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Trazado el gráfico de la función/ (\pi^(3)*cos(7x)*ln(x+5))/((3+x)*(x-e))+(\pi)/(12)

Gráfico de la función y = (\pi^(3)*cos(7x)*ln(x+5))/((3+x)*(x-e))+(\pi)/(12)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         3                         
       pi *cos(7*x)*log(x + 5)   pi
f(x) = ----------------------- + --
           (3 + x)*(x - E)       12
f(x)=π12+π3cos⁡(7x)log⁡(x+5)(x+3)(x−e)f{\left(x \right)} = \frac{\pi}{12} + \frac{\pi^{3} \cos{\left(7 x \right)} \log{\left(x + 5 \right)}}{\left(x + 3\right) \left(x - e\right)} 
f = pi/12 + ((pi^3*cos(7*x))*log(x + 5))/(((x + 3)*(x - E)))
Gráfico de la función
02468-8-6-4-2-1010-500500
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−3x_{1} = -3
x2=2.71828182845905x_{2} = 2.71828182845905
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:
π12+π3cos⁡(7x)log⁡(x+5)(x+3)(x−e)=0\frac{\pi}{12} + \frac{\pi^{3} \cos{\left(7 x \right)} \log{\left(x + 5 \right)}}{\left(x + 3\right) \left(x - e\right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=4.25753335463382x_{1} = 4.25753335463382
x2=2.01741819667147x_{2} = 2.01741819667147
x3=14.0533248872284x_{3} = 14.0533248872284
x4=12.2780308823307x_{4} = 12.2780308823307
x5=10.141973486048x_{5} = 10.141973486048
x6=8.3326260789725x_{6} = 8.3326260789725
x7=6.04365472661502x_{7} = 6.04365472661502
x8=−4.69973101528258x_{8} = -4.69973101528258
x9=0.218524153901069x_{9} = 0.218524153901069
x10=19.3262787742392x_{10} = 19.3262787742392
x11=18.3340713623404x_{11} = 18.3340713623404
x12=−2.01445041492265x_{12} = -2.01445041492265
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 ((pi^3*cos(7*x))*log(x + 5))/(((3 + x)*(x - E))) + pi/12.
π3cos⁡(0⋅7)log⁡(5)3(−e)+π12\frac{\pi^{3} \cos{\left(0 \cdot 7 \right)} \log{\left(5 \right)}}{3 \left(- e\right)} + \frac{\pi}{12}
Resultado:
f(0)=−π3log⁡(5)3e+π12f{\left(0 \right)} = - \frac{\pi^{3} \log{\left(5 \right)}}{3 e} + \frac{\pi}{12}
Punto:
(0, pi/12 - pi^3*exp(-1)*log(5)/3)
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
π3(−2x−3+e)log⁡(x+5)cos⁡(7x)(x+3)2(x−e)2+1(x+3)(x−e)(−7π3log⁡(x+5)sin⁡(7x)+π3cos⁡(7x)x+5)=0\frac{\pi^{3} \left(- 2 x - 3 + e\right) \log{\left(x + 5 \right)} \cos{\left(7 x \right)}}{\left(x + 3\right)^{2} \left(x - e\right)^{2}} + \frac{1}{\left(x + 3\right) \left(x - e\right)} \left(- 7 \pi^{3} \log{\left(x + 5 \right)} \sin{\left(7 x \right)} + \frac{\pi^{3} \cos{\left(7 x \right)}}{x + 5}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=8.07333647496712x_{1} = 8.07333647496712
x2=52.0599824431461x_{2} = 52.0599824431461
x3=−1.8022238393788x_{3} = -1.8022238393788
x4=54.3040055516743x_{4} = 54.3040055516743
x5=4.02174290240437x_{5} = 4.02174290240437
x6=12.563392288607x_{6} = 12.563392288607
x7=26.0289521380567x_{7} = 26.0289521380567
x8=70.0121165076394x_{8} = 70.0121165076394
x9=38.1469644317389x_{9} = 38.1469644317389
x10=10.3186474204251x_{10} = 10.3186474204251
x11=74.0513349501713x_{11} = 74.0513349501713
x12=76.295343927415x_{12} = 76.295343927415
x13=82.1297636529613x_{13} = 82.1297636529613
x14=12.1144698375243x_{14} = 12.1144698375243
x15=60.1384557413183x_{15} = 60.1384557413183
x16=48.0207341652299x_{16} = 48.0207341652299
x17=6.27600201658947x_{17} = 6.27600201658947
x18=0.00325421683906397x_{18} = 0.00325421683906397
x19=21.9895024350614x_{19} = 21.9895024350614
x20=61.9336687702985x_{20} = 61.9336687702985
x21=86.1689746797423x_{21} = 86.1689746797423
x22=98.2865977350379x_{22} = 98.2865977350379
x23=96.0425944361507x_{23} = 96.0425944361507
x24=32.3124080463883x_{24} = 32.3124080463883
x25=92.0033874566411x_{25} = 92.0033874566411
x26=72.256127136215x_{26} = 72.256127136215
x27=100.08180010051x_{27} = 100.08180010051
x28=87.9641789814419x_{28} = 87.9641789814419
x29=83.4761675470586x_{29} = 83.4761675470586
x30=59.6896523226666x_{30} = 59.6896523226666
x31=24.2336522964021x_{31} = 24.2336522964021
x32=42.1862448499456x_{32} = 42.1862448499456
x33=64.1776837058196x_{33} = 64.1776837058196
x34=17.9499279684132x_{34} = 17.9499279684132
x35=50.2647621330582x_{35} = 50.2647621330582
x36=28.2730577384618x_{36} = 28.2730577384618
x37=65.9728946695263x_{37} = 65.9728946695263
x38=34.1076624603019x_{38} = 34.1076624603019
x39=56.099222406278x_{39} = 56.099222406278
x40=90.2081838864455x_{40} = 90.2081838864455
x41=39.9422024370822x_{41} = 39.9422024370822
x42=2.28752323373221x_{42} = 2.28752323373221
x43=78.090550524266x_{43} = 78.090550524266
x44=30.0683299718735x_{44} = 30.0683299718735
x45=46.2255094633007x_{45} = 46.2255094633007
x46=68.2169072569131x_{46} = 68.2169072569131
x47=43.981475255152x_{47} = 43.981475255152
x48=94.2473915065267x_{48} = 94.2473915065267
x49=−3.89807610019929x_{49} = -3.89807610019929
x50=16.154494175718x_{50} = 16.154494175718
x51=20.1941562416054x_{51} = 20.1941562416054
Signos de extremos en los puntos:
                                                    3         
                    pi           2.56897222618036*pi          
(8.073336474967125, -- + ------------------------------------)
                    12   89.398771262936 - 11.0733364749671*E 

                                                      3         
                     pi            4.04405506732452*pi          
(52.059982443146104, -- + -------------------------------------)
                     12   2866.42171931012 - 55.0599824431461*E 

                                                       3         
                      pi            1.16104917264473*pi          
(-1.8022238393787962, -- + -------------------------------------)
                      12   -2.15866075091114 - 1.1977761606212*E 

                                                    3         
                    pi           4.08263228931646*pi          
(54.30400555167433, -- - ------------------------------------)
                    12   3111.8370356113 - 57.3040055516743*E 

                                                    3         
                    pi           2.18325235937609*pi          
(4.021742902404369, -- - ------------------------------------)
                    12   28.239644680253 - 7.02174290240437*E 

                                                     3         
                     pi           2.86519396007298*pi          
(12.563392288607037, -- + ------------------------------------)
                     12   195.529002663252 - 15.563392288607*E 

                                                      3         
                     pi            3.43475881593178*pi          
(26.028952138056738, -- + -------------------------------------)
                     12   755.593205819419 - 29.0289521380567*E 

                                                    3          
                    pi            4.3176210754271*pi           
(70.01211650763938, -- + -------------------------------------)
                    12   5111.73280740219 - 73.0121165076394*E 

                                                     3         
                    pi            3.76452946220998*pi          
(38.14696443173887, -- - -------------------------------------)
                    12   1569.63178865157 - 41.1469644317389*E 

                                                     3         
                     pi           2.72814145435153*pi          
(10.318647420425066, -- - ------------------------------------)
                     12   137.43042684832 - 13.3186474204251*E 

                                                     3         
                    pi            4.37007154596634*pi          
(74.05133495017131, -- - -------------------------------------)
                    12   5705.75421275298 - 77.0513349501713*E 

                                                    3         
                    pi           4.39806415983066*pi          
(76.29534392741502, -- + ------------------------------------)
                    12   6049.86553678479 - 79.295343927415*E 

                                                     3         
                    pi            4.46737693861186*pi          
(82.12976365296126, -- - -------------------------------------)
                    12   6991.68736865016 - 85.1297636529613*E 

                                                      3         
                     pi            2.83925488726611*pi          
(12.114469837524252, -- - -------------------------------------)
                     12   183.103788956858 - 15.1144698375243*E 

                                                     3         
                    pi            4.17647780924103*pi          
(60.13845574131828, -- + -------------------------------------)
                    12   3797.04922617445 - 63.1384557413183*E 

                                                    3         
                   pi            3.97062780913351*pi          
(48.0207341652299, -- - -------------------------------------)
                   12   2450.05311226337 - 51.0207341652299*E 

                                                      3         
                     pi            2.41961466936029*pi          
(6.2760020165894685, -- + -------------------------------------)
                     12   58.2162073620035 - 9.27600201658947*E 

                                                         3           
                       pi             1.60967081950616*pi            
(0.003254216839063967, -- + ----------------------------------------)
                       12   0.00977324044442755 - 3.00325421683906*E 

                                                      3         
                     pi            3.29522921084744*pi          
(21.989502435061436, -- - -------------------------------------)
                     12   549.506724646757 - 24.9895024350614*E 

                                                     3         
                    pi            4.20366669574458*pi          
(61.93366877029854, -- + -------------------------------------)
                    12   4021.58033365995 - 64.9336687702985*E 

                                                    3         
                    pi           4.51269478998863*pi          
(86.16897467974232, -- + ------------------------------------)
                    12   7683.5991213973 - 89.1689746797423*E 

                                                     3         
                    pi            4.63749186734505*pi          
(98.28659773503789, -- - -------------------------------------)
                    12   9955.11508753427 - 101.286597735038*E 

                                                     3         
                    pi            4.61552574239542*pi          
(96.04259443615067, -- + -------------------------------------)
                    12   9512.30772933537 - 99.0425944361507*E 

                                                     3         
                    pi            3.61921541106166*pi          
(32.31240804638829, -- + -------------------------------------)
                    12   1141.02893789546 - 35.3124080463883*E 

                                                     3         
                    pi            4.57472819964053*pi          
(92.00338745664106, -- - -------------------------------------)
                    12   8740.63346586674 - 95.0033874566411*E 

                                                    3         
                    pi           4.34709918559549*pi          
(72.25612713621504, -- - ------------------------------------)
                    12   5437.71629013352 - 75.256127136215*E 

                                                    3        
                     pi          4.65472383031407*pi         
(100.08180010051007, -- - ----------------------------------)
                     12   10316.61211166 - 103.08180010051*E 

                                                     3         
                    pi            4.53219509357381*pi          
(87.96417898144195, -- + -------------------------------------)
                    12   8001.58932082348 - 90.9641789814419*E 

                                                    3         
                   pi            4.48271222512188*pi          
(83.4761675470586, -- + -------------------------------------)
                   12   7218.69905098577 - 86.4761675470586*E 

                                                     3         
                    pi            4.16956348060399*pi          
(59.68965232266658, -- - -------------------------------------)
                    12   3741.92355136882 - 62.6896523226666*E 

                                                    3         
                   pi            3.37513667707299*pi          
(24.2336522964021, -- + -------------------------------------)
                   12   659.970860512121 - 27.2336522964021*E 

                                                    3         
                    pi           3.85403315811599*pi          
(42.18624484994561, -- + ------------------------------------)
                    12   1906.2379890894 - 45.1862448499456*E 

                                                     3         
                    pi            4.23664504278263*pi          
(64.17768370581959, -- - -------------------------------------)
                    12   4311.30813696168 - 67.1776837058196*E 

                                                      3         
                     pi            3.13299844107053*pi          
(17.949927968413167, -- + -------------------------------------)
                     12   376.049697976461 - 20.9499279684132*E 

                                                     3         
                     pi           4.01208448949026*pi          
(50.264762133058234, -- + ------------------------------------)
                     12   2677.3405986921 - 53.2647621330582*E 

                                                     3         
                    pi            3.50460815649998*pi          
(28.27305773846178, -- - -------------------------------------)
                    12   884.184967097779 - 31.2730577384618*E 

                                                    3        
                    pi           4.26226632861361*pi         
(65.97289466952634, -- - -----------------------------------)
                    12   4550.341515085 - 68.9728946695263*E 

                                                    3         
                   pi            3.66621791451667*pi          
(34.1076624603019, -- + -------------------------------------)
                   12   1265.65562588679 - 37.1076624603019*E 

                                                     3         
                     pi           4.11245703930087*pi          
(56.099222406277995, -- - ------------------------------------)
                     12   3315.42042180788 - 59.099222406278*E 

                                                     3         
                    pi            4.55604757917594*pi          
(90.20818388644552, -- - -------------------------------------)
                    12   8408.14099175011 - 93.2081838864455*E 

                                                    3         
                    pi           3.80530105699967*pi          
(39.94220243708221, -- - ------------------------------------)
                    12   1715.2061428361 - 42.9422024370822*E 

                                                     3         
                    pi            1.89467545119058*pi          
(2.287523233732214, -- - -------------------------------------)
                    12   12.0953322460613 - 5.28752323373221*E 

                                                    3         
                    pi           4.41990738048049*pi          
(78.09055052426596, -- + ------------------------------------)
                    12   6332.40573275573 - 81.090550524266*E 

                                                     3         
                    pi            3.55717300254862*pi          
(30.06832997187348, -- - -------------------------------------)
                    12   994.309457213086 - 33.0683299718735*E 

                                                    3         
                    pi           3.93617859946821*pi          
(46.22550946330068, -- - ------------------------------------)
                    12   2275.4742535316 - 49.2255094633007*E 

                                                     3         
                    pi            4.29339645997405*pi          
(68.21690725691315, -- + -------------------------------------)
                    12   4858.19715746903 - 71.2169072569131*E 

                                                    3         
                    pi           3.89137776729222*pi          
(43.98147525515202, -- + ------------------------------------)
                    12   2066.31459138501 - 46.981475255152*E 

                                                   3          
                    pi           4.5975986708236*pi           
(94.24739150652671, -- + ------------------------------------)
                    12   9165.3129803041 - 97.2473915065267*E 

                                                        3         
                      pi           0.0534328387682002*pi          
(-3.8980761001992934, -- - --------------------------------------)
                      12   3.50076898234705 + 0.898076100199293*E 

                                                     3         
                     pi           3.05146776138743*pi          
(16.154494175718003, -- + ------------------------------------)
                     12   309.431164600461 - 19.154494175718*E 

                                                      3         
                     pi            3.22635693349497*pi          
(20.194156241605413, -- - -------------------------------------)
                     12   468.386415035187 - 23.1941562416054*E


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=−1.8022238393788x_{1} = -1.8022238393788
x2=54.3040055516743x_{2} = 54.3040055516743
x3=4.02174290240437x_{3} = 4.02174290240437
x4=38.1469644317389x_{4} = 38.1469644317389
x5=10.3186474204251x_{5} = 10.3186474204251
x6=74.0513349501713x_{6} = 74.0513349501713
x7=82.1297636529613x_{7} = 82.1297636529613
x8=12.1144698375243x_{8} = 12.1144698375243
x9=48.0207341652299x_{9} = 48.0207341652299
x10=0.00325421683906397x_{10} = 0.00325421683906397
x11=21.9895024350614x_{11} = 21.9895024350614
x12=98.2865977350379x_{12} = 98.2865977350379
x13=92.0033874566411x_{13} = 92.0033874566411
x14=72.256127136215x_{14} = 72.256127136215
x15=100.08180010051x_{15} = 100.08180010051
x16=59.6896523226666x_{16} = 59.6896523226666
x17=64.1776837058196x_{17} = 64.1776837058196
x18=28.2730577384618x_{18} = 28.2730577384618
x19=65.9728946695263x_{19} = 65.9728946695263
x20=56.099222406278x_{20} = 56.099222406278
x21=90.2081838864455x_{21} = 90.2081838864455
x22=39.9422024370822x_{22} = 39.9422024370822
x23=30.0683299718735x_{23} = 30.0683299718735
x24=46.2255094633007x_{24} = 46.2255094633007
x25=−3.89807610019929x_{25} = -3.89807610019929
x26=20.1941562416054x_{26} = 20.1941562416054
Puntos máximos de la función:
x26=8.07333647496712x_{26} = 8.07333647496712
x26=52.0599824431461x_{26} = 52.0599824431461
x26=12.563392288607x_{26} = 12.563392288607
x26=26.0289521380567x_{26} = 26.0289521380567
x26=70.0121165076394x_{26} = 70.0121165076394
x26=76.295343927415x_{26} = 76.295343927415
x26=60.1384557413183x_{26} = 60.1384557413183
x26=6.27600201658947x_{26} = 6.27600201658947
x26=61.9336687702985x_{26} = 61.9336687702985
x26=86.1689746797423x_{26} = 86.1689746797423
x26=96.0425944361507x_{26} = 96.0425944361507
x26=32.3124080463883x_{26} = 32.3124080463883
x26=87.9641789814419x_{26} = 87.9641789814419
x26=83.4761675470586x_{26} = 83.4761675470586
x26=24.2336522964021x_{26} = 24.2336522964021
x26=42.1862448499456x_{26} = 42.1862448499456
x26=17.9499279684132x_{26} = 17.9499279684132
x26=50.2647621330582x_{26} = 50.2647621330582
x26=34.1076624603019x_{26} = 34.1076624603019
x26=2.28752323373221x_{26} = 2.28752323373221
x26=78.090550524266x_{26} = 78.090550524266
x26=68.2169072569131x_{26} = 68.2169072569131
x26=43.981475255152x_{26} = 43.981475255152
x26=94.2473915065267x_{26} = 94.2473915065267
x26=16.154494175718x_{26} = 16.154494175718
Decrece en los intervalos
[100.08180010051,∞)\left[100.08180010051, \infty\right)
Crece en los intervalos
(−∞,−3.89807610019929]\left(-\infty, -3.89807610019929\right]
Asíntotas verticales
Hay:
x1=−3x_{1} = -3
x2=2.71828182845905x_{2} = 2.71828182845905
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(π12+π3cos⁡(7x)log⁡(x+5)(x+3)(x−e))=π12\lim_{x \to -\infty}\left(\frac{\pi}{12} + \frac{\pi^{3} \cos{\left(7 x \right)} \log{\left(x + 5 \right)}}{\left(x + 3\right) \left(x - e\right)}\right) = \frac{\pi}{12}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=π12y = \frac{\pi}{12}
lim⁡x→∞(π12+π3cos⁡(7x)log⁡(x+5)(x+3)(x−e))=π12\lim_{x \to \infty}\left(\frac{\pi}{12} + \frac{\pi^{3} \cos{\left(7 x \right)} \log{\left(x + 5 \right)}}{\left(x + 3\right) \left(x - e\right)}\right) = \frac{\pi}{12}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=π12y = \frac{\pi}{12}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((pi^3*cos(7*x))*log(x + 5))/(((3 + x)*(x - E))) + pi/12, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(π12+π3cos⁡(7x)log⁡(x+5)(x+3)(x−e)x)=0\lim_{x \to -\infty}\left(\frac{\frac{\pi}{12} + \frac{\pi^{3} \cos{\left(7 x \right)} \log{\left(x + 5 \right)}}{\left(x + 3\right) \left(x - e\right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(π12+π3cos⁡(7x)log⁡(x+5)(x+3)(x−e)x)=0\lim_{x \to \infty}\left(\frac{\frac{\pi}{12} + \frac{\pi^{3} \cos{\left(7 x \right)} \log{\left(x + 5 \right)}}{\left(x + 3\right) \left(x - e\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:
π12+π3cos⁡(7x)log⁡(x+5)(x+3)(x−e)=π12+π3log⁡(5−x)cos⁡(7x)(3−x)(−x−e)\frac{\pi}{12} + \frac{\pi^{3} \cos{\left(7 x \right)} \log{\left(x + 5 \right)}}{\left(x + 3\right) \left(x - e\right)} = \frac{\pi}{12} + \frac{\pi^{3} \log{\left(5 - x \right)} \cos{\left(7 x \right)}}{\left(3 - x\right) \left(- x - e\right)}
- No
π12+π3cos⁡(7x)log⁡(x+5)(x+3)(x−e)=−π12−π3log⁡(5−x)cos⁡(7x)(3−x)(−x−e)\frac{\pi}{12} + \frac{\pi^{3} \cos{\left(7 x \right)} \log{\left(x + 5 \right)}}{\left(x + 3\right) \left(x - e\right)} = - \frac{\pi}{12} - \frac{\pi^{3} \log{\left(5 - x \right)} \cos{\left(7 x \right)}}{\left(3 - x\right) \left(- x - e\right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор