Sr Examen

Otras calculadoras

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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
            /x\        /x\
f(x) = 3*sin|-| + 4*cos|-|
            \2/        \5/
f(x)=3sin⁡(x2)+4cos⁡(x5)f{\left(x \right)} = 3 \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{5} \right)} 
f = 3*sin(x/2) + 4*cos(x/5)
Gráfico de la función
05-20-15-10-5201015-1010
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:
3sin⁡(x2)+4cos⁡(x5)=03 \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{5} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−83.3892412300318x_{1} = -83.3892412300318
x2=105.106317985356x_{2} = 105.106317985356
x3=−146.221094301828x_{3} = -146.221094301828
x4=−10.858538377662x_{4} = -10.858538377662
x5=42.2744649135599x_{5} = 42.2744649135599
x6=994.451110771072x_{6} = 994.451110771072
x7=−38.2474662612071x_{7} = -38.2474662612071
x8=87.4162398823847x_{8} = 87.4162398823847
x9=69.663392797105x_{9} = 69.663392797105
x10=−56.0003133464867x_{10} = -56.0003133464867
x11=24.5843868105888x_{11} = 24.5843868105888
x12=−73.6903914494578x_{12} = -73.6903914494578
x13=6.83153972530914x_{13} = 6.83153972530914
x14=51.9733146941339x_{14} = 51.9733146941339
x15=−20.5573881582359x_{15} = -20.5573881582359
x16=−101.079319333003x_{16} = -101.079319333003
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 3*sin(x/2) + 4*cos(x/5).
3sin⁡(02)+4cos⁡(05)3 \sin{\left(\frac{0}{2} \right)} + 4 \cos{\left(\frac{0}{5} \right)}
Resultado:
f(0)=4f{\left(0 \right)} = 4
Punto:
(0, 4)
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
−4sin⁡(x5)5+3cos⁡(x2)2=0- \frac{4 \sin{\left(\frac{x}{5} \right)}}{5} + \frac{3 \cos{\left(\frac{x}{2} \right)}}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−96.8521246486164x_{1} = -96.8521246486164
x2=28.8115814949753x_{2} = 28.8115814949753
x3=47.1238898038469x_{3} = 47.1238898038469
x4=83.8535401337678x_{4} = 83.8535401337678
x5=−66.744480776607x_{5} = -66.744480776607
x6=2.60434504092264x_{6} = 2.60434504092264
x7=98.1604073125049x_{7} = 98.1604073125049
x8=10.394239473926x_{8} = 10.394239473926
x9=−85.9429556398339x_{9} = -85.9429556398339
x10=78.5398163397448x_{10} = 78.5398163397448
x11=−90.3351519028827x_{11} = -90.3351519028827
x12=−66.7444807766067x_{12} = -66.7444807766067
x13=−27.5032988310868x_{13} = -27.5032988310868
x14=−41.8101660098239x_{14} = -41.8101660098239
x15=21.0216870619719x_{15} = 21.0216870619719
x16=−192.408186920199x_{16} = -192.408186920199
x17=39.7207505037578x_{17} = 39.7207505037578
x18=−78.5398163397448x_{18} = -78.5398163397448
x19=−60.2275080308732x_{19} = -60.2275080308732
x20=−34.0202715768206x_{20} = -34.0202715768206
x21=−3.91262770481113x_{21} = -3.91262770481113
x22=15.707963267949x_{22} = 15.707963267949
x23=−15.707963267949x_{23} = -15.707963267949
x24=73.2260925457219x_{24} = 73.2260925457219
x25=−235.619449019234x_{25} = -235.619449019234
x26=65.4361981127185x_{26} = 65.4361981127185
x27=91.6434345667712x_{27} = 91.6434345667712
x28=−23.111102568038x_{28} = -23.111102568038
x29=35.3285542407091x_{29} = 35.3285542407091
x30=−71.1366770396558x_{30} = -71.1366770396558
x31=54.527029103936x_{31} = 54.527029103936
x32=−52.4376135978699x_{32} = -52.4376135978699
x33=58.9192253669847x_{33} = 58.9192253669847
x34=−8.30482396785991x_{34} = -8.30482396785991
x35=−47.1238898038469x_{35} = -47.1238898038469
Signos de extremos en los puntos:
(-96.85212464861644, 6.36195940974127)

(28.811581494975293, 6.36195940974127)

(47.1238898038469, -7)

(83.8535401337678, -4.60030467360184)

(-66.74448077660699, 0.0567286054710046)

(2.60434504092264, 6.36195940974127)

(98.16040731250493, 0.0567286054709988)

(10.394239473926005, -4.60030467360185)

(-85.94295563983388, 2.18161665838957)

(78.53981633974483, -1)

(-90.33515190288267, 0.0567286054710108)

(-66.74448077660674, 0.0567286054710006)

(-27.503298831086806, 0.0567286054710054)

(-41.81016600982394, -4.60030467360185)

(21.021687061971928, -4.60030467360184)

(-192.40818692019872, 0.0567286054710046)

(39.72075050375784, 2.18161665838957)

(-78.53981633974483, -7)

(-60.227508030873224, 6.36195940974127)

(-34.02027157682057, 6.36195940974127)

(-3.9126277048111255, 0.0567286054710041)

(15.707963267948966, -1)

(-15.707963267948966, -7)

(73.22609254572187, -4.60030467360185)

(-235.61944901923448, -1)

(65.4361981127185, 6.36195940974127)

(91.64343456677116, 6.36195940974127)

(-23.11110256803802, 2.18161665838957)

(35.32855424070906, 0.0567286054710046)

(-71.13667703965578, 2.18161665838957)

(54.52702910393595, 2.18161665838957)

(-52.43761359786986, -4.60030467360184)

(58.91922536698474, 0.0567286054710059)

(-8.304823967859912, 2.18161665838957)

(-47.1238898038469, -1)


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=47.1238898038469x_{1} = 47.1238898038469
x2=83.8535401337678x_{2} = 83.8535401337678
x3=−66.744480776607x_{3} = -66.744480776607
x4=98.1604073125049x_{4} = 98.1604073125049
x5=10.394239473926x_{5} = 10.394239473926
x6=−90.3351519028827x_{6} = -90.3351519028827
x7=−66.7444807766067x_{7} = -66.7444807766067
x8=−27.5032988310868x_{8} = -27.5032988310868
x9=−41.8101660098239x_{9} = -41.8101660098239
x10=21.0216870619719x_{10} = 21.0216870619719
x11=−192.408186920199x_{11} = -192.408186920199
x12=−78.5398163397448x_{12} = -78.5398163397448
x13=−3.91262770481113x_{13} = -3.91262770481113
x14=−15.707963267949x_{14} = -15.707963267949
x15=73.2260925457219x_{15} = 73.2260925457219
x16=35.3285542407091x_{16} = 35.3285542407091
x17=−52.4376135978699x_{17} = -52.4376135978699
x18=58.9192253669847x_{18} = 58.9192253669847
Puntos máximos de la función:
x18=−96.8521246486164x_{18} = -96.8521246486164
x18=28.8115814949753x_{18} = 28.8115814949753
x18=2.60434504092264x_{18} = 2.60434504092264
x18=−85.9429556398339x_{18} = -85.9429556398339
x18=78.5398163397448x_{18} = 78.5398163397448
x18=39.7207505037578x_{18} = 39.7207505037578
x18=−60.2275080308732x_{18} = -60.2275080308732
x18=−34.0202715768206x_{18} = -34.0202715768206
x18=15.707963267949x_{18} = 15.707963267949
x18=−235.619449019234x_{18} = -235.619449019234
x18=65.4361981127185x_{18} = 65.4361981127185
x18=91.6434345667712x_{18} = 91.6434345667712
x18=−23.111102568038x_{18} = -23.111102568038
x18=−71.1366770396558x_{18} = -71.1366770396558
x18=54.527029103936x_{18} = 54.527029103936
x18=−8.30482396785991x_{18} = -8.30482396785991
x18=−47.1238898038469x_{18} = -47.1238898038469
Decrece en los intervalos
[98.1604073125049,∞)\left[98.1604073125049, \infty\right)
Crece en los intervalos
(−∞,−192.408186920199]\left(-\infty, -192.408186920199\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
−75sin⁡(x2)+16cos⁡(x5)100=0- \frac{75 \sin{\left(\frac{x}{2} \right)} + 16 \cos{\left(\frac{x}{5} \right)}}{100} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−56.4266830427416x_{1} = -56.4266830427416
x2=1488.7855091588x_{2} = 1488.7855091588
x3=75.7627711316745x_{3} = 75.7627711316745
x4=−37.8210965649522x_{4} = -37.8210965649522
x5=37.5555459234864x_{5} = 37.5555459234864
x6=94.6761491066091x_{6} = 94.6761491066091
x7=−12.2369619715925x_{7} = -12.2369619715925
x8=100.387398995282x_{8} = 100.387398995282
x9=−100.652949636748x_{9} = -100.652949636748
x10=31.8442960348133x_{10} = 31.8442960348133
x11=18.4850084760193x_{11} = 18.4850084760193
x12=−44.3468445957765x_{12} = -44.3468445957765
x13=12.9309180598786x_{13} = 12.9309180598786
x14=−333.338229923285x_{14} = -333.338229923285
x15=−75.0688150433883x_{15} = -75.0688150433883
x16=−30.9875570369826x_{16} = -30.9875570369826
x17=−49.9009350119173x_{17} = -49.9009350119173
x18=−63.2602225707112x_{18} = -63.2602225707112
x19=62.4034835728805x_{19} = 62.4034835728805
x20=6.40517002905429x_{20} = 6.40517002905429
x21=56.6922336842074x_{21} = 56.6922336842074
x22=132.068876172646x_{22} = 132.068876172646
x23=69.2370231008502x_{23} = 69.2370231008502
x24=−68.9714724593843x_{24} = -68.9714724593843
x25=−3675.23503520114x_{25} = -3675.23503520114
x26=−6.13961938758844x_{26} = -6.13961938758844
x27=43.6528885074904x_{27} = 43.6528885074904
x28=25.0107565068436x_{28} = 25.0107565068436
x29=−88.1081602201054x_{29} = -88.1081602201054
x30=−93.8194101087785x_{30} = -93.8194101087785
x31=−0.428369498915326x_{31} = -0.428369498915326
x32=1595.80708330174x_{32} = 1595.80708330174
x33=−25.2763071483095x_{33} = -25.2763071483095
x34=50.5948911002034x_{34} = 50.5948911002034
x35=−1156.24966244064x_{35} = -1156.24966244064
x36=87.8426095786395x_{36} = 87.8426095786395
x37=−19.1789645643055x_{37} = -19.1789645643055
x38=−82.0108176361013x_{38} = -82.0108176361013
x39=81.3168615478152x_{39} = 81.3168615478152

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
[1488.7855091588,∞)\left[1488.7855091588, \infty\right)
Convexa en los intervalos
(−∞,−3675.23503520114]\left(-\infty, -3675.23503520114\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(3sin⁡(x2)+4cos⁡(x5))=⟨−7,7⟩\lim_{x \to -\infty}\left(3 \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{5} \right)}\right) = \left\langle -7, 7\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−7,7⟩y = \left\langle -7, 7\right\rangle
lim⁡x→∞(3sin⁡(x2)+4cos⁡(x5))=⟨−7,7⟩\lim_{x \to \infty}\left(3 \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{5} \right)}\right) = \left\langle -7, 7\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−7,7⟩y = \left\langle -7, 7\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 3*sin(x/2) + 4*cos(x/5), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(3sin⁡(x2)+4cos⁡(x5)x)=0\lim_{x \to -\infty}\left(\frac{3 \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{5} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(3sin⁡(x2)+4cos⁡(x5)x)=0\lim_{x \to \infty}\left(\frac{3 \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{5} \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:
3sin⁡(x2)+4cos⁡(x5)=−3sin⁡(x2)+4cos⁡(x5)3 \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{5} \right)} = - 3 \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{5} \right)}
- No
3sin⁡(x2)+4cos⁡(x5)=3sin⁡(x2)−4cos⁡(x5)3 \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{5} \right)} = 3 \sin{\left(\frac{x}{2} \right)} - 4 \cos{\left(\frac{x}{5} \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор