This is a mathematical exercise set from a 2Bac (2nd-year Baccalaureate) level focused on analysis and sequences. Here’s a breakdown of the first exercise:

Exercice 1 Analysis

1. Given Function:
   $f(x) = x - 2\sqrt{x} + 2$
   The curve $(\mathcal{C}_f)$ represents $f$ in a standard orthonormal plane $(O, \vec{i}, \vec{j})$.

Questions Breakdown

1. Domain and Limit:

   • Find $D_f$, the domain of $f(x)$.
   • Calculate $\lim_{x \to +\infty} f(x)$.

2. Infinite Branches:

   • Study the behavior of the curve $(\mathcal{C}_f)$ as $x \to +\infty$.

3. Relative Position:

   • Compare the curve $(\mathcal{C}_f)$ to the line $\Delta : y = x$.

4. Right Derivability:

   • Examine whether $f$ is differentiable at $x = 0$ from the right-hand side.

5. Function Evaluation:

   • Calculate $f(x)$ for $x \in [0, +\infty]$.

6. Variation Table:

   • Determine the monotonicity of $f$ and create its variation table.

7. Curve Construction:

   • Sketch the line $\Delta$ and curve $(\mathcal{C}_f)$.

Sequence Analysis

• Given the sequence $(u_n)$ defined as: u_{n+1} = \frac{1}{3}u_n + n - 2 \\ u_0 = 1 \end{cases} \quad (n \in \mathbb{N})$$

8. Monotonicity:

   • Prove that $u_n$ is increasing for all $n \in \mathbb{N}$.

9. Boundedness:

   • Show that $u_n$ is bounded above.

10. Convergence:

    • Deduce whether the sequence $(u_n)$ converges.

Would you like detailed solutions for specific parts of this exercise? Let me know where you'd like to begin!

5 Related Questions

1. How do you determine the domain of a function involving square roots?
2. What steps are involved in analyzing the behavior of a curve as $x \to +\infty$?
3. How do you prove that a sequence is monotonic and bounded?
4. What methods are used to compare a curve to a line (relative position)?
5. How do you determine the convergence of a sequence defined recursively?

Tip:

For recursive sequences, always check monotonicity and boundedness to confirm convergence.