Exercise 1 -- Part A
Consider the sequence $\left( u _ { n } \right)$ defined for every natural integer $n$ by: $$u _ { n } = \int _ { 0 } ^ { n } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$ We will not attempt to calculate $u _ { n }$ as a function of $n$.
- a. Show that the sequence $(u_n)$ is increasing. b. Prove that for every real number $x \geqslant 0$, we have: $- x ^ { 2 } \leqslant - 2 x + 1$, then: $$\mathrm { e } ^ { - x ^ { 2 } } \leqslant \mathrm { e } ^ { - 2 x + 1 }$$ Deduce that for every natural integer $n$, we have: $u _ { n } < \frac { \mathrm { e } } { 2 }$. c. Prove that the sequence $(u_n)$ is convergent. We will not attempt to calculate its limit.
- In this question, we propose to obtain an approximate value of $u _ { 2 }$.
In the orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ below, we have drawn the curve $\mathscr{C}_f$ representing the function $f$ defined on the interval $[0;2]$ by $f(x) = \mathrm{e}^{-x^2}$, and the rectangle OABC where $\mathrm{A}(2;0)$, $\mathrm{B}(2;1)$ and $\mathrm{C}(0;1)$. We have shaded the region $\mathscr{D}$ between the curve $\mathscr{C}_f$, the horizontal axis, the vertical axis and the line with equation $x = 2$.
Consider the random experiment consisting of choosing a point $M$ at random inside the rectangle OABC. We admit that the probability $p$ that this point belongs to the region is: $p = \frac{\text{area of } \mathscr{D}}{\text{area of } \mathrm{OABC}}$. a. Justify that $u_2 = 2p$. b. Consider the following algorithm:
| L1 | Variables: $N, C$ integers; $X, Y, F$ real numbers |
| L2 | Input: Enter $N$ |
| L3 | Initialization: $C$ takes the value 0 |
| L4 | Processing: |
| L5 | For $k$ varying from 1 to $N$ |
| L6 | $X$ takes the value of a random number between 0 and 2 |
| L7 | $Y$ takes the value of a random number between 0 and 1 |
| L8 | If $Y \leqslant \mathrm{e}^{-X^2}$ then |
| L9 | $C$ takes the value $C + 1$ |
| L10 | End if |
| L11 | End for |
| L12 | Display $C$ |
| L13 | $F$ takes the value $C/N$ |
| L14 | Display $F$ |
i. What does the condition on line $L8$ allow us to test regarding the position of point $M(X;Y)$? ii. Interpret the value $F$ displayed by this algorithm. iii. What can we conjecture about the value of $F$ when $N$ becomes very large? c. By running this algorithm for $N = 10^6$, we obtain $C = 441138$.
We admit in this case that the value $F$ displayed by the algorithm is an approximate value of the probability $p$ to within $10^{-3}$. Deduce an approximate value of $u_2$ to within $10^{-2}$.
Part B
The sign, modeled by the region $\mathscr{D}$ defined in Part A, is cut from a rectangular sheet of 2 meters by 1 meter. It is represented in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$; the chosen unit is the meter.
For $x$ a real number belonging to the interval $[0;2]$, we denote:
- $M$ the point on the curve $\mathscr{C}_f$ with coordinates $(x; \mathrm{e}^{-x^2})$,
- $N$ the point with coordinates $(x; 0)$,
- $P$ the point with coordinates $\left(0; \mathrm{e}^{-x^2}\right)$,
- $A(x)$ the area of rectangle $ONMP$.
- Justify that for every real number $x$ in the interval $[0;2]$, we have: $A(x) = x\mathrm{e}^{-x^2}$.
- Determine the position of point $M$ on the curve $\mathscr{C}_f$ for which the area of rectangle $ONMP$ is maximum.
- The rectangle $ONMP$ of maximum area obtained in question 2. must be painted blue, and the rest of the sign in white. Determine, in $\mathrm{m}^2$ and to within $10^{-2}$, the measure of the surface to be painted blue and that of the surface to be painted white.