Mister Exam

Other calculators

Plotting a function graph/ (x-3)e^-x

Graphing y = (x-3)e^-x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                -x
f(x) = (x - 3)*E
f(x)=e−x(x−3)f{\left(x \right)} = e^{- x} \left(x - 3\right) 
f = E^(-x)*(x - 3)
The graph of the function
02468-8-6-4-2-1010-200000200000
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:
e−x(x−3)=0e^{- x} \left(x - 3\right) = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=3x_{1} = 3
Numerical solution
x1=99.432316424891x_{1} = 99.432316424891
x2=83.4917816149558x_{2} = 83.4917816149558
x3=43.9557499214057x_{3} = 43.9557499214057
x4=59.664342946604x_{4} = 59.664342946604
x5=113.396350396671x_{5} = 113.396350396671
x6=79.51136695866x_{6} = 79.51136695866
x7=38.1905363866884x_{7} = 38.1905363866884
x8=30.9653321772927x_{8} = 30.9653321772927
x9=65.6052138551392x_{9} = 65.6052138551392
x10=115.392040334004x_{10} = 115.392040334004
x11=93.4517230466241x_{11} = 93.4517230466241
x12=111.400841299949x_{12} = 111.400841299949
x13=63.6232240789579x_{13} = 63.6232240789579
x14=117.387900375534x_{14} = 117.387900375534
x15=105.415520933891x_{15} = 105.415520933891
x16=69.5733090128955x_{16} = 69.5733090128955
x17=49.8119589630405x_{17} = 49.8119589630405
x18=77.5221246603965x_{18} = 77.5221246603965
x19=85.4828412467504x_{19} = 85.4828412467504
x20=61.642856145511x_{20} = 61.642856145511
x21=101.426455152084x_{21} = 101.426455152084
x22=75.5336138177003x_{22} = 75.5336138177003
x23=53.7430576092052x_{23} = 53.7430576092052
x24=40.0970717014418x_{24} = 40.0970717014418
x25=42.020216210141x_{25} = 42.020216210141
x26=87.4744046501982x_{26} = 87.4744046501982
x27=119.383920620405x_{27} = 119.383920620405
x28=109.405524706139x_{28} = 109.405524706139
x29=89.466430197318x_{29} = 89.466430197318
x30=3x_{30} = 3
x31=51.7754697845928x_{31} = 51.7754697845928
x32=97.4384664647568x_{32} = 97.4384664647568
x33=91.4588807455217x_{33} = 91.4588807455217
x34=71.5591096232555x_{34} = 71.5591096232555
x35=55.714063380457x_{35} = 55.714063380457
x36=32.6626982028378x_{36} = 32.6626982028378
x37=103.420862702525x_{37} = 103.420862702525
x38=57.6879649775293x_{38} = 57.6879649775293
x39=36.3071598061728x_{39} = 36.3071598061728
x40=67.5886304003902x_{40} = 67.5886304003902
x41=107.410413305772x_{41} = 107.410413305772
x42=121.380091923383x_{42} = 121.380091923383
x43=73.545912319012x_{43} = 73.545912319012
x44=81.5012725708786x_{44} = 81.5012725708786
x45=47.853370487631x_{45} = 47.853370487631
x46=34.4578471962376x_{46} = 34.4578471962376
x47=95.444927247289x_{47} = 95.444927247289
x48=34.2046743865559x_{48} = 34.2046743865559
x49=45.9008089996782x_{49} = 45.9008089996782
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x - 3)*E^(-x).
−3e−0- 3 e^{- 0}
The result:
f(0)=−3f{\left(0 \right)} = -3
The point:
(0, -3)
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
−(x−3)e−x+e−x=0- \left(x - 3\right) e^{- x} + e^{- x} = 0
Solve this equation
The roots of this equation
x1=4x_{1} = 4
The values of the extrema at the points:
     -4 
(4, 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:
The function has no minima
Maxima of the function at points:
x1=4x_{1} = 4
Decreasing at intervals
(−∞,4]\left(-\infty, 4\right]
Increasing at intervals
[4,∞)\left[4, \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−5)e−x=0\left(x - 5\right) e^{- x} = 0
Solve this equation
The roots of this equation
x1=5x_{1} = 5

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
[5,∞)\left[5, \infty\right)
Convex at the intervals
(−∞,5]\left(-\infty, 5\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(e−x(x−3))=−∞\lim_{x \to -\infty}\left(e^{- x} \left(x - 3\right)\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(e−x(x−3))=0\lim_{x \to \infty}\left(e^{- x} \left(x - 3\right)\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 - 3)*E^(-x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x−3)e−xx)=∞\lim_{x \to -\infty}\left(\frac{\left(x - 3\right) e^{- x}}{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞((x−3)e−xx)=0\lim_{x \to \infty}\left(\frac{\left(x - 3\right) e^{- x}}{x}\right) = 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:
e−x(x−3)=(−x−3)exe^{- x} \left(x - 3\right) = \left(- x - 3\right) e^{x}
- No
e−x(x−3)=−(−x−3)exe^{- x} \left(x - 3\right) = - \left(- x - 3\right) e^{x}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator