Rate of Change. Lecture 7.

Transcription

1 Rate of Change. Lecture 7. Let s begin with a familiar example Example 1. We go for a drive and observe that between 1:00p.m. and 1:45p.m. we travel 42 miles. We then ask, What was our average speed during those 45 minutes? But speed or velocity is the rate of change of distance with respect to time. So we have asked to determine: distance time change in distance change in time So suppose that we measure distance in miles and time in hours. The desired answer is then, of course: distance time 42 miles 3/4 hours 56 miles per hour. That is, on average we changed distance by 56 miles every hour. Key Observation. Although we averaged 56 mph, we were rarely actually hitting 56 on our speedometer we were stuck behind a semi for several miles and were well over 70 on an open straight stretch. In other words, the distance traveled was not a linear function of time. But by claiming that we traveled at an average of 56 mph, we linearized that aspect of our trip. And linear functions are often more digestible than non-linear ones.

2 Example 2. Suppose that the price of widgets during one thirty day period is given by the function p f(t) where p is measured in dollars and t is days. We ask What is the average rate of change of price from the (end of the) 3rd through the 12th day? Again this is asking for the ratio of the change in price to the change in time: p price change in price time change in time. Suppose that the price function is linear, say f(p) 0.5p So for the period from the third through the twelfth day the average change in price is p f(12) f(3) 12 3 ( ) ( ) dollars per day. That is, the average rate of change of a linear function is just its slope! 2

3 Example 3. In a college dorm the number of people infected by a flu bug as a function of time is modeled quite well by N(t) (1.5) t students as a function of the number t of days since it was first observed. Student Health is later asked to report the average rate of infection for the two weeks from the 7th day to the 21st. Now based on the model the total number of reported cases through days 7 and 21 were N(7) 46.1 and N(21) 99.6, so what Student Health was asked for was the average rate N N(21) N(7) new cases per day. But that average is just the slope of the linear function f(x) through the points f(7) 46.1 and f(21) Untitled N(t) 21 7 N f(21) f(7)

4 In general, given a function y f(x), whether by means of a table, a graph, or a formula, The average rate of change of y with respect to x over some interval from x 1 to x 2 is the ratio of the change in y to the change in x: y x f(x 2) f(x 1 ) x 2 x 1. In other words this average rate of change is just the slope of the line through the two points (x 1, f(x 1 )) and (x 2, f(x 2 )). Example 4. The distance in feet above the ground of a bullet fired straight up in the air is given by s(t) 16t t, where t is measured in seconds. What is the average rate of climb of the bullet during the first 10 seconds? In the second ten seconds? Well, in the first ten it is s s(10) s(0) 10 0 and for the second ten s s(20) s(10) feet per second, 320 feet per second. 4

5 Bonus Questions. Here a couple of fun bonus questions about this last example. 1. Since this bullet is fired straight up and gravity hasn t yet given up, the bullet is going to be forced to stop climbing and fall back to the earth. How high will it go and how long will it take the bullet to get that high? 2. During the first 10 seconds of its trip the bullet climbed at an average rate of 640 feet per second. What was its average rate of climb in miles per hour? In meters per minute? This question is far from frivolous. Many people, unfortunately, are plagued by the difficulty of converting from one system of measurement to another and that shouldn t be! The whole secret to such conversions is just to understand what phrases like feet per second, miles per hour, and feet per mile mean. For example, So feet per second 640 ft sec feet second ft 3600 sec 640 sec hr ft ( miles and 5280 feet 1 mile. hr miles hr 5280ft ) mph. 5

6 Further Examples. 1. With the bullet in Example 4, above, what is the average rate of climb of the bullet from t 5 sec until h seconds later? What is the average rate of climb from an unspecified time t until h seconds later? 2. An investment of $1000 is earning 12% annual interest compounded continuously. What is the average annual rate of increase in value for the first 5 years of this investment? What rate of simple interest would produce the same increase in value over the five years? 3. A bike is coasting downhill in such a way that its distance from the top of the hill as a function of time in minutes is given as s(t) 0.25t 2 miles per minute. What is the average rate of change ( average velocity) of the bike between minute 1 and h minutes later? ( Can you convert this to miles per hour?) What is the limit of this average velocity as h approaches 0? Can you guess what that represents? 6