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Trazado el gráfico de la función/ sin(3*x)*cos(2*x)+cos(3*x)*sin(2*x)

Gráfico de la función y = sin(3*x)*cos(2*x)+cos(3*x)*sin(2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución analítica
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)}
Solución numérica
x1=−42.0973415581032x_{1} = -42.0973415581032
x2=45.867252742411x_{2} = 45.867252742411
x3=−15.707963267949x_{3} = -15.707963267949
x4=−57.8053048260522x_{4} = -57.8053048260522
x5=55.9203492338983x_{5} = 55.9203492338983
x6=8.16814089933346x_{6} = 8.16814089933346
x7=26.3893782901543x_{7} = 26.3893782901543
x8=21.9911485751286x_{8} = 21.9911485751286
x9=0x_{9} = 0
x10=−21.9911485751286x_{10} = -21.9911485751286
x11=−64.0884901332318x_{11} = -64.0884901332318
x12=28.2743338823081x_{12} = 28.2743338823081
x13=48.3805268652828x_{13} = 48.3805268652828
x14=−93.6194610769758x_{14} = -93.6194610769758
x15=72.2566310325652x_{15} = 72.2566310325652
x16=−35.8141562509236x_{16} = -35.8141562509236
x17=−47.7522083345649x_{17} = -47.7522083345649
x18=−3.76991118430775x_{18} = -3.76991118430775
x19=−76.026542216873x_{19} = -76.026542216873
x20=87.9645943005142x_{20} = 87.9645943005142
x21=−43.9822971502571x_{21} = -43.9822971502571
x22=50.2654824574367x_{22} = 50.2654824574367
x23=−74.1415866247191x_{23} = -74.1415866247191
x24=16.3362817986669x_{24} = 16.3362817986669
x25=189.752196276824x_{25} = 189.752196276824
x26=−89.8495498926681x_{26} = -89.8495498926681
x27=52.1504380495906x_{27} = 52.1504380495906
x28=18.2212373908208x_{28} = 18.2212373908208
x29=−98.0176907920015x_{29} = -98.0176907920015
x30=−81.6814089933346x_{30} = -81.6814089933346
x31=−77.9114978090269x_{31} = -77.9114978090269
x32=94.2477796076938x_{32} = 94.2477796076938
x33=38.3274303737955x_{33} = 38.3274303737955
x34=−91.734505484822x_{34} = -91.734505484822
x35=−20.1061929829747x_{35} = -20.1061929829747
x36=−32.0442450666159x_{36} = -32.0442450666159
x37=67.8584013175395x_{37} = 67.8584013175395
x38=30.159289474462x_{38} = 30.159289474462
x39=−13.8230076757951x_{39} = -13.8230076757951
x40=−10.0530964914873x_{40} = -10.0530964914873
x41=42.0973415581032x_{41} = 42.0973415581032
x42=−33.9292006587698x_{42} = -33.9292006587698
x43=−79.7964534011807x_{43} = -79.7964534011807
x44=−23.8761041672824x_{44} = -23.8761041672824
x45=65.9734457253857x_{45} = 65.9734457253857
x46=101.159283445591x_{46} = 101.159283445591
x47=98.0176907920015x_{47} = 98.0176907920015
x48=70.3716754404114x_{48} = 70.3716754404114
x49=20.1061929829747x_{49} = 20.1061929829747
x50=−54.0353936417444x_{50} = -54.0353936417444
x51=62.2035345410779x_{51} = 62.2035345410779
x52=32.0442450666159x_{52} = 32.0442450666159
x53=54.0353936417444x_{53} = 54.0353936417444
x54=−87.9645943005142x_{54} = -87.9645943005142
x55=77.9114978090269x_{55} = 77.9114978090269
x56=5.02654824574367x_{56} = 5.02654824574367
x57=−27.6460153515902x_{57} = -27.6460153515902
x58=99.9026463841554x_{58} = 99.9026463841554
x59=60.318578948924x_{59} = 60.318578948924
x60=11.3097335529233x_{60} = 11.3097335529233
x61=64.0884901332318x_{61} = 64.0884901332318
x62=11.9380520836412x_{62} = 11.9380520836412
x63=−11.9380520836412x_{63} = -11.9380520836412
x64=−65.9734457253857x_{64} = -65.9734457253857
x65=92.3628240155399x_{65} = 92.3628240155399
x66=43.9822971502571x_{66} = 43.9822971502571
x67=84.1946831162065x_{67} = 84.1946831162065
x68=−49.6371639267187x_{68} = -49.6371639267187
x69=−37.6991118430775x_{69} = -37.6991118430775
x70=−59.6902604182061x_{70} = -59.6902604182061
x71=33.9292006587698x_{71} = 33.9292006587698
x72=86.0796387083603x_{72} = 86.0796387083603
x73=40.2123859659494x_{73} = 40.2123859659494
x74=−96.1327351998477x_{74} = -96.1327351998477
x75=82.3097275240526x_{75} = 82.3097275240526
x76=−5.65486677646163x_{76} = -5.65486677646163
x77=89.2212313619501x_{77} = 89.2212313619501
x78=76.026542216873x_{78} = 76.026542216873
x79=−71.6283125018473x_{79} = -71.6283125018473
x80=−55.9203492338983x_{80} = -55.9203492338983
x81=6.28318530717959x_{81} = 6.28318530717959
x82=−45.867252742411x_{82} = -45.867252742411
x83=−69.7433569096934x_{83} = -69.7433569096934
x84=−1.88495559215388x_{84} = -1.88495559215388
x85=74.1415866247191x_{85} = 74.1415866247191
x86=10.0530964914873x_{86} = 10.0530964914873
x87=1.88495559215388x_{87} = 1.88495559215388
x88=23.8761041672824x_{88} = 23.8761041672824
x89=−25.7610597594363x_{89} = -25.7610597594363
x90=5.65486677646163x_{90} = 5.65486677646163
x91=−99.9026463841554x_{91} = -99.9026463841554
x92=−86.0796387083603x_{92} = -86.0796387083603
x93=−67.8584013175395x_{93} = -67.8584013175395
x94=−52.7787565803085x_{94} = -52.7787565803085
x95=96.1327351998477x_{95} = 96.1327351998477
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(3*x)*cos(2*x) + cos(3*x)*sin(2*x).
sin⁡(0⋅3)cos⁡(0⋅2)+sin⁡(0⋅2)cos⁡(0⋅3)\sin{\left(0 \cdot 3 \right)} \cos{\left(0 \cdot 2 \right)} + \sin{\left(0 \cdot 2 \right)} \cos{\left(0 \cdot 3 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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)sin⁡(3x)+5cos⁡(2x)cos⁡(3x)=0- 5 \sin{\left(2 x \right)} \sin{\left(3 x \right)} + 5 \cos{\left(2 x \right)} \cos{\left(3 x \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)}
Signos de extremos en los puntos:
 -pi      
(----, -1)
  2       

               ___   /        ___\ /      ___\ 
 -pi     5   \/ 5    |  1   \/ 5 | |1   \/ 5 | 
(----, - - + ----- + |- - - -----|*|- + -----|)
  10     8     8     \  4     4  / \4     4  / 

                    2         
         /      ___\      ___ 
 pi  5   |1   \/ 5 |    \/ 5  
(--, - + |- + -----|  - -----)
 10  8   \4     4  /      8   

 pi    
(--, 1)
 2     

       /             ___________                       ___________\       /       /             ___________                       ___________\\    /       /             ___________                       ___________\\      /       /             ___________                       ___________\\    /       /             ___________                       ___________\\ 
       |      ___   /       ___        ___     ____   /       ___ |       |       |      ___   /       ___        ___     ____   /       ___ ||    |       |      ___   /       ___        ___     ____   /       ___ ||      |       |      ___   /       ___        ___     ____   /       ___ ||    |       |      ___   /       ___        ___     ____   /       ___ || 
       |I   \/ 2 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 + \/ 5  |       |       |I   \/ 2 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 + \/ 5  ||    |       |I   \/ 2 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 + \/ 5  ||      |       |I   \/ 2 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 + \/ 5  ||    |       |I   \/ 2 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 + \/ 5  || 
(-I*log|- - -------------------- + ------- + ---------------------|, - cos|2*I*log|- - -------------------- + ------- + ---------------------||*sin|3*I*log|- - -------------------- + ------- + ---------------------|| - cos|3*I*log|- - -------------------- + ------- + ---------------------||*sin|2*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  || 
(-I*log|- - - ------- - -------------------- + ---------------------|, - cos|2*I*log|- - - ------- - -------------------- + ---------------------||*sin|3*I*log|- - - ------- - -------------------- + ---------------------|| - cos|3*I*log|- - - ------- - -------------------- + ---------------------||*sin|2*I*log|- - - ------- - -------------------- + ---------------------||)
       \  4      4               8                       8          /       \       \  4      4               8                       8          //    \       \  4      4               8                       8          //      \       \  4      4               8                       8          //    \       \  4      4               8                       8          // 

       /               ___________             ___________             ___________                      ___________\       /       /               ___________             ___________             ___________                      ___________\\    /       /               ___________             ___________             ___________                      ___________\\      /       /               ___________             ___________             ___________                      ___________\\    /       /               ___________             ___________             ___________                      ___________\\ 
       |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ |       |       |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ ||    |       |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ ||      |       |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ ||    |       |        ___   /       ___      ____   /       ___      ____   /       ___        ___     ___   /       ___ || 
       |  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  || 
(-I*log|- - - -------------------- - --------------------- - --------------------- + ------- + --------------------|, - cos|2*I*log|- - - -------------------- - --------------------- - --------------------- + ------- + --------------------||*sin|3*I*log|- - - -------------------- - --------------------- - --------------------- + ------- + --------------------|| - cos|3*I*log|- - - -------------------- - --------------------- - --------------------- + ------- + --------------------||*sin|2*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     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     \/ 2 *\/  5 - \/ 5  |       |       |I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     \/ 2 *\/  5 - \/ 5  ||    |       |I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     \/ 2 *\/  5 - \/ 5  ||      |       |I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     \/ 2 *\/  5 - \/ 5  ||    |       |I   I*\/ 5    \/ 2 *\/  5 + \/ 5     \/ 10 *\/  5 + \/ 5     \/ 10 *\/  5 - \/ 5     \/ 2 *\/  5 - \/ 5  || 
(-I*log|- - ------- - -------------------- - --------------------- - --------------------- + --------------------|, - cos|2*I*log|- - ------- - -------------------- - --------------------- - --------------------- + --------------------||*sin|3*I*log|- - ------- - -------------------- - --------------------- - --------------------- + --------------------|| - cos|3*I*log|- - ------- - -------------------- - --------------------- - --------------------- + --------------------||*sin|2*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   \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 - \/ 5  |       |       |I   \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 - \/ 5  ||    |       |I   \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 - \/ 5  ||      |       |I   \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 - \/ 5  ||    |       |I   \/ 2 *\/  5 + \/ 5     \/ 2 *\/  5 - \/ 5     \/ 10 *\/  5 + \/ 5     I*\/ 5    \/ 10 *\/  5 - \/ 5  || 
(-I*log|- - -------------------- - -------------------- - --------------------- + ------- + ---------------------|, - cos|2*I*log|- - -------------------- - -------------------- - --------------------- + ------- + ---------------------||*sin|3*I*log|- - -------------------- - -------------------- - --------------------- + ------- + ---------------------|| - cos|3*I*log|- - -------------------- - -------------------- - --------------------- + ------- + ---------------------||*sin|2*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  || 
(-I*log|- - - ------- - -------------------- - -------------------- - --------------------- + ---------------------|, - cos|2*I*log|- - - ------- - -------------------- - -------------------- - --------------------- + ---------------------||*sin|3*I*log|- - - ------- - -------------------- - -------------------- - --------------------- + ---------------------|| - cos|3*I*log|- - - ------- - -------------------- - -------------------- - --------------------- + ---------------------||*sin|2*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=−π2x_{1} = - \frac{\pi}{2}
x2=−π10x_{2} = - \frac{\pi}{10}
x3=atan⁡(2+25−25+5+105+5)x_{3} = \operatorname{atan}{\left(\frac{2 + 2 \sqrt{5}}{- \sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{10} \sqrt{\sqrt{5} + 5}} \right)}
x4=−π+atan⁡(4−45−105+5−105−5−25+5+25−5)x_{4} = - \pi + \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)}
x5=atan⁡(4+45−105+5−25+5−25−5+105−5)+πx_{5} = \operatorname{atan}{\left(\frac{4 + 4 \sqrt{5}}{- \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)} + \pi
Puntos máximos de la función:
x5=π10x_{5} = \frac{\pi}{10}
x5=π2x_{5} = \frac{\pi}{2}
x5=−atan⁡(−−25−2−25+5+105+5)x_{5} = - \operatorname{atan}{\left(- \frac{- 2 \sqrt{5} - 2}{- \sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{10} \sqrt{\sqrt{5} + 5}} \right)}
x5=atan⁡(−4+45−105+5−105−5−25+5+25−5)+πx_{5} = \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
x5=−π+atan⁡(−45−4−105+5−25+5−25−5+105−5)x_{5} = - \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)}
Decrece en los intervalos
[atan⁡(4+45−105+5−25+5−25−5+105−5)+π,∞)\left[\operatorname{atan}{\left(\frac{4 + 4 \sqrt{5}}{- \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)} + \pi, \infty\right)
Crece en los intervalos
(−∞,−π+atan⁡(4−45−105+5−105−5−25+5+25−5)]\left(-\infty, - \pi + \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)}\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)cos⁡(3x)+sin⁡(3x)cos⁡(2x))=0- 25 \left(\sin{\left(2 x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(2 x \right)}\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)}

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⁡(−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)
Convexa en los intervalos
(−∞,−9π5]\left(-\infty, - \frac{9 \pi}{5}\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(2x)cos⁡(3x)+sin⁡(3x)cos⁡(2x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(2 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)cos⁡(3x)+sin⁡(3x)cos⁡(2x))=⟨−2,2⟩\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(2 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 sin(3*x)*cos(2*x) + cos(3*x)*sin(2*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(2x)cos⁡(3x)+sin⁡(3x)cos⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(2 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)cos⁡(3x)+sin⁡(3x)cos⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(2 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)cos⁡(3x)+sin⁡(3x)cos⁡(2x)=−sin⁡(2x)cos⁡(3x)−sin⁡(3x)cos⁡(2x)\sin{\left(2 x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(2 x \right)} = - \sin{\left(2 x \right)} \cos{\left(3 x \right)} - \sin{\left(3 x \right)} \cos{\left(2 x \right)}
- No
sin⁡(2x)cos⁡(3x)+sin⁡(3x)cos⁡(2x)=sin⁡(2x)cos⁡(3x)+sin⁡(3x)cos⁡(2x)\sin{\left(2 x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(2 x \right)} = \sin{\left(2 x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(2 x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор