\documentclass[12pt]{article}
\usepackage{ammaths}
\begin{document}
\module{Examples of functions in real life}{01}{07}{1 period}
%\prereq{}
\object{\begin{itemize}
\item Discover the concept of function.
\item See that functions appear everywhere in physical phenomenons and human activities.
\end{itemize}}
\mater{\begin{itemize}
\item Four different texts about a practical example of function.
\item Slideshow with the four formulas and graphs.
\end{itemize}}
\modpart{Team work on one document}{25 mins}
Working in teams of four or five people, students work on a text about a practical example of function. Each team has to
prepare a presentation showing
\begin{itemize}
\itemb in what field the function appears ;
\itemb what it used for ;
\itemb the formula of the function ;
\itemb the graph of the function, with precise information on the axes.
\end{itemize}
\modpart{Oral presentations}{30 mins}
Each team goes to the board to present their function to the class.
\pagebreak
\moddocdis{Examples of functions in real life}{01}{07}{Document 1}
{\small\sf Below is a text introducing a practical example of a \emph{function}. Read the text carefully and then,
working as a team, prepare a presentation to explain to the class
\begin{itemize}
\itemb in what field the function appears ;
\itemb what it is used for ;
\itemb the formula of the function ;
\itemb how the graph of the function is drawn.
\end{itemize}
You may use your calculator to try out a few computations.}
\hrulefill
Any sound is a vibration, a series of variations of pressure, emitted by a source, transmitted in the air and received
by our ear. Musical sounds are periodic vibrations, whereas noise is made of random vibrations.
A pure sound is a simple, sinusoidal vibration. The number of actual vibrations during one second is called the
\emph{frequency}. The greater the frequency, the higher the sound. For example, the A (la) above middle C (do) on a
piano its usually set to a frequency of 440 Hz, which means $440$ vibrations in one second.
Of all musical instruments, only synthesisers can produce pure sounds. Other instruments, such as the guitar or the
violin, produce a mix of different pure sounds : a complex sound.
Let's look more closely at the example of the flute. When playing the note A with a 220 Hz frequency, a flute emits in
fact three simultaneous pure sound : the main one with a frequency of 220 Hz, a second one, more difficult to hear, with
a frequency of 440 Hz and the third one, easier to hear than the second but still behind the main sound, a frequency of
660Hz. The resulting sound can be expressed as a function $s$, where the variations of pressure $s(t)$ are given as a
function of the time $t$ in seconds :
$$s(t)=1\sin(220t)+0.1\sin(440t)+0.4\sin(660t).$$
The graph of this function is shown below :
{\scriptsize
\begin{center}
\psset{xunit=100.0cm,yunit=2cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-0.06,-1.32)(0.06,1.51)
\psaxes[xAxis=true,yAxis=true,Dx=0.01,Dy=0.5,ticksize=-2pt 0,subticks=2]{->}(0,0)(-0.05,-1.32)(0.05,1.51)
\psplot[plotpoints=200]{-0.04621451104100947}{0.0473186119873817}{sin(220*x)+0.1*sin(440*x)+0.4*sin(660*x)}
\end{pspicture*}
\end{center}}
\pagebreak
\moddocdis{Examples of functions in real life}{01}{07}{Document 2}
{\small\sf Below is a text introducing a practical example of a \emph{function}. Read the text carefully and then,
working as a team, prepare a presentation to explain to the class
\begin{itemize}
\itemb in what field the function appears ;
\itemb what it is used for ;
\itemb the formula of the function ;
\itemb how the graph of the function is drawn.
\end{itemize}
You may use your calculator to try out a few computations.}
\hrulefill
An \emph{income tax} is a tax levied by the government over the income of individuals, organisations or companies. In
France, the income tax paid by each family depends on the family quotient, which is equal to the net income of the
family (the total income minor various deductions) divided by the number of people in the family (children being counted
as 0.5 most of the time).
For a single person, the family quotient is therefore equal to the net income. The rule to compute the income tax $T$ in
this case is given by the government as the following formula, where the net income is $I$ :
$$T(I)=\left\{
\begin{array}{ll}
0 & \textrm{ if } I\leq 5687\\
I\times 0.055-312.79 & \textrm{ if } 5688\leq I\leq 11344\\
I\times 0.14-1277.03 & \textrm{ if } 11345\leq I\leq 25195\\
I\times 0.30-5308.23 & \textrm{ if } 25196\leq I\leq 67546\\
I\times 0.40-12062.83 & \textrm{ if } I>67546
\end{array}
\right.$$
This function can be graphed as shown below :
{\scriptsize
\begin{center}
\psset{xunit=1,yunit=1}
\begin{pspicture}(0,0)(10,6)
\psline(0,0)(0.5688,0)(1.1344,0.06)(2.5195,0.45)(6.7546,2.99)(10.0000,5.58)
\psline[linestyle=dashed](0.5688,0)(0.5688,0)
\psline[linestyle=dashed](1.1344,0)(1.1344,0.06)
\psline[linestyle=dashed](2.5195,0)(2.5195,0.45)
\psline[linestyle=dashed](6.7546,0)(6.7546,2.99)
\uput[d](0.5688,0){\rotatebox{90}{\scriptsize $5688$}}
\uput[d](1.1344,0){\rotatebox{90}{\scriptsize $11344$}}
\uput[d](2.5195,0){\rotatebox{90}{\scriptsize $25195$}}
\uput[d](6.7546,0){\rotatebox{90}{\scriptsize $67546$}}
\psdots[dotstyle=+](0,0.5)(0,1)(0,1.5)(0,2)(0,2.5)(0,3)(0,3.5)(0,4)(0,4.5)(0,5)(0,5.5)
\uput[l](0,0.5){$2000$}
\uput[l](0,1){$4000$}
\uput[l](0,1.5){$6000$}
\uput[l](0,2){$8000$}
\uput[l](0,2.5){$10000$}
\uput[l](0,3){$12000$}
\uput[l](0,3.5){$14000$}
\uput[l](0,4){$16000$}
\uput[l](0,4.5){$18000$}
\uput[l](0,5){$20000$}
\uput[l](0,5.5){$22000$}
\psline{->}(0,0)(10.5,0)
\psline{->}(0,0)(0,6)
\end{pspicture}
\end{center}}
\pagebreak
\moddocdis{Examples of functions in real life}{01}{07}{Document 3}
{\small\sf Below is a text introducing a practical example of a \emph{function}. Read the text carefully and then,
working as a team, prepare a presentation to explain to the class
\begin{itemize}
\itemb in what field the function appears ;
\itemb what it is used for ;
\itemb the formula of the function ;
\itemb how the graph of the function is drawn.
\end{itemize}
You may use your calculator to try out a few computations.}
\hrulefill
A volleyball is thrown into the air by a player. During its \emph{trajectory}, before it reaches the ground or another
player, the position of a volleyball can described by three coordinates, the abscissa $x$, the ordinate $y$ and the
altitude $z$ They all depend on the time $t$ elapsed since the moment the ball was thrown.
To simplify the problem, we neglect air friction, and we also suppose that the ball doesn't move sideways, so that the
coordinate $y$ is constant, always equal to $0$.
In these conditions, the trajectory depends only on three parameters : the initial speed of the ball, which is related
to the strength and technique of the player throwing it, the initial height of the ball, and the angle between the
ground and the direction of the ball when it is thrown. Suppose that the initial speed is $v_0=12$ meters per second,
the initial height is $2$ meters and the angle is $\alpha=30^\circ$. Then, the altitude $z$ of the ball as a function of
the abscissa $x$, is given by the formula $$z(x)=-0.04x^2+0.58x+2.$$
The graph of this function, shown below, gives a graphical representation of the trajectory, the ground being the
$x$-axis.
{\scriptsize
\begin{center}
\psset{xunit=0.5cm,yunit=1.0cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-1,-0.84)(19,4.68)
\psaxes[xAxis=true,yAxis=true,Dx=2,Dy=1,ticksize=-2pt 0,subticks=2]{->}(0,0)(-0.43,-0.84)(18.51,4.68)
\psplot[plotpoints=200]{0}{17.38}{-0.04*x^2+0.58*x+2}
\rput[bl](-1.62,0.31){$z$}
\end{pspicture*}
\end{center}
}
\pagebreak
\moddocdis{Examples of functions in real life}{01}{07}{Document 4}
{\small\sf Below is a text introducing a practical example of a \emph{function}. Read the text carefully and then,
working as a team, prepare a presentation to explain to the class
\begin{itemize}
\itemb in what field the function appears ;
\itemb what it is used for ;
\itemb the formula of the function ;
\itemb how the graph of the function is drawn.
\end{itemize}
You may use your calculator to try out a few computations.}
\hrulefill
\emph{Radioactive decay} is the process in which an unstable atomic nucleus loses energy by emitting ionising particles
and radiation. This decay, or loss of energy, results in an atom of one type, called the parent nuclide, transforming to
an atom of a different type, called the daughter nuclide. For example: a lead-214 atom (the ``parent'') emits radiation
and transforms to a bismuth-214 atom (the ``daughter''). This is a random process on the atomic level, in that it is
impossible to predict when a given atom will decay, but given a large number of similar atoms, the decay rate, on
average, is predictable.
The \emph{half-life} of a radioactive nuclide is the time it takes for half the quantity to transform. For the lead-214
atom, it's approximately $27$ minutes, or $1620$ seconds. This means that if at one moment we have $1$kg of lead-214,
then $27$ minutes later there will only be $500$g left, the other $500$g having decayed. Another $27$ minutes later,
only
$25$g will be left.
The number $N$ of atoms of lead-214 therefore depends on the time $t$ since it first appeared (for example when it's
generated by another radioactive substance, such as the polonium-218). If the initial weight of lead-214 is $1$kg, then
the weight after $t$ seconds is precisely given by the formula
$$N(t)=2^{-\frac{t}{1620}}.$$
The graph of this function is given below.
{\scriptsize
\begin{center}
\psset{xunit=0.0010cm,yunit=4.0cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-1000,-0.15)(10504.51,1.05)
\psaxes[xAxis=true,yAxis=true,Dx=1000,Dy=0.2,ticksize=-2pt 0,subticks=2]{->}(0,0)(-512.89,-0.07)(10504.51,1.05)
\psplot[plotpoints=200]{0}{10504.51037605458}{2^(-x/1620)}
\rput[bl](-247.87,1.16){$N$}
\end{pspicture*}
\end{center}}
\end{document}