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Trazado el gráfico de la función/ 5/sqrt(2)*sin(x)**2-5*sqrt(2)*sin(x)*cos(x)-5/sqrt(2)*cos(x)**2

Gráfico de la función y = 5/sqrt(2)*sin(x)**2-5*sqrt(2)*sin(x)*cos(x)-5/sqrt(2)*cos(x)**2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         5      2          ___                   5      2   
f(x) = -----*sin (x) - 5*\/ 2 *sin(x)*cos(x) - -----*cos (x)
         ___                                     ___        
       \/ 2                                    \/ 2
f(x)=(−52sin⁡(x)cos⁡(x)+52sin⁡2(x))−52cos⁡2(x)f{\left(x \right)} = \left(- 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)}\right) - \frac{5}{\sqrt{2}} \cos^{2}{\left(x \right)} 
f = -(5*sqrt(2))*sin(x)*cos(x) + (5/sqrt(2))*sin(x)^2 - 5/sqrt(2)*cos(x)^2
Gráfico de la función
02468-8-6-4-2-1010-1010
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:
(−52sin⁡(x)cos⁡(x)+52sin⁡2(x))−52cos⁡2(x)=0\left(- 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)}\right) - \frac{5}{\sqrt{2}} \cos^{2}{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−2atan⁡(−1+22−2+2)x_{1} = - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x2=2atan⁡(1+2+22+2)x_{2} = 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x3=2atan⁡(−2+1+22−2)x_{3} = 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
x4=2atan⁡(−22+2+1+2)x_{4} = 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}
Solución numérica
x1=−33.3794219443916x_{1} = -33.3794219443916
x2=−99.3528676697772x_{2} = -99.3528676697772
x3=76.5763209312512x_{3} = 76.5763209312512
x4=90.7134878724053x_{4} = 90.7134878724053
x5=27.8816348006094x_{5} = 27.8816348006094
x6=−184.175869316702x_{6} = -184.175869316702
x7=−66.3661448070844x_{7} = -66.3661448070844
x8=−30.2378292908018x_{8} = -30.2378292908018
x9=−19.2422550032375x_{9} = -19.2422550032375
x10=79.717913584841x_{10} = 79.717913584841
x11=−47.5165888855456x_{11} = -47.5165888855456
x12=−52.2289778659303x_{12} = -52.2289778659303
x13=−9.8174770424681x_{13} = -9.8174770424681
x14=10.6028752058656x_{14} = 10.6028752058656
x15=32.5940237809941x_{15} = 32.5940237809941
x16=42.0188017417635x_{16} = 42.0188017417635
x17=4.31968989868597x_{17} = 4.31968989868597
x18=56.1559686829176x_{18} = 56.1559686829176
x19=−3.53429173528852x_{19} = -3.53429173528852
x20=−67.9369411338793x_{20} = -67.9369411338793
x21=93.8550805259951x_{21} = 93.8550805259951
x22=54.5851723561227x_{22} = 54.5851723561227
x23=−8.24668071567321x_{23} = -8.24668071567321
x24=−89.9280897090078x_{24} = -89.9280897090078
x25=46.7311907221482x_{25} = 46.7311907221482
x26=12.1736715326604x_{26} = 12.1736715326604
x27=92.2842841992002x_{27} = 92.2842841992002
x28=70.2931356240716x_{28} = 70.2931356240716
x29=−96.2112750161874x_{29} = -96.2112750161874
x30=13.7444678594553x_{30} = 13.7444678594553
x31=48.3019870489431x_{31} = 48.3019870489431
x32=−69.5077374606742x_{32} = -69.5077374606742
x33=−1.96349540849362x_{33} = -1.96349540849362
x34=−91.4988860358027x_{34} = -91.4988860358027
x35=71.8639319508665x_{35} = 71.8639319508665
x36=−74.2201264410589x_{36} = -74.2201264410589
x37=49.872783375738x_{37} = 49.872783375738
x38=65.5807466436869x_{38} = 65.5807466436869
x39=−17.6714586764426x_{39} = -17.6714586764426
x40=−77.3617190946487x_{40} = -77.3617190946487
x41=87.5718952188155x_{41} = 87.5718952188155
x42=64.009950316892x_{42} = 64.009950316892
x43=−0.392699081698724x_{43} = -0.392699081698724
x44=−82.0741080750334x_{44} = -82.0741080750334
x45=−16.1006623496477x_{45} = -16.1006623496477
x46=62.4391539900971x_{46} = 62.4391539900971
x47=−60.0829594999048x_{47} = -60.0829594999048
x48=−88.3572933822129x_{48} = -88.3572933822129
x49=2.74889357189107x_{49} = 2.74889357189107
x50=−55.3705705195201x_{50} = -55.3705705195201
x51=84.4303025652257x_{51} = 84.4303025652257
x52=−44.3749962319558x_{52} = -44.3749962319558
x53=−97.7820713429823x_{53} = -97.7820713429823
x54=−23.9546439836222x_{54} = -23.9546439836222
x55=57.7267650097125x_{55} = 57.7267650097125
x56=−58.5121631731099x_{56} = -58.5121631731099
x57=35.7356164345839x_{57} = 35.7356164345839
x58=26.3108384738145x_{58} = 26.3108384738145
x59=−75.7909227678538x_{59} = -75.7909227678538
x60=40.4480054149686x_{60} = 40.4480054149686
x61=−31.8086256175967x_{61} = -31.8086256175967
x62=9.03207887907065x_{62} = 9.03207887907065
x63=−53.7997741927252x_{63} = -53.7997741927252
x64=−38.0918109247762x_{64} = -38.0918109247762
x65=−39.6626072515711x_{65} = -39.6626072515711
x66=21.5984494934298x_{66} = 21.5984494934298
x67=−25.5254403104171x_{67} = -25.5254403104171
x68=−36.5210145979813x_{68} = -36.5210145979813
x69=34.164820107789x_{69} = 34.164820107789
x70=18.45685683984x_{70} = 18.45685683984
x71=−83.6449044018282x_{71} = -83.6449044018282
x72=98.5674695063798x_{72} = 98.5674695063798
x73=86.0010988920206x_{73} = 86.0010988920206
x74=−45.9457925587507x_{74} = -45.9457925587507
x75=20.0276531666349x_{75} = 20.0276531666349
x76=43.5895980685584x_{76} = 43.5895980685584
x77=5.89048622548086x_{77} = 5.89048622548086
x78=−22.3838476568273x_{78} = -22.3838476568273
x79=−11.388273369263x_{79} = -11.388273369263
x80=68.7223392972767x_{80} = 68.7223392972767
x81=−80.5033117482384x_{81} = -80.5033117482384
x82=−41.233403578366x_{82} = -41.233403578366
x83=−61.6537558266997x_{83} = -61.6537558266997
x84=24.7400421470196x_{84} = 24.7400421470196
x85=100.138265833175x_{85} = 100.138265833175
x86=78.1471172580461x_{86} = 78.1471172580461
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 (5/sqrt(2))*sin(x)^2 - (5*sqrt(2))*sin(x)*cos(x) - 5/sqrt(2)*cos(x)^2.
−52cos⁡2(0)+(52sin⁡2(0)−52sin⁡(0)cos⁡(0))- \frac{5}{\sqrt{2}} \cos^{2}{\left(0 \right)} + \left(\frac{5}{\sqrt{2}} \sin^{2}{\left(0 \right)} - 5 \sqrt{2} \sin{\left(0 \right)} \cos{\left(0 \right)}\right)
Resultado:
f(0)=−522f{\left(0 \right)} = - \frac{5 \sqrt{2}}{2}
Punto:
(0, -5*sqrt(2)/2)
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
52sin⁡2(x)+52sin⁡(x)cos⁡(x)+1022sin⁡(x)cos⁡(x)−52cos⁡2(x)=05 \sqrt{2} \sin^{2}{\left(x \right)} + 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + 10 \frac{\sqrt{2}}{2} \sin{\left(x \right)} \cos{\left(x \right)} - 5 \sqrt{2} \cos^{2}{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=2atan⁡(−1+22−2+2)x_{1} = 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x2=−2atan⁡(1+2+22+2)x_{2} = - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x3=−2atan⁡(−2+1+22−2)x_{3} = - 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
x4=−2atan⁡(−22+2+1+2)x_{4} = - 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}
Signos de extremos en los puntos:
       /                      ___________\          /      /                      ___________\\   ___         /      /                      ___________\\   ___              /      /                      ___________\\    /      /                      ___________\\ 
       |       ___     ___   /       ___ |         2|      |       ___     ___   /       ___ || \/ 2         2|      |       ___     ___   /       ___ || \/ 2        ___    |      |       ___     ___   /       ___ ||    |      |       ___     ___   /       ___ || 
(2*atan\-1 + \/ 2  + \/ 2 *\/  2 - \/ 2  /, - 5*cos \2*atan\-1 + \/ 2  + \/ 2 *\/  2 - \/ 2  //*----- + 5*sin \2*atan\-1 + \/ 2  + \/ 2 *\/  2 - \/ 2  //*----- - 5*\/ 2 *cos\2*atan\-1 + \/ 2  + \/ 2 *\/  2 - \/ 2  //*sin\2*atan\-1 + \/ 2  + \/ 2 *\/  2 - \/ 2  //)
                                                                                                  2                                                         2                                                                                                           

        /                     ___________\          /      /                     ___________\\   ___         /      /                     ___________\\   ___              /      /                     ___________\\    /      /                     ___________\\ 
        |      ___     ___   /       ___ |         2|      |      ___     ___   /       ___ || \/ 2         2|      |      ___     ___   /       ___ || \/ 2        ___    |      |      ___     ___   /       ___ ||    |      |      ___     ___   /       ___ || 
(-2*atan\1 + \/ 2  + \/ 2 *\/  2 + \/ 2  /, - 5*cos \2*atan\1 + \/ 2  + \/ 2 *\/  2 + \/ 2  //*----- + 5*sin \2*atan\1 + \/ 2  + \/ 2 *\/  2 + \/ 2  //*----- + 5*\/ 2 *cos\2*atan\1 + \/ 2  + \/ 2 *\/  2 + \/ 2  //*sin\2*atan\1 + \/ 2  + \/ 2 *\/  2 + \/ 2  //)
                                                                                                 2                                                        2                                                                                                         

        /                     ___________\          /      /                     ___________\\   ___         /      /                     ___________\\   ___              /      /                     ___________\\    /      /                     ___________\\ 
        |      ___     ___   /       ___ |         2|      |      ___     ___   /       ___ || \/ 2         2|      |      ___     ___   /       ___ || \/ 2        ___    |      |      ___     ___   /       ___ ||    |      |      ___     ___   /       ___ || 
(-2*atan\1 - \/ 2  + \/ 2 *\/  2 - \/ 2  /, - 5*cos \2*atan\1 - \/ 2  + \/ 2 *\/  2 - \/ 2  //*----- + 5*sin \2*atan\1 - \/ 2  + \/ 2 *\/  2 - \/ 2  //*----- + 5*\/ 2 *cos\2*atan\1 - \/ 2  + \/ 2 *\/  2 - \/ 2  //*sin\2*atan\1 - \/ 2  + \/ 2 *\/  2 - \/ 2  //)
                                                                                                 2                                                        2                                                                                                         

        /                     ___________\          /      /                     ___________\\   ___         /      /                     ___________\\   ___              /      /                     ___________\\    /      /                     ___________\\ 
        |      ___     ___   /       ___ |         2|      |      ___     ___   /       ___ || \/ 2         2|      |      ___     ___   /       ___ || \/ 2        ___    |      |      ___     ___   /       ___ ||    |      |      ___     ___   /       ___ || 
(-2*atan\1 + \/ 2  - \/ 2 *\/  2 + \/ 2  /, - 5*cos \2*atan\1 + \/ 2  - \/ 2 *\/  2 + \/ 2  //*----- + 5*sin \2*atan\1 + \/ 2  - \/ 2 *\/  2 + \/ 2  //*----- + 5*\/ 2 *cos\2*atan\1 + \/ 2  - \/ 2 *\/  2 + \/ 2  //*sin\2*atan\1 + \/ 2  - \/ 2 *\/  2 + \/ 2  //)
                                                                                                 2                                                        2


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=−2atan⁡(1+2+22+2)x_{1} = - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x2=−2atan⁡(−22+2+1+2)x_{2} = - 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}
Puntos máximos de la función:
x2=2atan⁡(−1+22−2+2)x_{2} = 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x2=−2atan⁡(−2+1+22−2)x_{2} = - 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
Decrece en los intervalos
[−2atan⁡(1+2+22+2),−2atan⁡(−2+1+22−2)]∪[−2atan⁡(−22+2+1+2),∞)\left[- 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}, - 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}\right] \cup \left[- 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}, \infty\right)
Crece en los intervalos
(−∞,−2atan⁡(1+2+22+2)]\left(-\infty, - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}\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
102(−sin⁡2(x)+2sin⁡(x)cos⁡(x)+cos⁡2(x))=010 \sqrt{2} \left(- \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−2atan⁡(−1+22−2+2)x_{1} = - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x2=2atan⁡(1+2+22+2)x_{2} = 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x3=2atan⁡(−2+1+22−2)x_{3} = 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
x4=2atan⁡(−22+2+1+2)x_{4} = 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}

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
[2atan⁡(1+2+22+2),∞)\left[2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}, \infty\right)
Convexa en los intervalos
(−∞,2atan⁡(−22+2+1+2)]∪[2atan⁡(−2+1+22−2),2atan⁡(1+2+22+2)]\left(-\infty, 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}\right] \cup \left[2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}, 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((−52sin⁡(x)cos⁡(x)+52sin⁡2(x))−52cos⁡2(x))=2⟨−5,5⟩+2⟨−52,0⟩+2⟨0,52⟩\lim_{x \to -\infty}\left(\left(- 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)}\right) - \frac{5}{\sqrt{2}} \cos^{2}{\left(x \right)}\right) = \sqrt{2} \left\langle -5, 5\right\rangle + \sqrt{2} \left\langle - \frac{5}{2}, 0\right\rangle + \sqrt{2} \left\langle 0, \frac{5}{2}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=2⟨−5,5⟩+2⟨−52,0⟩+2⟨0,52⟩y = \sqrt{2} \left\langle -5, 5\right\rangle + \sqrt{2} \left\langle - \frac{5}{2}, 0\right\rangle + \sqrt{2} \left\langle 0, \frac{5}{2}\right\rangle
lim⁡x→∞((−52sin⁡(x)cos⁡(x)+52sin⁡2(x))−52cos⁡2(x))=2⟨−5,5⟩+2⟨−52,0⟩+2⟨0,52⟩\lim_{x \to \infty}\left(\left(- 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)}\right) - \frac{5}{\sqrt{2}} \cos^{2}{\left(x \right)}\right) = \sqrt{2} \left\langle -5, 5\right\rangle + \sqrt{2} \left\langle - \frac{5}{2}, 0\right\rangle + \sqrt{2} \left\langle 0, \frac{5}{2}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=2⟨−5,5⟩+2⟨−52,0⟩+2⟨0,52⟩y = \sqrt{2} \left\langle -5, 5\right\rangle + \sqrt{2} \left\langle - \frac{5}{2}, 0\right\rangle + \sqrt{2} \left\langle 0, \frac{5}{2}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (5/sqrt(2))*sin(x)^2 - (5*sqrt(2))*sin(x)*cos(x) - 5/sqrt(2)*cos(x)^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−52sin⁡(x)cos⁡(x)+52sin⁡2(x))−52cos⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{\left(- 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)}\right) - \frac{5}{\sqrt{2}} \cos^{2}{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((−52sin⁡(x)cos⁡(x)+52sin⁡2(x))−52cos⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{\left(- 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)}\right) - \frac{5}{\sqrt{2}} \cos^{2}{\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:
(−52sin⁡(x)cos⁡(x)+52sin⁡2(x))−52cos⁡2(x)=52sin⁡2(x)+52sin⁡(x)cos⁡(x)−52cos⁡2(x)\left(- 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)}\right) - \frac{5}{\sqrt{2}} \cos^{2}{\left(x \right)} = \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)} + 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} - \frac{5}{\sqrt{2}} \cos^{2}{\left(x \right)}
- No
(−52sin⁡(x)cos⁡(x)+52sin⁡2(x))−52cos⁡2(x)=−52sin⁡2(x)−52sin⁡(x)cos⁡(x)+52cos⁡2(x)\left(- 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)}\right) - \frac{5}{\sqrt{2}} \cos^{2}{\left(x \right)} = - \frac{5}{\sqrt{2}} \sin^{2}{\left(x \right)} - 5 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{5}{\sqrt{2}} \cos^{2}{\left(x \right)}
- No
es decir, función
no es
par ni impar
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