Mister Exam
Graphing y = cos(7x)+cos(3x)
The solution
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f(x) = cos(7*x) + cos(3*x)
$$f{\left(x \right)} = \cos{\left(3 x \right)} + \cos{\left(7 x \right)}$$
f = cos(3*x) + cos(7*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:
$$\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{9 \pi}{10}$$
$$x_{2} = - \frac{3 \pi}{4}$$
$$x_{3} = - \frac{\pi}{2}$$
$$x_{4} = - \frac{\pi}{4}$$
$$x_{5} = \frac{\pi}{10}$$
$$x_{6} = \frac{\pi}{4}$$
$$x_{7} = \frac{\pi}{2}$$
$$x_{8} = \frac{3 \pi}{4}$$
$$x_{9} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{10} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
$$x_{11} = - i \log{\left(- \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{12} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{13} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
$$x_{14} = - i \log{\left(- \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{5} i}{4} + \frac{i}{4} \right)}$$
Numerical solution
$$x_{1} = -9.73893722612836$$
$$x_{2} = -0.785398163397448$$
$$x_{3} = -87.6504350351552$$
$$x_{4} = 46.18141200777$$
$$x_{5} = 56.2345084992573$$
$$x_{6} = 26.0752190247953$$
$$x_{7} = 84.037603483527$$
$$x_{8} = 54.1924732744239$$
$$x_{9} = -68.329640215578$$
$$x_{10} = 93.9336203423348$$
$$x_{11} = 19.6349540849362$$
$$x_{12} = 66.2876049907446$$
$$x_{13} = 92.0486647501809$$
$$x_{14} = 33.6150413934108$$
$$x_{15} = -5.96902604182061$$
$$x_{16} = -41.7831822927443$$
$$x_{17} = -60.0044196835651$$
$$x_{18} = -51.8362787842316$$
$$x_{19} = -80.8960108299372$$
$$x_{20} = 88.2787535658732$$
$$x_{21} = -47.9092879672443$$
$$x_{22} = -49.9513231920777$$
$$x_{23} = 58.1194640914112$$
$$x_{24} = 60.0044196835651$$
$$x_{25} = -16.6504410640259$$
$$x_{26} = -49.3230046613598$$
$$x_{27} = 38.0132711084365$$
$$x_{28} = -69.9004365423729$$
$$x_{29} = 4.08407044966673$$
$$x_{30} = 95.1902574037707$$
$$x_{31} = -3.92699081698724$$
$$x_{32} = -55.7632696012188$$
$$x_{33} = -95.8185759344887$$
$$x_{34} = -17.9070781254618$$
$$x_{35} = 62.0464549083984$$
$$x_{36} = -93.9336203423348$$
$$x_{37} = 22.3053078404875$$
$$x_{38} = 80.1106126665397$$
$$x_{39} = 16.0221225333079$$
$$x_{40} = -31.7300858012569$$
$$x_{41} = 12.2522113490002$$
$$x_{42} = -77.7544181763474$$
$$x_{43} = 81.9955682586936$$
$$x_{44} = 91.8915851175014$$
$$x_{45} = -29.845130209103$$
$$x_{46} = 70.0575161750524$$
$$x_{47} = -61.8893752757189$$
$$x_{48} = -53.7212343763855$$
$$x_{49} = -81.9955682586936$$
$$x_{50} = 58.7477826221291$$
$$x_{51} = 44.2964564156161$$
$$x_{52} = 38.484510006475$$
$$x_{53} = 48.0663675999238$$
$$x_{54} = -39.8982267005904$$
$$x_{55} = -58.9048622548086$$
$$x_{56} = 2.19911485751286$$
$$x_{57} = 40.0553063332699$$
$$x_{58} = -85.7654794430014$$
$$x_{59} = 49.9513231920777$$
$$x_{60} = -43.6681378848981$$
$$x_{61} = -71.9424717672063$$
$$x_{62} = 14.1371669411541$$
$$x_{63} = 98.174770424681$$
$$x_{64} = 10.2101761241668$$
$$x_{65} = 100.216805649514$$
$$x_{66} = -16.0221225333079$$
$$x_{67} = -90.3207887907066$$
$$x_{68} = 0.314159265358979$$
$$x_{69} = 90.1637091580271$$
$$x_{70} = 71.9424717672063$$
$$x_{71} = -21.6769893097696$$
$$x_{72} = 78.2256570743859$$
$$x_{73} = 36.1283155162826$$
$$x_{74} = 74.6128255227576$$
$$x_{75} = -99.7455667514759$$
$$x_{76} = -33.7721210260903$$
$$x_{77} = 76.1836218495525$$
$$x_{78} = 18.0641577581413$$
$$x_{79} = -36.9137136796801$$
$$x_{80} = 27.9601746169492$$
$$x_{81} = 162.577419823272$$
$$x_{82} = 24.1902634326414$$
$$x_{83} = -19.7920337176157$$
$$x_{84} = 68.1725605828985$$
$$x_{85} = -75.712382951514$$
$$x_{86} = -63.7743308678728$$
$$x_{87} = -11.7809724509617$$
$$x_{88} = -7.85398163397448$$
$$x_{89} = 5.96902604182061$$
$$x_{90} = -38.0132711084365$$
$$x_{91} = -91.8915851175014$$
$$x_{92} = -73.8274273593601$$
$$x_{93} = 32.2013246992954$$
$$x_{94} = -97.7035315266426$$
$$x_{95} = 34.2433599241287$$
$$x_{96} = -27.9601746169492$$
$$x_{97} = -25.9181393921158$$
$$x_{98} = -65.6592864600267$$
so we need to solve the equation:
$$\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{9 \pi}{10}$$
$$x_{2} = - \frac{3 \pi}{4}$$
$$x_{3} = - \frac{\pi}{2}$$
$$x_{4} = - \frac{\pi}{4}$$
$$x_{5} = \frac{\pi}{10}$$
$$x_{6} = \frac{\pi}{4}$$
$$x_{7} = \frac{\pi}{2}$$
$$x_{8} = \frac{3 \pi}{4}$$
$$x_{9} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{10} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
$$x_{11} = - i \log{\left(- \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{12} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$x_{13} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}$$
$$x_{14} = - i \log{\left(- \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{5} i}{4} + \frac{i}{4} \right)}$$
Numerical solution
$$x_{1} = -9.73893722612836$$
$$x_{2} = -0.785398163397448$$
$$x_{3} = -87.6504350351552$$
$$x_{4} = 46.18141200777$$
$$x_{5} = 56.2345084992573$$
$$x_{6} = 26.0752190247953$$
$$x_{7} = 84.037603483527$$
$$x_{8} = 54.1924732744239$$
$$x_{9} = -68.329640215578$$
$$x_{10} = 93.9336203423348$$
$$x_{11} = 19.6349540849362$$
$$x_{12} = 66.2876049907446$$
$$x_{13} = 92.0486647501809$$
$$x_{14} = 33.6150413934108$$
$$x_{15} = -5.96902604182061$$
$$x_{16} = -41.7831822927443$$
$$x_{17} = -60.0044196835651$$
$$x_{18} = -51.8362787842316$$
$$x_{19} = -80.8960108299372$$
$$x_{20} = 88.2787535658732$$
$$x_{21} = -47.9092879672443$$
$$x_{22} = -49.9513231920777$$
$$x_{23} = 58.1194640914112$$
$$x_{24} = 60.0044196835651$$
$$x_{25} = -16.6504410640259$$
$$x_{26} = -49.3230046613598$$
$$x_{27} = 38.0132711084365$$
$$x_{28} = -69.9004365423729$$
$$x_{29} = 4.08407044966673$$
$$x_{30} = 95.1902574037707$$
$$x_{31} = -3.92699081698724$$
$$x_{32} = -55.7632696012188$$
$$x_{33} = -95.8185759344887$$
$$x_{34} = -17.9070781254618$$
$$x_{35} = 62.0464549083984$$
$$x_{36} = -93.9336203423348$$
$$x_{37} = 22.3053078404875$$
$$x_{38} = 80.1106126665397$$
$$x_{39} = 16.0221225333079$$
$$x_{40} = -31.7300858012569$$
$$x_{41} = 12.2522113490002$$
$$x_{42} = -77.7544181763474$$
$$x_{43} = 81.9955682586936$$
$$x_{44} = 91.8915851175014$$
$$x_{45} = -29.845130209103$$
$$x_{46} = 70.0575161750524$$
$$x_{47} = -61.8893752757189$$
$$x_{48} = -53.7212343763855$$
$$x_{49} = -81.9955682586936$$
$$x_{50} = 58.7477826221291$$
$$x_{51} = 44.2964564156161$$
$$x_{52} = 38.484510006475$$
$$x_{53} = 48.0663675999238$$
$$x_{54} = -39.8982267005904$$
$$x_{55} = -58.9048622548086$$
$$x_{56} = 2.19911485751286$$
$$x_{57} = 40.0553063332699$$
$$x_{58} = -85.7654794430014$$
$$x_{59} = 49.9513231920777$$
$$x_{60} = -43.6681378848981$$
$$x_{61} = -71.9424717672063$$
$$x_{62} = 14.1371669411541$$
$$x_{63} = 98.174770424681$$
$$x_{64} = 10.2101761241668$$
$$x_{65} = 100.216805649514$$
$$x_{66} = -16.0221225333079$$
$$x_{67} = -90.3207887907066$$
$$x_{68} = 0.314159265358979$$
$$x_{69} = 90.1637091580271$$
$$x_{70} = 71.9424717672063$$
$$x_{71} = -21.6769893097696$$
$$x_{72} = 78.2256570743859$$
$$x_{73} = 36.1283155162826$$
$$x_{74} = 74.6128255227576$$
$$x_{75} = -99.7455667514759$$
$$x_{76} = -33.7721210260903$$
$$x_{77} = 76.1836218495525$$
$$x_{78} = 18.0641577581413$$
$$x_{79} = -36.9137136796801$$
$$x_{80} = 27.9601746169492$$
$$x_{81} = 162.577419823272$$
$$x_{82} = 24.1902634326414$$
$$x_{83} = -19.7920337176157$$
$$x_{84} = 68.1725605828985$$
$$x_{85} = -75.712382951514$$
$$x_{86} = -63.7743308678728$$
$$x_{87} = -11.7809724509617$$
$$x_{88} = -7.85398163397448$$
$$x_{89} = 5.96902604182061$$
$$x_{90} = -38.0132711084365$$
$$x_{91} = -91.8915851175014$$
$$x_{92} = -73.8274273593601$$
$$x_{93} = 32.2013246992954$$
$$x_{94} = -97.7035315266426$$
$$x_{95} = 34.2433599241287$$
$$x_{96} = -27.9601746169492$$
$$x_{97} = -25.9181393921158$$
$$x_{98} = -65.6592864600267$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\cos{\left(3 x \right)} + \cos{\left(7 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\cos{\left(3 x \right)} + \cos{\left(7 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to -\infty}\left(\cos{\left(3 x \right)} + \cos{\left(7 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\cos{\left(3 x \right)} + \cos{\left(7 x \right)}\right) = \left\langle -2, 2\right\rangle$$
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 cos(7*x) + cos(3*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(3 x \right)} + \cos{\left(7 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\cos{\left(3 x \right)} + \cos{\left(7 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(3 x \right)} + \cos{\left(7 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\cos{\left(3 x \right)} + \cos{\left(7 x \right)}}{x}\right) = 0$$
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:
$$\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = \cos{\left(3 x \right)} + \cos{\left(7 x \right)}$$
- Yes
$$\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = - \cos{\left(3 x \right)} - \cos{\left(7 x \right)}$$
- No
so, the function
is
even
So, check:
$$\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = \cos{\left(3 x \right)} + \cos{\left(7 x \right)}$$
- Yes
$$\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = - \cos{\left(3 x \right)} - \cos{\left(7 x \right)}$$
- No
so, the function
is
even