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Trazado el gráfico de la función/ cos2xcos3x-sin2xsin3x

Gráfico de la función y = cos2xcos3x-sin2xsin3x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = cos(2*x)*cos(3*x) - sin(2*x)*sin(3*x)
f(x)=−sin⁡(2x)sin⁡(3x)+cos⁡(2x)cos⁡(3x)f{\left(x \right)} = - \sin{\left(2 x \right)} \sin{\left(3 x \right)} + \cos{\left(2 x \right)} \cos{\left(3 x \right)} 
f = -sin(2*x)*sin(3*x) + cos(2*x)*cos(3*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⁡(2x)sin⁡(3x)+cos⁡(2x)cos⁡(3x)=0- \sin{\left(2 x \right)} \sin{\left(3 x \right)} + \cos{\left(2 x \right)} \cos{\left(3 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π2x_{1} = - \frac{\pi}{2}
x2=−π10x_{2} = - \frac{\pi}{10}
x3=π10x_{3} = \frac{\pi}{10}
x4=π2x_{4} = \frac{\pi}{2}
x5=−ilog⁡(−25+58+105+58+i4+5i4)x_{5} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x6=−ilog⁡(−25+58+105+58−5i4−i4)x_{6} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
x7=−ilog⁡(−105+516−105−516−25+516+25−516−i4+5i4)x_{7} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x8=−ilog⁡(−105+516−105−516−25+516+25−516−5i4+i4)x_{8} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} + \frac{i}{4} \right)}
x9=−ilog⁡(−105+516−25+516−25−516+105−516+i4+5i4)x_{9} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x10=−ilog⁡(−105+516−25+516−25−516+105−516−5i4−i4)x_{10} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
Solución numérica
x1=−26.0752190247953x_{1} = -26.0752190247953
x2=−81.9955682586936x_{2} = -81.9955682586936
x3=−97.7035315266426x_{3} = -97.7035315266426
x4=92.0486647501809x_{4} = 92.0486647501809
x5=14.1371669411541x_{5} = 14.1371669411541
x6=58.1194640914112x_{6} = 58.1194640914112
x7=24.1902634326414x_{7} = 24.1902634326414
x8=17.9070781254618x_{8} = 17.9070781254618
x9=95.1902574037707x_{9} = 95.1902574037707
x10=38.0132711084365x_{10} = 38.0132711084365
x11=36.1283155162826x_{11} = 36.1283155162826
x12=−7.85398163397448x_{12} = -7.85398163397448
x13=−58.1194640914112x_{13} = -58.1194640914112
x14=−102.730079772386x_{14} = -102.730079772386
x15=46.18141200777x_{15} = 46.18141200777
x16=100.216805649514x_{16} = 100.216805649514
x17=27.3318560862312x_{17} = 27.3318560862312
x18=12.2522113490002x_{18} = 12.2522113490002
x19=−69.4291976443344x_{19} = -69.4291976443344
x20=−11.6238928182822x_{20} = -11.6238928182822
x21=0.314159265358979x_{21} = 0.314159265358979
x22=−75.712382951514x_{22} = -75.712382951514
x23=−36.1283155162826x_{23} = -36.1283155162826
x24=61.8893752757189x_{24} = 61.8893752757189
x25=32.3584043319749x_{25} = 32.3584043319749
x26=−65.6592864600267x_{26} = -65.6592864600267
x27=44.2964564156161x_{27} = 44.2964564156161
x28=−85.7654794430014x_{28} = -85.7654794430014
x29=−4.08407044966673x_{29} = -4.08407044966673
x30=−14.1371669411541x_{30} = -14.1371669411541
x31=80.1106126665397x_{31} = 80.1106126665397
x32=78.2256570743859x_{32} = 78.2256570743859
x33=−61.8893752757189x_{33} = -61.8893752757189
x34=56.2345084992573x_{34} = 56.2345084992573
x35=−93.9336203423348x_{35} = -93.9336203423348
x36=−92.0486647501809x_{36} = -92.0486647501809
x37=54.3495529071034x_{37} = 54.3495529071034
x38=90.1637091580271x_{38} = 90.1637091580271
x39=93.9336203423348x_{39} = 93.9336203423348
x40=72.5707902979242x_{40} = 72.5707902979242
x41=83.8805238508475x_{41} = 83.8805238508475
x42=−51.8362787842316x_{42} = -51.8362787842316
x43=−103.358398303104x_{43} = -103.358398303104
x44=−29.845130209103x_{44} = -29.845130209103
x45=60.0044196835651x_{45} = 60.0044196835651
x46=−39.8982267005904x_{46} = -39.8982267005904
x47=−60.0044196835651x_{47} = -60.0044196835651
x48=5.96902604182061x_{48} = 5.96902604182061
x49=−31.7300858012569x_{49} = -31.7300858012569
x50=7.22566310325652x_{50} = 7.22566310325652
x51=16.0221225333079x_{51} = 16.0221225333079
x52=21.0486707790516x_{52} = 21.0486707790516
x53=76.340701482232x_{53} = 76.340701482232
x54=−95.8185759344887x_{54} = -95.8185759344887
x55=65.0309679293087x_{55} = 65.0309679293087
x56=68.1725605828985x_{56} = 68.1725605828985
x57=70.0575161750524x_{57} = 70.0575161750524
x58=−83.8805238508475x_{58} = -83.8805238508475
x59=22.3053078404875x_{59} = 22.3053078404875
x60=−53.0929158456675x_{60} = -53.0929158456675
x61=66.2876049907446x_{61} = 66.2876049907446
x62=−49.9513231920777x_{62} = -49.9513231920777
x63=−5.96902604182061x_{63} = -5.96902604182061
x64=−43.6681378848981x_{64} = -43.6681378848981
x65=71.9424717672063x_{65} = 71.9424717672063
x66=48.0663675999238x_{66} = 48.0663675999238
x67=−9.73893722612836x_{67} = -9.73893722612836
x68=−63.7743308678728x_{68} = -63.7743308678728
x69=98.3318500573605x_{69} = 98.3318500573605
x70=88.2787535658732x_{70} = 88.2787535658732
x71=−17.9070781254618x_{71} = -17.9070781254618
x72=27.9601746169492x_{72} = 27.9601746169492
x73=34.2433599241287x_{73} = 34.2433599241287
x74=−16.0221225333079x_{74} = -16.0221225333079
x75=−76.9690200129499x_{75} = -76.9690200129499
x76=−73.8274273593601x_{76} = -73.8274273593601
x77=−19.7920337176157x_{77} = -19.7920337176157
x78=88.9070720965912x_{78} = 88.9070720965912
x79=81.9955682586936x_{79} = 81.9955682586936
x80=49.9513231920777x_{80} = 49.9513231920777
x81=−87.6504350351552x_{81} = -87.6504350351552
x82=26.0752190247953x_{82} = 26.0752190247953
x83=−21.6769893097696x_{83} = -21.6769893097696
x84=−71.9424717672063x_{84} = -71.9424717672063
x85=2.19911485751286x_{85} = 2.19911485751286
x86=−27.9601746169492x_{86} = -27.9601746169492
x87=10.3672557568463x_{87} = 10.3672557568463
x88=39.8982267005904x_{88} = 39.8982267005904
x89=−33.6150413934108x_{89} = -33.6150413934108
x90=−70.0575161750524x_{90} = -70.0575161750524
x91=−41.7831822927443x_{91} = -41.7831822927443
x92=4.08407044966673x_{92} = 4.08407044966673
x93=−53.7212343763855x_{93} = -53.7212343763855
x94=−80.1106126665397x_{94} = -80.1106126665397
x95=−38.0132711084365x_{95} = -38.0132711084365
x96=−48.0663675999238x_{96} = -48.0663675999238
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(2*x)*cos(3*x) - sin(2*x)*sin(3*x).
−sin⁡(0⋅2)sin⁡(0⋅3)+cos⁡(0⋅2)cos⁡(0⋅3)- \sin{\left(0 \cdot 2 \right)} \sin{\left(0 \cdot 3 \right)} + \cos{\left(0 \cdot 2 \right)} \cos{\left(0 \cdot 3 \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
−5sin⁡(2x)cos⁡(3x)−5sin⁡(3x)cos⁡(2x)=0- 5 \sin{\left(2 x \right)} \cos{\left(3 x \right)} - 5 \sin{\left(3 x \right)} \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=−9π5x_{2} = - \frac{9 \pi}{5}
x3=−8π5x_{3} = - \frac{8 \pi}{5}
x4=−6π5x_{4} = - \frac{6 \pi}{5}
x5=−πx_{5} = - \pi
x6=−4π5x_{6} = - \frac{4 \pi}{5}
x7=−2π5x_{7} = - \frac{2 \pi}{5}
x8=π5x_{8} = \frac{\pi}{5}
x9=2π5x_{9} = \frac{2 \pi}{5}
x10=4π5x_{10} = \frac{4 \pi}{5}
x11=πx_{11} = \pi
x12=6π5x_{12} = \frac{6 \pi}{5}
x13=8π5x_{13} = \frac{8 \pi}{5}
x14=2πx_{14} = 2 \pi
x15=−2ilog⁡(−105+58+25+58+i4+5i4)x_{15} = - 2 i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x16=−2ilog⁡(−25+58+105+58−5i4−i4)x_{16} = - 2 i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
x17=−2ilog⁡(−105−516+25−516+25+516+105+516+i4+5i4)x_{17} = - 2 i \log{\left(- \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x18=−2ilog⁡(−105+516−105−516−25+516+25−516−i4+5i4)x_{18} = - 2 i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x19=−2ilog⁡(−105+516−25+516−25−516+105−516−5i4−i4)x_{19} = - 2 i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
x20=−2ilog⁡(−25−516+25+516+105−516+105+516−5i4+i4)x_{20} = - 2 i \log{\left(- \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{5} i}{4} + \frac{i}{4} \right)}
Signos de extremos en los puntos:
(0, 1)

                ___   /        ___\ /      ___\ 
 -9*pi    5   \/ 5    |  1   \/ 5 | |1   \/ 5 | 
(-----, - - - ----- + |- - + -----|*|- - -----|)
   5      8     8     \  4     4  / \4     4  / 

                         2         
            /        ___\      ___ 
 -8*pi  5   |  1   \/ 5 |    \/ 5  
(-----, - + |- - - -----|  - -----)
   5    8   \  4     4  /      8   

                         2         
            /        ___\      ___ 
 -6*pi  5   |  1   \/ 5 |    \/ 5  
(-----, - + |- - + -----|  + -----)
   5    8   \  4     4  /      8   

(-pi, -1)

                         2         
            /        ___\      ___ 
 -4*pi  5   |  1   \/ 5 |    \/ 5  
(-----, - + |- - + -----|  + -----)
   5    8   \  4     4  /      8   

                         2         
            /        ___\      ___ 
 -2*pi  5   |  1   \/ 5 |    \/ 5  
(-----, - + |- - - -----|  - -----)
   5    8   \  4     4  /      8   

             ___   /        ___\ /      ___\ 
 pi    5   \/ 5    |  1   \/ 5 | |1   \/ 5 | 
(--, - - - ----- + |- - + -----|*|- - -----|)
 5     8     8     \  4     4  / \4     4  / 

                        2         
           /        ___\      ___ 
 2*pi  5   |  1   \/ 5 |    \/ 5  
(----, - + |- - - -----|  - -----)
  5    8   \  4     4  /      8   

                        2         
           /        ___\      ___ 
 4*pi  5   |  1   \/ 5 |    \/ 5  
(----, - + |- - + -----|  + -----)
  5    8   \  4     4  /      8   

(pi, -1)

                        2         
           /        ___\      ___ 
 6*pi  5   |  1   \/ 5 |    \/ 5  
(----, - + |- - + -----|  + -----)
  5    8   \  4     4  /      8   

                        2         
           /        ___\      ___ 
 8*pi  5   |  1   \/ 5 |    \/ 5  
(----, - + |- - - -----|  - -----)
  5    8   \  4     4  /      8   

(2*pi, 1)

         /              ___________                      ___________\     /       /              ___________                      ___________\\    /       /              ___________                      ___________\\      /       /              ___________                      ___________\\    /       /              ___________                      ___________\\ 
         |      ____   /       ___        ___     ___   /       ___ |     |       |      ____   /       ___        ___     ___   /       ___ ||    |       |      ____   /       ___        ___     ___   /       ___ ||      |       |      ____   /       ___        ___     ___   /       ___ ||    |       |      ____   /       ___        ___     ___   /       ___ || 
         |I   \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5  |     |       |I   \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5  ||    |       |I   \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5  ||      |       |I   \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5  ||    |       |I   \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5  || 
(-2*I*log|- - --------------------- + ------- + --------------------|, cos|4*I*log|- - --------------------- + ------- + --------------------||*cos|6*I*log|- - --------------------- + ------- + --------------------|| - sin|4*I*log|- - --------------------- + ------- + --------------------||*sin|6*I*log|- - --------------------- + ------- + --------------------||)
         \4             8                4               8          /     \       \4             8                4               8          //    \       \4             8                4               8          //      \       \4             8                4               8          //    \       \4             8                4               8          // 

         /                         ___________             ___________\     /       /                         ___________             ___________\\    /       /                         ___________             ___________\\      /       /                         ___________             ___________\\    /       /                         ___________             ___________\\ 
         |          ___     ___   /       ___      ____   /       ___ |     |       |          ___     ___   /       ___      ____   /       ___ ||    |       |          ___     ___   /       ___      ____   /       ___ ||      |       |          ___     ___   /       ___      ____   /       ___ ||    |       |          ___     ___   /       ___      ____   /       ___ || 
         |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5  |     |       |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5  ||    |       |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5  ||      |       |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5  ||    |       |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5  || 
(-2*I*log|- - - ------- - -------------------- + ---------------------|, cos|4*I*log|- - - ------- - -------------------- + ---------------------||*cos|6*I*log|- - - ------- - -------------------- + ---------------------|| - sin|4*I*log|- - - ------- - -------------------- + ---------------------||*sin|6*I*log|- - - ------- - -------------------- + ---------------------||)
         \  4      4               8                       8          /     \       \  4      4               8                       8          //    \       \  4      4               8                       8          //      \       \  4      4               8                       8          //    \       \  4      4               8                       8          // 

         /              ___________                      ___________            ___________             ___________\     /       /              ___________                      ___________            ___________             ___________\\    /       /              ___________                      ___________            ___________             ___________\\      /       /              ___________                      ___________            ___________             ___________\\    /       /              ___________                      ___________            ___________             ___________\\ 
         |      ____   /       ___        ___     ___   /       ___      ___   /       ___      ____   /       ___ |     |       |      ____   /       ___        ___     ___   /       ___      ___   /       ___      ____   /       ___ ||    |       |      ____   /       ___        ___     ___   /       ___      ___   /       ___      ____   /       ___ ||      |       |      ____   /       ___        ___     ___   /       ___      ___   /       ___      ____   /       ___ ||    |       |      ____   /       ___        ___     ___   /       ___      ___   /       ___      ____   /       ___ || 
         |I   \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5  |     |       |I   \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5  ||    |       |I   \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5  ||      |       |I   \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5  ||    |       |I   \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5  || 
(-2*I*log|- - --------------------- + ------- + -------------------- + -------------------- + ---------------------|, cos|4*I*log|- - --------------------- + ------- + -------------------- + -------------------- + ---------------------||*cos|6*I*log|- - --------------------- + ------- + -------------------- + -------------------- + ---------------------|| - sin|4*I*log|- - --------------------- + ------- + -------------------- + -------------------- + ---------------------||*sin|6*I*log|- - --------------------- + ------- + -------------------- + -------------------- + ---------------------||)
         \4             16               4               16                     16                      16         /     \       \4             16               4               16                     16                      16         //    \       \4             16               4               16                     16                      16         //      \       \4             16               4               16                     16                      16         //    \       \4             16               4               16                     16                      16         // 

         /               ___________             ___________             ___________                      ___________\     /       /               ___________             ___________             ___________                      ___________\\    /       /               ___________             ___________             ___________                      ___________\\      /       /               ___________             ___________             ___________                      ___________\\    /       /               ___________             ___________             ___________                      ___________\\ 
         |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ |     |       |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ ||    |       |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ ||      |       |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ ||    |       |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ || 
         |  I   \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 - \/ 5  |     |       |  I   \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 - \/ 5  ||    |       |  I   \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 - \/ 5  ||      |       |  I   \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 - \/ 5  ||    |       |  I   \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     I*\/ 5    \/ 2 *\/  5 - \/ 5  || 
(-2*I*log|- - - -------------------- - --------------------- - --------------------- + ------- + --------------------|, cos|4*I*log|- - - -------------------- - --------------------- - --------------------- + ------- + --------------------||*cos|6*I*log|- - - -------------------- - --------------------- - --------------------- + ------- + --------------------|| - sin|4*I*log|- - - -------------------- - --------------------- - --------------------- + ------- + --------------------||*sin|6*I*log|- - - -------------------- - --------------------- - --------------------- + ------- + --------------------||)
         \  4            16                      16                      16               4               16         /     \       \  4            16                      16                      16               4               16         //    \       \  4            16                      16                      16               4               16         //      \       \  4            16                      16                      16               4               16         //    \       \  4            16                      16                      16               4               16         // 

         /                         ___________            ___________             ___________             ___________\     /       /                         ___________            ___________             ___________             ___________\\    /       /                         ___________            ___________             ___________             ___________\\      /       /                         ___________            ___________             ___________             ___________\\    /       /                         ___________            ___________             ___________             ___________\\ 
         |          ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ |     |       |          ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ ||    |       |          ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ ||      |       |          ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ ||    |       |          ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ || 
         |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  |     |       |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  ||    |       |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  ||      |       |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  ||    |       |  I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  || 
(-2*I*log|- - - ------- - -------------------- - -------------------- - --------------------- + ---------------------|, cos|4*I*log|- - - ------- - -------------------- - -------------------- - --------------------- + ---------------------||*cos|6*I*log|- - - ------- - -------------------- - -------------------- - --------------------- + ---------------------|| - sin|4*I*log|- - - ------- - -------------------- - -------------------- - --------------------- + ---------------------||*sin|6*I*log|- - - ------- - -------------------- - -------------------- - --------------------- + ---------------------||)
         \  4      4               16                     16                      16                      16         /     \       \  4      4               16                     16                      16                      16         //    \       \  4      4               16                     16                      16                      16         //      \       \  4      4               16                     16                      16                      16         //    \       \  4      4               16                     16                      16                      16         // 

         /                       ___________            ___________             ___________             ___________\     /       /                       ___________            ___________             ___________             ___________\\    /       /                       ___________            ___________             ___________             ___________\\      /       /                       ___________            ___________             ___________             ___________\\    /       /                       ___________            ___________             ___________             ___________\\ 
         |        ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ |     |       |        ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ ||    |       |        ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ ||      |       |        ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ ||    |       |        ___     ___   /       ___      ___   /       ___      ____   /       ___      ____   /       ___ || 
         |I   I*\/ 5    \/ 2 *\/  5 - \/ 5     \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  |     |       |I   I*\/ 5    \/ 2 *\/  5 - \/ 5     \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  ||    |       |I   I*\/ 5    \/ 2 *\/  5 - \/ 5     \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  ||      |       |I   I*\/ 5    \/ 2 *\/  5 - \/ 5     \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  ||    |       |I   I*\/ 5    \/ 2 *\/  5 - \/ 5     \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5  || 
(-2*I*log|- - ------- - -------------------- + -------------------- + --------------------- + ---------------------|, cos|4*I*log|- - ------- - -------------------- + -------------------- + --------------------- + ---------------------||*cos|6*I*log|- - ------- - -------------------- + -------------------- + --------------------- + ---------------------|| - sin|4*I*log|- - ------- - -------------------- + -------------------- + --------------------- + ---------------------||*sin|6*I*log|- - ------- - -------------------- + -------------------- + --------------------- + ---------------------||)
         \4      4               16                     16                      16                      16         /     \       \4      4               16                     16                      16                      16         //    \       \4      4               16                     16                      16                      16         //      \       \4      4               16                     16                      16                      16         //    \       \4      4               16                     16                      16                      16         //


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=−9π5x_{1} = - \frac{9 \pi}{5}
x2=−πx_{2} = - \pi
x3=π5x_{3} = \frac{\pi}{5}
x4=πx_{4} = \pi
x5=2atan⁡(2+25−105+5+25+5)+2πx_{5} = 2 \operatorname{atan}{\left(\frac{2 + 2 \sqrt{5}}{- \sqrt{10} \sqrt{\sqrt{5} + 5} + \sqrt{2} \sqrt{\sqrt{5} + 5}} \right)} + 2 \pi
x6=−2atan⁡(−−25−2−25+5+105+5)x_{6} = - 2 \operatorname{atan}{\left(- \frac{- 2 \sqrt{5} - 2}{- \sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{10} \sqrt{\sqrt{5} + 5}} \right)}
x7=2atan⁡(4+45−105−5+25−5+25+5+105+5)x_{7} = 2 \operatorname{atan}{\left(\frac{4 + 4 \sqrt{5}}{- \sqrt{10} \sqrt{5 - \sqrt{5}} + \sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{10} \sqrt{\sqrt{5} + 5}} \right)}
x8=2atan⁡(−4+45−105+5−105−5−25+5+25−5)+2πx_{8} = 2 \operatorname{atan}{\left(\frac{-4 + 4 \sqrt{5}}{- \sqrt{10} \sqrt{\sqrt{5} + 5} - \sqrt{10} \sqrt{5 - \sqrt{5}} - \sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{2} \sqrt{5 - \sqrt{5}}} \right)} + 2 \pi
x9=−2π+2atan⁡(−45−4−105+5−25+5−25−5+105−5)x_{9} = - 2 \pi + 2 \operatorname{atan}{\left(\frac{- 4 \sqrt{5} - 4}{- \sqrt{10} \sqrt{\sqrt{5} + 5} - \sqrt{2} \sqrt{\sqrt{5} + 5} - \sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{10} \sqrt{5 - \sqrt{5}}} \right)}
x10=2atan⁡(4−45−25−5+25+5+105−5+105+5)x_{10} = 2 \operatorname{atan}{\left(\frac{4 - 4 \sqrt{5}}{- \sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{10} \sqrt{5 - \sqrt{5}} + \sqrt{10} \sqrt{\sqrt{5} + 5}} \right)}
Puntos máximos de la función:
x10=0x_{10} = 0
x10=−8π5x_{10} = - \frac{8 \pi}{5}
x10=−6π5x_{10} = - \frac{6 \pi}{5}
x10=−4π5x_{10} = - \frac{4 \pi}{5}
x10=−2π5x_{10} = - \frac{2 \pi}{5}
x10=2π5x_{10} = \frac{2 \pi}{5}
x10=4π5x_{10} = \frac{4 \pi}{5}
x10=6π5x_{10} = \frac{6 \pi}{5}
x10=8π5x_{10} = \frac{8 \pi}{5}
x10=2πx_{10} = 2 \pi
Decrece en los intervalos
[2atan⁡(−4+45−105+5−105−5−25+5+25−5)+2π,∞)\left[2 \operatorname{atan}{\left(\frac{-4 + 4 \sqrt{5}}{- \sqrt{10} \sqrt{\sqrt{5} + 5} - \sqrt{10} \sqrt{5 - \sqrt{5}} - \sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{2} \sqrt{5 - \sqrt{5}}} \right)} + 2 \pi, \infty\right)
Crece en los intervalos
(−∞,−9π5]\left(-\infty, - \frac{9 \pi}{5}\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
25(sin⁡(2x)sin⁡(3x)−cos⁡(2x)cos⁡(3x))=025 \left(\sin{\left(2 x \right)} \sin{\left(3 x \right)} - \cos{\left(2 x \right)} \cos{\left(3 x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−π2x_{1} = - \frac{\pi}{2}
x2=−π10x_{2} = - \frac{\pi}{10}
x3=π10x_{3} = \frac{\pi}{10}
x4=π2x_{4} = \frac{\pi}{2}
x5=−ilog⁡(−25+58+105+58+i4+5i4)x_{5} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x6=−ilog⁡(−25+58+105+58−5i4−i4)x_{6} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
x7=−ilog⁡(−105+516−105−516−25+516+25−516−i4+5i4)x_{7} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x8=−ilog⁡(−105+516−105−516−25+516+25−516−5i4+i4)x_{8} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} + \frac{i}{4} \right)}
x9=−ilog⁡(−105+516−25+516−25−516+105−516+i4+5i4)x_{9} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x10=−ilog⁡(−105+516−25+516−25−516+105−516−5i4−i4)x_{10} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \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
[atan⁡(−4+45−105+5−105−5−25+5+25−5)+π,∞)\left[\operatorname{atan}{\left(\frac{-4 + 4 \sqrt{5}}{- \sqrt{10} \sqrt{\sqrt{5} + 5} - \sqrt{10} \sqrt{5 - \sqrt{5}} - \sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{2} \sqrt{5 - \sqrt{5}}} \right)} + \pi, \infty\right)
Convexa en los intervalos
(−∞,−π+atan⁡(−45−4−105+5−25+5−25−5+105−5)]\left(-\infty, - \pi + \operatorname{atan}{\left(\frac{- 4 \sqrt{5} - 4}{- \sqrt{10} \sqrt{\sqrt{5} + 5} - \sqrt{2} \sqrt{\sqrt{5} + 5} - \sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{10} \sqrt{5 - \sqrt{5}}} \right)}\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−sin⁡(2x)sin⁡(3x)+cos⁡(2x)cos⁡(3x))=⟨−2,2⟩\lim_{x \to -\infty}\left(- \sin{\left(2 x \right)} \sin{\left(3 x \right)} + \cos{\left(2 x \right)} \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(−sin⁡(2x)sin⁡(3x)+cos⁡(2x)cos⁡(3x))=⟨−2,2⟩\lim_{x \to \infty}\left(- \sin{\left(2 x \right)} \sin{\left(3 x \right)} + \cos{\left(2 x \right)} \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(2*x)*cos(3*x) - sin(2*x)*sin(3*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−sin⁡(2x)sin⁡(3x)+cos⁡(2x)cos⁡(3x)x)=0\lim_{x \to -\infty}\left(\frac{- \sin{\left(2 x \right)} \sin{\left(3 x \right)} + \cos{\left(2 x \right)} \cos{\left(3 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−sin⁡(2x)sin⁡(3x)+cos⁡(2x)cos⁡(3x)x)=0\lim_{x \to \infty}\left(\frac{- \sin{\left(2 x \right)} \sin{\left(3 x \right)} + \cos{\left(2 x \right)} \cos{\left(3 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⁡(2x)sin⁡(3x)+cos⁡(2x)cos⁡(3x)=−sin⁡(2x)sin⁡(3x)+cos⁡(2x)cos⁡(3x)- \sin{\left(2 x \right)} \sin{\left(3 x \right)} + \cos{\left(2 x \right)} \cos{\left(3 x \right)} = - \sin{\left(2 x \right)} \sin{\left(3 x \right)} + \cos{\left(2 x \right)} \cos{\left(3 x \right)}
- Sí
−sin⁡(2x)sin⁡(3x)+cos⁡(2x)cos⁡(3x)=sin⁡(2x)sin⁡(3x)−cos⁡(2x)cos⁡(3x)- \sin{\left(2 x \right)} \sin{\left(3 x \right)} + \cos{\left(2 x \right)} \cos{\left(3 x \right)} = \sin{\left(2 x \right)} \sin{\left(3 x \right)} - \cos{\left(2 x \right)} \cos{\left(3 x \right)}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор