Mister Exam
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-
How to use it?
- Graphing y =:
- sin(1/x)
- 3x^4-6x^2+2
- cos(5x)*cos(x)
-
x^-1
- Integral of d{x}:
- cos(5x)*cos(x)
-
Identical expressions
- cos(5x)*cos(x)
- co sinus of e of (5x) multiply by co sinus of e of (x)
- cos(5x)cos(x)
- cos5xcosx
-
Similar expressions
- cos(5x)*cosx
Graphing y = cos(5x)*cos(x)
The solution
You have entered
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f(x) = cos(5*x)*cos(x)
$$f{\left(x \right)} = \cos{\left(x \right)} \cos{\left(5 x \right)}$$
f = cos(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:
$$\cos{\left(x \right)} \cos{\left(5 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = - \frac{\pi}{10}$$
$$x_{3} = \frac{\pi}{10}$$
$$x_{4} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 64.4026493525391$$
$$x_{2} = 93.9336203423348$$
$$x_{3} = -17.9070781254618$$
$$x_{4} = 68.1725605828985$$
$$x_{5} = 29.8451302545263$$
$$x_{6} = -83.8805238508475$$
$$x_{7} = -65.6592864600267$$
$$x_{8} = -85.7654794430014$$
$$x_{9} = 4.08407044966673$$
$$x_{10} = 61.8893752757189$$
$$x_{11} = -38.0132711084365$$
$$x_{12} = 34.2433599241287$$
$$x_{13} = 98.3318500573605$$
$$x_{14} = 83.8805238508475$$
$$x_{15} = -23.5619448943193$$
$$x_{16} = -71.9424717672063$$
$$x_{17} = -5.96902604182061$$
$$x_{18} = 90.1637091580271$$
$$x_{19} = -97.7035315266426$$
$$x_{20} = 86.3937979237947$$
$$x_{21} = 100.216805649514$$
$$x_{22} = -45.5530934328517$$
$$x_{23} = 70.0575161750524$$
$$x_{24} = -55.6061899685393$$
$$x_{25} = 2.19911485751286$$
$$x_{26} = -60.0044196835651$$
$$x_{27} = 44.2964564156161$$
$$x_{28} = -9.73893722612836$$
$$x_{29} = -14.1371668872424$$
$$x_{30} = 46.18141200777$$
$$x_{31} = -63.7743308678728$$
$$x_{32} = -67.5442419493278$$
$$x_{33} = 60.0044196835651$$
$$x_{34} = -29.8451301142458$$
$$x_{35} = 78.2256570743859$$
$$x_{36} = -53.7212343763855$$
$$x_{37} = -87.6504350351552$$
$$x_{38} = 92.0486647501809$$
$$x_{39} = 5.96902604182061$$
$$x_{40} = -93.9336203423348$$
$$x_{41} = 81.9955682586936$$
$$x_{42} = -33.6150413934108$$
$$x_{43} = -61.8893752757189$$
$$x_{44} = 76.340701482232$$
$$x_{45} = 42.4115007836476$$
$$x_{46} = 58.7477826221291$$
$$x_{47} = 38.0132711084365$$
$$x_{48} = -21.6769893097696$$
$$x_{49} = 17.9070781254618$$
$$x_{50} = -36.1283154601932$$
$$x_{51} = -58.1194640341413$$
$$x_{52} = -80.1106126089377$$
$$x_{53} = -51.8362786978016$$
$$x_{54} = 32.3584043319749$$
$$x_{55} = 24.1902634326414$$
$$x_{56} = 48.0663675999238$$
$$x_{57} = -39.8982267005904$$
$$x_{58} = -48.0663675999238$$
$$x_{59} = 39.2699081498428$$
$$x_{60} = 12.2522113490002$$
$$x_{61} = -81.9955682586936$$
$$x_{62} = 16.0221225333079$$
$$x_{63} = 20.4203522181614$$
$$x_{64} = -95.8185758695698$$
$$x_{65} = -70.0575161750524$$
$$x_{66} = -27.9601746169492$$
$$x_{67} = -92.0486647501809$$
$$x_{68} = 26.0752190247953$$
$$x_{69} = -77.5973385436679$$
$$x_{70} = -1.57079634217526$$
$$x_{71} = 22.3053078404875$$
$$x_{72} = -73.827427283678$$
$$x_{73} = 66.2876049907446$$
$$x_{74} = 49.9513231920777$$
$$x_{75} = -31.7300858012569$$
$$x_{76} = -49.9513231920777$$
$$x_{77} = 56.2345084992573$$
$$x_{78} = -43.6681378848981$$
$$x_{79} = 10.3672557568463$$
$$x_{80} = -41.7831822927443$$
$$x_{81} = -4.08407044966673$$
$$x_{82} = 27.9601746169492$$
$$x_{83} = 39.8982267005904$$
$$x_{84} = 88.2787535658732$$
$$x_{85} = -16.0221225333079$$
$$x_{86} = -75.712382951514$$
$$x_{87} = 95.8185759450348$$
$$x_{88} = 71.9424717672063$$
$$x_{89} = 7.85398168446658$$
$$x_{90} = -19.7920337176157$$
$$x_{91} = -89.5353904622891$$
$$x_{92} = -11.6238928182822$$
$$x_{93} = 73.8274273864408$$
$$x_{94} = -7.85398154718208$$
$$x_{95} = 0.314159265358979$$
$$x_{96} = 51.8362788222469$$
$$x_{97} = 54.3495529071034$$
so we need to solve the equation:
$$\cos{\left(x \right)} \cos{\left(5 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = - \frac{\pi}{10}$$
$$x_{3} = \frac{\pi}{10}$$
$$x_{4} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 64.4026493525391$$
$$x_{2} = 93.9336203423348$$
$$x_{3} = -17.9070781254618$$
$$x_{4} = 68.1725605828985$$
$$x_{5} = 29.8451302545263$$
$$x_{6} = -83.8805238508475$$
$$x_{7} = -65.6592864600267$$
$$x_{8} = -85.7654794430014$$
$$x_{9} = 4.08407044966673$$
$$x_{10} = 61.8893752757189$$
$$x_{11} = -38.0132711084365$$
$$x_{12} = 34.2433599241287$$
$$x_{13} = 98.3318500573605$$
$$x_{14} = 83.8805238508475$$
$$x_{15} = -23.5619448943193$$
$$x_{16} = -71.9424717672063$$
$$x_{17} = -5.96902604182061$$
$$x_{18} = 90.1637091580271$$
$$x_{19} = -97.7035315266426$$
$$x_{20} = 86.3937979237947$$
$$x_{21} = 100.216805649514$$
$$x_{22} = -45.5530934328517$$
$$x_{23} = 70.0575161750524$$
$$x_{24} = -55.6061899685393$$
$$x_{25} = 2.19911485751286$$
$$x_{26} = -60.0044196835651$$
$$x_{27} = 44.2964564156161$$
$$x_{28} = -9.73893722612836$$
$$x_{29} = -14.1371668872424$$
$$x_{30} = 46.18141200777$$
$$x_{31} = -63.7743308678728$$
$$x_{32} = -67.5442419493278$$
$$x_{33} = 60.0044196835651$$
$$x_{34} = -29.8451301142458$$
$$x_{35} = 78.2256570743859$$
$$x_{36} = -53.7212343763855$$
$$x_{37} = -87.6504350351552$$
$$x_{38} = 92.0486647501809$$
$$x_{39} = 5.96902604182061$$
$$x_{40} = -93.9336203423348$$
$$x_{41} = 81.9955682586936$$
$$x_{42} = -33.6150413934108$$
$$x_{43} = -61.8893752757189$$
$$x_{44} = 76.340701482232$$
$$x_{45} = 42.4115007836476$$
$$x_{46} = 58.7477826221291$$
$$x_{47} = 38.0132711084365$$
$$x_{48} = -21.6769893097696$$
$$x_{49} = 17.9070781254618$$
$$x_{50} = -36.1283154601932$$
$$x_{51} = -58.1194640341413$$
$$x_{52} = -80.1106126089377$$
$$x_{53} = -51.8362786978016$$
$$x_{54} = 32.3584043319749$$
$$x_{55} = 24.1902634326414$$
$$x_{56} = 48.0663675999238$$
$$x_{57} = -39.8982267005904$$
$$x_{58} = -48.0663675999238$$
$$x_{59} = 39.2699081498428$$
$$x_{60} = 12.2522113490002$$
$$x_{61} = -81.9955682586936$$
$$x_{62} = 16.0221225333079$$
$$x_{63} = 20.4203522181614$$
$$x_{64} = -95.8185758695698$$
$$x_{65} = -70.0575161750524$$
$$x_{66} = -27.9601746169492$$
$$x_{67} = -92.0486647501809$$
$$x_{68} = 26.0752190247953$$
$$x_{69} = -77.5973385436679$$
$$x_{70} = -1.57079634217526$$
$$x_{71} = 22.3053078404875$$
$$x_{72} = -73.827427283678$$
$$x_{73} = 66.2876049907446$$
$$x_{74} = 49.9513231920777$$
$$x_{75} = -31.7300858012569$$
$$x_{76} = -49.9513231920777$$
$$x_{77} = 56.2345084992573$$
$$x_{78} = -43.6681378848981$$
$$x_{79} = 10.3672557568463$$
$$x_{80} = -41.7831822927443$$
$$x_{81} = -4.08407044966673$$
$$x_{82} = 27.9601746169492$$
$$x_{83} = 39.8982267005904$$
$$x_{84} = 88.2787535658732$$
$$x_{85} = -16.0221225333079$$
$$x_{86} = -75.712382951514$$
$$x_{87} = 95.8185759450348$$
$$x_{88} = 71.9424717672063$$
$$x_{89} = 7.85398168446658$$
$$x_{90} = -19.7920337176157$$
$$x_{91} = -89.5353904622891$$
$$x_{92} = -11.6238928182822$$
$$x_{93} = 73.8274273864408$$
$$x_{94} = -7.85398154718208$$
$$x_{95} = 0.314159265358979$$
$$x_{96} = 51.8362788222469$$
$$x_{97} = 54.3495529071034$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\cos{\left(x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\cos{\left(x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\cos{\left(x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\cos{\left(x \right)} \cos{\left(5 x \right)}\right) = \left\langle -1, 1\right\rangle$$
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 cos(5*x)*cos(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \cos{\left(5 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \cos{\left(5 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \cos{\left(5 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \cos{\left(5 x \right)}}{x}\right) = 0$$
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:
$$\cos{\left(x \right)} \cos{\left(5 x \right)} = \cos{\left(x \right)} \cos{\left(5 x \right)}$$
- Yes
$$\cos{\left(x \right)} \cos{\left(5 x \right)} = - \cos{\left(x \right)} \cos{\left(5 x \right)}$$
- No
so, the function
is
even
So, check:
$$\cos{\left(x \right)} \cos{\left(5 x \right)} = \cos{\left(x \right)} \cos{\left(5 x \right)}$$
- Yes
$$\cos{\left(x \right)} \cos{\left(5 x \right)} = - \cos{\left(x \right)} \cos{\left(5 x \right)}$$
- No
so, the function
is
even