Examples of functions in real life
Transcription
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