Mister Exam
Graphing y = cos(2x)+cos(6x)
The solution
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f(x) = cos(2*x) + cos(6*x)
$$f{\left(x \right)} = \cos{\left(2 x \right)} + \cos{\left(6 x \right)}$$
f = cos(2*x) + cos(6*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:
$$\cos{\left(2 x \right)} + \cos{\left(6 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{7 \pi}{8}$$
$$x_{2} = - \frac{3 \pi}{4}$$
$$x_{3} = - \frac{3 \pi}{8}$$
$$x_{4} = - \frac{\pi}{4}$$
$$x_{5} = \frac{\pi}{8}$$
$$x_{6} = \frac{\pi}{4}$$
$$x_{7} = \frac{5 \pi}{8}$$
$$x_{8} = \frac{3 \pi}{4}$$
Numerical solution
$$x_{1} = 10.2101761241668$$
$$x_{2} = -64.009950316892$$
$$x_{3} = 68.329640215578$$
$$x_{4} = -91.8915851175014$$
$$x_{5} = -53.0143760293278$$
$$x_{6} = -42.0188017417635$$
$$x_{7} = -75.7909227678538$$
$$x_{8} = -71.8639319508665$$
$$x_{9} = 1.96349540849362$$
$$x_{10} = -3.92699081698724$$
$$x_{11} = 38.0918109247762$$
$$x_{12} = -11.7809724509617$$
$$x_{13} = 93.8550805259951$$
$$x_{14} = 98.174770424681$$
$$x_{15} = -23.9546439836222$$
$$x_{16} = -25.9181393921158$$
$$x_{17} = 8.24668071567321$$
$$x_{18} = 27.8816348006094$$
$$x_{19} = -79.717913584841$$
$$x_{20} = 12.1736715326604$$
$$x_{21} = 44.3749962319558$$
$$x_{22} = -53.7997741927252$$
$$x_{23} = -16.1006623496477$$
$$x_{24} = -35.7356164345839$$
$$x_{25} = 5.89048622548086$$
$$x_{26} = -33.7721210260903$$
$$x_{27} = -43.5895980685584$$
$$x_{28} = 42.0188017417635$$
$$x_{29} = 0.392699081698724$$
$$x_{30} = -38.0918109247762$$
$$x_{31} = -27.8816348006094$$
$$x_{32} = -1.96349540849362$$
$$x_{33} = 64.009950316892$$
$$x_{34} = -82.0741080750334$$
$$x_{35} = 27.4889357189107$$
$$x_{36} = 88.3572933822129$$
$$x_{37} = 60.0829594999048$$
$$x_{38} = 66.3661448070844$$
$$x_{39} = -31.8086256175967$$
$$x_{40} = 78.1471172580461$$
$$x_{41} = -69.9004365423729$$
$$x_{42} = -45.9457925587507$$
$$x_{43} = -57.7267650097125$$
$$x_{44} = 10.6028752058656$$
$$x_{45} = 56.941366846315$$
$$x_{46} = 82.0741080750334$$
$$x_{47} = -87.5718952188155$$
$$x_{48} = -77.7544181763474$$
$$x_{49} = 96.2112750161874$$
$$x_{50} = -9.8174770424681$$
$$x_{51} = 57.3340659280137$$
$$x_{52} = -89.9280897090078$$
$$x_{53} = -20.0276531666349$$
$$x_{54} = 40.0553063332699$$
$$x_{55} = -49.872783375738$$
$$x_{56} = 34.164820107789$$
$$x_{57} = 49.0873852123405$$
$$x_{58} = 23.9546439836222$$
$$x_{59} = -98.174770424681$$
$$x_{60} = -5.89048622548086$$
$$x_{61} = 70.2931356240716$$
$$x_{62} = 74.2201264410589$$
$$x_{63} = 149.618350127214$$
$$x_{64} = -30.6305283725005$$
$$x_{65} = -62.0464549083984$$
$$x_{66} = 86.0010988920206$$
$$x_{67} = 89.9280897090078$$
$$x_{68} = 84.037603483527$$
$$x_{69} = 16.1006623496477$$
$$x_{70} = 54.1924732744239$$
$$x_{71} = -99.7455667514759$$
$$x_{72} = 92.2842841992002$$
$$x_{73} = -97.7820713429823$$
$$x_{74} = 3.53429173528852$$
$$x_{75} = 71.8639319508665$$
$$x_{76} = 30.2378292908018$$
$$x_{77} = -67.9369411338793$$
$$x_{78} = 100.138265833175$$
$$x_{79} = 20.0276531666349$$
$$x_{80} = -75.0055246044563$$
$$x_{81} = 18.0641577581413$$
$$x_{82} = 22.3838476568273$$
$$x_{83} = 56.1559686829176$$
$$x_{84} = -55.7632696012188$$
$$x_{85} = -47.9092879672443$$
$$x_{86} = -65.5807466436869$$
$$x_{87} = -96.6039740978861$$
$$x_{88} = -21.5984494934298$$
$$x_{89} = 52.2289778659303$$
$$x_{90} = 62.0464549083984$$
$$x_{91} = 32.2013246992954$$
$$x_{92} = -60.0829594999048$$
$$x_{93} = -84.037603483527$$
$$x_{94} = 49.872783375738$$
$$x_{95} = 76.1836218495525$$
$$x_{96} = -86.0010988920206$$
$$x_{97} = -40.0553063332699$$
$$x_{98} = -93.8550805259951$$
$$x_{99} = -13.7444678594553$$
so we need to solve the equation:
$$\cos{\left(2 x \right)} + \cos{\left(6 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{7 \pi}{8}$$
$$x_{2} = - \frac{3 \pi}{4}$$
$$x_{3} = - \frac{3 \pi}{8}$$
$$x_{4} = - \frac{\pi}{4}$$
$$x_{5} = \frac{\pi}{8}$$
$$x_{6} = \frac{\pi}{4}$$
$$x_{7} = \frac{5 \pi}{8}$$
$$x_{8} = \frac{3 \pi}{4}$$
Numerical solution
$$x_{1} = 10.2101761241668$$
$$x_{2} = -64.009950316892$$
$$x_{3} = 68.329640215578$$
$$x_{4} = -91.8915851175014$$
$$x_{5} = -53.0143760293278$$
$$x_{6} = -42.0188017417635$$
$$x_{7} = -75.7909227678538$$
$$x_{8} = -71.8639319508665$$
$$x_{9} = 1.96349540849362$$
$$x_{10} = -3.92699081698724$$
$$x_{11} = 38.0918109247762$$
$$x_{12} = -11.7809724509617$$
$$x_{13} = 93.8550805259951$$
$$x_{14} = 98.174770424681$$
$$x_{15} = -23.9546439836222$$
$$x_{16} = -25.9181393921158$$
$$x_{17} = 8.24668071567321$$
$$x_{18} = 27.8816348006094$$
$$x_{19} = -79.717913584841$$
$$x_{20} = 12.1736715326604$$
$$x_{21} = 44.3749962319558$$
$$x_{22} = -53.7997741927252$$
$$x_{23} = -16.1006623496477$$
$$x_{24} = -35.7356164345839$$
$$x_{25} = 5.89048622548086$$
$$x_{26} = -33.7721210260903$$
$$x_{27} = -43.5895980685584$$
$$x_{28} = 42.0188017417635$$
$$x_{29} = 0.392699081698724$$
$$x_{30} = -38.0918109247762$$
$$x_{31} = -27.8816348006094$$
$$x_{32} = -1.96349540849362$$
$$x_{33} = 64.009950316892$$
$$x_{34} = -82.0741080750334$$
$$x_{35} = 27.4889357189107$$
$$x_{36} = 88.3572933822129$$
$$x_{37} = 60.0829594999048$$
$$x_{38} = 66.3661448070844$$
$$x_{39} = -31.8086256175967$$
$$x_{40} = 78.1471172580461$$
$$x_{41} = -69.9004365423729$$
$$x_{42} = -45.9457925587507$$
$$x_{43} = -57.7267650097125$$
$$x_{44} = 10.6028752058656$$
$$x_{45} = 56.941366846315$$
$$x_{46} = 82.0741080750334$$
$$x_{47} = -87.5718952188155$$
$$x_{48} = -77.7544181763474$$
$$x_{49} = 96.2112750161874$$
$$x_{50} = -9.8174770424681$$
$$x_{51} = 57.3340659280137$$
$$x_{52} = -89.9280897090078$$
$$x_{53} = -20.0276531666349$$
$$x_{54} = 40.0553063332699$$
$$x_{55} = -49.872783375738$$
$$x_{56} = 34.164820107789$$
$$x_{57} = 49.0873852123405$$
$$x_{58} = 23.9546439836222$$
$$x_{59} = -98.174770424681$$
$$x_{60} = -5.89048622548086$$
$$x_{61} = 70.2931356240716$$
$$x_{62} = 74.2201264410589$$
$$x_{63} = 149.618350127214$$
$$x_{64} = -30.6305283725005$$
$$x_{65} = -62.0464549083984$$
$$x_{66} = 86.0010988920206$$
$$x_{67} = 89.9280897090078$$
$$x_{68} = 84.037603483527$$
$$x_{69} = 16.1006623496477$$
$$x_{70} = 54.1924732744239$$
$$x_{71} = -99.7455667514759$$
$$x_{72} = 92.2842841992002$$
$$x_{73} = -97.7820713429823$$
$$x_{74} = 3.53429173528852$$
$$x_{75} = 71.8639319508665$$
$$x_{76} = 30.2378292908018$$
$$x_{77} = -67.9369411338793$$
$$x_{78} = 100.138265833175$$
$$x_{79} = 20.0276531666349$$
$$x_{80} = -75.0055246044563$$
$$x_{81} = 18.0641577581413$$
$$x_{82} = 22.3838476568273$$
$$x_{83} = 56.1559686829176$$
$$x_{84} = -55.7632696012188$$
$$x_{85} = -47.9092879672443$$
$$x_{86} = -65.5807466436869$$
$$x_{87} = -96.6039740978861$$
$$x_{88} = -21.5984494934298$$
$$x_{89} = 52.2289778659303$$
$$x_{90} = 62.0464549083984$$
$$x_{91} = 32.2013246992954$$
$$x_{92} = -60.0829594999048$$
$$x_{93} = -84.037603483527$$
$$x_{94} = 49.872783375738$$
$$x_{95} = 76.1836218495525$$
$$x_{96} = -86.0010988920206$$
$$x_{97} = -40.0553063332699$$
$$x_{98} = -93.8550805259951$$
$$x_{99} = -13.7444678594553$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\cos{\left(2 x \right)} + \cos{\left(6 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\cos{\left(2 x \right)} + \cos{\left(6 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to -\infty}\left(\cos{\left(2 x \right)} + \cos{\left(6 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\cos{\left(2 x \right)} + \cos{\left(6 x \right)}\right) = \left\langle -2, 2\right\rangle$$
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(2*x) + cos(6*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(2 x \right)} + \cos{\left(6 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)} + \cos{\left(6 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(2 x \right)} + \cos{\left(6 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)} + \cos{\left(6 x \right)}}{x}\right) = 0$$
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:
$$\cos{\left(2 x \right)} + \cos{\left(6 x \right)} = \cos{\left(2 x \right)} + \cos{\left(6 x \right)}$$
- Yes
$$\cos{\left(2 x \right)} + \cos{\left(6 x \right)} = - \cos{\left(2 x \right)} - \cos{\left(6 x \right)}$$
- No
so, the function
is
even
So, check:
$$\cos{\left(2 x \right)} + \cos{\left(6 x \right)} = \cos{\left(2 x \right)} + \cos{\left(6 x \right)}$$
- Yes
$$\cos{\left(2 x \right)} + \cos{\left(6 x \right)} = - \cos{\left(2 x \right)} - \cos{\left(6 x \right)}$$
- No
so, the function
is
even