Mister Exam
Graphing y = tan(sqrt(x^3))
The solution
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f(x) = tan\\/ x /
$$f{\left(x \right)} = \tan{\left(\sqrt{x^{3}} \right)}$$
f = tan(sqrt(x^3))
The graph of the function
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{3 \left(\tan^{2}{\left(\sqrt{x^{3}} \right)} + 1\right) \sqrt{x^{3}}}{2 x} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{3 \left(\tan^{2}{\left(\sqrt{x^{3}} \right)} + 1\right) \sqrt{x^{3}}}{2 x} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{3 \left(6 x \tan{\left(\sqrt{x^{3}} \right)} + \frac{\sqrt{x^{3}}}{x^{2}}\right) \left(\tan^{2}{\left(\sqrt{x^{3}} \right)} + 1\right)}{4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -26$$
$$x_{2} = 92.2428244626029$$
$$x_{3} = -20$$
$$x_{4} = 6.26927745240496$$
$$x_{5} = 44.0306045912742$$
$$x_{6} = -100$$
$$x_{7} = -52$$
$$x_{8} = -98$$
$$x_{9} = 58.109313442111$$
$$x_{10} = -46$$
$$x_{11} = -62$$
$$x_{12} = -94$$
$$x_{13} = 30.2655628057948$$
$$x_{14} = 52.1859675196535$$
$$x_{15} = 64.2681253460119$$
$$x_{16} = 34.320376023085$$
$$x_{17} = -30$$
$$x_{18} = 74.5765644784697$$
$$x_{19} = -88$$
$$x_{20} = 99.9350759285183$$
$$x_{21} = 42.1158371363077$$
$$x_{22} = 71.8843370513831$$
$$x_{23} = 11.8585944065254$$
$$x_{24} = 40.1564996359368$$
$$x_{25} = -76$$
$$x_{26} = -8.55358927481284$$
$$x_{27} = -44$$
$$x_{28} = -70$$
$$x_{29} = -78$$
$$x_{30} = -7.00058913350688$$
$$x_{31} = 86.2567902952232$$
$$x_{32} = 76.7436275805143$$
$$x_{33} = 66.0841266497851$$
$$x_{34} = -38$$
$$x_{35} = -80$$
$$x_{36} = 68.3829923388494$$
$$x_{37} = -92$$
$$x_{38} = 22.0684273449427$$
$$x_{39} = 28.3307856310302$$
$$x_{40} = 20.2469417441212$$
$$x_{41} = -56$$
$$x_{42} = 50.1362477622459$$
$$x_{43} = 81.9170861978907$$
$$x_{44} = -72$$
$$x_{45} = 62.4260925242595$$
$$x_{46} = -58$$
$$x_{47} = -60$$
$$x_{48} = 26.3274971573717$$
$$x_{49} = -68$$
$$x_{50} = 93.9792241997309$$
$$x_{51} = 36.0854076118897$$
$$x_{52} = 18.3394130124857$$
$$x_{53} = -24$$
$$x_{54} = 80.0552317475696$$
$$x_{55} = 78.1714680438355$$
$$x_{56} = 84.215087794644$$
$$x_{57} = -54$$
$$x_{58} = 14.1812858180921$$
$$x_{59} = 38.1481271522911$$
$$x_{60} = 60.2870278729972$$
$$x_{61} = 4.45624804800686$$
$$x_{62} = -64$$
$$x_{63} = 7.8475142235842$$
$$x_{64} = -84$$
$$x_{65} = -48$$
$$x_{66} = 32.1403393148926$$
$$x_{67} = -12$$
$$x_{68} = -10.0685939575807$$
$$x_{69} = -66$$
$$x_{70} = -90$$
$$x_{71} = -22$$
$$x_{72} = -28$$
$$x_{73} = -40$$
$$x_{74} = 46.213205416178$$
$$x_{75} = 47.7410908061906$$
$$x_{76} = 98.2519081306458$$
$$x_{77} = 9.9552234107877$$
$$x_{78} = -16$$
$$x_{79} = 56.1697953739808$$
$$x_{80} = -32$$
$$x_{81} = 96.1274371927986$$
$$x_{82} = 24.2448179213503$$
$$x_{83} = 16.3268232397407$$
$$x_{84} = -86$$
$$x_{85} = 2.12041215458852$$
$$x_{86} = -82$$
$$x_{87} = -36$$
$$x_{88} = -74$$
$$x_{89} = 90.2696199934722$$
$$x_{90} = 90.0490464633268$$
$$x_{91} = -18$$
$$x_{92} = 88.2746077056417$$
$$x_{93} = -34$$
$$x_{94} = 54.1961903309946$$
$$x_{95} = -14$$
$$x_{96} = -10.0443619772128$$
$$x_{97} = -96$$
$$x_{98} = -42$$
$$x_{99} = -50$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[99.9350759285183, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.12041215458852\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{3 \left(6 x \tan{\left(\sqrt{x^{3}} \right)} + \frac{\sqrt{x^{3}}}{x^{2}}\right) \left(\tan^{2}{\left(\sqrt{x^{3}} \right)} + 1\right)}{4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -26$$
$$x_{2} = 92.2428244626029$$
$$x_{3} = -20$$
$$x_{4} = 6.26927745240496$$
$$x_{5} = 44.0306045912742$$
$$x_{6} = -100$$
$$x_{7} = -52$$
$$x_{8} = -98$$
$$x_{9} = 58.109313442111$$
$$x_{10} = -46$$
$$x_{11} = -62$$
$$x_{12} = -94$$
$$x_{13} = 30.2655628057948$$
$$x_{14} = 52.1859675196535$$
$$x_{15} = 64.2681253460119$$
$$x_{16} = 34.320376023085$$
$$x_{17} = -30$$
$$x_{18} = 74.5765644784697$$
$$x_{19} = -88$$
$$x_{20} = 99.9350759285183$$
$$x_{21} = 42.1158371363077$$
$$x_{22} = 71.8843370513831$$
$$x_{23} = 11.8585944065254$$
$$x_{24} = 40.1564996359368$$
$$x_{25} = -76$$
$$x_{26} = -8.55358927481284$$
$$x_{27} = -44$$
$$x_{28} = -70$$
$$x_{29} = -78$$
$$x_{30} = -7.00058913350688$$
$$x_{31} = 86.2567902952232$$
$$x_{32} = 76.7436275805143$$
$$x_{33} = 66.0841266497851$$
$$x_{34} = -38$$
$$x_{35} = -80$$
$$x_{36} = 68.3829923388494$$
$$x_{37} = -92$$
$$x_{38} = 22.0684273449427$$
$$x_{39} = 28.3307856310302$$
$$x_{40} = 20.2469417441212$$
$$x_{41} = -56$$
$$x_{42} = 50.1362477622459$$
$$x_{43} = 81.9170861978907$$
$$x_{44} = -72$$
$$x_{45} = 62.4260925242595$$
$$x_{46} = -58$$
$$x_{47} = -60$$
$$x_{48} = 26.3274971573717$$
$$x_{49} = -68$$
$$x_{50} = 93.9792241997309$$
$$x_{51} = 36.0854076118897$$
$$x_{52} = 18.3394130124857$$
$$x_{53} = -24$$
$$x_{54} = 80.0552317475696$$
$$x_{55} = 78.1714680438355$$
$$x_{56} = 84.215087794644$$
$$x_{57} = -54$$
$$x_{58} = 14.1812858180921$$
$$x_{59} = 38.1481271522911$$
$$x_{60} = 60.2870278729972$$
$$x_{61} = 4.45624804800686$$
$$x_{62} = -64$$
$$x_{63} = 7.8475142235842$$
$$x_{64} = -84$$
$$x_{65} = -48$$
$$x_{66} = 32.1403393148926$$
$$x_{67} = -12$$
$$x_{68} = -10.0685939575807$$
$$x_{69} = -66$$
$$x_{70} = -90$$
$$x_{71} = -22$$
$$x_{72} = -28$$
$$x_{73} = -40$$
$$x_{74} = 46.213205416178$$
$$x_{75} = 47.7410908061906$$
$$x_{76} = 98.2519081306458$$
$$x_{77} = 9.9552234107877$$
$$x_{78} = -16$$
$$x_{79} = 56.1697953739808$$
$$x_{80} = -32$$
$$x_{81} = 96.1274371927986$$
$$x_{82} = 24.2448179213503$$
$$x_{83} = 16.3268232397407$$
$$x_{84} = -86$$
$$x_{85} = 2.12041215458852$$
$$x_{86} = -82$$
$$x_{87} = -36$$
$$x_{88} = -74$$
$$x_{89} = 90.2696199934722$$
$$x_{90} = 90.0490464633268$$
$$x_{91} = -18$$
$$x_{92} = 88.2746077056417$$
$$x_{93} = -34$$
$$x_{94} = 54.1961903309946$$
$$x_{95} = -14$$
$$x_{96} = -10.0443619772128$$
$$x_{97} = -96$$
$$x_{98} = -42$$
$$x_{99} = -50$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[99.9350759285183, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.12041215458852\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \tan{\left(\sqrt{x^{3}} \right)} = i$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = i$$
$$\lim_{x \to \infty} \tan{\left(\sqrt{x^{3}} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty} \tan{\left(\sqrt{x^{3}} \right)} = i$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = i$$
$$\lim_{x \to \infty} \tan{\left(\sqrt{x^{3}} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(sqrt(x^3)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(\sqrt{x^{3}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\sqrt{x^{3}} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(\sqrt{x^{3}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\sqrt{x^{3}} \right)}}{x}\right)$$
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:
$$\tan{\left(\sqrt{x^{3}} \right)} = \tan{\left(\sqrt{- x^{3}} \right)}$$
- No
$$\tan{\left(\sqrt{x^{3}} \right)} = - \tan{\left(\sqrt{- x^{3}} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(\sqrt{x^{3}} \right)} = \tan{\left(\sqrt{- x^{3}} \right)}$$
- No
$$\tan{\left(\sqrt{x^{3}} \right)} = - \tan{\left(\sqrt{- x^{3}} \right)}$$
- No
so, the function
not is
neither even, nor odd