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Trazado el gráfico de la función/ -12*sin(x)/(4*x-5)^2+3*cos(x)/(4*x-5)

Gráfico de la función y = -12*sin(x)/(4*x-5)^2+3*cos(x)/(4*x-5)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       -12*sin(x)   3*cos(x)
f(x) = ---------- + --------
                2   4*x - 5 
       (4*x - 5)
f(x)=(−1)12sin⁡(x)(4x−5)2+3cos⁡(x)4x−5f{\left(x \right)} = \frac{\left(-1\right) 12 \sin{\left(x \right)}}{\left(4 x - 5\right)^{2}} + \frac{3 \cos{\left(x \right)}}{4 x - 5} 
f = (-12*sin(x))/(4*x - 5)^2 + (3*cos(x))/(4*x - 5)
Gráfico de la función
02468-8-6-4-2-1010-2000020000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=1.25x_{1} = 1.25
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:
(−1)12sin⁡(x)(4x−5)2+3cos⁡(x)4x−5=0\frac{\left(-1\right) 12 \sin{\left(x \right)}}{\left(4 x - 5\right)^{2}} + \frac{3 \cos{\left(x \right)}}{4 x - 5} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=4.40549368077057x_{1} = 4.40549368077057
x2=98.9499335239915x_{2} = 98.9499335239915
x3=20.368093402004x_{3} = 20.368093402004
x4=−10.9135458958347x_{4} = -10.9135458958347
x5=−80.0983204688928x_{5} = -80.0983204688928
x6=−58.1026172287233x_{6} = -58.1026172287233
x7=−98.9501888982924x_{7} = -98.9501888982924
x8=−89.5243747480746x_{8} = -89.5243747480746
x9=54.9592548237107x_{9} = 54.9592548237107
x10=89.5240627559632x_{10} = 89.5240627559632
x11=−70.67193163692x_{11} = -70.67193163692
x12=86.3820520567989x_{12} = 86.3820520567989
x13=−39.2452189143973x_{13} = -39.2452189143973
x14=−64.3874153642423x_{14} = -64.3874153642423
x15=−42.3885893354801x_{15} = -42.3885893354801
x16=−83.2403702043979x_{16} = -83.2403702043979
x17=−4.54140529634423x_{17} = -4.54140529634423
x18=−92.6663359084136x_{18} = -92.6663359084136
x19=−7.74324193621266x_{19} = -7.74324193621266
x20=61.2443900632584x_{20} = 61.2443900632584
x21=67.5291554970894x_{21} = 67.5291554970894
x22=−95.8082732103145x_{22} = -95.8082732103145
x23=42.3871967100184x_{23} = 42.3871967100184
x24=26.6642097779614x_{24} = 26.6642097779614
x25=−45.5317208615507x_{25} = -45.5317208615507
x26=−54.9600829118353x_{26} = -54.9600829118353
x27=−114.659504649886x_{27} = -114.659504649886
x28=−1.18046337707441x_{28} = -1.18046337707441
x29=58.101876336132x_{29} = 58.101876336132
x30=70.6714309282998x_{30} = 70.6714309282998
x31=51.8165054251746x_{31} = 51.8165054251746
x32=32.9551927406921x_{32} = 32.9551927406921
x33=10.8922333244709x_{33} = 10.8922333244709
x34=−73.8141062009687x_{34} = -73.8141062009687
x35=−48.6746586267264x_{35} = -48.6746586267264
x36=−17.2246842404762x_{36} = -17.2246842404762
x37=17.2162090251713x_{37} = 17.2162090251713
x38=−36.1015492589898x_{38} = -36.1015492589898
x39=39.2435940186675x_{39} = 39.2435940186675
x40=−14.0719937158111x_{40} = -14.0719937158111
x41=−32.9574978288151x_{41} = -32.9574978288151
x42=48.6736027087111x_{42} = 48.6736027087111
x43=23.5170656824234x_{43} = 23.5170656824234
x44=7.70017150944394x_{44} = 7.70017150944394
x45=−51.8174370666795x_{45} = -51.8174370666795
x46=95.808000809027x_{46} = 95.808000809027
x47=64.3868121024294x_{47} = 64.3868121024294
x48=120.942962647521x_{48} = 120.942962647521
x49=73.813647226469x_{49} = 73.813647226469
x50=36.0996286776759x_{50} = 36.0996286776759
x51=−76.9562340053109x_{51} = -76.9562340053109
x52=−20.3741405593268x_{52} = -20.3741405593268
x53=76.9558117555725x_{53} = 76.9558117555725
x54=45.5305140180641x_{54} = 45.5305140180641
x55=−23.5215979966902x_{55} = -23.5215979966902
x56=−29.8129486293675x_{56} = -29.8129486293675
x57=14.0592564209629x_{57} = 14.0592564209629
x58=−61.2450568449008x_{58} = -61.2450568449008
x59=29.810130661807x_{59} = 29.810130661807
x60=−86.3823871628331x_{60} = -86.3823871628331
x61=80.0979307054563x_{61} = 80.0979307054563
x62=83.2400093168968x_{62} = 83.2400093168968
x63=−26.6677333361162x_{63} = -26.6677333361162
x64=−67.5297039037201x_{64} = -67.5297039037201
x65=92.666044718432x_{65} = 92.666044718432
x66=1094.84412536027x_{66} = 1094.84412536027
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 (-12*sin(x))/(4*x - 5)^2 + (3*cos(x))/(4*x - 5).
3cos⁡(0)−5+0⋅4+(−1)12sin⁡(0)(−5+0⋅4)2\frac{3 \cos{\left(0 \right)}}{-5 + 0 \cdot 4} + \frac{\left(-1\right) 12 \sin{\left(0 \right)}}{\left(-5 + 0 \cdot 4\right)^{2}}
Resultado:
f(0)=−35f{\left(0 \right)} = - \frac{3}{5}
Punto:
(0, -3/5)
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
−12(40−32x)sin⁡(x)(4x−5)4−3sin⁡(x)4x−5−24cos⁡(x)(4x−5)2=0- \frac{12 \left(40 - 32 x\right) \sin{\left(x \right)}}{\left(4 x - 5\right)^{4}} - \frac{3 \sin{\left(x \right)}}{4 x - 5} - \frac{24 \cos{\left(x \right)}}{\left(4 x - 5\right)^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
3(−cos⁡(x)+12sin⁡(x)4x−5+96cos⁡(x)(4x−5)2−384sin⁡(x)(4x−5)3)4x−5=0\frac{3 \left(- \cos{\left(x \right)} + \frac{12 \sin{\left(x \right)}}{4 x - 5} + \frac{96 \cos{\left(x \right)}}{\left(4 x - 5\right)^{2}} - \frac{384 \sin{\left(x \right)}}{\left(4 x - 5\right)^{3}}\right)}{4 x - 5} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1801.70172289517x_{1} = -1801.70172289517
x2=−10.7436520056636x_{2} = -10.7436520056636
x3=−61.2130160013781x_{3} = -61.2130160013781
x4=92.6441544927304x_{4} = 92.6441544927304
x5=32.891816523492x_{5} = 32.891816523492
x6=−86.3595506679355x_{6} = -86.3595506679355
x7=10.6734637134079x_{7} = 10.6734637134079
x8=−29.7482493910137x_{8} = -29.7482493910137
x9=−48.6345230296364x_{9} = -48.6345230296364
x10=−92.6450290819262x_{10} = -92.6450290819262
x11=−45.4888776445288x_{11} = -45.4888776445288
x12=−89.5023296088885x_{12} = -89.5023296088885
x13=−58.0688755802646x_{13} = -58.0688755802646
x14=98.9294526649244x_{14} = 98.9294526649244
x15=−83.2166833768914x_{15} = -83.2166833768914
x16=−26.5956611422569x_{16} = -26.5956611422569
x17=64.3550977071098x_{17} = 64.3550977071098
x18=39.1907824919094x_{18} = 39.1907824919094
x19=−70.6440985836036x_{19} = -70.6440985836036
x20=−13.9387809317578x_{20} = -13.9387809317578
x21=76.9293721657338x_{21} = 76.9293721657338
x22=164.915283535243x_{22} = 164.915283535243
x23=−95.7876568167277x_{23} = -95.7876568167277
x24=−32.898796429671x_{24} = -32.898796429671
x25=−39.1956891695905x_{25} = -39.1956891695905
x26=80.0725463603366x_{26} = 80.0725463603366
x27=83.215599148175x_{27} = 83.215599148175
x28=−54.9244494284112x_{28} = -54.9244494284112
x29=−111.499929553304x_{29} = -111.499929553304
x30=89.5013924623307x_{30} = 89.5013924623307
x31=−17.114914232902x_{31} = -17.114914232902
x32=36.0420173599822x_{32} = 36.0420173599822
x33=51.7768811533207x_{33} = 51.7768811533207
x34=−73.7874402117406x_{34} = -73.7874402117406
x35=7.34718515990664x_{35} = 7.34718515990664
x36=86.35854399961x_{36} = 86.35854399961
x37=54.9219568948526x_{37} = 54.9219568948526
x38=70.6425934403162x_{38} = 70.6425934403162
x39=−42.342645581177x_{39} = -42.342645581177
x40=−80.0737174778353x_{40} = -80.0737174778353
x41=−64.3569118752525x_{41} = -64.3569118752525
x42=−23.4402387009825x_{42} = -23.4402387009825
x43=−20.2807131093469x_{43} = -20.2807131093469
x44=61.211010300196x_{44} = 61.211010300196
x45=95.786838721002x_{45} = 95.786838721002
x46=13.8984622704874x_{46} = 13.8984622704874
x47=67.498947986623x_{47} = 67.498947986623
x48=23.4263884592527x_{48} = 23.4263884592527
x49=−67.5005968272328x_{49} = -67.5005968272328
x50=−76.9306410598324x_{50} = -76.9306410598324
x51=29.7396985824995x_{51} = 29.7396985824995
x52=48.6313418120943x_{52} = 48.6313418120943
x53=17.08858095508x_{53} = 17.08858095508
x54=−36.0478238755638x_{54} = -36.0478238755638
x55=26.5849383107225x_{55} = 26.5849383107225
x56=42.3384442389181x_{56} = 42.3384442389181
x57=−4.13297116044875x_{57} = -4.13297116044875
x58=−98.9302195716006x_{58} = -98.9302195716006
x59=20.2621165073649x_{59} = 20.2621165073649
x60=58.0666462864925x_{60} = 58.0666462864925
x61=45.4852395109622x_{61} = 45.4852395109622
x62=−51.7796865509126x_{62} = -51.7796865509126
x63=−127.211147690413x_{63} = -127.211147690413
x64=−7.50667012912818x_{64} = -7.50667012912818
x65=73.7860607534841x_{65} = 73.7860607534841
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=1.25x_{1} = 1.25

lim⁡x→1.25−(3(−cos⁡(x)+12sin⁡(x)4x−5+96cos⁡(x)(4x−5)2−384sin⁡(x)(4x−5)3)4x−5)=−∞\lim_{x \to 1.25^-}\left(\frac{3 \left(- \cos{\left(x \right)} + \frac{12 \sin{\left(x \right)}}{4 x - 5} + \frac{96 \cos{\left(x \right)}}{\left(4 x - 5\right)^{2}} - \frac{384 \sin{\left(x \right)}}{\left(4 x - 5\right)^{3}}\right)}{4 x - 5}\right) = -\infty
lim⁡x→1.25+(3(−cos⁡(x)+12sin⁡(x)4x−5+96cos⁡(x)(4x−5)2−384sin⁡(x)(4x−5)3)4x−5)=−∞\lim_{x \to 1.25^+}\left(\frac{3 \left(- \cos{\left(x \right)} + \frac{12 \sin{\left(x \right)}}{4 x - 5} + \frac{96 \cos{\left(x \right)}}{\left(4 x - 5\right)^{2}} - \frac{384 \sin{\left(x \right)}}{\left(4 x - 5\right)^{3}}\right)}{4 x - 5}\right) = -\infty
- los límites son iguales, es decir omitimos el punto correspondiente

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
[164.915283535243,∞)\left[164.915283535243, \infty\right)
Convexa en los intervalos
(−∞,−127.211147690413]\left(-\infty, -127.211147690413\right]
Asíntotas verticales
Hay:
x1=1.25x_{1} = 1.25
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((−1)12sin⁡(x)(4x−5)2+3cos⁡(x)4x−5)=0\lim_{x \to -\infty}\left(\frac{\left(-1\right) 12 \sin{\left(x \right)}}{\left(4 x - 5\right)^{2}} + \frac{3 \cos{\left(x \right)}}{4 x - 5}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((−1)12sin⁡(x)(4x−5)2+3cos⁡(x)4x−5)=0\lim_{x \to \infty}\left(\frac{\left(-1\right) 12 \sin{\left(x \right)}}{\left(4 x - 5\right)^{2}} + \frac{3 \cos{\left(x \right)}}{4 x - 5}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-12*sin(x))/(4*x - 5)^2 + (3*cos(x))/(4*x - 5), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−1)12sin⁡(x)(4x−5)2+3cos⁡(x)4x−5x)=0\lim_{x \to -\infty}\left(\frac{\frac{\left(-1\right) 12 \sin{\left(x \right)}}{\left(4 x - 5\right)^{2}} + \frac{3 \cos{\left(x \right)}}{4 x - 5}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((−1)12sin⁡(x)(4x−5)2+3cos⁡(x)4x−5x)=0\lim_{x \to \infty}\left(\frac{\frac{\left(-1\right) 12 \sin{\left(x \right)}}{\left(4 x - 5\right)^{2}} + \frac{3 \cos{\left(x \right)}}{4 x - 5}}{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:
(−1)12sin⁡(x)(4x−5)2+3cos⁡(x)4x−5=3cos⁡(x)−4x−5+12sin⁡(x)(−4x−5)2\frac{\left(-1\right) 12 \sin{\left(x \right)}}{\left(4 x - 5\right)^{2}} + \frac{3 \cos{\left(x \right)}}{4 x - 5} = \frac{3 \cos{\left(x \right)}}{- 4 x - 5} + \frac{12 \sin{\left(x \right)}}{\left(- 4 x - 5\right)^{2}}
- No
(−1)12sin⁡(x)(4x−5)2+3cos⁡(x)4x−5=−3cos⁡(x)−4x−5−12sin⁡(x)(−4x−5)2\frac{\left(-1\right) 12 \sin{\left(x \right)}}{\left(4 x - 5\right)^{2}} + \frac{3 \cos{\left(x \right)}}{4 x - 5} = - \frac{3 \cos{\left(x \right)}}{- 4 x - 5} - \frac{12 \sin{\left(x \right)}}{\left(- 4 x - 5\right)^{2}}
- No
es decir, función
no es
par ni impar
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