Examining Applied Rational Functions

Transcription

1 HiMAP Pull-Out Setion: Summer 1990 Eamining Applied Rational Funtions Flod Vest Referenes Environmental Protetion Agen. Gas Mileage Guide. (Copies an usuall e otained from a loal new ar dealer.) Information Please Almana New York: Simon and Shuster. United States, Motor Vehile Information and Cost Savings At U.S. Code, Title 15, Setion Vest, Flod. Fall "Driving Speed vs. Fuel Effiien," The AMATYC Journal. Vest, Flod. Jan "Seondar Shool Mathematis from the EPA Gas Mileage Guide," Mathematis Teaher: Wagner, Clifford. Fe "Determining Fuel Consumption An Eerise in Applied Mathematis," Mathematis Teaher:

2 HiMAP Pull-Out Setion: Summer Rational funtions our in amazing plaes. The even our in the area of omparing the fuel osts for automoiles. Some people plae great importane on inreased fuel effiien and the resultant redued fuel osts. Just how muh mone is saved improved gas mileage? For eample if gasoline osts $1.00 per gallon and ou drive 15,000 miles per ear, how muh would ou save improving gas mileage from 10 mpg to 20 mpg? The answer would e 15,000 (1.00) 15,000 (1.00) = $750 per ear Over a period of ten ears ou would save $7500. = 15,000 Would ou save as muh improving fuel effiien from 20 to 30 mpg? Let us onsider the general prolem implied here. Assume gasoline osts $1.00 per gallon and a person drives 15,000 miles per ear, and = annual savings in fuel osts from improved gas mileage from mpg to ( + 10) mpg. The equation for would e , whih simplifies to = 150, We have here a rational funtion. Atuall, a rational funtion is of the form = f()/g() where f() and g() are polnomials. A polnomial is an epression of the form a n n +a n 1 n a 1 + a 0, where n is a nonnegative integer and eah oeffiient a k is a real numer. Thus, for aove, f() = 150,000 and g() = ( + 10) = are ver simple polnomials. Rational funtions have eoti graphs. The an do suh things as "going off to infinit." The ma disappear at ertain plaes. The often have something alled asmptotes. As ou an see, suh eoti funtions an arise from real appliations. 150,000 Let us graph = for oth positive and negative values of. ( + 10) (Even though an analst's primar interest ma e in positive values of, there are times when he or she needs to eamine the funtion for negative values also.) We will alulate a few points programming a sientifi alulator to alulate the -values. You an see the graph is somewhat involved. We have dotted in a referene line at = 10. Graphs of rational funtions do interesting things. Eamining the graph, we see that the urve does not eist at = 0 and = 10. Looking at the equation 150,000 =, we see that the denominator is zero at = 0, ( + 10) and at = 10, and thus is undefined for these values. You an also see that the graph gets lose to the -ais as inreases to the "right." We sa that the graph approahes the -ais asmptotiall from aove as goes to infinit. Atuall the urve will never ross or touh the -ais as gets large, ut instead it alwas gets loser to the -ais. We areviate this writing, 0 +." This is read, "as goes to infinit, approahes the -ais from aove."

3 HiMAP Pull-Out Setion: Summer 1990 "The pressure of a gas at a onstant temperature varies inversel as the volume." Bole's Law 2 Looking at the urve where " goes to the left," ou an see that the graph approahes the -ais from aove. We areviate this 0 +." We sa that the urve approahes the -ais asmptotiall from aove, and that the -ais is an asmptote. You an now see that the -ais is also an asmptote. As approahes 0 from the "right," the urve runs alongside the -ais and "goes up." We write 0 + ;" whih is read, "as approahes 0 from the positive side, goes off to positive infinit." What does the urve do as approahes zero from the left? You an see that it goes down eside the -ais. We write The -ais is a vertial asmptote. Can ou find another vertial asmptote for the graph? You will oserve that 10 eause as approahes 10 from the "right," dereases rapidl. What happens 10, that is, as approahes 10 from the left? The urve runs up eside the = 10 asmptote, approahing it ut never touhing it and inreases positivel. So, +. Thus we see that rational funtions have eoti graphs. The do suh things as "going off to infinit." The ma disappear at ertain plaes. The an have asmptotes. The solutions to man applied prolems result in rational funtions. In our disussion of the graph, we have omitted the atual fuel effiien onsiderations. Eamine these doing You Tr Its #1 and #2. Let us eamine more losel the asmptoti ehavior of another rational funtion that is simpler than the one aove. This will give us some useful generalizations. Bole's Law from hemistr states that the pressure of a gas at a onstant temperature varies inversel as the volume. This means that = k / where k is a onstant. This is a rational funtion. If the units or properties of the gas were just right, the onstant k would e one. For purpose of simplifing the mathematis, let us assume that k = 1. This gives us the rational funtion = 1/. Let us onsider the graph of = 1/. If ou were to op and omplete the following tale and use it to plot some points, ou would get the following graph of = 1/ = On a piee of srath paper, op and omplete the following tale for = 1/ with values of ,000 10,000

4 3 HiMAP Pull-Out Setion: Summer 1990 Imagine these points eing plotted on our graph of = 1. Use this imagination to etend the graph so that as gets large, the urve approahes the -ais ( = 0) from aove. We areviate this 0 +," whih is read, "as goes to infinit, approahes 0 from the positive side." On a piee of srath paper, op and omplete the following tale for = 1 with values of ,000 Imagine these points eing plotted on the graph. Using this imagination, etend the graph so that as gets negativel small (goes to the left), the urve approahes the -ais ( = 0) from elow. We areviate this 0, whih is read, "as goes to negative infinit, approahes 0 from the negative side." Complete the following tale for Use our imagination to etend our graph so that as approahes 0 from the positive side (from the right), the urve goes up inreasingl fast eside the -ais. We write, 0 whih is read, "as approahes 0 from the positive side, goes to infinit." Complete the following tale for Using our imagination, etend our graph so that as approahes 0 from the negative side (from the left), the urve goes down inreasingl fast eside the -ais. To areviate this we write " whih is read, "as approahes 0 from the negative side, goes to negative infinit." From the aove disussion, the graph of = 1/ is (displaed at left). We an now form the following generalizations whih are useful in working with more omple rational funtions: , 0, Now let us stud the graph. We sa that the -ais is an asmptote. the urve approahes the -ais asmptotiall from aove. the urve approahes the -ais asmptotiall from elow. The -ais is also an asmptote. 0 +, the urve approahes the -ais asmptotiall. The urve gets loser and loser to the -ais ut does not touh it. The urve also approahes the -ais asmptotiall from the left 0. Do the You Tr Its #3 #7.

5 HiMAP Pull-Out Setion: Summer You Tr Its # 1 You Tr Its # 2 You Tr Its # 3 If gasoline osts $1.00 per gallon and ou drive 15,000 miles per ear, how muh would ou save inreasing gas mileage from 20 mpg to 30 mpg? Cop and omplete the following tale. Annual Savings from Improving Gas Mileage from mpg to + 10 mpg (Assuming gasoline osts $1.00 per gallon and one drives 15,000 miles/ear.) mpg (savings) Disuss this tale from the point of view of mone saved from improved gas mileage. You might give our own opinion of the omparison of the several advantages of small ars (more fuel effiient) and larger (less fuel effiient) ars. Consider suh items as fuel osts, power, omfort, safet, maintenane osts, and useful life of the ar. Do ou think that some people might put too muh importane on fuel effiien? Do ou think that ar uers are aware of the figures in this tale? Complete the steps elow and use the proedures and onepts in the aove disussion to graph = 1. Answer the questions with omplete sentenes. a d e Calulate and plot a few onvenient points and sketh the urve. What are the asmptotes? For what values of does the urve approah the ais from elow? For what values of does its approah the -ais from aove? Desrie the values of for whih the urve approahes the -ais asmptotiall. What is when = 0? Cop and omplete eah sentene. Build a tale of and values to illustrate eah sentene. 0 If If f g h i You Tr Its # 4 If ou, or a friend, or a teaher has a graphing alulator suh as a Casio 7000 or an HP28S, use it to see if the alulator an manage asmptotes graphing = 1/. Report the ode whih gives the graph and report on the results. If ou have a printer for a alulator, attah the printout to our work. You might prefer to do this projet with a mathematial software pakage on a omputer. You Tr Its # 5

6 5 HiMAP Pull-Out Setion: Summer 1990 You Tr Its #5 Amerians drive approimatel miles eah ear (as reported in Information Please Almana, 1977; p. 85). a d e f If m is the average gasoline mileage for all automoiles (in mpg), give an equation for g(m) whih is the national fuel onsumption in gallons for the ear. If m = 20 mpg, what is g(m)? Epress our answer in terms of units of illions of gallons. (Hint: 10 9 is one illion.) Eplain the meaning of g(20). Suppose m is 15 mpg this ear and that m will improve to 20 mpg in some future ear. How muh fuel will the nation e saving that ear? As we go from 20 to 25 mpg, will we save the same amount? Eplain. Graph g(m) for 10 m 45 in inrements of 5 with the g(m)-ais in units of illions of dollars. Write a disussion of the graph from several points of view. You Tr Its #6 Assume fuel osts $1.00 per gallon and the nation drives miles per ear and = annual savings (in units of illions of dollars) in fuel osts for improving mileage from m mpg to m + 5 mpg. a Write an equation for in terms of m. Use the equation and a alulator to op and omplete the following tale. You ma wish to program our alulator to do the alulations. m (in illions of dollars) Draw a graph whih shows how to read the figures in the aove tale from the graph ou drew in You Tr It #5. Write an eplanation of the graph. You Tr Its #7 a Graph = for all real numers. m(m + 5) Write a disussion of the graph from several points of view.

7 HiMAP Pull-Out Setion: Summer Some Answers to "You Tr Its" ,000 15,000 Savings from inreasing gas = (1.00) (1.00) = $250 mileage from 20 to 30 mpg (mpg) (savings) As gas mileage inreases, the savings resulting from inreasing gas mileage 10 mpg delines. The savings of $750 per ear from improvement from 10 mpg to 20 mpg might e onsidered some to e sustantial, ut the $50 per ear improvement from 50 mpg to 60 mpg might not e signifiant. 3 a d e This is the graph of = 1/. Your graph will e etter than ours sine ou laeled the units and aes. We alulated and printed our graph on a alulator. The -ais is horizontal with positive to the right. The -ais is vertial with positive upward. Eah tik represents one. the urve approahes the -ais from elow. the urve approahes the -ais from aove. 0 +, the urve approahes asmptotiall the negative half of the -ais from the right. 0, the urve approahes asmptotiall the positive half of the -ais from the left. When = 0, is undefined. f 5 a g (m) = /m. g (20) is the fuel onsumption for the nation for an average 20 mpg. g(20) = /20 = ( )/(2 10) = = 50 illion gallons a 3 =. m(m + 5) m (in illions of dollars) a Eah numeral on the leg of a "triangle" gives the annual savings in fuel osts from improving mileage an inrement of 5 mpg from the m-oordinate of the leg. As m, 0 +. As m 0 +,. As m 0,. As m 5 +,. As m 5,. As m, 0 +.