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Trazado el gráfico de la función/ (10*ln((x+5)^2)+5)/(300*((sin)^2)*(((x/3)+5)^3)+7)

Gráfico de la función y = (10*ln((x+5)^2)+5)/(300*((sin)^2)*(((x/3)+5)^3)+7)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
               /       2\      
         10*log\(x + 5) / + 5  
f(x) = ------------------------
                          3    
              2    /x    \     
       300*sin (x)*|- + 5|  + 7
                   \3    /
f(x)=10log⁡((x+5)2)+5(x3+5)3⋅300sin⁡2(x)+7f{\left(x \right)} = \frac{10 \log{\left(\left(x + 5\right)^{2} \right)} + 5}{\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7} 
f = (10*log((x + 5)^2) + 5)/((x/3 + 5)^3*(300*sin(x)^2) + 7)
Gráfico de la función
05-30-25-20-15-10-53010152025-500500
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:
10log⁡((x+5)2)+5(x3+5)3⋅300sin⁡2(x)+7=0\frac{10 \log{\left(\left(x + 5\right)^{2} \right)} + 5}{\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−5−e−14x_{1} = -5 - e^{- \frac{1}{4}}
x2=−5+e−14x_{2} = -5 + e^{- \frac{1}{4}}
Solución numérica
x1=−5.7788007830714x_{1} = -5.7788007830714
x2=−4.2211992169286x_{2} = -4.2211992169286
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 (10*log((x + 5)^2) + 5)/((300*sin(x)^2)*(x/3 + 5)^3 + 7).
5+10log⁡(52)(03+5)3⋅300sin⁡2(0)+7\frac{5 + 10 \log{\left(5^{2} \right)}}{\left(\frac{0}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(0 \right)} + 7}
Resultado:
f(0)=57+10log⁡(25)7f{\left(0 \right)} = \frac{5}{7} + \frac{10 \log{\left(25 \right)}}{7}
Punto:
(0, 5/7 + 10*log(25)/7)
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
(−600(x3+5)3sin⁡(x)cos⁡(x)−300(x3+5)2sin⁡2(x))(10log⁡((x+5)2)+5)((x3+5)3⋅300sin⁡2(x)+7)2+10(2x+10)(x+5)2((x3+5)3⋅300sin⁡2(x)+7)=0\frac{\left(- 600 \left(\frac{x}{3} + 5\right)^{3} \sin{\left(x \right)} \cos{\left(x \right)} - 300 \left(\frac{x}{3} + 5\right)^{2} \sin^{2}{\left(x \right)}\right) \left(10 \log{\left(\left(x + 5\right)^{2} \right)} + 5\right)}{\left(\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7\right)^{2}} + \frac{10 \left(2 x + 10\right)}{\left(x + 5\right)^{2} \left(\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7\right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−83.2727874973584x_{1} = -83.2727874973584
x2=95.8310897856742x_{2} = 95.8310897856742
x3=−39.3276550335433x_{3} = -39.3276550335433
x4=67.560889266173x_{4} = 67.560889266173
x5=70.7018927392371x_{5} = 70.7018927392371
x6=−36.1946474863643x_{6} = -36.1946474863643
x7=−33.0646303012334x_{7} = -33.0646303012334
x8=−80.1321829512664x_{8} = -80.1321829512664
x9=−76.9916785989834x_{9} = -76.9916785989834
x10=−13.3747250275116x_{10} = -13.3747250275116
x11=−17.7653088173889x_{11} = -17.7653088173889
x12=51.8566623761523x_{12} = 51.8566623761523
x13=−26.8229665926194x_{13} = -26.8229665926194
x14=−29.9394719831602x_{14} = -29.9394719831602
x15=−70.7110384364717x_{15} = -70.7110384364717
x16=−45.5989676823298x_{16} = -45.5989676823298
x17=−48.7362866272455x_{17} = -48.7362866272455
x18=98.9723464721323x_{18} = 98.9723464721323
x19=33.0145731944214x_{19} = 33.0145731944214
x20=89.5486353126801x_{20} = 89.5486353126801
x21=−89.5542464262811x_{21} = -89.5542464262811
x22=20.4570078590476x_{22} = 20.4570078590476
x23=86.4074410623569x_{23} = 86.4074410623569
x24=73.8429364199345x_{24} = 73.8429364199345
x25=61.2790218683648x_{25} = 61.2790218683648
x26=−61.2913758541987x_{26} = -61.2913758541987
x27=29.8747766975811x_{27} = 29.8747766975811
x28=1.62571032846112x_{28} = 1.62571032846112
x29=83.2662715234224x_{29} = 83.2662715234224
x30=−73.8512905214352x_{30} = -73.8512905214352
x31=23.5959433880961x_{31} = 23.5959433880961
x32=−95.8359736534704x_{32} = -95.8359736534704
x33=−23.7240583014129x_{33} = -23.7240583014129
x34=14.1804076736695x_{34} = 14.1804076736695
x35=36.1545700235369x_{35} = 36.1545700235369
x36=−51.8743358627882x_{36} = -51.8743358627882
x37=76.9840163471242x_{37} = 76.9840163471242
x38=−1.55694240961403x_{38} = -1.55694240961403
x39=92.6898521764109x_{39} = 92.6898521764109
x40=4.76797091488681x_{40} = 4.76797091488681
x41=64.4199305552876x_{41} = 64.4199305552876
x42=−42.4626302394958x_{42} = -42.4626302394958
x43=42.4350478667949x_{43} = 42.4350478667949
x44=−58.1519860167309x_{44} = -58.1519860167309
x45=80.1251290602825x_{45} = 80.1251290602825
x46=−92.6950805761372x_{46} = -92.6950805761372
x47=17.3184749964622x_{47} = 17.3184749964622
x48=54.9973800268033x_{48} = 54.9973800268033
x49=−86.4134789918403x_{49} = -86.4134789918403
x50=−7.49785025429627x_{50} = -7.49785025429627
x51=−64.4310462520234x_{51} = -64.4310462520234
x52=45.5754833964017x_{52} = 45.5754833964017
x53=−67.5709467881468x_{53} = -67.5709467881468
x54=−10.6252638359778x_{54} = -10.6252638359778
x55=−55.0129430380975x_{55} = -55.0129430380975
x56=39.2947365767581x_{56} = 39.2947365767581
x57=26.7352181076125x_{57} = 26.7352181076125
x58=48.7160262498578x_{58} = 48.7160262498578
x59=−15.6656497396484x_{59} = -15.6656497396484
x60=7.90562211078873x_{60} = 7.90562211078873
x61=−20.6690930522039x_{61} = -20.6690930522039
x62=58.1381693149369x_{62} = 58.1381693149369
x63=−98.9769190377406x_{63} = -98.9769190377406
x64=11.042834018472x_{64} = 11.042834018472
Signos de extremos en los puntos:
(-83.27278749735838, -2.60876054959853e-5)

(95.83108978567422, 6.43131332192689e-6)

(-39.32765503354333, -0.000474913916560134)

(67.56088926617302, 1.4507467909696e-5)

(70.70189273923708, 1.30910673431019e-5)

(-36.19464748636427, -0.000700794081610218)

(-33.06463030123335, -0.00110128677978278)

(-80.13218295126643, -2.97806040251224e-5)

(-76.99167859898338, -3.42187254884363e-5)

(-13.374725027511552, 1.48703204380161)

(-17.76530881738893, -0.316751283456898)

(51.85666237615227, 2.58540901594878e-5)

(-26.822966592619352, -0.0036838405646894)

(-29.93947198316022, -0.00188845157155698)

(-70.71103843647168, -4.62004275354222e-5)

(-45.59896768232981, -0.000248934113473829)

(-48.73628662724555, -0.000189168265992073)

(98.97234647213226, 5.95132014594302e-6)

(33.01457319442138, 6.32718558258914e-5)

(89.5486353126801, 7.56056840101053e-6)

(-89.55424642628113, -2.03676922605015e-5)

(20.45700785904762, 0.000140992043746909)

(86.4074410623569, 8.2269016993418e-6)

(73.84293641993447, 1.18552521496999e-5)

(61.27902186836483, 1.80284503322335e-5)

(-61.29137585419873, -7.77450917857276e-5)

(29.87477669758108, 7.57930653886531e-5)

(1.6257103284611183, 0.000840989400699168)

(83.26627152342245, 8.97508828438985e-6)

(-73.85129052143519, -3.96022732594355e-5)

(23.595943388096135, 0.000112938247003005)

(-95.83597365347038, -1.62222548986629e-5)

(-23.72405830141292, -0.00885941658165251)

(14.180407673669468, 0.000232529037135901)

(36.15457002353693, 5.33845281560915e-5)

(-51.87433586278823, -0.000147316137453507)

(76.98401634712424, 1.07719565849402e-5)

(-1.5569424096140303, 0.00110121732983566)

(92.68985217641092, 6.96512010822983e-6)

(4.767970914886807, 0.000591099487611004)

(64.4199305552876, 1.61389889798209e-5)

(-42.46263023949575, -0.00033750581154634)

(42.435047866794875, 3.90620699992672e-5)

(-58.15198601673087, -9.47043749469472e-5)

(80.1251290602825, 9.81821027414399e-6)

(-92.69508057613723, -1.81356163737875e-5)

(17.31847499646221, 0.00017920215327258)

(54.99738002680327, 2.28092784327236e-5)

(-86.41347899184032, -2.29885184762429e-5)

(-7.497850254296271, 0.00564613576665274)

(-64.43104625202345, -6.4654143251366e-5)

(45.57548339640173, 3.38139043165193e-5)

(-67.57094678814683, -5.43807485776194e-5)

(-10.625263835977757, 0.0484960050826985)

(-55.01294303809747, -0.000117095734849419)

(39.29473657675809, 4.5471552816923e-5)

(26.735218107612454, 9.18902101634538e-5)

(48.716026249857805, 2.94743639686436e-5)

(-15.66564973964842, 7.48349266640593)

(7.905622110788726, 0.000421625678784061)

(-20.66909305220391, -0.0316848431855068)

(58.13816931493692, 2.02293790751269e-5)

(-98.97691903774061, -1.45722597118256e-5)

(11.042834018472027, 0.000308976630604364)


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=95.8310897856742x_{1} = 95.8310897856742
x2=67.560889266173x_{2} = 67.560889266173
x3=70.7018927392371x_{3} = 70.7018927392371
x4=−13.3747250275116x_{4} = -13.3747250275116
x5=51.8566623761523x_{5} = 51.8566623761523
x6=98.9723464721323x_{6} = 98.9723464721323
x7=33.0145731944214x_{7} = 33.0145731944214
x8=89.5486353126801x_{8} = 89.5486353126801
x9=20.4570078590476x_{9} = 20.4570078590476
x10=86.4074410623569x_{10} = 86.4074410623569
x11=73.8429364199345x_{11} = 73.8429364199345
x12=61.2790218683648x_{12} = 61.2790218683648
x13=29.8747766975811x_{13} = 29.8747766975811
x14=1.62571032846112x_{14} = 1.62571032846112
x15=83.2662715234224x_{15} = 83.2662715234224
x16=23.5959433880961x_{16} = 23.5959433880961
x17=14.1804076736695x_{17} = 14.1804076736695
x18=36.1545700235369x_{18} = 36.1545700235369
x19=76.9840163471242x_{19} = 76.9840163471242
x20=−1.55694240961403x_{20} = -1.55694240961403
x21=92.6898521764109x_{21} = 92.6898521764109
x22=4.76797091488681x_{22} = 4.76797091488681
x23=64.4199305552876x_{23} = 64.4199305552876
x24=42.4350478667949x_{24} = 42.4350478667949
x25=80.1251290602825x_{25} = 80.1251290602825
x26=17.3184749964622x_{26} = 17.3184749964622
x27=54.9973800268033x_{27} = 54.9973800268033
x28=−7.49785025429627x_{28} = -7.49785025429627
x29=45.5754833964017x_{29} = 45.5754833964017
x30=−10.6252638359778x_{30} = -10.6252638359778
x31=39.2947365767581x_{31} = 39.2947365767581
x32=26.7352181076125x_{32} = 26.7352181076125
x33=48.7160262498578x_{33} = 48.7160262498578
x34=−15.6656497396484x_{34} = -15.6656497396484
x35=7.90562211078873x_{35} = 7.90562211078873
x36=58.1381693149369x_{36} = 58.1381693149369
x37=11.042834018472x_{37} = 11.042834018472
Puntos máximos de la función:
x37=−83.2727874973584x_{37} = -83.2727874973584
x37=−39.3276550335433x_{37} = -39.3276550335433
x37=−36.1946474863643x_{37} = -36.1946474863643
x37=−33.0646303012334x_{37} = -33.0646303012334
x37=−80.1321829512664x_{37} = -80.1321829512664
x37=−76.9916785989834x_{37} = -76.9916785989834
x37=−17.7653088173889x_{37} = -17.7653088173889
x37=−26.8229665926194x_{37} = -26.8229665926194
x37=−29.9394719831602x_{37} = -29.9394719831602
x37=−70.7110384364717x_{37} = -70.7110384364717
x37=−45.5989676823298x_{37} = -45.5989676823298
x37=−48.7362866272455x_{37} = -48.7362866272455
x37=−89.5542464262811x_{37} = -89.5542464262811
x37=−61.2913758541987x_{37} = -61.2913758541987
x37=−73.8512905214352x_{37} = -73.8512905214352
x37=−95.8359736534704x_{37} = -95.8359736534704
x37=−23.7240583014129x_{37} = -23.7240583014129
x37=−51.8743358627882x_{37} = -51.8743358627882
x37=−42.4626302394958x_{37} = -42.4626302394958
x37=−58.1519860167309x_{37} = -58.1519860167309
x37=−92.6950805761372x_{37} = -92.6950805761372
x37=−86.4134789918403x_{37} = -86.4134789918403
x37=−64.4310462520234x_{37} = -64.4310462520234
x37=−67.5709467881468x_{37} = -67.5709467881468
x37=−55.0129430380975x_{37} = -55.0129430380975
x37=−20.6690930522039x_{37} = -20.6690930522039
x37=−98.9769190377406x_{37} = -98.9769190377406
Decrece en los intervalos
[98.9723464721323,∞)\left[98.9723464721323, \infty\right)
Crece en los intervalos
[−17.7653088173889,−15.6656497396484]\left[-17.7653088173889, -15.6656497396484\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(10log⁡((x+5)2)+5(x3+5)3⋅300sin⁡2(x)+7)y = \lim_{x \to -\infty}\left(\frac{10 \log{\left(\left(x + 5\right)^{2} \right)} + 5}{\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(10log⁡((x+5)2)+5(x3+5)3⋅300sin⁡2(x)+7)y = \lim_{x \to \infty}\left(\frac{10 \log{\left(\left(x + 5\right)^{2} \right)} + 5}{\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (10*log((x + 5)^2) + 5)/((300*sin(x)^2)*(x/3 + 5)^3 + 7), 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→−∞(10log⁡((x+5)2)+5x((x3+5)3⋅300sin⁡2(x)+7))y = x \lim_{x \to -\infty}\left(\frac{10 \log{\left(\left(x + 5\right)^{2} \right)} + 5}{x \left(\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7\right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(10log⁡((x+5)2)+5x((x3+5)3⋅300sin⁡2(x)+7))y = x \lim_{x \to \infty}\left(\frac{10 \log{\left(\left(x + 5\right)^{2} \right)} + 5}{x \left(\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7\right)}\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:
10log⁡((x+5)2)+5(x3+5)3⋅300sin⁡2(x)+7=10log⁡((5−x)2)+5300(5−x3)3sin⁡2(x)+7\frac{10 \log{\left(\left(x + 5\right)^{2} \right)} + 5}{\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7} = \frac{10 \log{\left(\left(5 - x\right)^{2} \right)} + 5}{300 \left(5 - \frac{x}{3}\right)^{3} \sin^{2}{\left(x \right)} + 7}
- No
10log⁡((x+5)2)+5(x3+5)3⋅300sin⁡2(x)+7=−10log⁡((5−x)2)+5300(5−x3)3sin⁡2(x)+7\frac{10 \log{\left(\left(x + 5\right)^{2} \right)} + 5}{\left(\frac{x}{3} + 5\right)^{3} \cdot 300 \sin^{2}{\left(x \right)} + 7} = - \frac{10 \log{\left(\left(5 - x\right)^{2} \right)} + 5}{300 \left(5 - \frac{x}{3}\right)^{3} \sin^{2}{\left(x \right)} + 7}
- No
es decir, función
no es
par ni impar
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