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y=sinx*sqrt(sin^2x)+cosx*sqrt(cos^2x)
Trazado el gráfico de la función/ y=sinx*sqrt(sin^2x)+cosx*sqrt(cos^2x)

Gráfico de la función y = y=sinx*sqrt(sin^2x)+cosx*sqrt(cos^2x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                 _________             _________
                /    2                /    2    
f(x) = sin(x)*\/  sin (x)  + cos(x)*\/  cos (x)
f(x)=sin⁡2(x)sin⁡(x)+cos⁡2(x)cos⁡(x)f{\left(x \right)} = \sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} 
f = sqrt(sin(x)^2)*sin(x) + sqrt(cos(x)^2)*cos(x)
Gráfico de la función
02468-8-6-4-2-10102-2
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:
sin⁡2(x)sin⁡(x)+cos⁡2(x)cos⁡(x)=0\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π4x_{1} = - \frac{\pi}{4}
Solución numérica
x1=−38.484510006475x_{1} = -38.484510006475
x2=18.0641577581413x_{2} = 18.0641577581413
x3=−57.3340659280137x_{3} = -57.3340659280137
x4=30.6305283725005x_{4} = 30.6305283725005
x5=−60.4756585816035x_{5} = -60.4756585816035
x6=−47.9092879672443x_{6} = -47.9092879672443
x7=96.6039740978861x_{7} = 96.6039740978861
x8=52.621676947629x_{8} = 52.621676947629
x9=68.329640215578x_{9} = 68.329640215578
x10=8.63937979737193x_{10} = 8.63937979737193
x11=55.7632696012188x_{11} = 55.7632696012188
x12=62.0464549083984x_{12} = 62.0464549083984
x13=−91.8915851175014x_{13} = -91.8915851175014
x14=−98.174770424681x_{14} = -98.174770424681
x15=27.4889357189107x_{15} = 27.4889357189107
x16=−201.847327993144x_{16} = -201.847327993144
x17=49.4800842940392x_{17} = 49.4800842940392
x18=99.7455667514759x_{18} = 99.7455667514759
x19=71.4712328691678x_{19} = 71.4712328691678
x20=−3.92699081698724x_{20} = -3.92699081698724
x21=93.4623814442964x_{21} = 93.4623814442964
x22=−69.9004365423729x_{22} = -69.9004365423729
x23=−82.4668071567321x_{23} = -82.4668071567321
x24=−54.1924732744239x_{24} = -54.1924732744239
x25=84.037603483527x_{25} = 84.037603483527
x26=−19.6349540849362x_{26} = -19.6349540849362
x27=2.35619449019234x_{27} = 2.35619449019234
x28=−63.6172512351933x_{28} = -63.6172512351933
x29=24.3473430653209x_{29} = 24.3473430653209
x30=−10.2101761241668x_{30} = -10.2101761241668
x31=−79.3252145031423x_{31} = -79.3252145031423
x32=−16.4933614313464x_{32} = -16.4933614313464
x33=−76.1836218495525x_{33} = -76.1836218495525
x34=−13.3517687777566x_{34} = -13.3517687777566
x35=−25.9181393921158x_{35} = -25.9181393921158
x36=−35.3429173528852x_{36} = -35.3429173528852
x37=33.7721210260903x_{37} = 33.7721210260903
x38=5.49778714378214x_{38} = 5.49778714378214
x39=46.3384916404494x_{39} = 46.3384916404494
x40=627.533132554561x_{40} = 627.533132554561
x41=−85.6083998103219x_{41} = -85.6083998103219
x42=−32.2013246992954x_{42} = -32.2013246992954
x43=11.7809724509617x_{43} = 11.7809724509617
x44=21.2057504117311x_{44} = 21.2057504117311
x45=90.3207887907066x_{45} = 90.3207887907066
x46=−41.6261026600648x_{46} = -41.6261026600648
x47=77.7544181763474x_{47} = 77.7544181763474
x48=74.6128255227576x_{48} = 74.6128255227576
x49=40.0553063332699x_{49} = 40.0553063332699
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 sin(x)*sqrt(sin(x)^2) + cos(x)*sqrt(cos(x)^2).
sin⁡2(0)sin⁡(0)+cos⁡2(0)cos⁡(0)\sqrt{\sin^{2}{\left(0 \right)}} \sin{\left(0 \right)} + \sqrt{\cos^{2}{\left(0 \right)}} \cos{\left(0 \right)}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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
−2sin⁡(x)∣cos⁡(x)∣+2cos⁡(x)∣sin⁡(x)∣=0- 2 \sin{\left(x \right)} \left|{\cos{\left(x \right)}}\right| + 2 \cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=85.6233696319782x_{1} = 85.6233696319782
x2=66x_{2} = 66
x3=−31.234571006245x_{3} = -31.234571006245
x4=98x_{4} = 98
x5=−58.1354410675642x_{5} = -58.1354410675642
x6=−24x_{6} = -24
x7=−46x_{7} = -46
x8=22x_{8} = 22
x9=−74x_{9} = -74
x10=64x_{10} = 64
x11=−21.75x_{11} = -21.75
x12=36x_{12} = 36
x13=56.6792735086239x_{13} = 56.6792735086239
x14=51.8060555240722x_{14} = 51.8060555240722
x15=−8x_{15} = -8
x16=−2x_{16} = -2
x17=86x_{17} = 86
x18=−81.6753830585147x_{18} = -81.6753830585147
x19=−64.7389972250886x_{19} = -64.7389972250886
x20=0.815963668261716x_{20} = 0.815963668261716
x21=42x_{21} = 42
x22=−80.1244245561668x_{22} = -80.1244245561668
x23=−18x_{23} = -18
x24=−75.1852705444511x_{24} = -75.1852705444511
x25=73.7935652883146x_{25} = 73.7935652883146
x26=−77.1220108719966x_{26} = -77.1220108719966
x27=−64.8876357947581x_{27} = -64.8876357947581
x28=72.2569395086773x_{28} = 72.2569395086773
x29=−40.7475426173107x_{29} = -40.7475426173107
x30=−37.6953179870105x_{30} = -37.6953179870105
x31=22.1281485034953x_{31} = 22.1281485034953
x32=29.8182608971954x_{32} = 29.8182608971954
x33=−20.8151070724778x_{33} = -20.8151070724778
x34=38x_{34} = 38
x35=42.1996061034036x_{35} = 42.1996061034036
x36=−14.1581158829323x_{36} = -14.1581158829323
x37=−84x_{37} = -84
x38=10x_{38} = 10
x39=−28x_{39} = -28
x40=14x_{40} = 14
x41=−72x_{41} = -72
x42=45.0490290950343x_{42} = 45.0490290950343
x43=0x_{43} = 0
x44=−42.7758123652672x_{44} = -42.7758123652672
x45=12.720495165641x_{45} = 12.720495165641
x46=54x_{46} = 54
x47=−52x_{47} = -52
x48=−12x_{48} = -12
x49=−59.6854188636277x_{49} = -59.6854188636277
x50=32x_{50} = 32
x51=−53.21051316578x_{51} = -53.21051316578
x52=−36.1466670839032x_{52} = -36.1466670839032
x53=−40x_{53} = -40
x54=−86.7044131211503x_{54} = -86.7044131211503
x55=44x_{55} = 44
x56=63.6707650442113x_{56} = 63.6707650442113
x57=−6x_{57} = -6
x58=−50x_{58} = -50
x59=16x_{59} = 16
x60=−78x_{60} = -78
x61=−9.25753963078497x_{61} = -9.25753963078497
x62=−65.75x_{62} = -65.75
x63=4x_{63} = 4
x64=−56x_{64} = -56
x65=78.6598841410821x_{65} = 78.6598841410821
x66=−34x_{66} = -34
x67=28.2757180710112x_{67} = 28.2757180710112
x68=−62x_{68} = -62
x69=−15.7050892844756x_{69} = -15.7050892844756
x70=100.641230697572x_{70} = 100.641230697572
x71=−97.1587363628829x_{71} = -97.1587363628829
x72=−18.8136202642423x_{72} = -18.8136202642423
x73=−68x_{73} = -68
x74=95.7807719870693x_{74} = 95.7807719870693
x75=76x_{75} = 76
x76=−94x_{76} = -94
x77=66.996445860184x_{77} = 66.996445860184
x78=−43.75x_{78} = -43.75
x79=−96x_{79} = -96
x80=82x_{80} = 82
x81=6.28525877373941x_{81} = 6.28525877373941
x82=88.9393875730149x_{82} = 88.9393875730149
x83=7.83019844012797x_{83} = 7.83019844012797
x84=34.6994561309022x_{84} = 34.6994561309022
x85=−30x_{85} = -30
x86=48x_{86} = 48
x87=20x_{87} = 20
x88=58x_{88} = 58
x89=−87.75x_{89} = -87.75
x90=70x_{90} = 70
x91=23.0976732112262x_{91} = 23.0976732112262
x92=60x_{92} = 60
x93=50.2662808147101x_{93} = 50.2662808147101
x94=−90x_{94} = -90
x95=92x_{95} = 92
x96=88x_{96} = 88
x97=1.1428321394192x_{97} = 1.1428321394192
x98=94.25x_{98} = 94.25
x99=−100x_{99} = -100
x100=80x_{100} = 80
x101=26x_{101} = 26
Signos de extremos en los puntos:
(85.62336963197819, -1)

(66, -1)

(-31.234571006244988, 1)

(98, -1)

(-58.135441067564244, -1)

(-24, 1)

(-46, -1)

(22, -1)

(-74, 1)

(64, 1)

(-21.75, -1)

(36, -1)

(56.679273508623886, 1)

(51.806055524072164, 1)

(-8, -1)

(-2, -1)

(86, -1)

(-81.67538305851468, 1)

(-64.73899722508855, -1)

(0.8159636682617161, 1)

(42, -1)

(-80.12442455616677, 1)

(-18, 1)

(-75.18527054445106, 1)

(73.79356528831458, -1)

(-77.1220108719966, -1)

(-64.88763579475807, -1)

(72.2569395086773, -1)

(-40.747542617310664, -1)

(-37.69531798701046, 1)

(22.128148503495254, -1)

(29.818260897195415, -1)

(-20.81510707247777, -1)

(38, 1)

(42.199606103403646, -1)

(-14.158115882932258, -1)

(-84, -1)

(10, -1)

(-28, -1)

(14, 1)

(-72, -1)

(45.04902909503426, 1)

(0, 1)

(-42.77581236526724, 1)

(12.720495165641042, 1)

(54, -1)

(-52, -1)

(-12, 1)

(-59.68541886362766, -1)

(32, 1)

(-53.210513165779965, -1)

(-36.14666708390321, 1)

(-40, -1)

(-86.70441312115032, 1)

(44, 1)

(63.670765044211265, 1)

(-6, 1)

(-50, 1)

(16, -1)

(-78, -1)

(-9.257539630784969, -1)

(-65.75, -1)

(4, -1)

(-56, 1)

(78.65988414108214, -1)

(-34, -1)

(28.275718071011216, -1)

(-62, 1)

(-15.705089284475568, -1)

(100.64123069757201, 1)

(-97.1587363628829, -1)

(-18.81362026424233, 1)

(-68, 1)

(95.7807719870693, 1)

(76, 1)

(-94, 1)

(66.99644586018401, -1)

(-43.75, 1)

(-96, -1)

(82, 1)

(6.2852587737394146, 1)

(88.93938757301494, 1)

(7.830198440127966, 1)

(34.69945613090222, -1)

(-30, 1)

(48, -1)

(20, 1)

(58, 1)

(-87.75, 1)

(70, 1)

(23.097673211226223, -1)

(60, -1)

(50.26628081471005, 1)

(-90, -1)

(92, -1)

(88, 1)

(1.1428321394191958, 1)

(94.25, 1)

(-100, 1)

(80, -1)

(26, 1)


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=64x_{1} = 64
x2=42x_{2} = 42
x3=−75.1852705444511x_{3} = -75.1852705444511
x4=73.7935652883146x_{4} = 73.7935652883146
x5=45.0490290950343x_{5} = 45.0490290950343
x6=12.720495165641x_{6} = 12.720495165641
x7=63.6707650442113x_{7} = 63.6707650442113
x8=82x_{8} = 82
x9=−100x_{9} = -100
Puntos máximos de la función:
x9=56.6792735086239x_{9} = 56.6792735086239
x9=22.1281485034953x_{9} = 22.1281485034953
x9=0x_{9} = 0
x9=−42.7758123652672x_{9} = -42.7758123652672
x9=−86.7044131211503x_{9} = -86.7044131211503
x9=−50x_{9} = -50
x9=−15.7050892844756x_{9} = -15.7050892844756
x9=92x_{9} = 92
Decrece en los intervalos
[82,∞)\left[82, \infty\right)
Crece en los intervalos
(−∞,−100]\left(-\infty, -100\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
2(sin⁡2(x)sign⁡(cos⁡(x))−sin⁡(x)∣sin⁡(x)∣+cos⁡2(x)sign⁡(sin⁡(x))−cos⁡(x)∣cos⁡(x)∣)=02 \left(\sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)} - \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + \cos^{2}{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \left|{\cos{\left(x \right)}}\right|\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=66x_{1} = 66
x2=96.6039740978861x_{2} = 96.6039740978861
x3=98x_{3} = 98
x4=68.329640215578x_{4} = 68.329640215578
x5=−91.8915851175014x_{5} = -91.8915851175014
x6=−24x_{6} = -24
x7=−46x_{7} = -46
x8=99.7455667514759x_{8} = 99.7455667514759
x9=22x_{9} = 22
x10=−74x_{10} = -74
x11=64x_{11} = 64
x12=−3.92699081698724x_{12} = -3.92699081698724
x13=−21.75x_{13} = -21.75
x14=36x_{14} = 36
x15=−54.1924732744239x_{15} = -54.1924732744239
x16=−8x_{16} = -8
x17=−2x_{17} = -2
x18=86x_{18} = 86
x19=−10.2101761241668x_{19} = -10.2101761241668
x20=−76.1836218495525x_{20} = -76.1836218495525
x21=−25.9181393921158x_{21} = -25.9181393921158
x22=42x_{22} = 42
x23=−18x_{23} = -18
x24=77.7544181763474x_{24} = 77.7544181763474
x25=26.5349511925516x_{25} = 26.5349511925516
x26=−57.3340659280137x_{26} = -57.3340659280137
x27=48.4287393324485x_{27} = 48.4287393324485
x28=38x_{28} = 38
x29=−69.9004365423729x_{29} = -69.9004365423729
x30=−82.4668071567321x_{30} = -82.4668071567321
x31=2.35619449019234x_{31} = 2.35619449019234
x32=−84x_{32} = -84
x33=−19.6349540849362x_{33} = -19.6349540849362
x34=0.25x_{34} = 0.25
x35=24.3473430653209x_{35} = 24.3473430653209
x36=10x_{36} = 10
x37=−28x_{37} = -28
x38=14x_{38} = 14
x39=−72x_{39} = -72
x40=−85.6083998103219x_{40} = -85.6083998103219
x41=4.62961173837522x_{41} = 4.62961173837522
x42=90.3207887907066x_{42} = 90.3207887907066
x43=54x_{43} = 54
x44=−52x_{44} = -52
x45=−12x_{45} = -12
x46=74.6128255227576x_{46} = 74.6128255227576
x47=40.0553063332699x_{47} = 40.0553063332699
x48=32x_{48} = 32
x49=30.6305283725005x_{49} = 30.6305283725005
x50=8.63937979737193x_{50} = 8.63937979737193
x51=−40x_{51} = -40
x52=−98.174770424681x_{52} = -98.174770424681
x53=44x_{53} = 44
x54=−6x_{54} = -6
x55=−50x_{55} = -50
x56=16x_{56} = 16
x57=−78x_{57} = -78
x58=−65.75x_{58} = -65.75
x59=4x_{59} = 4
x60=−56x_{60} = -56
x61=−34x_{61} = -34
x62=84.037603483527x_{62} = 84.037603483527
x63=−62x_{63} = -62
x64=−63.6172512351933x_{64} = -63.6172512351933
x65=−35.3429173528852x_{65} = -35.3429173528852
x66=−68x_{66} = -68
x67=33.7721210260903x_{67} = 33.7721210260903
x68=76x_{68} = 76
x69=46.3384916404494x_{69} = 46.3384916404494
x70=−94x_{70} = -94
x71=−32.2013246992954x_{71} = -32.2013246992954
x72=11.7809724509617x_{72} = 11.7809724509617
x73=−41.6261026600648x_{73} = -41.6261026600648
x74=−43.75x_{74} = -43.75
x75=−96x_{75} = -96
x76=−17.2854412916667x_{76} = -17.2854412916667
x77=−38.484510006475x_{77} = -38.484510006475
x78=82x_{78} = 82
x79=18.0641577581413x_{79} = 18.0641577581413
x80=−60.4756585816035x_{80} = -60.4756585816035
x81=−47.9092879672443x_{81} = -47.9092879672443
x82=52.621676947629x_{82} = 52.621676947629
x83=55.7632696012188x_{83} = 55.7632696012188
x84=62.0464549083984x_{84} = 62.0464549083984
x85=−30x_{85} = -30
x86=70.3086442209404x_{86} = 70.3086442209404
x87=48x_{87} = 48
x88=−79.3252145031423x_{88} = -79.3252145031423
x89=−13.3517687777566x_{89} = -13.3517687777566
x90=20x_{90} = 20
x91=58x_{91} = 58
x92=−87.75x_{92} = -87.75
x93=70x_{93} = 70
x94=60x_{94} = 60
x95=92x_{95} = 92
x96=−90x_{96} = -90
x97=88x_{97} = 88
x98=94.25x_{98} = 94.25
x99=−100x_{99} = -100
x100=80x_{100} = 80
x101=26x_{101} = 26

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
[96.6039740978861,∞)\left[96.6039740978861, \infty\right)
Convexa en los intervalos
(−∞,−98.174770424681]\left(-\infty, -98.174770424681\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡2(x)sin⁡(x)+cos⁡2(x)cos⁡(x))=2⟨−1,1⟩∣⟨−1,1⟩∣\lim_{x \to -\infty}\left(\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}\right) = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=2⟨−1,1⟩∣⟨−1,1⟩∣y = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|
lim⁡x→∞(sin⁡2(x)sin⁡(x)+cos⁡2(x)cos⁡(x))=2⟨−1,1⟩∣⟨−1,1⟩∣\lim_{x \to \infty}\left(\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}\right) = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=2⟨−1,1⟩∣⟨−1,1⟩∣y = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)*sqrt(sin(x)^2) + cos(x)*sqrt(cos(x)^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡2(x)sin⁡(x)+cos⁡2(x)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡2(x)sin⁡(x)+cos⁡2(x)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}}{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:
sin⁡2(x)sin⁡(x)+cos⁡2(x)cos⁡(x)=−sin⁡(x)∣sin⁡(x)∣+cos⁡2(x)cos⁡(x)\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} = - \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}
- No
sin⁡2(x)sin⁡(x)+cos⁡2(x)cos⁡(x)=sin⁡(x)∣sin⁡(x)∣−cos⁡2(x)cos⁡(x)\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} = \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| - \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = y=sinx*sqrt(sin^2x)+cosx*sqrt(cos^2x)
    © Sr Examen – Калькулятор