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Trazado el gráfico de la función/ 2*cos(9-4*x)+sqrt(2*x)-3/7

Gráfico de la función y = 2*cos(9-4*x)+sqrt(2*x)-3/7

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                          _____   3
f(x) = 2*cos(9 - 4*x) + \/ 2*x  - -
                                  7
f(x)=(2x+2cos⁡(9−4x))−37f{\left(x \right)} = \left(\sqrt{2 x} + 2 \cos{\left(9 - 4 x \right)}\right) - \frac{3}{7} 
f = sqrt(2*x) + 2*cos(9 - 4*x) - 3/7
Gráfico de la función
02468-8-6-4-2-1010-1010
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:
(2x+2cos⁡(9−4x))−37=0\left(\sqrt{2 x} + 2 \cos{\left(9 - 4 x \right)}\right) - \frac{3}{7} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=0.251335943989918x_{1} = 0.251335943989918
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 2*cos(9 - 4*x) + sqrt(2*x) - 3/7.
(2cos⁡(9−0)+0⋅2)−37\left(2 \cos{\left(9 - 0 \right)} + \sqrt{0 \cdot 2}\right) - \frac{3}{7}
Resultado:
f(0)=2cos⁡(9)−37f{\left(0 \right)} = 2 \cos{\left(9 \right)} - \frac{3}{7}
Punto:
(0, -3/7 + 2*cos(9))
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
−8sin⁡(4x−9)+2x2x=0- 8 \sin{\left(4 x - 9 \right)} + \frac{\sqrt{2} \sqrt{x}}{2 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=54.0892834063863x_{1} = 54.0892834063863
x2=24.2456364584294x_{2} = 24.2456364584294
x3=69.7968870486663x_{3} = 69.7968870486663
x4=60.3723080646223x_{4} = 60.3723080646223
x5=90.2169207696331x_{5} = 90.2169207696331
x6=82.3630475360198x_{6} = 82.3630475360198
x7=3.83208817299133x_{7} = 3.83208817299133
x8=34.4475596366142x_{8} = 34.4475596366142
x9=39.9526078848436x_{9} = 39.9526078848436
x10=65.8645284221869x_{10} = 65.8645284221869
x11=94.1393076324715x_{11} = 94.1393076324715
x12=25.8162941079439x_{12} = 25.8162941079439
x13=72.1478349981642x_{13} = 72.1478349981642
x14=38.3818823808778x_{14} = 38.3818823808778
x15=17.9631773148193x_{15} = 17.9631773148193
x16=32.0990305869768x_{16} = 32.0990305869768
x17=68.2261209975607x_{17} = 68.2261209975607
x18=61.9430680994263x_{18} = 61.9430680994263
x19=86.2852245990039x_{19} = 86.2852245990039
x20=76.0799607800728x_{20} = 76.0799607800728
x21=10.1109318031739x_{21} = 10.1109318031739
x22=2.26469198465725x_{22} = 2.26469198465725
x23=14.0250714890693x_{23} = 14.0250714890693
x24=28.1639754169949x_{24} = 28.1639754169949
x25=87.8560422889261x_{25} = 87.8560422889261
x26=100.422565341385x_{26} = 100.422565341385
x27=78.4311266918072x_{27} = 78.4311266918072
x28=16.3926250887216x_{28} = 16.3926250887216
x29=6.1680916093823x_{29} = 6.1680916093823
x30=47.8062894591274x_{30} = 47.8062894591274
x31=90.9976759970707x_{31} = 90.9976759970707
x32=64.2936990327761x_{32} = 64.2936990327761
x33=98.0708073010212x_{33} = 98.0708073010212
x34=56.4395318794755x_{34} = 56.4395318794755
x35=58.0103683057417x_{35} = 58.0103683057417
x36=20.3092541485659x_{36} = 20.3092541485659
x37=46.2355469698698x_{37} = 46.2355469698698
x38=43.8727664729265x_{38} = 43.8727664729265
x39=80.0019476367872x_{39} = 80.0019476367872
x40=36.0184389879603x_{40} = 36.0184389879603
x41=21.8802298125211x_{41} = 21.8802298125211
x42=83.9338209709017x_{42} = 83.9338209709017
x43=51.7270118267972x_{43} = 51.7270118267972
x44=42.3019087632911x_{44} = 42.3019087632911
x45=50.1561677550782x_{45} = 50.1561677550782
Signos de extremos en los puntos:
(54.0892834063863, 11.9721767260237)

(24.245636458429427, 8.53467386488745)

(69.79688704866633, 13.386297554566)

(60.37230806462229, 12.5596846724716)

(90.21692076963312, 15.0039084266473)

(82.36304753601978, 14.4059001301765)

(3.8320881729913285, 4.33781371096655)

(34.447559636614244, 5.87196376251811)

(39.95260788484358, 10.5102047577523)

(65.86452842218691, 9.04887513842969)

(94.13930763247146, 11.2929770544756)

(25.816294107943943, 8.75670789736752)

(72.1478349981642, 9.58385012479324)

(38.38188238087779, 10.3327183097085)

(17.963177314819262, 7.56485335054724)

(32.09903058697677, 9.58355443016341)

(68.2261209975607, 13.2525914720361)

(61.94306809942629, 12.7017173594174)

(86.28522459900393, 10.7081263341344)

(76.07996078007275, 13.9066378573831)

(10.110931803173926, 6.06752839090999)

(2.264691984657253, 3.69621083691683)

(14.02507148906931, 2.86822425602854)

(28.163975416994877, 5.07690090715368)

(87.85604228892612, 10.8271609964825)

(100.42256534138511, 11.743490375782)

(78.43112669180718, 10.0959943751599)

(16.392625088721587, 7.29679249350779)

(6.1680916093823, 1.084985881547)

(47.80628945912742, 11.3494336296887)

(90.99767599707071, 11.0620797226861)

(64.29369903277615, 8.91118844471726)

(98.07080730102119, 15.57640565913)

(56.43953187947552, 8.19602292636717)

(58.01036830574175, 8.34285549443472)

(20.309254148565852, 3.94507840585862)

(46.2355469698698, 11.1874487514675)

(43.872766472926514, 6.93886521195164)

(80.00194763678718, 10.2207908416293)

(36.018438987960344, 6.05909963760818)

(21.880229812521087, 4.18695456454217)

(83.9338209709017, 14.5277100578039)

(51.72701182679721, 7.74281471994944)

(42.30190876329105, 6.76964661531981)

(50.15616775507822, 7.58718894132015)


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=34.4475596366142x_{1} = 34.4475596366142
x2=65.8645284221869x_{2} = 65.8645284221869
x3=94.1393076324715x_{3} = 94.1393076324715
x4=72.1478349981642x_{4} = 72.1478349981642
x5=86.2852245990039x_{5} = 86.2852245990039
x6=14.0250714890693x_{6} = 14.0250714890693
x7=28.1639754169949x_{7} = 28.1639754169949
x8=87.8560422889261x_{8} = 87.8560422889261
x9=100.422565341385x_{9} = 100.422565341385
x10=78.4311266918072x_{10} = 78.4311266918072
x11=6.1680916093823x_{11} = 6.1680916093823
x12=90.9976759970707x_{12} = 90.9976759970707
x13=64.2936990327761x_{13} = 64.2936990327761
x14=56.4395318794755x_{14} = 56.4395318794755
x15=58.0103683057417x_{15} = 58.0103683057417
x16=20.3092541485659x_{16} = 20.3092541485659
x17=43.8727664729265x_{17} = 43.8727664729265
x18=80.0019476367872x_{18} = 80.0019476367872
x19=36.0184389879603x_{19} = 36.0184389879603
x20=21.8802298125211x_{20} = 21.8802298125211
x21=51.7270118267972x_{21} = 51.7270118267972
x22=42.3019087632911x_{22} = 42.3019087632911
x23=50.1561677550782x_{23} = 50.1561677550782
Puntos máximos de la función:
x23=54.0892834063863x_{23} = 54.0892834063863
x23=24.2456364584294x_{23} = 24.2456364584294
x23=69.7968870486663x_{23} = 69.7968870486663
x23=60.3723080646223x_{23} = 60.3723080646223
x23=90.2169207696331x_{23} = 90.2169207696331
x23=82.3630475360198x_{23} = 82.3630475360198
x23=3.83208817299133x_{23} = 3.83208817299133
x23=39.9526078848436x_{23} = 39.9526078848436
x23=25.8162941079439x_{23} = 25.8162941079439
x23=38.3818823808778x_{23} = 38.3818823808778
x23=17.9631773148193x_{23} = 17.9631773148193
x23=32.0990305869768x_{23} = 32.0990305869768
x23=68.2261209975607x_{23} = 68.2261209975607
x23=61.9430680994263x_{23} = 61.9430680994263
x23=76.0799607800728x_{23} = 76.0799607800728
x23=10.1109318031739x_{23} = 10.1109318031739
x23=2.26469198465725x_{23} = 2.26469198465725
x23=16.3926250887216x_{23} = 16.3926250887216
x23=47.8062894591274x_{23} = 47.8062894591274
x23=98.0708073010212x_{23} = 98.0708073010212
x23=46.2355469698698x_{23} = 46.2355469698698
x23=83.9338209709017x_{23} = 83.9338209709017
Decrece en los intervalos
[100.422565341385,∞)\left[100.422565341385, \infty\right)
Crece en los intervalos
(−∞,6.1680916093823]\left(-\infty, 6.1680916093823\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
−(32cos⁡(4x−9)+24x32)=0- (32 \cos{\left(4 x - 9 \right)} + \frac{\sqrt{2}}{4 x^{\frac{3}{2}}}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=74.1139276217897x_{1} = 74.1139276217897
x2=44.2687923640162x_{2} = 44.2687923640162
x3=85.8949078715441x_{3} = 85.8949078715441
x4=67.8307416993827x_{4} = 67.8307416993827
x5=70.1869458313115x_{5} = 70.1869458313115
x6=77.2555205367391x_{6} = 77.2555205367391
x7=12.8528152616001x_{7} = 12.8528152616001
x8=23.8484257766642x_{8} = 23.8484257766642
x9=36.4148075379994x_{9} = 36.4148075379994
x10=58.40596249479x_{10} = 58.40596249479
x11=8.14036729886751x_{11} = 8.14036729886751
x12=18.3506974867687x_{12} = 18.3506974867687
x13=4.21381473281588x_{13} = 4.21381473281588
x14=12.0675429320517x_{14} = 12.0675429320517
x15=40.3418217046029x_{15} = 40.3418217046029
x16=59.9767590630875x_{16} = 59.9767590630875
x17=41.9126174311043x_{17} = 41.9126174311043
x18=89.0365003431236x_{18} = 89.0365003431236
x19=19.9214897407961x_{19} = 19.9214897407961
x20=80.3971134264092x_{20} = 80.3971134264092
x21=66.259945195731x_{21} = 66.259945195731
x22=4.99864641855277x_{22} = 4.99864641855277
x23=96.1050775942545x_{23} = 96.1050775942545
x24=62.3329651125599x_{24} = 62.3329651125599
x25=56.0497807751235x_{25} = 56.0497807751235
x26=71.7577420047115x_{26} = 71.7577420047115
x27=84.3241116421505x_{27} = 84.3241116421505
x28=37.2002304449985x_{28} = 37.2002304449985
x29=100.03207410379x_{29} = 100.03207410379
x30=78.0409267743193x_{30} = 78.0409267743193
x31=52.1227760356083x_{31} = 52.1227760356083
x32=63.9037612336877x_{32} = 63.9037612336877
x33=89.8218919741365x_{33} = 89.8218919741365
x34=1.85620870814728x_{34} = 1.85620870814728
x35=37.9856046363644x_{35} = 37.9856046363644
x36=15.9944246785141x_{36} = 15.9944246785141
x37=48.1958008140198x_{37} = 48.1958008140198
x38=81.9679098628164x_{38} = 81.9679098628164
x39=1249.85498261936x_{39} = 1249.85498261936
x40=22.2776268977494x_{40} = 22.2776268977494
x41=45.8395891686823x_{41} = 45.8395891686823
x42=92.1780928300775x_{42} = 92.1780928300775
x43=30.1316181008061x_{43} = 30.1316181008061
x44=34.0586395140538x_{44} = 34.0586395140538
x45=26.2046645746075x_{45} = 26.2046645746075
x46=99.2466703859405x_{46} = 99.2466703859405
x47=14.4236211090915x_{47} = 14.4236211090915
x48=88.2510955603283x_{48} = 88.2510955603283

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
[1249.85498261936,∞)\left[1249.85498261936, \infty\right)
Convexa en los intervalos
(−∞,4.21381473281588]\left(-\infty, 4.21381473281588\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((2x+2cos⁡(9−4x))−37)=⟨−177,117⟩+∞i\lim_{x \to -\infty}\left(\left(\sqrt{2 x} + 2 \cos{\left(9 - 4 x \right)}\right) - \frac{3}{7}\right) = \left\langle - \frac{17}{7}, \frac{11}{7}\right\rangle + \infty i
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−177,117⟩+∞iy = \left\langle - \frac{17}{7}, \frac{11}{7}\right\rangle + \infty i
lim⁡x→∞((2x+2cos⁡(9−4x))−37)=∞\lim_{x \to \infty}\left(\left(\sqrt{2 x} + 2 \cos{\left(9 - 4 x \right)}\right) - \frac{3}{7}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 2*cos(9 - 4*x) + sqrt(2*x) - 3/7, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((2x+2cos⁡(9−4x))−37x)=0\lim_{x \to -\infty}\left(\frac{\left(\sqrt{2 x} + 2 \cos{\left(9 - 4 x \right)}\right) - \frac{3}{7}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((2x+2cos⁡(9−4x))−37x)=0\lim_{x \to \infty}\left(\frac{\left(\sqrt{2 x} + 2 \cos{\left(9 - 4 x \right)}\right) - \frac{3}{7}}{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:
(2x+2cos⁡(9−4x))−37=2−x+2cos⁡(4x+9)−37\left(\sqrt{2 x} + 2 \cos{\left(9 - 4 x \right)}\right) - \frac{3}{7} = \sqrt{2} \sqrt{- x} + 2 \cos{\left(4 x + 9 \right)} - \frac{3}{7}
- No
(2x+2cos⁡(9−4x))−37=−2−x−2cos⁡(4x+9)+37\left(\sqrt{2 x} + 2 \cos{\left(9 - 4 x \right)}\right) - \frac{3}{7} = - \sqrt{2} \sqrt{- x} - 2 \cos{\left(4 x + 9 \right)} + \frac{3}{7}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор