Examples of functions in real life

0
Season 01 • Episode 07 • Examples of functions in real life
Examples of functions in real life
Objectives :
• Disover the onept of funtion.
• See that funtions appear everywhere
Season
Episode
Time frame
01
07
1 period
in physial phenomenons and human ativities.
Materials :
• Four dierent texts about a pratial example of funtion.
• Slideshow with the four formulas and graphs.
1 – Team work on one document
25 mins
Working in teams of four or ve people, students work on a text about a pratial example
of funtion. Eah team has to prepare a presentation showing
•
•
•
•
in what eld the funtion appears ;
what it used for ;
the formula of the funtion ;
the graph of the funtion, with preise information on the axes.
2 – Oral presentations
Eah team goes to the board to present their funtion to the lass.
30 mins
Examples of functions in real life
Season
Episode
Document
01
07
Document 1
Below is a text introducing a practical example of a function. Read the text carefully
and then, working as a team, prepare a presentation to explain to the class
• in what field the function appears ;
• what it is used for ;
• the formula of the function ;
• how the graph of the function is drawn.
You may use your calculator to try out a few computations.
Any sound is a vibration, a series of variations of pressure, emitted by a soure, transmitted
in the air and reeived by our ear. Musial sounds are periodi vibrations, whereas noise
is made of random vibrations.
A pure sound is a simple, sinusoidal vibration. The number of atual vibrations during
one seond is alled the frequeny. The greater the frequeny, the higher the sound. For
example, the A (la) above middle C (do) on a piano its usually set to a frequeny of 440
Hz, whih means
440
vibrations in one seond.
Of all musial instruments, only synthesisers an produe pure sounds. Other instruments,
suh as the guitar or the violin, produe a mix of dierent pure sounds : a omplex sound.
Let's look more losely at the example of the ute. When playing the note A with a 220
Hz frequeny, a ute emits in fat three simultaneous pure sound : the main one with a
frequeny of 220 Hz, a seond one, more diult to hear, with a frequeny of 440 Hz and
the third one, easier to hear than the seond but still behind the main sound, a frequeny
of 660Hz. The resulting sound an be expressed as a funtion
pressure
s(t)
are given as a funtion of the time
t
s,
where the variations of
in seonds :
s(t) = 1 sin(220t) + 0.1 sin(440t) + 0.4 sin(660t).
The graph of this funtion is shown below :
1.0
0.5
−0.05 −0.04 −0.03 −0.02 −0.01
−0.5
−1.0
0.01
0.02
0.03
0.04
Examples of functions in real life
Season
Episode
Document
01
07
Document 2
Below is a text introducing a practical example of a function. Read the text carefully
and then, working as a team, prepare a presentation to explain to the class
• in what field the function appears ;
• what it is used for ;
• the formula of the function ;
• how the graph of the function is drawn.
You may use your calculator to try out a few computations.
An inome tax is a tax levied by the government over the inome of individuals, organisations or ompanies. In Frane, the inome tax paid by eah family depends on the family
quotient, whih is equal to the net inome of the family (the total inome minor various
dedutions) divided by the number of people in the family (hildren being ounted as 0.5
most of the time).
For a single person, the family quotient is therefore equal to the net inome. The rule
to ompute the inome tax
T
in this ase is given by the government as the following
formula, where the net inome is
I
:

0




 I × 0.055 − 312.79
I × 0.14 − 1277.03
T (I) =


I
× 0.30 − 5308.23



I × 0.40 − 12062.83
if
if
if
if
if
I 6 5687
5688 6 I 6 11344
11345 6 I 6 25195
25196 6 I 6 67546
I > 67546
This funtion an be graphed as shown below :
22000 +
20000 +
18000 +
16000 +
14000 +
12000 +
10000 +
8000 +
6000 +
4000 +
67546
25195
11344
5688
2000 +
Season
Episode
Document
Examples of functions in real life
01
07
Document 3
Below is a text introducing a practical example of a function. Read the text carefully
and then, working as a team, prepare a presentation to explain to the class
• in what field the function appears ;
• what it is used for ;
• the formula of the function ;
• how the graph of the function is drawn.
You may use your calculator to try out a few computations.
A volleyball is thrown into the air by a player. During its trajetory, before it reahes the
ground or another player, the position of a volleyball an desribed by three oordinates,
the absissa
x,
the ordinate
y
and the altitude
z
They all depend on the time
t
elapsed
sine the moment the ball was thrown.
To simplify the problem, we neglet air frition, and we also suppose that the ball doesn't
move sideways, so that the oordinate
y
is onstant, always equal to
0.
In these onditions, the trajetory depends only on three parameters : the initial speed
of the ball, whih is related to the strength and tehnique of the player throwing it, the
initial height of the ball, and the angle between the ground and the diretion of the ball
when it is thrown. Suppose that the initial speed is v0 = 12 meters per seond, the initial
◦
height is 2 meters and the angle is α = 30 . Then, the altitude z of the ball as a funtion
of the absissa
x,
is given by the formula
z(x) = −0.04x2 + 0.58x + 2.
The graph of this funtion, shown below, gives a graphial representation of the trajetory,
the ground being the
x-axis.
4
3
2
1
z
2
4
6
8
10
12
14
16
18
Season
Episode
Document
Examples of functions in real life
01
07
Document 4
Below is a text introducing a practical example of a function. Read the text carefully
and then, working as a team, prepare a presentation to explain to the class
• in what field the function appears ;
• what it is used for ;
• the formula of the function ;
• how the graph of the function is drawn.
You may use your calculator to try out a few computations.
Radioative deay is the proess in whih an unstable atomi nuleus loses energy by
emitting ionising partiles and radiation. This deay, or loss of energy, results in an atom
of one type, alled the parent nulide, transforming to an atom of a dierent type, alled
the daughter nulide. For example : a lead-214 atom (the parent) emits radiation and
transforms to a bismuth-214 atom (the daughter). This is a random proess on the
atomi level, in that it is impossible to predit when a given atom will deay, but given a
large number of similar atoms, the deay rate, on average, is preditable.
The half-life of a radioative nulide is the time it takes for half the quantity to transform.
For the lead-214 atom, it's approximately
if at one moment we have
1kg
27
minutes, or
of lead-214, then
27
1620
seonds. This means that
minutes later there will only be
500g
500g having deayed. Another 27 minutes later, only 25g will be left.
The number N of atoms of lead-214 therefore depends on the time t sine it rst appeared
left, the other
(for example when it's generated by another radioative substane, suh as the polonium218). If the initial weight of lead-214 is
1kg,
then the weight after
t
seonds is preisely
given by the formula
t
N(t) = 2− 1620 .
The graph of this funtion is given below.
1.0
0.8
0.6
0.4
0.2
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000