2.4 Using Raes of Change o Creae a Graphical Model YOU WILL NEED graphing calculaor or graphing sofware GOAL Represen verbal descripions of raes of change using graphs. LEARN ABOUT he Mah Today Seve walked o his par-ime job. As he sared walking, he sped up for 3 min. Then he walked a a consan pace for anoher 2 min. When he realized ha he would be early for work, he slowed down. His walk ended and he came o a complee sop once he reached his desinaion 10 min afer he sared.? Wha would he speed versus ime graph of Seve s walk o work look like? EXAMPLE 1 Represening he siuaion wih a graph Creae a speed versus ime graph for Seve s walk o work. Soluion A: Assuming ha he changed speed a a consan rae Because Seve was speeding up, his speed increased as ime increased. His speed increased a a consan rae, so he graph should be a sraigh line wih a posiive slope ha begins a (0, 0) and ends a x 3. Beween 3 min and min, Seve walked a he same rae, so his speed did no change. The graph should be a horizonal line ha connecs o he firs line. 0 2 4 6 8 10 (min) Afer min, Seve slowed down a a consan rae, decreasing his speed as ime increased, so you migh draw a sraigh line wih a negaive slope ha begins a x and ends a x 10. 96 2.4 Using Raes of Change o Creae a Graphical Model
2.4 Soluion B: Assuming ha he walked a a variable speed Because Seve was speeding up, his speed increased as ime increased. His speed increased a a variable rae, so you migh draw an increasing curve ha sars a (0, 0) and ends a x 3. Beween 3 min and min, Seve walked a he same rae, so his speed did no change. The graph should be a horizonal line ha connecs o he firs line. 0 2 4 6 8 10 (min) Afer min, Seve slowed down a a variable rae, decreasing his speed as ime increased, so you migh draw a decreasing curve ha begins a x and ends a x 10. Reflecing A. Which deails in he given descripion were mos imporan for deermining he shape of he graph? B. Are hese he only wo graphs ha could represen Seve s walk o work? Explain. APPLY he Mah EXAMPLE 2 Represening he siuaion wih a graph A flask, a beaker, and a graduaed cylinder are being filled wih waer. The rae a which he waer flows from he ap is he same when filling all hree conainers. Draw possible waer level versus ime graphs for he hree conainers. Chaper 2 97
Soluion Heigh (cm) Heigh (cm) Heigh (cm) Graduaed Cylinder (s) Beaker (s) Flask (s) Since he conainers are being filled wih waer, he heigh of he waer in he conainers increases as ime increases. All he graphs should be increasing curves. Boh he graduaed cylinder and he beaker have a consan diameer so he waer level increases a a consan rae. The waer level will rise he fases in he conainer wih he smalles diameer. The waer level in he graduaed cylinder increases faser han he waer level in he beaker, so he slope of he line for he graduaed cylinder mus be greaer han he slope of he line for he beaker. The diameer of he flask varies, so he waer level will increase a differen raes. As he waer level rises, he diameer of each crosssecion ges smaller, causing he waer level o increase more rapidly. So he graph mus be nonlinear. EXAMPLE 3 Using a graph o deermine he rae of change A cyclis is observed moving a a speed of 10 m>s. She begins o slow down a a consan rae and, 4 s laer, is a a speed of m>s. She coninues o slow down a a differen consan rae and finally comes o a sop 6 s laer. a) Skech a graph of speed versus ime. b) Wha is he average rae of change of he cyclis s speed in he firs 4 s? c) Esimae he insananeous rae of change in speed a 3 s. 98 2.4 Using Raes of Change o Creae a Graphical Model
2.4 Soluion a) 10 0 (m/s) (s) 10 The cyclis begins a 10 m>s and slows down a a consan rae. Place a poin a (0, 10) and anoher a (4, ), and connec he poins wih a sraigh line. Because he cyclis sops a 10 s, her speed is 0 m>s. Place anoher poin a (10, 0), and connec i o he previous poin wih a sraigh line. b) Average rae of change 2 10 4 2 0 21.2 The cyclis s speed is decreasing a he rae of 1.2 m>s 2. c) The equaion of he line for he firs par of he graph is y 2 4 x 1 10, since he slope is 2 and he 4 y-inercep is 10. When x 3.1, y 6.12. When x 2.9, y 6.37. Average rae of change Dy Dx 6.12 2 6.37 3.1 2 2.9 21.2 m>s 2 The insananeous rae of change is he same as he average rae of change calculaed in par b). This should no be surprising, since he angen line a x 3 has he same slope as he secan line on he inerval 0 # x # 4. To deermine he average rae of change of he cyclis s speed in he firs 4 s, calculae he slope of he secan line beween (0, 10) and (4, ). Subsiue x 3.1 and x 2.9 ino he equaion of he line. These values are close o x 3, bu on opposie sides of i. Deermine he corresponding y-values. Calculae he average rae of change beween hese poins, and esimae he insananeous rae of change based on your answer. Recall ha he rae of change of a linear funcion is consan, so he rae of change a 3 s will be same as he rae of change a any ime beween 0 s and 4 s. Chaper 2 99
EXAMPLE 4 Using reasoning o represen and analyze a siuaion Adam and his friend are esing a moion sensor. Adam sands 0. m in fron of he sensor and hen walks 4 m away from i a a consan rae for 10 s. Nex, Adam walks 1 m oward he sensor for s and hen wais here for anoher s. a) Draw a disance versus ime graph for Adam s moion sensor walk. b) Wha is he average rae of change in his disance in he firs 10 s? c) Wha are he insananeous raes of change a 1 s and 7 s? d) Wha is he average rae of change in he nex s? e) Wha are he insananeous raes of change a 12 s and 14 s? f) Wha is he insananeous rae of change a 18 s? g) Draw a speed versus ime graph for Adam s moion sensor walk. Soluion a) 4 3 2 1 Disance (m) The graph begins wih a sraigh line since he rae a which Adam walks is consan. The graph has a posiive slope since he walks away from he sensor, and his disance from he sensor increases as ime increases. Adam sars 0. m from he sensor. Use (0, 0.) as he disance inercep. Adam walks 4 m away from he sensor a a consan rae for 10 s, so use he poin (10, 4.). 0 2 4 6 8 10 12 14 16 1820 (s) Adam hen walks oward he sensor. The line has a negaive slope, because his disance from he sensor decreases as ime increases. The line is no very seep because he is walking slowly. The graph ends wih a horizonal line ha has a slope of 0 because Adam is no moving. The slope indicaes ha his disance from he sensor does no change. b) Average rae of change slope of secan 4. 2 0. 10 2 0 0.4 Adam s disance from he sensor is increasing, on average, by 0.4 m>s. Calculae he slope of he secan line beween he poins (0, 0.) and (10, 4.). 100 2.4 Using Raes of Change o Creae a Graphical Model
2.4 c) Insananeous rae of change slope of angen 0.4 Adam s disance from he sensor is increasing. He is moving away from he sensor a a rae of 0.4 m>s. Esimae he slopes of he angen lines a 1 s and 7 s. Boh of he angen lines have he same slope as he secan line since he graph is linear on he inerval 0 # # 10. d) Average rae of change slope of secan 3. 2 4. 1 2 10 20.2 Adam s disance from he sensor is decreasing. He is moving oward he sensor a a rae of 0.2 m>s. e) Insananeous rae of change slope of angen 20.2 Adam s disance from he sensor is decreasing by 0.2 m>s a 12 s and 14 s. Calculae he slope of he secan line beween he poins (10, 4.) and (1, 3.). Esimae he slope of he angen lines a 12 s and 14 s. As in par c), boh of hese angen lines have he same slope as he secan line since he graph is linear on he inerval 10, # 1. f) Insananeous rae of change slope of angen 0 Adam s disance from he sensor is no changing a 18 s. Esimae he slope of he angen line a 18 s. Again, he angen line has he same slope as he secan line since he graph is linear on he inerval 1, # 20. Since he line is horizonal, is slope is 0. g) Average speed (m/s) 0. 0.4 0.3 0.2 0.1 0 2 4 6 8 10 12 14 16 18 20 (s) There are hree differen speeds a which Adam walks, over hree differen inervals of ime. Using he previous calculaions, 0.4 m>s when 0 # # 10 0.2 m>s when 10, # 1 0 m>s when 1, # 20 Noe ha speed is a non-negaive quaniy. Dd ` D ` Chaper 2 101
In Summary Key Ideas In a problem ha involves movemen, a possible graph shows displacemen (disance, heigh, or deph) versus ime. Disance, heigh, or deph is he dependen variable, and ime is he independen variable. The rae of change in hese relaionships is speed: ` change in displacemen `, change in ime Dd S ` D ` On a displacemen (disance, heigh, or deph) versus ime graph, he magniude of he slope of a secan line represens he average speed on he corresponding inerval. The magniude of he slope of a angen line represens he insananeous speed a a specific poin. As a resul, observing how he slopes of angen lines change a differen poins on a graph gives you insigh ino how he speed changes over ime. Need o Know When he rae of change of displacemen (or speed) is consan: Displacemen Displacemen Displacemen An increasing line indicaes ha displacemen increases as ime increases. A decreasing line indicaes ha displacemen decreases as ime increases. A horizonal line indicaes ha here is no change in displacemen as ime increases. (coninued) 102 2.4 Using Raes of Change o Creae a Graphical Model
2.4 When he rae of change of displacemen (or speed) is variable, an increasing curve indicaes ha displacemen increases as ime increases. Displacemen The speed is increasing as he ime increases. or Displacemen The speed is decreasing as he ime increases. A decreasing curve indicaes ha displacemen decreases as ime increases. Displacemen The speed is decreasing as he ime increases. or Displacemen The speed is increasing as he ime increases. CHECK Your Undersanding 1. The following graphs show disance versus ime. Mach each graph wih he descripion given below. a) d b) d c) d A Disance is decreasing B There is no change in C Disance is increasing over ime. disance over ime. over ime. 2. Which of he graphs in quesion 1 show ha he speed is consan? Explain. 3. Jan sands m away from a moion sensor and hen walks 4 m oward i a a consan rae for s. Then she walks 2 m away from he locaion where she changed direcion a a variable rae for he nex 3 s. She sops and wais a his locaion for 2 s. Draw a disance versus ime graph o show Jan s moion sensor walk. Chaper 2 103
PRACTISING 4. Rachel climbed M. Fuji while in Japan. There are 10 levels o he mounain. She was able o drive o Level, where she began her climb. She walked a a consan rae for 40 min from Level o Level 6. She slowed slighly bu hen coninued a a consan rae for a oal of 90 min from Level 6 o Level 7. She decided o ea and res here, which ook approximaely 2 h. From Level 7 o Level 8, a 40 min rip, she ravelled a a consan rae. Coninuing on o Level 9, a 4 min rip, she decreased slighly o a new consan rae. During mos of he 1 h she ook o reach Level 10, he op of M. Fuji, she mainained a consan rae. As she neared he op, however, she acceleraed. a) Using he informaion given and he following able, draw an elevaion versus ime graph o describe Rachel s climb. Level 6 7 8 9 10 Elevaion (m) 2100 2400 2700 3100 3400 3740 b) Calculae Rachel s average speed over each par of her climb. c) Draw a speed versus ime graph o describe Rachel s climb.. The conainers shown are being filled wih waer a a consan rae. K Draw a graph of he waer level versus ime for each conainer. a) a 2 L plasic pop bole b) a vase 6. John is riding a bicycle a a consan cruising speed along a fla road. He slows down as he climbs a hill. A he op of he hill, he speeds up, back o his consan cruising speed on a fla road. He hen acceleraes down he hill. He comes o anoher hill and coass o a sop as he sars o climb. a) Skech a possible graph o show John s speed versus ime, and anoher graph o show his disance ravelled versus ime. b) Skech a possible graph of John s elevaion (in relaion o his saring poin) versus ime. 104 2.4 Using Raes of Change o Creae a Graphical Model
7. A swimming pool is 0 m long. Kommy swims from one end of he A pool o he oher end in 4 s. He ress for 10 s and hen akes s o swim back o his saring poin. a) Use he informaion given o find he average speed for Kommy s firs lengh of he pool. b) Wha is he average speed for Kommy s second lengh of he pool? c) If you were o graph Kommy s disance versus ime for his firs and second lenghs of he pool, how would he wo graphs compare? How is his relaed o Kommy s speed? d) Draw a disance versus ime graph for Kommy s swim. e) Wha is Kommy s speed a ime 0? f) Draw a speed versus ime graph for Kommy s swim. 2.4 8. The following graphs show speed versus ime. Mach each graph wih he descripion given below. a) s c) s b) s d) s A The rae a which he C The rae a which he speed speed increases is increasing decreases is decreasing as as ime increases. ime increases. B The rae a which he D The rae a which he speed speed increases is decreasing decreases is increasing as as ime increases. ime increases. 9. A jockey is warming up a horse. Whenever he jockey has he horse T accelerae or decelerae, she does so a a nonconsan rae a firs slowly and hen more quickly. The jockey begins by having he horse ro around he rack a a consan rae. She hen increases he rae o a caner and allows he horse o caner a a consan rae for several laps. Nex, she slowly begins o decrease he speed of he horse o a ro and hen o a walk. To finish, he jockey walks he horse around he rack once. Draw a speed versus ime graph o represen his siuaion. Chaper 2 10
10. a) Describe how you would walk oward or away from a moion sensor C deecor o give each disance versus ime graph shown below. i) ii) 8 d 8 d Disance (m) b) For each par of each graph, deermine he speed a which you mus walk. 11. A cross-counry runner is raining for a marahon. His raining program requires him o run a differen speeds for differen lenghs of ime. His program also requires him o accelerae and decelerae a a consan rae. Today he begins by jogging for 10 min a a rae of miles per hour. He hen spends 1 min acceleraing o a rae of 10 miles per hour. He says a his rae for min. He hen deceleraes for 1 min o a rae of 7 miles per hour. He says a his rae for 30 min. Finally, o cool down, he deceleraes for 2 min o a rae of 3 miles per hour. He says a his rae for a final 10 min and hen sops. a) Make a speed versus ime graph o represen his siuaion. b) Wha is he insananeous rae of change in he runner s speed a 10. min? c) Calculae he runner s average rae a which he changed speeds from minue 11 o minue 49. d) Explain why your answer for par c) does no accuraely represen he runner's raining schedule from minue 11 o minue 49. 12. Creae a scenario ha could be used o creae eiher a disance versus ime graph or a speed versus ime graph. Exchange your scenario wih a parner and creae he corresponding graph. Exending 6 4 2 0 2 4 6 8 (s) 0 2 4 6 8 (s) 13. Two women are running on he same rack. One has jus finished her workou and is deceleraing a firs slowly and hen more quickly as she comes o a complee sop. The oher woman is jus saring her workou and is acceleraing a firs quickly and hen more slowly as she reaches her arge speed. Use one graph o illusrae he raes of boh women. 14. A graph displays changes in rae of speed versus ime. The graph has sraigh lines from poin o poin. If he graph had been drawn o display changes in disance versus ime, how would i be differen? Disance (m) 6 4 2 106 2.4 Using Raes of Change o Creae a Graphical Model