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Trazado el gráfico de la función/ (exp((-x)/2)/2-cos(x)*sin(sin(x)))/x-(cos(sin(x))-e^((-x)/2))/x^2

Gráfico de la función y = (exp((-x)/2)/2-cos(x)*sin(sin(x)))/x-(cos(sin(x))-e^((-x)/2))/x^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        -x                                           
        ---                                          
         2                                        -x 
       e                                          ---
       ---- - cos(x)*sin(sin(x))                   2 
        2                          cos(sin(x)) - E   
f(x) = ------------------------- - ------------------
                   x                        2        
                                           x
f(x)=−−e(−1)x2+cos⁡(sin⁡(x))x2+e(−1)x22−sin⁡(sin⁡(x))cos⁡(x)xf{\left(x \right)} = - \frac{- e^{\frac{\left(-1\right) x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}} + \frac{\frac{e^{\frac{\left(-1\right) x}{2}}}{2} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x} 
f = -(-E^((-x)/2) + cos(sin(x)))/x^2 + (exp((-x)/2)/2 - sin(sin(x))*cos(x))/x
Gráfico de la función
02468-8-6-4-2-10105-10
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:
−−e(−1)x2+cos⁡(sin⁡(x))x2+e(−1)x22−sin⁡(sin⁡(x))cos⁡(x)x=0- \frac{- e^{\frac{\left(-1\right) x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}} + \frac{\frac{e^{\frac{\left(-1\right) x}{2}}}{2} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=20.4517633182518x_{1} = 20.4517633182518
x2=37.6725610881152x_{2} = 37.6725610881152
x3=59.6735009919361x_{3} = 59.6735009919361
x4=1.2935145217858x_{4} = 1.2935145217858
x5=105.249454856867x_{5} = 105.249454856867
x6=92.6839114698133x_{6} = 92.6839114698133
x7=45.5671881401567x_{7} = 45.5671881401567
x8=226.190249972607x_{8} = 226.190249972607
x9=86.401230015273x_{9} = 86.401230015273
x10=15.6441816493174x_{10} = 15.6441816493174
x11=34.5285495599042x_{11} = 34.5285495599042
x12=80.1186275923176x_{12} = 80.1186275923176
x13=48.707871555818x_{13} = 48.707871555818
x14=29.8666412571427x_{14} = 29.8666412571427
x15=12.4872541864278x_{15} = 12.4872541864278
x16=51.8486651663544x_{16} = 51.8486651663544
x17=14.1819924110663x_{17} = 14.1819924110663
x18=78.5270811909166x_{18} = 78.5270811909166
x19=73.8361243776383x_{19} = 73.8361243776383
x20=67.5537480546902x_{20} = 67.5537480546902
x21=42.426639401039x_{21} = 42.426639401039
x22=−1.73371630155173x_{22} = -1.73371630155173
x23=83.2599178065512x_{23} = 83.2599178065512
x24=4.76977107733803x_{24} = 4.76977107733803
x25=26.7275777377294x_{25} = 26.7275777377294
x26=28.2389073182524x_{26} = 28.2389073182524
x27=23.5891854106957x_{27} = 23.5891854106957
x28=21.9455590447597x_{28} = 21.9455590447597
x29=81.6691638577428x_{29} = 81.6691638577428
x30=50.2455775793592x_{30} = 50.2455775793592
x31=95.8252769757041x_{31} = 95.8252769757041
x32=−2.12466883462839x_{32} = -2.12466883462839
x33=58.1305115060264x_{33} = 58.1305115060264
x34=89.5425619034202x_{34} = 89.5425619034202
x35=70.6949182400329x_{35} = 70.6949182400329
x36=36.1460864338985x_{36} = 36.1460864338985
x37=43.9595450359395x_{37} = 43.9595450359395
x38=64.4126190823488x_{38} = 64.4126190823488
x39=94.2371676849837x_{39} = 94.2371676849837
x40=87.9532241306717x_{40} = 87.9532241306717
x41=56.5309765021226x_{41} = 56.5309765021226
x42=65.9582834654203x_{42} = 65.9582834654203
x43=72.2427879360818x_{43} = 72.2427879360818
x44=6.15068665036262x_{44} = 6.15068665036262
x45=7.92129717826965x_{45} = 7.92129717826965
x46=100.521016418248x_{46} = 100.521016418248
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 (exp((-x)/2)/2 - cos(x)*sin(sin(x)))/x - (cos(sin(x)) - E^((-x)/2))/x^2.
−sin⁡(sin⁡(0))cos⁡(0)+e(−1)0220−−e(−1)02+cos⁡(sin⁡(0))02\frac{- \sin{\left(\sin{\left(0 \right)} \right)} \cos{\left(0 \right)} + \frac{e^{\frac{\left(-1\right) 0}{2}}}{2}}{0} - \frac{- e^{\frac{\left(-1\right) 0}{2}} + \cos{\left(\sin{\left(0 \right)} \right)}}{0^{2}}
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
−e(−1)x24+sin⁡(x)sin⁡(sin⁡(x))−cos⁡2(x)cos⁡(sin⁡(x))x−e(−1)x22−sin⁡(sin⁡(x))cos⁡(x)x2+−e(−1)x22+sin⁡(sin⁡(x))cos⁡(x)x2−2(e(−1)x2−cos⁡(sin⁡(x)))x3=0\frac{- \frac{e^{\frac{\left(-1\right) x}{2}}}{4} + \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{x} - \frac{\frac{e^{\frac{\left(-1\right) x}{2}}}{2} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x^{2}} + \frac{- \frac{e^{\frac{\left(-1\right) x}{2}}}{2} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \left(e^{\frac{\left(-1\right) x}{2}} - \cos{\left(\sin{\left(x \right)} \right)}\right)}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1.93007408048574x_{1} = -1.93007408048574
x2=63.5665984080535x_{2} = 63.5665984080535
x3=60.42458359052x_{3} = 60.42458359052
x4=33.8009961986888x_{4} = 33.8009961986888
x5=82.4179991548889x_{5} = 82.4179991548889
x6=38.428471870656x_{6} = 38.428471870656
x7=30.6580596186008x_{7} = 30.6580596186008
x8=25.8552584606865x_{8} = 25.8552584606865
x9=88.7016222038868x_{9} = 88.7016222038868
x10=69.850511674875x_{10} = 69.850511674875
x11=−0.033790285342516x_{11} = -0.033790285342516
x12=66.708572569087x_{12} = 66.708572569087
x13=19.5651215987813x_{13} = 19.5651215987813
x14=93.4999487345468x_{14} = 93.4999487345468
x15=84.074602601828x_{15} = 84.074602601828
x16=21.2270546596334x_{16} = 21.2270546596334
x17=71.5072467334144x_{17} = 71.5072467334144
x18=40.0862646696593x_{18} = 40.0862646696593
x19=49.5132102460951x_{19} = 49.5132102460951
x20=99.7834541642334x_{20} = 99.7834541642334
x21=68.3653525223029x_{21} = 68.3653525223029
x22=76.1343024153111x_{22} = 76.1343024153111
x23=41.571106470485x_{23} = 41.571106470485
x24=18.0820731824244x_{24} = 18.0820731824244
x25=62.0814749554559x_{25} = 62.0814749554559
x26=44.7135898561596x_{26} = 44.7135898561596
x27=8.63282034357175x_{27} = 8.63282034357175
x28=11.787104732733x_{28} = 11.787104732733
x29=32.1425638107126x_{29} = 32.1425638107126
x30=80.9327917405466x_{30} = 80.9327917405466
x31=54.1404029068618x_{31} = 54.1404029068618
x32=3.75092855882849x_{32} = 3.75092855882849
x33=27.5148353953567x_{33} = 27.5148353953567
x34=52.6553585097603x_{34} = 52.6553585097603
x35=46.3709894042637x_{35} = 46.3709894042637
x36=22.7106982434412x_{36} = 22.7106982434412
x37=77.7909636231305x_{37} = 77.7909636231305
x38=2.10531699821158x_{38} = 2.10531699821158
x39=16.4179171000477x_{39} = 16.4179171000477
x40=55.7974460756187x_{40} = 55.7974460756187
x41=24.3712203656572x_{41} = 24.3712203656572
x42=5.45882698165754x_{42} = 5.45882698165754
x43=98.1269500433719x_{43} = 98.1269500433719
x44=90.358179623441x_{44} = 90.358179623441
x45=47.8559526964791x_{45} = 47.8559526964791
x46=74.6491161193322x_{46} = 74.6491161193322
x47=85.5598188583576x_{47} = 85.5598188583576
x48=10.1126591305442x_{48} = 10.1126591305442
x49=96.6417065655218x_{49} = 96.6417065655218
x50=91.8434108974037x_{50} = 91.8434108974037
Signos de extremos en los puntos:
(-1.930074080485738, 0.0121751516814136)

(63.5665984080535, -0.00744622077607571)

(60.42458359051995, -0.00784397087030914)

(33.80099619868885, 0.0129581474369869)

(82.41799915488893, -0.00570901229687562)

(38.42847187065602, -0.0125259108790993)

(30.65805961860079, 0.0142095100231356)

(25.8552584606865, -0.0189982130888589)

(88.70162220388683, -0.0052970299080708)

(69.850511674875, -0.00676054480107385)

(-0.033790285342516, -0.625703609307118)

(66.70857256908701, -0.00708684389380194)

(19.56512159878134, -0.0255986674849128)

(93.49994873454683, 0.00484194971729688)

(84.07460260182802, 0.00537364521121259)

(21.22705465963345, 0.0199919644163597)

(71.50724673341442, 0.00629526569944311)

(40.086264669659336, 0.0110165534044349)

(49.51321024609514, 0.00899384597663131)

(99.78345416423338, 0.00454230860768394)

(68.36535252230293, 0.0065772498871008)

(76.13430241531114, -0.00619045812313403)

(41.57110647048502, -0.0115421226076137)

(18.082073182424363, 0.0231197755627614)

(62.08147495545586, 0.00722440396103394)

(44.71358985615962, -0.0107014744736439)

(8.632820343571748, 0.0439379011930541)

(11.787104732733045, 0.0336545377659331)

(32.14256381071264, -0.0150988467750716)

(80.93279174054663, 0.0055778014565613)

(54.14040290686175, -0.00878209518815057)

(3.750928558828491, -0.146823697689196)

(27.514835395356723, 0.0157273307719214)

(52.65535850976031, 0.00847499047606231)

(46.370989404263696, 0.00958028815515183)

(22.71069824344117, -0.0218119300888545)

(77.79096362313051, 0.00579807605090685)

(2.1053169982115834, 0.197975479737667)

(16.41791710004775, -0.0309615043409198)

(55.797446075618666, 0.00801268502117623)

(24.37122036565722, 0.017606342342308)

(5.458826981657544, 0.066585519095165)

(98.12695004337189, -0.00477963222892754)

(90.35817962344102, 0.00500709584366417)

(47.85595269647905, -0.0099748680247489)

(74.64911611933218, 0.00603645530716869)

(85.55981885835757, -0.00549531264268704)

(10.11265913054425, -0.0528076153572385)

(96.64170656552182, 0.00468734668104291)

(91.84341089740369, -0.00511255424090387)


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=63.5665984080535x_{1} = 63.5665984080535
x2=60.42458359052x_{2} = 60.42458359052
x3=82.4179991548889x_{3} = 82.4179991548889
x4=38.428471870656x_{4} = 38.428471870656
x5=25.8552584606865x_{5} = 25.8552584606865
x6=88.7016222038868x_{6} = 88.7016222038868
x7=69.850511674875x_{7} = 69.850511674875
x8=−0.033790285342516x_{8} = -0.033790285342516
x9=66.708572569087x_{9} = 66.708572569087
x10=19.5651215987813x_{10} = 19.5651215987813
x11=76.1343024153111x_{11} = 76.1343024153111
x12=41.571106470485x_{12} = 41.571106470485
x13=44.7135898561596x_{13} = 44.7135898561596
x14=32.1425638107126x_{14} = 32.1425638107126
x15=54.1404029068618x_{15} = 54.1404029068618
x16=3.75092855882849x_{16} = 3.75092855882849
x17=22.7106982434412x_{17} = 22.7106982434412
x18=16.4179171000477x_{18} = 16.4179171000477
x19=98.1269500433719x_{19} = 98.1269500433719
x20=47.8559526964791x_{20} = 47.8559526964791
x21=85.5598188583576x_{21} = 85.5598188583576
x22=10.1126591305442x_{22} = 10.1126591305442
x23=91.8434108974037x_{23} = 91.8434108974037
Puntos máximos de la función:
x23=−1.93007408048574x_{23} = -1.93007408048574
x23=33.8009961986888x_{23} = 33.8009961986888
x23=30.6580596186008x_{23} = 30.6580596186008
x23=93.4999487345468x_{23} = 93.4999487345468
x23=84.074602601828x_{23} = 84.074602601828
x23=21.2270546596334x_{23} = 21.2270546596334
x23=71.5072467334144x_{23} = 71.5072467334144
x23=40.0862646696593x_{23} = 40.0862646696593
x23=49.5132102460951x_{23} = 49.5132102460951
x23=99.7834541642334x_{23} = 99.7834541642334
x23=68.3653525223029x_{23} = 68.3653525223029
x23=18.0820731824244x_{23} = 18.0820731824244
x23=62.0814749554559x_{23} = 62.0814749554559
x23=8.63282034357175x_{23} = 8.63282034357175
x23=11.787104732733x_{23} = 11.787104732733
x23=80.9327917405466x_{23} = 80.9327917405466
x23=27.5148353953567x_{23} = 27.5148353953567
x23=52.6553585097603x_{23} = 52.6553585097603
x23=46.3709894042637x_{23} = 46.3709894042637
x23=77.7909636231305x_{23} = 77.7909636231305
x23=2.10531699821158x_{23} = 2.10531699821158
x23=55.7974460756187x_{23} = 55.7974460756187
x23=24.3712203656572x_{23} = 24.3712203656572
x23=5.45882698165754x_{23} = 5.45882698165754
x23=90.358179623441x_{23} = 90.358179623441
x23=74.6491161193322x_{23} = 74.6491161193322
x23=96.6417065655218x_{23} = 96.6417065655218
Decrece en los intervalos
[98.1269500433719,∞)\left[98.1269500433719, \infty\right)
Crece en los intervalos
(−∞,−0.033790285342516]\left(-\infty, -0.033790285342516\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→−∞(−−e(−1)x2+cos⁡(sin⁡(x))x2+e(−1)x22−sin⁡(sin⁡(x))cos⁡(x)x)=−∞\lim_{x \to -\infty}\left(- \frac{- e^{\frac{\left(-1\right) x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}} + \frac{\frac{e^{\frac{\left(-1\right) x}{2}}}{2} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(−−e(−1)x2+cos⁡(sin⁡(x))x2+e(−1)x22−sin⁡(sin⁡(x))cos⁡(x)x)=0\lim_{x \to \infty}\left(- \frac{- e^{\frac{\left(-1\right) x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}} + \frac{\frac{e^{\frac{\left(-1\right) x}{2}}}{2} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x}\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 (exp((-x)/2)/2 - cos(x)*sin(sin(x)))/x - (cos(sin(x)) - E^((-x)/2))/x^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−−e(−1)x2+cos⁡(sin⁡(x))x2+e(−1)x22−sin⁡(sin⁡(x))cos⁡(x)xx)=∞\lim_{x \to -\infty}\left(\frac{- \frac{- e^{\frac{\left(-1\right) x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}} + \frac{\frac{e^{\frac{\left(-1\right) x}{2}}}{2} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x}}{x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞(−−e(−1)x2+cos⁡(sin⁡(x))x2+e(−1)x22−sin⁡(sin⁡(x))cos⁡(x)xx)=0\lim_{x \to \infty}\left(\frac{- \frac{- e^{\frac{\left(-1\right) x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}} + \frac{\frac{e^{\frac{\left(-1\right) x}{2}}}{2} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x}}{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:
−−e(−1)x2+cos⁡(sin⁡(x))x2+e(−1)x22−sin⁡(sin⁡(x))cos⁡(x)x=−ex22+sin⁡(sin⁡(x))cos⁡(x)x−−ex2+cos⁡(sin⁡(x))x2- \frac{- e^{\frac{\left(-1\right) x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}} + \frac{\frac{e^{\frac{\left(-1\right) x}{2}}}{2} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x} = - \frac{\frac{e^{\frac{x}{2}}}{2} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x} - \frac{- e^{\frac{x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}}
- No
−−e(−1)x2+cos⁡(sin⁡(x))x2+e(−1)x22−sin⁡(sin⁡(x))cos⁡(x)x=ex22+sin⁡(sin⁡(x))cos⁡(x)x+−ex2+cos⁡(sin⁡(x))x2- \frac{- e^{\frac{\left(-1\right) x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}} + \frac{\frac{e^{\frac{\left(-1\right) x}{2}}}{2} - \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x} = \frac{\frac{e^{\frac{x}{2}}}{2} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x} + \frac{- e^{\frac{x}{2}} + \cos{\left(\sin{\left(x \right)} \right)}}{x^{2}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор