Mister Exam
Graphing y = sin(2x)-cos(3x)
The solution
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f(x) = sin(2*x) - cos(3*x)
$$f{\left(x \right)} = \sin{\left(2 x \right)} - \cos{\left(3 x \right)}$$
f = sin(2*x) - cos(3*x)
The graph of the function
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:
$$\sin{\left(2 x \right)} - \cos{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{10}$$
$$x_{3} = \frac{\pi}{2}$$
$$x_{4} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
$$x_{5} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{6} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
Numerical solution
$$x_{1} = -43.6681378848981$$
$$x_{2} = -23.5619449019235$$
$$x_{3} = 61.261056745001$$
$$x_{4} = -29.845130209103$$
$$x_{5} = 78.2256570743859$$
$$x_{6} = -9.73893722612836$$
$$x_{7} = 16.6504410640259$$
$$x_{8} = 99.5884871187965$$
$$x_{9} = -12.2522113490002$$
$$x_{10} = 36.1283155162826$$
$$x_{11} = -87.6504350351552$$
$$x_{12} = -76.340701482232$$
$$x_{13} = 23.5619449019235$$
$$x_{14} = 4.08407044966673$$
$$x_{15} = -97.7035315266426$$
$$x_{16} = 94.5619388730528$$
$$x_{17} = 38.0132711084365$$
$$x_{18} = -91.420346219463$$
$$x_{19} = -80.1106126665397$$
$$x_{20} = 86.3937979737193$$
$$x_{21} = 24.1902634326414$$
$$x_{22} = 64.4026493985908$$
$$x_{23} = -95.8185759344887$$
$$x_{24} = 81.9955682586936$$
$$x_{25} = -34.8716784548467$$
$$x_{26} = -66.2876049907446$$
$$x_{27} = -93.9336203423348$$
$$x_{28} = -88.9070720965912$$
$$x_{29} = -33.6150413934108$$
$$x_{30} = 75.712382951514$$
$$x_{31} = 27.9601746169492$$
$$x_{32} = 34.2433599241287$$
$$x_{33} = -26.0752190247953$$
$$x_{34} = 60.632738214283$$
$$x_{35} = 73.1991088286422$$
$$x_{36} = -51.8362787842316$$
$$x_{37} = 58.1194640914112$$
$$x_{38} = -73.8274273593601$$
$$x_{39} = 51.8362787842316$$
$$x_{40} = 40.5265452313083$$
$$x_{41} = -39.8982267005904$$
$$x_{42} = 29.845130209103$$
$$x_{43} = -45.553093477052$$
$$x_{44} = 80.1106126665397$$
$$x_{45} = 31.7300858012569$$
$$x_{46} = -58.1194640914112$$
$$x_{47} = 11.6238928182822$$
$$x_{48} = -100.216805649514$$
$$x_{49} = -47.4380490692059$$
$$x_{50} = 79.4822941358218$$
$$x_{51} = -36.1283155162826$$
$$x_{52} = 0.314159265358979$$
$$x_{53} = 68.1725605828985$$
$$x_{54} = -63.7743308678728$$
$$x_{55} = 54.3495529071034$$
$$x_{56} = -5.96902604182061$$
$$x_{57} = 44.2964564156161$$
$$x_{58} = -67.5442420521806$$
$$x_{59} = 88.2787535658732$$
$$x_{60} = 92.0486647501809$$
$$x_{61} = -7.85398163397448$$
$$x_{62} = 10.3672557568463$$
$$x_{63} = 98.3318500573605$$
$$x_{64} = -71.3141532364883$$
$$x_{65} = -19.7920337176157$$
$$x_{66} = -22.3053078404875$$
$$x_{67} = -56.2345084992573$$
$$x_{68} = 7.85398163397448$$
$$x_{69} = -14.1371669411541$$
$$x_{70} = -83.8805238508475$$
$$x_{71} = -77.5973385436679$$
$$x_{72} = -70.0575161750524$$
$$x_{73} = -3.45575191894877$$
$$x_{74} = 61.8893752757189$$
$$x_{75} = -60.0044196835651$$
$$x_{76} = 48.0663675999238$$
$$x_{77} = 71.9424717672063$$
$$x_{78} = 14.1371669411541$$
$$x_{79} = 55.6061899685393$$
$$x_{80} = -0.942477796076938$$
$$x_{81} = -32.3584043319749$$
$$x_{82} = 20.4203522483337$$
$$x_{83} = -27.3318560862312$$
$$x_{84} = -49.9513231920777$$
$$x_{85} = 95.8185759344887$$
$$x_{86} = -53.7212343763855$$
$$x_{87} = 84.5088423815654$$
$$x_{88} = -76.9690200129499$$
$$x_{89} = -16.0221225333079$$
$$x_{90} = 17.9070781254618$$
so we need to solve the equation:
$$\sin{\left(2 x \right)} - \cos{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{10}$$
$$x_{3} = \frac{\pi}{2}$$
$$x_{4} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
$$x_{5} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{6} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
Numerical solution
$$x_{1} = -43.6681378848981$$
$$x_{2} = -23.5619449019235$$
$$x_{3} = 61.261056745001$$
$$x_{4} = -29.845130209103$$
$$x_{5} = 78.2256570743859$$
$$x_{6} = -9.73893722612836$$
$$x_{7} = 16.6504410640259$$
$$x_{8} = 99.5884871187965$$
$$x_{9} = -12.2522113490002$$
$$x_{10} = 36.1283155162826$$
$$x_{11} = -87.6504350351552$$
$$x_{12} = -76.340701482232$$
$$x_{13} = 23.5619449019235$$
$$x_{14} = 4.08407044966673$$
$$x_{15} = -97.7035315266426$$
$$x_{16} = 94.5619388730528$$
$$x_{17} = 38.0132711084365$$
$$x_{18} = -91.420346219463$$
$$x_{19} = -80.1106126665397$$
$$x_{20} = 86.3937979737193$$
$$x_{21} = 24.1902634326414$$
$$x_{22} = 64.4026493985908$$
$$x_{23} = -95.8185759344887$$
$$x_{24} = 81.9955682586936$$
$$x_{25} = -34.8716784548467$$
$$x_{26} = -66.2876049907446$$
$$x_{27} = -93.9336203423348$$
$$x_{28} = -88.9070720965912$$
$$x_{29} = -33.6150413934108$$
$$x_{30} = 75.712382951514$$
$$x_{31} = 27.9601746169492$$
$$x_{32} = 34.2433599241287$$
$$x_{33} = -26.0752190247953$$
$$x_{34} = 60.632738214283$$
$$x_{35} = 73.1991088286422$$
$$x_{36} = -51.8362787842316$$
$$x_{37} = 58.1194640914112$$
$$x_{38} = -73.8274273593601$$
$$x_{39} = 51.8362787842316$$
$$x_{40} = 40.5265452313083$$
$$x_{41} = -39.8982267005904$$
$$x_{42} = 29.845130209103$$
$$x_{43} = -45.553093477052$$
$$x_{44} = 80.1106126665397$$
$$x_{45} = 31.7300858012569$$
$$x_{46} = -58.1194640914112$$
$$x_{47} = 11.6238928182822$$
$$x_{48} = -100.216805649514$$
$$x_{49} = -47.4380490692059$$
$$x_{50} = 79.4822941358218$$
$$x_{51} = -36.1283155162826$$
$$x_{52} = 0.314159265358979$$
$$x_{53} = 68.1725605828985$$
$$x_{54} = -63.7743308678728$$
$$x_{55} = 54.3495529071034$$
$$x_{56} = -5.96902604182061$$
$$x_{57} = 44.2964564156161$$
$$x_{58} = -67.5442420521806$$
$$x_{59} = 88.2787535658732$$
$$x_{60} = 92.0486647501809$$
$$x_{61} = -7.85398163397448$$
$$x_{62} = 10.3672557568463$$
$$x_{63} = 98.3318500573605$$
$$x_{64} = -71.3141532364883$$
$$x_{65} = -19.7920337176157$$
$$x_{66} = -22.3053078404875$$
$$x_{67} = -56.2345084992573$$
$$x_{68} = 7.85398163397448$$
$$x_{69} = -14.1371669411541$$
$$x_{70} = -83.8805238508475$$
$$x_{71} = -77.5973385436679$$
$$x_{72} = -70.0575161750524$$
$$x_{73} = -3.45575191894877$$
$$x_{74} = 61.8893752757189$$
$$x_{75} = -60.0044196835651$$
$$x_{76} = 48.0663675999238$$
$$x_{77} = 71.9424717672063$$
$$x_{78} = 14.1371669411541$$
$$x_{79} = 55.6061899685393$$
$$x_{80} = -0.942477796076938$$
$$x_{81} = -32.3584043319749$$
$$x_{82} = 20.4203522483337$$
$$x_{83} = -27.3318560862312$$
$$x_{84} = -49.9513231920777$$
$$x_{85} = 95.8185759344887$$
$$x_{86} = -53.7212343763855$$
$$x_{87} = 84.5088423815654$$
$$x_{88} = -76.9690200129499$$
$$x_{89} = -16.0221225333079$$
$$x_{90} = 17.9070781254618$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$3 \sin{\left(3 x \right)} + 2 \cos{\left(2 x \right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$3 \sin{\left(3 x \right)} + 2 \cos{\left(2 x \right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- 4 \sin{\left(2 x \right)} + 9 \cos{\left(3 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
$$x_{3} = i \left(\log{\left(18 \right)} - \log{\left(- \sqrt{4 \sqrt{85} + 235} - 2 i + \sqrt{85} i \right)}\right)$$
$$x_{4} = - i \log{\left(- \frac{\sqrt{235 - 4 \sqrt{85}}}{18} - \frac{i \left(2 + \sqrt{85}\right)}{18} \right)}$$
$$x_{5} = - i \log{\left(\frac{\sqrt{235 - 4 \sqrt{85}}}{18} - \frac{i \left(2 + \sqrt{85}\right)}{18} \right)}$$
$$x_{6} = - i \log{\left(\frac{\sqrt{4 \sqrt{85} + 235}}{18} - \frac{i \left(2 - \sqrt{85}\right)}{18} \right)}$$
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
$$\left[\frac{\pi}{2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \pi + \operatorname{atan}{\left(- \frac{- \sqrt{85} - 2}{\sqrt{235 - 4 \sqrt{85}}} \right)}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- 4 \sin{\left(2 x \right)} + 9 \cos{\left(3 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
$$x_{3} = i \left(\log{\left(18 \right)} - \log{\left(- \sqrt{4 \sqrt{85} + 235} - 2 i + \sqrt{85} i \right)}\right)$$
$$x_{4} = - i \log{\left(- \frac{\sqrt{235 - 4 \sqrt{85}}}{18} - \frac{i \left(2 + \sqrt{85}\right)}{18} \right)}$$
$$x_{5} = - i \log{\left(\frac{\sqrt{235 - 4 \sqrt{85}}}{18} - \frac{i \left(2 + \sqrt{85}\right)}{18} \right)}$$
$$x_{6} = - i \log{\left(\frac{\sqrt{4 \sqrt{85} + 235}}{18} - \frac{i \left(2 - \sqrt{85}\right)}{18} \right)}$$
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
$$\left[\frac{\pi}{2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \pi + \operatorname{atan}{\left(- \frac{- \sqrt{85} - 2}{\sqrt{235 - 4 \sqrt{85}}} \right)}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} - \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} - \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} - \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} - \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 2\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(2*x) - cos(3*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} - \cos{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} - \cos{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} - \cos{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} - \cos{\left(3 x \right)}}{x}\right) = 0$$
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:
$$\sin{\left(2 x \right)} - \cos{\left(3 x \right)} = - \sin{\left(2 x \right)} - \cos{\left(3 x \right)}$$
- No
$$\sin{\left(2 x \right)} - \cos{\left(3 x \right)} = \sin{\left(2 x \right)} + \cos{\left(3 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(2 x \right)} - \cos{\left(3 x \right)} = - \sin{\left(2 x \right)} - \cos{\left(3 x \right)}$$
- No
$$\sin{\left(2 x \right)} - \cos{\left(3 x \right)} = \sin{\left(2 x \right)} + \cos{\left(3 x \right)}$$
- No
so, the function
not is
neither even, nor odd