Mister Exam
Graphing y = tan(x)+(5/2)
The solution
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f(x) = tan(x) + 5/2
$$f{\left(x \right)} = \tan{\left(x \right)} + \frac{5}{2}$$
f = tan(x) + 5/2
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:
$$\tan{\left(x \right)} + \frac{5}{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \operatorname{atan}{\left(\frac{5}{2} \right)}$$
Numerical solution
$$x_{1} = -45.1725870999396$$
$$x_{2} = -114.287625478915$$
$$x_{3} = 80.4911190436521$$
$$x_{4} = -57.7389577142988$$
$$x_{5} = -70.305328328658$$
$$x_{6} = -42.0309944463498$$
$$x_{7} = -4.33188260327232$$
$$x_{8} = 441.774274206478$$
$$x_{9} = -117.429218132505$$
$$x_{10} = -340.48229653738$$
$$x_{11} = 77.3495263900623$$
$$x_{12} = -86.0132915966069$$
$$x_{13} = -151.986737321993$$
$$x_{14} = 23.9424512790358$$
$$x_{15} = -13.7566605640417$$
$$x_{16} = -1.19028994968253$$
$$x_{17} = -10.6150679104519$$
$$x_{18} = -7.47347525686212$$
$$x_{19} = 36.508821893395$$
$$x_{20} = 83.6327116972419$$
$$x_{21} = 67.9247484292929$$
$$x_{22} = -29.4646238319907$$
$$x_{23} = 14.5176733182664$$
$$x_{24} = 102.482267618781$$
$$x_{25} = 74.2079337364725$$
$$x_{26} = 96.1990823116011$$
$$x_{27} = 27.0840439326256$$
$$x_{28} = -26.3230311784009$$
$$x_{29} = -20.0398458712213$$
$$x_{30} = -51.4557724071192$$
$$x_{31} = -98.5796622109661$$
$$x_{32} = 1.95130270390726$$
$$x_{33} = 89.9158970044215$$
$$x_{34} = -73.4469209822478$$
$$x_{35} = 155.889342729807$$
$$x_{36} = -76.5885136358376$$
$$x_{37} = -82.8716989430172$$
$$x_{38} = 259.56190029827$$
$$x_{39} = -35.7478091391703$$
$$x_{40} = -79.7301062894274$$
$$x_{41} = 61.6415631221133$$
$$x_{42} = 58.4999704685235$$
$$x_{43} = 30.2256365862154$$
$$x_{44} = 86.7743043508317$$
$$x_{45} = 45.9335998541644$$
$$x_{46} = 52.216785161344$$
$$x_{47} = -155.128329975582$$
$$x_{48} = 8.23448801108685$$
$$x_{49} = -64.0221430214784$$
so we need to solve the equation:
$$\tan{\left(x \right)} + \frac{5}{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \operatorname{atan}{\left(\frac{5}{2} \right)}$$
Numerical solution
$$x_{1} = -45.1725870999396$$
$$x_{2} = -114.287625478915$$
$$x_{3} = 80.4911190436521$$
$$x_{4} = -57.7389577142988$$
$$x_{5} = -70.305328328658$$
$$x_{6} = -42.0309944463498$$
$$x_{7} = -4.33188260327232$$
$$x_{8} = 441.774274206478$$
$$x_{9} = -117.429218132505$$
$$x_{10} = -340.48229653738$$
$$x_{11} = 77.3495263900623$$
$$x_{12} = -86.0132915966069$$
$$x_{13} = -151.986737321993$$
$$x_{14} = 23.9424512790358$$
$$x_{15} = -13.7566605640417$$
$$x_{16} = -1.19028994968253$$
$$x_{17} = -10.6150679104519$$
$$x_{18} = -7.47347525686212$$
$$x_{19} = 36.508821893395$$
$$x_{20} = 83.6327116972419$$
$$x_{21} = 67.9247484292929$$
$$x_{22} = -29.4646238319907$$
$$x_{23} = 14.5176733182664$$
$$x_{24} = 102.482267618781$$
$$x_{25} = 74.2079337364725$$
$$x_{26} = 96.1990823116011$$
$$x_{27} = 27.0840439326256$$
$$x_{28} = -26.3230311784009$$
$$x_{29} = -20.0398458712213$$
$$x_{30} = -51.4557724071192$$
$$x_{31} = -98.5796622109661$$
$$x_{32} = 1.95130270390726$$
$$x_{33} = 89.9158970044215$$
$$x_{34} = -73.4469209822478$$
$$x_{35} = 155.889342729807$$
$$x_{36} = -76.5885136358376$$
$$x_{37} = -82.8716989430172$$
$$x_{38} = 259.56190029827$$
$$x_{39} = -35.7478091391703$$
$$x_{40} = -79.7301062894274$$
$$x_{41} = 61.6415631221133$$
$$x_{42} = 58.4999704685235$$
$$x_{43} = 30.2256365862154$$
$$x_{44} = 86.7743043508317$$
$$x_{45} = 45.9335998541644$$
$$x_{46} = 52.216785161344$$
$$x_{47} = -155.128329975582$$
$$x_{48} = 8.23448801108685$$
$$x_{49} = -64.0221430214784$$
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
$$\tan^{2}{\left(x \right)} + 1 = 0$$
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
$$\tan^{2}{\left(x \right)} + 1 = 0$$
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
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
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:
$$y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} + \frac{5}{2}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(x \right)} + \frac{5}{2}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} + \frac{5}{2}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(x \right)} + \frac{5}{2}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x) + 5/2, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + \frac{5}{2}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + \frac{5}{2}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + \frac{5}{2}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + \frac{5}{2}}{x}\right)$$
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:
$$\tan{\left(x \right)} + \frac{5}{2} = \frac{5}{2} - \tan{\left(x \right)}$$
- No
$$\tan{\left(x \right)} + \frac{5}{2} = \tan{\left(x \right)} - \frac{5}{2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(x \right)} + \frac{5}{2} = \frac{5}{2} - \tan{\left(x \right)}$$
- No
$$\tan{\left(x \right)} + \frac{5}{2} = \tan{\left(x \right)} - \frac{5}{2}$$
- No
so, the function
not is
neither even, nor odd