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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                    _________
f(x) = cos(7*x) - \/ 3 - 2*x
f(x)=−3−2x+cos⁡(7x)f{\left(x \right)} = - \sqrt{3 - 2 x} + \cos{\left(7 x \right)} 
f = -sqrt(3 - 2*x) + cos(7*x)
Gráfico de la función
02468-8-6-4-2-10100-10
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:
−3−2x+cos⁡(7x)=0- \sqrt{3 - 2 x} + \cos{\left(7 x \right)} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 cos(7*x) - sqrt(3 - 2*x).
−3−0+cos⁡(0⋅7)- \sqrt{3 - 0} + \cos{\left(0 \cdot 7 \right)}
Resultado:
f(0)=1−3f{\left(0 \right)} = 1 - \sqrt{3}
Punto:
(0, 1 - sqrt(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
−7sin⁡(7x)+13−2x=0- 7 \sin{\left(7 x \right)} + \frac{1}{\sqrt{3 - 2 x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−48.0235383853828x_{1} = -48.0235383853828
x2=−92.005277232862x_{2} = -92.005277232862
x3=−93.8004589115684x_{3} = -93.8004589115684
x4=−41.7404970244311x_{4} = -41.7404970244311
x5=−15.7114420179181x_{5} = -15.7114420179181
x6=−53.8539343907798x_{6} = -53.8539343907798
x7=−70.0109297534926x_{7} = -70.0109297534926
x8=−39.9453482510258x_{8} = -39.9453482510258
x9=−83.9269650952228x_{9} = -83.9269650952228
x10=−8.53173701229299x_{10} = -8.53173701229299
x11=−26.027588502925x_{11} = -26.027588502925
x12=−27.8228698479462x_{12} = -27.8228698479462
x13=−30.0720980721532x_{13} = -30.0720980721532
x14=−45.7753940754083x_{14} = -45.7753940754083
x15=−49.8186979987095x_{15} = -49.8186979987095
x16=−57.8969371054026x_{16} = -57.8969371054026
x17=−79.8846135343777x_{17} = -79.8846135343777
x18=−4.0453205320747x_{18} = -4.0453205320747
x19=−71.8061465818746x_{19} = -71.8061465818746
x20=−35.9015563027617x_{20} = -35.9015563027617
x21=−31.8672238234273x_{21} = -31.8672238234273
x22=0.011843010557566x_{22} = 0.011843010557566
x23=−81.6798266934498x_{23} = -81.6798266934498
x24=−73.1558991269434x_{24} = -73.1558991269434
x25=−59.6921052352214x_{25} = -59.6921052352214
x26=−19.7440226621522x_{26} = -19.7440226621522
x27=−61.9324432264544x_{27} = -61.9324432264544
x28=−97.8367233022134x_{28} = -97.8367233022134
x29=−65.9752025474228x_{29} = -65.9752025474228
x30=−33.2086727603554x_{30} = -33.2086727603554
x31=−5.83971418262412x_{31} = -5.83971418262412
x32=−89.7582794629219x_{32} = -89.7582794629219
x33=−1.78723239967825x_{33} = -1.78723239967825
x34=−43.9801572467789x_{34} = -43.9801572467789
x35=−9.86929648411665x_{35} = -9.86929648411665
x36=−13.9164432056044x_{36} = -13.9164432056044
x37=−52.0587063468326x_{37} = -52.0587063468326
x38=−67.7703754344952x_{38} = -67.7703754344952
x39=−37.6968067820758x_{39} = -37.6968067820758
x40=−32.3110425528725x_{40} = -32.3110425528725
x41=−75.8486634986945x_{41} = -75.8486634986945
x42=−96.0415142363773x_{42} = -96.0415142363773
x43=−87.9630685789546x_{43} = -87.9630685789546
x44=−17.9486855045077x_{44} = -17.9486855045077
x45=−63.7276641383059x_{45} = -63.7276641383059
x46=−23.7892141694748x_{46} = -23.7892141694748
x47=−74.0534870767904x_{47} = -74.0534870767904
x48=−21.9941259993555x_{48} = -21.9941259993555
x49=−85.722144745496x_{49} = -85.722144745496
Signos de extremos en los puntos:
(-48.023538385382814, -10.9521367587898)

(-92.00527723286199, -14.6751256695285)

(-93.8004589115684, -14.8057755869935)

(-41.740497024431086, -10.2993977973246)

(-15.711442017918086, -6.86680536594253)

(-53.8539343907798, -9.52187286040916)

(-70.0109297534926, -10.9592460489313)

(-39.945348251025806, -10.1043096836513)

(-83.92696509522281, -14.0710507928455)

(-8.531737012292988, -5.47871822560782)

(-26.027588502924985, -6.42010294753443)

(-27.822869847946222, -6.65822465636248)

(-30.072098072153224, -8.94617063203695)

(-45.77539407540834, -8.72383088484785)

(-49.81869799870947, -11.1309121742386)

(-57.89693710540264, -11.899174358736)

(-79.88461353437769, -11.758167060605)

(-4.045320532074696, -4.32934092475383)

(-71.80614658187456, -11.1084259376817)

(-35.9015563027617, -7.649015690589)

(-31.867223823427263, -9.16896255772696)

(0.011843010557566003, -0.728634018924508)

(-81.67982669344983, -11.8981097726265)

(-73.15589912694338, -13.2192523521769)

(-59.69210523522139, -12.0626563552676)

(-19.74402266215221, -5.51852565138521)

(-61.93244322645437, -10.2635118159474)

(-97.83672330221337, -13.0952082116244)

(-65.97520254742278, -12.6167400012389)

(-33.20867276035545, -7.33185424620093)

(-5.839714182624125, -4.83067884391322)

(-89.75827946292192, -12.5099248548129)

(-1.787232399678253, -1.56562520200588)

(-43.98015724677889, -8.53742388958547)

(-9.869296484116655, -3.76894893211706)

(-13.91644320560442, -6.55240583695575)

(-52.058706346832636, -9.34984949874665)

(-67.77037543449524, -12.7702598504057)

(-37.696806782075804, -7.8541469771012)

(-32.31104255287251, -7.223415779042)

(-75.84866349869448, -13.4376720539526)

(-96.04151423637734, -12.967264931317)

(-87.96306857895465, -12.3763845226288)

(-17.948685504507697, -5.23703805800253)

(-63.72766413830591, -10.4217824852578)

(-23.789214169474842, -8.1116494915633)

(-74.05348707679036, -13.2924901422487)

(-21.994125999355497, -7.85458054981262)

(-85.72214474549598, -14.2076774749311)


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=−48.0235383853828x_{1} = -48.0235383853828
x2=−92.005277232862x_{2} = -92.005277232862
x3=−93.8004589115684x_{3} = -93.8004589115684
x4=−41.7404970244311x_{4} = -41.7404970244311
x5=−15.7114420179181x_{5} = -15.7114420179181
x6=−39.9453482510258x_{6} = -39.9453482510258
x7=−83.9269650952228x_{7} = -83.9269650952228
x8=−8.53173701229299x_{8} = -8.53173701229299
x9=−30.0720980721532x_{9} = -30.0720980721532
x10=−49.8186979987095x_{10} = -49.8186979987095
x11=−57.8969371054026x_{11} = -57.8969371054026
x12=−4.0453205320747x_{12} = -4.0453205320747
x13=−31.8672238234273x_{13} = -31.8672238234273
x14=−73.1558991269434x_{14} = -73.1558991269434
x15=−59.6921052352214x_{15} = -59.6921052352214
x16=−65.9752025474228x_{16} = -65.9752025474228
x17=−5.83971418262412x_{17} = -5.83971418262412
x18=−13.9164432056044x_{18} = -13.9164432056044
x19=−67.7703754344952x_{19} = -67.7703754344952
x20=−75.8486634986945x_{20} = -75.8486634986945
x21=−23.7892141694748x_{21} = -23.7892141694748
x22=−74.0534870767904x_{22} = -74.0534870767904
x23=−21.9941259993555x_{23} = -21.9941259993555
x24=−85.722144745496x_{24} = -85.722144745496
Puntos máximos de la función:
x24=−53.8539343907798x_{24} = -53.8539343907798
x24=−70.0109297534926x_{24} = -70.0109297534926
x24=−26.027588502925x_{24} = -26.027588502925
x24=−27.8228698479462x_{24} = -27.8228698479462
x24=−45.7753940754083x_{24} = -45.7753940754083
x24=−79.8846135343777x_{24} = -79.8846135343777
x24=−71.8061465818746x_{24} = -71.8061465818746
x24=−35.9015563027617x_{24} = -35.9015563027617
x24=0.011843010557566x_{24} = 0.011843010557566
x24=−81.6798266934498x_{24} = -81.6798266934498
x24=−19.7440226621522x_{24} = -19.7440226621522
x24=−61.9324432264544x_{24} = -61.9324432264544
x24=−97.8367233022134x_{24} = -97.8367233022134
x24=−33.2086727603554x_{24} = -33.2086727603554
x24=−89.7582794629219x_{24} = -89.7582794629219
x24=−1.78723239967825x_{24} = -1.78723239967825
x24=−43.9801572467789x_{24} = -43.9801572467789
x24=−9.86929648411665x_{24} = -9.86929648411665
x24=−52.0587063468326x_{24} = -52.0587063468326
x24=−37.6968067820758x_{24} = -37.6968067820758
x24=−32.3110425528725x_{24} = -32.3110425528725
x24=−96.0415142363773x_{24} = -96.0415142363773
x24=−87.9630685789546x_{24} = -87.9630685789546
x24=−17.9486855045077x_{24} = -17.9486855045077
x24=−63.7276641383059x_{24} = -63.7276641383059
Decrece en los intervalos
[−4.0453205320747,∞)\left[-4.0453205320747, \infty\right)
Crece en los intervalos
(−∞,−93.8004589115684]\left(-\infty, -93.8004589115684\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
−49cos⁡(7x)+1(3−2x)32=0- 49 \cos{\left(7 x \right)} + \frac{1}{\left(3 - 2 x\right)^{\frac{3}{2}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−47.7970852515864x_{1} = -47.7970852515864
x2=−17.7275707709715x_{2} = -17.7275707709715
x3=−99.8577654789786x_{3} = -99.8577654789786
x4=−87.7401960479584x_{4} = -87.7401960479584
x5=−21.7667399153285x_{5} = -21.7667399153285
x6=−59.9146620351336x_{6} = -59.9146620351336
x7=−13.6883505767845x_{7} = -13.6883505767845
x8=−35.679521112563x_{8} = -35.679521112563
x9=−72.0322299225854x_{9} = -72.0322299225854
x10=−11.8931511587129x_{10} = -11.8931511587129
x11=−15.9323769052506x_{11} = -15.9323769052506
x12=−7.85401766392059x_{12} = -7.85401766392059
x13=−76.0714236206429x_{13} = -76.0714236206429
x14=−91.7793842357248x_{14} = -91.7793842357248
x15=−25.8059468784425x_{15} = -25.8059468784425
x16=−67.9930427819922x_{16} = -67.9930427819922
x17=−83.7010029599709x_{17} = -83.7010029599709
x18=−32.537918721594x_{18} = -32.537918721594
x19=−95.8185770081499x_{19} = -95.8185770081499
x20=−80.5594130037071x_{20} = -80.5594130037071
x21=−73.827425782725x_{21} = -73.827425782725
x22=−29.8451243354824x_{22} = -29.8451243354824
x23=−94.0233812365067x_{23} = -94.0233812365067
x24=−24.0107518521786x_{24} = -24.0107518521786
x25=−89.9841883998313x_{25} = -89.9841883998313
x26=−63.9538485015614x_{26} = -63.9538485015614
x27=−77.8666193717961x_{27} = -77.8666193717961
x28=−6.05883543156724x_{28} = -6.05883543156724
x29=−69.7882385172584x_{29} = -69.7882385172584
x30=−37.9235071541682x_{30} = -37.9235071541682
x31=−51.8362814304567x_{31} = -51.8362814304567
x32=−39.7187032252782x_{32} = -39.7187032252782
x33=−50.0410857678559x_{33} = -50.0410857678559
x34=−2.01943916009825x_{34} = -2.01943916009825
x35=−98.0625696494844x_{35} = -98.0625696494844
x36=−65.7490443810328x_{36} = -65.7490443810328
x37=−61.7098577465988x_{37} = -61.7098577465988
x38=−41.9627054703247x_{38} = -41.9627054703247
x39=−81.9058071153771x_{39} = -81.9058071153771
x40=−28.0499279901373x_{40} = -28.0499279901373
x41=−43.7579010604731x_{41} = -43.7579010604731
x42=−33.8843256608808x_{42} = -33.8843256608808
x43=−3.81470695109503x_{43} = -3.81470695109503
x44=0.223684607533485x_{44} = 0.223684607533485
x45=−85.9450002837517x_{45} = -85.9450002837517
x46=−46.0018892791311x_{46} = -46.0018892791311
x47=−55.8754669670827x_{47} = -55.8754669670827
x48=−19.9715429376717x_{48} = -19.9715429376717

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
[0.223684607533485,∞)\left[0.223684607533485, \infty\right)
Convexa en los intervalos
(−∞,−95.8185770081499]\left(-\infty, -95.8185770081499\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−3−2x+cos⁡(7x))=−∞\lim_{x \to -\infty}\left(- \sqrt{3 - 2 x} + \cos{\left(7 x \right)}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(−3−2x+cos⁡(7x))=⟨−1,1⟩−∞i\lim_{x \to \infty}\left(- \sqrt{3 - 2 x} + \cos{\left(7 x \right)}\right) = \left\langle -1, 1\right\rangle - \infty i
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1,1⟩−∞iy = \left\langle -1, 1\right\rangle - \infty i
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(7*x) - sqrt(3 - 2*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−3−2x+cos⁡(7x)x)=0\lim_{x \to -\infty}\left(\frac{- \sqrt{3 - 2 x} + \cos{\left(7 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−3−2x+cos⁡(7x)x)=0\lim_{x \to \infty}\left(\frac{- \sqrt{3 - 2 x} + \cos{\left(7 x \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:
−3−2x+cos⁡(7x)=−2x+3+cos⁡(7x)- \sqrt{3 - 2 x} + \cos{\left(7 x \right)} = - \sqrt{2 x + 3} + \cos{\left(7 x \right)}
- No
−3−2x+cos⁡(7x)=2x+3−cos⁡(7x)- \sqrt{3 - 2 x} + \cos{\left(7 x \right)} = \sqrt{2 x + 3} - \cos{\left(7 x \right)}
- No
es decir, función
no es
par ni impar
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