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Trazado el gráfico de la función/ 5*x*cos(x)-5*sin(x)-3*cos(x)+11

Gráfico de la función y = 5*x*cos(x)-5*sin(x)-3*cos(x)+11

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = 5*x*cos(x) - 5*sin(x) - 3*cos(x) + 11
f(x)=((5xcos⁡(x)−5sin⁡(x))−3cos⁡(x))+11f{\left(x \right)} = \left(\left(5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 3 \cos{\left(x \right)}\right) + 11 
f = (5*x)*cos(x) - 5*sin(x) - 3*cos(x) + 11
Gráfico de la función
02468-8-6-4-2-1010-100100
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:
((5xcos⁡(x)−5sin⁡(x))−3cos⁡(x))+11=0\left(\left(5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 3 \cos{\left(x \right)}\right) + 11 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−98.9722211633103x_{1} = -98.9722211633103
x2=−54.9994602493402x_{2} = -54.9994602493402
x3=92.6422161875555x_{3} = 92.6422161875555
x4=−11.0987870729828x_{4} = -11.0987870729828
x5=−76.927743733045x_{5} = -76.927743733045
x6=3.63456029572582x_{6} = 3.63456029572582
x7=95.8311780170773x_{7} = 95.8311780170773
x8=98.9276239663429x_{8} = 98.9276239663429
x9=83.2667232240061x_{9} = 83.2667232240061
x10=14.225638614699x_{10} = 14.225638614699
x11=−70.6409157528489x_{11} = -70.6409157528489
x12=42.3348215282261x_{12} = 42.3348215282261
x13=−67.5618503922841x_{13} = -67.5618503922841
x14=−7.45604393306802x_{14} = -7.45604393306802
x15=26.7494840852553x_{15} = 26.7494840852553
x16=−42.4393949120741x_{16} = -42.4393949120741
x17=54.9189580113136x_{17} = 54.9189580113136
x18=−58.0649153146759x_{18} = -58.0649153146759
x19=36.0380092808625x_{19} = 36.0380092808625
x20=−89.4998740173775x_{20} = -89.4998740173775
x21=−92.6898476587112x_{21} = -92.6898476587112
x22=−80.1254797782395x_{22} = -80.1254797782395
x23=−23.611579206207x_{23} = -23.611579206207
x24=−26.5858119589135x_{24} = -26.5858119589135
x25=−64.3533820482344x_{25} = -64.3533820482344
x26=2.5720459603393x_{26} = 2.5720459603393
x27=−73.843549400265x_{27} = -73.843549400265
x28=10.6776857441286x_{28} = 10.6776857441286
x29=−45.483651036011x_{29} = -45.483651036011
x30=39.3009325855665x_{30} = 39.3009325855665
x31=−86.4075914044314x_{31} = -86.4075914044314
x32=−13.9166148084075x_{32} = -13.9166148084075
x33=86.3564824686678x_{33} = 86.3564824686678
x34=−32.8911661397078x_{34} = -32.8911661397078
x35=80.0703453934248x_{35} = 80.0703453934248
x36=−17.3458029968401x_{36} = -17.3458029968401
x37=−29.8845300882719x_{37} = -29.8845300882719
x38=48.6280553171916x_{38} = 48.6280553171916
x39=23.4216983894966x_{39} = 23.4216983894966
x40=20.480840740071x_{40} = 20.480840740071
x41=−4.93549555513284x_{41} = -4.93549555513284
x42=−48.7190259203714x_{42} = -48.7190259203714
x43=70.7029553133234x_{43} = 70.7029553133234
x44=17.08456051547x_{44} = 17.08456051547
x45=−61.2804532313404x_{45} = -61.2804532313404
x46=−95.7853754956168x_{46} = -95.7853754956168
x47=−20.2669616515902x_{47} = -20.2669616515902
x48=61.2082571226657x_{48} = 61.2082571226657
x49=51.8596964783245x_{49} = 51.8596964783245
x50=−83.2140249721979x_{50} = -83.2140249721979
x51=274.877690164582x_{51} = 274.877690164582
x52=64.4214557321206x_{52} = 64.4214557321206
x53=67.496405754421x_{53} = 67.496405754421
x54=8.01829795287702x_{54} = 8.01829795287702
x55=29.7352838378684x_{55} = 29.7352838378684
x56=89.548882954501x_{56} = 89.548882954501
x57=−36.1609791433337x_{57} = -36.1609791433337
x58=73.7837009043018x_{58} = 73.7837009043018
x59=58.1403243251852x_{59} = 58.1403243251852
x60=45.5797832160117x_{60} = 45.5797832160117
x61=76.9847322211426x_{61} = 76.9847322211426
x62=33.0237623788463x_{62} = 33.0237623788463
x63=−39.1894794603212x_{63} = -39.1894794603212
x64=−51.7751787593182x_{64} = -51.7751787593182
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (5*x)*cos(x) - 5*sin(x) - 3*cos(x) + 11.
(−3cos⁡(0)+(0⋅5cos⁡(0)−5sin⁡(0)))+11\left(- 3 \cos{\left(0 \right)} + \left(0 \cdot 5 \cos{\left(0 \right)} - 5 \sin{\left(0 \right)}\right)\right) + 11
Resultado:
f(0)=8f{\left(0 \right)} = 8
Punto:
(0, 8)
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
−5xsin⁡(x)+3sin⁡(x)=0- 5 x \sin{\left(x \right)} + 3 \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=35x_{2} = \frac{3}{5}
x3=πx_{3} = \pi
Signos de extremos en los puntos:
(0, 8)

(3/5, 11 - 5*sin(3/5))

(pi, 14 - 5*pi)


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=0x_{1} = 0
x2=πx_{2} = \pi
Puntos máximos de la función:
x2=35x_{2} = \frac{3}{5}
Decrece en los intervalos
[0,35]∪[π,∞)\left[0, \frac{3}{5}\right] \cup \left[\pi, \infty\right)
Crece en los intervalos
(−∞,0]∪[35,π]\left(-\infty, 0\right] \cup \left[\frac{3}{5}, \pi\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
−5xcos⁡(x)−5sin⁡(x)+3cos⁡(x)=0- 5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)} + 3 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−45.5747469515189x_{1} = -45.5747469515189
x2=48.7154664769241x_{2} = 48.7154664769241
x3=−11.0809753498408x_{3} = -11.0809753498408
x4=64.4183175986539x_{4} = 64.4183175986539
x5=36.1564324109877x_{5} = 36.1564324109877
x6=−54.9958564503977x_{6} = -54.9958564503977
x7=−95.8289458928569x_{7} = -95.8289458928569
x8=29.8792707944442x_{8} = 29.8792707944442
x9=−86.4052910215308x_{9} = -86.4052910215308
x10=−23.6032382011184x_{10} = -23.6032382011184
x11=−39.2949687396327x_{11} = -39.2949687396327
x12=86.4054517172713x_{12} = 86.4054517172713
x13=−64.4180285354548x_{13} = -64.4180285354548
x14=80.1231869525253x_{14} = 80.1231869525253
x15=−1.94516822769241x_{15} = -1.94516822769241
x16=−48.7149611738873x_{16} = -48.7149611738873
x17=−61.2772163755161x_{17} = -61.2772163755161
x18=58.1368425099087x_{18} = 58.1368425099087
x19=51.8557863017468x_{19} = 51.8557863017468
x20=−76.981908902113x_{20} = -76.981908902113
x21=−98.9702114148046x_{21} = -98.9702114148046
x22=20.4706353441694x_{22} = 20.4706353441694
x23=−80.1230000759427x_{23} = -80.1230000759427
x24=11.0906104690472x_{24} = 11.0906104690472
x25=−33.0164614236917x_{25} = -33.0164614236917
x26=83.2643018503341x_{26} = 83.2643018503341
x27=45.5753242286093x_{27} = 45.5753242286093
x28=14.2105078047203x_{28} = 14.2105078047203
x29=0.295541826625746x_{29} = 0.295541826625746
x30=54.9962529864659x_{30} = 54.9962529864659
x31=61.2775358169501x_{31} = 61.2775358169501
x32=89.5466328501838x_{32} = 89.5466328501838
x33=−51.8553403111838x_{33} = -51.8553403111838
x34=92.6878420508855x_{34} = 92.6878420508855
x35=−83.2641288042697x_{35} = -83.2641288042697
x36=−70.6998590594156x_{36} = -70.6998590594156
x37=67.5591754150481x_{37} = 67.5591754150481
x38=23.605385638108x_{38} = 23.605385638108
x39=76.9821113356993x_{39} = 76.9821113356993
x40=−4.89248333217413x_{40} = -4.89248333217413
x41=−89.5464832284687x_{41} = -89.5464832284687
x42=102.111612012768x_{42} = 102.111612012768
x43=−29.877929070978x_{43} = -29.877929070978
x44=−58.1364876390888x_{44} = -58.1364876390888
x45=73.8410800508079x_{45} = 73.8410800508079
x46=−73.8408600340127x_{46} = -73.8408600340127
x47=17.338431423236x_{47} = 17.338431423236
x48=−92.6877023968617x_{48} = -92.6877023968617
x49=98.9703339043433x_{49} = 98.9703339043433
x50=95.8290765429349x_{50} = 95.8290765429349
x51=2.14518169202759x_{51} = 2.14518169202759
x52=26.7417718676335x_{52} = 26.7417718676335
x53=−67.5589125951703x_{53} = -67.5589125951703
x54=−36.15551560738x_{54} = -36.15551560738
x55=42.4353994744485x_{55} = 42.4353994744485
x56=70.7000990533064x_{56} = 70.7000990533064
x57=−42.4347336864523x_{57} = -42.4347336864523
x58=−14.2046110168158x_{58} = -14.2046110168158
x59=−26.7400975719042x_{59} = -26.7400975719042
x60=−17.3344604946717x_{60} = -17.3344604946717
x61=−7.97014055927956x_{61} = -7.97014055927956
x62=−20.4677824898438x_{62} = -20.4677824898438
x63=4.93890638841786x_{63} = 4.93890638841786
x64=39.2957450538463x_{64} = 39.2957450538463
x65=7.98850946199004x_{65} = 7.98850946199004
x66=33.0175605622193x_{66} = 33.0175605622193

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
[102.111612012768,∞)\left[102.111612012768, \infty\right)
Convexa en los intervalos
(−∞,−95.8289458928569]\left(-\infty, -95.8289458928569\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((5xcos⁡(x)−5sin⁡(x))−3cos⁡(x))+11)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(\left(5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 3 \cos{\left(x \right)}\right) + 11\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(((5xcos⁡(x)−5sin⁡(x))−3cos⁡(x))+11)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(\left(5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 3 \cos{\left(x \right)}\right) + 11\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (5*x)*cos(x) - 5*sin(x) - 3*cos(x) + 11, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(((5xcos⁡(x)−5sin⁡(x))−3cos⁡(x))+11x)y = x \lim_{x \to -\infty}\left(\frac{\left(\left(5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 3 \cos{\left(x \right)}\right) + 11}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(((5xcos⁡(x)−5sin⁡(x))−3cos⁡(x))+11x)y = x \lim_{x \to \infty}\left(\frac{\left(\left(5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 3 \cos{\left(x \right)}\right) + 11}{x}\right)
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:
((5xcos⁡(x)−5sin⁡(x))−3cos⁡(x))+11=−5xcos⁡(x)+5sin⁡(x)−3cos⁡(x)+11\left(\left(5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 3 \cos{\left(x \right)}\right) + 11 = - 5 x \cos{\left(x \right)} + 5 \sin{\left(x \right)} - 3 \cos{\left(x \right)} + 11
- No
((5xcos⁡(x)−5sin⁡(x))−3cos⁡(x))+11=5xcos⁡(x)−5sin⁡(x)+3cos⁡(x)−11\left(\left(5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right) - 3 \cos{\left(x \right)}\right) + 11 = 5 x \cos{\left(x \right)} - 5 \sin{\left(x \right)} + 3 \cos{\left(x \right)} - 11
- No
es decir, función
no es
par ni impar
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