Mister Exam
Graphing y = sin(4*x)*cos(5*x)
The solution
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f(x) = sin(4*x)*cos(5*x)
$$f{\left(x \right)} = \sin{\left(4 x \right)} \cos{\left(5 x \right)}$$
f = sin(4*x)*cos(5*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:
$$\sin{\left(4 x \right)} \cos{\left(5 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{10}$$
$$x_{3} = \frac{\pi}{10}$$
Numerical solution
$$x_{1} = -43.9822971502571$$
$$x_{2} = -25.9181393921158$$
$$x_{3} = 56.5486677646163$$
$$x_{4} = -7.85398160320836$$
$$x_{5} = 24.1902634326414$$
$$x_{6} = -55.7632696012188$$
$$x_{7} = 72.2566310325652$$
$$x_{8} = -97.7035315266426$$
$$x_{9} = -41.7831822927443$$
$$x_{10} = -85.7654794430014$$
$$x_{11} = 14.765485471872$$
$$x_{12} = 38.0132711084365$$
$$x_{13} = -75.712382951514$$
$$x_{14} = -11.7809724509617$$
$$x_{15} = -61.8893752757189$$
$$x_{16} = -27.9601746169492$$
$$x_{17} = -21.9911485751286$$
$$x_{18} = 28.5884931476671$$
$$x_{19} = 76.1836218495525$$
$$x_{20} = -51.8362787041329$$
$$x_{21} = 73.1991088286422$$
$$x_{22} = 2.19911485751286$$
$$x_{23} = -71.9424717672063$$
$$x_{24} = -58.1194640633422$$
$$x_{25} = -31.7300858012569$$
$$x_{26} = 92.0486647501809$$
$$x_{27} = -77.7544181763474$$
$$x_{28} = 48.0663675999238$$
$$x_{29} = 81.9955682586936$$
$$x_{30} = -53.7212343763855$$
$$x_{31} = -63.7743308678728$$
$$x_{32} = -66.2876049907446$$
$$x_{33} = 32.2013246992954$$
$$x_{34} = 50.2654824574367$$
$$x_{35} = -9.73893722612836$$
$$x_{36} = -5.96902604182061$$
$$x_{37} = 78.8539756051038$$
$$x_{38} = 51.8362786498421$$
$$x_{39} = 86.3937979933609$$
$$x_{40} = -3.92699081698724$$
$$x_{41} = 34.2433599241287$$
$$x_{42} = -39.8982267005904$$
$$x_{43} = -87.9645943005142$$
$$x_{44} = -81.9955682586936$$
$$x_{45} = 0$$
$$x_{46} = -47.9092879672443$$
$$x_{47} = 64.4026494609915$$
$$x_{48} = 44.924774946334$$
$$x_{49} = 42.4115009489444$$
$$x_{50} = -65.9734457253857$$
$$x_{51} = 18.8495559215388$$
$$x_{52} = 78.2256570743859$$
$$x_{53} = 7.85398160566834$$
$$x_{54} = 100.216805649514$$
$$x_{55} = -83.8805238508475$$
$$x_{56} = 94.2477796076938$$
$$x_{57} = 60.0044196835651$$
$$x_{58} = 54.1924732744239$$
$$x_{59} = 12.2522113490002$$
$$x_{60} = 91.420346219463$$
$$x_{61} = 21.9911485751286$$
$$x_{62} = 70.0575161750524$$
$$x_{63} = 16.0221225333079$$
$$x_{64} = -2.82743338823081$$
$$x_{65} = -93.9336203423348$$
$$x_{66} = 4.08407044966673$$
$$x_{67} = -38.0132711084365$$
$$x_{68} = -17.9070781254618$$
$$x_{69} = -44.924774946334$$
$$x_{70} = 84.037603483527$$
$$x_{71} = -95.8185758707203$$
$$x_{72} = 90.1637091580271$$
$$x_{73} = -73.8274272865168$$
$$x_{74} = 46.18141200777$$
$$x_{75} = -49.9513231920777$$
$$x_{76} = 10.2101761241668$$
$$x_{77} = 38.6415896391545$$
$$x_{78} = -29.8451301291032$$
$$x_{79} = -69.9004365423729$$
$$x_{80} = -80.1106126319001$$
$$x_{81} = -33.7721210260903$$
$$x_{82} = 66.7588438887831$$
$$x_{83} = 43.9822971502571$$
$$x_{84} = -91.8915851175014$$
$$x_{85} = 68.1725605828985$$
$$x_{86} = 40.0553063332699$$
$$x_{87} = -14.1371669407303$$
$$x_{88} = 26.0752190247953$$
$$x_{89} = -60.0044196835651$$
$$x_{90} = 98.174770424681$$
$$x_{91} = 18.0641577581413$$
$$x_{92} = -36.1283154987097$$
$$x_{93} = -16.0221225333079$$
$$x_{94} = 62.0464549083984$$
$$x_{95} = -19.7920337176157$$
$$x_{96} = 29.8451301331324$$
$$x_{97} = -99.7455667514759$$
so we need to solve the equation:
$$\sin{\left(4 x \right)} \cos{\left(5 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{10}$$
$$x_{3} = \frac{\pi}{10}$$
Numerical solution
$$x_{1} = -43.9822971502571$$
$$x_{2} = -25.9181393921158$$
$$x_{3} = 56.5486677646163$$
$$x_{4} = -7.85398160320836$$
$$x_{5} = 24.1902634326414$$
$$x_{6} = -55.7632696012188$$
$$x_{7} = 72.2566310325652$$
$$x_{8} = -97.7035315266426$$
$$x_{9} = -41.7831822927443$$
$$x_{10} = -85.7654794430014$$
$$x_{11} = 14.765485471872$$
$$x_{12} = 38.0132711084365$$
$$x_{13} = -75.712382951514$$
$$x_{14} = -11.7809724509617$$
$$x_{15} = -61.8893752757189$$
$$x_{16} = -27.9601746169492$$
$$x_{17} = -21.9911485751286$$
$$x_{18} = 28.5884931476671$$
$$x_{19} = 76.1836218495525$$
$$x_{20} = -51.8362787041329$$
$$x_{21} = 73.1991088286422$$
$$x_{22} = 2.19911485751286$$
$$x_{23} = -71.9424717672063$$
$$x_{24} = -58.1194640633422$$
$$x_{25} = -31.7300858012569$$
$$x_{26} = 92.0486647501809$$
$$x_{27} = -77.7544181763474$$
$$x_{28} = 48.0663675999238$$
$$x_{29} = 81.9955682586936$$
$$x_{30} = -53.7212343763855$$
$$x_{31} = -63.7743308678728$$
$$x_{32} = -66.2876049907446$$
$$x_{33} = 32.2013246992954$$
$$x_{34} = 50.2654824574367$$
$$x_{35} = -9.73893722612836$$
$$x_{36} = -5.96902604182061$$
$$x_{37} = 78.8539756051038$$
$$x_{38} = 51.8362786498421$$
$$x_{39} = 86.3937979933609$$
$$x_{40} = -3.92699081698724$$
$$x_{41} = 34.2433599241287$$
$$x_{42} = -39.8982267005904$$
$$x_{43} = -87.9645943005142$$
$$x_{44} = -81.9955682586936$$
$$x_{45} = 0$$
$$x_{46} = -47.9092879672443$$
$$x_{47} = 64.4026494609915$$
$$x_{48} = 44.924774946334$$
$$x_{49} = 42.4115009489444$$
$$x_{50} = -65.9734457253857$$
$$x_{51} = 18.8495559215388$$
$$x_{52} = 78.2256570743859$$
$$x_{53} = 7.85398160566834$$
$$x_{54} = 100.216805649514$$
$$x_{55} = -83.8805238508475$$
$$x_{56} = 94.2477796076938$$
$$x_{57} = 60.0044196835651$$
$$x_{58} = 54.1924732744239$$
$$x_{59} = 12.2522113490002$$
$$x_{60} = 91.420346219463$$
$$x_{61} = 21.9911485751286$$
$$x_{62} = 70.0575161750524$$
$$x_{63} = 16.0221225333079$$
$$x_{64} = -2.82743338823081$$
$$x_{65} = -93.9336203423348$$
$$x_{66} = 4.08407044966673$$
$$x_{67} = -38.0132711084365$$
$$x_{68} = -17.9070781254618$$
$$x_{69} = -44.924774946334$$
$$x_{70} = 84.037603483527$$
$$x_{71} = -95.8185758707203$$
$$x_{72} = 90.1637091580271$$
$$x_{73} = -73.8274272865168$$
$$x_{74} = 46.18141200777$$
$$x_{75} = -49.9513231920777$$
$$x_{76} = 10.2101761241668$$
$$x_{77} = 38.6415896391545$$
$$x_{78} = -29.8451301291032$$
$$x_{79} = -69.9004365423729$$
$$x_{80} = -80.1106126319001$$
$$x_{81} = -33.7721210260903$$
$$x_{82} = 66.7588438887831$$
$$x_{83} = 43.9822971502571$$
$$x_{84} = -91.8915851175014$$
$$x_{85} = 68.1725605828985$$
$$x_{86} = 40.0553063332699$$
$$x_{87} = -14.1371669407303$$
$$x_{88} = 26.0752190247953$$
$$x_{89} = -60.0044196835651$$
$$x_{90} = 98.174770424681$$
$$x_{91} = 18.0641577581413$$
$$x_{92} = -36.1283154987097$$
$$x_{93} = -16.0221225333079$$
$$x_{94} = 62.0464549083984$$
$$x_{95} = -19.7920337176157$$
$$x_{96} = 29.8451301331324$$
$$x_{97} = -99.7455667514759$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(4 x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(4 x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(4 x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(4 x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(4*x)*cos(5*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(4 x \right)} \cos{\left(5 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(4 x \right)} \cos{\left(5 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(4 x \right)} \cos{\left(5 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(4 x \right)} \cos{\left(5 x \right)}}{x}\right) = 0$$
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:
$$\sin{\left(4 x \right)} \cos{\left(5 x \right)} = - \sin{\left(4 x \right)} \cos{\left(5 x \right)}$$
- No
$$\sin{\left(4 x \right)} \cos{\left(5 x \right)} = \sin{\left(4 x \right)} \cos{\left(5 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(4 x \right)} \cos{\left(5 x \right)} = - \sin{\left(4 x \right)} \cos{\left(5 x \right)}$$
- No
$$\sin{\left(4 x \right)} \cos{\left(5 x \right)} = \sin{\left(4 x \right)} \cos{\left(5 x \right)}$$
- No
so, the function
not is
neither even, nor odd