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Plotting a function graph/ sin^10x

Graphing y = sin^10x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          10   
f(x) = sin  (x)
f(x)=sin⁡10(x)f{\left(x \right)} = \sin^{10}{\left(x \right)} 
f = sin(x)^10
The graph of the function
02468-8-6-4-2-101002
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:
sin⁡10(x)=0\sin^{10}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=−50.2324868643099x_{1} = -50.2324868643099
x2=−78.4770869973023x_{2} = -78.4770869973023
x3=−37.7050845774162x_{3} = -37.7050845774162
x4=−72.2246959176148x_{4} = -72.2246959176148
x5=0x_{5} = 0
x6=−75.4384341141264x_{6} = -75.4384341141264
x7=−65.9767997500511x_{7} = -65.9767997500511
x8=78.5063807665967x_{8} = 78.5063807665967
x9=59.7284836024868x_{9} = 59.7284836024868
x10=34.5221271875178x_{10} = 34.5221271875178
x11=12.5300063809925x_{11} = 12.5300063809925
x12=21.9922667177402x_{12} = 21.9922667177402
x13=65.9767997500161x_{13} = 65.9767997500161
x14=28.2714767215875x_{14} = 28.2714767215875
x15=84.7607022787199x_{15} = 84.7607022787199
x16=−15.7128978925504x_{16} = -15.7128978925504
x17=−31.4542226523405x_{17} = -31.4542226523405
x18=100.498513288837x_{18} = 100.498513288837
x19=−9.46211027265062x_{19} = -9.46211027265062
x20=72.2558523684046x_{20} = 72.2558523684046
x21=81.7206614302856x_{21} = 81.7206614302856
x22=−94.2169095008228x_{22} = -94.2169095008228
x23=−81.6894562305931x_{23} = -81.6894562305931
x24=−97.4205438221578x_{24} = -97.4205438221578
x25=6.26279749505248x_{25} = 6.26279749505248
x26=56.5142520241778x_{26} = 56.5142520241778
x27=−97.4305329626649x_{27} = -97.4305329626649
x28=−34.4931251477632x_{28} = -34.4931251477632
x29=6.2792892384702x_{29} = 6.2792892384702
x30=−59.6972707217566x_{30} = -59.6972707217566
x31=15.7441111366748x_{31} = 15.7441111366748
x32=−43.9845333346793x_{32} = -43.9845333346793
x33=−28.2402825408958x_{33} = -28.2402825408958
x34=−56.4851017371817x_{34} = -56.4851017371817
x35=50.2636644628673x_{35} = 50.2636644628673
x36=−21.9922667177402x_{36} = -21.9922667177402
x37=−6.24808314337863x_{37} = -6.24808314337863
x38=−100.469080855946x_{38} = -100.469080855946
x39=87.9603746328622x_{39} = 87.9603746328622
x40=40.7768097416968x_{40} = 40.7768097416968
x41=−72.2774086724237x_{41} = -72.2774086724237
x42=62.7687524265754x_{42} = 62.7687524265754
x43=94.2480403442937x_{43} = 94.2480403442937
x44=37.7363001125115x_{44} = 37.7363001125115
x45=87.9690658629265x_{45} = 87.9690658629265
x46=−12.5716694317032x_{46} = -12.5716694317032
x47=6.2418172277331x_{47} = 6.2418172277331
x48=−87.9690658638002x_{48} = -87.9690658638002
x49=−53.4463306387129x_{49} = -53.4463306387129
x50=18.7848743105356x_{50} = 18.7848743105356
x51=43.9845333346789x_{51} = 43.9845333346789
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)^10.
sin⁡10(0)\sin^{10}{\left(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
10sin⁡9(x)cos⁡(x)=010 \sin^{9}{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
The values of the extrema at the points:
(0, 0)

 -pi     
(----, 1)
  2      

 pi    
(--, 1)
 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:
Minima of the function at points:
x1=0x_{1} = 0
Maxima of the function at points:
x1=−π2x_{1} = - \frac{\pi}{2}
x1=π2x_{1} = \frac{\pi}{2}
Decreasing at intervals
(−∞,−π2]∪[0,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \infty\right)
Increasing at intervals
(−∞,0]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{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
10(−sin⁡2(x)+9cos⁡2(x))sin⁡8(x)=010 \left(- \sin^{2}{\left(x \right)} + 9 \cos^{2}{\left(x \right)}\right) \sin^{8}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−2atan⁡(13−103)x_{2} = - 2 \operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt{10}}{3} \right)}
x3=2atan⁡(13−103)x_{3} = 2 \operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt{10}}{3} \right)}
x4=−2atan⁡(13+103)x_{4} = - 2 \operatorname{atan}{\left(\frac{1}{3} + \frac{\sqrt{10}}{3} \right)}
x5=2atan⁡(13+103)x_{5} = 2 \operatorname{atan}{\left(\frac{1}{3} + \frac{\sqrt{10}}{3} \right)}

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
[2atan⁡(13+103),∞)\left[2 \operatorname{atan}{\left(\frac{1}{3} + \frac{\sqrt{10}}{3} \right)}, \infty\right)
Convex at the intervals
(−∞,2atan⁡(13−103)]∪[−2atan⁡(13−103),2atan⁡(13+103)]\left(-\infty, 2 \operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt{10}}{3} \right)}\right] \cup \left[- 2 \operatorname{atan}{\left(\frac{1}{3} - \frac{\sqrt{10}}{3} \right)}, 2 \operatorname{atan}{\left(\frac{1}{3} + \frac{\sqrt{10}}{3} \right)}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞sin⁡10(x)=⟨0,1⟩\lim_{x \to -\infty} \sin^{10}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡x→∞sin⁡10(x)=⟨0,1⟩\lim_{x \to \infty} \sin^{10}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)^10, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡10(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{10}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡10(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{10}{\left(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:
sin⁡10(x)=sin⁡10(x)\sin^{10}{\left(x \right)} = \sin^{10}{\left(x \right)}
- Yes
sin⁡10(x)=−sin⁡10(x)\sin^{10}{\left(x \right)} = - \sin^{10}{\left(x \right)}
- No
so, the function
is
even
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