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Trazado el gráfico de la función/ 1-x*sin(1/x)

Gráfico de la función y = 1-x*sin(1/x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                /1\
f(x) = 1 - x*sin|-|
                \x/
f(x)=−xsin⁡(1x)+1f{\left(x \right)} = - x \sin{\left(\frac{1}{x} \right)} + 1 
f = -x*sin(1/x) + 1
Gráfico de la función
02468-8-6-4-2-101002
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
−xsin⁡(1x)+1=0- x \sin{\left(\frac{1}{x} \right)} + 1 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=405225.527998708x_{1} = 405225.527998708
x2=455780.401008261x_{2} = 455780.401008261
x3=364781.678189971x_{3} = 364781.678189971
x4=476002.36469548x_{4} = 476002.36469548
x5=−433993.65110348x_{5} = -433993.65110348
x6=435558.445020876x_{6} = 435558.445020876
x7=465891.381952058x_{7} = 465891.381952058
x8=−413771.704666834x_{8} = -413771.704666834
x9=−383438.804822281x_{9} = -383438.804822281
x10=486113.349126226x_{10} = 486113.349126226
x11=385003.596701728x_{11} = 385003.596701728
x12=374892.635718381x_{12} = 374892.635718381
x13=−423882.676690473x_{13} = -423882.676690473
x14=−403660.735212108x_{14} = -403660.735212108
x15=−464326.587117811x_{15} = -464326.587117811
x16=−474437.569593925x_{16} = -474437.569593925
x17=−494659.539553244x_{17} = -494659.539553244
x18=−373327.844348658x_{18} = -373327.844348658
x19=−454215.606459339x_{19} = -454215.606459339
x20=445669.421986579x_{20} = 445669.421986579
x21=−444104.627742657x_{21} = -444104.627742657
x22=425447.470257743x_{22} = 425447.470257743
x23=−393549.768524296x_{23} = -393549.768524296
x24=395114.560874761x_{24} = 395114.560874761
x25=506335.322645352x_{25} = 506335.322645352
x26=496224.335141153x_{26} = 496224.335141153
x27=−363216.887373036x_{27} = -363216.887373036
x28=415336.497858046x_{28} = 415336.497858046
x29=−484548.553773896x_{29} = -484548.553773896
x30=−504770.526835857x_{30} = -504770.526835857
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 1 - x*sin(1/x).
−0sin⁡(10)+1- 0 \sin{\left(\frac{1}{0} \right)} + 1
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
−sin⁡(1x)+cos⁡(1x)x=0- \sin{\left(\frac{1}{x} \right)} + \frac{\cos{\left(\frac{1}{x} \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−33838.8069987955x_{1} = -33838.8069987955
x2=27189.2682985119x_{2} = 27189.2682985119
x3=−20277.3091678955x_{3} = -20277.3091678955
x4=−37229.2008029351x_{4} = -37229.2008029351
x5=10237.7072403896x_{5} = 10237.7072403896
x6=8542.66546353133x_{6} = 8542.66546353133
x7=−12649.1268015143x_{7} = -12649.1268015143
x8=−38924.3992166982x_{8} = -38924.3992166982
x9=17018.2064665824x_{9} = 17018.2064665824
x10=38208.0321713516x_{10} = 38208.0321713516
x11=23798.8964040481x_{11} = 23798.8964040481
x12=34817.6371394969x_{12} = 34817.6371394969
x13=−10954.0235496446x_{13} = -10954.0235496446
x14=32274.8438867786x_{14} = 32274.8438867786
x15=40750.8308380171x_{15} = 40750.8308380171
x16=36512.8341467568x_{16} = 36512.8341467568
x17=24646.4882156886x_{17} = 24646.4882156886
x18=39903.2310836223x_{18} = 39903.2310836223
x19=22951.3055101033x_{19} = 22951.3055101033
x20=16170.6287133314x_{20} = 16170.6287133314
x21=−16886.9773640064x_{21} = -16886.9773640064
x22=−10106.4851636254x_{22} = -10106.4851636254
x23=−35534.0033488874x_{23} = -35534.0033488874
x24=−32991.2092909147x_{24} = -32991.2092909147
x25=17865.7867327802x_{25} = 17865.7867327802
x26=25494.0808534902x_{26} = 25494.0808534902
x27=−28753.2265249185x_{27} = -28753.2265249185
x28=−24515.2570410387x_{28} = -24515.2570410387
x29=−42314.798461984x_{29} = -42314.798461984
x30=35665.2355068933x_{30} = 35665.2355068933
x31=−32143.6119273287x_{31} = -32143.6119273287
x32=21256.1269144408x_{32} = 21256.1269144408
x33=39055.6315237518x_{33} = 39055.6315237518
x34=−40619.5984700161x_{34} = -40619.5984700161
x35=−21972.4849235058x_{35} = -21972.4849235058
x36=33970.0390649632x_{36} = 33970.0390649632
x37=−18582.1393820539x_{37} = -18582.1393820539
x38=19560.9534972764x_{38} = 19560.9534972764
x39=−29600.822198113x_{39} = -29600.822198113
x40=15323.0538900739x_{40} = 15323.0538900739
x41=−27905.6313719175x_{41} = -27905.6313719175
x42=−30448.418348057x_{42} = -30448.418348057
x43=28036.8629745587x_{43} = 28036.8629745587
x44=−31296.0149360127x_{44} = -31296.0149360127
x45=29732.0539618849x_{45} = 29732.0539618849
x46=18713.369170442x_{46} = 18713.369170442
x47=−21124.8963893094x_{47} = -21124.8963893094
x48=−9258.95883733077x_{48} = -9258.95883733077
x49=−27058.0367879985x_{49} = -27058.0367879985
x50=−23667.6653660369x_{50} = -23667.6653660369
x51=−7563.95858775919x_{51} = -7563.95858775919
x52=−15191.825714434x_{52} = -15191.825714434
x53=30579.6501825125x_{53} = 30579.6501825125
x54=−26210.4428283739x_{54} = -26210.4428283739
x55=12780.3528304711x_{55} = 12780.3528304711
x56=37360.4330405425x_{56} = 37360.4330405425
x57=13627.9152205591x_{57} = 13627.9152205591
x58=−39771.9987451226x_{58} = -39771.9987451226
x59=−19429.7234310854x_{59} = -19429.7234310854
x60=−14344.2549267547x_{60} = -14344.2549267547
x61=−17734.5572628217x_{61} = -17734.5572628217
x62=−36381.6019475619x_{62} = -36381.6019475619
x63=28884.4582116784x_{63} = 28884.4582116784
x64=22103.7156394291x_{64} = 22103.7156394291
x65=6847.70624844332x_{65} = 6847.70624844332
x66=−34686.4050257305x_{66} = -34686.4050257305
x67=−11801.5713963528x_{67} = -11801.5713963528
x68=42446.0308837659x_{68} = 42446.0308837659
x69=33122.4413057753x_{69} = 33122.4413057753
x70=−41467.1983793308x_{70} = -41467.1983793308
x71=−25362.8495556371x_{71} = -25362.8495556371
x72=−38076.7998978638x_{72} = -38076.7998978638
x73=−8411.44821783076x_{73} = -8411.44821783076
x74=7695.1721243119x_{74} = 7695.1721243119
x75=−6716.49789803616x_{75} = -6716.49789803616
x76=14475.4825115465x_{76} = 14475.4825115465
x77=−16039.4000376645x_{77} = -16039.4000376645
x78=−22820.0746241929x_{78} = -22820.0746241929
x79=20408.539477891x_{79} = 20408.539477891
x80=31427.2468354985x_{80} = 31427.2468354985
x81=9390.17882712825x_{81} = 9390.17882712825
x82=−13496.6883408414x_{82} = -13496.6883408414
x83=41598.4307750453x_{83} = 41598.4307750453
x84=11085.2472504735x_{84} = 11085.2472504735
x85=11932.7963857739x_{85} = 11932.7963857739
x86=26341.6742377005x_{86} = 26341.6742377005
Signos de extremos en los puntos:
(-33838.80699879548, 1.45552236929802e-10)

(27189.26829851187, 2.25451657343001e-10)

(-20277.309167895473, 4.05347977405768e-10)

(-37229.200802935076, 1.20249032953268e-10)

(10237.707240389647, 1.59016921852384e-9)

(8542.665463531333, 2.28382057709808e-9)

(-12649.126801514269, 1.04166408831929e-9)

(-38924.399216698235, 1.1000311772591e-10)

(17018.206466582393, 5.7546789555829e-10)

(38208.03217135156, 1.14166676112859e-10)

(23798.896404048144, 2.94262614275453e-10)

(34817.63713949693, 1.37483469053734e-10)

(-10954.023549644575, 1.38899736157327e-9)

(32274.8438867786, 1.60000235283064e-10)

(40750.83083801706, 1.00363384269997e-10)

(36512.83414675676, 1.25013666085749e-10)

(24646.48821568855, 2.74371303454757e-10)

(39903.23108362226, 1.04672492895475e-10)

(22951.305510103328, 3.1639824094043e-10)

(16170.628713331358, 6.37374819589809e-10)

(-16886.977364006372, 5.84446713247644e-10)

(-10106.485163625435, 1.63173063949529e-9)

(-35534.00334888743, 1.31995858687617e-10)

(-32991.20929091468, 1.53127177604517e-10)

(17865.786732780187, 5.22161092142426e-10)

(25494.080853490228, 2.56430654488327e-10)

(-28753.226524918457, 2.01593186588411e-10)

(-24515.257041038723, 2.77316725139087e-10)

(-42314.79846198405, 9.30816534960854e-11)

(35665.235506893325, 1.31026189897909e-10)

(-32143.611927328704, 1.613092992514e-10)

(21256.126914440752, 3.68876040823807e-10)

(39055.631523751756, 1.09265041459139e-10)

(-40619.59847001615, 1.01013086784008e-10)

(-21972.484923505832, 3.4521563385681e-10)

(33970.03906496316, 1.44429801451906e-10)

(-18582.139382053876, 4.82677786628471e-10)

(19560.953497276427, 4.35580793656243e-10)

(-29600.82219811303, 1.90213400586003e-10)

(15323.053890073896, 7.09836078804926e-10)

(-27905.63137191755, 2.14025130951256e-10)

(-30448.418348056977, 1.79770753838682e-10)

(28036.86297455866, 2.12026285417721e-10)

(-31296.0149360127, 1.70165215251927e-10)

(29732.053961884885, 1.88537963019542e-10)

(18713.369170442, 4.75931849486244e-10)

(-21124.896389309444, 3.73473252324175e-10)

(-9258.958837330767, 1.94412619336504e-9)

(-27058.03678799853, 2.27643792705123e-10)

(-23667.665366036923, 2.97534774595931e-10)

(-7563.9585877591935, 2.9130670098354e-9)

(-15191.82571443399, 7.22152337928605e-10)

(30579.650182512534, 1.78231207570434e-10)

(-26210.442828373863, 2.42605269207274e-10)

(12780.35283047108, 1.02038266724946e-9)

(37360.433040542535, 1.1940559652146e-10)

(13627.915220559116, 8.97407925748439e-10)

(-39771.99874512261, 1.05364383884421e-10)

(-19429.72343108544, 4.41484404589687e-10)

(-14344.254926754675, 8.10014610941323e-10)

(-17734.557262821672, 5.29917332237062e-10)

(-36381.60194756191, 1.25917165583189e-10)

(28884.45821167836, 1.99765426422971e-10)

(22103.715639429116, 3.41128569836258e-10)

(6847.706248443317, 3.55433615872869e-9)

(-34686.405025730484, 1.38525746429252e-10)

(-11801.571396352814, 1.19665521935985e-9)

(42446.030883765874, 9.2506891036237e-11)

(33122.441305775326, 1.51916146329256e-10)

(-41467.19837933084, 9.69256896965476e-11)

(-25362.84955563705, 2.59091192944538e-10)

(-38076.799897863806, 1.14954934460343e-10)

(-8411.448217830764, 2.35563069050926e-9)

(7695.172124311905, 2.8145700214921e-9)

(-6716.4978980361575, 3.69456221172015e-9)

(14475.482511546521, 7.95394528019244e-10)

(-16039.400037664549, 6.47847109291888e-10)

(-22820.074624192912, 3.20047655044675e-10)

(20408.539477890954, 4.00151911605917e-10)

(31427.246835498532, 1.68747127382574e-10)

(9390.178827128255, 1.89017068663588e-9)

(-13496.688340841416, 9.14943676377789e-10)

(41598.43077504528, 9.63151780553062e-11)

(11085.247250473549, 1.35630695563549e-9)

(11932.796385773858, 1.1704809343982e-9)

(26341.6742377005, 2.40193864797789e-10)


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=17018.2064665824x_{1} = 17018.2064665824
x2=40750.8308380171x_{2} = 40750.8308380171
x3=−29600.822198113x_{3} = -29600.822198113
x4=28036.8629745587x_{4} = 28036.8629745587
x5=−21124.8963893094x_{5} = -21124.8963893094
x6=−23667.6653660369x_{6} = -23667.6653660369
x7=−39771.9987451226x_{7} = -39771.9987451226
x8=6847.70624844332x_{8} = 6847.70624844332
x9=42446.0308837659x_{9} = 42446.0308837659
x10=41598.4307750453x_{10} = 41598.4307750453
Puntos máximos de la función:
x10=−33838.8069987955x_{10} = -33838.8069987955
x10=−37229.2008029351x_{10} = -37229.2008029351
x10=8542.66546353133x_{10} = 8542.66546353133
x10=38208.0321713516x_{10} = 38208.0321713516
x10=34817.6371394969x_{10} = 34817.6371394969
x10=−10954.0235496446x_{10} = -10954.0235496446
x10=−28753.2265249185x_{10} = -28753.2265249185
x10=−9258.95883733077x_{10} = -9258.95883733077
x10=−7563.95858775919x_{10} = -7563.95858775919
x10=−14344.2549267547x_{10} = -14344.2549267547
x10=−34686.4050257305x_{10} = -34686.4050257305
x10=−22820.0746241929x_{10} = -22820.0746241929
x10=−13496.6883408414x_{10} = -13496.6883408414
x10=11932.7963857739x_{10} = 11932.7963857739
Decrece en los intervalos
[42446.0308837659,∞)\left[42446.0308837659, \infty\right)
Crece en los intervalos
(−∞,−39771.9987451226]\left(-\infty, -39771.9987451226\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
sin⁡(1x)x3=0\frac{\sin{\left(\frac{1}{x} \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1πx_{1} = \frac{1}{\pi}
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(sin⁡(1x)x3)=⟨−∞,∞⟩\lim_{x \to 0^-}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x^{3}}\right) = \left\langle -\infty, \infty\right\rangle
lim⁡x→0+(sin⁡(1x)x3)=⟨−∞,∞⟩\lim_{x \to 0^+}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x^{3}}\right) = \left\langle -\infty, \infty\right\rangle
- los límites son iguales, es decir omitimos el punto correspondiente

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
[1π,∞)\left[\frac{1}{\pi}, \infty\right)
Convexa en los intervalos
(−∞,1π]\left(-\infty, \frac{1}{\pi}\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−xsin⁡(1x)+1)=0\lim_{x \to -\infty}\left(- x \sin{\left(\frac{1}{x} \right)} + 1\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(−xsin⁡(1x)+1)=0\lim_{x \to \infty}\left(- x \sin{\left(\frac{1}{x} \right)} + 1\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 1 - x*sin(1/x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−xsin⁡(1x)+1x)=0\lim_{x \to -\infty}\left(\frac{- x \sin{\left(\frac{1}{x} \right)} + 1}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−xsin⁡(1x)+1x)=0\lim_{x \to \infty}\left(\frac{- x \sin{\left(\frac{1}{x} \right)} + 1}{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:
−xsin⁡(1x)+1=−xsin⁡(1x)+1- x \sin{\left(\frac{1}{x} \right)} + 1 = - x \sin{\left(\frac{1}{x} \right)} + 1
- Sí
−xsin⁡(1x)+1=xsin⁡(1x)−1- x \sin{\left(\frac{1}{x} \right)} + 1 = x \sin{\left(\frac{1}{x} \right)} - 1
- No
es decir, función
es
par
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