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e^(-x)*sin(2*x)
Plotting a function graph/ e^(-x)*sin(2*x)

Graphing y = e^(-x)*sin(2*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        -x         
f(x) = e  *sin(2*x)
f(x)=e−xsin⁡(2x)f{\left(x \right)} = e^{- x} \sin{\left(2 x \right)} 
f = sin(2*x)/E^x
The graph of the function
0102030405060708090-10-2500025000
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−xsin⁡(2x)=0e^{- x} \sin{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
Numerical solution
x1=72.2566310325652x_{1} = 72.2566310325652
x2=94.2477796076938x_{2} = 94.2477796076938
x3=−3.14159265358979x_{3} = -3.14159265358979
x4=36.1283155162826x_{4} = 36.1283155162826
x5=40.8407044966673x_{5} = 40.8407044966673
x6=98.9601685880785x_{6} = 98.9601685880785
x7=−15.707963267949x_{7} = -15.707963267949
x8=−31.4159265358979x_{8} = -31.4159265358979
x9=6.28318530717959x_{9} = 6.28318530717959
x10=62.8318530717959x_{10} = 62.8318530717959
x11=81.6814089933346x_{11} = 81.6814089933346
x12=100.530964914873x_{12} = 100.530964914873
x13=48.6946861306418x_{13} = 48.6946861306418
x14=50.2654824574367x_{14} = 50.2654824574367
x15=84.8230016469244x_{15} = 84.8230016469244
x16=−17.2787595947439x_{16} = -17.2787595947439
x17=61.261056745001x_{17} = 61.261056745001
x18=−9.42477796076938x_{18} = -9.42477796076938
x19=76.9690200129499x_{19} = 76.9690200129499
x20=−1.5707963267949x_{20} = -1.5707963267949
x21=15.707963267949x_{21} = 15.707963267949
x22=97.3893722612836x_{22} = 97.3893722612836
x23=73.8274273593601x_{23} = 73.8274273593601
x24=51.8362787842316x_{24} = 51.8362787842316
x25=78.5398163397448x_{25} = 78.5398163397448
x26=−6.28318530717959x_{26} = -6.28318530717959
x27=12.5663706143592x_{27} = 12.5663706143592
x28=−23.5619449019235x_{28} = -23.5619449019235
x29=70.6858347057703x_{29} = 70.6858347057703
x30=58.1194640914112x_{30} = 58.1194640914112
x31=56.5486677646163x_{31} = 56.5486677646163
x32=−28.2743338823081x_{32} = -28.2743338823081
x33=18.8495559215388x_{33} = 18.8495559215388
x34=7.85398163397448x_{34} = 7.85398163397448
x35=59.6902604182061x_{35} = 59.6902604182061
x36=−25.1327412287183x_{36} = -25.1327412287183
x37=−21.9911485751286x_{37} = -21.9911485751286
x38=86.3937979737193x_{38} = 86.3937979737193
x39=10.9955742875643x_{39} = 10.9955742875643
x40=−7.85398163397448x_{40} = -7.85398163397448
x41=92.6769832808989x_{41} = 92.6769832808989
x42=80.1106126665397x_{42} = 80.1106126665397
x43=43.9822971502571x_{43} = 43.9822971502571
x44=28.2743338823081x_{44} = 28.2743338823081
x45=4.71238898038469x_{45} = 4.71238898038469
x46=32.9867228626928x_{46} = 32.9867228626928
x47=20.4203522483337x_{47} = 20.4203522483337
x48=64.4026493985908x_{48} = 64.4026493985908
x49=42.4115008234622x_{49} = 42.4115008234622
x50=65.9734457253857x_{50} = 65.9734457253857
x51=−14.1371669411541x_{51} = -14.1371669411541
x52=26.7035375555132x_{52} = 26.7035375555132
x53=34.5575191894877x_{53} = 34.5575191894877
x54=95.8185759344887x_{54} = 95.8185759344887
x55=0x_{55} = 0
x56=14.1371669411541x_{56} = 14.1371669411541
x57=54.9778714378214x_{57} = 54.9778714378214
x58=21.9911485751286x_{58} = 21.9911485751286
x59=87.9645943005142x_{59} = 87.9645943005142
x60=37.6991118430775x_{60} = 37.6991118430775
x61=−29.845130209103x_{61} = -29.845130209103
x62=29.845130209103x_{62} = 29.845130209103
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(2*x)/E^x.
e(−1)0sin⁡(2⋅0)e^{\left(-1\right) 0} \sin{\left(2 \cdot 0 \right)}
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
−e−xsin⁡(2x)+2e−xcos⁡(2x)=0- e^{- x} \sin{\left(2 x \right)} + 2 e^{- x} \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=atan⁡(2)2x_{1} = \frac{\operatorname{atan}{\left(2 \right)}}{2}
The values of the extrema at the points:
                   -atan(2)  
                   --------- 
              ___      2     
 atan(2)  2*\/ 5 *e          
(-------, ------------------)
    2             5


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=atan⁡(2)2x_{1} = \frac{\operatorname{atan}{\left(2 \right)}}{2}
Decreasing at intervals
(−∞,atan⁡(2)2]\left(-\infty, \frac{\operatorname{atan}{\left(2 \right)}}{2}\right]
Increasing at intervals
[atan⁡(2)2,∞)\left[\frac{\operatorname{atan}{\left(2 \right)}}{2}, \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
−(3sin⁡(2x)+4cos⁡(2x))e−x=0- \left(3 \sin{\left(2 x \right)} + 4 \cos{\left(2 x \right)}\right) e^{- x} = 0
Solve this equation
The roots of this equation
x1=−atan⁡(43)2x_{1} = - \frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{2}

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
(−∞,−atan⁡(43)2]\left(-\infty, - \frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{2}\right]
Convex at the intervals
[−atan⁡(43)2,∞)\left[- \frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{2}, \infty\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−xsin⁡(2x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(e^{- x} \sin{\left(2 x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(e−xsin⁡(2x))=0\lim_{x \to \infty}\left(e^{- x} \sin{\left(2 x \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 sin(2*x)/E^x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(e−xsin⁡(2x)x)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{e^{- x} \sin{\left(2 x \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(e−xsin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{e^{- x} \sin{\left(2 x \right)}}{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−xsin⁡(2x)=−exsin⁡(2x)e^{- x} \sin{\left(2 x \right)} = - e^{x} \sin{\left(2 x \right)}
- No
e−xsin⁡(2x)=exsin⁡(2x)e^{- x} \sin{\left(2 x \right)} = e^{x} \sin{\left(2 x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = e^(-x)*sin(2*x)
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