Mister Exam

Graphing y = z*cosz

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(z) = z*cos(z)
f(z)=zcos(z)f{\left(z \right)} = z \cos{\left(z \right)}
f = z*cos(z)
The graph of the function
02468-8-6-4-2-1010-2020
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis Z at f = 0
so we need to solve the equation:
zcos(z)=0z \cos{\left(z \right)} = 0
Solve this equation
The points of intersection with the axis Z:

Analytical solution
z1=0z_{1} = 0
z2=π2z_{2} = - \frac{\pi}{2}
z3=π2z_{3} = \frac{\pi}{2}
Numerical solution
z1=7.85398163397448z_{1} = 7.85398163397448
z2=73.8274273593601z_{2} = -73.8274273593601
z3=54.9778714378214z_{3} = -54.9778714378214
z4=73.8274273593601z_{4} = 73.8274273593601
z5=0z_{5} = 0
z6=26.7035375555132z_{6} = -26.7035375555132
z7=1.5707963267949z_{7} = -1.5707963267949
z8=95.8185759344887z_{8} = -95.8185759344887
z9=39.2699081698724z_{9} = -39.2699081698724
z10=4.71238898038469z_{10} = -4.71238898038469
z11=14.1371669411541z_{11} = 14.1371669411541
z12=10.9955742875643z_{12} = 10.9955742875643
z13=58.1194640914112z_{13} = 58.1194640914112
z14=70.6858347057703z_{14} = 70.6858347057703
z15=36.1283155162826z_{15} = -36.1283155162826
z16=54.9778714378214z_{16} = 54.9778714378214
z17=23.5619449019235z_{17} = 23.5619449019235
z18=92.6769832808989z_{18} = -92.6769832808989
z19=86.3937979737193z_{19} = -86.3937979737193
z20=10.9955742875643z_{20} = -10.9955742875643
z21=92.6769832808989z_{21} = 92.6769832808989
z22=39.2699081698724z_{22} = 39.2699081698724
z23=32.9867228626928z_{23} = -32.9867228626928
z24=98.9601685880785z_{24} = 98.9601685880785
z25=36.1283155162826z_{25} = 36.1283155162826
z26=7.85398163397448z_{26} = -7.85398163397448
z27=58.1194640914112z_{27} = -58.1194640914112
z28=67.5442420521806z_{28} = -67.5442420521806
z29=61.261056745001z_{29} = -61.261056745001
z30=26.7035375555132z_{30} = 26.7035375555132
z31=86.3937979737193z_{31} = 86.3937979737193
z32=48.6946861306418z_{32} = -48.6946861306418
z33=51.8362787842316z_{33} = 51.8362787842316
z34=42.4115008234622z_{34} = -42.4115008234622
z35=89.5353906273091z_{35} = -89.5353906273091
z36=98.9601685880785z_{36} = -98.9601685880785
z37=14.1371669411541z_{37} = -14.1371669411541
z38=80.1106126665397z_{38} = 80.1106126665397
z39=64.4026493985908z_{39} = -64.4026493985908
z40=95.8185759344887z_{40} = 95.8185759344887
z41=114.668131856027z_{41} = -114.668131856027
z42=1.5707963267949z_{42} = 1.5707963267949
z43=45.553093477052z_{43} = 45.553093477052
z44=17.2787595947439z_{44} = -17.2787595947439
z45=4.71238898038469z_{45} = 4.71238898038469
z46=48.6946861306418z_{46} = 48.6946861306418
z47=76.9690200129499z_{47} = 76.9690200129499
z48=45.553093477052z_{48} = -45.553093477052
z49=20.4203522483337z_{49} = 20.4203522483337
z50=17.2787595947439z_{50} = 17.2787595947439
z51=83.2522053201295z_{51} = -83.2522053201295
z52=20.4203522483337z_{52} = -20.4203522483337
z53=80.1106126665397z_{53} = -80.1106126665397
z54=61.261056745001z_{54} = 61.261056745001
z55=32.9867228626928z_{55} = 32.9867228626928
z56=64.4026493985908z_{56} = 64.4026493985908
z57=23.5619449019235z_{57} = -23.5619449019235
z58=29.845130209103z_{58} = 29.845130209103
z59=42.4115008234622z_{59} = 42.4115008234622
z60=89.5353906273091z_{60} = 89.5353906273091
z61=51.8362787842316z_{61} = -51.8362787842316
z62=70.6858347057703z_{62} = -70.6858347057703
z63=83.2522053201295z_{63} = 83.2522053201295
z64=67.5442420521806z_{64} = 67.5442420521806
z65=29.845130209103z_{65} = -29.845130209103
z66=76.9690200129499z_{66} = -76.9690200129499
z67=114.668131856027z_{67} = 114.668131856027
The points of intersection with the Y axis coordinate
The graph crosses Y axis when z equals 0:
substitute z = 0 to z*cos(z).
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
ddzf(z)=0\frac{d}{d z} f{\left(z \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddzf(z)=\frac{d}{d z} f{\left(z \right)} =
the first derivative
zsin(z)+cos(z)=0- z \sin{\left(z \right)} + \cos{\left(z \right)} = 0
Solve this equation
The roots of this equation
z1=12.6452872238566z_{1} = -12.6452872238566
z2=72.270467060309z_{2} = -72.270467060309
z3=72.270467060309z_{3} = 72.270467060309
z4=84.8347887180423z_{4} = -84.8347887180423
z5=9.52933440536196z_{5} = 9.52933440536196
z6=78.5525459842429z_{6} = 78.5525459842429
z7=6.43729817917195z_{7} = -6.43729817917195
z8=37.7256128277765z_{8} = 37.7256128277765
z9=18.90240995686z_{9} = -18.90240995686
z10=56.5663442798215z_{10} = -56.5663442798215
z11=37.7256128277765z_{11} = -37.7256128277765
z12=22.0364967279386z_{12} = 22.0364967279386
z13=69.1295029738953z_{13} = -69.1295029738953
z14=28.309642854452z_{14} = -28.309642854452
z15=47.145097736761z_{15} = 47.145097736761
z16=97.3996388790738z_{16} = -97.3996388790738
z17=25.1724463266467z_{17} = -25.1724463266467
z18=59.7070073053355z_{18} = -59.7070073053355
z19=9.52933440536196z_{19} = -9.52933440536196
z20=6.43729817917195z_{20} = 6.43729817917195
z21=53.4257904773947z_{21} = -53.4257904773947
z22=28.309642854452z_{22} = 28.309642854452
z23=91.1171613944647z_{23} = 91.1171613944647
z24=25.1724463266467z_{24} = 25.1724463266467
z25=15.7712848748159z_{25} = -15.7712848748159
z26=100.540910786842z_{26} = -100.540910786842
z27=44.0050179208308z_{27} = 44.0050179208308
z28=65.9885986984904z_{28} = -65.9885986984904
z29=81.6936492356017z_{29} = 81.6936492356017
z30=116.247530303932z_{30} = -116.247530303932
z31=91.1171613944647z_{31} = -91.1171613944647
z32=47.145097736761z_{32} = -47.145097736761
z33=147.661626855354z_{33} = -147.661626855354
z34=3.42561845948173z_{34} = -3.42561845948173
z35=50.2853663377737z_{35} = -50.2853663377737
z36=94.2583883450399z_{36} = -94.2583883450399
z37=97.3996388790738z_{37} = 97.3996388790738
z38=87.9759605524932z_{38} = -87.9759605524932
z39=94.2583883450399z_{39} = 94.2583883450399
z40=75.4114834888481z_{40} = 75.4114834888481
z41=15.7712848748159z_{41} = 15.7712848748159
z42=53.4257904773947z_{42} = 53.4257904773947
z43=69.1295029738953z_{43} = 69.1295029738953
z44=87.9759605524932z_{44} = 87.9759605524932
z45=75.4114834888481z_{45} = -75.4114834888481
z46=81.6936492356017z_{46} = -81.6936492356017
z47=3.42561845948173z_{47} = 3.42561845948173
z48=62.8477631944545z_{48} = 62.8477631944545
z49=40.8651703304881z_{49} = -40.8651703304881
z50=59.7070073053355z_{50} = 59.7070073053355
z51=78.5525459842429z_{51} = -78.5525459842429
z52=40.8651703304881z_{52} = 40.8651703304881
z53=12.6452872238566z_{53} = 12.6452872238566
z54=34.5864242152889z_{54} = -34.5864242152889
z55=31.4477146375462z_{55} = -31.4477146375462
z56=34.5864242152889z_{56} = 34.5864242152889
z57=22.0364967279386z_{57} = -22.0364967279386
z58=31.4477146375462z_{58} = 31.4477146375462
z59=50.2853663377737z_{59} = 50.2853663377737
z60=56.5663442798215z_{60} = 56.5663442798215
z61=44.0050179208308z_{61} = -44.0050179208308
z62=100.540910786842z_{62} = 100.540910786842
z63=65.9885986984904z_{63} = 65.9885986984904
z64=62.8477631944545z_{64} = -62.8477631944545
z65=0.86033358901938z_{65} = 0.86033358901938
z66=18.90240995686z_{66} = 18.90240995686
z67=84.8347887180423z_{67} = 84.8347887180423
z68=0.86033358901938z_{68} = -0.86033358901938
The values of the extrema at the points:
(-12.645287223856643, -12.6059312978927)

(-72.27046706030896, 72.2635495982494)

(72.27046706030896, -72.2635495982494)

(-84.83478871804229, 84.8288955236568)

(9.529334405361963, -9.47729425947979)

(78.55254598424293, -78.5461815917343)

(-6.437298179171947, -6.36100394483385)

(37.7256128277765, 37.71236621281)

(-18.902409956860023, -18.876013697969)

(-56.56634427982152, -56.5575071728762)

(-37.7256128277765, -37.71236621281)

(22.036496727938566, -22.0138420791585)

(-69.12950297389526, -69.1222713069218)

(-28.30964285445201, 28.2919975390943)

(47.14509773676103, -47.1344957575419)

(-97.39963887907376, 97.3945057956234)

(-25.172446326646664, -25.1526068178715)

(-59.70700730533546, 59.6986348402658)

(-9.529334405361963, 9.47729425947979)

(6.437298179171947, 6.36100394483385)

(-53.42579047739466, 53.4164341598961)

(28.30964285445201, -28.2919975390943)

(91.11716139446474, -91.1116744496469)

(25.172446326646664, 25.1526068178715)

(-15.771284874815882, 15.7396769621337)

(-100.54091078684232, -100.535938055826)

(44.005017920830845, 43.9936599791065)

(-65.98859869849039, 65.9810229367917)

(81.69364923560168, 81.6875294965246)

(-116.2475303039321, 116.243229375987)

(-91.11716139446474, 91.1116744496469)

(-47.14509773676103, 47.1344957575419)

(-147.66162685535437, 147.658240851742)

(-3.4256184594817283, 3.2883713955909)

(-50.28536633777365, -50.2754260353972)

(-94.25838834503986, -94.2530842251087)

(97.39963887907376, -97.3945057956234)

(-87.97596055249322, -87.9702777324248)

(94.25838834503986, 94.2530842251087)

(75.41148348884815, 75.4048540732019)

(15.771284874815882, -15.7396769621337)

(53.42579047739466, -53.4164341598961)

(69.12950297389526, 69.1222713069218)

(87.97596055249322, 87.9702777324248)

(-75.41148348884815, -75.4048540732019)

(-81.69364923560168, -81.6875294965246)

(3.4256184594817283, -3.2883713955909)

(62.84776319445445, 62.8398089721545)

(-40.86517033048807, 40.8529404645174)

(59.70700730533546, -59.6986348402658)

(-78.55254598424293, 78.5461815917343)

(40.86517033048807, -40.8529404645174)

(12.645287223856643, 12.6059312978927)

(-34.58642421528892, 34.5719767335884)

(-31.447714637546234, -31.4318272785346)

(34.58642421528892, -34.5719767335884)

(-22.036496727938566, 22.0138420791585)

(31.447714637546234, 31.4318272785346)

(50.28536633777365, 50.2754260353972)

(56.56634427982152, 56.5575071728762)

(-44.005017920830845, -43.9936599791065)

(100.54091078684232, 100.535938055826)

(65.98859869849039, -65.9810229367917)

(-62.84776319445445, -62.8398089721545)

(0.8603335890193797, 0.561096338191045)

(18.902409956860023, 18.876013697969)

(84.83478871804229, -84.8288955236568)

(-0.8603335890193797, -0.561096338191045)


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:
z1=12.6452872238566z_{1} = -12.6452872238566
z2=72.270467060309z_{2} = 72.270467060309
z3=9.52933440536196z_{3} = 9.52933440536196
z4=78.5525459842429z_{4} = 78.5525459842429
z5=6.43729817917195z_{5} = -6.43729817917195
z6=18.90240995686z_{6} = -18.90240995686
z7=56.5663442798215z_{7} = -56.5663442798215
z8=37.7256128277765z_{8} = -37.7256128277765
z9=22.0364967279386z_{9} = 22.0364967279386
z10=69.1295029738953z_{10} = -69.1295029738953
z11=47.145097736761z_{11} = 47.145097736761
z12=25.1724463266467z_{12} = -25.1724463266467
z13=28.309642854452z_{13} = 28.309642854452
z14=91.1171613944647z_{14} = 91.1171613944647
z15=100.540910786842z_{15} = -100.540910786842
z16=50.2853663377737z_{16} = -50.2853663377737
z17=94.2583883450399z_{17} = -94.2583883450399
z18=97.3996388790738z_{18} = 97.3996388790738
z19=87.9759605524932z_{19} = -87.9759605524932
z20=15.7712848748159z_{20} = 15.7712848748159
z21=53.4257904773947z_{21} = 53.4257904773947
z22=75.4114834888481z_{22} = -75.4114834888481
z23=81.6936492356017z_{23} = -81.6936492356017
z24=3.42561845948173z_{24} = 3.42561845948173
z25=59.7070073053355z_{25} = 59.7070073053355
z26=40.8651703304881z_{26} = 40.8651703304881
z27=31.4477146375462z_{27} = -31.4477146375462
z28=34.5864242152889z_{28} = 34.5864242152889
z29=44.0050179208308z_{29} = -44.0050179208308
z30=65.9885986984904z_{30} = 65.9885986984904
z31=62.8477631944545z_{31} = -62.8477631944545
z32=84.8347887180423z_{32} = 84.8347887180423
z33=0.86033358901938z_{33} = -0.86033358901938
Maxima of the function at points:
z33=72.270467060309z_{33} = -72.270467060309
z33=84.8347887180423z_{33} = -84.8347887180423
z33=37.7256128277765z_{33} = 37.7256128277765
z33=28.309642854452z_{33} = -28.309642854452
z33=97.3996388790738z_{33} = -97.3996388790738
z33=59.7070073053355z_{33} = -59.7070073053355
z33=9.52933440536196z_{33} = -9.52933440536196
z33=6.43729817917195z_{33} = 6.43729817917195
z33=53.4257904773947z_{33} = -53.4257904773947
z33=25.1724463266467z_{33} = 25.1724463266467
z33=15.7712848748159z_{33} = -15.7712848748159
z33=44.0050179208308z_{33} = 44.0050179208308
z33=65.9885986984904z_{33} = -65.9885986984904
z33=81.6936492356017z_{33} = 81.6936492356017
z33=116.247530303932z_{33} = -116.247530303932
z33=91.1171613944647z_{33} = -91.1171613944647
z33=47.145097736761z_{33} = -47.145097736761
z33=147.661626855354z_{33} = -147.661626855354
z33=3.42561845948173z_{33} = -3.42561845948173
z33=94.2583883450399z_{33} = 94.2583883450399
z33=75.4114834888481z_{33} = 75.4114834888481
z33=69.1295029738953z_{33} = 69.1295029738953
z33=87.9759605524932z_{33} = 87.9759605524932
z33=62.8477631944545z_{33} = 62.8477631944545
z33=40.8651703304881z_{33} = -40.8651703304881
z33=78.5525459842429z_{33} = -78.5525459842429
z33=12.6452872238566z_{33} = 12.6452872238566
z33=34.5864242152889z_{33} = -34.5864242152889
z33=22.0364967279386z_{33} = -22.0364967279386
z33=31.4477146375462z_{33} = 31.4477146375462
z33=50.2853663377737z_{33} = 50.2853663377737
z33=56.5663442798215z_{33} = 56.5663442798215
z33=100.540910786842z_{33} = 100.540910786842
z33=0.86033358901938z_{33} = 0.86033358901938
z33=18.90240995686z_{33} = 18.90240995686
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
d2dz2f(z)=0\frac{d^{2}}{d z^{2}} f{\left(z \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dz2f(z)=\frac{d^{2}}{d z^{2}} f{\left(z \right)} =
the second derivative
(zcos(z)+2sin(z))=0- (z \cos{\left(z \right)} + 2 \sin{\left(z \right)}) = 0
Solve this equation
The roots of this equation
z1=29.9118938695518z_{1} = 29.9118938695518
z2=48.7357007949054z_{2} = 48.7357007949054
z3=83.2762171649775z_{3} = 83.2762171649775
z4=95.839441141233z_{4} = 95.839441141233
z5=86.4169374541167z_{5} = 86.4169374541167
z6=51.8748140534268z_{6} = -51.8748140534268
z7=36.1835330907526z_{7} = 36.1835330907526
z8=0z_{8} = 0
z9=51.8748140534268z_{9} = 51.8748140534268
z10=67.573830670859z_{10} = 67.573830670859
z11=33.0471686947054z_{11} = 33.0471686947054
z12=8.09616360322292z_{12} = 8.09616360322292
z13=26.7780870755585z_{13} = -26.7780870755585
z14=5.08698509410227z_{14} = 5.08698509410227
z15=55.0142096788381z_{15} = -55.0142096788381
z16=92.6985552433969z_{16} = -92.6985552433969
z17=61.2936749662429z_{17} = 61.2936749662429
z18=11.17270586833z_{18} = 11.17270586833
z19=20.5175229099417z_{19} = 20.5175229099417
z20=36.1835330907526z_{20} = -36.1835330907526
z21=23.6463238196036z_{21} = -23.6463238196036
z22=58.153842078645z_{22} = -58.153842078645
z23=20.5175229099417z_{23} = -20.5175229099417
z24=70.7141100665485z_{24} = 70.7141100665485
z25=45.5969279840735z_{25} = 45.5969279840735
z26=14.2763529183365z_{26} = 14.2763529183365
z27=42.458570771699z_{27} = 42.458570771699
z28=5.08698509410227z_{28} = -5.08698509410227
z29=29.9118938695518z_{29} = -29.9118938695518
z30=98.9803718651523z_{30} = 98.9803718651523
z31=42.458570771699z_{31} = -42.458570771699
z32=80.1355651940744z_{32} = -80.1355651940744
z33=89.5577188827244z_{33} = -89.5577188827244
z34=11.17270586833z_{34} = -11.17270586833
z35=2.2889297281034z_{35} = 2.2889297281034
z36=48.7357007949054z_{36} = -48.7357007949054
z37=17.3932439645948z_{37} = -17.3932439645948
z38=92.6985552433969z_{38} = 92.6985552433969
z39=39.3207281322521z_{39} = 39.3207281322521
z40=39.3207281322521z_{40} = -39.3207281322521
z41=83.2762171649775z_{41} = -83.2762171649775
z42=73.8545010149048z_{42} = 73.8545010149048
z43=58.153842078645z_{43} = 58.153842078645
z44=8.09616360322292z_{44} = -8.09616360322292
z45=76.9949898891676z_{45} = -76.9949898891676
z46=64.4336791037316z_{46} = 64.4336791037316
z47=64.4336791037316z_{47} = -64.4336791037316
z48=89.5577188827244z_{48} = 89.5577188827244
z49=55.0142096788381z_{49} = 55.0142096788381
z50=33.0471686947054z_{50} = -33.0471686947054
z51=67.573830670859z_{51} = -67.573830670859
z52=80.1355651940744z_{52} = 80.1355651940744
z53=76.9949898891676z_{53} = 76.9949898891676
z54=70.7141100665485z_{54} = -70.7141100665485
z55=61.2936749662429z_{55} = -61.2936749662429
z56=17.3932439645948z_{56} = 17.3932439645948
z57=26.7780870755585z_{57} = 26.7780870755585
z58=14.2763529183365z_{58} = -14.2763529183365
z59=98.9803718651523z_{59} = -98.9803718651523
z60=23.6463238196036z_{60} = 23.6463238196036
z61=86.4169374541167z_{61} = -86.4169374541167
z62=73.8545010149048z_{62} = -73.8545010149048
z63=45.5969279840735z_{63} = -45.5969279840735
z64=2.2889297281034z_{64} = -2.2889297281034
z65=95.839441141233z_{65} = -95.839441141233

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 z->+oo and z->-oo
limz(zcos(z))=,\lim_{z \to -\infty}\left(z \cos{\left(z \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
limz(zcos(z))=,\lim_{z \to \infty}\left(z \cos{\left(z \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 z*cos(z), divided by z at z->+oo and z ->-oo
limzcos(z)=1,1\lim_{z \to -\infty} \cos{\left(z \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=1,1zy = \left\langle -1, 1\right\rangle z
limzcos(z)=1,1\lim_{z \to \infty} \cos{\left(z \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=1,1zy = \left\langle -1, 1\right\rangle z
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-z) и f = -f(-z).
So, check:
zcos(z)=zcos(z)z \cos{\left(z \right)} = - z \cos{\left(z \right)}
- No
zcos(z)=zcos(z)z \cos{\left(z \right)} = z \cos{\left(z \right)}
- Yes
so, the function
is
odd