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Trazado el gráfico de la función/ (-2-sqrt(2)*sqrt(pi)*erf(x*sqrt(2)/2)/2)*exp(x^1)

Gráfico de la función y = (-2-sqrt(2)*sqrt(pi)*erf(x*sqrt(2)/2)/2)*exp(x^1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /                     /    ___\\      
       |       ___   ____    |x*\/ 2 ||      
       |     \/ 2 *\/ pi *erf|-------||  / 1\
       |                     \   2   /|  \x /
f(x) = |-2 - -------------------------|*e    
       \                 2            /
f(x)=(−2πerf⁡(2x2)2−2)ex1f{\left(x \right)} = \left(- \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{x^{1}} 
f = (-(sqrt(2)*sqrt(pi))*erf((sqrt(2)*x)/2)/2 - 2)*exp(x^1)
Gráfico de la función
02468-8-6-4-2-1010-100000100000
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 - (sqrt(2)*sqrt(pi))*erf((x*sqrt(2))/2)/2)*exp(x^1).
(−2−2πerf⁡(022)2)e01\left(-2 - \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{0 \sqrt{2}}{2} \right)}}{2}\right) e^{0^{1}}
Resultado:
f(0)=−2f{\left(0 \right)} = -2
Punto:
(0, -2)
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
(−2πerf⁡(2x2)2−2)ex1−ex1e−x22=0\left(- \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{x^{1}} - e^{x^{1}} e^{- \frac{x^{2}}{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−60.8720030830002x_{1} = -60.8720030830002
x2=−84.8720030830002x_{2} = -84.8720030830002
x3=−110.872003083x_{3} = -110.872003083
x4=−104.872003083x_{4} = -104.872003083
x5=−50.8720030830002x_{5} = -50.8720030830002
x6=−32.8720030830002x_{6} = -32.8720030830002
x7=−82.8720030830002x_{7} = -82.8720030830002
x8=−116.872003083x_{8} = -116.872003083
x9=−106.872003083x_{9} = -106.872003083
x10=−70.8720030830002x_{10} = -70.8720030830002
x11=−108.872003083x_{11} = -108.872003083
x12=−54.8720030830002x_{12} = -54.8720030830002
x13=−38.8720030830002x_{13} = -38.8720030830002
x14=−98.8720030830002x_{14} = -98.8720030830002
x15=−74.8720030830002x_{15} = -74.8720030830002
x16=−40.8720030830002x_{16} = -40.8720030830002
x17=−48.8720030830002x_{17} = -48.8720030830002
x18=−118.872003083x_{18} = -118.872003083
x19=−94.8720030830002x_{19} = -94.8720030830002
x20=−30.8720030830002x_{20} = -30.8720030830002
x21=−66.8720030830002x_{21} = -66.8720030830002
x22=−44.8720030830002x_{22} = -44.8720030830002
x23=−42.8720030830002x_{23} = -42.8720030830002
x24=−36.8720030830002x_{24} = -36.8720030830002
x25=−64.8720030830002x_{25} = -64.8720030830002
x26=−34.8720030830002x_{26} = -34.8720030830002
x27=−62.8720030830002x_{27} = -62.8720030830002
x28=−80.8720030830002x_{28} = -80.8720030830002
x29=−112.872003083x_{29} = -112.872003083
x30=−102.872003083x_{30} = -102.872003083
x31=−96.8720030830002x_{31} = -96.8720030830002
x32=−88.8720030830002x_{32} = -88.8720030830002
x33=−120.872003083x_{33} = -120.872003083
x34=−28.8720030829998x_{34} = -28.8720030829998
x35=−76.8720030830002x_{35} = -76.8720030830002
x36=−72.8720030830002x_{36} = -72.8720030830002
x37=−56.8720030830002x_{37} = -56.8720030830002
x38=−52.8720030830002x_{38} = -52.8720030830002
x39=−68.8720030830002x_{39} = -68.8720030830002
x40=−58.8720030830002x_{40} = -58.8720030830002
x41=−86.8720030830002x_{41} = -86.8720030830002
x42=−92.8720030830002x_{42} = -92.8720030830002
x43=−46.8720030830002x_{43} = -46.8720030830002
x44=−100.872003083x_{44} = -100.872003083
x45=−114.872003083x_{45} = -114.872003083
x46=−78.8720030830002x_{46} = -78.8720030830002
x47=−90.8720030830002x_{47} = -90.8720030830002
Signos de extremos en los puntos:
                                                                     ___   ____    /                   ___\ 
(-60.872003083000195, -7.32242537443395e-27 + 1.83060634360849e-27*\/ 2 *\/ pi *erf\30.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-84.8720030830002, -2.76431409789314e-37 + 6.91078524473285e-38*\/ 2 *\/ pi *erf\42.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-110.8720030830002, -1.41231268276666e-48 + 3.53078170691665e-49*\/ 2 *\/ pi *erf\55.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-104.8720030830002, -5.69767601643042e-46 + 1.42441900410761e-46*\/ 2 *\/ pi *erf\52.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-50.872003083000195, -1.61287152044994e-22 + 4.03217880112485e-23*\/ 2 *\/ pi *erf\25.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-32.872003083000195, -1.05901094255222e-14 + 2.64752735638056e-15*\/ 2 *\/ pi *erf\16.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-82.8720030830002, -2.04256719443973e-36 + 5.10641798609932e-37*\/ 2 *\/ pi *erf\41.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-116.8720030830002, -3.50077313654137e-51 + 8.75193284135341e-52*\/ 2 *\/ pi *erf\58.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-106.8720030830002, -7.71096597474066e-47 + 1.92774149368517e-47*\/ 2 *\/ pi *erf\53.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-70.8720030830002, -3.32437597690338e-31 + 8.31093994225845e-32*\/ 2 *\/ pi *erf\35.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-108.8720030830002, -1.04356576421941e-47 + 2.60891441054853e-48*\/ 2 *\/ pi *erf\54.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-54.872003083000195, -2.95407723424848e-24 + 7.38519308562119e-25*\/ 2 *\/ pi *erf\27.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-38.872003083000195, -2.62502567896482e-17 + 6.56256419741205e-18*\/ 2 *\/ pi *erf\19.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-98.8720030830002, -2.29860656102102e-43 + 5.74651640255255e-44*\/ 2 *\/ pi *erf\49.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-74.8720030830002, -6.08880699233452e-33 + 1.52220174808363e-33*\/ 2 *\/ pi *erf\37.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-40.872003083000195, -3.55258593766085e-18 + 8.88146484415213e-19*\/ 2 *\/ pi *erf\20.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-48.872003083000195, -1.19175981449722e-21 + 2.97939953624305e-22*\/ 2 *\/ pi *erf\24.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-118.8720030830002, -4.73778123980951e-52 + 1.18444530995238e-52*\/ 2 *\/ pi *erf\59.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-94.8720030830002, -1.25499665885795e-41 + 3.13749164714488e-42*\/ 2 *\/ pi *erf\47.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-30.87200308300019, -7.82509126389984e-14 + 1.95627281597496e-14*\/ 2 *\/ pi *erf\15.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-66.8720030830002, -1.81504778353551e-29 + 4.53761945883878e-30*\/ 2 *\/ pi *erf\33.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-44.872003083000195, -6.50678811553913e-20 + 1.62669702888478e-20*\/ 2 *\/ pi *erf\22.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-42.872003083000195, -4.80790224095739e-19 + 1.20197556023935e-19*\/ 2 *\/ pi *erf\21.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-36.872003083000195, -1.93964620030046e-16 + 4.84911550075114e-17*\/ 2 *\/ pi *erf\18.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-64.8720030830002, -1.34114898947836e-28 + 3.35287247369591e-29*\/ 2 *\/ pi *erf\32.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-34.872003083000195, -1.43321545860978e-15 + 3.58303864652444e-16*\/ 2 *\/ pi *erf\17.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-62.872003083000195, -9.90982512027978e-28 + 2.47745628006995e-28*\/ 2 *\/ pi *erf\31.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-80.8720030830002, -1.50926435855505e-35 + 3.77316089638764e-36*\/ 2 *\/ pi *erf\40.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-112.8720030830002, -1.91135736940886e-49 + 4.77839342352216e-50*\/ 2 *\/ pi *erf\56.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-102.8720030830002, -4.21004477189361e-45 + 1.0525111929734e-45*\/ 2 *\/ pi *erf\51.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-96.8720030830002, -1.69845328287544e-42 + 4.24613320718859e-43*\/ 2 *\/ pi *erf\48.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-88.8720030830002, -5.06301787920477e-39 + 1.26575446980119e-39*\/ 2 *\/ pi *erf\44.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-120.8720030830002, -6.4118896600273e-53 + 1.60297241500682e-53*\/ 2 *\/ pi *erf\60.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-28.87200308299975, -5.78200383282335e-13 + 1.44550095820584e-13*\/ 2 *\/ pi *erf\14.4360015414999*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-76.8720030830002, -8.24030418880661e-34 + 2.06007604720165e-34*\/ 2 *\/ pi *erf\38.4360015415001*\/ 2 /)

                                                                  ___   ____    /                   ___\ 
(-72.8720030830002, -4.4990536441921e-32 + 1.12476341104803e-32*\/ 2 *\/ pi *erf\36.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-56.872003083000195, -3.99790879199847e-25 + 9.99477197999618e-26*\/ 2 *\/ pi *erf\28.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-52.872003083000195, -2.18278424044359e-23 + 5.45696060110897e-24*\/ 2 *\/ pi *erf\26.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-68.8720030830002, -2.45640005872765e-30 + 6.14100014681912e-31*\/ 2 *\/ pi *erf\34.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-58.872003083000195, -5.41058118719257e-26 + 1.35264529679814e-26*\/ 2 *\/ pi *erf\29.4360015415001*\/ 2 /)

                                                                  ___   ____    /                   ___\ 
(-86.8720030830002, -3.7410923139333e-38 + 9.35273078483324e-39*\/ 2 *\/ pi *erf\43.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-92.8720030830002, -9.27324071627194e-41 + 2.31831017906799e-41*\/ 2 *\/ pi *erf\46.4360015415001*\/ 2 /)

                                                                     ___   ____    /                   ___\ 
(-46.872003083000195, -8.80598012577114e-21 + 2.20149503144278e-21*\/ 2 *\/ pi *erf\23.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-100.8720030830002, -3.11082569985316e-44 + 7.77706424963289e-45*\/ 2 *\/ pi *erf\50.4360015415001*\/ 2 /)

                                                                    ___   ____    /                   ___\ 
(-114.8720030830002, -2.58674090955336e-50 + 6.46685227388339e-51*\/ 2 *\/ pi *erf\57.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-78.8720030830002, -1.11520390134799e-34 + 2.78800975336997e-35*\/ 2 *\/ pi *erf\39.4360015415001*\/ 2 /)

                                                                   ___   ____    /                   ___\ 
(-90.8720030830002, -6.85204958714212e-40 + 1.71301239678553e-40*\/ 2 *\/ pi *erf\45.4360015415001*\/ 2 /)


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico
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
(xe−x22−2πerf⁡(2x2)2−2−2e−x22)ex1=0\left(x e^{- \frac{x^{2}}{2}} - \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2 - 2 e^{- \frac{x^{2}}{2}}\right) e^{x^{1}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−60.8720030830002x_{1} = -60.8720030830002
x2=−84.8720030830002x_{2} = -84.8720030830002
x3=−110.872003083x_{3} = -110.872003083
x4=−104.872003083x_{4} = -104.872003083
x5=−50.8720030830002x_{5} = -50.8720030830002
x6=−32.8720030830002x_{6} = -32.8720030830002
x7=−82.8720030830002x_{7} = -82.8720030830002
x8=−116.872003083x_{8} = -116.872003083
x9=−106.872003083x_{9} = -106.872003083
x10=−70.8720030830002x_{10} = -70.8720030830002
x11=−108.872003083x_{11} = -108.872003083
x12=−54.8720030830002x_{12} = -54.8720030830002
x13=−38.8720030830002x_{13} = -38.8720030830002
x14=−98.8720030830002x_{14} = -98.8720030830002
x15=−74.8720030830002x_{15} = -74.8720030830002
x16=−40.8720030830002x_{16} = -40.8720030830002
x17=−48.8720030830002x_{17} = -48.8720030830002
x18=−118.872003083x_{18} = -118.872003083
x19=−94.8720030830002x_{19} = -94.8720030830002
x20=−30.8720030830002x_{20} = -30.8720030830002
x21=−66.8720030830002x_{21} = -66.8720030830002
x22=−44.8720030830002x_{22} = -44.8720030830002
x23=−42.8720030830002x_{23} = -42.8720030830002
x24=−36.8720030830002x_{24} = -36.8720030830002
x25=−64.8720030830002x_{25} = -64.8720030830002
x26=−34.8720030830002x_{26} = -34.8720030830002
x27=−28.8720030829963x_{27} = -28.8720030829963
x28=−62.8720030830002x_{28} = -62.8720030830002
x29=−80.8720030830002x_{29} = -80.8720030830002
x30=−112.872003083x_{30} = -112.872003083
x31=−102.872003083x_{31} = -102.872003083
x32=−96.8720030830002x_{32} = -96.8720030830002
x33=−88.8720030830002x_{33} = -88.8720030830002
x34=−120.872003083x_{34} = -120.872003083
x35=−76.8720030830002x_{35} = -76.8720030830002
x36=−72.8720030830002x_{36} = -72.8720030830002
x37=−56.8720030830002x_{37} = -56.8720030830002
x38=−52.8720030830002x_{38} = -52.8720030830002
x39=−68.8720030830002x_{39} = -68.8720030830002
x40=−58.8720030830002x_{40} = -58.8720030830002
x41=−86.8720030830002x_{41} = -86.8720030830002
x42=−92.8720030830002x_{42} = -92.8720030830002
x43=−46.8720030830002x_{43} = -46.8720030830002
x44=−100.872003083x_{44} = -100.872003083
x45=−114.872003083x_{45} = -114.872003083
x46=−78.8720030830002x_{46} = -78.8720030830002
x47=−90.8720030830002x_{47} = -90.8720030830002

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:
No tiene corvaduras en todo el eje numérico
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((−2πerf⁡(2x2)2−2)ex1)=0\lim_{x \to -\infty}\left(\left(- \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{x^{1}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((−2πerf⁡(2x2)2−2)ex1)=−∞\lim_{x \to \infty}\left(\left(- \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{x^{1}}\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 - (sqrt(2)*sqrt(pi))*erf((x*sqrt(2))/2)/2)*exp(x^1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−2πerf⁡(2x2)2−2)ex1x)=0\lim_{x \to -\infty}\left(\frac{\left(- \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{x^{1}}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((−2πerf⁡(2x2)2−2)ex1x)=−∞\lim_{x \to \infty}\left(\frac{\left(- \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{x^{1}}}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
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:
(−2πerf⁡(2x2)2−2)ex1=(2πerf⁡(2x2)2−2)e−x\left(- \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{x^{1}} = \left(\frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{- x}
- No
(−2πerf⁡(2x2)2−2)ex1=−(2πerf⁡(2x2)2−2)e−x\left(- \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{x^{1}} = - \left(\frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} - 2\right) e^{- x}
- No
es decir, función
no es
par ni impar
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