Mister Exam
Graphing y = sin(3x)*cos(x/2)
The solution
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/x\
f(x) = sin(3*x)*cos|-|
\2/
$$f{\left(x \right)} = \sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}$$
f = sin(3*x)*cos(x/2)
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:
$$\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \pi$$
$$x_{3} = \pi$$
Numerical solution
$$x_{1} = -55.5014702134197$$
$$x_{2} = 90.0589894029074$$
$$x_{3} = -15.7079628703011$$
$$x_{4} = 9.42477810324019$$
$$x_{5} = -90.0589894029074$$
$$x_{6} = -73.3038285837618$$
$$x_{7} = 74.3510261349584$$
$$x_{8} = 17.8023583703422$$
$$x_{9} = -6.28318530717959$$
$$x_{10} = -85.870199198121$$
$$x_{11} = 26.1799387799149$$
$$x_{12} = -83.7758040957278$$
$$x_{13} = 34.5575190899152$$
$$x_{14} = 92.1533845053006$$
$$x_{15} = -59.6902604567951$$
$$x_{16} = 100.530964914873$$
$$x_{17} = 48.1710873550435$$
$$x_{18} = 70.162235930172$$
$$x_{19} = 94.2477796076938$$
$$x_{20} = 78.5398162363142$$
$$x_{21} = -24.0855436775217$$
$$x_{22} = -79.5870138909414$$
$$x_{23} = 80.634211442138$$
$$x_{24} = -87.9645943005142$$
$$x_{25} = 46.0766922526503$$
$$x_{26} = 0$$
$$x_{27} = 30.3687289847013$$
$$x_{28} = 15.7079633743983$$
$$x_{29} = 50.2654824574367$$
$$x_{30} = -92.1533845053006$$
$$x_{31} = 85.870199198121$$
$$x_{32} = 41.8879020478639$$
$$x_{33} = -9.42477776812281$$
$$x_{34} = -53.4070752386382$$
$$x_{35} = 59.6902605018059$$
$$x_{36} = 32.4631240870945$$
$$x_{37} = 21.991148585147$$
$$x_{38} = -41.8879020478639$$
$$x_{39} = 63.8790506229925$$
$$x_{40} = 87.9645943005142$$
$$x_{41} = -15.7079632961481$$
$$x_{42} = -19.8967534727354$$
$$x_{43} = 4.18879020478639$$
$$x_{44} = -72.2566309213882$$
$$x_{45} = 39.7935069454707$$
$$x_{46} = -32.4631240870945$$
$$x_{47} = 38.7463093942741$$
$$x_{48} = -43.9822971502571$$
$$x_{49} = -37.6991118430775$$
$$x_{50} = -96.342174710087$$
$$x_{51} = 28.2743338653134$$
$$x_{52} = 34.5575191693199$$
$$x_{53} = -99.4837673636768$$
$$x_{54} = 76.4454212373516$$
$$x_{55} = 6.28318530717959$$
$$x_{56} = -39.7935069454707$$
$$x_{57} = -17.8023583703422$$
$$x_{58} = -28.2743337726325$$
$$x_{59} = -70.162235930172$$
$$x_{60} = -68.0678408277789$$
$$x_{61} = 65.9734457524635$$
$$x_{62} = -94.2477796076938$$
$$x_{63} = -13.6135681655558$$
$$x_{64} = 61.7846555205993$$
$$x_{65} = 59.6902605192517$$
$$x_{66} = 43.9822971502571$$
$$x_{67} = -4.18879020478639$$
$$x_{68} = 13.6135681655558$$
$$x_{69} = 96.342174710087$$
$$x_{70} = 52.3598775598299$$
$$x_{71} = -48.1710873550435$$
$$x_{72} = -50.2654824574367$$
$$x_{73} = -46.0766922526503$$
$$x_{74} = -35.6047167406843$$
$$x_{75} = -21.9911485865022$$
$$x_{76} = -97.389372325484$$
$$x_{77} = -109.955742875469$$
$$x_{78} = 55.5014702134197$$
$$x_{79} = -57.5958653158129$$
$$x_{80} = 72.2566310277261$$
$$x_{81} = -9.42477805346027$$
$$x_{82} = 24.0855436775217$$
$$x_{83} = -61.7846555205993$$
$$x_{84} = -65.9734457654802$$
$$x_{85} = -26.1799387799149$$
$$x_{86} = -2.0943951023932$$
$$x_{87} = 16.7551608191456$$
$$x_{88} = 98.4365698124802$$
$$x_{89} = -63.8790506229925$$
$$x_{90} = 2.0943951023932$$
$$x_{91} = 68.0678408277789$$
$$x_{92} = -53.4070751925979$$
$$x_{93} = 19.8967534727354$$
$$x_{94} = -81.6814089933346$$
$$x_{95} = 8.37758040957278$$
so we need to solve the equation:
$$\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \pi$$
$$x_{3} = \pi$$
Numerical solution
$$x_{1} = -55.5014702134197$$
$$x_{2} = 90.0589894029074$$
$$x_{3} = -15.7079628703011$$
$$x_{4} = 9.42477810324019$$
$$x_{5} = -90.0589894029074$$
$$x_{6} = -73.3038285837618$$
$$x_{7} = 74.3510261349584$$
$$x_{8} = 17.8023583703422$$
$$x_{9} = -6.28318530717959$$
$$x_{10} = -85.870199198121$$
$$x_{11} = 26.1799387799149$$
$$x_{12} = -83.7758040957278$$
$$x_{13} = 34.5575190899152$$
$$x_{14} = 92.1533845053006$$
$$x_{15} = -59.6902604567951$$
$$x_{16} = 100.530964914873$$
$$x_{17} = 48.1710873550435$$
$$x_{18} = 70.162235930172$$
$$x_{19} = 94.2477796076938$$
$$x_{20} = 78.5398162363142$$
$$x_{21} = -24.0855436775217$$
$$x_{22} = -79.5870138909414$$
$$x_{23} = 80.634211442138$$
$$x_{24} = -87.9645943005142$$
$$x_{25} = 46.0766922526503$$
$$x_{26} = 0$$
$$x_{27} = 30.3687289847013$$
$$x_{28} = 15.7079633743983$$
$$x_{29} = 50.2654824574367$$
$$x_{30} = -92.1533845053006$$
$$x_{31} = 85.870199198121$$
$$x_{32} = 41.8879020478639$$
$$x_{33} = -9.42477776812281$$
$$x_{34} = -53.4070752386382$$
$$x_{35} = 59.6902605018059$$
$$x_{36} = 32.4631240870945$$
$$x_{37} = 21.991148585147$$
$$x_{38} = -41.8879020478639$$
$$x_{39} = 63.8790506229925$$
$$x_{40} = 87.9645943005142$$
$$x_{41} = -15.7079632961481$$
$$x_{42} = -19.8967534727354$$
$$x_{43} = 4.18879020478639$$
$$x_{44} = -72.2566309213882$$
$$x_{45} = 39.7935069454707$$
$$x_{46} = -32.4631240870945$$
$$x_{47} = 38.7463093942741$$
$$x_{48} = -43.9822971502571$$
$$x_{49} = -37.6991118430775$$
$$x_{50} = -96.342174710087$$
$$x_{51} = 28.2743338653134$$
$$x_{52} = 34.5575191693199$$
$$x_{53} = -99.4837673636768$$
$$x_{54} = 76.4454212373516$$
$$x_{55} = 6.28318530717959$$
$$x_{56} = -39.7935069454707$$
$$x_{57} = -17.8023583703422$$
$$x_{58} = -28.2743337726325$$
$$x_{59} = -70.162235930172$$
$$x_{60} = -68.0678408277789$$
$$x_{61} = 65.9734457524635$$
$$x_{62} = -94.2477796076938$$
$$x_{63} = -13.6135681655558$$
$$x_{64} = 61.7846555205993$$
$$x_{65} = 59.6902605192517$$
$$x_{66} = 43.9822971502571$$
$$x_{67} = -4.18879020478639$$
$$x_{68} = 13.6135681655558$$
$$x_{69} = 96.342174710087$$
$$x_{70} = 52.3598775598299$$
$$x_{71} = -48.1710873550435$$
$$x_{72} = -50.2654824574367$$
$$x_{73} = -46.0766922526503$$
$$x_{74} = -35.6047167406843$$
$$x_{75} = -21.9911485865022$$
$$x_{76} = -97.389372325484$$
$$x_{77} = -109.955742875469$$
$$x_{78} = 55.5014702134197$$
$$x_{79} = -57.5958653158129$$
$$x_{80} = 72.2566310277261$$
$$x_{81} = -9.42477805346027$$
$$x_{82} = 24.0855436775217$$
$$x_{83} = -61.7846555205993$$
$$x_{84} = -65.9734457654802$$
$$x_{85} = -26.1799387799149$$
$$x_{86} = -2.0943951023932$$
$$x_{87} = 16.7551608191456$$
$$x_{88} = 98.4365698124802$$
$$x_{89} = -63.8790506229925$$
$$x_{90} = 2.0943951023932$$
$$x_{91} = 68.0678408277789$$
$$x_{92} = -53.4070751925979$$
$$x_{93} = 19.8967534727354$$
$$x_{94} = -81.6814089933346$$
$$x_{95} = 8.37758040957278$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}\right) = \left\langle -1, 1\right\rangle$$
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(3*x)*cos(x/2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}}{x}\right) = 0$$
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:
$$\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)} = - \sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}$$
- No
$$\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)} = \sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)} = - \sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}$$
- No
$$\sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)} = \sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd