Mister Exam
Graphing y = tan((pi*x)/4)
The solution
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/pi*x\
f(x) = tan|----|
\ 4 /
$$f{\left(x \right)} = \tan{\left(\frac{\pi x}{4} \right)}$$
f = tan((pi*x)/4)
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:
$$\tan{\left(\frac{\pi x}{4} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 52$$
$$x_{2} = -64$$
$$x_{3} = 76$$
$$x_{4} = 48$$
$$x_{5} = 88$$
$$x_{6} = 60$$
$$x_{7} = 28$$
$$x_{8} = 92$$
$$x_{9} = -100$$
$$x_{10} = 36$$
$$x_{11} = 20$$
$$x_{12} = 84$$
$$x_{13} = -44$$
$$x_{14} = 44$$
$$x_{15} = 80$$
$$x_{16} = -72$$
$$x_{17} = 8$$
$$x_{18} = -92$$
$$x_{19} = 100$$
$$x_{20} = 64$$
$$x_{21} = -96$$
$$x_{22} = 24$$
$$x_{23} = 96$$
$$x_{24} = -76$$
$$x_{25} = 4$$
$$x_{26} = -28$$
$$x_{27} = -84$$
$$x_{28} = 32$$
$$x_{29} = -56$$
$$x_{30} = -68$$
$$x_{31} = -60$$
$$x_{32} = -36$$
$$x_{33} = -12$$
$$x_{34} = 56$$
$$x_{35} = -16$$
$$x_{36} = 40$$
$$x_{37} = -52$$
$$x_{38} = -88$$
$$x_{39} = 16$$
$$x_{40} = -8$$
$$x_{41} = -24$$
$$x_{42} = 68$$
$$x_{43} = -80$$
$$x_{44} = -20$$
$$x_{45} = -32$$
$$x_{46} = -4$$
$$x_{47} = -48$$
$$x_{48} = 0$$
$$x_{49} = 72$$
$$x_{50} = -40$$
$$x_{51} = 12$$
so we need to solve the equation:
$$\tan{\left(\frac{\pi x}{4} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 52$$
$$x_{2} = -64$$
$$x_{3} = 76$$
$$x_{4} = 48$$
$$x_{5} = 88$$
$$x_{6} = 60$$
$$x_{7} = 28$$
$$x_{8} = 92$$
$$x_{9} = -100$$
$$x_{10} = 36$$
$$x_{11} = 20$$
$$x_{12} = 84$$
$$x_{13} = -44$$
$$x_{14} = 44$$
$$x_{15} = 80$$
$$x_{16} = -72$$
$$x_{17} = 8$$
$$x_{18} = -92$$
$$x_{19} = 100$$
$$x_{20} = 64$$
$$x_{21} = -96$$
$$x_{22} = 24$$
$$x_{23} = 96$$
$$x_{24} = -76$$
$$x_{25} = 4$$
$$x_{26} = -28$$
$$x_{27} = -84$$
$$x_{28} = 32$$
$$x_{29} = -56$$
$$x_{30} = -68$$
$$x_{31} = -60$$
$$x_{32} = -36$$
$$x_{33} = -12$$
$$x_{34} = 56$$
$$x_{35} = -16$$
$$x_{36} = 40$$
$$x_{37} = -52$$
$$x_{38} = -88$$
$$x_{39} = 16$$
$$x_{40} = -8$$
$$x_{41} = -24$$
$$x_{42} = 68$$
$$x_{43} = -80$$
$$x_{44} = -20$$
$$x_{45} = -32$$
$$x_{46} = -4$$
$$x_{47} = -48$$
$$x_{48} = 0$$
$$x_{49} = 72$$
$$x_{50} = -40$$
$$x_{51} = 12$$
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
$$\frac{\pi \left(\tan^{2}{\left(\frac{\pi x}{4} \right)} + 1\right)}{4} = 0$$
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
$$\frac{\pi \left(\tan^{2}{\left(\frac{\pi x}{4} \right)} + 1\right)}{4} = 0$$
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
$$\frac{\pi^{2} \left(\tan^{2}{\left(\frac{\pi x}{4} \right)} + 1\right) \tan{\left(\frac{\pi x}{4} \right)}}{8} = 0$$
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
$$\frac{\pi^{2} \left(\tan^{2}{\left(\frac{\pi x}{4} \right)} + 1\right) \tan{\left(\frac{\pi x}{4} \right)}}{8} = 0$$
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
$$\lim_{x \to -\infty} \tan{\left(\frac{\pi x}{4} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(\frac{\pi x}{4} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty} \tan{\left(\frac{\pi x}{4} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(\frac{\pi x}{4} \right)} = \left\langle -\infty, \infty\right\rangle$$
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((pi*x)/4), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(\frac{\pi x}{4} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\frac{\pi x}{4} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(\frac{\pi x}{4} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\frac{\pi x}{4} \right)}}{x}\right)$$
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:
$$\tan{\left(\frac{\pi x}{4} \right)} = - \tan{\left(\frac{\pi x}{4} \right)}$$
- No
$$\tan{\left(\frac{\pi x}{4} \right)} = \tan{\left(\frac{\pi x}{4} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(\frac{\pi x}{4} \right)} = - \tan{\left(\frac{\pi x}{4} \right)}$$
- No
$$\tan{\left(\frac{\pi x}{4} \right)} = \tan{\left(\frac{\pi x}{4} \right)}$$
- No
so, the function
not is
neither even, nor odd