Consider the function $f$ defined on $\mathbb { R }$ by $$f ( x ) = x \mathrm { e } ^ { - x ^ { 2 } + 1 }$$ Let ( $\mathscr { C }$ ) denote the representative curve of $f$ in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ).
a. Show that for every real $x$, $f ( x ) = \frac { \mathrm { e } } { x } \times \frac { x ^ { 2 } } { \mathrm { e } ^ { x ^ { 2 } } }$. b. Deduce the limit of $f ( x )$ as $x$ tends to $+ \infty$.
For every real $x$, consider the points $M$ and $N$ on the curve $( \mathscr { C } )$ with abscissae $x$ and $- x$ respectively. a. Show that the point O is the midpoint of the segment $[ M N ]$. b. What can be deduced about the curve $( \mathscr { C } )$ ?
Study the variations of the function $f$ on the interval $[ 0 ; + \infty [$.
a. Show that the equation $f ( x ) = 0.5$ admits on $[ 0 ; + \infty [$ exactly two solutions denoted $\alpha$ and $\beta$ (with $\alpha < \beta$ ). b. Deduce the solutions on $[ 0 ; + \infty [$ of the inequality $f ( x ) \geqslant 0.5$. c. Give an approximate value to $10 ^ { - 2 }$ of $\alpha$ and $\beta$.
Let $A$ be a strictly positive real number. Set $I _ { A } = \int _ { 0 } ^ { A } f ( x ) \mathrm { d } x$. a. Justify that $I _ { A } = \frac { 1 } { 2 } \left( \mathrm { e } - \mathrm { e } ^ { - A ^ { 2 } + 1 } \right)$. b. Calculate the limit of $I _ { A }$ as $A$ tends to $+ \infty$.
As illustrated in the graph below, we are interested in the shaded part of the plane which is bounded by:
the curve $( \mathscr { C } )$ on $\mathbb { R }$ and the curve $\left( \mathscr { C } ^ { \prime } \right)$ symmetric to $( \mathscr { C } )$ with respect to the x-axis;
the circle with centre $\Omega \left( \frac { \sqrt { 2 } } { 2 } ; 0 \right)$ and radius 0.5 and its symmetric with respect to the y-axis.
It is admitted that the disk with centre $\Omega \left( \frac { \sqrt { 2 } } { 2 } ; 0 \right)$ and radius 0.5 and its symmetric with respect to the y-axis are situated entirely between the curve ( $\mathscr { C }$ ) and the curve ( $\mathscr { C } ^ { \prime }$ ). Determine an approximate value in square units to the nearest hundredth of the area of this shaded part of the plane.
Consider the function $f$ defined on $\mathbb { R }$ by
$$f ( x ) = x \mathrm { e } ^ { - x ^ { 2 } + 1 }$$
Let ( $\mathscr { C }$ ) denote the representative curve of $f$ in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ).
\begin{enumerate}
\item a. Show that for every real $x$, $f ( x ) = \frac { \mathrm { e } } { x } \times \frac { x ^ { 2 } } { \mathrm { e } ^ { x ^ { 2 } } }$.\\
b. Deduce the limit of $f ( x )$ as $x$ tends to $+ \infty$.
\item For every real $x$, consider the points $M$ and $N$ on the curve $( \mathscr { C } )$ with abscissae $x$ and $- x$ respectively.\\
a. Show that the point O is the midpoint of the segment $[ M N ]$.\\
b. What can be deduced about the curve $( \mathscr { C } )$ ?
\item Study the variations of the function $f$ on the interval $[ 0 ; + \infty [$.
\item a. Show that the equation $f ( x ) = 0.5$ admits on $[ 0 ; + \infty [$ exactly two solutions denoted $\alpha$ and $\beta$ (with $\alpha < \beta$ ).\\
b. Deduce the solutions on $[ 0 ; + \infty [$ of the inequality $f ( x ) \geqslant 0.5$.\\
c. Give an approximate value to $10 ^ { - 2 }$ of $\alpha$ and $\beta$.
\item Let $A$ be a strictly positive real number. Set $I _ { A } = \int _ { 0 } ^ { A } f ( x ) \mathrm { d } x$.\\
a. Justify that $I _ { A } = \frac { 1 } { 2 } \left( \mathrm { e } - \mathrm { e } ^ { - A ^ { 2 } + 1 } \right)$.\\
b. Calculate the limit of $I _ { A }$ as $A$ tends to $+ \infty$.
\item As illustrated in the graph below, we are interested in the shaded part of the plane which is bounded by:
\begin{itemize}
\item the curve $( \mathscr { C } )$ on $\mathbb { R }$ and the curve $\left( \mathscr { C } ^ { \prime } \right)$ symmetric to $( \mathscr { C } )$ with respect to the x-axis;
\item the circle with centre $\Omega \left( \frac { \sqrt { 2 } } { 2 } ; 0 \right)$ and radius 0.5 and its symmetric with respect to the y-axis.
\end{itemize}
It is admitted that the disk with centre $\Omega \left( \frac { \sqrt { 2 } } { 2 } ; 0 \right)$ and radius 0.5 and its symmetric with respect to the y-axis are situated entirely between the curve ( $\mathscr { C }$ ) and the curve ( $\mathscr { C } ^ { \prime }$ ).\\
Determine an approximate value in square units to the nearest hundredth of the area of this shaded part of the plane.
\end{enumerate}