Mister Exam
Graphing y = sin(1000*x)*sqrt(3)/3*(10+sin(5-|x|)*|x|)
The solution
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sin(1000*x)*\/ 3
f(x) = -----------------*(10 + sin(5 - |x|)*|x|)
3
$$f{\left(x \right)} = \frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right)$$
f = ((sqrt(3)*sin(1000*x))/3)*(sin(5 - |x|)*|x| + 10)
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:
$$\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -99.10154025749$$
$$x_{2} = 40.2469434851388$$
$$x_{3} = -67.9777818383759$$
$$x_{4} = -51.9430929344536$$
$$x_{5} = 5.74597296341573$$
$$x_{6} = 89.7898596322499$$
$$x_{7} = -64.5345962900415$$
$$x_{8} = -8.61110546348962$$
$$x_{9} = 99.3486060187879$$
$$x_{10} = 86.8336209452219$$
$$x_{11} = -17.7499984927823$$
$$x_{12} = 61.7228708650787$$
$$x_{13} = 23.4080068618976$$
$$x_{14} = -33.7249971362864$$
$$x_{15} = 58.2294198342868$$
$$x_{16} = -61.7114299857102$$
$$x_{17} = -24.2279625444845$$
$$x_{18} = -58.219995056326$$
$$x_{19} = -26.5935818126376$$
$$x_{20} = 45.6159253301238$$
$$x_{21} = -67.1546845631354$$
$$x_{22} = 18.1498936480029$$
$$x_{23} = -83.4187097307698$$
$$x_{24} = -81.5463205092303$$
$$x_{25} = -67.979492029775$$
$$x_{26} = 82.253178856288$$
$$x_{27} = -89.7144614085637$$
$$x_{28} = 67.9809234310295$$
$$x_{29} = 36.684377415968$$
$$x_{30} = -80.5227336754012$$
$$x_{31} = -30.5676965194287$$
$$x_{32} = -7.69690200129499$$
$$x_{33} = -58.2325614269404$$
$$x_{34} = -49.1870300318029$$
$$x_{35} = -55.4522519285134$$
$$x_{36} = 0$$
$$x_{37} = 30.2723868099912$$
$$x_{38} = -53.8154821559932$$
$$x_{39} = -23.772431609714$$
$$x_{40} = 20.1897831111952$$
$$x_{41} = -15.9844234214649$$
$$x_{42} = -13.5842466341223$$
$$x_{43} = 74.2641087382091$$
$$x_{44} = -97.3799474833228$$
$$x_{45} = 61.7114299857102$$
$$x_{46} = 36.7063685645431$$
$$x_{47} = -71.0188435270509$$
$$x_{48} = -26.6030065905984$$
$$x_{49} = 24.2499536930596$$
$$x_{50} = 9.62583989059913$$
$$x_{51} = 21.6707061244624$$
$$x_{52} = -49.7219869283657$$
$$x_{53} = 12.2394291586115$$
$$x_{54} = 47.9438454864338$$
$$x_{55} = -73.7677370989419$$
$$x_{56} = 74.2501287922262$$
$$x_{57} = -55.4585351138206$$
$$x_{58} = 80.5190197115064$$
$$x_{59} = -39.6594656589175$$
$$x_{60} = -61.7134460871179$$
$$x_{61} = -43.7466777012379$$
$$x_{62} = -21.903183980828$$
$$x_{63} = -75.7437988780499$$
$$x_{64} = 50.8403939130436$$
$$x_{65} = -95.9976467157433$$
$$x_{66} = -45.7227394803459$$
$$x_{67} = -6.0004419683565$$
$$x_{68} = -94.2666291636153$$
$$x_{69} = -12.4124325743333$$
$$x_{70} = 31.89030702659$$
$$x_{71} = 4.88831816898572$$
$$x_{72} = -36.0780500338252$$
$$x_{73} = 12.2396449783858$$
$$x_{74} = 32.6819883752946$$
$$x_{75} = 42.9392883892653$$
$$x_{76} = 61.7134460871179$$
$$x_{77} = 94.1629566060469$$
$$x_{78} = -85.7780458136157$$
$$x_{79} = 70.831792937014$$
$$x_{80} = -20.1533168727785$$
$$x_{81} = 51.9305265638393$$
$$x_{82} = 98.2501686483672$$
$$x_{83} = 96.0018329812683$$
$$x_{84} = 45.5028279945946$$
$$x_{85} = -13.5999545973902$$
$$x_{86} = 32.572032632419$$
$$x_{87} = -36.6906606012752$$
$$x_{88} = 78.0874269976279$$
$$x_{89} = -74.609683930104$$
so we need to solve the equation:
$$\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -99.10154025749$$
$$x_{2} = 40.2469434851388$$
$$x_{3} = -67.9777818383759$$
$$x_{4} = -51.9430929344536$$
$$x_{5} = 5.74597296341573$$
$$x_{6} = 89.7898596322499$$
$$x_{7} = -64.5345962900415$$
$$x_{8} = -8.61110546348962$$
$$x_{9} = 99.3486060187879$$
$$x_{10} = 86.8336209452219$$
$$x_{11} = -17.7499984927823$$
$$x_{12} = 61.7228708650787$$
$$x_{13} = 23.4080068618976$$
$$x_{14} = -33.7249971362864$$
$$x_{15} = 58.2294198342868$$
$$x_{16} = -61.7114299857102$$
$$x_{17} = -24.2279625444845$$
$$x_{18} = -58.219995056326$$
$$x_{19} = -26.5935818126376$$
$$x_{20} = 45.6159253301238$$
$$x_{21} = -67.1546845631354$$
$$x_{22} = 18.1498936480029$$
$$x_{23} = -83.4187097307698$$
$$x_{24} = -81.5463205092303$$
$$x_{25} = -67.979492029775$$
$$x_{26} = 82.253178856288$$
$$x_{27} = -89.7144614085637$$
$$x_{28} = 67.9809234310295$$
$$x_{29} = 36.684377415968$$
$$x_{30} = -80.5227336754012$$
$$x_{31} = -30.5676965194287$$
$$x_{32} = -7.69690200129499$$
$$x_{33} = -58.2325614269404$$
$$x_{34} = -49.1870300318029$$
$$x_{35} = -55.4522519285134$$
$$x_{36} = 0$$
$$x_{37} = 30.2723868099912$$
$$x_{38} = -53.8154821559932$$
$$x_{39} = -23.772431609714$$
$$x_{40} = 20.1897831111952$$
$$x_{41} = -15.9844234214649$$
$$x_{42} = -13.5842466341223$$
$$x_{43} = 74.2641087382091$$
$$x_{44} = -97.3799474833228$$
$$x_{45} = 61.7114299857102$$
$$x_{46} = 36.7063685645431$$
$$x_{47} = -71.0188435270509$$
$$x_{48} = -26.6030065905984$$
$$x_{49} = 24.2499536930596$$
$$x_{50} = 9.62583989059913$$
$$x_{51} = 21.6707061244624$$
$$x_{52} = -49.7219869283657$$
$$x_{53} = 12.2394291586115$$
$$x_{54} = 47.9438454864338$$
$$x_{55} = -73.7677370989419$$
$$x_{56} = 74.2501287922262$$
$$x_{57} = -55.4585351138206$$
$$x_{58} = 80.5190197115064$$
$$x_{59} = -39.6594656589175$$
$$x_{60} = -61.7134460871179$$
$$x_{61} = -43.7466777012379$$
$$x_{62} = -21.903183980828$$
$$x_{63} = -75.7437988780499$$
$$x_{64} = 50.8403939130436$$
$$x_{65} = -95.9976467157433$$
$$x_{66} = -45.7227394803459$$
$$x_{67} = -6.0004419683565$$
$$x_{68} = -94.2666291636153$$
$$x_{69} = -12.4124325743333$$
$$x_{70} = 31.89030702659$$
$$x_{71} = 4.88831816898572$$
$$x_{72} = -36.0780500338252$$
$$x_{73} = 12.2396449783858$$
$$x_{74} = 32.6819883752946$$
$$x_{75} = 42.9392883892653$$
$$x_{76} = 61.7134460871179$$
$$x_{77} = 94.1629566060469$$
$$x_{78} = -85.7780458136157$$
$$x_{79} = 70.831792937014$$
$$x_{80} = -20.1533168727785$$
$$x_{81} = 51.9305265638393$$
$$x_{82} = 98.2501686483672$$
$$x_{83} = 96.0018329812683$$
$$x_{84} = 45.5028279945946$$
$$x_{85} = -13.5999545973902$$
$$x_{86} = 32.572032632419$$
$$x_{87} = -36.6906606012752$$
$$x_{88} = 78.0874269976279$$
$$x_{89} = -74.609683930104$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right)\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right)\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right)\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right)\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of ((sin(1000*x)*sqrt(3))/3)*(10 + sin(5 - |x|)*|x|), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sqrt{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) \sin{\left(1000 x \right)}}{3 x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sqrt{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) \sin{\left(1000 x \right)}}{3 x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sqrt{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) \sin{\left(1000 x \right)}}{3 x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sqrt{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) \sin{\left(1000 x \right)}}{3 x}\right)$$
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:
$$\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) = - \frac{\sqrt{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) \sin{\left(1000 x \right)}}{3}$$
- No
$$\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) = \frac{\sqrt{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) \sin{\left(1000 x \right)}}{3}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) = - \frac{\sqrt{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) \sin{\left(1000 x \right)}}{3}$$
- No
$$\frac{\sqrt{3} \sin{\left(1000 x \right)}}{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) = \frac{\sqrt{3} \left(\sin{\left(5 - \left|{x}\right| \right)} \left|{x}\right| + 10\right) \sin{\left(1000 x \right)}}{3}$$
- No
so, the function
not is
neither even, nor odd