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e^x*cos(x)
Plotting a function graph/ e^x*cos(x)

Graphing y = e^x*cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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        x       
f(x) = e *cos(x)
f(x)=excos⁡(x)f{\left(x \right)} = e^{x} \cos{\left(x \right)} 
f = E^x*cos(x)
The graph of the function
05-55-50-45-40-35-30-25-20-15-10-510-2000020000
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:
excos⁡(x)=0e^{x} \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=−54.9778714378214x_{1} = -54.9778714378214
x2=−58.1194640914112x_{2} = -58.1194640914112
x3=10.9955742875643x_{3} = 10.9955742875643
x4=7.85398163397448x_{4} = 7.85398163397448
x5=1.5707963267949x_{5} = 1.5707963267949
x6=−67.5442420521806x_{6} = -67.5442420521806
x7=26.7035375555132x_{7} = 26.7035375555132
x8=−45.553093477052x_{8} = -45.553093477052
x9=−36.1283155162826x_{9} = -36.1283155162826
x10=−4.71238898038469x_{10} = -4.71238898038469
x11=−95.8185759344887x_{11} = -95.8185759344887
x12=−7.85398163397448x_{12} = -7.85398163397448
x13=−20.4203522483337x_{13} = -20.4203522483337
x14=−76.9690200129499x_{14} = -76.9690200129499
x15=−89.5353906273091x_{15} = -89.5353906273091
x16=−48.6946861306418x_{16} = -48.6946861306418
x17=4.71238898038469x_{17} = 4.71238898038469
x18=−42.4115008234622x_{18} = -42.4115008234622
x19=−70.6858347057703x_{19} = -70.6858347057703
x20=−51.8362787842316x_{20} = -51.8362787842316
x21=−39.2699081698724x_{21} = -39.2699081698724
x22=−80.1106126665397x_{22} = -80.1106126665397
x23=−64.4026493985908x_{23} = -64.4026493985908
x24=−10.9955742875643x_{24} = -10.9955742875643
x25=−61.261056745001x_{25} = -61.261056745001
x26=−105.243353895258x_{26} = -105.243353895258
x27=20.4203522483337x_{27} = 20.4203522483337
x28=−14.1371669411541x_{28} = -14.1371669411541
x29=−26.7035375555132x_{29} = -26.7035375555132
x30=−29.845130209103x_{30} = -29.845130209103
x31=29.845130209103x_{31} = 29.845130209103
x32=−86.3937979737193x_{32} = -86.3937979737193
x33=−32.9867228626928x_{33} = -32.9867228626928
x34=−83.2522053201295x_{34} = -83.2522053201295
x35=23.5619449019235x_{35} = 23.5619449019235
x36=−1.5707963267949x_{36} = -1.5707963267949
x37=−17.2787595947439x_{37} = -17.2787595947439
x38=14.1371669411541x_{38} = 14.1371669411541
x39=−98.9601685880785x_{39} = -98.9601685880785
x40=−23.5619449019235x_{40} = -23.5619449019235
x41=−73.8274273593601x_{41} = -73.8274273593601
x42=17.2787595947439x_{42} = 17.2787595947439
x43=−92.6769832808989x_{43} = -92.6769832808989
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to E^x*cos(x).
e0cos⁡(0)e^{0} \cos{\left(0 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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
−exsin⁡(x)+excos⁡(x)=0- e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
The values of the extrema at the points:
            pi 
            -- 
       ___  4  
 pi  \/ 2 *e   
(--, ---------)
 4       2


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=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,π4]\left(-\infty, \frac{\pi}{4}\right]
Increasing at intervals
[π4,∞)\left[\frac{\pi}{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
−2exsin⁡(x)=0- 2 e^{x} \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi

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
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Convex at the intervals
[0,π]\left[0, \pi\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(excos⁡(x))=0\lim_{x \to -\infty}\left(e^{x} \cos{\left(x \right)}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(excos⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(e^{x} \cos{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of E^x*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(excos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{e^{x} \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(excos⁡(x)x)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{e^{x} \cos{\left(x \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
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:
excos⁡(x)=e−xcos⁡(x)e^{x} \cos{\left(x \right)} = e^{- x} \cos{\left(x \right)}
- No
excos⁡(x)=−e−xcos⁡(x)e^{x} \cos{\left(x \right)} = - e^{- x} \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = e^x*cos(x)
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