Theme: Exponential function Main topics covered: Sequences; Functions, Logarithm function. This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A correct answer earns one point. An incorrect answer, a multiple answer, or no answer to a question earns or loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
A container initially containing 1 litre of water is left in the sun. Every hour, the volume of water decreases by $15 \%$. After how many whole hours does the volume of water become less than a quarter of a litre? a. 2 hours b. 8 hours. c. 9 hours d. 13 hours
We consider the function $f$ defined on the interval $] 0$; $+ \infty [ \operatorname { by } f ( x ) = 4 \ln ( 3 x )$. For every real $x$ in the interval $] 0$; $+ \infty [$, we have: a. $f ( 2 x ) = f ( x ) + \ln ( 24 )$ b. $f ( 2 x ) = f ( x ) + \ln ( 16 )$ c. $f ( 2 x ) = \ln ( 2 ) + f ( x )$ d. $f ( 2 x ) = 2 f ( x )$
We consider the function $g$ defined on the interval $] 1 ; + \infty [$ by: $$g ( x ) = \frac { \ln ( x ) } { x - 1 } .$$ We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathscr { C } _ { g }$ 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. In the rest of the exercise, we consider the function $h$ defined on the interval ]0;2] by: $$h ( x ) = x ^ { 2 } [ 1 + 2 \ln ( x ) ] .$$ We denote $\mathscr { C } _ { h }$ the representative curve of $h$ in a coordinate system of the plane. We assume that $h$ is twice differentiable on the interval ]0; 2]. We denote $h ^ { \prime }$ its derivative and $h ^ { \prime \prime }$ its second derivative. We assume that, for every real $x$ in the interval ] 0 ; 2], we have: $$h ^ { \prime } ( x ) = 4 x ( 1 + \ln ( x ) ) .$$
On the interval $\left. ] \frac { 1 } { \mathrm { e } } ; 2 \right]$, the function $h$ equals zero: a. exactly 0 times. b. exactly 1 time. c. exactly 2 times. d. exactly 3 times.
An equation of the tangent line to $\mathscr { C } _ { h }$ at the point with abscissa $\sqrt { \mathrm { e } }$ is: a. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x$ b. $y = ( 6 \sqrt { \mathrm { e } } ) \cdot x + 2 \mathrm { e }$ c. $y = 6 \mathrm { e } ^ { \frac { x } { 2 } }$ d. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x - 4 \mathrm { e }$.
On the interval $] 0 ; 2 ]$, the number of inflection points of the curve $\mathscr { C } _ { h }$ is equal to: a. 0 b. 1 c. 2 d. 3
We consider the sequence $\left( u _ { n } \right)$ defined for every natural number $n$ by $$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 3 \quad \text { and } \quad u _ { 0 } = 6 .$$ We can affirm that: a. the sequence $\left( u _ { n } \right)$ is strictly increasing. b. the sequence $( u _ { n } )$ is strictly decreasing. c. the sequence $( u _ { n } )$ is not monotonic. d. the sequence $( u _ { n } )$ is constant.
\section*{Exercise 2 — 6 points}
Theme: Exponential function\\
Main topics covered: Sequences; Functions, Logarithm function.\\
This exercise is a multiple choice questionnaire.\\
For each of the following questions, only one of the four proposed answers is correct.\\
A correct answer earns one point. An incorrect answer, a multiple answer, or no answer to a question earns or loses no points.\\
To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
\begin{enumerate}
\item A container initially containing 1 litre of water is left in the sun. Every hour, the volume of water decreases by $15 \%$. After how many whole hours does the volume of water become less than a quarter of a litre?\\
a. 2 hours\\
b. 8 hours.\\
c. 9 hours\\
d. 13 hours
\item We consider the function $f$ defined on the interval $] 0$; $+ \infty [ \operatorname { by } f ( x ) = 4 \ln ( 3 x )$. For every real $x$ in the interval $] 0$; $+ \infty [$, we have:\\
a. $f ( 2 x ) = f ( x ) + \ln ( 24 )$\\
b. $f ( 2 x ) = f ( x ) + \ln ( 16 )$\\
c. $f ( 2 x ) = \ln ( 2 ) + f ( x )$\\
d. $f ( 2 x ) = 2 f ( x )$
\item We consider the function $g$ defined on the interval $] 1 ; + \infty [$ by:
$$g ( x ) = \frac { \ln ( x ) } { x - 1 } .$$
We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathscr { C } _ { g }$ 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.
In the rest of the exercise, we consider the function $h$ defined on the interval ]0;2] by:
$$h ( x ) = x ^ { 2 } [ 1 + 2 \ln ( x ) ] .$$
We denote $\mathscr { C } _ { h }$ the representative curve of $h$ in a coordinate system of the plane.\\
We assume that $h$ is twice differentiable on the interval ]0; 2].\\
We denote $h ^ { \prime }$ its derivative and $h ^ { \prime \prime }$ its second derivative.\\
We assume that, for every real $x$ in the interval ] 0 ; 2], we have:
$$h ^ { \prime } ( x ) = 4 x ( 1 + \ln ( x ) ) .$$
\item On the interval $\left. ] \frac { 1 } { \mathrm { e } } ; 2 \right]$, the function $h$ equals zero:\\
a. exactly 0 times.\\
b. exactly 1 time.\\
c. exactly 2 times.\\
d. exactly 3 times.
\item An equation of the tangent line to $\mathscr { C } _ { h }$ at the point with abscissa $\sqrt { \mathrm { e } }$ is:\\
a. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x$\\
b. $y = ( 6 \sqrt { \mathrm { e } } ) \cdot x + 2 \mathrm { e }$\\
c. $y = 6 \mathrm { e } ^ { \frac { x } { 2 } }$\\
d. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x - 4 \mathrm { e }$.
\item On the interval $] 0 ; 2 ]$, the number of inflection points of the curve $\mathscr { C } _ { h }$ is equal to:\\
a. 0\\
b. 1\\
c. 2\\
d. 3
\item We consider the sequence $\left( u _ { n } \right)$ defined for every natural number $n$ by
$$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 3 \quad \text { and } \quad u _ { 0 } = 6 .$$
We can affirm that:\\
a. the sequence $\left( u _ { n } \right)$ is strictly increasing.\\
b. the sequence $( u _ { n } )$ is strictly decreasing.\\
c. the sequence $( u _ { n } )$ is not monotonic.\\
d. the sequence $( u _ { n } )$ is constant.
\end{enumerate}