Worksheet 2 Answers (AC 1

Average Rates of Change (Active Calculus 1.1) Solutions Learning Objectives: You should be able to... Compute an average rate of change, interpret it in the context of a situation, and represent it graphically. Describe the difference between an average rate of change and an instantaneous rate of change. Represent instantaneous rate of change graphically. Today we are focusing on the following definition: The average rate of change of a function f from x = a to x = b , denoted AV [ a,b ] , is given by AV [ a,b ] = f ( b ) − f ( a ) b − a . 1. Warmup. In 2013, Diana Nyad became the first person to swim the 110-mile Florida Straits unaided. Other swimmers questioned whether her swim was truly unaided; one of the things they looked at was her speed along the swim. 1 In response, Nyad released GPS data about her 53-hour swim. Here’s a very small sample of the data: Time (hours) 0 5 7 15 31 32 40 Distance (miles) 0 6 . 67 8 . 90 18 . 72 54 . 85 58 . 72 82 . 88 Find Nyad’s average speed between 5 and 7 hours into her swim. Also find her average speed between 31 and 32 hours into her swim. Why might these numbers have raised suspicion? Answer . AV [5 , 7] = 1 . 115 miles per hour; AV [31 , 32] = 3 . 870 miles per hour This indicates that Nyad was swimming significantly faster later in the swim, which is unlikely when she should have been much more tired. Nonetheless, they determined that Nyad was “unaided”: the increase in speed was due to the tides. 2. (Adapted from Active Calculus 1.1.2 ) Consider the position function s ( t ) = 64 − 16( t − 1) 2 , where s ( t ) is the height of a ball at time t . (a) Compute the average velocity of the ball on each of the following time intervals: 1 See the New York Times article “Celebration Gives Way to Questions and Doubts Afer a Record Swim” Adapted from materials created by the Cornell Active Learning in Math team and Harvard University’s Math 1a. 1/ 4