1.3 Representations of Functions 41 31. Two sisters, Nina and Lori, part on a street corner. Lori saunters due north at a rate of 150 feet per minute and Nina jogs off due east at a rate of 320 feet per minute. Assuming they maintain their speeds and directions, express the distance between the sisters as a function of the number of minutes since they parted. 32. Below are graphs of f , g, h, and j. Use the following clues to match the function with its graph. . h(0) = g(0) . h(x) > g(x) for x<0 . h(2) = f(2)=j(2) . j(x)=2f(x)for x>0 y x For Problems 33 through 35, if the interval is written using inequalities, write it using interval notation; if it is expressed in interval notation, rewrite it using inequalities. In all cases, indicate the interval on the number line. 33. (a) −1 ≤ x ≤ 3 (b) (−2, −1] 34. (a) −7 ≤ x<−5 (b) (−π, π) 35. (a) −1 <x (b) (−∞,3) 36. Let f(x)=2x 2 +x.Find the following. (a) f(3) (b) f(2x) (c) f(1+x) (d) f( 1 x ) (e) 1 f(x) 37. A right circular cylinder is inscribed in a sphere of radius 5. (a) Express the volume of the cylinder as a function of its radius, r. (b) Express the surface area of the cylinder as a function of its radius, r. 38. The height of a right circular cone is one third of the diameter of the base. (a) Express its volume as a function of its height, h. (b) Express its volume as a function of r, the radius of its base. 42 CHAPTER 1 Functions Are Lurking Everywhere x r 39. A vitamin capsule is constructed from a cylinder of length x centimeters and radius r centimeters, capped on either end by a hemisphere, as shown at left. Suppose that the length of the cylinder is equal to three times the diameter of the hemispherical caps. (a) Express the volume of the vitamin capsule as a function of x. Your strategy should be to begin by expressing the volume as a function of both x and r. (b) Express the surface area of the vitamin capsule as a function of x. 40. Calibration function for a bucket: Consider the bucket drawn below. The bucket walls can be thought of as a slice of a right circular cone, as shown on the right. right circular cone bucket bucket 24" 16" 16" 8" 12" 12" Let C be the calibration function. C takes as input a volume of liquid and produces as output the height of the liquid in the bucket. You will need to know how to find the volume of a cone in order to do this problem. Refer to the geometry summary in Appendix B: Geometric Formulas if you do not know the formula. (a) What is the domain of C? (b) What is the range of C? 41. Late at night a caped man is standing by a 14-foot-high street lamp on a highway. Denote his height by H . The length of the shadow he casts is a function of his distance from the lamppost. Write a formula for this function, where s is the distance between the man and the lamppost. 42. Assume that f is a function with domain (−∞, ∞). Which of the following statements is true for every such function f and all p, w, and z in the domain of f ? If a statement is not true for every function, find a function for which it is false. (Hint: The constant functions are good functions to use as a first check.) (a) f(2)f (3) = f(6) (b) f(p)+f(p)=2f(p) (c) f(4+5)=f(4)+f(5) (d) f(w)f(w)=  f(w)  2 (e) f(z)f(z)=f(z 2 ) (f) f(x) 5[f(x)] 2 = 1 5f(x) , where f(x)=0 (g) f(1)f (7) = f(7) 1.3 Representations of Functions 43 43. Translate each of these English sentences into a mathematical sentence, i.e., an equa- tion. Then comment on the validity of the statement, making qualifications if necessary. You will need to define your variables. The first example has been worked for you. Example: The cost of broccoli is proportional to its weight. Answer: Let C = the cost of broccoli and B = the weight of broccoli. C = kB for some constant k. We can emphasize that cost is a function of weight by writing C(B) = kB. Validity: In general this is true, because broccoli is usually sold by the pound. The proportionality constant k is the price per pound; at any given time and store it is fixed. The proportionality constant may vary with the season and the specific store. (a) The cost of a piece of sculpture is proportional to its weight. (b) The rate at which money grows in a savings account is proportional to the amount of money in the account. (c) The rate at which a population grows is proportional to the size of the population. (d) The total distance you travel is proportional to the time you spend traveling. 44. A gardener has a fixed length of fence that she will use to fence off a rectangular chili pepper garden. Express the area of the garden as a function of the length of one side of the garden. If you have trouble, reread the “Portable Strategies for Problem Solving” listed in this chapter. We’ve also included the following advice geared specifically toward this particular problem. . Give the length of fencing a name, such as L. (We don’t know what L is, but we know that it is fixed, so L is a constant, not a variable.) . Draw a picture of the garden. Call the length of one side of the fence s. How can you express the length of the adjacent side in terms of L and s? . What expression gives the area enclosed by the fence? 45. Cathy will fence off a circular pen for her rabbits. Express the area of the rabbit pen as a function of the length of fencing she uses. 46. A commuter rides his bicycle to the train station, takes the subway downtown and then walks from the subway station to his office. He bikes at an average speed of B miles per hour and can walk M miles in H hours. The subway ride takes R minutes. The commuter bikes X miles and walks Y miles to get to work. Assume that X, Y , B, H , M, and R are all constants. The amount of time it takes the commuter to get to work varies with how long he has to spend at the subway station locking his bike and waiting for the next train. Denote this time by w, where w is in hours. Express the time it takes for his commute as a function of w. Specify whether your answer is in minutes or in hours. 47. A rectangular piece of thick cardboard measuring 10 inches by 6 inches is being used to make an open-top box for raspberries. The box is constructed by cutting out squares 44 CHAPTER 1 Functions Are Lurking Everywhere x inches by x inches from each of the corners of the cardboard sheet and then folding up the sides, as shown in the acccompanying figure. Express the volume of the box as a function of x. x x xx x x x x x 6" 10" 48. A new mosque is being built in the Turkish town of Iznik. Under construction is an archway whose structure will be a rectangle capped by a semicircle, as shown in the figure below. The distance from the highest point of the arch to the floor is 7 meters. Denote the width of the archway by w and the height of the vertical wall of the rectangle y. The width and height are given in meters. (a) Express y, the height of the side wall, as a function of w, the width of the archway. (b) Express the area enclosed by the archway as a function of w. (c) Express the perimeter of the archway (excluding the floor) as a function of y. y w 49. During road construction gravel is being poured onto the ground from the top of a tall truck. The gravel falls into a conical pile whose height is always equal to half of its radius. Express the amount of gravel in the pile as a function of its (a) height. (b) radius. 1.3 Representations of Functions 45 50. In Durham, England, the circular plots of land at the center of the roundabouts 17 are often meticulously planted. Along the 8-meter diameter of one such circular plot is a line of yellow tulips. Rows of purple tulips, pansies, and marigolds are neatly planted parallel to the center row of yellow tulips. Each row of flowers extends from one side of the plot to the other. Express the length of a row of flowers as a function of its distance from the center line of yellow tulips. 51. (a) A bead maker has a collection of wooden spheres 2 centimeters in diameter and is making beads by drilling holes through the center of each sphere. The length of the bead is a function of the diameter of the hole he drills. Find a formula for this function. 2 cm length of bead diameter of hole If you are stuck, begin by trying to express half the length of the bead as a function of the radius of the hole drilled. (b) More generally, suppose he works with spherical beads of radius R. Again, the length of the bead is a function of the diameter of the hole he drills. Find a formula for this function. In the following problems, demonstrate your use of the portable strategies for problem solving described in this chapter. What simpler questions are you asking yourself? What concrete example can you give to convince your friends and relatives that you are right? Write this up clearly, so a reader can follow your train of thought easily. 52. At the Central Perk coffeehouse in Manhattan, Rachel serves c cups of coffee and d desserts per hour. The coffee costs a dollars per cup, and the desserts cost b dollars each. She averages a tip of k cents per dollar of the customers’ bills (excluding taxes). In addition, she makes a fixed wage of F dollars per hour. Consider c, d, a, b, k, and F as constants. Express Rachel’s earnings as a function of h, the number of hours she works. (In actuality, Rachel’s earnings are not a function of the hours she puts in. Other considerations complicate the situation. For instance, business is slow at certain times of the day, and some customers tip more generously than others. Nevertheless, by using the information provided, we can make a mathematical model of the situation that gives us a reasonably accurate picture.) 17 “Roundabout” is a British term for traffic circle. 46 CHAPTER 1 Functions Are Lurking Everywhere 53. I work an h-hour day. I spend 1/w of these h hours on the road and the remainder in consultation. I receive A dollars per hour as a consultant. I receive no money when I’m on the road. In fact, each day I pay G dollars in gas and tolls and I estimate that each day costs C cents in wear and tear on the car. I have no other expenses. Express my daily profit as a function of h, the number of hours I work. (A, w, G, and C are all constants.) (Note: If w = 5, I spend 1/5 of my workday on the road and the rest of my workday in consultation.) 54. Two bears, Bruno and Lollipop, discover a patch of huckleberries one morning. The patch covers an area of A acres and there are X bushels of huckleberries per acre. Bruno eats B bushels of huckleberries per hour; Lollipop can devour L bushels of huckleberries in C hours. Express your answers to parts (a) and (b) in terms of any or all of the constants A, X, B, L, and C. (a) Express the number of bushels of huckleberries the two bears eat as a function of t, the number of hours they have been eating. (b) In t hours, how many acres of huckleberries can the two bears together finish off? (c) Assuming that after T hours the bears have not yet finished the berry patch, how many hours longer does it take them to finish all the huckleberries in the patch? Express your answers in terms of any or all of the constants A, X, B, L, C, and T . If you are having difficulty, use this time-tested technique: Give the quantity you are looking for a name. (Avoid the letters already standing for something else.) 55. A manufacturer is packaging oatmeal in cylindrical containers. She needs the volume of the container to be 88 cubic inches in order to hold 18 ounces of rolled oats. Given this requirement, the height of the cylinder will depend upon the radius, r, selected. (a) Express the height, h, of the cylindrical oatmeal container with volume 88 cubic inches as a function of r. (b) Suppose that the lid, bottom, and sides of the container are all made of cardboard. (The lid will be attached to the container with tape.) Express the number of square inches of cardboard used in the container as a function of r, the radius of the container. (To figure out how much cardboard is used in the sides of the container, imagine cutting along the height and rolling the sheet out into a rectangle. The height of the rectangle will be h. What will its length be?) (c) When making the containers, the lid and the base are cut from squares of cardboard, 2r by 2r, and the excess cardboard is tossed into a recycling bin. Assume that the company must pay full price for the excess cardboard it uses. If cardboard costs k cents per square inch, express the cost of the cardboard for the container as a function of r.(kis a constant.) (d) Suppose the manufacturer decides to switch to plastic lids and bottoms to eliminate the taping problems. Assume that custom-made plastic lids and bottoms cost 7k cents per square inch. Express the cost of the container as a function of r. 56. A typist can type W words per minute. On average, each computer illustration takes C minutes to create and I minutes to insert. (a) What is the estimated amount of time it will take for this typist to create a document N words long and containing Z illustrations? 1.3 Representations of Functions 47 (b) The typist is paid $13 per hour for typing and a flat rate of $10 per picture. The cost of getting a document typed is a function of its length and the number of pictures. Write a function that gives a good estimate of the cost of getting a document of x words typed, assuming that the ratio of illustrations to words is 1:1000. (c) Given the document described in part (b), express the typist’s wages per hour as a function of x. 57. Filene’s Basement regularly marks down its merchandise. A discounted item now costs D dollars. This is after a p-percent markdown. Express the initial price of the item in terms of p and D. Try out your answer in the concrete case of an item that now costs $100 after a 20-percent markdown. Why should your answer not be $120? 58. Amir and Omar are tiling an area measuring A square meters. They lay down N tiles per square meter. Omar can put down q tiles in r hours while it takes Amir m minutes to lay one tile. A, N, q, r, and m are constants. (a) Give the number of tiles Amir and Omar can put down as a function of t, the number of hours they work together. (b) How many square meters can they tile in t hours? (c) After H hours of working with Omar, Amir leaves. The job is not yet done. How many hours will it take Omar to finish the job alone? Express the answer in terms of any or all of the constants A, N, q, r, m, and H . 59. Sam Wright plays the role of Sebastian the crab in the Disney film “The Little Mer- maid.” He spent H hours working on the production. P percent of this time was spent on the taping; of the remaining time, 1/n was spent on rehearsal and the rest on dub- bing and looping. Sam was paid D dollars per hour for each of the H hours spent on the production. (a) If Disney had changed the contract so that they paid for taping, dubbing, and looping but not rehearsal, how much would Sam’s pay have been? (b) If Disney were paying for taping only, and Sam wanted to earn the same thing he would have under the original contract, how much would he need to charge per hour? 2 CHAPTER Characterizing Functions and Introducing Rates of Change 2.1 FEATURES OF A FUNCTION: POSITIVE/NEGATIVE, INCREASING/DECREASING, CONTINUOUS/DISCONTINUOUS Just as you might characterize a fish by type, behavior, and habitat, so too can you charac- terize a function by type and behavior. Throughout this book we will build up a “type” cat- egorization of groups of functions (polynomial, rational, exponential, logarithmic, trigono- metric); for now we will focus on some basic behavioral descriptions that can be used to characterize functions. In characterizing a function we direct our attention to the values of the function—the outputs. Positive/Negative We say f is positive at a if the output value f(a)is positive; f is negative at a if the output value f(a)is negative. Graphically, this means that f is positive wherever its graph lies above the horizontal axis; f is negative wherever its graph lies below the horizontal axis. The graph of f can change sign from positive to negative (or vice versa) by either cutting through the horizontal axis or leaping over the horizontal axis. Frequently in our study of calculus we will uncover a wealth of information behind simple sign information. 49 50 CHAPTER 2 Characterizing Functions and Introducing Rates of Change Suppose y is a function of x. We label the horizontal (input) axis the x-axis and the vertical (output) axis the y-axis. Wherever the graph of f meets with the x-axis the value of the function is zero. The x-values at which the graph of f meets the x-axis are the zeros of f . If f(a)= 0,then a is called a zero of f or an x-intercept of f . We say that a is a root of the equation f(x)= 0.For example, if f(x)= (x − 1) 2 as in Figure 2.1(a), x = 1isa root of the equation f(x)=0. Each of the functions in Figure 2.1 has a zero at x = 1. As illustrated in Figure 2.1, the sign of f may change on either side of a zero or it may remain the same on either side of a zero. (There is a lack of consensus on terminology; sometimes one speaks of roots of functions and zeros of equations.) (a) f (x)=(x–1) 2 (b) f (x)=–(x–1) 2 (c) f (x ) = (x – 1) (d) f (x) = – (x – 1) The sign of f remains the same on either side of the zero of f The sign of f changes on either side of the zero of f –3 –2 –1 1 2 3 –3 –2 –1123 –3 –2 –1 1 2 3 –3 –2 –1123 –3 –2 –1 1 2 3 –3 –2 –1123 –3 –2 –1 1 2 3 –3 –2 –1123 ff ff Figure 2.1 If x = 0 is in the domain of the function f , then the graph of f will intersect the y-axis. The y-value (output) corresponding to x = 0, denoted f(0),iscalled the y-intercept of f . 1 If f is a function, then it can have at most one y-intercept. (Why?) There can be any number of x-intercepts because several different inputs can be assigned an output of 0. 2 The question of whether f(x)(the output) is positive or negative is completely different from the question of whether the x (the input) is positive or negative. EXERCISE 2.1 The following two graphs give a “happiness index” as a function of outdoor temperature (in degrees Celsius). When the index is positive the person is happy, and when the index is negative the person is unhappy. Can you guess which person is a skier and which is a bicyclist? 1 Keep x-intercepts and y-intercepts straight in your mind: i. x-intercepts are x-values (inputs) where the output y is zero. A point where the graph crosses the x-axis is of the form (a,0) where the number a is the x-intercept. ii. the y-intercept is the y-value (output) where x is zero. A point where the graph crosses the y-axis is of the form (0, b) where the number b is the y-intercept. 2 A function that is 1-to-1 can cross the x-axis at most once. . answers to parts (a) and (b) in terms of any or all of the constants A, X, B, L, and C. (a) Express the number of bushels of huckleberries the two bears eat as a function of t, the number of hours. manufacturer decides to switch to plastic lids and bottoms to eliminate the taping problems. Assume that custom-made plastic lids and bottoms cost 7k cents per square inch. Express the cost of. below. The distance from the highest point of the arch to the floor is 7 meters. Denote the width of the archway by w and the height of the vertical wall of the rectangle y. The width and height are