The graph in the figure, representing the continuous function $y = f(x)$, is the union of the parabolic arc $\Gamma_{1}$, the circular arc $\Gamma_{2}$ and the hyperbolic arc $\Gamma_{3}$.
a) Write an analytical expression of the function $f$ defined piecewise on the interval $[-2; 2]$, using the equations:
$$y = a(x + 2)^{2} \quad x^{2} + y^{2} + b = 0 \quad x^{2} - y^{2} + c = 0$$
and identify the appropriate values for the real parameters $a, b, c$.
Study the differentiability of the function $f$ and write the equations of any tangent lines at the points with abscissa
$$x = -2 \quad x = 0 \quad x = 1 \quad x = 2$$
b) Starting from the graph of the function $f$, deduce that of its derivative $f^{\prime}$ and identify the intervals of concavity and convexity of $F(x) = \int_{-2}^{x} f(t) dt$.
c) Consider the function $y = \frac{1}{4}(x + 2)^{2}$, defined on the interval $[-2; 0]$, of which $\Gamma_{1}$ is the representative graph. Explain why it is invertible and write the analytical expression of its inverse function $h$. Study the differentiability of $h$ and sketch its graph.
d) Let $S$ be the bounded region in the second quadrant, between the graph $\Gamma_{1}$ and the coordinate axes. Determine the value of the real parameter $k$ so that the line with equation $x = k$ divides $S$ into two equivalent regions.

Given a real parameter $a$, with $a \neq 0$, consider the function $f_{a}$ defined as follows:
$$f_{a}(x) = \frac{x^{2} - ax}{x^{2} - a}$$
whose graph will be denoted by $\Omega_{a}$.
a) As the parameter $a$ varies, determine the domain of $f_{a}$, study any discontinuities and write the equations of all its asymptotes.
b) Show that, for $a \neq 1$, all graphs $\Omega_{a}$ intersect their horizontal asymptote at the same point and share the same tangent line at the origin.
c) As $a < 1$ varies, identify the intervals of monotonicity of the function $f_{a}$. Study the function $f_{-1}(x)$ and sketch its graph $\Omega_{-1}$.
d) Determine the area of the bounded region between the graph $\Omega_{-1}$, the line tangent to it at the origin and the line $x = \sqrt{3}$.

Let $ABC$ be a right triangle with right angle at $A$. Let $O$ be the center of the square $BCDE$ constructed on the hypotenuse, on the opposite side from vertex $A$.
Prove that $O$ is equidistant from the lines $AB$ and $AC$.

A biased die, with faces numbered from 1 to 6, has the property that each even face appears with probability twice that of each odd face. Calculate the probabilities of obtaining, by rolling the die once, respectively:
－a prime number
－a number at least 3
－a number at most 3

Consider the line $r$ passing through the two points $A(1, -2, 0)$ and $B(2, 3, -1)$, determine the Cartesian equation of the spherical surface with center $C(1, -6, 7)$ and tangent to $r$.

Among all rectangular parallelepipeds with square base and volume $V$, determine whether the one with minimum total area also has minimum diagonal length.

Determine the equation of the tangent line to the curve with equation $y = \sqrt{25 - x^{2}}$ at its point with abscissa 3, using two different methods.

Determine the values of the real parameters $a$ and $b$ so that:
$$\lim_{x \rightarrow 0} \frac{\operatorname{sen} x - (ax^{3} + bx)}{x^{3}} = 1$$

Consider the function:
$$f(x) = \begin{cases} -1 + \arctan x & x < 0 \\ ax + b & x \geq 0 \end{cases}$$
Determine for which values of the real parameters $a, b$ the function is differentiable. Establish whether there exists an interval of $\mathbb{R}$ in which the function $f$ satisfies the hypotheses of Rolle's theorem. Justify your answer.

Given the function $f_{a}(x) = x^{5} - 5ax + a$, defined on the set of real numbers, determine for which values of the parameter $a > 0$ the function has three distinct real zeros.