Below is the graphical representation $\mathscr { C } _ { g }$ in an orthogonal coordinate system of a function $g$ defined and continuous on $\mathbb { R }$. The curve $\mathscr { C } _ { g }$ is symmetric with respect to the $y$-axis and lies in the half-plane $y > 0$.
For all $t \in \mathbb { R }$ we define: $$G ( t ) = \int _ { 0 } ^ { t } g ( u ) \mathrm { d } u$$
Part A
The justifications of the answers to the following questions may be based on graphical considerations.
- Is the function $G$ increasing on $[ 0 ; + \infty [$ ? Justify.
- Justify graphically the inequality $G ( 1 ) \leqslant 0.9$.
- Is the function $G$ positive on $\mathbb { R }$ ? Justify.
In the rest of the problem, the function $g$ is defined on $\mathbb { R }$ by $g ( u ) = \mathrm { e } ^ { - u ^ { 2 } }$.
Part B
- Study of $g$ a. Determine the limits of the function $g$ at the boundaries of its domain. b. Calculate the derivative of $g$ and deduce the table of variations of $g$ on $\mathbb { R }$. c. Specify the maximum of $g$ on $\mathbb { R }$. Deduce that $g ( 1 ) \leqslant 1$.
- We denote $E$ the set of points $M$ located between the curve $\mathscr { C } _ { g }$, the $x$-axis and the lines with equations $x = 0$ and $x = 1$. We call $I$ the area of this set. We recall that: $$I = G ( 1 ) = \int _ { 0 } ^ { 1 } g ( u ) \mathrm { d } u$$ We wish to estimate the area $I$ by the method called ``Monte-Carlo'' described below.
- We choose a point $M ( x ; y )$ by randomly drawing its coordinates $x$ and $y$ independently according to the uniform distribution on the interval $[ 0 ; 1 ]$. It is admitted that the probability that the point $M$ belongs to the set $E$ is equal to $I$.
- We repeat $n$ times the experiment of choosing a point $M$ at random. We count the number $c$ of points belonging to the set $E$ among the $n$ points obtained.
- The frequency $f = \frac { c } { n }$ is an estimate of the value of $I$. a. The figure below illustrates the method presented for $n = 100$. Determine the value of $f$ corresponding to this graph. b. The execution of the algorithm below uses the Monte-Carlo method described previously to determine a value of the number $f$. Copy and complete this algorithm. $f , x$ and $y$ are real numbers, $n , c$ and $i$ are natural integers. ALEA is a function that randomly generates a number between 0 and 1. \begin{verbatim} $c \leftarrow 0$ For $i$ varying from 1 to $n$ do: $x \leftarrow$ ALEA $y \leftarrow$ ALEA If $y \leqslant \ldots$ then $c \leftarrow \ldots$ end If end For $f \leftarrow \ldots$ \end{verbatim} c. An execution of the algorithm for $n = 1000$ gives $f = 0.757$. Deduce a confidence interval, at the 95\% confidence level, for the exact value of $I$.
Part C
We recall that the function $g$ is defined on $\mathbb { R }$ by $g ( u ) = \mathrm { e } ^ { - u ^ { 2 } }$ and that the function $G$ is defined on $\mathbb { R }$ by: $$G ( t ) = \int _ { 0 } ^ { t } g ( u ) \mathrm { d } u$$ We propose to determine an upper bound for $G ( t )$ for $t \geqslant 1$.
- A preliminary result. It is admitted that, for all real $u \geqslant 1$, we have $g ( u ) \leqslant \frac { 1 } { u ^ { 2 } }$. Deduce that, for all real $t \geqslant 1$, we have: $$\int _ { 1 } ^ { t } g ( u ) \mathrm { d } u \leqslant 1 - \frac { 1 } { t }$$
- Show that, for all real $t \geqslant 1$, $$G ( t ) \leqslant 2 - \frac { 1 } { t }$$ What can we say about the possible limit of $G ( t )$ as $t$ tends to $+ \infty$ ?