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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=1.5707963267949x_{1} = 1.5707963267949
x2=39.2699081698724x_{2} = 39.2699081698724
x3=−39.2699081698724x_{3} = -39.2699081698724
x4=887.499924639117x_{4} = 887.499924639117
x5=14.1371669411541x_{5} = 14.1371669411541
x6=−70.6858347057703x_{6} = -70.6858347057703
x7=32.9867228626928x_{7} = 32.9867228626928
x8=61.261056745001x_{8} = 61.261056745001
x9=36.1283155162826x_{9} = 36.1283155162826
x10=−61.261056745001x_{10} = -61.261056745001
x11=−7.85398163397448x_{11} = -7.85398163397448
x12=−36.1283155162826x_{12} = -36.1283155162826
x13=−48.6946861306418x_{13} = -48.6946861306418
x14=7.85398163397448x_{14} = 7.85398163397448
x15=70.6858347057703x_{15} = 70.6858347057703
x16=−80.1106126665397x_{16} = -80.1106126665397
x17=42.4115008234622x_{17} = 42.4115008234622
x18=51.8362787842316x_{18} = 51.8362787842316
x19=−98.9601685880785x_{19} = -98.9601685880785
x20=334.579617607313x_{20} = 334.579617607313
x21=80.1106126665397x_{21} = 80.1106126665397
x22=−64.4026493985908x_{22} = -64.4026493985908
x23=−29.845130209103x_{23} = -29.845130209103
x24=98.9601685880785x_{24} = 98.9601685880785
x25=83.2522053201295x_{25} = 83.2522053201295
x26=−89.5353906273091x_{26} = -89.5353906273091
x27=−86.3937979737193x_{27} = -86.3937979737193
x28=−10.9955742875643x_{28} = -10.9955742875643
x29=17.2787595947439x_{29} = 17.2787595947439
x30=−26.7035375555132x_{30} = -26.7035375555132
x31=−54.9778714378214x_{31} = -54.9778714378214
x32=89.5353906273091x_{32} = 89.5353906273091
x33=58.1194640914112x_{33} = 58.1194640914112
x34=−92.6769832808989x_{34} = -92.6769832808989
x35=4.71238898038469x_{35} = 4.71238898038469
x36=20.4203522483337x_{36} = 20.4203522483337
x37=73.8274273593601x_{37} = 73.8274273593601
x38=−67.5442420521806x_{38} = -67.5442420521806
x39=45.553093477052x_{39} = 45.553093477052
x40=−58.1194640914112x_{40} = -58.1194640914112
x41=−51.8362787842316x_{41} = -51.8362787842316
x42=−83.2522053201295x_{42} = -83.2522053201295
x43=−76.9690200129499x_{43} = -76.9690200129499
x44=−14.1371669411541x_{44} = -14.1371669411541
x45=−73.8274273593601x_{45} = -73.8274273593601
x46=92.6769832808989x_{46} = 92.6769832808989
x47=−20.4203522483337x_{47} = -20.4203522483337
x48=64.4026493985908x_{48} = 64.4026493985908
x49=95.8185759344887x_{49} = 95.8185759344887
x50=67.5442420521806x_{50} = 67.5442420521806
x51=54.9778714378214x_{51} = 54.9778714378214
x52=48.6946861306418x_{52} = 48.6946861306418
x53=−4.71238898038469x_{53} = -4.71238898038469
x54=76.9690200129499x_{54} = 76.9690200129499
x55=−45.553093477052x_{55} = -45.553093477052
x56=−95.8185759344887x_{56} = -95.8185759344887
x57=29.845130209103x_{57} = 29.845130209103
x58=26.7035375555132x_{58} = 26.7035375555132
x59=−17.2787595947439x_{59} = -17.2787595947439
x60=−32.9867228626928x_{60} = -32.9867228626928
x61=−1.5707963267949x_{61} = -1.5707963267949
x62=23.5619449019235x_{62} = 23.5619449019235
x63=10.9955742875643x_{63} = 10.9955742875643
x64=86.3937979737193x_{64} = 86.3937979737193
x65=−23.5619449019235x_{65} = -23.5619449019235
x66=−42.4115008234622x_{66} = -42.4115008234622
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=37.6858450405302x_{1} = 37.6858450405302
x2=−12.5264763376692x_{2} = -12.5264763376692
x3=−78.5334497119428x_{3} = -78.5334497119428
x4=−188.492906601895x_{4} = -188.492906601895
x5=25.1128337203766x_{5} = 25.1128337203766
x6=−21.9683925318703x_{6} = -21.9683925318703
x7=−69.1078034322536x_{7} = -69.1078034322536
x8=47.1132774827275x_{8} = 47.1132774827275
x9=−59.6818828624266x_{9} = -59.6818828624266
x10=−131.94310195667x_{10} = -131.94310195667
x11=−91.1006985770946x_{11} = -91.1006985770946
x12=−100.525991117835x_{12} = -100.525991117835
x13=−47.1132774827275x_{13} = -47.1132774827275
x14=94.2424741940464x_{14} = 94.2424741940464
x15=−15.6760783451944x_{15} = -15.6760783451944
x16=56.5398246709304x_{16} = 56.5398246709304
x17=−72.2497107001058x_{17} = -72.2497107001058
x18=−87.9589098892909x_{18} = -87.9589098892909
x19=81.6752872670354x_{19} = 81.6752872670354
x20=40.8284587489214x_{20} = 40.8284587489214
x21=−84.817106677999x_{21} = -84.817106677999
x22=−75.3915917440781x_{22} = -75.3915917440781
x23=−62.8238944845809x_{23} = -62.8238944845809
x24=179.067989026352x_{24} = 179.067989026352
x25=−1146.6808825192x_{25} = -1146.6808825192
x26=34.5430455066495x_{26} = 34.5430455066495
x27=−2.97508632168828x_{27} = -2.97508632168828
x28=−9.37147510585595x_{28} = -9.37147510585595
x29=50.2555336325565x_{29} = 50.2555336325565
x30=−37.6858450405302x_{30} = -37.6858450405302
x31=97.3842380053013x_{31} = 97.3842380053013
x32=43.9709264903445x_{32} = 43.9709264903445
x33=59.6818828624266x_{33} = 59.6818828624266
x34=72.2497107001058x_{34} = 72.2497107001058
x35=−94.2424741940464x_{35} = -94.2424741940464
x36=1683.89336539322x_{36} = 1683.89336539322
x37=78.5334497119428x_{37} = 78.5334497119428
x38=−53.397711687542x_{38} = -53.397711687542
x39=21.9683925318703x_{39} = 21.9683925318703
x40=28.2566407733299x_{40} = 28.2566407733299
x41=−25.1128337203766x_{41} = -25.1128337203766
x42=−18.8229989180076x_{42} = -18.8229989180076
x43=12.5264763376692x_{43} = 12.5264763376692
x44=−28.2566407733299x_{44} = -28.2566407733299
x45=−6.20274981679304x_{45} = -6.20274981679304
x46=62.8238944845809x_{46} = 62.8238944845809
x47=791.680717136973x_{47} = 791.680717136973
x48=−40.8284587489214x_{48} = -40.8284587489214
x49=2.97508632168828x_{49} = 2.97508632168828
x50=87.9589098892909x_{50} = 87.9589098892909
x51=53.397711687542x_{51} = 53.397711687542
x52=−34.5430455066495x_{52} = -34.5430455066495
x53=15.6760783451944x_{53} = 15.6760783451944
x54=−50.2555336325565x_{54} = -50.2555336325565
x55=31.4000043168626x_{55} = 31.4000043168626
x56=69.1078034322536x_{56} = 69.1078034322536
x57=−56.5398246709304x_{57} = -56.5398246709304
x58=9.37147510585595x_{58} = 9.37147510585595
x59=−65.9658661929102x_{59} = -65.9658661929102
x60=75.3915917440781x_{60} = 75.3915917440781
x61=−31.4000043168626x_{61} = -31.4000043168626
x62=6.20274981679304x_{62} = 6.20274981679304
x63=91.1006985770946x_{63} = 91.1006985770946
x64=65.9658661929102x_{64} = 65.9658661929102
x65=−97.3842380053013x_{65} = -97.3842380053013
x66=84.817106677999x_{66} = 84.817106677999
x67=100.525991117835x_{67} = 100.525991117835
x68=−43.9709264903445x_{68} = -43.9709264903445
x69=18.8229989180076x_{69} = 18.8229989180076
x70=−81.6752872670354x_{70} = -81.6752872670354
Signos de extremos en los puntos:
(37.68584504053022, 0.16288183381049)

(-12.5264763376692, -0.282318830106324*I)

(-78.53344971194282, 0.112840203476897*I)

(-188.49290660189519, -0.0728368182892935*I)

(25.112833720376596, 0.199510646718215)

(-21.968392531870297, 0.213298795668094*I)

(-69.10780343225363, -0.120288771309422*I)

(47.11327748272753, -0.145681325876889)

(-59.681882862426576, 0.129438509013877*I)

(-131.94310195666966, -0.0870569678200158*I)

(-91.10069857709462, 0.104768953369684*I)

(-100.52599111783519, -0.0997368037242384*I)

(-47.11327748272753, 0.145681325876889*I)

(94.24247419404638, 0.103007903504495)

(-15.676078345194368, 0.252441346243332*I)

(56.53982467093041, 0.132985959193641)

(-72.24971070010584, 0.117644477250395*I)

(-87.95890988929088, -0.106623531852143*I)

(81.67528726703536, 0.110648753785148)

(40.8284587489214, -0.15648976674518)

(-84.817106677999, 0.108580222480823*I)

(-75.39159174407808, -0.115167248976248*I)

(-62.82389448458093, -0.126160621108934*I)

(179.06798902635177, -0.0747290272027069)

(-1146.6808825191963, 0.0295310352979044*I)

(34.54304550664949, -0.170127373179912)

(-2.9750863216882792, 0.571744401877857*I)

(-9.371475105855954, 0.326196105910348*I)

(50.255533632556485, 0.141054375396673)

(-37.68584504053022, -0.16288183381049*I)

(97.38423800530128, -0.101332776087448)

(43.97092649034452, 0.150795754903091)

(59.681882862426576, -0.129438509013877)

(72.24971070010584, -0.117644477250395)

(-94.24247419404638, -0.103007903504495*I)

(1683.8933653932215, 0.0243692794377349)

(78.53344971194282, -0.112840203476897)

(-53.39771168754203, 0.136842071089773*I)

(21.968392531870297, -0.213298795668094)

(28.256640773329945, -0.18809261922504)

(-25.112833720376596, -0.199510646718215*I)

(-18.822998918007553, -0.230410584140235*I)

(12.5264763376692, 0.282318830106324)

(-28.256640773329945, 0.18809261922504*I)

(-6.202749816793043, -0.400222440722691*I)

(62.82389448458093, 0.126160621108934)

(791.6807171369735, 0.035540610191994)

(-40.8284587489214, 0.15648976674518*I)

(2.9750863216882792, -0.571744401877857)

(87.95890988929088, 0.106623531852143)

(53.39771168754203, -0.136842071089773)

(-34.54304550664949, 0.170127373179912*I)

(15.676078345194368, -0.252441346243332)

(-50.255533632556485, -0.141054375396673*I)

(31.400004316862624, 0.178435019744746)

(69.10780343225363, 0.120288771309422)

(-56.53982467093041, -0.132985959193641*I)

(9.371475105855954, -0.326196105910348)

(-65.96586619291024, 0.123119796833232*I)

(75.39159174407808, 0.115167248976248)

(-31.400004316862624, -0.178435019744746*I)

(6.202749816793043, 0.400222440722691)

(91.10069857709462, -0.104768953369684)

(65.96586619291024, -0.123119796833232)

(-97.38423800530128, 0.101332776087448*I)

(84.817106677999, -0.108580222480823)

(100.52599111783519, 0.0997368037242384)

(-43.97092649034452, -0.150795754903091*I)

(18.822998918007553, 0.230410584140235)

(-81.67528726703536, -0.110648753785148*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=47.1132774827275x_{1} = 47.1132774827275
x2=40.8284587489214x_{2} = 40.8284587489214
x3=179.067989026352x_{3} = 179.067989026352
x4=34.5430455066495x_{4} = 34.5430455066495
x5=97.3842380053013x_{5} = 97.3842380053013
x6=59.6818828624266x_{6} = 59.6818828624266
x7=72.2497107001058x_{7} = 72.2497107001058
x8=78.5334497119428x_{8} = 78.5334497119428
x9=21.9683925318703x_{9} = 21.9683925318703
x10=28.2566407733299x_{10} = 28.2566407733299
x11=2.97508632168828x_{11} = 2.97508632168828
x12=53.397711687542x_{12} = 53.397711687542
x13=15.6760783451944x_{13} = 15.6760783451944
x14=9.37147510585595x_{14} = 9.37147510585595
x15=91.1006985770946x_{15} = 91.1006985770946
x16=65.9658661929102x_{16} = 65.9658661929102
x17=84.817106677999x_{17} = 84.817106677999
Puntos máximos de la función:
x17=37.6858450405302x_{17} = 37.6858450405302
x17=25.1128337203766x_{17} = 25.1128337203766
x17=94.2424741940464x_{17} = 94.2424741940464
x17=56.5398246709304x_{17} = 56.5398246709304
x17=81.6752872670354x_{17} = 81.6752872670354
x17=50.2555336325565x_{17} = 50.2555336325565
x17=43.9709264903445x_{17} = 43.9709264903445
x17=1683.89336539322x_{17} = 1683.89336539322
x17=12.5264763376692x_{17} = 12.5264763376692
x17=62.8238944845809x_{17} = 62.8238944845809
x17=791.680717136973x_{17} = 791.680717136973
x17=87.9589098892909x_{17} = 87.9589098892909
x17=31.4000043168626x_{17} = 31.4000043168626
x17=69.1078034322536x_{17} = 69.1078034322536
x17=75.3915917440781x_{17} = 75.3915917440781
x17=6.20274981679304x_{17} = 6.20274981679304
x17=100.525991117835x_{17} = 100.525991117835
x17=18.8229989180076x_{17} = 18.8229989180076
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=−23.5193947727501x_{1} = -23.5193947727501
x2=4.48478972694075x_{2} = 4.48478972694075
x3=67.5294323412782x_{3} = 67.5294323412782
x4=−29.8115704591729x_{4} = -29.8115704591729
x5=−86.3822208710534x_{5} = -86.3822208710534
x6=−39.2444199463872x_{6} = -39.2444199463872
x7=58.1022509299587x_{7} = 58.1022509299587
x8=42.3879037163074x_{8} = 42.3879037163074
x9=26.6660146634262x_{9} = 26.6660146634262
x10=92.6661913350059x_{10} = 92.6661913350059
x11=89.5242198850226x_{11} = 89.5242198850226
x12=−4.48478972694075x_{12} = -4.48478972694075
x13=80.0981271693078x_{13} = 80.0981271693078
x14=168.069256951157x_{14} = 168.069256951157
x15=−26.6660146634262x_{15} = -26.6660146634262
x16=−17.2206080366588x_{16} = -17.2206080366588
x17=−83.2401911702273x_{17} = -83.2401911702273
x18=−2134.7117396669x_{18} = -2134.7117396669
x19=76.9560246644883x_{19} = 76.9560246644883
x20=−48.6741377261064x_{20} = -48.6741377261064
x21=23.5193947727501x_{21} = 23.5193947727501
x22=−70.6716835864667x_{22} = -70.6716835864667
x23=−92.6661913350059x_{23} = -92.6661913350059
x24=−45.5311260653452x_{24} = -45.5311260653452
x25=−80.0981271693078x_{25} = -80.0981271693078
x26=−1101.12731692271x_{26} = -1101.12731692271
x27=64.3871167803208x_{27} = 64.3871167803208
x28=17.2206080366588x_{28} = 17.2206080366588
x29=−20.3712140812955x_{29} = -20.3712140812955
x30=−42.3879037163074x_{30} = -42.3879037163074
x31=−36.1006062941223x_{31} = -36.1006062941223
x32=−64.3871167803208x_{32} = -64.3871167803208
x33=−95.8081379339785x_{33} = -95.8081379339785
x34=−51.8169770950755x_{34} = -51.8169770950755
x35=−54.9596737689483x_{35} = -54.9596737689483
x36=83.2401911702273x_{36} = 83.2401911702273
x37=14.0659233939484x_{37} = 14.0659233939484
x38=86.3822208710534x_{38} = 86.3822208710534
x39=7.72360366302726x_{39} = 7.72360366302726
x40=20.3712140812955x_{40} = 20.3712140812955
x41=29.8115704591729x_{41} = 29.8115704591729
x42=−61.2447269949324x_{42} = -61.2447269949324
x43=1829.97717426108x_{43} = 1829.97717426108
x44=73.8138787355599x_{44} = 73.8138787355599
x45=−32.9563680725012x_{45} = -32.9563680725012
x46=−58.1022509299587x_{46} = -58.1022509299587
x47=98.9500620501441x_{47} = 98.9500620501441
x48=39.2444199463872x_{48} = 39.2444199463872
x49=−7.72360366302726x_{49} = -7.72360366302726
x50=70.6716835864667x_{50} = 70.6716835864667
x51=61.2447269949324x_{51} = 61.2447269949324
x52=−73.8138787355599x_{52} = -73.8138787355599
x53=−98.9500620501441x_{53} = -98.9500620501441
x54=32.9563680725012x_{54} = 32.9563680725012
x55=51.8169770950755x_{55} = 51.8169770950755
x56=54.9596737689483x_{56} = 54.9596737689483
x57=45.5311260653452x_{57} = 45.5311260653452
x58=−10.9035394436736x_{58} = -10.9035394436736
x59=10.9035394436736x_{59} = 10.9035394436736
x60=36.1006062941223x_{60} = 36.1006062941223
x61=−67.5294323412782x_{61} = -67.5294323412782
x62=−76.9560246644883x_{62} = -76.9560246644883
x63=48.6741377261064x_{63} = 48.6741377261064
x64=95.8081379339785x_{64} = 95.8081379339785
x65=−89.5242198850226x_{65} = -89.5242198850226
x66=−14.0659233939484x_{66} = -14.0659233939484
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