1.2 A CATALOG OF ESSENTIAL FUNCTIONS

Transcription

1 SETION. A ATALOG OF ESSENTIAL FUNTIONS. A ATALOG OF ESSENTIAL FUNTIONS V Pla he Video V EXAMPLE A Table liss he average carbon dioide level in he amosphere, measured in pars per million a Mauna Loa Observaor from 98 o. Use he daa in Table o find a model for he carbon dioide level. SOLUTION We use he daa in Table o make he scaer plo in Figure, where represens ime (in ears) and represens he O level (in pars per million, ppm). TABLE 37 O level O level Year (in ppm) Year (in ppm) Noice ha he daa poins appear o lie close o a sraigh line, so i s naural o choose a linear model in his case. Bu here are man possible lines ha approimae hese daa poins, so which one should we use? From he graph, i appears ha one possibili is he line ha passes hrough he firs and las daa poins. The slope of his line is and is equaion is or FIGURE Scaer plo for he average O level Equaion gives one possible linear model for he carbon dioide level; i is graphed in Figure FIGURE Linear model hrough firs and las daa poins Thomson Brooks-ole coprigh 7

2 SETION. A ATALOG OF ESSENTIAL FUNTIONS A compuer or graphing calculaor finds he regression line b he mehod of leas squares, which is o minimize he sum of he squares of he verical disances beween he daa poins and he line. Alhough our model fis he daa reasonabl well, i gives values higher han mos of he acual O levels. A beer linear model is obained b a procedure from saisics called linear regression. If we use a graphing calculaor, we ener he daa from Table ino he daa edior and choose he linear regression command. (Wih Maple we use he fi[leassquare] command in he sas package; wih Mahemaica we use he Fi command.) The machine gives he slope and -inercep of he regression line as m.559 b So our leas squares model for he O level is In Figure 3 we graph he regression line as well as he daa poins. omparing wih Figure, we see ha i gives a beer fi han our previous linear model FIGURE 3 The regression line V Pla he Video V EXAMPLE B Use he linear model given b Equaion o esimae he average O level for 987 and o predic he level for he ear. According o his model, when will he O level eceed 4 pars per million? SOLUTION Using Equaion wih 987, we esimae ha he average O level in 987 was This is an eample of inerpolaion because we have esimaed a value beween observed values. (In fac, he Mauna Loa Observaor repored ha he average O level in 987 was ppm, so our esimae is quie accurae.) Wih, we ge So we predic ha he average O level in he ear will be ppm. This is an eample of erapolaion because we have prediced a value ouside he region of observaions. onsequenl, we are far less cerain abou he accurac of our predicion. Using Equaion, we see ha he O level eceeds 4 ppm when Thomson Brooks-ole coprigh 7

3 SETION. A ATALOG OF ESSENTIAL FUNTIONS 3 Solving his inequali, we ge We herefore predic ha he O level will eceed 4 ppm b he ear 9. This predicion is somewha risk because i involves a ime quie remoe from our observaions. Time (seconds) TABLE Heigh (meers) EXAMPLE A ball is dropped from he upper observaion deck of he N Tower, 45 m above he ground, and is heigh h above he ground is recorded a -second inervals in Table. Find a model o fi he daa and use he model o predic he ime a which he ball his he ground. SOLUTION We draw a scaer plo of he daa in Figure 4 and observe ha a linear model is inappropriae. Bu i looks as if he daa poins migh lie on a parabola, so we r a quadraic model insead. Using a graphing calculaor or compuer algebra ssem (which uses he leas squares mehod), we obain he following quadraic model: 3 h (meers) 4 h h (seconds) FIGURE 4 Scaer plo for a falling ball FIGURE 5 Quadraic model for a falling ball In Figure 5 we plo he graph of Equaion 3 ogeher wih he daa poins and see ha he quadraic model gives a ver good fi. The ball his he ground when h, so we solve he quadraic equaion The quadraic formula gives s The posiive roo is 9.67, so we predic ha he ball will hi he ground afer abou 9.7 seconds. EXAMPLE D Skech he graph of he funcion f () 6. SOLUTION ompleing he square, we wrie he equaion of he graph as 6 3 Thomson Brooks-ole coprigh 7

4 4 SETION. A ATALOG OF ESSENTIAL FUNTIONS This means we obain he desired graph b saring wih he parabola and shifing 3 unis o he lef and hen uni upward (see Figure 6). (_3, ) _3 _ FIGURE 6 (a) = (b) =(+3)@+ EXAMPLE E Skech he graph of he funcion sin. SOLUTION We obain he graph of sin from ha of sin b compressing horizonall b a facor of (see Figures 7 and 8). Thus, whereas he period of sin is, he period of sin is. =sin =sin 4 FIGURE 7 FIGURE 8 EXAMPLE F Figure 9 shows graphs of he number of hours of daligh as funcions of he ime of he ear a several laiudes. Given ha Philadelphia is locaed a approimael 4 N laiude, find a funcion ha models he lengh of daligh a Philadelphia FIGURE 9 Graph of he lengh of daligh from March hrough December a various laiudes Hours N 5 N 4 N 3 N N Mar. Apr. Ma June Jul Aug. Sep. Oc. Nov. Dec. Source: Lucia. Harrison, Daligh, Twiligh, Darkness and Time (New York: Silver, Burde, 935) page 4. Thomson Brooks-ole coprigh 7

5 SETION. A ATALOG OF ESSENTIAL FUNTIONS 5 SOLUTION Noice ha each curve resembles a shifed and sreched sine funcion. B looking a he blue curve we see ha, a he laiude of Philadelphia, daligh lass abou 4.8 hours on June and 9. hours on December, so he ampliude of he curve (he facor b which we have o srech he sine curve vericall) is B wha facor do we need o srech he sine curve horizonall if we measure he ime in das? Because here are abou 365 das in a ear, he period of our model should be 365. Bu he period of sin is, so he horizonal sreching facor is c 365. We also noice ha he curve begins is ccle on March, he 8h da of he ear, so we have o shif he curve 8 unis o he righ. In addiion, we shif i unis upward. Therefore, we model he lengh of daligh in Philadelphia on he h da of he ear b he funcion L.8 sin Anoher wa o solve 4 : _ EXAMPLE G If f s and s4, find he funcions f, f, f, and f. SOLUTION The domain of f s is,. The domain of s4 consiss of all numbers such ha 4, ha is, 4. Taking square roos of boh sides, we ge, or, so he domain of is he inerval,. The inersecion of he domains of f and is,,, Thus, according o he definiions, we have f s s4 f s s4 f s s4 s4 3 f s s4 4 Noice ha he domain of f is he inerval, ; we have o eclude because. EXAMPLE H Find f h if f,, and h 3. SOLUTION f h f h f 3 f Thomson Brooks-ole coprigh 7