Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 24 doc

5.4 Interpreting the Derivative: Meaning and Notation 211 ◆ EXAMPLE 5.12 Burning Calories While Bicycling: Suppose C(s) gives the number of calories used per mile of bicycle riding (measured in calories/mile) as a function of speed s (measured in miles/hour). 13 Then C  (s), also written dC ds , is the function that gives the instantaneous rate of change of calories/mile with respect to speed. dC ds = lim s→0 C s so the units for dC ds are calories/mi mi/hr . C  (12) = 7 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 C  (12) = 7 does not give us any information about the calories being burned by riding 1 mile at 12 mph. C(12) would tell us that. ◆ EXERCISE 5.6 Suppose we know not only that C  (12) = 7 but also that C(12) = 22. Then we can estimate that C(13) ≈ 29. 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 Great Fire of London erupted in 1666, destroying almost half of London. 15 Cloistered in his small hometown, Newton developed calculus in order to understand physical phenomena. The language of fluxions he used to explain his ideas reflected his scientific perspective; his terminology is no longer in common usage. Newton used the notation ˙y 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 Newton’s; part of his genius was his introduction of very useful notation, notation still very much in use. Leibniz’s notation for the derivative, dy dx , gives the derivative the appearance of a fraction. It is not a fraction; rather, it is the limit of the ratio y x as x → 0. As a mental model, however, it can be useful to think of dx as an infinitesimally small change in x and dy as the corresponding change in y, where the d reminds us of delta. This notion is not, as stated, entirely accurate (what can be meant by an infinitesimally small change in x?) 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 “derivative” was introduced by Joseph Lagrange (1736–1813). It was Lagrange who introduced the notation f  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 Burton’s The History of Mathematics—An 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 f  (1). PROBLEMS FOR SECTION 5.4 1. Which of the following are equal to g  (3)? A sketch with labeled points will be useful. (a) lim h→0 g(3 + h) − g(3) h (b) lim x→0 g(x) − g(3) x − 3 (c) lim x→3 g(x) − g(3) x − 3 (d) lim s→3 g(3) − g(s) 3 − s (e) lim x→3 g(3 + x) − g(3) x (f) lim x→0 g(3 + x) − g(3) x (g) lim x→0 g(3) − g(3 + x) −x 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 r cm. (a) If the radius increases by 0.5 cm, what is the corresponding increase in volume? What is V r ? (b) If the radius of the orange increases by r, what is the corresponding increase in volume? What is V r ? (Please simplify your answer.) (c) Show that lim r→0 V r = 4πr 2 .Conclude that for r very small V ≈ (4πr 2 )r. (d) The surface area of a sphere is 4πr 2 . Explain, in terms of an orange, why the approximation V ≈ (4πr 2 )r make sense. 3. Let F(t) be the number of fish in a pond at time t, where t is given in years. We’ll denote by C the carrying capacity of the pond; C is a constant that tells us how many fish the pond can support. Suppose at time t = 0 the fish population is small. At first the fish population will grow at an increasing rate, but eventually the fish compete for limited resources and the number of fish levels out at the carrying capacity of the pond. A graph of F(t)is drawn below. (a) Sketch the graph of F  (t), the slope function, versus time. (b) Interpret the slope of F as a rate of change. t F C 5.4 Interpreting the Derivative: Meaning and Notation 213 4. The curve below is an indifference curve, which shows combinations of food and clothing giving equal satisfaction, among which the household is indifferent. The slope of the tangent line T is called the marginal rate of substitution at the point b. 10 20 30 10 20 30 a b c T Quantity of Food Per Week Quantity of Clothing Per Week (a) Explain in terms of food and clothing what it would mean to know that the slope of the line T was −2. (b) The slope of the tangent line at a is a negative number of larger magnitude than the slope of the tangent line at b. 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 B(a) 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 dB da ? What are its units? (b) What is the numerical value of dB da ? (c) Write an equation for B(a). (d) Prior to major league baseball’s 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 A(p) gives the number of pounds of apples sold as a function of the price (in dollars) per pound. (a) What are the units of dA dp ? (b) Do you expect dA dp to be positive? Why or why not? (c) Interpret the statement A  (0.88) =−5. 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 A(t) gives the average number of acres of an American farm t years after 1940. A(t) is an increasing function. (a) What are the units of dA dt ? (b) Average farm size increased much more dramatically in the 50s than in the 80s. Which is larger, A(12) or A(43)? Which do you think is larger, A  (12) or A  (43)? 8. A baked apple is taken out of the oven and put into the refrigerator. The refrigerator is kept at a constant temperature. Newton’s 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 T be the temperature of the apple at time t, where t is measured in minutes and t = 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 T versus t. (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 apple’s temperature dropped from 110 degrees to 70 degrees in twenty minutes, the average rate of change of temperature over the first twenty minutes is −40 degrees 20 minutes or −2 degrees minute . 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 balloon’s altitude varies throughout the trip. Suppose that A(t), is the function that gives the balloon’s height (in feet) above the ground at time t, where t is the time from the start of the trip measured in hours. (a) Suppose that at time t = 4 hours A  (4) is 70. Interpret what A  (4) = 70 tells us in words. (b) Let f be the function that takes as input x, where x is the balloon’s horizontal distance from the mouth of the river (x 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 f(1000) = 4 then the balloon has taken 4 hours to travel 1000 feet up the river bank. i. Let h(x) = A(f (x)), where f and A are the functions given above. Describe in words the input and output of the function h. ii. Interpret the statement h(700) = 100 in words. iii. Interpret the statement h  (700) = 60 in words. 5.4 Interpreting the Derivative: Meaning and Notation 215 10. Suppose that C(s) gives the number of calories that an average adult burns by walking at a steady speed of s miles per hour for one hour. (a) What are the units of dC ds ? (b) Do you expect dC ds to be positive? Why or why not? (c) Interpret the statement C  (3) = 25. Hint: If you are having difficulties 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? 6 CHAPTER The Quadratics: A Profile of a Prominent Family of Functions 6.1 A PROFILE OF QUADRATICS FROM A CALCULUS PERSPECTIVE Face to Face with Quadratics Quadratic functions arise naturally in a large variety of situations. For instance, if you are tossing a dart, firing a cannon, throwing a football, or contemplating bungee jumping you have to concern yourself with the physics of falling bodies, and the laws of physics are such that the height of a falling object can be modeled with a quadratic function of time. An economist looking at the relationship between revenue and price from the perspective of a monopolist can end up with a quadratic model. In our introductory example our goal is to construct a garden with the largest possible area, subject to certain constraints. This area can be expressed as the product of two linear functions, resulting in a quadratic. In many situations, whether planning a garden or running a business, we make decisions based on optimizing (maximizing or minimizing) some quantity and these optimization problems sometimes bring us face to face with quadratic functions. The quadratics form a family whose members have predictable behavior and characteristics. In Example 6.1 we investigate the behavior of one particular quadratic, but quadratics are such an elemental part of the mathematical landscape that we will find it useful to become very well acquainted with the whole family of quadratic functions so we know what kind of behavior to expect from its members. ◆ EXAMPLE 6.1 A gardener has 30 feet of fencing that she will use to fence off a rectangular tomato plot along the south wall of her house. What should the dimensions of the garden be in order for her to have the most area to grow her tomatoes? 217 218 CHAPTER 6 The Quadratics: A Profile of a Prominent Family of Functions SOLUTION Let’s begin by drawing a picture of the tomato plot. Because the plot is alongside the house, she needs fencing for three sides of the plot only. She wants to maximize the area of the plot. Our strategy is to express the area, A, as a function of one variable. We know that area = (width)(length). Suppose we call the width of the garden w feet and the length . Can we express the area just in terms of w? If the width is w, why can’t the length be anything the gardener likes? Because she has only 30 feet of fencing. The amount of fencing constrains her options. Once she chooses w,  is determined and therefore the area is determined. (width + width + length) = 30 w + w +  = 30 2w +  = 30  = 30 − 2w So A(w) = (width)(length) = w(30 − 2w). w house length w Figure 6.1 What value of w will make A as big as possible? We can investigate this by looking at the graph of A(w). If we were to do this mindlessly using a graphing calculator with the default range and domain of [−10, 10], we would get a picture that looks like Figure 6.2(a) and provides little information. To get useful information from the graph we need to pick a reasonable domain and range. What values of w are reasonable? Certainly w, the width, can’t be negative, and 2w, twice the width, can’t be greater than the 30 feet of fencing. Therefore, 0 <w<15. Simply adjusting the domain to [0, 15] doesn’t help much (see Figure 6.2b), but by also adjusting the range we can make some headway. Figure 6.2(c) shows the graph with the range of [0, 150]. Use of the trace key (or some more sophisticated calculator feature) indicates that the high point has coordinates of approximately (7.5, 112.5); when w ≈ 7.5 feet, then A ≈ 112.5 feet. The gardener should use about 7.5 feet of fencing for the width of the plot and the remainder, (30 − 2(7.5)) feet = 15 feet, for the length. This will give a plot of 112.5 square feet for her tomatoes. 10 10 –10 (a) –10 15 (b) 10 –10 150 15 (c) Figure 6.2 ◆ 6.1 A Profile of Quadratics from a Calculus Perspective 219 Notice that in the example just completed the area of the garden could be expressed as the product of two linear functions, as w(30 − 2w) or 30w − 2w 2 . This expression is a quadratic, due to the w 2 term, and the graph that we looked at is a parabola. Similarly, there are other quantities that can be expressed as the product of two expressions; for example, revenue = (price per item) · (number of items sold). If the two factors are linear functions of the same variable (or are modeled by linear functions), then the product will be a quadratic. Definitions A function f(x) is quadratic if it can be expressed in the form f(x)=ax 2 + bx + c, where a, b, and c are constants and a = 0. The graph of a quadratic function is called a parabola. A Calculus Perspective There are several approaches we can take to studying quadratic functions. Let’s begin by examining their rates of change. We compute the derivative of a quadratic using the limit definition of derivative. f  (x) = lim h→0 f(x +h) − f(x) h = lim h→0 a(x + h) 2 + b(x + h) + c − (ax 2 + bx + c) h Using the function f(x)=ax 2 + bx + c = lim h→0 a(x 2 + 2hx + h 2 ) + bx + bh + c − ax 2 − bx − c h Expanding in order to simplify = lim h→0 ax 2 + 2ahx + ah 2 + bx + bh + c − ax 2 − bx − c h = lim h→0 2ahx + ah 2 + bh h = lim h→0 h(2ax + ah + b) h Factoring out an h = lim h→0 2ax + ah + b Because h = 0 we can say h/h = 1. =2ax + b Iff(x)=ax 2 + bx + c, then f  (x) = 2ax + b. Observations i. The derivative of a quadratic function is a linear function. ii. This formula for f  is valid regardless of the values of a, b, and c. We can, for example, set b = 0 and c = 0 and conclude that if f(x)=ax 2 then f  (x) = 2ax. Similarly, we know that the derivative of bx is b, and the derivative of the constant c is 0. iii. The value of c does not affect the value of the derivative. This makes geometric sense; c only shifts the graph vertically and therefore has no effect on the slope at any given x-value. ◆ EXAMPLE 6.2 Find the derivatives of the following functions. 220 CHAPTER 6 The Quadratics: A Profile of a Prominent Family of Functions i. f(x)=−3x 2 −6x ii. g(x) = −x 2 √ 7 + π 2 3 iii. h(x) = (2x − 1)(x − 4) 2 ANSWER i. f  (x) =−6x−6 ii. g  (x) = −2 √ 7 x iii. h  (x) = 2x −4.5 (Multiply out, then differentiate.) ◆ We have shown that if f is a quadratic function, then f  is linear. Let’s use this information about f  to see what we can say about the graph of f itself. We’ll refer back to the derivatives of the functions in Example 6.2. ◆ EXAMPLE 6.3 Use the derivatives f  , g  , and h  found in Example 6.2 to get information about the graphs of f , g, and h. In particular, determine the x-coordinate of any turning point. The sign of the derivative function tells us where the original functions are increasing and where they are decreasing. i. f  (x) =−6x−6 ii. g  (x) = −2 √ 7 x iii. h  (x) = 2x − 4.5 f ′ x –1 – 6 g′ x h′ x 2.25 – 4.5 f is increasing for x<−1 gis decreasing for x>0 his increasing for x>9/4 f is decreasing for x>−1 gis increasing for x<0 his decreasing for x<9/4 graph of f graph of f ′ + – –1 graph of g graph of g′ + – 0 graph of h graph of h′ – + 2.25 parabolas with derivative f  parabolas with derivative g  parabolas with derivative h  x –1 x π 2 /3 x 4 1 2 / Figure 6.3 . taken out of the oven and put into the refrigerator. The refrigerator is kept at a constant temperature. Newton’s Law of Cooling says that the difference between the temperature of the apple and the. regardless of the values of a, b, and c. We can, for example, set b = 0 and c = 0 and conclude that if f(x)=ax 2 then f  (x) = 2ax. Similarly, we know that the derivative of bx is b, and the derivative. food and clothing what it would mean to know that the slope of the line T was −2. (b) The slope of the tangent line at a is a negative number of larger magnitude than the slope of the tangent