Mister Exam
Graphing y = arccosx-sqrt(1-0.3*x^3)
The solution
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f(x) = acos(x) - / 1 - ----
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$$f{\left(x \right)} = - \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)}$$
f = -sqrt(1 - 3*x^3/10) + acos(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:
$$- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.562925952858039$$
$$x_{2} = 2.1414535556325$$
so we need to solve the equation:
$$- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.562925952858039$$
$$x_{2} = 2.1414535556325$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)}\right) = - \infty i$$
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 acos(x) - sqrt(1 - 3*x^3/10), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)}}{x}\right) = - \infty i$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)}}{x}\right) = - \infty i$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)} = - \sqrt{\frac{3 x^{3}}{10} + 1} + \operatorname{acos}{\left(- x \right)}$$
- No
$$- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)} = \sqrt{\frac{3 x^{3}}{10} + 1} - \operatorname{acos}{\left(- x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)} = - \sqrt{\frac{3 x^{3}}{10} + 1} + \operatorname{acos}{\left(- x \right)}$$
- No
$$- \sqrt{1 - \frac{3 x^{3}}{10}} + \operatorname{acos}{\left(x \right)} = \sqrt{\frac{3 x^{3}}{10} + 1} - \operatorname{acos}{\left(- x \right)}$$
- No
so, the function
not is
neither even, nor odd