626855354x_{9} = -147.661626855354
x10=25.1724463266467x_{10} = 25.1724463266467
x11=78.5525459842429x_{11} = 78.5525459842429
x12=−78.5525459842429x_{12} = -78.5525459842429
x13=87.9759605524932x_{13} = 87.9759605524932
x14=−22.0364967279386x_{14} = -22.0364967279386
x15=65.9885986984904x_{15} = 65.9885986984904
x16=−3.42561845948173x_{16} = -3.42561845948173
x17=−87.9759605524932x_{17} = -87.9759605524932
x18=−47.145097736761x_{18} = -47.145097736761
x19=−84.8347887180423x_{19} = -84.8347887180423
x20=31.4477146375462x_{20} = 31.4477146375462
x21=−62.8477631944545x_{21} = -62.8477631944545
x22=−6.43729817917195x_{22} = -6.43729817917195
x23=−94.2583883450399x_{23} = -94.2583883450399
x24=6.43729817917195x_{24} = 6.43729817917195
x25=47.145097736761x_{25} = 47.145097736761
x26=62.8477631944545x_{26} = 62.8477631944545
x27=40.8651703304881x_{27} = 40.8651703304881
x28=75.4114834888481x_{28} = 75.4114834888481
x29=−28.309642854452x_{29} = -28.309642854452
x30=100.540910786842x_{30} = 100.540910786842
x31=−50.2853663377737x_{31} = -50.2853663377737
x32=−116.247530303932x_{32} = -116.247530303932
x33=44.0050179208308x_{33} = 44.0050179208308
x34=50.2853663377737x_{34} = 50.2853663377737
x35=81.6936492356017x_{35} = 81.6936492356017
x36=53.4257904773947x_{36} = 53.4257904773947
x37=91.1171613944647x_{37} = 91.1171613944647
x38=22.0364967279386x_{38} = 22.0364967279386
x39=−56.5663442798215x_{39} = -56.5663442798215
x40=12.6452872238566x_{40} = 12.6452872238566
x41=−9.52933440536196x_{41} = -9.52933440536196
x42=15.7712848748159x_{42} = 15.7712848748159
x43=−91.1171613944647x_{43} = -91.1171613944647
x44=0.86033358901938x_{44} = 0.86033358901938
x45=−69.1295029738953x_{45} = -69.1295029738953
x46=3.42561845948173x_{46} = 3.42561845948173
x47=−0.86033358901938x_{47} = -0.86033358901938
x48=−31.4477146375462x_{48} = -31.4477146375462
x49=94.2583883450399x_{49} = 94.2583883450399
x50=−72.270467060309x_{50} = -72.270467060309
x51=37.7256128277765x_{51} = 37.7256128277765
x52=28.309642854452x_{52} = 28.309642854452
x53=−44.0050179208308x_{53} = -44.0050179208308
x54=−34.5864242152889x_{54} = -34.5864242152889
x55=−75.4114834888481x_{55} = -75.4114834888481
x56=−25.1724463266467x_{56} = -25.1724463266467
x57=18.90240995686x_{57} = 18.90240995686
x58=−53.4257904773947x_{58} = -53.4257904773947
x59=72.270467060309x_{59} = 72.270467060309
x60=34.5864242152889x_{60} = 34.5864242152889
x61=−100.540910786842x_{61} = -100.540910786842
x62=−37.7256128277765x_{62} = -37.7256128277765
x63=84.8347887180423x_{63} = 84.8347887180423
x64=−40.8651703304881x_{64} = -40.8651703304881
x65=−12.6452872238566x_{65} = -12.6452872238566
x66=−15.7712848748159x_{66} = -15.7712848748159
x67=−97.3996388790738x_{67} = -97.3996388790738
x68=69.1295029738953x_{68} = 69.1295029738953
The values of the extrema at the points:
(56.56634427982152, 56.5575071728762)

(9.529334405361963, -9.47729425947979)

(-81.69364923560168, -81.6875294965246)

(97.39963887907376, -97.3945057956234)

(-59.70700730533546, 59.6986348402658)

(-18.902409956860023, -18.876013697969)

(-65.98859869849039, 65.9810229367917)

(59.70700730533546, -59.6986348402658)

(-147.66162685535437, 147.658240851742)

(25.172446326646664, 25.1526068178715)

(78.55254598424293, -78.5461815917343)

(-78.55254598424293, 78.5461815917343)

(87.97596055249322, 87.9702777324248)

(-22.036496727938566, 22.0138420791585)

(65.98859869849039, -65.9810229367917)

(-3.4256184594817283, 3.2883713955909)

(-87.97596055249322, -87.9702777324248)

(-47.14509773676103, 47.1344957575419)

(-84.83478871804229, 84.8288955236568)

(31.447714637546234, 31.4318272785346)

(-62.84776319445445, -62.8398089721545)

(-6.437298179171947, -6.36100394483385)

(-94.25838834503986, -94.2530842251087)

(6.437298179171947, 6.36100394483385)

(47.14509773676103, -47.1344957575419)

(62.84776319445445, 62.8398089721545)

(40.86517033048807, -40.8529404645174)

(75.41148348884815, 75.4048540732019)

(-28.30964285445201, 28.2919975390943)

(100.54091078684232, 100.535938055826)

(-50.28536633777365, -50.2754260353972)

(-116.2475303039321, 116.243229375987)

(44.005017920830845, 43.9936599791065)

(50.28536633777365, 50.2754260353972)

(81.69364923560168, 81.6875294965246)

(53.42579047739466, -53.4164341598961)

(91.11716139446474, -91.1116744496469)

(22.036496727938566, -22.0138420791585)

(-56.56634427982152, -56.5575071728762)

(12.645287223856643, 12.6059312978927)

(-9.529334405361963, 9.47729425947979)

(15.771284874815882, -15.7396769621337)

(-91.11716139446474, 91.1116744496469)

(0.8603335890193797, 0.561096338191045)

(-69.12950297389526, -69.1222713069218)

(3.4256184594817283, -3.2883713955909)

(-0.8603335890193797, -0.561096338191045)

(-31.447714637546234, -31.4318272785346)

(94.25838834503986, 94.2530842251087)

(-72.27046706030896, 72.2635495982494)

(37.7256128277765, 37.71236621281)

(28.30964285445201, -28.2919975390943)

(-44.005017920830845, -43.9936599791065)

(-34.58642421528892, 34.5719767335884)

(-75.41148348884815, -75.4048540732019)

(-25.172446326646664, -25.1526068178715)

(18.902409956860023, 18.876013697969)

(-53.42579047739466, 53.4164341598961)

(72.27046706030896, -72.2635495982494)

(34.58642421528892, -34.5719767335884)

(-100.54091078684232, -100.535938055826)

(-37.7256128277765, -37.71236621281)

(84.83478871804229, -84.8288955236568)

(-40.86517033048807, 40.8529404645174)

(-12.645287223856643, -12.6059312978927)

(-15.771284874815882, 15.7396769621337)

(-97.39963887907376, 97.3945057956234)

(69.12950297389526, 69.1222713069218)


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=9.52933440536196x_{1} = 9.52933440536196
x2=−81.6936492356017x_{2} = -81.6936492356017
x3=97.3996388790738x_{3} = 97.3996388790738
x4=−18.90240995686x_{4} = -18.90240995686
x5=59.7070073053355x_{5} = 59.7070073053355
x6=78.5525459842429x_{6} = 78.5525459842429
x7=65.9885986984904x_{7} = 65.9885986984904
x8=−87.9759605524932x_{8} = -87.9759605524932
x9=−62.8477631944545x_{9} = -62.8477631944545
x10=−6.43729817917195x_{10} = -6.43729817917195
x11=−94.2583883450399x_{11} = -94.2583883450399
x12=47.145097736761x_{12} = 47.145097736761
x13=40.8651703304881x_{13} = 40.8651703304881
x14=−50.2853663377737x_{14} = -50.2853663377737
x15=53.4257904773947x_{15} = 53.4257904773947
x16=91.1171613944647x_{16} = 91.1171613944647
x17=22.0364967279386x_{17} = 22.0364967279386
x18=−56.5663442798215x_{18} = -56.5663442798215
x19=15.7712848748159x_{19} = 15.7712848748159
x20=−69.1295029738953x_{20} = -69.1295029738953
x21=3.42561845948173x_{21} = 3.42561845948173
x22=−0.86033358901938x_{22} = -0.86033358901938
x23=−31.4477146375462x_{23} = -31.4477146375462
x24=28.309642854452x_{24} = 28.309642854452
x25=−44.0050179208308x_{25} = -44.0050179208308
x26=−75.4114834888481x_{26} = -75.4114834888481
x27=−25.1724463266467x_{27} = -25.1724463266467
x28=72.270467060309x_{28} = 72.270467060309
x29=34.5864242152889x_{29} = 34.5864242152889
x30=−100.540910786842x_{30} = -100.540910786842
x31=−37.7256128277765x_{31} = -37.7256128277765
x32=84.8347887180423x_{32} = 84.8347887180423
x33=−12.6452872238566x_{33} = -12.6452872238566
Maxima of the function at points:
x33=56.5663442798215x_{33} = 56.5663442798215
x33=−59.7070073053355x_{33} = -59.7070073053355
x33=−65.9885986984904x_{33} = -65.9885986984904
x33=−147.661626855354x_{33} = -147.661626855354
x33=25.1724463266467x_{33} = 25.1724463266467
x33=−78.5525459842429x_{33} = -78.5525459842429
x33=87.9759605524932x_{33} = 87.9759605524932
x33=−22.0364967279386x_{33} = -22.0364967279386
x33=−3.42561845948173x_{33} = -3.42561845948173
x33=−47.145097736761x_{33} = -47.145097736761
x33=−84.8347887180423x_{33} = -84.8347887180423
x33=31.4477146375462x_{33} = 31.4477146375462
x33=6.43729817917195x_{33} = 6.43729817917195
x33=62.8477631944545x_{33} = 62.8477631944545
x33=75.4114834888481x_{33} = 75.4114834888481
x33=−28.309642854452x_{33} = -28.309642854452
x33=100.540910786842x_{33} = 100.540910786842
x33=−116.247530303932x_{33} = -116.247530303932
x33=44.0050179208308x_{33} = 44.0050179208308
x33=50.2853663377737x_{33} = 50.2853663377737
x33=81.6936492356017x_{33} = 81.6936492356017
x33=12.6452872238566x_{33} = 12.6452872238566
x33=−9.52933440536196x_{33} = -9.52933440536196
x33=−91.1171613944647x_{33} = -91.1171613944647
x33=0.86033358901938x_{33} = 0.86033358901938
x33=94.2583883450399x_{33} = 94.2583883450399
x33=−72.270467060309x_{33} = -72.270467060309
x33=37.7256128277765x_{33} = 37.7256128277765
x33=−34.5864242152889x_{33} = -34.5864242152889
x33=18.90240995686x_{33} = 18.90240995686
x33=−53.4257904773947x_{33} = -53.4257904773947
x33=−40.8651703304881x_{33} = -40.8651703304881
x33=−15.7712848748159x_{33} = -15.7712848748159
x33=−97.3996388790738x_{33} = -97.3996388790738
x33=69.1295029738953x_{33} = 69.1295029738953
Decreasing at intervals
[97.3996388790738,∞)\left[97.3996388790738, \infty\right)
Increasing at intervals
(−∞,−100.540910786842]\left(-\infty, -100.540910786842\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
−(xcos⁡(x)+2sin⁡(x))=0- (x \cos{\left(x \right)} + 2 \sin{\left(x \right)}) = 0
Solve this equation
The roots of this equation
x1=95.839441141233x_{1} = 95.839441141233
x2=−76.9949898891676x_{2} = -76.9949898891676
x3=−89.5577188827244x_{3} = -89.5577188827244
x4=14.2763529183365x_{4} = 14.2763529183365
x5=−23.6463238196036x_{5} = -23.6463238196036
x6=33.0471686947054x_{6} = 33.0471686947054
x7=−67.573830670859x_{7} = -67.573830670859
x8=−86.4169374541167x_{8} = -86.4169374541167
x9=−29.9118938695518x_{9} = -29.9118938695518
x10=−39.3207281322521x_{10} = -39.3207281322521
x11=26.7780870755585x_{11} = 26.7780870755585
x12=86.4169374541167x_{12} = 86.4169374541167
x13=−33.0471686947054x_{13} = -33.0471686947054
x14=39.3207281322521x_{14} = 39.3207281322521
x15=−58.153842078645x_{15} = -58.153842078645
x16=55.0142096788381x_{16} = 55.0142096788381
x17=−55.0142096788381x_{17} = -55.0142096788381
x18=−45.5969279840735x_{18} = -45.5969279840735
x19=−20.5175229099417x_{19} = -20.5175229099417
x20=76.9949898891676x_{20} = 76.9949898891676
x21=29.9118938695518x_{21} = 29.9118938695518
x22=2.2889297281034x_{22} = 2.2889297281034
x23=−48.7357007949054x_{23} = -48.7357007949054
x24=−95.839441141233x_{24} = -95.839441141233
x25=83.2762171649775x_{25} = 83.2762171649775
x26=−2.2889297281034x_{26} = -2.2889297281034
x27=−42.458570771699x_{27} = -42.458570771699
x28=8.09616360322292x_{28} = 8.09616360322292
x29=−83.2762171649775x_{29} = -83.2762171649775
x30=51.8748140534268x_{30} = 51.8748140534268
x31=48.7357007949054x_{31} = 48.7357007949054
x32=20.5175229099417x_{32} = 20.5175229099417
x33=58.153842078645x_{33} = 58.153842078645
x34=61.2936749662429x_{34} = 61.2936749662429
x35=−80.1355651940744x_{35} = -80.1355651940744
x36=0x_{36} = 0
x37=98.9803718651523x_{37} = 98.9803718651523
x38=23.6463238196036x_{38} = 23.6463238196036
x39=−92.6985552433969x_{39} = -92.6985552433969
x40=−14.2763529183365x_{40} = -14.2763529183365
x41=−17.3932439645948x_{41} = -17.3932439645948
x42=70.7141100665485x_{42} = 70.7141100665485
x43=−64.4336791037316x_{43} = -64.4336791037316
x44=36.1835330907526x_{44} = 36.1835330907526
x45=11.17270586833x_{45} = 11.17270586833
x46=−61.2936749662429x_{46} = -61.2936749662429
x47=80.1355651940744x_{47} = 80.1355651940744
x48=−11.17270586833x_{48} = -11.17270586833
x49=45.5969279840735x_{49} = 45.5969279840735
x50=−98.9803718651523x_{50} = -98.9803718651523
x51=−26.7780870755585x_{51} = -26.7780870755585
x52=73.8545010149048x_{52} = 73.8545010149048
x53=17.3932439645948x_{53} = 17.3932439645948
x54=42.458570771699x_{54} = 42.458570771699
x55=5.08698509410227x_{55} = 5.08698509410227
x56=−51.8748140534268x_{56} = -51.8748140534268
x57=−8.09616360322292x_{57} = -8.09616360322292
x58=−5.08698509410227x_{58} = -5.08698509410227
x59=−73.8545010149048x_{59} = -73.8545010149048
x60=89.5577188827244x_{60} = 89.5577188827244
x61=64.4336791037316x_{61} = 64.4336791037316
x62=67.573830670859x_{62} = 67.573830670859
x63=−70.7141100665485x_{63} = -70.7141100665485
x64=−36.1835330907526x_{64} = -36.1835330907526
x65=92.6985552433969x_{65} = 92.6985552433969

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
[95.839441141233,∞)\left[95.839441141233, \infty\right)
Convex at the intervals
(−∞,−95.839441141233]\left(-\infty, -95.839441141233\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xcos⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \cos{\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→∞(xcos⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \cos{\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*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞cos⁡(x)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\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→∞cos⁡(x)=⟨−1,1⟩\lim_{x \to \infty} \cos{\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:
xcos⁡(x)=−xcos⁡(x)x \cos{\left(x \right)} = - x \cos{\left(x \right)}
- No
xcos⁡(x)=xcos⁡(x)x \cos{\left(x \right)} = x \cos{\left(x \right)}
- Yes
so, the function
is
odd
    © Mister Exam – Calculator