Mister Exam
Graphing y = 14*asin(8^x)
The solution
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/ x\ f(x) = 14*asin\8 /
$$f{\left(x \right)} = 14 \operatorname{asin}{\left(8^{x} \right)}$$
f = 14*asin(8^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:
$$14 \operatorname{asin}{\left(8^{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -15.9649870906711$$
$$x_{2} = -67.964987090676$$
$$x_{3} = -61.964987090676$$
$$x_{4} = -85.964987090676$$
$$x_{5} = -19.964987090676$$
$$x_{6} = -59.964987090676$$
$$x_{7} = -41.964987090676$$
$$x_{8} = -95.964987090676$$
$$x_{9} = -57.964987090676$$
$$x_{10} = -45.964987090676$$
$$x_{11} = -75.964987090676$$
$$x_{12} = -81.964987090676$$
$$x_{13} = -53.964987090676$$
$$x_{14} = -91.964987090676$$
$$x_{15} = -23.964987090676$$
$$x_{16} = -49.964987090676$$
$$x_{17} = -105.964987090676$$
$$x_{18} = -97.964987090676$$
$$x_{19} = -87.964987090676$$
$$x_{20} = -107.964987090676$$
$$x_{21} = -51.964987090676$$
$$x_{22} = -43.964987090676$$
$$x_{23} = -69.964987090676$$
$$x_{24} = -21.964987090676$$
$$x_{25} = -73.964987090676$$
$$x_{26} = -77.964987090676$$
$$x_{27} = -79.964987090676$$
$$x_{28} = -37.964987090676$$
$$x_{29} = -99.964987090676$$
$$x_{30} = -27.964987090676$$
$$x_{31} = -35.964987090676$$
$$x_{32} = -39.964987090676$$
$$x_{33} = -55.964987090676$$
$$x_{34} = -65.964987090676$$
$$x_{35} = -71.964987090676$$
$$x_{36} = -109.964987090676$$
$$x_{37} = -25.964987090676$$
$$x_{38} = -83.964987090676$$
$$x_{39} = -31.964987090676$$
$$x_{40} = -93.964987090676$$
$$x_{41} = -17.964987090676$$
$$x_{42} = -101.964987090676$$
$$x_{43} = -103.964987090676$$
$$x_{44} = -47.964987090676$$
$$x_{45} = -63.964987090676$$
$$x_{46} = -33.964987090676$$
$$x_{47} = -89.964987090676$$
$$x_{48} = -29.964987090676$$
so we need to solve the equation:
$$14 \operatorname{asin}{\left(8^{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -15.9649870906711$$
$$x_{2} = -67.964987090676$$
$$x_{3} = -61.964987090676$$
$$x_{4} = -85.964987090676$$
$$x_{5} = -19.964987090676$$
$$x_{6} = -59.964987090676$$
$$x_{7} = -41.964987090676$$
$$x_{8} = -95.964987090676$$
$$x_{9} = -57.964987090676$$
$$x_{10} = -45.964987090676$$
$$x_{11} = -75.964987090676$$
$$x_{12} = -81.964987090676$$
$$x_{13} = -53.964987090676$$
$$x_{14} = -91.964987090676$$
$$x_{15} = -23.964987090676$$
$$x_{16} = -49.964987090676$$
$$x_{17} = -105.964987090676$$
$$x_{18} = -97.964987090676$$
$$x_{19} = -87.964987090676$$
$$x_{20} = -107.964987090676$$
$$x_{21} = -51.964987090676$$
$$x_{22} = -43.964987090676$$
$$x_{23} = -69.964987090676$$
$$x_{24} = -21.964987090676$$
$$x_{25} = -73.964987090676$$
$$x_{26} = -77.964987090676$$
$$x_{27} = -79.964987090676$$
$$x_{28} = -37.964987090676$$
$$x_{29} = -99.964987090676$$
$$x_{30} = -27.964987090676$$
$$x_{31} = -35.964987090676$$
$$x_{32} = -39.964987090676$$
$$x_{33} = -55.964987090676$$
$$x_{34} = -65.964987090676$$
$$x_{35} = -71.964987090676$$
$$x_{36} = -109.964987090676$$
$$x_{37} = -25.964987090676$$
$$x_{38} = -83.964987090676$$
$$x_{39} = -31.964987090676$$
$$x_{40} = -93.964987090676$$
$$x_{41} = -17.964987090676$$
$$x_{42} = -101.964987090676$$
$$x_{43} = -103.964987090676$$
$$x_{44} = -47.964987090676$$
$$x_{45} = -63.964987090676$$
$$x_{46} = -33.964987090676$$
$$x_{47} = -89.964987090676$$
$$x_{48} = -29.964987090676$$
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
$$\frac{14 \cdot 8^{x} \log{\left(8 \right)}}{\sqrt{1 - 8^{2 x}}} = 0$$
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
$$\frac{14 \cdot 8^{x} \log{\left(8 \right)}}{\sqrt{1 - 8^{2 x}}} = 0$$
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
$$- \frac{14 \cdot 8^{x} \left(\frac{8^{2 x}}{8^{2 x} - 1} - 1\right) \log{\left(8 \right)}^{2}}{\sqrt{1 - 8^{2 x}}} = 0$$
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
$$- \frac{14 \cdot 8^{x} \left(\frac{8^{2 x}}{8^{2 x} - 1} - 1\right) \log{\left(8 \right)}^{2}}{\sqrt{1 - 8^{2 x}}} = 0$$
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
$$\lim_{x \to -\infty}\left(14 \operatorname{asin}{\left(8^{x} \right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(14 \operatorname{asin}{\left(8^{x} \right)}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(14 \operatorname{asin}{\left(8^{x} \right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(14 \operatorname{asin}{\left(8^{x} \right)}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 14*asin(8^x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{14 \operatorname{asin}{\left(8^{x} \right)}}{x}\right) = 0$$
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:
$$y = x \lim_{x \to \infty}\left(\frac{14 \operatorname{asin}{\left(8^{x} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{14 \operatorname{asin}{\left(8^{x} \right)}}{x}\right) = 0$$
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:
$$y = x \lim_{x \to \infty}\left(\frac{14 \operatorname{asin}{\left(8^{x} \right)}}{x}\right)$$
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:
$$14 \operatorname{asin}{\left(8^{x} \right)} = 14 \operatorname{asin}{\left(8^{- x} \right)}$$
- No
$$14 \operatorname{asin}{\left(8^{x} \right)} = - 14 \operatorname{asin}{\left(8^{- x} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$14 \operatorname{asin}{\left(8^{x} \right)} = 14 \operatorname{asin}{\left(8^{- x} \right)}$$
- No
$$14 \operatorname{asin}{\left(8^{x} \right)} = - 14 \operatorname{asin}{\left(8^{- x} \right)}$$
- No
so, the function
not is
neither even, nor odd