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Trazado el gráfico de la función/ (cos(x))/(sqrt(x))

Gráfico de la función y = (cos(x))/(sqrt(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)
f(x) = ------
         ___ 
       \/ x
f(x)=cos⁡(x)xf{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sqrt{x}} 
f = cos(x)/sqrt(x)
Gráfico de la función
02468-8-6-4-2-10105-5
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
cos⁡(x)x=0\frac{\cos{\left(x \right)}}{\sqrt{x}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Solución numérica
x1=−29.845130209103x_{1} = -29.845130209103
x2=−67.5442420521806x_{2} = -67.5442420521806
x3=−70.6858347057703x_{3} = -70.6858347057703
x4=64.4026493985908x_{4} = 64.4026493985908
x5=−36.1283155162826x_{5} = -36.1283155162826
x6=−92.6769832808989x_{6} = -92.6769832808989
x7=−61.261056745001x_{7} = -61.261056745001
x8=1.5707963267949x_{8} = 1.5707963267949
x9=−76.9690200129499x_{9} = -76.9690200129499
x10=−98.9601685880785x_{10} = -98.9601685880785
x11=−95.8185759344887x_{11} = -95.8185759344887
x12=29.845130209103x_{12} = 29.845130209103
x13=80.1106126665397x_{13} = 80.1106126665397
x14=−64.4026493985908x_{14} = -64.4026493985908
x15=36.1283155162826x_{15} = 36.1283155162826
x16=73.8274273593601x_{16} = 73.8274273593601
x17=32.9867228626928x_{17} = 32.9867228626928
x18=−4.71238898038469x_{18} = -4.71238898038469
x19=−39.2699081698724x_{19} = -39.2699081698724
x20=26.7035375555132x_{20} = 26.7035375555132
x21=−7.85398163397448x_{21} = -7.85398163397448
x22=95.8185759344887x_{22} = 95.8185759344887
x23=−17.2787595947439x_{23} = -17.2787595947439
x24=−10.9955742875643x_{24} = -10.9955742875643
x25=98.9601685880785x_{25} = 98.9601685880785
x26=−86.3937979737193x_{26} = -86.3937979737193
x27=92.6769832808989x_{27} = 92.6769832808989
x28=−48.6946861306418x_{28} = -48.6946861306418
x29=54.9778714378214x_{29} = 54.9778714378214
x30=45.553093477052x_{30} = 45.553093477052
x31=23.5619449019235x_{31} = 23.5619449019235
x32=76.9690200129499x_{32} = 76.9690200129499
x33=−89.5353906273091x_{33} = -89.5353906273091
x34=334.579617607313x_{34} = 334.579617607313
x35=4.71238898038469x_{35} = 4.71238898038469
x36=−26.7035375555132x_{36} = -26.7035375555132
x37=−80.1106126665397x_{37} = -80.1106126665397
x38=7.85398163397448x_{38} = 7.85398163397448
x39=14.1371669411541x_{39} = 14.1371669411541
x40=86.3937979737193x_{40} = 86.3937979737193
x41=−45.553093477052x_{41} = -45.553093477052
x42=−83.2522053201295x_{42} = -83.2522053201295
x43=70.6858347057703x_{43} = 70.6858347057703
x44=83.2522053201295x_{44} = 83.2522053201295
x45=48.6946861306418x_{45} = 48.6946861306418
x46=−20.4203522483337x_{46} = -20.4203522483337
x47=51.8362787842316x_{47} = 51.8362787842316
x48=10.9955742875643x_{48} = 10.9955742875643
x49=20.4203522483337x_{49} = 20.4203522483337
x50=89.5353906273091x_{50} = 89.5353906273091
x51=17.2787595947439x_{51} = 17.2787595947439
x52=58.1194640914112x_{52} = 58.1194640914112
x53=61.261056745001x_{53} = 61.261056745001
x54=−32.9867228626928x_{54} = -32.9867228626928
x55=−51.8362787842316x_{55} = -51.8362787842316
x56=−14.1371669411541x_{56} = -14.1371669411541
x57=−58.1194640914112x_{57} = -58.1194640914112
x58=−42.4115008234622x_{58} = -42.4115008234622
x59=−54.9778714378214x_{59} = -54.9778714378214
x60=−1.5707963267949x_{60} = -1.5707963267949
x61=42.4115008234622x_{61} = 42.4115008234622
x62=39.2699081698724x_{62} = 39.2699081698724
x63=67.5442420521806x_{63} = 67.5442420521806
x64=887.499924639117x_{64} = 887.499924639117
x65=−23.5619449019235x_{65} = -23.5619449019235
x66=−73.8274273593601x_{66} = -73.8274273593601
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en cos(x)/sqrt(x).
cos⁡(0)0\frac{\cos{\left(0 \right)}}{\sqrt{0}}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
−sin⁡(x)x−cos⁡(x)2x32=0- \frac{\sin{\left(x \right)}}{\sqrt{x}} - \frac{\cos{\left(x \right)}}{2 x^{\frac{3}{2}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−28.2566407733299x_{1} = -28.2566407733299
x2=−47.1132774827275x_{2} = -47.1132774827275
x3=34.5430455066495x_{3} = 34.5430455066495
x4=87.9589098892909x_{4} = 87.9589098892909
x5=−75.3915917440781x_{5} = -75.3915917440781
x6=−81.6752872670354x_{6} = -81.6752872670354
x7=−9.37147510585595x_{7} = -9.37147510585595
x8=62.8238944845809x_{8} = 62.8238944845809
x9=−2.97508632168828x_{9} = -2.97508632168828
x10=−100.525991117835x_{10} = -100.525991117835
x11=−131.94310195667x_{11} = -131.94310195667
x12=21.9683925318703x_{12} = 21.9683925318703
x13=−12.5264763376692x_{13} = -12.5264763376692
x14=−62.8238944845809x_{14} = -62.8238944845809
x15=1683.89336539322x_{15} = 1683.89336539322
x16=−34.5430455066495x_{16} = -34.5430455066495
x17=−40.8284587489214x_{17} = -40.8284587489214
x18=−69.1078034322536x_{18} = -69.1078034322536
x19=791.680717136973x_{19} = 791.680717136973
x20=2.97508632168828x_{20} = 2.97508632168828
x21=53.397711687542x_{21} = 53.397711687542
x22=50.2555336325565x_{22} = 50.2555336325565
x23=−21.9683925318703x_{23} = -21.9683925318703
x24=40.8284587489214x_{24} = 40.8284587489214
x25=94.2424741940464x_{25} = 94.2424741940464
x26=−84.817106677999x_{26} = -84.817106677999
x27=81.6752872670354x_{27} = 81.6752872670354
x28=−72.2497107001058x_{28} = -72.2497107001058
x29=25.1128337203766x_{29} = 25.1128337203766
x30=−37.6858450405302x_{30} = -37.6858450405302
x31=−18.8229989180076x_{31} = -18.8229989180076
x32=179.067989026352x_{32} = 179.067989026352
x33=−15.6760783451944x_{33} = -15.6760783451944
x34=6.20274981679304x_{34} = 6.20274981679304
x35=31.4000043168626x_{35} = 31.4000043168626
x36=75.3915917440781x_{36} = 75.3915917440781
x37=56.5398246709304x_{37} = 56.5398246709304
x38=−6.20274981679304x_{38} = -6.20274981679304
x39=28.2566407733299x_{39} = 28.2566407733299
x40=72.2497107001058x_{40} = 72.2497107001058
x41=−56.5398246709304x_{41} = -56.5398246709304
x42=84.817106677999x_{42} = 84.817106677999
x43=−87.9589098892909x_{43} = -87.9589098892909
x44=100.525991117835x_{44} = 100.525991117835
x45=12.5264763376692x_{45} = 12.5264763376692
x46=15.6760783451944x_{46} = 15.6760783451944
x47=18.8229989180076x_{47} = 18.8229989180076
x48=−91.1006985770946x_{48} = -91.1006985770946
x49=−43.9709264903445x_{49} = -43.9709264903445
x50=65.9658661929102x_{50} = 65.9658661929102
x51=−188.492906601895x_{51} = -188.492906601895
x52=78.5334497119428x_{52} = 78.5334497119428
x53=−25.1128337203766x_{53} = -25.1128337203766
x54=9.37147510585595x_{54} = 9.37147510585595
x55=−1146.6808825192x_{55} = -1146.6808825192
x56=−97.3842380053013x_{56} = -97.3842380053013
x57=−53.397711687542x_{57} = -53.397711687542
x58=−59.6818828624266x_{58} = -59.6818828624266
x59=47.1132774827275x_{59} = 47.1132774827275
x60=−50.2555336325565x_{60} = -50.2555336325565
x61=69.1078034322536x_{61} = 69.1078034322536
x62=91.1006985770946x_{62} = 91.1006985770946
x63=−31.4000043168626x_{63} = -31.4000043168626
x64=37.6858450405302x_{64} = 37.6858450405302
x65=43.9709264903445x_{65} = 43.9709264903445
x66=−94.2424741940464x_{66} = -94.2424741940464
x67=−65.9658661929102x_{67} = -65.9658661929102
x68=59.6818828624266x_{68} = 59.6818828624266
x69=97.3842380053013x_{69} = 97.3842380053013
x70=−78.5334497119428x_{70} = -78.5334497119428
Signos de extremos en los puntos:
(-28.256640773329945, 0.18809261922504*I)

(-47.11327748272753, 0.145681325876889*I)

(34.54304550664949, -0.170127373179912)

(87.95890988929088, 0.106623531852143)

(-75.39159174407808, -0.115167248976248*I)

(-81.67528726703536, -0.110648753785148*I)

(-9.371475105855954, 0.326196105910348*I)

(62.82389448458093, 0.126160621108934)

(-2.9750863216882792, 0.571744401877857*I)

(-100.52599111783519, -0.0997368037242384*I)

(-131.94310195666966, -0.0870569678200158*I)

(21.968392531870297, -0.213298795668094)

(-12.5264763376692, -0.282318830106324*I)

(-62.82389448458093, -0.126160621108934*I)

(1683.8933653932215, 0.0243692794377349)

(-34.54304550664949, 0.170127373179912*I)

(-40.8284587489214, 0.15648976674518*I)

(-69.10780343225363, -0.120288771309422*I)

(791.6807171369735, 0.035540610191994)

(2.9750863216882792, -0.571744401877857)

(53.39771168754203, -0.136842071089773)

(50.255533632556485, 0.141054375396673)

(-21.968392531870297, 0.213298795668094*I)

(40.8284587489214, -0.15648976674518)

(94.24247419404638, 0.103007903504495)

(-84.817106677999, 0.108580222480823*I)

(81.67528726703536, 0.110648753785148)

(-72.24971070010584, 0.117644477250395*I)

(25.112833720376596, 0.199510646718215)

(-37.68584504053022, -0.16288183381049*I)

(-18.822998918007553, -0.230410584140235*I)

(179.06798902635177, -0.0747290272027069)

(-15.676078345194368, 0.252441346243332*I)

(6.202749816793043, 0.400222440722691)

(31.400004316862624, 0.178435019744746)

(75.39159174407808, 0.115167248976248)

(56.53982467093041, 0.132985959193641)

(-6.202749816793043, -0.400222440722691*I)

(28.256640773329945, -0.18809261922504)

(72.24971070010584, -0.117644477250395)

(-56.53982467093041, -0.132985959193641*I)

(84.817106677999, -0.108580222480823)

(-87.95890988929088, -0.106623531852143*I)

(100.52599111783519, 0.0997368037242384)

(12.5264763376692, 0.282318830106324)

(15.676078345194368, -0.252441346243332)

(18.822998918007553, 0.230410584140235)

(-91.10069857709462, 0.104768953369684*I)

(-43.97092649034452, -0.150795754903091*I)

(65.96586619291024, -0.123119796833232)

(-188.49290660189519, -0.0728368182892935*I)

(78.53344971194282, -0.112840203476897)

(-25.112833720376596, -0.199510646718215*I)

(9.371475105855954, -0.326196105910348)

(-1146.6808825191963, 0.0295310352979044*I)

(-97.38423800530128, 0.101332776087448*I)

(-53.39771168754203, 0.136842071089773*I)

(-59.681882862426576, 0.129438509013877*I)

(47.11327748272753, -0.145681325876889)

(-50.255533632556485, -0.141054375396673*I)

(69.10780343225363, 0.120288771309422)

(91.10069857709462, -0.104768953369684)

(-31.400004316862624, -0.178435019744746*I)

(37.68584504053022, 0.16288183381049)

(43.97092649034452, 0.150795754903091)

(-94.24247419404638, -0.103007903504495*I)

(-65.96586619291024, 0.123119796833232*I)

(59.681882862426576, -0.129438509013877)

(97.38423800530128, -0.101332776087448)

(-78.53344971194282, 0.112840203476897*I)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=34.5430455066495x_{1} = 34.5430455066495
x2=21.9683925318703x_{2} = 21.9683925318703
x3=2.97508632168828x_{3} = 2.97508632168828
x4=53.397711687542x_{4} = 53.397711687542
x5=40.8284587489214x_{5} = 40.8284587489214
x6=179.067989026352x_{6} = 179.067989026352
x7=28.2566407733299x_{7} = 28.2566407733299
x8=72.2497107001058x_{8} = 72.2497107001058
x9=84.817106677999x_{9} = 84.817106677999
x10=15.6760783451944x_{10} = 15.6760783451944
x11=65.9658661929102x_{11} = 65.9658661929102
x12=78.5334497119428x_{12} = 78.5334497119428
x13=9.37147510585595x_{13} = 9.37147510585595
x14=47.1132774827275x_{14} = 47.1132774827275
x15=91.1006985770946x_{15} = 91.1006985770946
x16=59.6818828624266x_{16} = 59.6818828624266
x17=97.3842380053013x_{17} = 97.3842380053013
Puntos máximos de la función:
x17=87.9589098892909x_{17} = 87.9589098892909
x17=62.8238944845809x_{17} = 62.8238944845809
x17=1683.89336539322x_{17} = 1683.89336539322
x17=791.680717136973x_{17} = 791.680717136973
x17=50.2555336325565x_{17} = 50.2555336325565
x17=94.2424741940464x_{17} = 94.2424741940464
x17=81.6752872670354x_{17} = 81.6752872670354
x17=25.1128337203766x_{17} = 25.1128337203766
x17=6.20274981679304x_{17} = 6.20274981679304
x17=31.4000043168626x_{17} = 31.4000043168626
x17=75.3915917440781x_{17} = 75.3915917440781
x17=56.5398246709304x_{17} = 56.5398246709304
x17=100.525991117835x_{17} = 100.525991117835
x17=12.5264763376692x_{17} = 12.5264763376692
x17=18.8229989180076x_{17} = 18.8229989180076
x17=69.1078034322536x_{17} = 69.1078034322536
x17=37.6858450405302x_{17} = 37.6858450405302
x17=43.9709264903445x_{17} = 43.9709264903445
Decrece en los intervalos
[179.067989026352,∞)\left[179.067989026352, \infty\right)
Crece en los intervalos
(−∞,2.97508632168828]\left(-\infty, 2.97508632168828\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
−cos⁡(x)+sin⁡(x)x+3cos⁡(x)4x2x=0\frac{- \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{3 \cos{\left(x \right)}}{4 x^{2}}}{\sqrt{x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=67.5294323412782x_{1} = 67.5294323412782
x2=−92.6661913350059x_{2} = -92.6661913350059
x3=4.48478972694075x_{3} = 4.48478972694075
x4=−95.8081379339785x_{4} = -95.8081379339785
x5=−7.72360366302726x_{5} = -7.72360366302726
x6=−17.2206080366588x_{6} = -17.2206080366588
x7=23.5193947727501x_{7} = 23.5193947727501
x8=−14.0659233939484x_{8} = -14.0659233939484
x9=−1101.12731692271x_{9} = -1101.12731692271
x10=−29.8115704591729x_{10} = -29.8115704591729
x11=−67.5294323412782x_{11} = -67.5294323412782
x12=−4.48478972694075x_{12} = -4.48478972694075
x13=80.0981271693078x_{13} = 80.0981271693078
x14=−45.5311260653452x_{14} = -45.5311260653452
x15=−89.5242198850226x_{15} = -89.5242198850226
x16=83.2401911702273x_{16} = 83.2401911702273
x17=36.1006062941223x_{17} = 36.1006062941223
x18=86.3822208710534x_{18} = 86.3822208710534
x19=42.3879037163074x_{19} = 42.3879037163074
x20=−10.9035394436736x_{20} = -10.9035394436736
x21=−23.5193947727501x_{21} = -23.5193947727501
x22=−61.2447269949324x_{22} = -61.2447269949324
x23=17.2206080366588x_{23} = 17.2206080366588
x24=73.8138787355599x_{24} = 73.8138787355599
x25=−80.0981271693078x_{25} = -80.0981271693078
x26=61.2447269949324x_{26} = 61.2447269949324
x27=98.9500620501441x_{27} = 98.9500620501441
x28=29.8115704591729x_{28} = 29.8115704591729
x29=−58.1022509299587x_{29} = -58.1022509299587
x30=−2134.7117396669x_{30} = -2134.7117396669
x31=7.72360366302726x_{31} = 7.72360366302726
x32=168.069256951157x_{32} = 168.069256951157
x33=−83.2401911702273x_{33} = -83.2401911702273
x34=64.3871167803208x_{34} = 64.3871167803208
x35=−98.9500620501441x_{35} = -98.9500620501441
x36=26.6660146634262x_{36} = 26.6660146634262
x37=48.6741377261064x_{37} = 48.6741377261064
x38=−51.8169770950755x_{38} = -51.8169770950755
x39=−42.3879037163074x_{39} = -42.3879037163074
x40=−39.2444199463872x_{40} = -39.2444199463872
x41=−64.3871167803208x_{41} = -64.3871167803208
x42=20.3712140812955x_{42} = 20.3712140812955
x43=1829.97717426108x_{43} = 1829.97717426108
x44=32.9563680725012x_{44} = 32.9563680725012
x45=51.8169770950755x_{45} = 51.8169770950755
x46=−86.3822208710534x_{46} = -86.3822208710534
x47=−32.9563680725012x_{47} = -32.9563680725012
x48=45.5311260653452x_{48} = 45.5311260653452
x49=92.6661913350059x_{49} = 92.6661913350059
x50=−20.3712140812955x_{50} = -20.3712140812955
x51=54.9596737689483x_{51} = 54.9596737689483
x52=−73.8138787355599x_{52} = -73.8138787355599
x53=−26.6660146634262x_{53} = -26.6660146634262
x54=58.1022509299587x_{54} = 58.1022509299587
x55=89.5242198850226x_{55} = 89.5242198850226
x56=76.9560246644883x_{56} = 76.9560246644883
x57=10.9035394436736x_{57} = 10.9035394436736
x58=−76.9560246644883x_{58} = -76.9560246644883
x59=−48.6741377261064x_{59} = -48.6741377261064
x60=−70.6716835864667x_{60} = -70.6716835864667
x61=70.6716835864667x_{61} = 70.6716835864667
x62=−36.1006062941223x_{62} = -36.1006062941223
x63=14.0659233939484x_{63} = 14.0659233939484
x64=95.8081379339785x_{64} = 95.8081379339785
x65=−54.9596737689483x_{65} = -54.9596737689483
x66=39.2444199463872x_{66} = 39.2444199463872
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(−cos⁡(x)+sin⁡(x)x+3cos⁡(x)4x2x)=−∞i\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{3 \cos{\left(x \right)}}{4 x^{2}}}{\sqrt{x}}\right) = - \infty i
lim⁡x→0+(−cos⁡(x)+sin⁡(x)x+3cos⁡(x)4x2x)=∞\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{3 \cos{\left(x \right)}}{4 x^{2}}}{\sqrt{x}}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[1829.97717426108,∞)\left[1829.97717426108, \infty\right)
Convexa en los intervalos
(−∞,7.72360366302726]\left(-\infty, 7.72360366302726\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{\sqrt{x}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{\sqrt{x}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x)/sqrt(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)xx)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{\sqrt{x} x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x)xx)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{\sqrt{x} x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
cos⁡(x)x=cos⁡(x)−x\frac{\cos{\left(x \right)}}{\sqrt{x}} = \frac{\cos{\left(x \right)}}{\sqrt{- x}}
- No
cos⁡(x)x=−cos⁡(x)−x\frac{\cos{\left(x \right)}}{\sqrt{x}} = - \frac{\cos{\left(x \right)}}{\sqrt{- x}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор