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  • Graphing y =:
  • 2x^4-4x^2+1
  • е^(1/2-x)
  • x/(x²-4)
  • (x)/(x^2-4) (x)/(x^2-4)
  • Integral of d{x}:
  • x*exp((-x^2)/2)

  • Identical expressions

  • x*exp((-x^ two)/ two)
  • x multiply by exponent of (( minus x squared ) divide by 2)
  • x multiply by exponent of (( minus x to the power of two) divide by two)
  • x*exp((-x2)/2)
  • x*exp-x2/2
  • x*exp((-x²)/2)
  • x*exp((-x to the power of 2)/2)
  • xexp((-x^2)/2)
  • xexp((-x2)/2)
  • xexp-x2/2
  • xexp-x^2/2
  • x*exp((-x^2) divide by 2)

  • Similar expressions

  • x*exp((x^2)/2)
Plotting a function graph/ x*exp((-x^2)/2)

Graphing y = x*exp((-x^2)/2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            2 
          -x  
          ----
           2  
f(x) = x*e
f(x)=xe(−1)x22f{\left(x \right)} = x e^{\frac{\left(-1\right) x^{2}}{2}} 
f = x*exp((-x^2)/2)
The graph of the function
02468-8-6-4-2-10101-1
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
xe(−1)x22=0x e^{\frac{\left(-1\right) x^{2}}{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=70.5388712182941x_{1} = 70.5388712182941
x2=−48.4076116917235x_{2} = -48.4076116917235
x3=28.9620890166341x_{3} = 28.9620890166341
x4=−8.82065718477295x_{4} = -8.82065718477295
x5=−30.6488081209205x_{5} = -30.6488081209205
x6=−15.3593372555059x_{6} = -15.3593372555059
x7=0x_{7} = 0
x8=34.8392502079753x_{8} = 34.8392502079753
x9=74.5233591950223x_{9} = 74.5233591950223
x10=−42.4653421912683x_{10} = -42.4653421912683
x11=−70.2800343339193x_{11} = -70.2800343339193
x12=25.0766626196802x_{12} = 25.0766626196802
x13=−34.5735053916839x_{13} = -34.5735053916839
x14=17.465102009593x_{14} = 17.465102009593
x15=−22.8793491121891x_{15} = -22.8793491121891
x16=−62.3160212977879x_{16} = -62.3160212977879
x17=−64.3061861108853x_{17} = -64.3061861108853
x18=52.6378406446428x_{18} = 52.6378406446428
x19=13.8305646213614x_{19} = 13.8305646213614
x20=84.4909972102885x_{20} = 84.4909972102885
x21=−52.3764545425553x_{21} = -52.3764545425553
x22=86.4854219951314x_{22} = 86.4854219951314
x23=9.02289694337605x_{23} = 9.02289694337605
x24=−10.247960314217x_{24} = -10.247960314217
x25=−44.4443699549213x_{25} = -44.4443699549213
x26=94.4654798903143x_{26} = 94.4654798903143
x27=92.4701421572582x_{27} = 92.4701421572582
x28=−24.8076741592432x_{28} = -24.8076741592432
x29=88.4800986212492x_{29} = 88.4800986212492
x30=40.7523965392307x_{30} = 40.7523965392307
x31=−32.6088462722306x_{31} = -32.6088462722306
x32=96.4610108264976x_{32} = 96.4610108264976
x33=7.83657923190551x_{33} = 7.83657923190551
x34=−72.2722807950237x_{34} = -72.2722807950237
x35=12.1040591364533x_{35} = 12.1040591364533
x36=76.5162106782208x_{36} = 76.5162106782208
x37=−78.2513955147154x_{37} = -78.2513955147154
x38=46.6877762829739x_{38} = 46.6877762829739
x39=−90.2179499984874x_{39} = -90.2179499984874
x40=−46.4252005910588x_{40} = -46.4252005910588
x41=−94.2086933283192x_{41} = -94.2086933283192
x42=−40.4883834667848x_{42} = -40.4883834667848
x43=−80.2451271074651x_{43} = -80.2451271074651
x44=−56.3497113142373x_{44} = -56.3497113142373
x45=23.1487959086869x_{45} = 23.1487959086869
x46=68.5473054354995x_{46} = 68.5473054354995
x47=−19.0686408223086x_{47} = -19.0686408223086
x48=78.5094259967043x_{48} = 78.5094259967043
x49=90.4750104191475x_{49} = 90.4750104191475
x50=−26.7467597908737x_{50} = -26.7467597908737
x51=−84.2334824940775x_{51} = -84.2334824940775
x52=−20.9648808652053x_{52} = -20.9648808652053
x53=−92.2132213430942x_{53} = -92.2132213430942
x54=−54.362591820007x_{54} = -54.362591820007
x55=−82.2391633180928x_{55} = -82.2391633180928
x56=−60.3265078154327x_{56} = -60.3265078154327
x57=19.3379596244699x_{57} = 19.3379596244699
x58=72.5309015703522x_{58} = 72.5309015703522
x59=82.4968425477036x_{59} = 82.4968425477036
x60=32.8752332341382x_{60} = 32.8752332341382
x61=−38.5138150497611x_{61} = -38.5138150497611
x62=58.5981051512238x_{62} = 58.5981051512238
x63=21.2345123835376x_{63} = 21.2345123835376
x64=48.6697660202548x_{64} = 48.6697660202548
x65=56.6104161729572x_{65} = 56.6104161729572
x66=−86.228064985273x_{66} = -86.228064985273
x67=42.7288452402976x_{67} = 42.7288452402976
x68=100.45260622169x_{68} = 100.45260622169
x69=30.9158612891769x_{69} = 30.9158612891769
x70=−17.1969684851483x_{70} = -17.1969684851483
x71=27.0151495573047x_{71} = 27.0151495573047
x72=98.4567232107033x_{72} = 98.4567232107033
x73=−36.5420283825472x_{73} = -36.5420283825472
x74=44.7073945322353x_{74} = 44.7073945322353
x75=62.5758396530863x_{75} = 62.5758396530863
x76=−74.2649444149412x_{76} = -74.2649444149412
x77=−76.2579924560991x_{77} = -76.2579924560991
x78=−28.6943596043799x_{78} = -28.6943596043799
x79=−58.3377124249816x_{79} = -58.3377124249816
x80=−98.2001902620909x_{80} = -98.2001902620909
x81=66.5562459808845x_{81} = 66.5562459808845
x82=64.5657397621909x_{82} = 64.5657397621909
x83=−88.2228929179074x_{83} = -88.2228929179074
x84=36.8071620516744x_{84} = 36.8071620516744
x85=60.5866054706585x_{85} = 60.5866054706585
x86=−7.6886044567655x_{86} = -7.6886044567655
x87=−68.2882415884721x_{87} = -68.2882415884721
x88=15.6247818795334x_{88} = 15.6247818795334
x89=38.7783713446935x_{89} = 38.7783713446935
x90=−66.2969435075443x_{90} = -66.2969435075443
x91=50.6531744084895x_{91} = 50.6531744084895
x92=80.5029780991381x_{92} = 80.5029780991381
x93=54.623627287895x_{93} = 54.623627287895
x94=−13.5703851876112x_{94} = -13.5703851876112
x95=−100.196193168581x_{95} = -100.196193168581
x96=10.4810551180426x_{96} = 10.4810551180426
x97=−96.2043534607085x_{97} = -96.2043534607085
x98=−50.3914159606663x_{98} = -50.3914159606663
x99=−11.8535751456397x_{99} = -11.8535751456397
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*exp((-x^2)/2).
0e(−1)0220 e^{\frac{\left(-1\right) 0^{2}}{2}}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−x2e(−1)x22+e(−1)x22=0- x^{2} e^{\frac{\left(-1\right) x^{2}}{2}} + e^{\frac{\left(-1\right) x^{2}}{2}} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1
x2=1x_{2} = 1
The values of the extrema at the points:
       -1/2 
(-1, -e    )

     -1/2 
(1, e    )


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=−1x_{1} = -1
Maxima of the function at points:
x1=1x_{1} = 1
Decreasing at intervals
[−1,1]\left[-1, 1\right]
Increasing at intervals
(−∞,−1]∪[1,∞)\left(-\infty, -1\right] \cup \left[1, \infty\right)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
x(x2−3)e−x22=0x \left(x^{2} - 3\right) e^{- \frac{x^{2}}{2}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−3x_{2} = - \sqrt{3}
x3=3x_{3} = \sqrt{3}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[−3,0]∪[3,∞)\left[- \sqrt{3}, 0\right] \cup \left[\sqrt{3}, \infty\right)
Convex at the intervals
(−∞,−3]∪[0,3]\left(-\infty, - \sqrt{3}\right] \cup \left[0, \sqrt{3}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xe(−1)x22)=0\lim_{x \to -\infty}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(xe(−1)x22)=0\lim_{x \to \infty}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*exp((-x^2)/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞e(−1)x22=0\lim_{x \to -\infty} e^{\frac{\left(-1\right) x^{2}}{2}} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞e(−1)x22=0\lim_{x \to \infty} e^{\frac{\left(-1\right) x^{2}}{2}} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
xe(−1)x22=−xe(−1)x22x e^{\frac{\left(-1\right) x^{2}}{2}} = - x e^{\frac{\left(-1\right) x^{2}}{2}}
- No
xe(−1)x22=xe(−1)x22x e^{\frac{\left(-1\right) x^{2}}{2}} = x e^{\frac{\left(-1\right) x^{2}}{2}}
- Yes
so, the function
is
odd
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