average rate of change

Transcription

1 average rate of change Module 2 : Investigation 5 MAT 170 Precalculus August 31, 2016

2 question 1 A car is driving away from a crosswalk. The distance d (in feet) of the car from the crosswalk t seconds since the car started moving is given by the formula d = t (g) Using the above formula, determine the d when t = 0, 1, 1.25, 2, 2.6, 3, and 4 : t d (a) As the value of t increases from 1 second to 3 seconds, what is d? (c) True of False : The car travels at a constant speed as the value of t increases from 1 second to 3 seconds. Explain your reasoning. 2

3 question 1 - solutions (g) Using the above formula, determine the d when t = 0, 1, 2, 3, and 4 : t d (a) As the value of t increases from 1 second to 3 seconds, what is d? When t = 1, we have d = 4.5. When t = 3, we have d = Therefore, as t increases from 1 second to 3 seconds d = = 8. 3

4 question 1 - solutions (c) True of False : The car travels at a constant speed as the value of t increases from 1 second to 3 seconds. Explain your reasoning. FALSE. If the car was traveling at a constant rate, then for any two intervals of time between 1 second to 3 seconds of equal length, the car should travel the same distance. As t increases from 1 second to 2 seconds, the change in d is d = = 3 feet. As t increases from 2 seconds to 3 seconds, the change in d is d = = 5 feet. The car travelled further from 2 seconds to 3 seconds than it did from 1 second to 2 seconds, so the car could not have been traveling at a constant rate. 4

5 question 1 (continued) A car is driving away from a crosswalk. The distance d (in feet) of the car from the crosswalk t seconds since the car started moving is given by the formula d = t (b) Illustrate on a graph : (i) t increasing from 1 to 3 seconds (i.e. t) (ii) the change in d as t increases from 1 to 3 seconds (i.e. d) (d) Suppose a second car wants to cover the same distance during this same time interval, from t = 1 to t = 3 seconds, but wants to maintain a constant speed. On your graph from part (a), graph the distance this second car has traveled with respect to the time in seconds. (e) What does the slope of the line you drew in part (d) represent in the given situation? 5

6 question 1 (continued) - possible solutions (b) Illustrate on a graph : (i) t increasing from 1 to 3 seconds (i.e. t) 15 (ii) the change in d as t increases from 1 to 3 seconds (i.e. d) 10 d 5 t

7 question 1 (continued) - possible solutions (d) Suppose a second car wants to cover the same distance during this same time interval, from t = 1 to t = 3 seconds, but wants to maintain a constant speed. On your graph from part (a), graph the distance (in feet) this second car has traveled from t = 1 to t = t d

8 question 1 (continued) - possible solutions (e) What does the slope of the line you drew in part (d) represent in the given situation? 15 The slope is 4, which represents the constant speed needed to travel 8 feet in 2 seconds. 10 d 5 t

9 average speed Definition : Average Speed The constant speed needed to travel some specific distance in some specific amount of time. The distance between ASU and the University of Arizona is 109 miles. If I make this trip in 1.25 hours, then my average speed is = 87.2 miles per hour. In other words, in order to travel at a constant speed from ASU to the University of Arizona and get there in 1.25 hours, my constant speed would need to be 87.2 miles per hour. 9

10 question 2 The distance d (in feet) of a car north of an intersection t seconds after it started moving is given by the formula d = 2t (a) Let d 1 denote the distance (in feet) the car has traveled when the time is t 1 = 2 seconds. What is the value of d 1? (b) Let d 2 denote the distance (in feet) the car has traveled when the time is t 2 = 3.5 seconds. What is the value of d 2? (c) Explain what d 1 d 2 t 1 t 2 represents in the context of this situation. 10

11 question 2 - possible solutions (a) Let d 1 denote the distance (in feet) the car has traveled when the time is t 1 = 2 seconds. What is the value of d 1? d 1 = 2(2) = 11 (b) Let d 2 denote the distance (in feet) the car has traveled when the time is t 2 = 3.5 seconds. What is the value of d 2? d 2 = 2(3.5) = 27.5 (c) Explain what d 1 d 2 t 1 t 2 represents in the context of this situation. This represents the constant rate of change needed to travel d 2 d 1 = 16.5 feet in t 2 t 1 = 1.5 seconds. 11

12 average rate of change Definition : Average rate of change Suppose quantity y can be express in terms of the variable x, that is, we have y = some algebraic expression involving x The average rate of change of y with respect to x over any interval x 1 to x 2 is y 2 y 1 x 2 x 1, where y = y 1 when x = x 1, and y = y 2 when x = x 2. (Notice that x 2 x 1 = x and y 2 y 1 = y.) 12

13 average rate of change Graphically, the average rate of change represents the slope of the straight line connecting the two points (x 1, y 1 ) and (x 2, y 2 ). 13

14 question 4 Marcos traveled in his car from Phoenix to Flagstaff, a distance of 144 miles. (a) Determine the amount of time required for Marcos to travel from Phoenix to Flagstaff if he drove at a constant speed of 64 miles per hour. (b) Construct a graph to represent Marcos distance from Phoenix in terms of time (in hours) since he left Phoenix. (c) Joni left Phoenix and arrived in Flagstaff at exactly the same times as Marcos, traveling the same route, but she did not travel at a constant rate. (i) True or False : Joni covered the same distance in the same amount of time as Marcos. Explain your reasoning. (ii) On your graph from part (b), construct a possible graph to represent Joni s distance travelled in miles in terms of the number of hours since she left Phoenix. 14

15 question 4 - possible solutions (a) Determine the amount of time required for Marcos to travel from Phoenix to Flagstaff if he drove at a constant speed of 64 miles per hour. Let t denote the amount of time (in hours) that it took Marcos to get from Phoenix to Flagstaff. The we have the formula 64t = 144, which implies t = = 2.25 hours. 15

16 question 4 - possible solutions (b) Construct a graph to represent Marcos distance from Phoenix in terms of time (in hours) since he left Phoenix Distance in miles since leaving Phoenix Time in number of hours since leaving Phoenix 16

17 question 4 - possible solutions (c) (i) True or False : Joni covered the same distance in the same amount of time as Marcos. Explain your reasoning. TRUE. She took the same route as Marcos, so she travelled the same distance (144 miles). Her trip began and ended at the same time as Marcos, so she traveled the same amount of time (2.25 hours). 17

18 question 4 - possible solutions (c) (ii) On your graph from part (b), construct a possible graph to represent Joni s distance travelled in miles in terms of the number of hours since she left Phoenix. 125 Distance in miles since leaving Phoenix Time in number of hours since leaving Phoenix 18

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