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- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
3/(x^2+1)
-
(1/3)^x
-
x/(x^3+2)
-
y=2x-3
-
Expresiones idénticas
- y=sinx*sqrt(sin^2x)+cosx*sqrt(cos^2x)
- y es igual a seno de x multiplicar por raíz cuadrada de ( seno de al cuadrado x) más coseno de x multiplicar por raíz cuadrada de ( coseno de al cuadrado x)
- y=sinx*√(sin^2x)+cosx*√(cos^2x)
- y=sinx*sqrt(sin2x)+cosx*sqrt(cos2x)
- y=sinx*sqrtsin2x+cosx*sqrtcos2x
- y=sinx*sqrt(sin²x)+cosx*sqrt(cos²x)
- y=sinx*sqrt(sin en el grado 2x)+cosx*sqrt(cos en el grado 2x)
- y=sinxsqrt(sin^2x)+cosxsqrt(cos^2x)
- y=sinxsqrt(sin2x)+cosxsqrt(cos2x)
- y=sinxsqrtsin2x+cosxsqrtcos2x
- y=sinxsqrtsin^2x+cosxsqrtcos^2x
-
Expresiones semejantes
- y=sinx*sqrt(sin^2x)-cosx*sqrt(cos^2x)
-
Expresiones con funciones
- sinx
- sinx+3/4(cos+1)
- sinx-x+1
- sinx/1+cos(x)
- sinx+tgx
- sinx-(24+x)/100
- Raíz cuadrada sqrt
- sqrt(xˆ2-9)
- sqrt(x),x
- sqrt(x^2-x^4)
- sqrt(x)-5*x^2
- sqrt(x(3-x^2))
- Seno sin
- sin(x)/1+cos(x)
- sin(-x+3)
- sin(x)/(-5-2*sin(x))
- sin(4*x)^(2)
- sin(2*x)*log(x)
- cosx
- cosx-(cosx)^2
- cosx/(x^2)
- cosx*cos(2x)
- cosx/(1-2sinx)
- cosx/2+1/2
- Raíz cuadrada sqrt
- sqrt(xˆ2-9)
- sqrt(x),x
- sqrt(x^2-x^4)
- sqrt(x)-5*x^2
- sqrt(x(3-x^2))
- Coseno cos
- cos(x),x
- cos(x)-sqrt(3)*sin(x)
- cos(3x)+3cos(x)
- cos(x)*e^x
- cos(x)+sin(2*x)/2
Gráfico de la función y = y=sinx*sqrt(sin^2x)+cosx*sqrt(cos^2x)
Solución
Ha introducido
[src]
_________ _________
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f(x) = sin(x)*\/ sin (x) + cos(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
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:
$$\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \frac{\pi}{4}$$
Solución numérica
$$x_{1} = -38.484510006475$$
$$x_{2} = 18.0641577581413$$
$$x_{3} = -57.3340659280137$$
$$x_{4} = 30.6305283725005$$
$$x_{5} = -60.4756585816035$$
$$x_{6} = -47.9092879672443$$
$$x_{7} = 96.6039740978861$$
$$x_{8} = 52.621676947629$$
$$x_{9} = 68.329640215578$$
$$x_{10} = 8.63937979737193$$
$$x_{11} = 55.7632696012188$$
$$x_{12} = 62.0464549083984$$
$$x_{13} = -91.8915851175014$$
$$x_{14} = -98.174770424681$$
$$x_{15} = 27.4889357189107$$
$$x_{16} = -201.847327993144$$
$$x_{17} = 49.4800842940392$$
$$x_{18} = 99.7455667514759$$
$$x_{19} = 71.4712328691678$$
$$x_{20} = -3.92699081698724$$
$$x_{21} = 93.4623814442964$$
$$x_{22} = -69.9004365423729$$
$$x_{23} = -82.4668071567321$$
$$x_{24} = -54.1924732744239$$
$$x_{25} = 84.037603483527$$
$$x_{26} = -19.6349540849362$$
$$x_{27} = 2.35619449019234$$
$$x_{28} = -63.6172512351933$$
$$x_{29} = 24.3473430653209$$
$$x_{30} = -10.2101761241668$$
$$x_{31} = -79.3252145031423$$
$$x_{32} = -16.4933614313464$$
$$x_{33} = -76.1836218495525$$
$$x_{34} = -13.3517687777566$$
$$x_{35} = -25.9181393921158$$
$$x_{36} = -35.3429173528852$$
$$x_{37} = 33.7721210260903$$
$$x_{38} = 5.49778714378214$$
$$x_{39} = 46.3384916404494$$
$$x_{40} = 627.533132554561$$
$$x_{41} = -85.6083998103219$$
$$x_{42} = -32.2013246992954$$
$$x_{43} = 11.7809724509617$$
$$x_{44} = 21.2057504117311$$
$$x_{45} = 90.3207887907066$$
$$x_{46} = -41.6261026600648$$
$$x_{47} = 77.7544181763474$$
$$x_{48} = 74.6128255227576$$
$$x_{49} = 40.0553063332699$$
o sea hay que resolver la ecuación:
$$\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \frac{\pi}{4}$$
Solución numérica
$$x_{1} = -38.484510006475$$
$$x_{2} = 18.0641577581413$$
$$x_{3} = -57.3340659280137$$
$$x_{4} = 30.6305283725005$$
$$x_{5} = -60.4756585816035$$
$$x_{6} = -47.9092879672443$$
$$x_{7} = 96.6039740978861$$
$$x_{8} = 52.621676947629$$
$$x_{9} = 68.329640215578$$
$$x_{10} = 8.63937979737193$$
$$x_{11} = 55.7632696012188$$
$$x_{12} = 62.0464549083984$$
$$x_{13} = -91.8915851175014$$
$$x_{14} = -98.174770424681$$
$$x_{15} = 27.4889357189107$$
$$x_{16} = -201.847327993144$$
$$x_{17} = 49.4800842940392$$
$$x_{18} = 99.7455667514759$$
$$x_{19} = 71.4712328691678$$
$$x_{20} = -3.92699081698724$$
$$x_{21} = 93.4623814442964$$
$$x_{22} = -69.9004365423729$$
$$x_{23} = -82.4668071567321$$
$$x_{24} = -54.1924732744239$$
$$x_{25} = 84.037603483527$$
$$x_{26} = -19.6349540849362$$
$$x_{27} = 2.35619449019234$$
$$x_{28} = -63.6172512351933$$
$$x_{29} = 24.3473430653209$$
$$x_{30} = -10.2101761241668$$
$$x_{31} = -79.3252145031423$$
$$x_{32} = -16.4933614313464$$
$$x_{33} = -76.1836218495525$$
$$x_{34} = -13.3517687777566$$
$$x_{35} = -25.9181393921158$$
$$x_{36} = -35.3429173528852$$
$$x_{37} = 33.7721210260903$$
$$x_{38} = 5.49778714378214$$
$$x_{39} = 46.3384916404494$$
$$x_{40} = 627.533132554561$$
$$x_{41} = -85.6083998103219$$
$$x_{42} = -32.2013246992954$$
$$x_{43} = 11.7809724509617$$
$$x_{44} = 21.2057504117311$$
$$x_{45} = 90.3207887907066$$
$$x_{46} = -41.6261026600648$$
$$x_{47} = 77.7544181763474$$
$$x_{48} = 74.6128255227576$$
$$x_{49} = 40.0553063332699$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- 2 \sin{\left(x \right)} \left|{\cos{\left(x \right)}}\right| + 2 \cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 85.6233696319782$$
$$x_{2} = 66$$
$$x_{3} = -31.234571006245$$
$$x_{4} = 98$$
$$x_{5} = -58.1354410675642$$
$$x_{6} = -24$$
$$x_{7} = -46$$
$$x_{8} = 22$$
$$x_{9} = -74$$
$$x_{10} = 64$$
$$x_{11} = -21.75$$
$$x_{12} = 36$$
$$x_{13} = 56.6792735086239$$
$$x_{14} = 51.8060555240722$$
$$x_{15} = -8$$
$$x_{16} = -2$$
$$x_{17} = 86$$
$$x_{18} = -81.6753830585147$$
$$x_{19} = -64.7389972250886$$
$$x_{20} = 0.815963668261716$$
$$x_{21} = 42$$
$$x_{22} = -80.1244245561668$$
$$x_{23} = -18$$
$$x_{24} = -75.1852705444511$$
$$x_{25} = 73.7935652883146$$
$$x_{26} = -77.1220108719966$$
$$x_{27} = -64.8876357947581$$
$$x_{28} = 72.2569395086773$$
$$x_{29} = -40.7475426173107$$
$$x_{30} = -37.6953179870105$$
$$x_{31} = 22.1281485034953$$
$$x_{32} = 29.8182608971954$$
$$x_{33} = -20.8151070724778$$
$$x_{34} = 38$$
$$x_{35} = 42.1996061034036$$
$$x_{36} = -14.1581158829323$$
$$x_{37} = -84$$
$$x_{38} = 10$$
$$x_{39} = -28$$
$$x_{40} = 14$$
$$x_{41} = -72$$
$$x_{42} = 45.0490290950343$$
$$x_{43} = 0$$
$$x_{44} = -42.7758123652672$$
$$x_{45} = 12.720495165641$$
$$x_{46} = 54$$
$$x_{47} = -52$$
$$x_{48} = -12$$
$$x_{49} = -59.6854188636277$$
$$x_{50} = 32$$
$$x_{51} = -53.21051316578$$
$$x_{52} = -36.1466670839032$$
$$x_{53} = -40$$
$$x_{54} = -86.7044131211503$$
$$x_{55} = 44$$
$$x_{56} = 63.6707650442113$$
$$x_{57} = -6$$
$$x_{58} = -50$$
$$x_{59} = 16$$
$$x_{60} = -78$$
$$x_{61} = -9.25753963078497$$
$$x_{62} = -65.75$$
$$x_{63} = 4$$
$$x_{64} = -56$$
$$x_{65} = 78.6598841410821$$
$$x_{66} = -34$$
$$x_{67} = 28.2757180710112$$
$$x_{68} = -62$$
$$x_{69} = -15.7050892844756$$
$$x_{70} = 100.641230697572$$
$$x_{71} = -97.1587363628829$$
$$x_{72} = -18.8136202642423$$
$$x_{73} = -68$$
$$x_{74} = 95.7807719870693$$
$$x_{75} = 76$$
$$x_{76} = -94$$
$$x_{77} = 66.996445860184$$
$$x_{78} = -43.75$$
$$x_{79} = -96$$
$$x_{80} = 82$$
$$x_{81} = 6.28525877373941$$
$$x_{82} = 88.9393875730149$$
$$x_{83} = 7.83019844012797$$
$$x_{84} = 34.6994561309022$$
$$x_{85} = -30$$
$$x_{86} = 48$$
$$x_{87} = 20$$
$$x_{88} = 58$$
$$x_{89} = -87.75$$
$$x_{90} = 70$$
$$x_{91} = 23.0976732112262$$
$$x_{92} = 60$$
$$x_{93} = 50.2662808147101$$
$$x_{94} = -90$$
$$x_{95} = 92$$
$$x_{96} = 88$$
$$x_{97} = 1.1428321394192$$
$$x_{98} = 94.25$$
$$x_{99} = -100$$
$$x_{100} = 80$$
$$x_{101} = 26$$
Signos de extremos en los puntos:
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:
$$x_{1} = 64$$
$$x_{2} = 42$$
$$x_{3} = -75.1852705444511$$
$$x_{4} = 73.7935652883146$$
$$x_{5} = 45.0490290950343$$
$$x_{6} = 12.720495165641$$
$$x_{7} = 63.6707650442113$$
$$x_{8} = 82$$
$$x_{9} = -100$$
Puntos máximos de la función:
$$x_{9} = 56.6792735086239$$
$$x_{9} = 22.1281485034953$$
$$x_{9} = 0$$
$$x_{9} = -42.7758123652672$$
$$x_{9} = -86.7044131211503$$
$$x_{9} = -50$$
$$x_{9} = -15.7050892844756$$
$$x_{9} = 92$$
Decrece en los intervalos
$$\left[82, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- 2 \sin{\left(x \right)} \left|{\cos{\left(x \right)}}\right| + 2 \cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 85.6233696319782$$
$$x_{2} = 66$$
$$x_{3} = -31.234571006245$$
$$x_{4} = 98$$
$$x_{5} = -58.1354410675642$$
$$x_{6} = -24$$
$$x_{7} = -46$$
$$x_{8} = 22$$
$$x_{9} = -74$$
$$x_{10} = 64$$
$$x_{11} = -21.75$$
$$x_{12} = 36$$
$$x_{13} = 56.6792735086239$$
$$x_{14} = 51.8060555240722$$
$$x_{15} = -8$$
$$x_{16} = -2$$
$$x_{17} = 86$$
$$x_{18} = -81.6753830585147$$
$$x_{19} = -64.7389972250886$$
$$x_{20} = 0.815963668261716$$
$$x_{21} = 42$$
$$x_{22} = -80.1244245561668$$
$$x_{23} = -18$$
$$x_{24} = -75.1852705444511$$
$$x_{25} = 73.7935652883146$$
$$x_{26} = -77.1220108719966$$
$$x_{27} = -64.8876357947581$$
$$x_{28} = 72.2569395086773$$
$$x_{29} = -40.7475426173107$$
$$x_{30} = -37.6953179870105$$
$$x_{31} = 22.1281485034953$$
$$x_{32} = 29.8182608971954$$
$$x_{33} = -20.8151070724778$$
$$x_{34} = 38$$
$$x_{35} = 42.1996061034036$$
$$x_{36} = -14.1581158829323$$
$$x_{37} = -84$$
$$x_{38} = 10$$
$$x_{39} = -28$$
$$x_{40} = 14$$
$$x_{41} = -72$$
$$x_{42} = 45.0490290950343$$
$$x_{43} = 0$$
$$x_{44} = -42.7758123652672$$
$$x_{45} = 12.720495165641$$
$$x_{46} = 54$$
$$x_{47} = -52$$
$$x_{48} = -12$$
$$x_{49} = -59.6854188636277$$
$$x_{50} = 32$$
$$x_{51} = -53.21051316578$$
$$x_{52} = -36.1466670839032$$
$$x_{53} = -40$$
$$x_{54} = -86.7044131211503$$
$$x_{55} = 44$$
$$x_{56} = 63.6707650442113$$
$$x_{57} = -6$$
$$x_{58} = -50$$
$$x_{59} = 16$$
$$x_{60} = -78$$
$$x_{61} = -9.25753963078497$$
$$x_{62} = -65.75$$
$$x_{63} = 4$$
$$x_{64} = -56$$
$$x_{65} = 78.6598841410821$$
$$x_{66} = -34$$
$$x_{67} = 28.2757180710112$$
$$x_{68} = -62$$
$$x_{69} = -15.7050892844756$$
$$x_{70} = 100.641230697572$$
$$x_{71} = -97.1587363628829$$
$$x_{72} = -18.8136202642423$$
$$x_{73} = -68$$
$$x_{74} = 95.7807719870693$$
$$x_{75} = 76$$
$$x_{76} = -94$$
$$x_{77} = 66.996445860184$$
$$x_{78} = -43.75$$
$$x_{79} = -96$$
$$x_{80} = 82$$
$$x_{81} = 6.28525877373941$$
$$x_{82} = 88.9393875730149$$
$$x_{83} = 7.83019844012797$$
$$x_{84} = 34.6994561309022$$
$$x_{85} = -30$$
$$x_{86} = 48$$
$$x_{87} = 20$$
$$x_{88} = 58$$
$$x_{89} = -87.75$$
$$x_{90} = 70$$
$$x_{91} = 23.0976732112262$$
$$x_{92} = 60$$
$$x_{93} = 50.2662808147101$$
$$x_{94} = -90$$
$$x_{95} = 92$$
$$x_{96} = 88$$
$$x_{97} = 1.1428321394192$$
$$x_{98} = 94.25$$
$$x_{99} = -100$$
$$x_{100} = 80$$
$$x_{101} = 26$$
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:
$$x_{1} = 64$$
$$x_{2} = 42$$
$$x_{3} = -75.1852705444511$$
$$x_{4} = 73.7935652883146$$
$$x_{5} = 45.0490290950343$$
$$x_{6} = 12.720495165641$$
$$x_{7} = 63.6707650442113$$
$$x_{8} = 82$$
$$x_{9} = -100$$
Puntos máximos de la función:
$$x_{9} = 56.6792735086239$$
$$x_{9} = 22.1281485034953$$
$$x_{9} = 0$$
$$x_{9} = -42.7758123652672$$
$$x_{9} = -86.7044131211503$$
$$x_{9} = -50$$
$$x_{9} = -15.7050892844756$$
$$x_{9} = 92$$
Decrece en los intervalos
$$\left[82, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$2 \left(\sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)} - \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + \cos^{2}{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \left|{\cos{\left(x \right)}}\right|\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 66$$
$$x_{2} = 96.6039740978861$$
$$x_{3} = 98$$
$$x_{4} = 68.329640215578$$
$$x_{5} = -91.8915851175014$$
$$x_{6} = -24$$
$$x_{7} = -46$$
$$x_{8} = 99.7455667514759$$
$$x_{9} = 22$$
$$x_{10} = -74$$
$$x_{11} = 64$$
$$x_{12} = -3.92699081698724$$
$$x_{13} = -21.75$$
$$x_{14} = 36$$
$$x_{15} = -54.1924732744239$$
$$x_{16} = -8$$
$$x_{17} = -2$$
$$x_{18} = 86$$
$$x_{19} = -10.2101761241668$$
$$x_{20} = -76.1836218495525$$
$$x_{21} = -25.9181393921158$$
$$x_{22} = 42$$
$$x_{23} = -18$$
$$x_{24} = 77.7544181763474$$
$$x_{25} = 26.5349511925516$$
$$x_{26} = -57.3340659280137$$
$$x_{27} = 48.4287393324485$$
$$x_{28} = 38$$
$$x_{29} = -69.9004365423729$$
$$x_{30} = -82.4668071567321$$
$$x_{31} = 2.35619449019234$$
$$x_{32} = -84$$
$$x_{33} = -19.6349540849362$$
$$x_{34} = 0.25$$
$$x_{35} = 24.3473430653209$$
$$x_{36} = 10$$
$$x_{37} = -28$$
$$x_{38} = 14$$
$$x_{39} = -72$$
$$x_{40} = -85.6083998103219$$
$$x_{41} = 4.62961173837522$$
$$x_{42} = 90.3207887907066$$
$$x_{43} = 54$$
$$x_{44} = -52$$
$$x_{45} = -12$$
$$x_{46} = 74.6128255227576$$
$$x_{47} = 40.0553063332699$$
$$x_{48} = 32$$
$$x_{49} = 30.6305283725005$$
$$x_{50} = 8.63937979737193$$
$$x_{51} = -40$$
$$x_{52} = -98.174770424681$$
$$x_{53} = 44$$
$$x_{54} = -6$$
$$x_{55} = -50$$
$$x_{56} = 16$$
$$x_{57} = -78$$
$$x_{58} = -65.75$$
$$x_{59} = 4$$
$$x_{60} = -56$$
$$x_{61} = -34$$
$$x_{62} = 84.037603483527$$
$$x_{63} = -62$$
$$x_{64} = -63.6172512351933$$
$$x_{65} = -35.3429173528852$$
$$x_{66} = -68$$
$$x_{67} = 33.7721210260903$$
$$x_{68} = 76$$
$$x_{69} = 46.3384916404494$$
$$x_{70} = -94$$
$$x_{71} = -32.2013246992954$$
$$x_{72} = 11.7809724509617$$
$$x_{73} = -41.6261026600648$$
$$x_{74} = -43.75$$
$$x_{75} = -96$$
$$x_{76} = -17.2854412916667$$
$$x_{77} = -38.484510006475$$
$$x_{78} = 82$$
$$x_{79} = 18.0641577581413$$
$$x_{80} = -60.4756585816035$$
$$x_{81} = -47.9092879672443$$
$$x_{82} = 52.621676947629$$
$$x_{83} = 55.7632696012188$$
$$x_{84} = 62.0464549083984$$
$$x_{85} = -30$$
$$x_{86} = 70.3086442209404$$
$$x_{87} = 48$$
$$x_{88} = -79.3252145031423$$
$$x_{89} = -13.3517687777566$$
$$x_{90} = 20$$
$$x_{91} = 58$$
$$x_{92} = -87.75$$
$$x_{93} = 70$$
$$x_{94} = 60$$
$$x_{95} = 92$$
$$x_{96} = -90$$
$$x_{97} = 88$$
$$x_{98} = 94.25$$
$$x_{99} = -100$$
$$x_{100} = 80$$
$$x_{101} = 26$$
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
$$\left[96.6039740978861, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -98.174770424681\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$2 \left(\sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)} - \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + \cos^{2}{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} - \cos{\left(x \right)} \left|{\cos{\left(x \right)}}\right|\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 66$$
$$x_{2} = 96.6039740978861$$
$$x_{3} = 98$$
$$x_{4} = 68.329640215578$$
$$x_{5} = -91.8915851175014$$
$$x_{6} = -24$$
$$x_{7} = -46$$
$$x_{8} = 99.7455667514759$$
$$x_{9} = 22$$
$$x_{10} = -74$$
$$x_{11} = 64$$
$$x_{12} = -3.92699081698724$$
$$x_{13} = -21.75$$
$$x_{14} = 36$$
$$x_{15} = -54.1924732744239$$
$$x_{16} = -8$$
$$x_{17} = -2$$
$$x_{18} = 86$$
$$x_{19} = -10.2101761241668$$
$$x_{20} = -76.1836218495525$$
$$x_{21} = -25.9181393921158$$
$$x_{22} = 42$$
$$x_{23} = -18$$
$$x_{24} = 77.7544181763474$$
$$x_{25} = 26.5349511925516$$
$$x_{26} = -57.3340659280137$$
$$x_{27} = 48.4287393324485$$
$$x_{28} = 38$$
$$x_{29} = -69.9004365423729$$
$$x_{30} = -82.4668071567321$$
$$x_{31} = 2.35619449019234$$
$$x_{32} = -84$$
$$x_{33} = -19.6349540849362$$
$$x_{34} = 0.25$$
$$x_{35} = 24.3473430653209$$
$$x_{36} = 10$$
$$x_{37} = -28$$
$$x_{38} = 14$$
$$x_{39} = -72$$
$$x_{40} = -85.6083998103219$$
$$x_{41} = 4.62961173837522$$
$$x_{42} = 90.3207887907066$$
$$x_{43} = 54$$
$$x_{44} = -52$$
$$x_{45} = -12$$
$$x_{46} = 74.6128255227576$$
$$x_{47} = 40.0553063332699$$
$$x_{48} = 32$$
$$x_{49} = 30.6305283725005$$
$$x_{50} = 8.63937979737193$$
$$x_{51} = -40$$
$$x_{52} = -98.174770424681$$
$$x_{53} = 44$$
$$x_{54} = -6$$
$$x_{55} = -50$$
$$x_{56} = 16$$
$$x_{57} = -78$$
$$x_{58} = -65.75$$
$$x_{59} = 4$$
$$x_{60} = -56$$
$$x_{61} = -34$$
$$x_{62} = 84.037603483527$$
$$x_{63} = -62$$
$$x_{64} = -63.6172512351933$$
$$x_{65} = -35.3429173528852$$
$$x_{66} = -68$$
$$x_{67} = 33.7721210260903$$
$$x_{68} = 76$$
$$x_{69} = 46.3384916404494$$
$$x_{70} = -94$$
$$x_{71} = -32.2013246992954$$
$$x_{72} = 11.7809724509617$$
$$x_{73} = -41.6261026600648$$
$$x_{74} = -43.75$$
$$x_{75} = -96$$
$$x_{76} = -17.2854412916667$$
$$x_{77} = -38.484510006475$$
$$x_{78} = 82$$
$$x_{79} = 18.0641577581413$$
$$x_{80} = -60.4756585816035$$
$$x_{81} = -47.9092879672443$$
$$x_{82} = 52.621676947629$$
$$x_{83} = 55.7632696012188$$
$$x_{84} = 62.0464549083984$$
$$x_{85} = -30$$
$$x_{86} = 70.3086442209404$$
$$x_{87} = 48$$
$$x_{88} = -79.3252145031423$$
$$x_{89} = -13.3517687777566$$
$$x_{90} = 20$$
$$x_{91} = 58$$
$$x_{92} = -87.75$$
$$x_{93} = 70$$
$$x_{94} = 60$$
$$x_{95} = 92$$
$$x_{96} = -90$$
$$x_{97} = 88$$
$$x_{98} = 94.25$$
$$x_{99} = -100$$
$$x_{100} = 80$$
$$x_{101} = 26$$
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
$$\left[96.6039740978861, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -98.174770424681\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}\right) = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty}\left(\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}\right) = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to -\infty}\left(\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}\right) = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty}\left(\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}\right) = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 2 \left\langle -1, 1\right\rangle \left|{\left\langle -1, 1\right\rangle}\right|$$
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
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}}{x}\right) = 0$$
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:
$$\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} = - \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}$$
- No
$$\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} = \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| - \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} = - \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}$$
- No
$$\sqrt{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} + \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)} = \sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right| - \sqrt{\cos^{2}{\left(x \right)}} \cos{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Gráfico