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Plotting a function graph/ sin3x-cos2x

Graphing y = sin3x-cos2x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(3*x) - cos(2*x)
f(x)=sin⁡(3x)−cos⁡(2x)f{\left(x \right)} = \sin{\left(3 x \right)} - \cos{\left(2 x \right)} 
f = sin(3*x) - cos(2*x)
The graph of the function
02468-8-6-4-2-10105-5
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⁡(3x)−cos⁡(2x)=0\sin{\left(3 x \right)} - \cos{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π10x_{1} = \frac{\pi}{10}
x2=π2x_{2} = \frac{\pi}{2}
x3=−ilog⁡(−25+58+105+58−5i4−i4)x_{3} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
x4=−ilog⁡(−105+516−105−516−25+516+25−516−i4+5i4)x_{4} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x5=−ilog⁡(−105+516−25+516−25−516+105−516−5i4−i4)x_{5} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
Numerical solution
x1=21.6769893097696x_{1} = 21.6769893097696
x2=−76.340701482232x_{2} = -76.340701482232
x3=−63.7743308678728x_{3} = -63.7743308678728
x4=−3.45575191894877x_{4} = -3.45575191894877
x5=24.1902634326414x_{5} = 24.1902634326414
x6=−70.0575161750524x_{6} = -70.0575161750524
x7=51.836278889862x_{7} = 51.836278889862
x8=45.5530935642345x_{8} = 45.5530935642345
x9=−36.1283154253128x_{9} = -36.1283154253128
x10=−49.9513231920777x_{10} = -49.9513231920777
x11=−57.4911455606932x_{11} = -57.4911455606932
x12=68.1725605828985x_{12} = 68.1725605828985
x13=763.092855556961x_{13} = 763.092855556961
x14=−9.73893722612836x_{14} = -9.73893722612836
x15=78.2256570743859x_{15} = 78.2256570743859
x16=6.59734457253857x_{16} = 6.59734457253857
x17=27.9601746169492x_{17} = 27.9601746169492
x18=−98.960168969851x_{18} = -98.960168969851
x19=−42.4115008665293x_{19} = -42.4115008665293
x20=14.1371670842217x_{20} = 14.1371670842217
x21=84.5088423815654x_{21} = 84.5088423815654
x22=70.6858346085267x_{22} = 70.6858346085267
x23=94.5619388730528x_{23} = 94.5619388730528
x24=−3082.21655243695x_{24} = -3082.21655243695
x25=−100.216805649514x_{25} = -100.216805649514
x26=58.1194641300388x_{26} = 58.1194641300388
x27=−46.18141200777x_{27} = -46.18141200777
x28=−90.1637091580271x_{28} = -90.1637091580271
x29=−5.96902604182061x_{29} = -5.96902604182061
x30=−77.5973385436679x_{30} = -77.5973385436679
x31=48.0663675999238x_{31} = 48.0663675999238
x32=95.8185759411754x_{32} = 95.8185759411754
x33=−60.0044196835651x_{33} = -60.0044196835651
x34=−17.2787596488556x_{34} = -17.2787596488556
x35=4.08407044966673x_{35} = 4.08407044966673
x36=98.3318500573605x_{36} = 98.3318500573605
x37=30.473448739821x_{37} = 30.473448739821
x38=44.2964564156161x_{38} = 44.2964564156161
x39=26.7035374759771x_{39} = 26.7035374759771
x40=7.85398173263687x_{40} = 7.85398173263687
x41=−29.8451300991422x_{41} = -29.8451300991422
x42=61.8893752757189x_{42} = 61.8893752757189
x43=64.4026493135056x_{43} = 64.4026493135056
x44=11.6238928182822x_{44} = 11.6238928182822
x45=−42.41150070898x_{45} = -42.41150070898
x46=88.2787535658732x_{46} = 88.2787535658732
x47=−12.2522113490002x_{47} = -12.2522113490002
x48=20.4203521560512x_{48} = 20.4203521560512
x49=−19.7920337176157x_{49} = -19.7920337176157
x50=1.57079642986981x_{50} = 1.57079642986981
x51=−87.6504350351552x_{51} = -87.6504350351552
x52=−61.2610567621223x_{52} = -61.2610567621223
x53=−39.8982267005904x_{53} = -39.8982267005904
x54=97.0752129959246x_{54} = 97.0752129959246
x55=−2.19911485751286x_{55} = -2.19911485751286
x56=95.8185760463912x_{56} = 95.8185760463912
x57=−93.9336203423348x_{57} = -93.9336203423348
x58=340.862803294903x_{58} = 340.862803294903
x59=85.7654794430014x_{59} = 85.7654794430014
x60=54.3495529071034x_{60} = 54.3495529071034
x61=−16.0221225333079x_{61} = -16.0221225333079
x62=92.0486647501809x_{62} = 92.0486647501809
x63=71.9424717672063x_{63} = 71.9424717672063
x64=50.5796417227957x_{64} = 50.5796417227957
x65=20.4203523376058x_{65} = 20.4203523376058
x66=−53.7212343763855x_{66} = -53.7212343763855
x67=−86.3937978510561x_{67} = -86.3937978510561
x68=34.2433599241287x_{68} = 34.2433599241287
x69=−43.6681378848981x_{69} = -43.6681378848981
x70=−56.2345084992573x_{70} = -56.2345084992573
x71=−80.1106125840384x_{71} = -80.1106125840384
x72=−23.5619448484475x_{72} = -23.5619448484475
x73=0.314159265358979x_{73} = 0.314159265358979
x74=−22.3053078404875x_{74} = -22.3053078404875
x75=−37.3849525777185x_{75} = -37.3849525777185
x76=38.0132711084365x_{76} = 38.0132711084365
x77=−66.2876049907446x_{77} = -66.2876049907446
x78=−83.8805238508475x_{78} = -83.8805238508475
x79=74.4557458900781x_{79} = 74.4557458900781
x80=−97.7035315266426x_{80} = -97.7035315266426
x81=10.3672557568463x_{81} = 10.3672557568463
x82=−67.5442421572865x_{82} = -67.5442421572865
x83=−36.1283156488921x_{83} = -36.1283156488921
x84=41.7831822927443x_{84} = 41.7831822927443
x85=−23.5619450008864x_{85} = -23.5619450008864
x86=83.2522057062651x_{86} = 83.2522057062651
x87=81.9955682586936x_{87} = 81.9955682586936
x88=−26.0752190247953x_{88} = -26.0752190247953
x89=17.9070781254618x_{89} = 17.9070781254618
x90=−81.3672497279756x_{90} = -81.3672497279756
x91=65.6592864600267x_{91} = 65.6592864600267
x92=−73.8274272806455x_{92} = -73.8274272806455
x93=89.5353906921091x_{93} = 89.5353906921091
x94=−13.5088484104361x_{94} = -13.5088484104361
x95=−33.6150413934108x_{95} = -33.6150413934108
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(3*x) - cos(2*x).
−cos⁡(0⋅2)+sin⁡(0⋅3)- \cos{\left(0 \cdot 2 \right)} + \sin{\left(0 \cdot 3 \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
2sin⁡(2x)+3cos⁡(3x)=02 \sin{\left(2 x \right)} + 3 \cos{\left(3 x \right)} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
x3=i(log⁡(6)−log⁡(−25−210+i+10i))x_{3} = i \left(\log{\left(6 \right)} - \log{\left(- \sqrt{25 - 2 \sqrt{10}} + i + \sqrt{10} i \right)}\right)
x4=i(log⁡(6)−log⁡(25−210+i+10i))x_{4} = i \left(\log{\left(6 \right)} - \log{\left(\sqrt{25 - 2 \sqrt{10}} + i + \sqrt{10} i \right)}\right)
x5=i(log⁡(6)−log⁡(−210+25−10i+i))x_{5} = i \left(\log{\left(6 \right)} - \log{\left(- \sqrt{2 \sqrt{10} + 25} - \sqrt{10} i + i \right)}\right)
x6=i(log⁡(6)−log⁡(210+25−10i+i))x_{6} = i \left(\log{\left(6 \right)} - \log{\left(\sqrt{2 \sqrt{10} + 25} - \sqrt{10} i + i \right)}\right)
The values of the extrema at the points:
 -pi     
(----, 2)
  2      

 pi    
(--, 0)
 2     

   /     /       _______________           \         \       /    /     /       _______________           \         \\      /    /     /       _______________           \         \\ 
   |     |      /          ____        ____|         |       |    |     |      /          ____        ____|         ||      |    |     |      /          ____        ____|         || 
(I*\- log\I - \/  25 - 2*\/ 10   + I*\/ 10 / + log(6)/, - cos\2*I*\- log\I - \/  25 - 2*\/ 10   + I*\/ 10 / + log(6)// + sin\3*I*\- log\I - \/  25 - 2*\/ 10   + I*\/ 10 / + log(6)//)

   /     /       _______________           \         \       /    /     /       _______________           \         \\      /    /     /       _______________           \         \\ 
   |     |      /          ____        ____|         |       |    |     |      /          ____        ____|         ||      |    |     |      /          ____        ____|         || 
(I*\- log\I + \/  25 - 2*\/ 10   + I*\/ 10 / + log(6)/, - cos\2*I*\- log\I + \/  25 - 2*\/ 10   + I*\/ 10 / + log(6)// + sin\3*I*\- log\I + \/  25 - 2*\/ 10   + I*\/ 10 / + log(6)//)

   /     /       _______________           \         \       /    /     /       _______________           \         \\      /    /     /       _______________           \         \\ 
   |     |      /          ____        ____|         |       |    |     |      /          ____        ____|         ||      |    |     |      /          ____        ____|         || 
(I*\- log\I - \/  25 + 2*\/ 10   - I*\/ 10 / + log(6)/, - cos\2*I*\- log\I - \/  25 + 2*\/ 10   - I*\/ 10 / + log(6)// + sin\3*I*\- log\I - \/  25 + 2*\/ 10   - I*\/ 10 / + log(6)//)

   /     /       _______________           \         \       /    /     /       _______________           \         \\      /    /     /       _______________           \         \\ 
   |     |      /          ____        ____|         |       |    |     |      /          ____        ____|         ||      |    |     |      /          ____        ____|         || 
(I*\- log\I + \/  25 + 2*\/ 10   - I*\/ 10 / + log(6)/, - cos\2*I*\- log\I + \/  25 + 2*\/ 10   - I*\/ 10 / + log(6)// + sin\3*I*\- log\I + \/  25 + 2*\/ 10   - I*\/ 10 / + log(6)//)


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=π2x_{1} = \frac{\pi}{2}
x2=−π−atan⁡(1−10210+25)x_{2} = - \pi - \operatorname{atan}{\left(\frac{1 - \sqrt{10}}{\sqrt{2 \sqrt{10} + 25}} \right)}
x3=atan⁡(1−10210+25)x_{3} = \operatorname{atan}{\left(\frac{1 - \sqrt{10}}{\sqrt{2 \sqrt{10} + 25}} \right)}
Maxima of the function at points:
x3=−π2x_{3} = - \frac{\pi}{2}
x3=π−atan⁡(1+1025−210)x_{3} = \pi - \operatorname{atan}{\left(\frac{1 + \sqrt{10}}{\sqrt{25 - 2 \sqrt{10}}} \right)}
x3=atan⁡(1+1025−210)x_{3} = \operatorname{atan}{\left(\frac{1 + \sqrt{10}}{\sqrt{25 - 2 \sqrt{10}}} \right)}
Decreasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−π−atan⁡(1−10210+25)]\left(-\infty, - \pi - \operatorname{atan}{\left(\frac{1 - \sqrt{10}}{\sqrt{2 \sqrt{10} + 25}} \right)}\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
−9sin⁡(3x)+4cos⁡(2x)=0- 9 \sin{\left(3 x \right)} + 4 \cos{\left(2 x \right)} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(3x)−cos⁡(2x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\sin{\left(3 x \right)} - \cos{\left(2 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(sin⁡(3x)−cos⁡(2x))=⟨−2,2⟩\lim_{x \to \infty}\left(\sin{\left(3 x \right)} - \cos{\left(2 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(3*x) - cos(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(3x)−cos⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)} - \cos{\left(2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(3x)−cos⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)} - \cos{\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:
sin⁡(3x)−cos⁡(2x)=−sin⁡(3x)−cos⁡(2x)\sin{\left(3 x \right)} - \cos{\left(2 x \right)} = - \sin{\left(3 x \right)} - \cos{\left(2 x \right)}
- No
sin⁡(3x)−cos⁡(2x)=sin⁡(3x)+cos⁡(2x)\sin{\left(3 x \right)} - \cos{\left(2 x \right)} = \sin{\left(3 x \right)} + \cos{\left(2 x \right)}
- No
so, the function
not is
neither even, nor odd
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