Chapter 3 The Derivative In developing the S-I-R model in chapter 1 we took the idea of the rate of change of a population as intuitively clear. The rate at which one quantity changes with respect to another is a central concept of calculus and leads to a broad range of insights. The chief purpose of this chapter is to develop a fuller understanding—both analytic and geometric—of the connection between a function and its rate of change. To do this we will introduce the concept of the derivative of a function. 3.1 Rates of Change The Changing Time of Sunrise ◦ The sun rises at different times, depending on the date and location. At 40 The time of sunrise N latitude (New York, Beijing, and Madrid are about at this latitude) in the is a function year 1990, for instance, the sun rose at 7:16 on January 23, 5:58 on March 24, 4:52 on July 25. Clearly the time of sunrise is a function of the date. If we represent the time of sunrise by T (in hours and minutes) and the date by d (the day of the year), we can express this functional relation in the form T = T (d). For example, from the table above we find T (23) = 7:16. It is not obvious from the table, but it is also true that the the rate at which the time of sunrise is changing is different at different times of the year—T ′ varies as d varies. 101 Copyright 1994, 2008 Five Colleges, Inc. DVI file created at 11:46, 21 May 2008 102 CHAPTER 3. THE DERIVATIVE We can see how the rate varies by looking at some further data for sunrise at the same latitude, taken from The Nautical Almanac for the Year 1990 : date time date time date time January 20 7:18 March 21 6:02 July 22 4:49 23 7:16 24 5:58 25 4:52 26 7:14 27 5:53 28 4:54 Let’s use this table to estimate the rate at which the time of sunrise is Calculate the rate changing on January 23. We’ll use the times three days earlier and three using earlier and days later, and compare them. On January 26 the sun rose 4 minutes earlier later dates than on January 20. This is a change of 4 minutes in 6 days, so the rate of change is − minutes 4 .67 minutes per day. − 6 days ≈ − We say this is the rate at which sunrise is changing on January 23, and we write minutes T ′(23) .67 . ≈ − day The rate is negative because the time of sunrise is decreasing—the sun is rising earlier each day. Similarly, we find that around March 24 the time of sunrise is changing approximately 9/6 = 1.5 minutes per day, and around July 25 the rate − − is 5/6 +.8 minutes per day. The last value is positive, since the time of sunrise≈ is increasing—the sun is rising later each day in July. Since March 24 is the 83rd day of the year and July 25 is the 206th, using our notation for rate of change we can write minutes minutes T ′(83) 1.5 ; T ′(206) .8 . ≈ − day ≈ day Notice that, in each case, we have calculated the rate on a given day by using times shortly before and shortly after that day. We will continue this pattern wherever possible. In particular, you should follow it when you do the exercises about a falling object, at the end of the section. Once we have the rates, we can estimate the time of sunrise for dates not given in the table. For instance, January 28 is five days after January 23, so the total change in the time of sunrise from January 23 to January 28 should be approximately minutes ∆T .67 5 days = 3.35 minutes. ≈ − day × − Copyright 1994, 2008 Five Colleges, Inc. DVI file created at 11:46, 21 May 2008 3.1. RATES OF CHANGE 103 In whole numbers, then, the sun rose 3 minutes earlier on January 28 than on January 23. Since sunrise was at 7:16 on the 23rd, it was at 7:13 on the 28th. By letting the change in the number of days be negative, we can use this same reasoning to tell us the time of sunrise on days shortly before the given dates. For example, March 18 is 6 days away from March 24, so the change − in the time of sunrise should be minutes ∆T 1.5 6 days = +9 minutes. ≈ − day × − Therefore, we can estimate that sunrise occurred at 5:58 + 0:09 = 6:07 on March 18. Changing Rates Suppose instead of using the tabulated values for March we tried to use our January data to predict the time of sunrise in March. Now March 24 is 60 days after January 23, so the change in the time of sunrise should be approximately minutes ∆T .67 60 days = 40.2 minutes, ≈ − day × − and we would conclude that sunrise on March 24 should be at about 7:16 0:40 = 6:37, which is more than half an hour later than the actual time! − This is a problem we met often in estimating future values in the S-I-R model. implicitly assume that the time of sunrise changes at the fixed rate Predictions over of .67 minutes per day over the entire 60-day time-span. But this turns out long time spans − not to be true: the rate actually varies, and the variation is too great for us are less reliable to get a useful estimate. Only with a much smaller time-span does the rate not vary too much. Here is the same lesson in another context. Suppose you are travelling in a car along a busy road at rush hour and notice that you are going 50 miles per hour. You would be fairly confident that in the next 30 seconds (1/120 of an hour) you will travel about miles 1 5 ∆ position 50 hour = mile = 2200 feet. ≈ hour × 120 12 The actual value ought to be within 50 feet of this, making the estimate accurate to within about 2% or 3%. On the other hand, if you wanted to Copyright 1994, 2008 Five Colleges, Inc. DVI file created at 11:46, 21 May 2008 104 CHAPTER 3. THE DERIVATIVE estimate how far you would go in the next 30 minutes, your speed would probably fluctuate too much for the calculation miles 1 ∆ position 50 hours = 25 miles ≈ hour × 2 to have the same level of reliability. Other Rates, Other Units In the S-I-R model the rates we analyzed were population growth rates. They told us how the three populations changed over time, in units of persons per day. If we were studying the growth of a colony of mold, measuring its size by its weight (in grams), we could describe its population growth rate in units of grams per hour. In discussing the motion of an automobile, the rate we consider is the velocity (in miles per hour), which tells us how the distance from some starting point changes over time. We also pay attention to the rate at which velocity changes over time. This is called acceleration, and can be measured in miles per hour per hour. While many rates do involve changes with respect to time, other rates do not. Two examples are the survival rate for a disease (survivors per thousand infected persons) and the dose rate for a medicine (milligrams per Examples of rates pound of body weight). Other common rates are the annual birth rate and the annual death rate, which might have values like 19.3 live births per 1,000 population and 12.4 deaths per 1,000 population. Any quantity expressed as a percentage, such as an interest rate or an unemployment rate, is a rate of a similar sort. An unemployment rate of 5%, for instance, means 5 unemployed workers per 100 workers. There are many other examples of rates in the economic world that make use of a variety of units—exchange rates (e.g., francs per dollar), marginal return (e.g., dollars of profit per dollar of change in price). Sometimes we even want to know the rate of change of one rate with The rate of change respect to another rate. For example, automobile fuel economy (in miles per of a rate gallon—the first rate) changes with speed (in miles per hour—the second rate), and we can measure the rate of change of fuel economy with speed. Take a car that goes 22 miles per gallon of fuel at 50 miles per hour, but only 19 miles per gallon at 60 miles per hour. Then its fuel economy is changing Copyright 1994, 2008 Five Colleges, Inc. DVI file created at 11:46, 21 May 2008 3.1. RATES OF CHANGE 105 approximately at the rate ∆ fuel economy 19 22 miles per gallon = − ∆ speed 60 50 miles per hour − = .3 miles per gallon per mile per hour. − Exercises A falling object. These questions deal with an object that is held motion- less 10,000 feet above the surface of the ocean and then dropped. Start a clock ticking the moment it is dropped, and let D be the number of feet it has fallen after the clock has run t seconds. The following table shows some of the values of t and D.