2.4 Reading a Graph to Get Information About a Function 91
fromeasttowest.Weusenoonasourbenchmarktime; nooncorrespondstotime 0. Therefore time 2 is 10:00 a.m.
position position position position
—2 —1 1 2 t —2 —1 1 2 t —3 —2 —1 1 2 3 t —2 —1 1 2 t
TRIP I TRIP II TRIP III TRIP IV
Answer parts (a), (b), and (c) for each of the trips corresponding to graphs I, II, III, and IV.
(a) For what values of is velocity positive? When is travel from west to east? (b) For what values of is velocity negative? When is travel from east to west?
(c) To which trip do each of the following velocity graphs correspond? (Be sure your answer to part (c) agrees with your answers to parts (a) and (b).)
velocity velocity velocity velocity
—3 —2 —1 1 2 3 t —2 —1 1 2 t —3 —2 1 2 t —3 —2 1 2 t
A B C D
When you have completed your work on this problem, compare your answers with those of one of your classmates. If you disagree about an answer, each of you should discuss your reasoning and see if you can come to a consensus on the answers.
5. Look back at the Þgures for Problem 4. What characteristic of the graph of position versus time determines the sign of the velocity?
6. An ape with budding consciousness throws a bone straight up into the air from a height of 2 feet.15 From the seven graphs that follow pick out the one that could be the boneÕs
(a) height versus time, (b) velocity versus time, (c) speed versus time.
15 Problem by Eric Brussel.
92 CHAPTER 2 Characterizing Functions and Introducing Rates of Change
(a) (b) (c) (d)
t t t t
(e) (f) (g)
t t t
7. The velocity of an object is given in miles per hour by % 25 63 22 1 over the time interval 2 2, where is measured in hours. Use your graphing calculator to answer the following questions.
(a) Sketch a graph of the velocity function over the time interval 2 2.
(b) Approximately when does the object change direction? Please give answers that are off by no more than 0.05. (Either use the ÒzoomÓ feature of your calculator or change the domain until you can answer this question. If your calculator has an equationsolver, usethataswellandcomparetheanswersyouarriveatgraphically with the answers you get using the equation solver.)
(c) On the interval 2 2, approximately when is the object going the fastest? How fast is it going at that time? (Give your answer accurate to within 0.1.)
(d) When on the interval 0 2 is the velocity most negative? (Give an answer accuratetowithin0.1.)Whenyouzoominonthegraphhere,whatdoyouobserve?
8. The displacement of an object is given by 25 63 22 1 miles over the time interval 2 2 where is measured in hours.
(a) Approximately when does the object change direction? Please give answers accu-rate to within 0.1. When you zoom in on the graph here, what do you observe?
(b) Approximately when is the objectÕs velocity positive? Negative? (c) Approximate the objectÕs velocity at time 0.
9. Attime 0 threejoggersstartatthesameplaceandjogonastraightroadfor6miles. They all take 1 hour to complete the run. Jogger A starts out quickly and slows down throughoutthehour.JoggerBstartsoutslowlyandpicksupspeedthroughoutthehour. Jogger C runs at a constant rate throughout. On the same set of axes, graph distance traveled versus time for each jogger. Clearly label which curve corresponds to which jogger. Be sure your picture reßects all the information given in this problem.
2.4 Reading a Graph to Get Information About a Function 93
10. A baseball ÒdiamondÓ is actually a square with sides 90 feet long. Several of the fastest players in history have been said to circle the diamond in approximately 13 seconds.
(a) Sketch a plausible graph of speed as a function of time for such a dash around the bases. Label the point at which the player touches each of the bases on your graph. (Keep in mind that your player will probably need to slow down as he approaches each base in order to make the necessary 90-degree turn.)
(b) Sketch a graph of his acceleration as a function of time. Again, label the point at which he touches each base.
11. Before restrictions were placed on the distance that a backstroker could travel under-water in a race, Harvard swimmer David Berkoff set an American record for the event by employing the following strategy. In a 100-meter race in a 50-meter pool, Berkoff wouldswimmostoftheÞrst50metersunderwater(wherethedrageffectofturbulence was lower) then come up for air and swim on the waterÕs surface (at a slightly lower speed) until the turn. He would then use a similar approach to the second 50 meters, but could not stay underwater as long due to the cumulative oxygen deprivation caused by the time underwater.
Assume that Berkoff is swimming a 100-meter race in HarvardÕs Blodgett pool (which runs 50 meters east to west). He starts on the east end, makes the 50-meter turn atthewestend,andÞnishestheraceattheeastend.Sketchagraphofhisvelocity,taking east-to-west travel to have a positive velocity and west-to-east a negative velocity.
12. Below are graphs of position versus time corresponding to three trips. To be realistic the graphs ought to be drawn with smooth curves; to make things simpler the situation is approximated by a model using straight lines. The trips are all taken along the Massachusetts Turnpike (Route 90) a road running east-west. Positions are given relative to the town of Sturbridge. Positive values of position indicate that we are east of Sturbridge, and negative values indicate positions west of Sturbridge. Thus, positive velocity indicates that we are traveling from west to east; negative velocity indicates that we are traveling from east to west. For each trip do the following:
position (in miles)
position (in miles)
position (in miles)
40 (3, 40)
1 2 3 4 5
1 2 3 4 5
(in hours) 0
1 2 3
94 CHAPTER 2 Characterizing Functions and Introducing Rates of Change
(a) Describe the trip in words. Include where the trip started and ended and how fast (and in what direction) we traveled.
(b) Graph velocity as a function of time.
(c) Graph speed as a function of time. (Note: Speed is always nonnegative (zero or positive); velocity may be zero, positive, or negative depending on direction.)
13. The annual Ironman triathlon held in Hawaii consists of a 2.4-mile swim followed by a 112-milebicycleride,andÞnallya26.2-milerun.Oneentrantcanswimapproximately 3 miles per hour, bike approximately 18 miles per hour, and run about 9 miles per hour. In addition, during each portion of the event, she slows down toward the end as she gets tired. Sketch a possible graph of her distance as a function of time.
14. Below is a graph that gives information about a boat trip. The boat is traveling on a narrowriver.Thetripbeginsat7:00a.m.ataboathouse.Toberealisticthegraphought to be drawn with smooth curves; to make things simpler the situation is approximated by a model using straight lines.
km. from boathouse
B (5.5, 225)
0 1 2 3 4 5 6
time 7 8
(in hrs. past 7:00 am)
(a) How fast is the boat going between 7:00 a.m. and 8:00 a.m.?
(b) At what time do you think that the boaters stopped to go Þshing? (c) What happens after they Þsh?
(d) How can you tell that the boat is going at a steady speed between 9:00 a.m. and 11:30 a.m.?
(e) How fast is the boat going between 10 a.m. and 11 a.m.?
(f) Using the information given above, sketch a graph to show how the speed of the boat varies with time. Label your vertical axis speed (in kilometers per hour) and your horizontal axis time (in hours past 7:00 a.m.). Consider the portion of the trip beginning at 7:00 a.m. and ending at 2:00 p.m.
2.5 The Real Number System: An Excursion 95
2.5 THE REAL NUMBER SYSTEM: AN EXCURSION
The Development of the Real Number System
The concept of the real number system did not emerge fully formed from the ancient world; ithadaverylongandtumultuousgestationperiodovertensofthousandsofyears,beginning with early counting systems.16 Tallying systems using notches in bones and sticks (or knots in ropes) gave way to symbolic number systems, beginning with counting numbers: 1, 2, 3, . . . .WehavetenÞngerstocounton; theancientEgyptians,17 alongwithpeopleofmany other ancient civilizations, counted by tensÑas we do today. The Babylonians, on the other hand,hadawell-developedplace-valuenumbersystembasedon60.18 Welloverathousand years elapsed between the use of the BabyloniansÕ symbolic number system and the Þrst evidence of the use of zero. Zero was Þrst introduced not as a number in its own right but simply as a positional place-holder.19 It took about another thousand years for zero to gain acceptance as a number.
The Chinese mathematician Liu Hui used negative numbers around 260 a.d.20, but the famous Arab algebraists of the 800s such as al-Khowarizmi, whose work laid the foundations of Western EuropeÕs understanding of algebra, avoided negative numbers. While the Hindu mathematician Bhaskara put negative and positive roots of equations on equal footing in the early 1100s, Europeans were skeptical.21 As late as the mid-1500s, when the Italian mathematicians Cardan and Tartaglia were battling over the solution of cubic equations, Cardan, while farsighted in terms of recognizing negative roots, referred to them as Þctitious.22
The use of fractions dates back to the ancient Egyptians, although their fractions were alwaysreciprocalsofcountingnumbers(14,butnot34).Positiverationalnumbersgained acceptance early, but irrational numbers were a cause of great consternation for well over a millennium. The Greek Pythagorean school (around 530 b.c.) held the mystical belief that the universe is governed by ratios of positive integers. Yet it was Pythagoras (or one of his followers) who proved the Pythagorean Theorem, which tells us that the diagonal of a square with sides 1 has length 2. Not only that, but the Pythagoreans proved that 2 is irrational: that is, they proved that 2 cannot be expressed in the form ! where ! and are integers. One possibly apocryphal story says that the Pythagoreans punished with
16 For some fascinating details and a very readable account, consult The History of Mathematics: An Introduction, by David Burton, McGraw-Hill Companies, Inc. 1997.
17 Due in part to their methods of record keeping and their hot dry climate, the ancient Egyptians and Babylonians have left modern historians more evidence of their mathematical development than have the people of other ancient civilizations. The writings of the Greek historian Herodotus (around 450 b.c.) have helped establish records of life in this part of the world. The climate of regions such as China and India has contributed to the disintegration of evidence. In addition, much destruction of ancient work throughout the world has been deliberately carried out in the course of political and religious crusades.
18 Perhaps the base of 60 was selected because it has so many proper divisors (a position advocated by Theon of Alexandria, father of Hypatia, the Þrst famous woman mathematician), or because a year was thought to be 360 days. For more information abouttheBabyloniansystem,seeBurtonÕsbook,Section1.3,orreadHowardEvesÕAnIntroductiontotheHistoryofMathematics, Saunders College Publishing, 1990.
19 The Mayans used zero in this manner in the Þrst century a.d. (The Story of Mathematics, by Lloyd Motz and Jefferson Weaver, Plenum Press, 1993, p. 33). Circa 150 a.d. the astronomer Ptolemy used the symbol ÒoÓ as a place-marker in his work. ThesymbolcamefromtheÞrstletteroftheGreekwordforÒnothingÓ(Burton, p.23).TheHindususedadotasazeroplace-holder in the Þfth century a.d.; this dot later metamorphosized into a small circle. Arabic uses a dot for zero.
20 Burton, pp. 157-58. 21 Burton, p. 173.
22 Burton, p. 294.