Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • -x^2+5x-4
  • |x^2+8x+12|
  • (x^2-4)/(2x+5)
  • -x^2-2x
  • Identical expressions

  • xsinx/(four +tg^ two (x))
  • x sinus of x divide by (4 plus tg squared (x))
  • x sinus of x divide by (four plus tg to the power of two (x))
  • xsinx/(4+tg2(x))
  • xsinx/4+tg2x
  • xsinx/(4+tg²(x))
  • xsinx/(4+tg to the power of 2(x))
  • xsinx/4+tg^2x
  • xsinx divide by (4+tg^2(x))
  • Similar expressions

  • xsinx/(4-tg^2(x))

Graphing y = xsinx/(4+tg^2(x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src]
         x*sin(x) 
f(x) = -----------
              2   
       4 + tan (x)
f(x)=xsin(x)tan2(x)+4f{\left(x \right)} = \frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}
f = (x*sin(x))/(tan(x)^2 + 4)
The graph of the function
02468-8-6-4-2-10102.5-2.5
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)tan2(x)+4=0\frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4} = 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=50.2654824574367x_{1} = -50.2654824574367
x2=23.5619448833633x_{2} = -23.5619448833633
x3=3.14159265358979x_{3} = -3.14159265358979
x4=23.5619449773569x_{4} = -23.5619449773569
x5=51.836278862996x_{5} = 51.836278862996
x6=72.2566310325652x_{6} = 72.2566310325652
x7=43.9822971502571x_{7} = -43.9822971502571
x8=37.6991118430775x_{8} = -37.6991118430775
x9=7.85398152870528x_{9} = -7.85398152870528
x10=34.5575191894877x_{10} = 34.5575191894877
x11=45.5530935521522x_{11} = -45.5530935521522
x12=94.2477796076938x_{12} = -94.2477796076938
x13=56.5486677646163x_{13} = -56.5486677646163
x14=21.9911485751286x_{14} = -21.9911485751286
x15=50.2654824574367x_{15} = 50.2654824574367
x16=15.707963267949x_{16} = -15.707963267949
x17=21.9911485751286x_{17} = 21.9911485751286
x18=87.9645943005142x_{18} = -87.9645943005142
x19=72.2566310325652x_{19} = -72.2566310325652
x20=59.6902604182061x_{20} = 59.6902604182061
x21=84.8230016469244x_{21} = 84.8230016469244
x22=58.1194640203505x_{22} = -58.1194640203505
x23=80.1106125976626x_{23} = -80.1106125976626
x24=9.42477796076938x_{24} = -9.42477796076938
x25=31.4159265358979x_{25} = 31.4159265358979
x26=9.42477796076938x_{26} = 9.42477796076938
x27=42.4115006809338x_{27} = 42.4115006809338
x28=7.85398171361397x_{28} = 7.85398171361397
x29=65.9734457253857x_{29} = -65.9734457253857
x30=73.8274274384211x_{30} = 73.8274274384211
x31=15.707963267949x_{31} = 15.707963267949
x32=29.8451301097218x_{32} = -29.8451301097218
x33=28.2743338823081x_{33} = 28.2743338823081
x34=94.2477796076938x_{34} = 94.2477796076938
x35=37.6991118430775x_{35} = 37.6991118430775
x36=1.57079626923918x_{36} = 1.57079626923918
x37=6.28318530717959x_{37} = 6.28318530717959
x38=20.42035217694x_{38} = 20.42035217694
x39=53.4070751110265x_{39} = -53.4070751110265
x40=42.4115007525711x_{40} = 42.4115007525711
x41=6.28318530717959x_{41} = -6.28318530717959
x42=75.398223686155x_{42} = -75.398223686155
x43=73.8274272831926x_{43} = -73.8274272831926
x44=64.4026493290031x_{44} = 64.4026493290031
x45=89.5353907014139x_{45} = -89.5353907014139
x46=0x_{46} = 0
x47=1.5707964157513x_{47} = -1.5707964157513
x48=95.8185759189139x_{48} = 95.8185759189139
x49=34.5575191894877x_{49} = -34.5575191894877
x50=97.3893722612836x_{50} = -97.3893722612836
x51=28.2743338823081x_{51} = -28.2743338823081
x52=36.128315443513x_{52} = -36.128315443513
x53=81.6814089933346x_{53} = 81.6814089933346
x54=100.530964914873x_{54} = 100.530964914873
x55=47.1238898038469x_{55} = 47.1238898038469
x56=86.3937979058448x_{56} = 86.3937979058448
x57=14.1371670350454x_{57} = 14.1371670350454
x58=95.8185760136054x_{58} = 95.8185760136054
x59=78.5398163397448x_{59} = 78.5398163397448
x60=12.5663706143592x_{60} = 12.5663706143592
x61=67.544242126936x_{61} = -67.544242126936
x62=87.9645943005142x_{62} = 87.9645943005142
x63=29.8451302874799x_{63} = 29.8451302874799
x64=100.530964914873x_{64} = -100.530964914873
x65=78.5398163397448x_{65} = -78.5398163397448
x66=43.9822971502571x_{66} = 43.9822971502571
x67=31.4159265358979x_{67} = -31.4159265358979
x68=81.6814089933346x_{68} = -81.6814089933346
x69=56.5486677646163x_{69} = 56.5486677646163
x70=3.14159265358979x_{70} = 3.14159265358979
x71=14.1371668678372x_{71} = -14.1371668678372
x72=59.6902604182061x_{72} = -59.6902604182061
x73=65.9734457253857x_{73} = 65.9734457253857
x74=51.8362786963082x_{74} = -51.8362786963082
x75=95.818575869486x_{75} = -95.818575869486
x76=47.1238898038469x_{76} = -47.1238898038469
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))/(4 + tan(x)^2).
0sin(0)tan2(0)+4\frac{0 \sin{\left(0 \right)}}{\tan^{2}{\left(0 \right)} + 4}
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
x(2tan2(x)+2)sin(x)tan(x)(tan2(x)+4)2+xcos(x)+sin(x)tan2(x)+4=0- \frac{x \left(2 \tan^{2}{\left(x \right)} + 2\right) \sin{\left(x \right)} \tan{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 4\right)^{2}} + \frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4} = 0
Solve this equation
The roots of this equation
x1=454.703366375065x_{1} = 454.703366375065
x2=14.1371669411541x_{2} = 14.1371669411541
x3=69.9460058299699x_{3} = 69.9460058299699
x4=80.1106126665397x_{4} = -80.1106126665397
x5=7.85398163397448x_{5} = -7.85398163397448
x6=11.7561606649435x_{6} = 11.7561606649435
x7=90.2804556957995x_{7} = -90.2804556957995
x8=55.7243617726486x_{8} = -55.7243617726486
x9=10.2725607549115x_{9} = 10.2725607549115
x10=67.5442420521806x_{10} = 67.5442420521806
x11=60.5216851226385x_{11} = 60.5216851226385
x12=76.2289489691309x_{12} = -76.2289489691309
x13=80.1106126665397x_{13} = 80.1106126665397
x14=49.4416503630997x_{14} = -49.4416503630997
x15=62.0071694687546x_{15} = 62.0071694687546
x16=77.7144563954775x_{16} = 77.7144563954775
x17=29.845130209103x_{17} = -29.845130209103
x18=19.6879744764593x_{18} = -19.6879744764593
x19=62.0071694687546x_{19} = -62.0071694687546
x20=70.6858347057703x_{20} = 70.6858347057703
x21=33.735647763708x_{21} = 33.735647763708
x22=2.40467079662226x_{22} = 2.40467079662226
x23=69.9460058299699x_{23} = -69.9460058299699
x24=68.2900468139916x_{24} = 68.2900468139916
x25=24.3132699931952x_{25} = 24.3132699931952
x26=0x_{26} = 0
x27=46.3003430565376x_{27} = 46.3003430565376
x28=85.65343015591x_{28} = -85.65343015591
x29=5.49385370718343x_{29} = -5.49385370718343
x30=76.2289489691309x_{30} = 76.2289489691309
x31=45.553093477052x_{31} = 45.553093477052
x32=54.238892641524x_{32} = -54.238892641524
x33=10.2725607549115x_{33} = -10.2725607549115
x34=83.997442131108x_{34} = -83.997442131108
x35=82.5119289102023x_{35} = 82.5119289102023
x36=99.7050166950322x_{36} = -99.7050166950322
x37=58.1194640914112x_{37} = 58.1194640914112
x38=7.13941996437343x_{38} = -7.13941996437343
x39=38.5324692900873x_{39} = 38.5324692900873
x40=4.01872293357174x_{40} = 4.01872293357174
x41=63.663110358029x_{41} = -63.663110358029
x42=91.9364513523705x_{42} = -91.9364513523705
x43=83.997442131108x_{43} = 83.997442131108
x44=11.7561606649435x_{44} = -11.7561606649435
x45=51.8362787842316x_{45} = -51.8362787842316
x46=73.8274273593601x_{46} = -73.8274273593601
x47=47.9562027081837x_{47} = -47.9562027081837
x48=4.71238898038469x_{48} = 4.71238898038469
x49=18.0330900698586x_{49} = 18.0330900698586
x50=95.8185759344887x_{50} = -95.8185759344887
x51=25.9686624297459x_{51} = -25.9686624297459
x52=98.2194935076171x_{52} = -98.2194935076171
x53=1.5707963267949x_{53} = 1.5707963267949
x54=16.5483332246309x_{54} = 16.5483332246309
x55=54.238892641524x_{55} = 54.238892641524
x56=90.2804556957995x_{56} = 90.2804556957995
x57=92.6769832808989x_{57} = 92.6769832808989
x58=73.0874722075759x_{58} = 73.0874722075759
x59=25.9686624297459x_{59} = 25.9686624297459
x60=40.0178633797347x_{60} = -40.0178633797347
x61=68.2900468139916x_{61} = -68.2900468139916
x62=58.865755516954x_{62} = -58.865755516954
x63=55.7243617726486x_{63} = 55.7243617726486
x64=18.0330900698586x_{64} = -18.0330900698586
x65=32.2503175803598x_{65} = 32.2503175803598
x66=33.735647763708x_{66} = -33.735647763708
x67=32.2503175803598x_{67} = -32.2503175803598
x68=5.49385370718343x_{68} = 5.49385370718343
x69=27.453878484238x_{69} = 27.453878484238
x70=40.0178633797347x_{70} = 40.0178633797347
x71=87.138945811754x_{71} = 87.138945811754
x72=36.1283155162826x_{72} = 36.1283155162826
x73=93.4219711563462x_{73} = -93.4219711563462
x74=4.01872293357174x_{74} = -4.01872293357174
x75=99.7050166950322x_{75} = 99.7050166950322
x76=16.5483332246309x_{76} = -16.5483332246309
x77=0.997817540674201x_{77} = 0.997817540674201
x78=91.9364513523705x_{78} = 91.9364513523705
x79=36.8767140943411x_{79} = -36.8767140943411
x80=27.453878484238x_{80} = -27.453878484238
x81=77.7144563954775x_{81} = -77.7144563954775
x82=98.2194935076171x_{82} = 98.2194935076171
x83=71.4315058489604x_{83} = -71.4315058489604
x84=41.6736614988808x_{84} = -41.6736614988808
x85=47.9562027081837x_{85} = 47.9562027081837
x86=46.3003430565376x_{86} = -46.3003430565376
The values of the extrema at the points:
(454.7033663750651, 64.582180890724)

(14.137166941154069, 4.29347676987172e-30)

(69.9460058299699, 9.93432901594521)

(-80.11061266653972, -1.92283264304371e-27)

(-7.853981633974483, 7.36192861774987e-31)

(11.756160664943506, -1.66847511356391)

(-90.28045569579949, 12.8225080937804)

(-55.72436177264862, -7.91435195205804)

(10.27256075491152, -1.45763299397885)

(67.54424205218055, -1.3132184568469e-27)

(60.52168512263849, -8.59574594780144)

(-76.2289489691309, 10.8267221900235)

(80.11061266653972, -1.92283264304371e-27)

(-49.441650363099725, -7.02197513700988)

(62.0071694687546, -8.80673558504463)

(77.71445639547751, 11.0377136292074)

(-29.845130209103036, -1.12127665170554e-29)

(-19.687974476459267, 2.79557707373584)

(-62.0071694687546, -8.80673558504463)

(70.68583470577035, 6.77618297499813e-29)

(33.73564776370796, 4.7910880047921)

(2.4046707966222645, 0.335020556538555)

(-69.9460058299699, 9.93432901594521)

(68.29004681399161, -9.69912415120835)

(24.313269993195227, -3.45264222691292)

(0, 0)

(46.30034305653763, 6.57579015556219)

(-85.65343015590999, -12.1653166830211)

(-5.493853707183429, -0.777532226748693)

(76.2289489691309, 10.8267221900235)

(45.553093477052, 1.74530768724744e-35)

(-54.238892641523954, -7.70336350275712)

(-10.27256075491152, -1.45763299397885)

(-83.99744213110796, 11.9301098752818)

(82.51192891020227, 11.7191179837181)

(-99.70501669503224, -14.1611083032433)

(58.119464091411174, 1.39112146798308e-29)

(-7.139419964373433, 1.012038154624)

(38.53246929008733, 5.47244715183819)

(4.018722933571741, -0.567362924067202)

(-63.66311035802902, 9.04193923563947)

(-91.93645135237047, -13.0577154126865)

(83.99744213110796, 11.9301098752818)

(-11.756160664943506, -1.66847511356391)

(-51.83627878423159, 3.09398107171563e-30)

(-73.82742735936014, -4.43565443427593e-28)

(-47.956202708183724, -6.81098836459494)

(4.71238898038469, -1.59017658143397e-31)

(18.033090069858616, -2.56044421510836)

(-95.81857593448869, 3.676520165044e-28)

(-25.968662429745905, 3.68780913408032)

(-98.21949350761713, -13.9501156336138)

(1.5707963267948966, 5.8895428941999e-33)

(16.54833322463088, -2.34951143571411)

(54.238892641523954, -7.70336350275712)

(90.28045569579949, 12.8225080937804)

(92.6769832808989, -2.69152684487792e-27)

(73.08747220757594, -10.3805252333763)

(25.968662429745905, 3.68780913408032)

(-40.01786337973469, 5.68342974137886)

(-68.29004681399161, -9.69912415120835)

(-58.86575551695403, 8.36054305299913)

(55.72436177264862, -7.91435195205804)

(-18.033090069858616, -2.56044421510836)

(32.2503175803598, 4.58011040647008)

(-33.73564776370796, 4.7910880047921)

(-32.2503175803598, 4.58011040647008)

(5.493853707183429, -0.777532226748693)

(27.453878484238032, 3.89877781647259)

(40.01786337973469, 5.68342974137886)

(87.13894581175403, -12.3763087646776)

(36.12831551628262, -3.66424875021481e-28)

(-93.42197115634625, -13.2687078181799)

(-4.018722933571741, -0.567362924067202)

(99.70501669503224, -14.1611083032433)

(-16.54833322463088, -2.34951143571411)

(0.9978175406742009, 0.130960306067674)

(91.93645135237047, -13.0577154126865)

(-36.87671409434108, -5.23725594007668)

(-27.453878484238032, 3.89877781647259)

(-77.71445639547751, 11.0377136292074)

(98.21949350761713, -13.9501156336138)

(-71.43150584896037, 10.1453198768279)

(-41.67366149888085, -5.91862382854916)

(47.956202708183724, -6.81098836459494)

(-46.30034305653763, 6.57579015556219)


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=14.1371669411541x_{1} = 14.1371669411541
x2=7.85398163397448x_{2} = -7.85398163397448
x3=11.7561606649435x_{3} = 11.7561606649435
x4=55.7243617726486x_{4} = -55.7243617726486
x5=10.2725607549115x_{5} = 10.2725607549115
x6=60.5216851226385x_{6} = 60.5216851226385
x7=49.4416503630997x_{7} = -49.4416503630997
x8=62.0071694687546x_{8} = 62.0071694687546
x9=62.0071694687546x_{9} = -62.0071694687546
x10=70.6858347057703x_{10} = 70.6858347057703
x11=68.2900468139916x_{11} = 68.2900468139916
x12=24.3132699931952x_{12} = 24.3132699931952
x13=0x_{13} = 0
x14=85.65343015591x_{14} = -85.65343015591
x15=5.49385370718343x_{15} = -5.49385370718343
x16=45.553093477052x_{16} = 45.553093477052
x17=54.238892641524x_{17} = -54.238892641524
x18=10.2725607549115x_{18} = -10.2725607549115
x19=99.7050166950322x_{19} = -99.7050166950322
x20=58.1194640914112x_{20} = 58.1194640914112
x21=4.01872293357174x_{21} = 4.01872293357174
x22=91.9364513523705x_{22} = -91.9364513523705
x23=11.7561606649435x_{23} = -11.7561606649435
x24=51.8362787842316x_{24} = -51.8362787842316
x25=47.9562027081837x_{25} = -47.9562027081837
x26=18.0330900698586x_{26} = 18.0330900698586
x27=95.8185759344887x_{27} = -95.8185759344887
x28=98.2194935076171x_{28} = -98.2194935076171
x29=1.5707963267949x_{29} = 1.5707963267949
x30=16.5483332246309x_{30} = 16.5483332246309
x31=54.238892641524x_{31} = 54.238892641524
x32=73.0874722075759x_{32} = 73.0874722075759
x33=68.2900468139916x_{33} = -68.2900468139916
x34=55.7243617726486x_{34} = 55.7243617726486
x35=18.0330900698586x_{35} = -18.0330900698586
x36=5.49385370718343x_{36} = 5.49385370718343
x37=87.138945811754x_{37} = 87.138945811754
x38=93.4219711563462x_{38} = -93.4219711563462
x39=4.01872293357174x_{39} = -4.01872293357174
x40=99.7050166950322x_{40} = 99.7050166950322
x41=16.5483332246309x_{41} = -16.5483332246309
x42=91.9364513523705x_{42} = 91.9364513523705
x43=36.8767140943411x_{43} = -36.8767140943411
x44=98.2194935076171x_{44} = 98.2194935076171
x45=41.6736614988808x_{45} = -41.6736614988808
x46=47.9562027081837x_{46} = 47.9562027081837
Maxima of the function at points:
x46=454.703366375065x_{46} = 454.703366375065
x46=69.9460058299699x_{46} = 69.9460058299699
x46=80.1106126665397x_{46} = -80.1106126665397
x46=90.2804556957995x_{46} = -90.2804556957995
x46=67.5442420521806x_{46} = 67.5442420521806
x46=76.2289489691309x_{46} = -76.2289489691309
x46=80.1106126665397x_{46} = 80.1106126665397
x46=77.7144563954775x_{46} = 77.7144563954775
x46=29.845130209103x_{46} = -29.845130209103
x46=19.6879744764593x_{46} = -19.6879744764593
x46=33.735647763708x_{46} = 33.735647763708
x46=2.40467079662226x_{46} = 2.40467079662226
x46=69.9460058299699x_{46} = -69.9460058299699
x46=46.3003430565376x_{46} = 46.3003430565376
x46=76.2289489691309x_{46} = 76.2289489691309
x46=83.997442131108x_{46} = -83.997442131108
x46=82.5119289102023x_{46} = 82.5119289102023
x46=7.13941996437343x_{46} = -7.13941996437343
x46=38.5324692900873x_{46} = 38.5324692900873
x46=63.663110358029x_{46} = -63.663110358029
x46=83.997442131108x_{46} = 83.997442131108
x46=73.8274273593601x_{46} = -73.8274273593601
x46=4.71238898038469x_{46} = 4.71238898038469
x46=25.9686624297459x_{46} = -25.9686624297459
x46=90.2804556957995x_{46} = 90.2804556957995
x46=92.6769832808989x_{46} = 92.6769832808989
x46=25.9686624297459x_{46} = 25.9686624297459
x46=40.0178633797347x_{46} = -40.0178633797347
x46=58.865755516954x_{46} = -58.865755516954
x46=32.2503175803598x_{46} = 32.2503175803598
x46=33.735647763708x_{46} = -33.735647763708
x46=32.2503175803598x_{46} = -32.2503175803598
x46=27.453878484238x_{46} = 27.453878484238
x46=40.0178633797347x_{46} = 40.0178633797347
x46=36.1283155162826x_{46} = 36.1283155162826
x46=0.997817540674201x_{46} = 0.997817540674201
x46=27.453878484238x_{46} = -27.453878484238
x46=77.7144563954775x_{46} = -77.7144563954775
x46=71.4315058489604x_{46} = -71.4315058489604
x46=46.3003430565376x_{46} = -46.3003430565376
Decreasing at intervals
[99.7050166950322,)\left[99.7050166950322, \infty\right)
Increasing at intervals
(,99.7050166950322]\left(-\infty, -99.7050166950322\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=limx(xsin(x)tan2(x)+4)y = \lim_{x \to -\infty}\left(\frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=limx(xsin(x)tan2(x)+4)y = \lim_{x \to \infty}\left(\frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x*sin(x))/(4 + tan(x)^2), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlimx(sin(x)tan2(x)+4)y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlimx(sin(x)tan2(x)+4)y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}\right)
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)tan2(x)+4=xsin(x)tan2(x)+4\frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4} = \frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}
- Yes
xsin(x)tan2(x)+4=xsin(x)tan2(x)+4\frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4} = - \frac{x \sin{\left(x \right)}}{\tan^{2}{\left(x \right)} + 4}
- No
so, the function
is
even