5.4 Interpreting the Derivative: Meaning and Notation 211
Burning Calories While Bicycling: Suppose gives the number of calories used per
mile of bicycle riding (measured in calories/mile) as a function of speed (measured in miles/hour).13 Then , also written , is the function that gives the instantaneous rate
of change of calories/mile with respect to speed.
lim0 so the units for are calories/mi.
127 tells us that when the speed of the cyclist is 12 mph, the number of calories used per mile is increasing at a rate of 7 calories/mile per mph. Practically speaking, this means that if you currently ride at a pace of 12 mph, increasing your speed by 1 mph will result in your burning approximately 7 more calories for every mile you ride.14 Notice that
12 7 does not give us any information about the calories being burned by riding 1 mile at 12 mph. 12 would tell us that.
Suppose we know not only that 127 but also that 1222. Then we can estimate that 1329. Explain the reasoning behind this.
A Dash of History. During the decade from 1665 to 1675, about a century before the American War of Independence, both Isaac Newton in England and Gottfried Leibniz in Continental Europe developed the ideas of calculus.
Newton began his work during years of turmoil. The years 1665—1666 were the years of the Great Plague, which wiped out about one quarter of the population of London. The GreatFireofLondoneruptedin1666, destroyingalmosthalfofLondon.15 Cloisteredinhis small hometown, Newton developed calculus in order to understand physical phenomena. The language of ßuxions he used to explain his ideas reßected his scientiÞc perspective; his terminology is no longer in common usage. Newton used the notation to denote a derivative. Although we will not adopt this notation, it is still used by some physicists and engineers.
Around the same time that Newton did his work Leibniz was also developing calculus. The language and notation he used in his work differed from NewtonÕs; part of his genius
was his introduction of very useful notation, notation still very much in use.
LeibnizÕs notation for the derivative, , gives the derivative the appearance of a fraction. It is not a fraction; rather, it is the limit of the ratio as 0. As a mental
model, however, it can be useful to think of as an inÞnitesimally small change in and as the corresponding change in , where the reminds us of delta. This notion is not, as stated, entirely accurate (what can be meant by an inÞnitesimally small change in ?) but it was essentially the model used by Leibniz; part of the genius of his notation is that the mental model generally does not lead us astray.
Well over a century after Newton and Leibniz did their work the word ÒderivativeÓ was introduced by Joseph Lagrange (1736—1813). It was Lagrange who introduced the notation emphasizing that the derivative is a function.
13 For our model we hold all other factors (such as the bicyclist, weather conditions, terrain, and the bicycle itself) constant. 14 We say approximately because we are interpreting the rate of change over an interval, albeit a small one, and the function
is probably not perfectly linear.
15 These facts are from David BurtonÕs The History of MathematicsAn Introduction, The McGraw-Hill Companies, Inc., 1997; p.349.
212 CHAPTER 5 The Derivative Function
Answers to Selected Exercises
ANSWERS TO EXERCISE 5.5 a, b, c, and d are equal to 1.
P R O B L E M S F O R S E C T I O N 5 . 4
1. Which of the following are equal to 3? A sketch with labeled points will be useful.
(a) lim 3 3 0
(b) lim 3 0
(c) lim 3 3
(d) lim 3 3
(f) lim 3 3 0
(e) lim 3 3 3
(g) lim 33 0
2. An orange is growing on a tree. Assume that the orange is always spherical, and that it has not yet reached its mature size. Its current radius is cm.
(a) If the radius increases by 0.5 cm, what is the corresponding increase in volume? What is ?
(b) If the radius of the orange increases by , what is the corresponding increase in volume? What is ? (Please simplify your answer.)
(c) Showthatlim0 42.Concludethatfor verysmall 42. (d) The surface area of a sphere is 42. Explain, in terms of an orange, why the
approximation 42 make sense.
3. Let be the number of Þsh in a pond at time , where is given in years. WeÕll denote by the carrying capacity of the pond; is a constant that tells us how many Þsh the pond can support. Suppose at time 0 the Þsh population is small. At Þrst the Þsh population will grow at an increasing rate, but eventually the Þsh compete for limited resources and the number of Þsh levels out at the carrying capacity of the pond. A graph of is drawn below.
(a) Sketch the graph of , the slope function, versus time.
(b) Interpret the slope of as a rate of change.
5.4 Interpreting the Derivative: Meaning and Notation 213
4. The curve below is an indifference curve, which shows combinations of food and clothinggivingequalsatisfaction, amongwhichthehouseholdisindifferent.Theslope of the tangent line is called the marginal rate of substitution at the point .
10 20 30 Quantity of Food Per Week
(a) Explain in terms of food and clothing what it would mean to know that the slope of the line was 2.
(b) Theslopeofthetangentlineat isanegativenumberoflargermagnitudethanthe slope of the tangent line at . What does this mean in terms of food and clothing?
5. This problem deals with the effect of altitude on how far a batted ball will travel. The drag resistance on the ball is proportional to the density of the air, i.e., the barometric pressure if the temperature is held constant. Let us take as an example a 400-foot home run in Yankee Stadium, which is approximately at sea level. On average, an increase in altitude of 275 feet would increase the length of this drive by 2 feet. (Adair, Robert K. The Physics of Baseball. New York: Harper & Row, 1990.)
Let be the distance this ball would travel as a function of the altitude of the ballpark in which it is hit. Assume the relationship between altitude and distance is linear.
(a) What is the meaning of ? What are its units?
(b) What is the numerical value of ? (c) Write an equation for .
(d) Prior to major league baseballÕs 1993 expansion into Denver, Atlanta, which has an altitude of 1050 feet, was the highest city in the majors. How far would this 400-foot Yankee Stadium drive travel in Atlanta?
(e) How far would it travel in Denver (altitude 5280 feet)?
6. Suppose that gives the number of pounds of apples sold as a function of the price (in dollars) per pound.
(a) What are the units of ?
(b) Do you expect to be positive? Why or why not? (c) Interpret the statement 0.885.
7. Between 1940 and 1995 the size of the average farm in America increased from 174 acres to 469 acres. (Facts from the World Almanac and Book of Facts 1997.) Suppose
214 CHAPTER 5 The Derivative Function
that gives the average number of acres of an American farm years after 1940. is an increasing function.
(a) What are the units of ?
(b) Average farm size increased much more dramatically in the 50s than in the 80s. Which is larger, 12 or 43?
Which do you think is larger, 12 or 43?
8. A baked apple is taken out of the oven and put into the refrigerator. The refrigerator is kept at a constant temperature. NewtonÕs Law of Cooling says that the difference between the temperature of the apple and the temperature of the refrigerator decreases at a rate proportional to itself. That is, the apple cools down most rapidly at the outset of its stay in the refrigerator, and cools increasingly slowly as time goes by. You have the following pieces of information: At the moment the apple is put in the refrigerator its temperature is 110 degrees and is dropping at a rate of 4 degrees per minute. Twenty minutes later the temperature of the apple is 70 degrees.
(a) Let be the temperature of the apple at time , where is measured in minutes and 0 is when the apple is put in the refrigerator. Express the three bits of information provided above in functional notation. Sketch a graph of versus .
(b) Using the same set of axes as you did in part (a), draw the cooling curve the baked apple would have if it were cooling linearly, with the initial temperature of 110 degrees and initial rate of cooling of 4 degrees per minute. What would the temperature of the apple be after 15 minutes using this linear model? In reality, is the temperature more or less?
(c) Since the appleÕs temperature dropped from 110 degrees to 70 degrees in twenty
minutes, the average rate of change of temperature over the Þrst twenty minutes is 40 degrees or 2 degrees. Using the same set of axes as you did in parts (a) and
(b), draw the cooling curve the baked apple would have if it were cooling linearly, with the initial temperature of 110 degrees and rate of cooling of 2 degrees per minute. What would the temperature of the apple be after 15 minutes using this linear model? In reality, is the temperature more or less?
9. A hot-air balloonist is taking a balloon trip up a river valley. The trip begins at the mouth of the river. The balloonÕs altitude varies throughout the trip.
Suppose that , is the function that gives the balloonÕs height (in feet) above the ground at time , where is the time from the start of the trip measured in hours.
(a) Suppose that at time 4 hours 4 is 70. Interpret what 470 tells us in words.
(b) Let be the function that takes as input , where is the balloonÕs horizontal distance from the mouth of the river ( measured in feet) and gives as output the time it has taken the balloon to make it from the mouth of the river to this point. In other words, if 1000 4 then the balloon has taken 4 hours to travel 1000 feet up the river bank.
i. Let , where and are the functions given above. Describe in words the input and output of the function .
ii. Interpret the statement 700100 in words. iii. Interpret the statement 70060 in words.
5.4 Interpreting the Derivative: Meaning and Notation 215
10. Suppose that gives the number of calories that an average adult burns by walking at a steady speed of miles per hour for one hour.
(a) What are the units of ?
(b) Do you expect to be positive? Why or why not? (c) Interpret the statement 325.
Hint: If you are having difÞculties with this problem, consider sketching a graph. What are the labels on the axes? (That is, what are the independent and dependent variables?) Thinking about these variables, what should the graph look like? How do your assumptions about the graph relate to the questions posed above?