Exercise 4 (7 points) — Main topics covered: sequences, functions, antiderivatives. This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers or the absence of an answer to a question neither awards nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
Consider the sequence $(u_n)$ defined for all natural number $n$ by $$u_n = \frac{(-1)^n}{n+1}.$$ We can affirm that: a. the sequence $(u_n)$ diverges to $+\infty$. b. the sequence $(u_n)$ diverges to $-\infty$. c. the sequence $(u_n)$ has no limit. d. the sequence $(u_n)$ converges. In questions 2 and 3, we consider two sequences $(v_n)$ and $(w_n)$ satisfying the relation: $$w_n = \mathrm{e}^{-2v_n} + 2.$$
Let $a$ be a strictly positive real number. We have $v_0 = \ln(a)$. a. $w_0 = \dfrac{1}{a^2} + 2$ b. $w_0 = \dfrac{1}{a^2 + 2}$ c. $w_0 = -2a + 2$ d. $w_0 = \dfrac{1}{-2a} + 2$
We know that the sequence $(v_n)$ is increasing. We can affirm that the sequence $(w_n)$ is: a. decreasing and bounded above by 3. b. decreasing and bounded below by 2. c. increasing and bounded above by 3. d. increasing and bounded below by 2.
Consider the sequence $(a_n)$ defined as follows: $$a_0 = 2 \text{ and, for all natural number } n, \quad a_{n+1} = \frac{1}{3}a_n + \frac{8}{3}.$$ For all natural number $n$, we have: a. $a_n = 4 \times \left(\dfrac{1}{3}\right)^n - 2$ b. $a_n = -\dfrac{2}{3^n} + 4$ c. $a_n = 4 - \left(\dfrac{1}{3}\right)^n$ d. $a_n = 2 \times \left(\dfrac{1}{3}\right)^n + \dfrac{8n}{3}$
Consider a sequence $(b_n)$ such that, for all natural number $n$, we have: $$b_{n+1} = b_n + \ln\left(\frac{2}{(b_n)^2 + 3}\right)$$ We can affirm that: a. the sequence $(b_n)$ is increasing. b. the sequence $(b_n)$ is decreasing. c. the sequence $(b_n)$ is not monotone. d. the direction of variation of the sequence $(b_n)$ depends on $b_0$.
Consider the function $g$ defined on the interval $]0; +\infty[$ by: $$g(x) = \frac{\mathrm{e}^x}{x}$$ We denote $\mathcal{C}_g$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathcal{C}_g$ has: a. a vertical asymptote and a horizontal asymptote. b. a vertical asymptote and no horizontal asymptote. c. no vertical asymptote and a horizontal asymptote. d. no vertical asymptote and no horizontal asymptote.
Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{x^2+1}$$ Let $F$ be an antiderivative on $\mathbb{R}$ of the function $f$. For all real $x$, we have: a. $F(x) = \dfrac{1}{2}x^2\mathrm{e}^{x^2+1}$ b. $F(x) = \left(1 + 2x^2\right)\mathrm{e}^{x^2+1}$ c. $F(x) = \mathrm{e}^{x^2+1}$ d. $F(x) = \dfrac{1}{2}\mathrm{e}^{x^2+1}$
\textbf{Exercise 4} (7 points) — Main topics covered: sequences, functions, antiderivatives.
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers or the absence of an answer to a question neither awards nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
\begin{enumerate}
\item Consider the sequence $(u_n)$ defined for all natural number $n$ by
$$u_n = \frac{(-1)^n}{n+1}.$$
We can affirm that:\\
a. the sequence $(u_n)$ diverges to $+\infty$.\\
b. the sequence $(u_n)$ diverges to $-\infty$.\\
c. the sequence $(u_n)$ has no limit.\\
d. the sequence $(u_n)$ converges.
In questions 2 and 3, we consider two sequences $(v_n)$ and $(w_n)$ satisfying the relation:
$$w_n = \mathrm{e}^{-2v_n} + 2.$$
\item Let $a$ be a strictly positive real number. We have $v_0 = \ln(a)$.\\
a. $w_0 = \dfrac{1}{a^2} + 2$\\
b. $w_0 = \dfrac{1}{a^2 + 2}$\\
c. $w_0 = -2a + 2$\\
d. $w_0 = \dfrac{1}{-2a} + 2$
\item We know that the sequence $(v_n)$ is increasing. We can affirm that the sequence $(w_n)$ is:\\
a. decreasing and bounded above by 3.\\
b. decreasing and bounded below by 2.\\
c. increasing and bounded above by 3.\\
d. increasing and bounded below by 2.
\item Consider the sequence $(a_n)$ defined as follows:
$$a_0 = 2 \text{ and, for all natural number } n, \quad a_{n+1} = \frac{1}{3}a_n + \frac{8}{3}.$$
For all natural number $n$, we have:\\
a. $a_n = 4 \times \left(\dfrac{1}{3}\right)^n - 2$\\
b. $a_n = -\dfrac{2}{3^n} + 4$\\
c. $a_n = 4 - \left(\dfrac{1}{3}\right)^n$\\
d. $a_n = 2 \times \left(\dfrac{1}{3}\right)^n + \dfrac{8n}{3}$
\item Consider a sequence $(b_n)$ such that, for all natural number $n$, we have:
$$b_{n+1} = b_n + \ln\left(\frac{2}{(b_n)^2 + 3}\right)$$
We can affirm that:\\
a. the sequence $(b_n)$ is increasing.\\
b. the sequence $(b_n)$ is decreasing.\\
c. the sequence $(b_n)$ is not monotone.\\
d. the direction of variation of the sequence $(b_n)$ depends on $b_0$.
\item Consider the function $g$ defined on the interval $]0; +\infty[$ by:
$$g(x) = \frac{\mathrm{e}^x}{x}$$
We denote $\mathcal{C}_g$ the representative curve of the function $g$ in an orthogonal coordinate system.\\
The curve $\mathcal{C}_g$ has:\\
a. a vertical asymptote and a horizontal asymptote.\\
b. a vertical asymptote and no horizontal asymptote.\\
c. no vertical asymptote and a horizontal asymptote.\\
d. no vertical asymptote and no horizontal asymptote.
\item Let $f$ be the function defined on $\mathbb{R}$ by
$$f(x) = x\mathrm{e}^{x^2+1}$$
Let $F$ be an antiderivative on $\mathbb{R}$ of the function $f$. For all real $x$, we have:\\
a. $F(x) = \dfrac{1}{2}x^2\mathrm{e}^{x^2+1}$\\
b. $F(x) = \left(1 + 2x^2\right)\mathrm{e}^{x^2+1}$\\
c. $F(x) = \mathrm{e}^{x^2+1}$\\
d. $F(x) = \dfrac{1}{2}\mathrm{e}^{x^2+1}$
\end{enumerate}