Mister Exam
Graphing y = 3^tg(x)
The solution
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tan(x) f(x) = 3
$$f{\left(x \right)} = 3^{\tan{\left(x \right)}}$$
f = 3^tan(x)
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:
$$3^{\tan{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 95.8584736273053$$
$$x_{2} = 36.1642338403311$$
$$x_{3} = -14.0993498451125$$
$$x_{4} = -80.0748686325117$$
$$x_{5} = 29.8839393722148$$
$$x_{6} = -51.8040152296396$$
$$x_{7} = -58.0829356280364$$
$$x_{8} = 51.87549652732$$
$$x_{9} = 14.1723345642698$$
$$x_{10} = -95.7890396514303$$
$$x_{11} = -7.81966061582318$$
$$x_{12} = 58.1560516643718$$
$$x_{13} = 7.89232677984328$$
$$x_{14} = 73.867005766353$$
$$x_{15} = -29.8117696742404$$
$$x_{16} = 80.1477997799319$$
$$x_{17} = -73.7964261193455$$
$$x_{18} = -36.0911002771662$$
so we need to solve the equation:
$$3^{\tan{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 95.8584736273053$$
$$x_{2} = 36.1642338403311$$
$$x_{3} = -14.0993498451125$$
$$x_{4} = -80.0748686325117$$
$$x_{5} = 29.8839393722148$$
$$x_{6} = -51.8040152296396$$
$$x_{7} = -58.0829356280364$$
$$x_{8} = 51.87549652732$$
$$x_{9} = 14.1723345642698$$
$$x_{10} = -95.7890396514303$$
$$x_{11} = -7.81966061582318$$
$$x_{12} = 58.1560516643718$$
$$x_{13} = 7.89232677984328$$
$$x_{14} = 73.867005766353$$
$$x_{15} = -29.8117696742404$$
$$x_{16} = 80.1477997799319$$
$$x_{17} = -73.7964261193455$$
$$x_{18} = -36.0911002771662$$
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
$$3^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} = 0$$
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
$$3^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} = 0$$
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
$$3^{\tan{\left(x \right)}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} + 2 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$3^{\tan{\left(x \right)}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} + 2 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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:
$$y = \lim_{x \to -\infty} 3^{\tan{\left(x \right)}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} 3^{\tan{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} 3^{\tan{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} 3^{\tan{\left(x \right)}}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3^tan(x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{3^{\tan{\left(x \right)}}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{3^{\tan{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{3^{\tan{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{3^{\tan{\left(x \right)}}}{x}\right)$$
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:
$$3^{\tan{\left(x \right)}} = 3^{- \tan{\left(x \right)}}$$
- No
$$3^{\tan{\left(x \right)}} = - 3^{- \tan{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$3^{\tan{\left(x \right)}} = 3^{- \tan{\left(x \right)}}$$
- No
$$3^{\tan{\left(x \right)}} = - 3^{- \tan{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd