Chapter 1 - 5E-FM.qk 11:09 AM Page 1 CA L C U L U S...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5E-FM.qk 1/19/06 11:09 AM Page 1 CA L C U L U S 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 2 By the time you finish this course, you will be able to explain the formation and location of rainbows, compute the force exerted by water on a dam, analyze the population cycles of predators and prey, and calculate the escape velocity of a rocket. A P review of Calculus 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 3 Calculus is fundamentally different from the mathematics that you have studied previously. Calculus is less static and more dynamic. It is concerned with change and motion; it deals with quantities that approach other quantities. For that reason it may be useful to have an overview of the subject before beginning its intensive study. Here we give a glimpse of some of the main ideas of calculus by showing how the concept of a limit arises when we attempt to solve a variety of problems. A¡ A™ A£ A¢ A∞ The Area Problem The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the “method of exhaustion.” They knew how to find the area A of any polygon by dividing it into triangles as in Figure 1 and adding the areas of these triangles. It is a much more difficult problem to find the area of a curved figure. The Greek method of exhaustion was to inscribe polygons in the figure and circumscribe polygons about the figure and then let the number of sides of the polygons increase. Figure 2 illustrates this process for the special case of a circle with inscribed regular polygons. A=A¡+A™+A£+A¢+A∞ FIGURE 1 A£ FIGURE 2 A¢ A∞ Aß A¶ A¡™ The Preview Module is a numerical and pictorial investigation of the approximation of the area of a circle by inscribed and circumscribed polygons. Let An be the area of the inscribed polygon with n sides. As n increases, it appears that An becomes closer and closer to the area of the circle. We say that the area of the circle is the limit of the areas of the inscribed polygons, and we write A nl lim An The Greeks themselves did not use limits explicitly. However, by indirect reasoning, Eudoxus (fifth century B.C.) used exhaustion to prove the familiar formula for the area of a circle: A r 2. We will use a similar idea in Chapter 5 to find areas of regions of the type shown in Figure 3. We will approximate the desired area A by areas of rectangles (as in Figure 4), let the width of the rectangles decrease, and then calculate A as the limit of these sums of areas of rectangles. y (1, 1) y (1, 1) y (1, 1) y (1, 1) y=≈ A 0 1 x 0 1 4 1 2 3 4 1 x 0 1 x 0 1 n 1 x FIGURE 3 FIGURE 4 3 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 4 4 ❙❙❙❙ A PREVIEW OF CALCULUS Is it possible to fill a circle with rectangles? Try it for yourself. Resources / Module 1 / Area / Rectangles in Circles y The area problem is the central problem in the branch of calculus called integral calculus. The techniques that we will develop in Chapter 5 for finding areas will also enable us to compute the volume of a solid, the length of a curve, the force of water against a dam, the mass and center of gravity of a rod, and the work done in pumping water out of a tank. t y=ƒ P The Tangent Problem Consider the problem of trying to find an equation of the tangent line t to a curve with equation y f x at a given point P. (We will give a precise definition of a tangent line in Chapter 2. For now you can think of it as a line that touches the curve at P as in Figure 5.) Since we know that the point P lies on the tangent line, we can find the equation of t if we know its slope m. The problem is that we need two points to compute the slope and we know only one point, P, on t . To get around the problem we first find an approximation to m by taking a nearby point Q on the curve and computing the slope mPQ of the secant line PQ. From Figure 6 we see that 1 0 x FIGURE 5 The tangent line at P y mPQ fx x fa a t Q { x, ƒ} ƒ-f(a) x-a P { a, f(a)} Now imagine that Q moves along the curve toward P as in Figure 7. You can see that the secant line rotates and approaches the tangent line as its limiting position. This means that the slope mPQ of the secant line becomes closer and closer to the slope m of the tangent line. We write m Q lP lim mPQ 0 a x x and we say that m is the limit of mPQ as Q approaches P along the curve. Since x approaches a as Q approaches P, we could also use Equation 1 to write fx x fa a FIGURE 6 The secant line PQ y 2 m xla lim t Q P 0 x FIGURE 7 Secant lines approaching the tangent line Specific examples of this procedure will be given in Chapter 2. The tangent problem has given rise to the branch of calculus called differential calculus, which was not invented until more than 2000 years after integral calculus. The main ideas behind differential calculus are due to the French mathematician Pierre Fermat (1601–1665) and were developed by the English mathematicians John Wallis (1616–1703), Isaac Barrow (1630–1677), and Isaac Newton (1642–1727) and the German mathematician Gottfried Leibniz (1646–1716). The two branches of calculus and their chief problems, the area problem and the tangent problem, appear to be very different, but it turns out that there is a very close connection between them. The tangent problem and the area problem are inverse problems in a sense that will be described in Chapter 5. Velocity When we look at the speedometer of a car and read that the car is traveling at 48 mi h, what does that information indicate to us? We know that if the velocity remains constant, then after an hour we will have traveled 48 mi. But if the velocity of the car varies, what does it mean to say that the velocity at a given instant is 48 mi h? 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 5 A P REVIEW OF CALCULUS ❙❙❙❙ 5 In order to analyze this question, let’s examine the motion of a car that travels along a straight road and assume that we can measure the distance traveled by the car (in feet) at l-second intervals as in the following chart: t d Time elapsed (s) Distance (ft) 0 0 1 2 2 10 3 25 4 43 5 78 As a first step toward finding the velocity after 2 seconds have elapsed, we find the average velocity during the time interval 2 t 4: average velocity distance traveled time elapsed 43 4 10 2 16.5 ft s Similarly, the average velocity in the time interval 2 average velocity 25 3 10 2 t 3 is 15 ft s We have the feeling that the velocity at the instant t 2 can’t be much different from the average velocity during a short time interval starting at t 2. So let’s imagine that the distance traveled has been measured at 0.l-second time intervals as in the following chart: t d 2.0 10.00 2.1 11.02 2.2 12.16 2.3 13.45 2.4 14.96 2.5 16.80 Then we can compute, for instance, the average velocity over the time interval 2, 2.5 : average velocity 16.80 2.5 10.00 2 13.6 ft s d The results of such calculations are shown in the following chart: Time interval Average velocity (ft s) Q { t, f(t)} 2, 3 15.0 2, 2.5 13.6 2, 2.4 12.4 2, 2.3 11.5 2, 2.2 10.8 2, 2.1 10.2 20 10 0 1 2 P { 2, f(2)} 3 4 5 t The average velocities over successively smaller intervals appear to be getting closer to a number near 10, and so we expect that the velocity at exactly t 2 is about 10 ft s. In Chapter 2 we will define the instantaneous velocity of a moving object as the limiting value of the average velocities over smaller and smaller time intervals. In Figure 8 we show a graphical representation of the motion of the car by plotting the distance traveled as a function of time. If we write d f t , then f t is the number of feet traveled after t seconds. The average velocity in the time interval 2, t is average velocity distance traveled time elapsed ft t f2 2 FIGURE 8 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 6 6 ❙❙❙❙ A PREVIEW OF CALCULUS which is the same as the slope of the secant line PQ in Figure 8. The velocity v when t is the limiting value of this average velocity as t approaches 2; that is, v 2 lim tl2 ft t f2 2 and we recognize from Equation 2 that this is the same as the slope of the tangent line to the curve at P. Thus, when we solve the tangent problem in differential calculus, we are also solving problems concerning velocities. The same techniques also enable us to solve problems involving rates of change in all of the natural and social sciences. The Limit of a Sequence In the fifth century B.C. the Greek philosopher Zeno of Elea posed four problems, now known as Zeno’s paradoxes, that were intended to challenge some of the ideas concerning space and time that were held in his day. Zeno’s second paradox concerns a race between the Greek hero Achilles and a tortoise that has been given a head start. Zeno argued, as follows, that Achilles could never pass the tortoise: Suppose that Achilles starts at position a 1 and the tortoise starts at position t1 . (See Figure 9.) When Achilles reaches the point a 2 t1, the tortoise is farther ahead at position t2. When Achilles reaches a 3 t2, the tortoise is at t3. This process continues indefinitely and so it appears that the tortoise will always be ahead! But this defies common sense. a¡ Achilles FIGURE 9 tortoise t¡ t™ t£ t¢ a™ a£ a¢ a∞ ... ... One way of explaining this paradox is with the idea of a sequence. The successive positions of Achilles a 1, a 2 , a 3 , . . . or the successive positions of the tortoise t1, t2 , t3 , . . . form what is known as a sequence. In general, a sequence a n is a set of numbers written in a definite order. For instance, the sequence {1, 1 , 1 , 1 , 1 , . . .} 2345 can be described by giving the following formula for the nth term: an a¢ a£ 0 a™ a¡ 1 1 n (a) 1 We can visualize this sequence by plotting its terms on a number line as in Figure 10(a) or by drawing its graph as in Figure 10(b). Observe from either picture that the terms of the sequence a n 1 n are becoming closer and closer to 0 as n increases. In fact, we can find terms as small as we please by making n large enough. We say that the limit of the sequence is 0, and we indicate this by writing lim 1 n 0 12345678 n nl (b) FIGURE 10 In general, the notation nl lim a n L 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 7 A P REVIEW OF CALCULUS ❙❙❙❙ 7 is used if the terms a n approach the number L as n becomes large. This means that the numbers a n can be made as close as we like to the number L by taking n sufficiently large. The concept of the limit of a sequence occurs whenever we use the decimal representation of a real number. For instance, if a1 a2 a3 a4 a5 a6 a7 3.1 3.14 3.141 3.1415 3.14159 3.141592 3.1415926 then nl lim a n The terms in this sequence are rational approximations to . Let’s return to Zeno’s paradox. The successive positions of Achilles and the tortoise form sequences a n and tn , where a n tn for all n. It can be shown that both sequences have the same limit: nl lim a n p nl lim tn It is precisely at this point p that Achilles overtakes the tortoise. The Sum of a Series Watch a movie of Zeno’s attempt to reach the wall. Resources / Module 1 / Introduction / Zeno’s Paradox Another of Zeno’s paradoxes, as passed on to us by Aristotle, is the following: “A man standing in a room cannot walk to the wall. In order to do so, he would first have to go half the distance, then half the remaining distance, and then again half of what still remains. This process can always be continued and can never be ended.” (See Figure 11.) FIGURE 11 1 2 1 4 1 8 1 16 Of course, we know that the man can actually reach the wall, so this suggests that perhaps the total distance can be expressed as the sum of infinitely many smaller distances as follows: 3 1 1 2 1 4 1 8 1 16 1 2n 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 8 8 ❙❙❙❙ A PREVIEW OF CALCULUS Zeno was arguing that it doesn’t make sense to add infinitely many numbers together. But there are other situations in which we implicitly use infinite sums. For instance, in decimal notation, the symbol 0.3 0.3333 . . . means 3 10 3 100 3 1000 3 10,000 and so, in some sense, it must be true that 3 10 3 100 3 1000 3 10,000 1 3 More generally, if dn denotes the nth digit in the decimal representation of a number, then 0.d1 d2 d3 d4 . . . d1 10 d2 10 2 d3 10 3 dn 10 n Therefore, some infinite sums, or infinite series as they are called, have a meaning. But we must define carefully what the sum of an infinite series is. Returning to the series in Equation 3, we denote by sn the sum of the first n terms of the series. Thus s1 s2 s3 s4 s5 s6 s7 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0.5 1 4 1 4 1 4 1 4 1 4 1 4 0.75 1 8 1 8 1 8 1 8 1 8 0.875 1 16 1 16 1 16 1 16 0.9375 1 32 1 32 1 32 0.96875 1 64 1 64 0.984375 1 128 0.9921875 s10 1 2 1 4 1 1024 0.99902344 s16 1 2 1 4 1 2 16 0.99998474 Observe that as we add more and more terms, the partial sums become closer and closer to 1. In fact, it can be shown that by taking n large enough (that is, by adding sufficiently many terms of the series), we can make the partial sum sn as close as we please to the number 1. It therefore seems reasonable to say that the sum of the infinite series is 1 and to write 1 2 1 4 1 8 1 2n 1 5E-Preview (pp 02-09) 1/17/06 11:44 AM Page 9 A P REVIEW OF CALCULUS ❙❙❙❙ 9 In other words, the reason the sum of the series is 1 is that nl lim sn 1 In Chapter 12 we will discuss these ideas further. We will then use Newton’s idea of combining infinite series with differential and integral calculus. Summary We have seen that the concept of a limit arises in trying to find the area of a region, the slope of a tangent to a curve, the velocity of a car, or the sum of an infinite series. In each case the common theme is the calculation of a quantity as the limit of other, easily calculated quantities. It is this basic idea of a limit that sets calculus apart from other areas of mathematics. In fact, we could define calculus as the part of mathematics that deals with limits. Sir Isaac Newton invented his version of calculus in order to explain the motion of the planets around the Sun. Today calculus is used in calculating the orbits of satellites and spacecraft, in predicting population sizes, in estimating how fast coffee prices rise, in forecasting weather, in measuring the cardiac output of the heart, in calculating life insurance premiums, and in a great variety of other areas. We will explore some of these uses of calculus in this book. In order to convey a sense of the power of the subject, we end this preview with a list of some of the questions that you will be able to answer using calculus: rays from Sun 1. How can we explain the fact, illustrated in Figure 12, that the angle of elevation 2. 3. 4. 5. 138° rays from Sun 42° observer FIGURE 12 6. 7. 8. 9. 10. from an observer up to the highest point in a rainbow is 42°? (See page 232.) How can we explain the shapes of cans on supermarket shelves? (See page 288.) Where is the best place to sit in a movie theater? (See page 485.) How far away from an airport should a pilot start descent? (See page 197.) How can we fit curves together to design shapes to represent letters on a laser printer? (See page 705.) Where should an infielder position himself to catch a baseball thrown by an outfielder and relay it to home plate? (See page 658.) Does a ball thrown upward take longer to reach its maximum height or to fall back to its original height? (See page 646.) How can we explain the fact that planets and satellites move in elliptical orbits? (See page 916.) How can we distribute water flow among turbines at a hydroelectric station so as to maximize the total energy production? (See page 1009.) If a marble, a squash ball, a steel bar, and a lead pipe roll down a slope, which of them reaches the bottom first? (See page 1075.) 5E-01(pp 10-19) 1/17/06 11:56 AM Page 10 CHAPTER 1 A graphical representation of a function––here the number of hours of daylight as a function of the time of year at various latitudes–– is often the most natural and convenient way to represent the function. F unctions and Models 5E-01(pp 10-19) 1/17/06 11:56 AM Page 11 The fundamental objects that we deal with in calculus are functions. This chapter prepares the way for calculus by discussing the basic ideas concerning functions, their graphs, and ways of transforming and combining them. We stress that a function can be represented in different ways: by an equation, in a table, by a graph, or in words. We look at the main types of functions that occur in calculus and describe the process of using these functions as mathematical models of real-world phenomena. We also discuss the use of graphing calculators and graphing software for computers. |||| 1.1 Four Ways to Represent a Function Functions arise whenever one quantity depends on another. Consider the following four situations. A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A r 2. With each positive number r there is associated one value of A, and we say that A is a function of r. B. The human population of the world P depends on the time t. The table gives estimates of the world population P t at time t, for certain years. For instance, P 1950 2,560,000,000 Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Population (millions) 1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 But for each value of the time t there is a corresponding value of P, and we say that P is a function of t. C. The cost C of mailing a first-class letter depends on the weight w of the letter. Although there is no simple formula that connects w and C, the post office has a rule for determining C when w is known. D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a. a {cm/s@} 100 50 5 10 15 20 25 30 t (seconds) FIGURE 1 _50 Calif. Dept. of Mines and Geology Vertical ground acceleration during the Northridge earthquake 11 5E-01(pp 10-19) 1/17/06 11:56 AM Page 12 12 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS Each of these examples describes a rule whereby, given a number (r, t, w, or t), another number ( A, P, C, or a) is assigned. In each case we say that the second number is a function of the first number. A function f is a rule that assigns to each element x in a set A exactly one element, called f x , in a set B. We usually consider functions for which the sets A and B are sets of real numbers. The set A is called the domain of the function. The number f x is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of f x as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable. It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f x according to the rule of the function. Thus, we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. The preprogrammed functions in a calculator are good examples of a function as a machine. For example, the square root key on your calculator computes such a function. You press the key labeled s (or s x ) and enter the input x. If x 0, then x is not in the domain of this function; that is, x is not an acceptable input, and the calculator will indicate an error. If x 0, then an approximation to s x will appear in the display. Thus, the s x key on your calculator is not quite the same as the exact mathematical function f defined by f x s x. Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow connects an element of A to an element of B. The arrow indicates that f x is associated with x, f a is associated with a, and so on. The most common method for visualizing a function is its graph. If f is a function with domain A, then its graph is the set of ordered pairs x, f x x A x (input) FIGURE 2 f ƒ (output) Machine diagram for a function ƒ x a ƒ f(a) A FIGURE 3 f B Arrow diagram for ƒ (Notice that these are input-output pairs.) In other words, the graph of f consists of all points x, y in the coordinate plane such that y f x and x is in the domain of f . The graph of a function f gives us a useful picture of the behavior or “life history” of a function. Since the y-coordinate of any point x, y on the graph is y f x , we can read the value of f x from the graph as being the height of the graph above the point x (see Figure 4). The graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis as in Figure 5. y { x, ƒ} y ƒ f (2) f (1) 0 1 2 x x range y ƒ(x) 0 x domain FIGURE 4 FIGURE 5 5E-01(pp 10-19) 1/17/06 11:56 AM Page 13 S ECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ❙❙❙❙ 13 EXAMPLE 1 The graph of a function f is shown in Figure 6. (a) Find the values of f 1 and f 5 . (b) What are the domain and range of f ? y 1 0 1 x F IGURE 6 SOLUTION |||| The notation for intervals is given in Appendix A. (a) We see from Figure 6 that the point 1, 3 lies on the graph of f , so the value of f at 1 is f 1 3. (In other words, the point on the graph that lies above x 1 is 3 units above the x-axis.) When x 5, the graph lies about 0.7 unit below the x-axis, so we estimate that f5 0.7. (b) We see that f x is defined when 0 x 7, so the domain of f is the closed interval 0, 7 . Notice that f takes on all values from 2 to 4, so the range of f is y 2 y 4 2, 4 EXAMPLE 2 Sketch the graph and find the domain and range of each function. (a) f x SOLUTION y 2x 1 (b) t x x2 y=2x-1 0 -1 1 2 x FIGURE 7 (a) The equation of the graph is y 2 x 1, and we recognize this as being the equation of a line with slope 2 and y-intercept 1. (Recall the slope-intercept form of the equation of a line: y m x b. See Appendix B.) This enables us to sketch the graph of f in Figure 7. The expression 2 x 1 is defined for all real numbers, so the domain of f is the set of all real numbers, which we denote by . The graph shows that the range is also . (b) Since t 2 2 2 4 and t 1 1 2 1, we could plot the points 2, 4 and 1, 1 , together with a few other points on the graph, and join them to produce the graph (Figure 8). The equation of the graph is y x 2, which represents a parabola (see Appendix C). The domain of t is . The range of t consists of all values of t x , that is, all numbers of the form x 2. But x 2 0 for all numbers x and any positive number y is a square. So the range of t is y y 0 0, . This can also be seen from Figure 8. y (2, 4) y=≈ (_1, 1) 1 0 1 x FIGURE 8 5E-01(pp 10-19) 1/17/06 11:56 AM Page 14 14 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS Representations of Functions There are four possible ways to represent a function: ■ ■ ■ ■ verbally numerically visually algebraically (by a description in words) (by a table of values) (by a graph) (by an explicit formula) If a single function can be represented in all four ways, it is often useful to go from one representation to another to gain additional insight into the function. (In Example 2, for instance, we started with algebraic formulas and then obtained the graphs.) But certain functions are described more naturally by one method than by another. With this in mind, let’s reexamine the four situations that we considered at the beginning of this section. A. The most useful representation of the area of a circle as a function of its radius is r 2, though it is possible to compile a table of probably the algebraic formula A r values or to sketch a graph (half a parabola). Because a circle has to have a positive 0, , and the range is also 0, . radius, the domain is r r 0 B. We are given a description of the function in words: P t is the human population of the world at time t. The table of values of world population on page 11 provides a convenient representation of this function. If we plot these values, we get the graph (called a scatter plot) in Figure 9. It too is a useful representation; the graph allows us to absorb all the data at once. What about a formula? Of course, it’s impossible to devise an explicit formula that gives the exact human population P t at any time t. But it is possible to find an expression for a function that approximates P t . In fact, using methods explained in Section 1.2, we obtain the approximation Pt ft 0.008079266 1.013731 t and Figure 10 shows that it is a reasonably good “fit.” The function f is called a mathematical model for population growth. In other words, it is a function with an explicit formula that approximates the behavior of our given function. We will see, however, that the ideas of calculus can be applied to a table of values; an explicit formula is not necessary. P 6x10' P 6x10 ' 1900 1920 1940 1960 1980 2000 t 1900 1920 1940 1960 1980 2000 t FIGURE 9 FIGURE 10 5E-01(pp 10-19) 1/17/06 11:56 AM Page 15 S ECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ❙❙❙❙ 15 |||| A function defined by a table of values is called a tabular function. w (ounces) C w (dollars) 0.37 0.60 0.83 1.06 1.29 0 1 2 3 4 w w w w w 1 2 3 4 5 The function P is typical of the functions that arise whenever we attempt to apply calculus to the real world. We start with a verbal description of a function. Then we may be able to construct a table of values of the function, perhaps from instrument readings in a scientific experiment. Even though we don’t have complete knowledge of the values of the function, we will see throughout the book that it is still possible to perform the operations of calculus on such a function. C. Again the function is described in words: C w is the cost of mailing a first-class letter with weight w. The rule that the U.S. Postal Service used as of 2002 is as follows: The cost is 37 cents for up to one ounce, plus 23 cents for each successive ounce up to 11 ounces. The table of values shown in the margin is the most convenient representation for this function, though it is possible to sketch a graph (see Example 10). D. The graph shown in Figure 1 is the most natural representation of the vertical acceleration function a t . It’s true that a table of values could be compiled, and it is even possible to devise an approximate formula. But everything a geologist needs to know—amplitudes and patterns—can be seen easily from the graph. (The same is true for the patterns seen in electrocardiograms of heart patients and polygraphs for liedetection.) Figures 11 and 12 show the graphs of the north-south and east-west accelerations for the Northridge earthquake; when used in conjunction with Figure 1, they provide a great deal of information about the earthquake. a {cm/s@} 200 a {cm/s@} 400 200 100 5 _200 10 15 20 25 30 t (seconds) _100 5 10 15 20 25 30 t (seconds) _400 Calif. Dept. of Mines and Geology _200 Calif. Dept. of Mines and Geology FIGURE 11 North-south acceleration for the Northridge earthquake FIGURE 12 East-west acceleration for the Northridge earthquake In the next example we sketch the graph of a function that is defined verbally. EXAMPLE 3 When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on. SOLUTION The initial temperature of the running water is close to room temperature because of the water that has been sitting in the pipes. When the water from the hotwater tank starts coming out, T increases quickly. In the next phase, T is constant at the temperature of the heated water in the tank. When the tank is drained, T decreases to the temperature of the water supply. This enables us to make the rough sketch of T as a function of t in Figure 13. T 0 t FIGURE 13 5E-01(pp 10-19) 1/17/06 11:56 AM Page 16 16 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS A more accurate graph of the function in Example 3 could be obtained by using a thermometer to measure the temperature of the water at 10-second intervals. In general, scientists collect experimental data and use them to sketch the graphs of functions, as the next example illustrates. t 0 2 4 6 8 Ct 0.0800 0.0570 0.0408 0.0295 0.0210 EXAMPLE 4 The data shown in the margin come from an experiment on the lactonization of hydroxyvaleric acid at 25 C. They give the concentration C t of this acid (in moles per liter) after t minutes. Use these data to draw an approximation to the graph of the concentration function. Then use this graph to estimate the concentration after 5 minutes. SOLUTION We plot the five points corresponding to the data from the table in Figure 14. The curve-fitting methods of Section 1.2 could be used to choose a model and graph it. But the data points in Figure 14 look quite well behaved, so we simply draw a smooth curve through them by hand as in Figure 15. C( t ) 0.08 0.06 0.04 0.02 0 1 2345678 t C (t ) 0.08 0.06 0.04 0.02 0 1 2345678 t FIGURE 14 FIGURE 15 Then we use the graph to estimate that the concentration after 5 minutes is C5 0.035 mole liter In the following example we start with a verbal description of a function in a physical situation and obtain an explicit algebraic formula. The ability to do this is a useful skill in solving calculus problems that ask for the maximum or minimum values of quantities. EXAMPLE 5 A rectangular storage container with an open top has a volume of 10 m3. The length of its base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the width of the base. SOLUTION We draw a diagram as in Figure 16 and introduce notation by letting w and 2w be the width and length of the base, respectively, and h be the height. The area of the base is 2w w 2w 2, so the cost, in dollars, of the material for the base is 10 2w 2 . Two of the sides have area wh and the other two have area 2wh, so the 2 2wh . The total cost is therefore cost of the material for the sides is 6 2 wh w 2w FIGURE 16 h C 10 2w 2 6 2 wh 2 2wh 20w 2 36wh To express C as a function of w alone, we need to eliminate h and we do so by using the fact that the volume is 10 m3. Thus w 2w h 10 5 w2 which gives h 10 2w 2 5E-01(pp 10-19) 1/17/06 11:56 AM Page 17 S ECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ❙❙❙❙ 17 |||| In setting up applied functions as in Example 5, it may be useful to review the principles of problem solving as discussed on page 58, particularly Step 1: Understand the Problem. Substituting this into the expression for C, we have C Therefore, the equation Cw expresses C as a function of w. EXAMPLE 6 Find the domain of each function. 20w 2 36w 5 w 2 20w 2 180 w 20w 2 180 w w 0 (a) f x SOLUTION |||| If a function is given by a formula and the domain is not stated explicitly, the convention is that the domain is the set of all numbers for which the formula makes sense and defines a real number. sx 2 (b) t x 1 x 2 x (a) Because the square root of a negative number is not defined (as a real number), the domain of f consists of all values of x such that x 2 0. This is equivalent to x 2, so the domain is the interval 2, . (b) Since 1 1 tx 2 x x xx 1 and division by 0 is not allowed, we see that t x is not defined when x Thus, the domain of t is xx 0, x 1 0 or x 1. which could also be written in interval notation as ,0 0, 1 1, The graph of a function is a curve in the xy-plane. But the question arises: Which curves in the xy-plane are graphs of functions? This is answered by the following test. The Vertical Line Test A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. The reason for the truth of the Vertical Line Test can be seen in Figure 17. If each vertical line x a intersects a curve only once, at a, b , then exactly one functional value b. But if a line x a intersects the curve twice, at a, b and a, c , is defined by f a then the curve can’t represent a function because a function can’t assign two different values to a. y x=a (a, b) y (a, c) x=a (a, b) 0 a x 0 a x FIGURE 17 5E-01(pp 10-19) 1/17/06 11:56 AM Page 18 18 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS For example, the parabola x y 2 2 shown in Figure 18(a) is not the graph of a function of x because, as you can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions of x. Notice that the equation x y 2 2 implies y 2 x 2, so y s x 2. Thus, the upper and lower halves of the parabola are the graphs of the functions f x s x 2 [from Example 6(a)] and tx s x 2. [See Figures 18(b) and (c).] We observe that if we reverse the roles of x and y, then the equation x h y y 2 2 does define x as a function of y (with y as the independent variable and x as the dependent variable) and the parabola now appears as the graph of the function h. y y y _2 (_2, 0) 0 x _2 0 x 0 x FIGURE 18 (a) x=¥-2 (b) y=œ„„„„ x+2 (c) y=_ œ„„„„ x+2 Piecewise Defined Functions The functions in the following four examples are defined by different formulas in different parts of their domains. EXAMPLE 7 A function f is defined by fx 1 x2 x if x if x 1 1 Evaluate f 0 , f 1 , and f 2 and sketch the graph. SOLUTION Remember that a function is a rule. For this particular function the rule is the following: First look at the value of the input x. If it happens that x 1, then the value of f x is 1 x. On the other hand, if x 1, then the value of f x is x 2. Since 0 Since 1 y 1, we have f 0 1, we have f 1 1, we have f 2 1 1 22 0 1 4. 1. 0. Since 2 1 x 1 FIGURE 19 How do we draw the graph of f ? We observe that if x 1, then f x 1 x, so the part of the graph of f that lies to the left of the vertical line x 1 must coincide with the line y 1 x, which has slope 1 and y-intercept 1. If x 1, then f x x 2, so the part of the graph of f that lies to the right of the line x 1 must coincide with the graph of y x 2, which is a parabola. This enables us to sketch the graph in Figure l9. The solid dot indicates that the point 1, 0 is included on the graph; the open dot indicates that the point 1, 1 is excluded from the graph. 5E-01(pp 10-19) 1/17/06 11:57 AM Page 19 S ECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ❙❙❙❙ 19 The next example of a piecewise defined function is the absolute value function. Recall that the absolute value of a number a, denoted by a , is the distance from a to 0 on the real number line. Distances are always positive or 0, so we have |||| For a more extensive review of absolute values, see Appendix A. a For example, 3 3 3 3 0 0 for every number a 0 s2 1 s2 1 3 3 In general, we have a a a a if a if a 0 0 (Remember that if a is negative, then a is positive.) x. EXAMPLE 8 Sketch the graph of the absolute value function f x y y=| x | SOLUTION From the preceding discussion we know that x 0 x x if x x if x 0 0 FIGURE 20 Using the same method as in Example 7, we see that the graph of f coincides with the x to the left of the line y x to the right of the y-axis and coincides with the line y y-axis (see Figure 20). EXAMPLE 9 Find a formula for the function f graphed in Figure 21. y 1 0 1 x F IGURE 21 SOLUTION The line through 0, 0 and 1, 1 has slope m 1 and y-intercept b 0, so its equation is y x. Thus, for the part of the graph of f that joins 0, 0 to 1, 1 , we have fx |||| Point-slope form of the equation of a line: x if 0 x 1 The line through 1, 1 and 2, 0 has slope m y So we have fx 2 x 0 1x 2 1, so its point-slope form is or y 2 x y See Appendix B. y1 mx x1 if 1 x 2 20 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS We also see that the graph of f coincides with the x-axis for x tion together, we have the following three-piece formula for f : x 2 0 if 0 x if 1 if x x x 2 1 2 2. Putting this informa- fx C 1 EXAMPLE 10 In Example C at the beginning of this section we considered the cost C w of mailing a first-class letter with weight w. In effect, this is a piecewise defined function because, from the table of values, we have Cw 0.37 if 0 0.60 if 1 0.83 if 2 1.06 if 3 w w w w 1 2 3 4 0 1 2 3 4 5 w FIGURE 22 The graph is shown in Figure 22. You can see why functions similar to this one are called step functions—they jump from one value to the next. Such functions will be studied in Chapter 2. Symmetry y If a function f satisfies f x f x for every number x in its domain, then f is called an even function. For instance, the function f x x 2 is even because ƒ f (_x) _x 0 x f x x x 2 x2 fx FIGURE 23 The geometric significance of an even function is that its graph is symmetric with respect to the y-axis (see Figure 23). This means that if we have plotted the graph of f for x 0, we obtain the entire graph simply by reflecting about the y-axis. If f satisfies f x f x for every number x in its domain, then f is called an odd function. For example, the function f x x 3 is odd because f x x 3 An even function x3 fx y The graph of an odd function is symmetric about the origin (see Figure 24). If we already have the graph of f for x 0, we can obtain the entire graph by rotating through 180 about the origin. ƒ x x _x 0 EXAMPLE 11 Determine whether each of the following functions is even, odd, or neither even nor odd. (a) f x x5 x (b) t x 1 x4 (c) h x 2x x 2 SOLUTION 5 FIGURE 24 (a) f x x x 5 x x x 5 1 5x 5 x x An odd function fx Therefore, f is an odd function. (b) So t is even. t x 1 x 4 1 x4 tx S ECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION 2 ❙❙❙❙ 21 (c) Since h x nor odd. h h x and h x x 2 x x 2x x2 h x , we conclude that h is neither even The graphs of the functions in Example 11 are shown in Figure 25. Notice that the graph of h is symmetric neither about the y-axis nor about the origin. y 1 y y f 1 g 1 1 h _1 _1 1 x x 1 x FIGURE 25 (a) ( b) (c) Increasing and Decreasing Functions The graph shown in Figure 26 rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing on the interval a, b , decreasing on b, c , and increasing again on c, d . Notice that if x 1 and x 2 are any two numbers between a and b with x 1 x 2, then f x 1 f x 2 . We use this as the defining property of an increasing function. y B y=ƒ C f(x™) A f(x ¡) D 0 a x¡ x™ b c d x FIGURE 26 A function f is called increasing on an interval I if y f x1 y=≈ f x2 whenever x 1 x 2 in I It is called decreasing on I if f x1 f x2 whenever x 1 x 2 in I 0 x FIGURE 27 In the definition of an increasing function it is important to realize that the inequality f x1 f x 2 must be satisfied for every pair of numbers x 1 and x 2 in I with x 1 x 2. You can see from Figure 27 that the function f x x 2 is decreasing on the interval , 0 and increasing on the interval 0, . 22 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS |||| 1.1 (a) (b) (c) (d) (e) (f) Exercises 5–8 |||| Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. 1. The graph of a function f is given. State the value of f 1 . Estimate the value of f 2 . For what values of x is f x 2? Estimate the values of x such that f x State the domain and range of f. On what interval is f increasing? y 5. y 1 0 1 x 6. y 1 0 1 x 0. 7. 1 0 1 x y 1 0 1 x 8. y 1 0 1 x ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 2. The graphs of f and t are given. (a) State the values of f 4 and t 3 . (b) For what values of x is f x tx? (c) Estimate the solution of the equation f x (d) On what interval is f decreasing? (e) State the domain and range of f. (f) State the domain and range of t. y 9. The graph shown gives the weight of a certain person as a 1. function of age. Describe in words how this persons weight varies over time. What do you think happened when this person was 30 years old? 200 Weight (pounds) 150 100 50 0 10 20 30 40 50 60 70 Age (years) g f 2 0 2 x 10. The graph shown gives a salesman’s distance from his home as a function of time on a certain day. Describe in words what the graph indicates about his travels on this day. 3. Figures 1, 11, and 12 were recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use them to estimate the ranges of the vertical, north-south, and east-west ground acceleration functions at USC during the Northridge earthquake. 4. In this section we discussed examples of ordinary, everyday Distance from home (miles) 8 A.M. 10 NOON 2 4 6 P.M. Time (hours) functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function. 11. You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time. S ECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ❙❙❙❙ 23 12. Sketch a rough graph of the number of hours of daylight as a 23–27 |||| Find the domain of the function. x 3x st 1 4 sx 2 ■ ■ function of the time of year. 13. Sketch a rough graph of the outdoor temperature as a function 23. f x 25. f t 27. h x ■ ■ 1 st 5x ■ ■ 24. f x 26. t u 5x x2 su 3x 4 2 u s4 of time during a typical spring day. 14. You place a frozen pie in an oven and bake it for an hour. Then 3 you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time. 15. A homeowner mows the lawn every Wednesday afternoon. ■ ■ ■ ■ ■ ■ Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period. 16. An airplane flies from an airport and lands an hour later at 28. Find the domain and range and sketch the graph of the function hx 29–40 |||| s4 x 2. Find the domain and sketch the graph of the function. 5 t2 sx 3x x x x 1 if x if x if x if x 1 1 0 0 1 1 6t 5 x 30. F x 32. H t 34. F x 36. t x 1 2 another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let x t be the horizontal distance traveled and y t be the altitude of the plane. (a) Sketch a possible graph of x t . (b) Sketch a possible graph of y t . (c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocity. 17. The number N (in thousands) of cellular phone subscribers in 29. f x 31. f t 33. t x 35. G x x t2 t 3 4 2 2x x x2 1 Malaysia is shown in the table. (Midyear estimates are given.) t N 1991 132 1993 304 1995 873 1997 2461 37. f x 38. f x 2x 3 3x x x2 (a) Use the data to sketch a rough graph of N as a function of t. (b) Use your graph to estimate the number of cell-phone subscribers in Malaysia at midyear in 1994 and 1996. 18. Temperature readings T (in °F) were recorded every two hours 39. f x 2 if x if x from midnight to 2:00 P.M. in Dallas on June 2, 2001. The time t was measured in hours from midnight. t T 0 73 2 73 4 70 6 69 8 72 10 81 12 88 14 91 40. f x 1 if x 1 1 3x 2 if x 7 2 x if x 1 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 41–46 |||| Find an expression for the function whose graph is the given curve. (a) Use the readings to sketch a rough graph of T as a function of t. (b) Use your graph to estimate the temperature at 11:00 A.M. 19. If f x 41. The line segment joining the points 42. The line segment joining the points 43. The bottom half of the parabola x 44. The top half of the circle x 45. y 2, 1 and 4, 3, y y 2 6 2 and 6, 3 1 1 y 2 fa x 2, find f 2 , f 2 , f a , f a , 3x 1 , 2 f a , f 2a , f a 2 , [ f a ] 2, and f a h . 2 0 1 2 20. A spherical balloon with radius r inches has volume 4 3 Vr 3 r . Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r 1 inches. 46. 21–22 Find f 2 where h 0. |||| h,f x h , and fx h h x x ■ fx , 1 0 1 x 1 0 1 x 21. f x ■ ■ x ■ x2 ■ ■ ■ 22. f x ■ ■ 1 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 24 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS 47–51 |||| Find a formula for the described function and state its 55. In a certain country, income tax is assessed as follows. There is domain. 47. A rectangle has perimeter 20 m. Express the area of the rect- angle as a function of the length of one of its sides. 48. A rectangle has area 16 m2. Express the perimeter of the rect- angle as a function of the length of one of its sides. 49. Express the area of an equilateral triangle as a function of the length of a side. 50. Express the surface area of a cube as a function of its volume. 51. An open rectangular box with volume 2 m3 has a square base. no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. (a) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of $14,000? On $26,000? (c) Sketch the graph of the total assessed tax T as a function of the income I. 56. The functions in Example 10 and Exercises 54 and 55(a) are Express the surface area of the box as a function of the length of a side of the base. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life. 57–58 |||| Graphs of f and t are shown. Decide whether each function is even, odd, or neither. Explain your reasoning. 52. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window. 57. f y g 58. y f x g x ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 59. (a) If the point 5, 3 is on the graph of an even function, what x 53. A box with an open top is to be constructed from a rectangular other point must also be on the graph? (b) If the point 5, 3 is on the graph of an odd function, what other point must also be on the graph? 60. A function f has domain piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x. 20 x x 12 x x x x x x 5, 5 and a portion of its graph is shown. (a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd. y _5 0 5 x 61–66 |||| Determine whether f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. 54. A taxi company charges two dollars for the first mile (or part of 61. f x 63. f x 65. f x ■ ■ x x2 x3 ■ 2 62. f x x x4 3 a mile) and 20 cents for each succeeding tenth of a mile (or part). Express the cost C (in dollars) of a ride as a function of the distance x traveled (in miles) for 0 x 2, and sketch the graph of this function. x x ■ ■ ■ 64. f x 66. f x ■ ■ 4x 2 2x 2 ■ ■ 3x 3 ■ 1 ■ SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ❙❙❙❙ 25 |||| 1.2 Mathematical Models: A Catalog of Essential Functions A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. Figure 1 illustrates the process of mathematical modeling. Given a real-world problem, our first task is to formulate a mathematical model by identifying and naming the independent and dependent variables and making assumptions that simplify the phenomenon enough to make it mathematically tractable. We use our knowledge of the physical situation and our mathematical skills to obtain equations that relate the variables. In situations where there is no physical law to guide us, we may need to collect data (either from a library or the Internet or by conducting our own experiments) and examine the data in the form of a table in order to discern patterns. From this numerical representation of a function we may wish to obtain a graphical representation by plotting the data. The graph might even suggest a suitable algebraic formula in some cases. Real-world problem Formulate Mathematical model Test Solve FIGURE 1 The modeling process Real-world predictions Interpret Mathematical conclusions The second stage is to apply the mathematics that we know (such as the calculus that will be developed throughout this book) to the mathematical model that we have formulated in order to derive mathematical conclusions. Then, in the third stage, we take those mathematical conclusions and interpret them as information about the original real-world phenomenon by way of offering explanations or making predictions. The final step is to test our predictions by checking against new real data. If the predictions don’t compare well with reality, we need to refine our model or to formulate a new model and start the cycle again. A mathematical model is never a completely accurate representation of a physical situation—it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say. There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs of these functions and give examples of situations appropriately modeled by such functions. Linear Models |||| The coordinate geometry of lines is reviewed in Appendix B. When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for 26 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS the function as y fx mx b where m is the slope of the line and b is the y-intercept. A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear function f x 3x 2 and a table of sample values. Notice that whenever x increases by 0.1, the value of f x increases by 0.3. So f x increases three times as fast as x. Thus, the slope of the graph y 3x 2, namely 3, can be interpreted as the rate of change of y with respect to x. y x y=3x-2 fx 3x 1.0 1.3 1.6 1.9 2.2 2.5 2 0 _2 x 1.0 1.1 1.2 1.3 1.4 1.5 FIGURE 2 EXAMPLE 1 (a) As dry air moves upward, it expands and cools. If the ground temperature is 20 C and the temperature at a height of 1 km is 10 C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? SOLUTION (a) Because we are assuming that T is a linear function of h, we can write T We are given that T 20 when h 20 0, so m0 b 1, so 20 10 and the required linear function is 20 10 C km, and this represents b mh b In other words, the y-intercept is b 20. We are also given that T 10 when h T 10 20 10 m1 20 10h T=_10h+20 The slope of the line is therefore m T 10 0 1 3 h (b) The graph is sketched in Figure 3. The slope is m the rate of change of temperature with respect to height. (c) At a height of h 2.5 km, the temperature is T 10 2.5 20 FIGURE 3 5C If there is no physical law or principle to help us formulate a model, we construct an empirical model, which is based entirely on collected data. We seek a curve that “fits” the data in the sense that it captures the basic trend of the data points. S ECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ❙❙❙❙ 27 TABLE 1 Year 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 CO2 level (in ppm) 338.7 341.1 344.4 347.2 351.5 354.2 356.4 358.9 362.6 366.6 369.4 EXAMPLE 2 Table 1 lists the average carbon dioxide level in the atmosphere, measured in parts per million at Mauna Loa Observatory from 1980 to 2000. Use the data in Table 1 to find a model for the carbon dioxide level. SOLUTION We use the data in Table 1 to make the scatter plot in Figure 4, where t represents time (in years) and C represents the CO2 level (in parts per million, ppm). C 370 360 350 340 FIGURE 4 Scatter plot for the average CO™ level 1980 1985 1990 1995 2000 t Notice that the data points appear to lie close to a straight line, so it’s natural to choose a linear model in this case. But there are many possible lines that approximate these data points, so which one should we use? From the graph, it appears that one possibility is the line that passes through the first and last data points. The slope of this line is 369.4 2000 and its equation is C or 1 338.7 1980 30.7 20 1.535 338.7 1.535 t 1980 C 1.535 t 2700.6 Equation 1 gives one possible linear model for the carbon dioxide level; it is graphed in Figure 5. C 370 360 350 340 FIGURE 5 Linear model through first and last data points 1980 1985 1990 1995 2000 t Although our model fits the data reasonably well, it gives values higher than most of the actual CO2 levels. A better linear model is obtained by a procedure from statistics 28 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS |||| A computer or graphing calculator finds the regression line by the method of least squares, which is to minimize the sum of the squares of the vertical distances between the data points and the line. The details are explained in Section 15.7. called linear regression. If we use a graphing calculator, we enter the data from Table 1 into the data editor and choose the linear regression command. (With Maple we use the fit[leastsquare] command in the stats package; with Mathematica we use the Fit command.) The machine gives the slope and y-intercept of the regression line as m 1.53818 b 2707.25 So our least squares model for the CO2 level is 2 C 1.53818t 2707.25 In Figure 6 we graph the regression line as well as the data points. Comparing with Figure 5, we see that it gives a better fit than our previous linear model. C 370 360 350 340 FIGURE 6 1980 1985 1990 1995 2000 t The regression line EXAMPLE 3 Use the linear model given by Equation 2 to estimate the average CO2 level for 1987 and to predict the level for the year 2010. According to this model, when will the CO2 level exceed 400 parts per million? SOLUTION Using Equation 2 with t 1987, we estimate that the average CO2 level in 1987 2707.25 349.11 was C 1987 1.53818 1987 This is an example of interpolation because we have estimated a value between observed values. (In fact, the Mauna Loa Observatory reported that the average CO2 level in 1987 was 348.93 ppm, so our estimate is quite accurate.) With t 2010, we get C 2010 1.53818 2010 2707.25 384.49 So we predict that the average CO2 level in the year 2010 will be 384.5 ppm. This is an example of extrapolation because we have predicted a value outside the region of observations. Consequently, we are far less certain about the accuracy of our prediction. Using Equation 2, we see that the CO2 level exceeds 400 ppm when 1.53818 t Solving this inequality, we get t 3107.25 1.53818 2020.08 2707.25 400 S ECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ❙❙❙❙ 29 We therefore predict that the CO2 level will exceed 400 ppm by the year 2020. This prediction is somewhat risky because it involves a time quite remote from our observations. Polynomials A function P is called a polynomial if Px an x n an 1 x n 1 a2 x 2 a1 x a0 where n is a nonnegative integer and the numbers a 0 , a 1, a 2 , . . . , a n are constants called the coefficients of the polynomial. The domain of any polynomial is , . If the leading coefficient a n 0, then the degree of the polynomial is n. For example, the function Px 2 x 6 x 4 2 x 3 s2 5 is a polynomial of degree 6. A polynomial of degree 1 is of the form P x m x b and so it is a linear function. A polynomial of degree 2 is of the form P x a x 2 b x c and is called a quadratic function. Its graph is always a parabola obtained by shifting the parabola y a x 2, as we will see in the next section. The parabola opens upward if a 0 and downward if a 0. (See Figure 7.) y 2 y 2 0 1 x 1 x FIGURE 7 The graphs of quadratic functions are parabolas. (a) y=≈+x+1 (b) y=_2≈+3x+1 A polynomial of degree 3 is of the form Px ax 3 bx 2 cx d and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see later why the graphs have these shapes. y y y 1 0 1 x 2 1 x 20 1 x FIGURE 8 (a) y=˛-x+1 (b) y=x$-3≈+x (c) y=3x%-25˛+60x 30 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS Polynomials are commonly used to model various quantities that occur in the natural and social sciences. For instance, in Section 3.4 we will explain why economists often use a polynomial P x to represent the cost of producing x units of a commodity. In the following example we use a quadratic function to model the fall of a ball. T ABLE 2 Time (seconds) 0 1 2 3 4 5 6 7 8 9 Height (meters) 450 445 431 408 375 332 279 216 143 61 EXAMPLE 4 A ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground, and its height h above the ground is recorded at 1-second intervals in Table 2. Find a model to fit the data and use the model to predict the time at which the ball hits the ground. SOLUTION We draw a scatter plot of the data in Figure 9 and observe that a linear model is inappropriate. But it looks as if the data points might lie on a parabola, so we try a quadratic model instead. Using a graphing calculator or computer algebra system (which uses the least squares method), we obtain the following quadratic model: 3 h (meters) 400 h 449.36 0.96t h 400 4.90t 2 200 200 0 2 4 6 8 t (seconds) 0 2 4 6 8 t FIGURE 9 FIGURE 10 Scatter plot for a falling ball Quadratic model for a falling ball In Figure 10 we plot the graph of Equation 3 together with the data points and see that the quadratic model gives a very good fit. The ball hits the ground when h 0, so we solve the quadratic equation 4.90t 2 The quadratic formula gives t The positive root is t 9.7 seconds. 0.96 s 0.96 2 2 0.96t 449.36 0 4 4.90 449.36 4.90 9.67, so we predict that the ball will hit the ground after about Power Functions A function of the form f x consider several cases. (i) a x a, where a is a constant, is called a power function. We n, where n is a positive integer The graphs of f x x n for n 1, 2, 3, 4, and 5 are shown in Figure 11. (These are polynomials with only one term.) We already know the shape of the graphs of y x (a line through the origin with slope 1) and y x 2 [a parabola, see Example 2(b) in Section 1.1]. S ECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ❙❙❙❙ 31 y 1 y=x y 1 y=≈ y 1 y=x # y 1 y=x$ y 1 y=x% 0 1 x 0 1 x 0 1 x 0 1 x 0 1 x FIGURE 11 Graphs of ƒ=x n for n=1, 2, 3, 4, 5 The general shape of the graph of f x x n depends on whether n is even or odd. n If n is even, then f x x is an even function and its graph is similar to the parabola y x 2. If n is odd, then f x x n is an odd function and its graph is similar to that 3 of y x . Notice from Figure 12, however, that as n increases, the graph of y x n becomes flatter near 0 and steeper when x 1. (If x is small, then x 2 is smaller, x 3 is 4 even smaller, x is smaller still, and so on.) y y y=x $ y=x ^ y=≈ (_1, 1) (1, 1) 0 x (1, 1) y=x # y=x % 0 x (_1, _1) FIGURE 12 Families of power functions (ii) a 1 n, where n is a positive integer n The function f x x 1 n sx is a root function. For n 2 it is the square root function f x sx, whose domain is 0, and whose graph is the upper half of the n parabola x y 2. [See Figure 13(a).] For other even values of n, the graph of y sx is 3 similar to that of y sx. For n 3 we have the cube root function f x sx whose domain is (recall that every real number has a cube root) and whose graph is shown in n 3 Figure 13(b). The graph of y sx for n odd n 3 is similar to that of y sx. y y (1, 1) 0 x 0 (1, 1) x FIGURE 13 Graphs of root functions x (a) ƒ=œ„ (b) ƒ=œx #„ 5E-01(pp 32-45) 1/17/06 12:10 PM Page 32 32 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS y (iii) a y=Δ 1 0 1 x 1 The graph of the reciprocal function f x x 1 1 x is shown in Figure 14. Its graph has the equation y 1 x, or xy 1, and is a hyperbola with the coordinate axes as its asymptotes. This function arises in physics and chemistry in connection with Boyle’s Law, which says that, when the temperature is constant, the volume V of a gas is inversely proportional to the pressure P: V C P FIGURE 14 The reciprocal function where C is a constant. Thus, the graph of V as a function of P (see Figure 15) has the same general shape as the right half of Figure 14. V FIGURE 15 Volume as a function of pressure at constant temperature 0 P Another instance in which a power function is used to model a physical phenomenon is discussed in Exercise 22. Rational Functions y A rational function f is a ratio of two polynomials: fx 2 x 20 0 Px Qx where P and Q are polynomials. The domain consists of all values of x such that Q x 0. A simple example of a rational function is the function f x 1 x, whose domain is x x 0 ; this is the reciprocal function graphed in Figure 14. The function fx is a rational function with domain x x 2x 4 x 2 x2 4 1 FIGURE 16 ƒ= 2x$-≈+1 ≈-4 2 . Its graph is shown in Figure 16. Algebraic Functions A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. Here are two more examples: fx sx 2 1 tx x4 x 16 x 2 sx x 3 2 sx 1 5E-01(pp 32-45) 1/17/06 12:10 PM Page 33 S ECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ❙❙❙❙ 33 When we sketch algebraic functions in Chapter 4, we will see that their graphs can assume a variety of shapes. Figure 17 illustrates some of the possibilities. y y y 2 1 x 1 0 5 x 1 0 1 x FIGURE 17 (a) ƒ=x œ„„„„ x+3 $ ≈-25 (b) ©=œ„„„„„„ (c) h(x)=x@?#(x-2)@ An example of an algebraic function occurs in the theory of relativity. The mass of a particle with velocity v is m fv m0 s1 v2 c2 where m 0 is the rest mass of the particle and c a vacuum. 3.0 10 5 km s is the speed of light in Trigonometric Functions Trigonometry and the trigonometric functions are reviewed on Reference Page 2 and also in Appendix D. In calculus the convention is that radian measure is always used (except when otherwise indicated). For example, when we use the function f x sin x, it is understood that sin x means the sine of the angle whose radian measure is x. Thus, the graphs of the sine and cosine functions are as shown in Figure 18. y _ _π π 2 y 3π 2 1 0 _1 π 2 _π 2π 5π 2 _ π 2 1 π 0 _1 π 2 3π 2 3π 2π 5π 2 π 3π x x (a) ƒ=sin x FIGURE 18 (b) ©=cos x Notice that for both the sine and cosine functions the domain is is the closed interval 1, 1 . Thus, for all values of x, we have 1 or, in terms of absolute values, sin x 1 cos x 1 sin x 1 1 cos x 1 , and the range 5E-01(pp 32-45) 1/17/06 12:10 PM Page 34 34 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS Also, the zeros of the sine function occur at the integer multiples of ; that is, sin x 0 when x n n an integer An important property of the sine and cosine functions is that they are periodic functions and have period 2 . This means that, for all values of x, sin x 2 sin x cos x 2 cos x The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves. For instance, in Example 4 in Section 1.3 we will see that a reasonable model for the number of hours of daylight in Philadelphia t days after January 1 is given by the function Lt y 12 2.8 sin 2 t 365 80 The tangent function is related to the sine and cosine functions by the equation tan x sin x cos x 1 _ 3π _π π _ 2 2 0 π 2 π 3π 2 x and its graph is shown in Figure 19. It is undefined whenever cos x 0, that is, when x 2, 3 2, . . . . Its range is , . Notice that the tangent function has period : tan x tan x for all x F IGURE 19 y= tan x The remaining three trigonometric functions (cosecant, secant, and cotangent) are the reciprocals of the sine, cosine, and tangent functions. Their graphs are shown in Appendix D. Exponential Functions The exponential functions are the functions of the form f x a x, where the base a is a x x positive constant. The graphs of y 2 and y 0.5 are shown in Figure 20. In both cases the domain is , and the range is 0, . y y 1 0 1 x 1 0 1 x FIGURE 20 (a) y=2® (b) y=(0.5)® Exponential functions will be studied in detail in Chapter 7, and we will see that they are useful for modeling many natural phenomena, such as population growth (if a 1) and radioactive decay (if a 1 . 5E-01(pp 32-45) 1/17/06 12:10 PM Page 35 S ECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ❙❙❙❙ 35 Logarithmic Functions log a x, where the base a is a positive constant, are the The logarithmic functions f x inverse functions of the exponential functions. They will be studied in Chapter 7. Figure 21 shows the graphs of four logarithmic functions with various bases. In each case the , , and the function increases slowly when x 1. domain is 0, , the range is y y=log ™ x y=log £ x y=log ∞ x y=log ¡¸ x 1 x 1 0 FIGURE 21 Transcendental Functions These are functions that are not algebraic. The set of transcendental functions includes the trigonometric, inverse trigonometric, exponential, and logarithmic functions, but it also includes a vast number of other functions that have never been named. In Chapter 12 we will study transcendental functions that are defined as sums of infinite series. EXAMPLE 5 Classify the following functions as one of the types of functions that we have discussed. 5x x5 (a) f x (b) t x 1x 1 t 5t 4 (c) h x (d) u t 1 sx SOLUTION 5 x is an exponential function. (The x is the exponent.) (a) f x x 5 is a power function. (The x is the base.) We could also consider it to be a (b) t x polynomial of degree 5. 1x (c) h x is an algebraic function. 1 sx 1 t 5t 4 is a polynomial of degree 4. (d) u t |||| 1.2 1–2 Exercises (e) s x 2. (a) y |||| Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. tan 2 x x x 10 x 2t 6 ■ ■ (f) t x (b) y (d) y x log10 x x2 sx 1 6 6 1. (a) f x 5 sx (b) t x x4 (d) r x s1 x2 x3 x2 1 x ■ (c) y (e) y ■ x 10 cos ■ ■ (c) h x x9 t4 ■ ■ (f) y ■ sin ■ ■ ■ 5E-01(pp 32-45) 1/17/06 12:10 PM Page 36 36 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS 3–4 |||| Match each equation with its graph. Explain your choices. (Don’t use a computer or graphing calculator.) (b) What do the slope, the y-intercept, and the x-intercept of the graph represent? 9. The relationship between the Fahrenheit F and Celsius C 3. (a) y x2 (b) y x5 y (c) y g h x8 temperature scales is given by the linear function F 9 C 32. 5 (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the F-intercept and what does it represent? 10. Jason leaves Detroit at 2:00 P.M. and drives at a constant speed 0 x f west along I-96. He passes Ann Arbor, 40 mi from Detroit, at 2:50 P.M. (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent? 11. Biologists have noticed that the chirping rate of crickets of a 4. (a) y (c) y 3x x3 y (b) y (d) y 3x 3 sx F certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70 F and 173 chirps per minute at 80 F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. g 12. The manager of a furniture factory finds that it costs $2200 to f x G ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. (a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent? 13. At the surface of the ocean, the water pressure is the same as 5. (a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that f2 1 and sketch several members of the family. (c) Which function belongs to both families? 6. What do all members of the family of linear functions the air pressure above the water, 15 lb in2. Below the surface, the water pressure increases by 4.34 lb in2 for every 10 ft of descent. (a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 100 lb in2 ? 14. The monthly cost of driving a car depends on the number of fx 1 m x 3 have in common? Sketch several members of the family. 7. What do all members of the family of linear functions fx c x have in common? Sketch several members of the family. 8. The manager of a weekend flea market knows from past expe- rience that if he charges x dollars for a rental space at the flea market, then the number y of spaces he can rent is given by the equation y 200 4 x. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can’t be negative quantities.) miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the y-intercept represent? (e) Why does a linear function give a suitable model in this situation? 5E-01(pp 32-45) 1/17/06 12:10 PM Page 37 S ECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ❙❙❙❙ 37 15–16 |||| For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. Temperature (°F) 50 55 60 65 70 75 80 85 90 x Chirping rate (chirps min) 20 46 79 91 113 140 173 198 211 15. (a) y (b) y 0 x 0 16. (a) y (b) y (a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the chirping rate at 100 F. ; 19. The table gives the winning heights for the Olympic pole vault competitions in the 20th century. Year 0 ■ ■ ■ ■ ■ ■ Height (ft) 10.83 11.48 12.17 12.96 13.42 12.96 13.77 14.15 14.27 14.10 14.92 Year 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 Height (ft) 14.96 15.42 16.73 17.71 18.04 18.04 18.96 18.85 19.77 19.02 19.42 x ■ 0 ■ ■ ■ ■ x ■ ; 17. The table shows (lifetime) peptic ulcer rates (per 100 population) for various family incomes as reported by the 1989 National Health Interview Survey. Ulcer rate (per 100 population) 14.1 13.0 13.4 12.5 12.0 12.4 10.5 9.4 8.2 Income $4,000 $6,000 $8,000 $12,000 $16,000 $20,000 $30,000 $45,000 $60,000 1900 1904 1908 1912 1920 1924 1928 1932 1936 1948 1952 (a) Make a scatter plot and decide whether a linear model is appropriate. (b) Find and graph the regression line. (c) Use the linear model to predict the height of the winning pole vault at the 2000 Olympics and compare with the winning height of 19.36 feet. (d) Is it reasonable to use the model to predict the winning height at the 2100 Olympics? (a) Make a scatter plot of these data and decide whether a linear model is appropriate. (b) Find and graph a linear model using the first and last data points. (c) Find and graph the least squares regression line. (d) Use the linear model in part (c) to estimate the ulcer rate for an income of $25,000. (e) According to the model, how likely is someone with an income of $80,000 to suffer from peptic ulcers? (f) Do you think it would be reasonable to apply the model to someone with an income of $200,000? ; 20. A study by the U. S. Office of Science and Technology in 1972 estimated the cost (in 1972 dollars) to reduce automobile emissions by certain percentages: Reduction in emissions (%) 50 55 60 65 70 Cost per car (in $) 45 55 62 70 80 Reduction in emissions (%) 75 80 85 90 95 Cost per car (in $) 90 100 200 375 600 ; 18. Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures. Find a model that captures the “diminishing returns” trend of these data. 5E-01(pp 32-45) 1/17/06 12:10 PM Page 38 38 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS ; 21. Use the data in the table to model the population of the world in the 20th century by a cubic function. Then use your model to estimate the population in the year 1925. Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Population (millions) 1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 distance from Earth to the Sun) and their periods T (time of revolution in years). Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto d 0.387 0.723 1.000 1.523 5.203 9.541 19.190 30.086 39.507 T 0.241 0.615 1.000 1.881 11.861 29.457 84.008 164.784 248.350 ; 22. The table shows the mean (average) distances d of the planets from the Sun (taking the unit of measurement to be the (a) Fit a power model to the data. (b) Kepler’s Third Law of Planetary Motion states that “The square of the period of revolution of a planet is proportional to the cube of its mean distance from the Sun.” Does your model corroborate Kepler’s Third Law? |||| 1.3 New Functions from Old Functions In this section we start with the basic functions we discussed in Section 1.2 and obtain new functions by shifting, stretching, and reflecting their graphs. We also show how to combine pairs of functions by the standard arithmetic operations and by composition. Transformations of Functions By applying certain transformations to the graph of a given function we can obtain the graphs of certain related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs. Let’s first consider translations. If c is a positive number, then the graph of y f x c is just the graph of y f x shifted upward a distance of c units (because each y-coordinate is increased by the same number c). Likewise, if t x f x c , where c 0, then the value of t at x is the same as the value of f at x c (c units to the left of x). Therefore, the graph of y f x c is just the graph of y f x shifted c units to the right (see Figure 1). Vertical and Horizontal Shifts Suppose c y y y y f f f f x x x x 0. To obtain the graph of c, shift the graph of y f x a distance c units upward c, shift the graph of y f x a distance c units downward c , shift the graph of y f x a distance c units to the right c , shift the graph of y f x a distance c units to the left Now let’s consider the stretching and reflecting transformations. If c 1, then the graph of y c f x is the graph of y f x stretched by a factor of c in the vertical direction (because each y-coordinate is multiplied by the same number c). The graph of 5E-01(pp 32-45) 1/17/06 12:11 PM Page 39 SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ❙❙❙❙ 39 y y y=ƒ+c y=cƒ (c>1) y=f(x+c) c y =ƒ y=f(x-c) y=f(_x) y=ƒ y= 1 ƒ c x 0 x c 0 c c y=ƒ-c y=_ƒ FIGURE 1 F IGURE 2 Translating the graph of ƒ Stretching and reflecting the graph of ƒ y f x is the graph of y f x reflected about the x-axis because the point x, y is replaced by the point x, y . (See Figure 2 and the following chart, where the results of other stretching, compressing, and reflecting transformations are also given.) In Module 1.3 you can see the effect of combining the transformations of this section. Vertical and Horizontal Stretching and Reflecting Suppose c y y y y y y 1. To obtain the graph of c f x , stretch the graph of y f x vertically by a factor of c 1 c f x , compress the graph of y f x vertically by a factor of c f c x , compress the graph of y f x horizontally by a factor of c f x c , stretch the graph of y f x horizontally by a factor of c f x , reflect the graph of y f x about the x-axis f x , reflect the graph of y f x about the y-axis Figure 3 illustrates these stretching transformations when applied to the cosine function with c 2. For instance, in order to get the graph of y 2 cos x we multiply the y-coordinate of each point on the graph of y cos x by 2. This means that the graph of y cos x gets stretched vertically by a factor of 2. y 2 1 0 y 2 1 y=Ł 2 x 2 y=2 Ł x y=Ł x 1 y= Ł 2 x x 1 0 x y=Ł x FIGURE 3 y=Ł 2x 5E-01(pp 32-45) 1/17/06 12:11 PM Page 40 40 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS EXAMPLE 1 Given the graph of y y sx 2, y s x, y s x, use transformations to graph y 2 s x, and y s x. sx 2, SOLUTION The graph of the square root function y s x, obtained from Figure 13 in Section 1.2, is shown in Figure 4(a). In the other parts of the figure we sketch y s x 2 by shifting 2 units downward, y s x 2 by shifting 2 units to the right, y s x by reflecting about the x-axis, y 2 s x by stretching vertically by a factor of 2, and y s x by reflecting about the y-axis. y y y y y y 1 0 1 x 0 _2 x 0 2 x 0 x 0 x 0 x (a) y=œ„ x FIGURE 4 (b) y=œ„-2 x (c) y=œ„„„„ x-2 (d) y=_ œ„ x (e) y=2 œ„ x (f ) y=œ„„ _x EXAMPLE 2 Sketch the graph of the function f ( x) x2 6x 10. SOLUTION Completing the square, we write the equation of the graph as y x2 6x 10 x 3 2 1 x 2 and shifting This means we obtain the desired graph by starting with the parabola y 3 units to the left and then 1 unit upward (see Figure 5). y y (_3, 1) 0 x _3 _1 1 0 x FIGURE 5 (a) y=≈ (b) y=(x+3)@+1 EXAMPLE 3 Sketch the graphs of the following functions. (a) y SOLUTION sin 2 x (b) y 1 sin x (a) We obtain the graph of y sin 2 x from that of y sin x by compressing horizontally by a factor of 2 (see Figures 6 and 7). Thus, whereas the period of y sin x is 2 , the period of y sin 2 x is 2 2 . y y y=sin x 1 0 π 2 y=sin 2x 1 π x 0π π 4 2 π x FIGURE 6 FIGURE 7 5E-01(pp 32-45) 1/17/06 12:11 PM Page 41 S ECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ❙❙❙❙ 41 (b) To obtain the graph of y 1 sin x, we again start with y sin x. We reflect about the x-axis to get the graph of y sin x and then we shift 1 unit upward to get y 1 sin x. (See Figure 8.) y 2 1 0 π 2 y=1-sin x π FIGURE 8 3π 2 2π x EXAMPLE 4 Figure 9 shows graphs of the number of hours of daylight as functions of the time of the year at several latitudes. Given that Philadelphia is located at approximately 40 N latitude, find a function that models the length of daylight at Philadelphia. 20 18 16 14 12 Hours 10 8 6 4 60° N 50° N 40° N 30° N 20° N FIGURE 9 2 0 Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Graph of the length of daylight from March 21 through December 21 at various latitudes Source: Lucia C. Harrison, Daylight, Twilight, Darkness and Time (New York: Silver, Burdett, 1935) page 40. SOLUTION Notice that each curve resembles a shifted and stretched sine function. By looking at the blue curve we see that, at the latitude of Philadelphia, daylight lasts about 14.8 hours on June 21 and 9.2 hours on December 21, so the amplitude of the curve (the factor by which we have to stretch the sine curve vertically) is 1 14.8 9.2 2.8. 2 By what factor do we need to stretch the sine curve horizontally if we measure the time t in days? Because there are about 365 days in a year, the period of our model should be 365. But the period of y sin t is 2 , so the horizontal stretching factor is c 2 365. We also notice that the curve begins its cycle on March 21, the 80th day of the year, so we have to shift the curve 80 units to the right. In addition, we shift it 12 units upward. Therefore, we model the length of daylight in Philadelphia on the t th day of the year by the function Lt 12 2.8 sin 2 t 365 80 y y Another transformation of some interest is taking the absolute value of a function. If f x , then according to the definition of absolute value, y f x when f x 0 and f x when f x 0. This tells us how to get the graph of y f x from the graph 5E-01(pp 32-45) 1/17/06 12:11 PM Page 42 42 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS of y f x : The part of the graph that lies above the x-axis remains the same; the part that lies below the x-axis is reflected about the x-axis. EXAMPLE 5 Sketch the graph of the function y SOLUTION We first graph the parabola y 2 x2 1. x 2 1 in Figure 10(a) by shifting the parabola y x downward 1 unit. We see that the graph lies below the x-axis when 1 x 1, so we reflect that part of the graph about the x-axis to obtain the graph of y x2 1 in Figure 10(b). y y _1 0 1 x _1 0 1 x FIGURE 10 (a) y=≈-1 (b) y=| ≈-1 | Combinations of Functions Two functions f and t can be combined to form new functions f t, f t, f t, and f t in a manner similar to the way we add, subtract, multiply, and divide real numbers. If we define the sum f t by the equation 1 f tx fx tx then the right side of Equation 1 makes sense if both f x and t x are defined, that is, if x belongs to the domain of f and also to the domain of t. If the domain of f is A and the domain of t is B, then the domain of f t is the intersection of these domains, that is, A B. Notice that the sign on the left side of Equation 1 stands for the operation of addition of functions, but the sign on the right side of the equation stands for addition of the numbers f x and t x . Similarly, we can define the difference f t and the product f t, and their domains are also A B. But in defining the quotient f t we must remember not to divide by 0. Algebra of Functions Let f and t be functions with domains A and B. Then the functions f f f t, f tx tx ft x f t x t, f t, and f t are defined as follows: fx fx f xtx fx tx tx tx domain domain domain domain A A A x B B B A B tx 0 5E-01(pp 32-45) 1/17/06 12:11 PM Page 43 S ECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ❙❙❙❙ 43 EXAMPLE 6 If f x s x and t x s4 x 2, find the functions f t, f t, f t, and f t. 0: |||| Another way to solve 4 x 2 x2 x 0 2 _2 + 2 - s x is 0, . The domain of t x s4 x 2 consists 2 2 of all numbers x such that 4 x 0, that is, x 4. Taking square roots of both sides, we get x 2, or 2 x 2, so the domain of t is the interval 2, 2 . The intersection of the domains of f and t is SOLUTION The domain of f x 0, Thus, according to the definitions, we have f f tx tx ft x f t x sx sx s4 s4 x 2 2, 2 0, 2 x2 x2 s4x x 4 x2 x 3 0 0 0 0 x x x x 2 2 2 2 2 because s x s4 sx s4 x 2 Notice that the domain of f t is the interval 0, 2 ; we have to exclude x t2 0. The graph of the function f t is obtained from the graphs of f and t by graphical addition. This means that we add corresponding y-coordinates as in Figure 11. Figure 12 shows the result of using this procedure to graph the function f t from Example 6. y 5 4 y y=(f+g)(x) y=( f+g)(x) ©=œ„„„„„ 4-≈ y=© 2 3 1.5 f (a)+g(a) 2 1 1 f(a) y=ƒ a g(a) 0.5 x ƒ=œ„ x _2 _1 0 1 2 x FIGURE 11 FIGURE 12 Composition of Functions There is another way of combining two functions to get a new function. For example, suppose that y f u x 2 1. Since y is a function of u and u is, su and u t x in turn, a function of x, it follows that y is ultimately a function of x. We compute this by substitution: y fu f tx f x2 1 sx 2 1 5E-01(pp 32-45) 1/17/06 12:11 PM Page 44 44 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS The procedure is called composition because the new function is composed of the two given functions f and t. In general, given any two functions f and t, we start with a number x in the domain of t and find its image t x . If this number t x is in the domain of f , then we can calculate the value of f t x . The result is a new function h x f t x obtained by substituting t into f . It is called the composition (or composite) of f and t and is denoted by f t (“ f circle t ”). Definition Given two functions f and t, the composite function f t (also called the composition of f and t) is defined by f tx f tx The domain of f t is the set of all x in the domain of t such that t x is in the domain of f . In other words, f t x is defined whenever both t x and f t x are defined. The best way to picture f t is by either a machine diagram (Figure 13) or an arrow diagram (Figure 14). FIGURE 13 The f • g machine is composed of the g machine (first) and then the f machine. x (input) g g(x) f f{ ©} (output) f•g f g FIGURE 14 Arrow diagram for f • g x © f{ ©} EXAMPLE 7 If f x SOLUTION We have x 2 and t x x 3, find the composite functions f t and t f . f tx tfx f tx tfx fx t x2 3 x2 x 3 3 2 | NOTE You can see from Example 7 that, in general, f t t f . Remember, the notation f t means that the function t is applied first and then f is applied second. In Example 7, f t is the function that first subtracts 3 and then squares; t f is the function that first squares and then subtracts 3. ■ EXAMPLE 8 If f x (a) f t SOLUTION s x and t x s2 (b) t f (c) f f x, find each function and its domain. (d) t t (a) f tx f tx x 0 f (s2 xx x) 2 ss2 x ,2 . 4 s2 x The domain of f t is x 2 5E-01(pp 32-45) 1/17/06 12:12 PM Page 45 S ECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ❙❙❙❙ 45 (b) If 0 a b, then a 2 b 2. tfx tfx t(s x ) s2 sx For s x to be defined we must have x 0. For s2 sx to be defined we must have 2 s x 0, that is, s x 2, or x 4. Thus, we have 0 x 4, so the domain of t f is the closed interval 0, 4 . (c) ffx . ttx t(s2 x) ffx f (sx ) ssx 4 sx The domain of f f is 0, (d) t tx s2 s2 x x 0. This 2. Thus, This expression is defined when 2 x 0, that is, x 2, and 2 s2 latter inequality is equivalent to s2 x 2, or 2 x 4, that is, x 2 x 2, so the domain of t t is the closed interval 2, 2 . It is possible to take the composition of three or more functions. For instance, the composite function f t h is found by first applying h, then t, and then f as follows: f t hx EXAMPLE 9 Find f SOLUTION f thx x 10, and h x f tx 3 10 t h if f x f t hx xx 1 ,t x x 3. f thx fx 3 x 3 3 10 10 x 1 So far we have used composition to build complicated functions from simpler ones. But in calculus it is often useful to be able to decompose a complicated function into simpler ones, as in the following example. EXAMPLE 10 Given F x cos2 x 2 9 , find functions f , t, and h such that F f t h. cos x 9 , the formula for F says: First add 9, then take the SOLUTION Since F x cosine of the result, and finally square. So we let hx Then f t hx f thx cos x 9 f tx 2 x 9 tx cos x fx x2 9 f cos x 9 Fx |||| 1.3 Exercises (d) (e) (f) (g) (h) Shift 3 units to the left. Reflect about the x-axis. Reflect about the y-axis. Stretch vertically by a factor of 3. Shrink vertically by a factor of 3. 1. Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. 5E-01(pp 46-55) 1/17/06 12:06 PM Page 46 46 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS 2. Explain how the following graphs are obtained from the graph 6. y 3 of y (a) y (c) y (e) y f x. 5f x fx f 5x (b) y (d) y (f) y fx 5 5f x 5f x 3 3. The graph of y f x is given. Match each equation with its graph and give reasons for your choices. (a) y f x 4 (b) y f x 3 (c) y 1 f x (d) y fx 4 3 (e) y 2 f x 6 @ y 6 0 2 5 x 7. _4 y _1 0 _1 x ! 3 f $ # _2.5 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ _6 _3 0 3 6 x 8. (a) How is the graph of y % _3 4. The graph of f is given. Draw the graphs of the following functions. (a) y f x (c) y 2f x 2 sin x related to the graph of y sin x ? Use your answer and Figure 6 to sketch the graph of y 2 sin x. (b) How is the graph of y 1 sx related to the graph of y sx ? Use your answer and Figure 4(a) to sketch the graph of y 1 sx. 4 y (b) y (d) y fx 1 2 4 fx 3 9–24 |||| Graph the function, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 9. y 11. y 1 0 1 x x3 x 1 1 2 10. y 12. y 14. y 16. y 18. y 1 x2 x2 4x 3 13. y 15. y 17. y 19. y 21. y 23. y ■ ■ 2 cos x 4 sin 3x 1 x x 1 4 2 3 sx sin x 2 sx 1 2 5. The graph of f is given. Use it to graph the following functions. (a) y f 2 x (c) y f x y 1 0 1 3 8x 1 ■ ■ ■ ■ 4 3 1 4 ■ ■ ■ (b) y (d) y f ( 2 x) fx 1 x2 2 20. y 22. y 24. y ■ ■ x 1 tan x 4 x 2 ■ sin x 2x x 25. The city of New Orleans is located at latitude 30 N. Use 2 6–7 |||| The graph of y s3x x is given. Use transformations to create a function whose graph is as shown. Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. Use the fact that on March 31 the Sun rises at 5:51 A.M. and sets at 6:18 P.M. in New Orleans to check the accuracy of your model. 26. A variable star is one whose brightness alternately increases y 1.5 y=œ„„„„„„ 3x-≈ 0 3 x and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by 0.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time. 5E-01(pp 46-55) 1/17/06 12:07 PM Page 47 S ECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ❙❙❙❙ 47 27. (a) How is the graph of y (b) Sketch the graph of y (c) Sketch the graph of y f ( x ) related to the graph of f ? sin x . sx. 38. f x 39. f x 40. f x ■ ■ 1 x s2 x ■ 3 x, 1 , x tx tx 5x 2 x x x2 ■ 3x 1 2 1 ■ 2 28. Use the given graph of f to sketch the graph of y 1f x. Which features of f are the most important in sketching y 1 f x ? Explain how they are used. y 1 0 1 x 3, ■ tx ■ ■ ■ ■ ■ ■ 41–44 |||| Find f t h. x 2x 1, 1, tx tx 2x, x, x 2 2 41. f x 42. f x 43. f x 44. f x ■ ■ hx hx 2, x 1 hx 1 x x sx 3 3 ■ ■ ■ ■ s x 1, t x 2 , tx x1 ■ ■ ■ cos x, h x ■ ■ ■ 45–50 29–30 |||| |||| Express the function in the form f t. x2 x x2 scos t ■ ■ ■ ■ Use graphical addition to sketch the graph of f t. 45. F x 47. G x 1 2 10 46. F x 48. G x 50. u t ■ ■ sin( s x ) 1 x 1 ■ 29. y 4 3 tan t tan t ■ ■ ■ g 49. u t f 0 x ■ ■ 51–53 |||| Express the function in the form f t h. 1 4 ■ 51. H x 30. y 3x ■ 2 52. H x 3 ssx 1 ■ ■ ■ 53. H x f ■ ■ sec (sx ) ■ ■ ■ ■ ■ 54. Use the table to evaluate each expression. 0 g x (a) f t 1 (d) t t 1 x 1 3 6 (b) t f 1 (e) t f 3 2 1 3 3 4 2 4 2 1 (c) f f 1 (f) f t 6 5 2 2 6 5 3 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ fx tx 31–32 |||| Find f x 3 t, f 2x , x, ■ t, f t, and f t and state their domains. tx 3x s1 ■ ■ 31. f x 32. f x ■ ■ 2 2 1 x ■ ■ ■ ■ ■ s1 ■ tx ■ 55. Use the given graphs of f and t to evaluate each expression, 33–34 |||| Use the graphs of f and t and the method of graphical addition to sketch the graph of f t. or explain why it is undefined. (a) f t 2 (b) t f 0 (d) t f 6 (e) t t y 2 (c) f t 0 (f) f f 4 33. f x ■ ■ x, ■ tx ■ 1x ■ ■ 34. f x ■ ■ x 3, ■ tx ■ ■ x2 ■ g 2 f 35–40 |||| Find the functions f t, t f , f f , and t t and their 2x 2 1 x, x, 3 domains. 35. f x 36. f x 37. f x tx tx 1 3x 1x sx 2 0 2 x sin x, tx 5E-01(pp 46-55) 1/17/06 12:07 PM Page 48 48 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS 56. Use the given graphs of f and t to estimate the value of f t x for x 5, 4, 3, . . . , 5. Use these estimates to sketch a rough graph of f t. y g 1 0 1 x (b) Sketch the graph of the voltage V t in a circuit if the switch is turned on at time t 0 and 120 volts are applied instantaneously to the circuit. Write a formula for V t in terms of H t . (c) Sketch the graph of the voltage V t in a circuit if the switch is turned on at time t 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V t in terms of H t . (Note that starting at t 5 corresponds to a translation.) 60. The Heaviside function defined in Exercise 59 can also be used f 57. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm s. (a) Express the radius r of this circle as a function of the time t (in seconds). (b) If A is the area of this circle as a function of the radius, find A r and interpret it. 58. An airplane is flying at a speed of 350 mi h at an altitude of one mile and passes directly over a radar station at time t 0. (a) Express the horizontal distance d (in miles) that the plane has flown as a function of t. (b) Express the distance s between the plane and the radar station as a function of d. (c) Use composition to express s as a function of t. 59. The Heaviside function H is defined by to define the ramp function y ctH t , which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y tH t . (b) Sketch the graph of the voltage V t in a circuit if the switch is turned on at time t 0 and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for V t in terms of H t for t 60. (c) Sketch the graph of the voltage V t in a circuit if the switch is turned on at time t 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for V t in terms of H t for t 32. 61. (a) If t x 2 x 1 and h x 4 x 2 4 x 7, find a function f such that f t h. (Think about what operations you would have to perform on the formula for t to end up with the formula for h.) (b) If f x 3x 2 3x 2, find a function 3x 5 and h x t such that f t h. x h. f t. Is h always an 4 and h x 4x 1, find a function t such that tf Ht 0 1 if t if t 0 0 62. If f x It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. 63. Suppose t is an even function and let h even function? 64. Suppose t is an odd function and let h f t. Is h always an odd function? What if f is odd? What if f is even? |||| 1.4 Graphing Calculators and Computers In this section we assume that you have access to a graphing calculator or a computer with graphing software. We will see that the use of such a device enables us to graph more complicated functions and to solve more complex problems than would otherwise be possible. We also point out some of the pitfalls that can occur with these machines. Graphing calculators and computers can give very accurate graphs of functions. But we will see in Chapter 4 that only through the use of calculus can we be sure that we have uncovered all the interesting aspects of a graph. A graphing calculator or computer displays a rectangular portion of the graph of a function in a display window or viewing screen, which we refer to as a viewing rectangle. The default screen often gives an incomplete or misleading picture, so it is important to choose the viewing rectangle with care. If we choose the x-values to range from a minimum value of Xmin a to a maximum value of Xmax b and the y-values to range from 5E-01(pp 46-55) 1/17/06 12:07 PM Page 49 S ECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS ❙❙❙❙ 49 (a, d ) y=d ( b, d ) a minimum of Ymin lies in the rectangle c to a maximum of Ymax d, then the visible portion of the graph x=a x=b a, b c, d x, y a x b, c y d (a, c) y=c ( b, c ) FIGURE 1 The viewing rectangle a, b by c, d shown in Figure 1. We refer to this rectangle as the a, b by c, d viewing rectangle. The machine draws the graph of a function f much as you would. It plots points of the form x, f x for a certain number of equally spaced values of x between a and b. If an x-value is not in the domain of f , or if f x lies outside the viewing rectangle, it moves on to the next x-value. The machine connects each point to the preceding plotted point to form a representation of the graph of f . EXAMPLE 1 Draw the graph of the function f x x2 3 in each of the following view- 2 ing rectangles. (a) 2, 2 by 2, 2 (c) 10, 10 by 5, 30 (b) (d) 4, 4 by 4, 4 50, 50 by 100, 1000 _2 2 _2 (a) _2, 2 by _2, 2 4 S OLUTION For part (a) we select the range by setting X min 2, X max 2, Y min 2, and Y max 2. The resulting graph is shown in Figure 2(a). The display window is blank! A moment’s thought provides the explanation: Notice that x 2 0 for all x, so x 2 3 3 for all x. Thus, the range of the function f x x 2 3 is 3, . This means that the graph of f lies entirely outside the viewing rectangle 2, 2 by 2, 2 . The graphs for the viewing rectangles in parts (b), (c), and (d) are also shown in Figure 2. Observe that we get a more complete picture in parts (c) and (d), but in part (d) it is not clear that the y-intercept is 3. 30 1000 _4 4 _10 _4 _5 10 _50 _100 50 (b) _4, 4 by _4, 4 FIGURE 2 Graphs of ƒ=≈+3 (c) _10, 10 by _5, 30 (d) _50, 50 by _100, 1000 We see from Example 1 that the choice of a viewing rectangle can make a big difference in the appearance of a graph. Sometimes it’s necessary to change to a larger viewing rectangle to obtain a more complete picture, a more global view, of the graph. In the next example we see that knowledge of the domain and range of a function sometimes provides us with enough information to select a good viewing rectangle. EXAMPLE 2 Determine an appropriate viewing rectangle for the function fx s8 2 x 2 and use it to graph f . SOLUTION The expression for f x is defined when 8 2x 2 0 &? &? 2x 2 x 8 2 &? x 2 &? 2 4 x 2 5E-01(pp 46-55) 1/17/06 12:07 PM Page 50 50 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS 4 Therefore, the domain of f is the interval 0 s8 2x 2 2, 2 . Also, s8 2 s2 2.83 _3 _1 3 so the range of f is the interval [0, 2 s2 ]. We choose the viewing rectangle so that the x-interval is somewhat larger than the domain and the y-interval is larger than the range. Taking the viewing rectangle to be 3, 3 by 1, 4 , we get the graph shown in Figure 3. EXAMPLE 3 Graph the function y FIGURE 3 x3 150x. 5 _5 5 _5 FIGURE 4 SOLUTION Here the domain is , the set of all real numbers. That doesn’t help us choose a viewing rectangle. Let’s experiment. If we start with the viewing rectangle 5, 5 by 5, 5 , we get the graph in Figure 4. It appears blank, but actually the graph is so nearly vertical that it blends in with the y-axis. If we change the viewing rectangle to 20, 20 by 20, 20 , we get the picture shown in Figure 5(a). The graph appears to consist of vertical lines, but we know that can’t be correct. If we look carefully while the graph is being drawn, we see that the graph leaves the screen and reappears during the graphing process. This indicates that we need to see more in the vertical direction, so we change the viewing rectangle to 20, 20 by 500, 500 . The resulting graph is shown in Figure 5(b). It still doesn’t quite reveal all the main features of the function, so we try 20, 20 by 1000, 1000 in Figure 5(c). Now we are more confident that we have arrived at an appropriate viewing rectangle. In Chapter 4 we will be able to see that the graph shown in Figure 5(c) does indeed reveal all the main features of the function. 500 1000 20 _20 20 _20 20 _20 20 _20 _500 _1000 (a) FIGURE 5 ( b) (c) ƒ=˛-150x EXAMPLE 4 Graph the function f x sin 50 x in an appropriate viewing rectangle. SOLUTION Figure 6(a) shows the graph of f produced by a graphing calculator using the viewing rectangle 12, 12 by 1.5, 1.5 . At first glance the graph appears to be reasonable. But if we change the viewing rectangle to the ones shown in the following parts of Figure 6, the graphs look very different. Something strange is happening. In order to explain the big differences in appearance of these graphs and to find an appropriate viewing rectangle, we need to find the period of the function y sin 50 x. We know that the function y sin x has period 2 and the graph of y sin 50 x is compressed horizontally by a factor of 50, so the period of y sin 50 x is 2 50 25 0.126 5E-01(pp 46-55) 1/17/06 12:07 PM Page 51 S ECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS ❙❙❙❙ 51 1.5 1.5 _12 12 _10 10 |||| The appearance of the graphs in Figure 6 depends on the machine used. The graphs you get with your own graphing device might not look like these figures, but they will also be quite inaccurate. _1.5 _1.5 (a) 1.5 (b) 1.5 _9 9 _6 6 FIGURE 6 Graphs of ƒ=sin 50x in four viewing rectangles _1.5 _1.5 (c) (d) 1.5 _.25 .25 This suggests that we should deal only with small values of x in order to show just a few oscillations of the graph. If we choose the viewing rectangle 0.25, 0.25 by 1.5, 1.5 , we get the graph shown in Figure 7. Now we see what went wrong in Figure 6. The oscillations of y sin 50 x are so rapid that when the calculator plots points and joins them, it misses most of the maximum and minimum points and therefore gives a very misleading impression of the graph. We have seen that the use of an inappropriate viewing rectangle can give a misleading impression of the graph of a function. In Examples 1 and 3 we solved the problem by changing to a larger viewing rectangle. In Example 4 we had to make the viewing rectangle smaller. In the next example we look at a function for which there is no single viewing rectangle that reveals the true shape of the graph. EXAMPLE 5 Graph the function f x _1.5 FIGURE 7 ƒ=sin 50x sin x 1 100 cos 100 x. SOLUTION Figure 8 shows the graph of f produced by a graphing calculator with viewing rectangle 6.5, 6.5 by 1.5, 1.5 . It looks much like the graph of y sin x, but perhaps with some bumps attached. If we zoom in to the viewing rectangle 0.1, 0.1 by 0.1, 0.1 , we can see much more clearly the shape of these bumps in Figure 9. The 1 reason for this behavior is that the second term, 100 cos 100 x, is very small in comparison with the first term, sin x. Thus, we really need two graphs to see the true nature of this function. 1.5 0.1 _6.5 6.5 _0.1 0.1 _1.5 _0.1 FIGURE 8 FIGURE 9 5E-01(pp 46-55) 1/17/06 12:07 PM Page 52 52 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS EXAMPLE 6 Draw the graph of the function y 1 1 x . SOLUTION Figure 10(a) shows the graph produced by a graphing calculator with viewing rectangle 9, 9 by 9, 9 . In connecting successive points on the graph, the calculator produced a steep line segment from the top to the bottom of the screen. That line segment is not truly part of the graph. Notice that the domain of the function y 1 1 x is x x 1 . We can eliminate the extraneous near-vertical line by experimenting with a change of scale. When we change to the smaller viewing rectangle 4.7, 4.7 by 4.7, 4.7 on this particular calculator, we obtain the much better graph in Figure 10(b). |||| Another way to avoid the extraneous line is to change the graphing mode on the calculator so that the dots are not connected. Alternatively, we could zoom in using the Zoom Decimal mode. _9 9 4.7 9 _4.7 4.7 FIGURE 10 y= 1 1-x _9 _4.7 (a) (b) 3 sx. EXAMPLE 7 Graph the function y SOLUTION Some graphing devices display the graph shown in Figure 11, whereas others produce a graph like that in Figure 12. We know from Section 1.2 (Figure 13) that the graph in Figure 12 is correct, so what happened in Figure 11? The explanation is that some machines compute the cube root of x using a logarithm, which is not defined if x is negative, so only the right half of the graph is produced. 2 2 _3 3 _3 3 _2 _2 FIGURE 11 FIGURE 12 You should experiment with your own machine to see which of these two graphs is produced. If you get the graph in Figure 11, you can obtain the correct picture by graphing the function x fx x 13 x 3 Notice that this function is equal to sx (except when x 0). To understand how the expression for a function relates to its graph, it’s helpful to graph a family of functions, that is, a collection of functions whose equations are related. In the next example we graph members of a family of cubic polynomials. 5E-01(pp 46-55) 1/17/06 12:07 PM Page 53 S ECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS ❙❙❙❙ 53 x 3 c x for various values of the number c. How does the graph change when c is changed? EXAMPLE 8 Graph the function y x 3 c x for c 2, 1, 0, 1, and 2. We see that, for positive values of c, the graph increases from left to right with no maximum or minimum points (peaks or valleys). When c 0, the curve is flat at the origin. When c is negative, the curve has a maximum point and a minimum point. As c decreases, the maximum point becomes higher and the minimum point lower. SOLUTION Figure 13 shows the graphs of y (a) y=˛+2x FIGURE 13 (b) y=˛+x (c) y=˛ (d) y=˛-x (e) y=˛-2x Several members of the family of functions y=˛+cx , all graphed in the viewing rectangle _2, 2 by _2.5, 2.5 EXAMPLE 9 Find the solution of the equation cos x x correct to two decimal places. SOLUTION The solutions of the equation cos x x are the x-coordinates of the points of intersection of the curves y cos x and y x. From Figure 14(a) we see that there is only one solution and it lies between 0 and 1. Zooming in to the viewing rectangle 0, 1 by 0, 1 , we see from Figure 14(b) that the root lies between 0.7 and 0.8. So we zoom in further to the viewing rectangle 0.7, 0.8 by 0.7, 0.8 in Figure 14(c). By moving the cursor to the intersection point of the two curves, or by inspection and the fact that the x-scale is 0.01, we see that the root of the equation is about 0.74. (Many calculators have a built-in intersection feature.) 1.5 y=x y=Ł x 1 y=Ł x 5 y=x y=x y=Ł x 1 0.8 0.8 _5 FIGURE 14 _1.5 0 0.7 Locating the roots of cos x=x (a) _5, 5 by _1.5, 1.5 x-scale=1 (b) 0, 1 by 0, 1 x-scale=0.1 (c) 0.7 , 0.8 by 0.7 , 0.8 x-scale=0.01 |||| 1.4 ; Exercises 2. Use a graphing calculator or computer to determine which of 1. Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f x x 4 2. (a) 2, 2 by 2, 2 (b) 0, 4 by 0, 4 (c) 4, 4 by 4, 4 (d) 8, 8 by 4, 40 (e) 40, 40 by 80, 800 the given viewing rectangles produces the most appropriate graph of the function f x x 2 7x 6. (a) 5, 5 by 5, 5 (b) 0, 10 by 20, 100 (c) 15, 8 by 20, 100 (d) 10, 3 by 100, 20 5E-01(pp 46-55) 1/17/06 12:08 PM Page 54 54 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS 3. Use a graphing calculator or computer to determine which of 26. Use graphs to determine which of the functions the given viewing rectangles produces the most appropriate graph of the function f x 10 25x x 3. (a) 4, 4 by 4, 4 (b) 10, 10 by 10, 10 (c) 20, 20 by 100, 100 (d) 100, 100 by 200, 200 4. Use a graphing calculator or computer to determine which of fx x4 100 x 3 and t x x 3 is eventually larger. x 3 27. For what values of x is it true that sin x 28. Graph the polynomials P x 5 0.1? 3x 5x 2 x and Qx 3x 5 on the same screen, first using the viewing rectangle 2, 2 by [ 2, 2] and then changing to 10, 10 by 10,000, 10,000 . What do you observe from these graphs? n fx sx, where n is a positive integer. 4 6 (a) Graph the functions y sx, y sx, and y sx on the same screen using the viewing rectangle 1, 4 by 1, 3 . 3 5 (b) Graph the functions y x, y sx, and y sx on the same screen using the viewing rectangle 3, 3 by 2, 2 . (See Example 7.) 3 4 5 (c) Graph the functions y sx, y sx, y sx, and y sx on the same screen using the viewing rectangle 1, 3 by 1, 2 . (d) What conclusions can you make from these graphs? the given viewing rectangles produces the most appropriate graph of the function f x s8 x x 2 . (a) 4, 4 by 4, 4 (b) 5, 5 by 0, 100 (c) 10, 10 by 10, 40 (d) 2, 10 by 2, 6 5–18 |||| Determine an appropriate viewing rectangle for the given function and use it to draw the graph. 29. In this exercise we consider the family of root functions 5. f x 6. f x 7. f x 9. f x 11. f x 13. f x 15. f x 17. y ■ ■ 5 x3 20 x 30 x 2 3 x2 200 x 2 0.01x s81 x2 4 x x4 100 x 5 8. f x 10. f x 12. f x 14. f x 16. y 18. y xx s0.1 x x x 2 6x 20 9 30. In this exercise we consider the family of functions 100 cos 100 x sin x 40 3 cos x 2 ■ ■ ■ ■ 3 sin 120 x tan 25x x2 0.02 sin 50 x ■ ■ ■ ■ fx 1 x n, where n is a positive integer. (a) Graph the functions y 1 x and y 1 x 3 on the same screen using the viewing rectangle 3, 3 by 3, 3 . (b) Graph the functions y 1 x 2 and y 1 x 4 on the same screen using the same viewing rectangle as in part (a). (c) Graph all of the functions in parts (a) and (b) on the same screen using the viewing rectangle 1, 3 by 1, 3 . (d) What conclusions can you make from these graphs? 31. Graph the function f x 32. Graph the function f x 33. Graph the function y ■ ■ x 4 c x 2 x for several values of c. How does the graph change when c changes? s1 cx 2 for various values of c. Describe how changing the value of c affects the graph. x n 2 x, x 0, for n 1, 2, 3, 4, 5, and 6. How does the graph change as n increases? 19. Graph the ellipse 4 x 2 2y 2 1 by graphing the functions whose graphs are the upper and lower halves of the ellipse. 9x 2 1 by graphing the functions whose graphs are the upper and lower branches of the hyperbola. |||| 20. Graph the hyperbola y 2 34. The curves with equations 21–23 Find all solutions of the equation correct to two decimal y 9x 2 places. 21. x 3 x sc x2 4 ■ 0 ■ ■ 22. x 3 4x ■ 1 ■ ■ ■ 23. x 2 ■ ■ sin x ■ ■ ■ are called bullet-nose curves. Graph some of these curves to see why. What happens as c increases? 35. What happens to the graph of the equation y 2 24. We saw in Example 9 that the equation cos x x has exactly cx 3 x 2 as one solution. (a) Use a graph to show that the equation cos x 0.3x has three solutions and find their values correct to two decimal places. (b) Find an approximate value of m such that the equation cos x m x has exactly two solutions. 25. Use graphs to determine which of the functions f x c varies? 36. This exercise explores the effect of the inner function t on a 10 x 2 and t x x 3 10 is eventually larger (that is, larger when x is very large). composite function y f t x . (a) Graph the function y sin( s x ) using the viewing rectangle 0, 400 by 1.5, 1.5 . How does this graph differ from the graph of the sine function? (b) Graph the function y sin x 2 using the viewing rectangle 5, 5 by 1.5, 1.5 . How does this graph differ from the graph of the sine function? 5E-01(pp 46-55) 1/17/06 12:08 PM Page 55 C HAPTER 1 REVIEW ❙❙❙❙ 55 37. The figure shows the graphs of y sin 96 x and y displayed by a TI-83 graphing calculator. sin 2 x as explain its appearance, we replot the curve in dot mode in the second graph. 0 2π 0 2π 0 2π 0 2π y=sin 96x y=sin 2x The first graph is inaccurate. Explain why the two graphs appear identical. [Hint: The TI-83’s graphing window is 95 pixels wide. What specific points does the calculator plot?] 38. The first graph in the figure is that of y sin 45 x as displayed by a TI-83 graphing calculator. It is inaccurate and so, to help What two sine curves does the calculator appear to be plotting? Show that each point on the graph of y sin 45 x that the TI-83 chooses to plot is in fact on one of these two curves. (The TI-83’s graphing window is 95 pixels wide.) |||| 1 Review ■ CONCEPT CHECK ■ 1. (a) What is a function? What are its domain and range? 8. Draw, by hand, a rough sketch of the graph of each function. (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function? 2. Discuss four ways of representing a function. Illustrate your (a) y (c) y (e) y sin x 2x x (b) y (d) y (f) y tan x 1x sx 9. Suppose that f has domain A and t has domain B. discussion with examples. 3. (a) What is an even function? How can you tell if a function is (a) What is the domain of f t ? (b) What is the domain of f t ? (c) What is the domain of f t ? 10. How is the composite function f t defined? What is its even by looking at its graph? (b) What is an odd function? How can you tell if a function is odd by looking at its graph? 4. What is an increasing function? 5. What is a mathematical model? 6. Give an example of each type of function. domain? 11. Suppose the graph of f is given. Write an equation for each of (a) Linear function (c) Exponential function (e) Polynomial of degree 5 functions. x (a) f x (c) h x x3 (b) Power function (d) Quadratic function (f) Rational function 7. Sketch by hand, on the same axes, the graphs of the following (b) t x (d) j x x2 x4 the graphs that are obtained from the graph of f as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Reflect about the x-axis. (f) Reflect about the y-axis. (g) Stretch vertically by a factor of 2. (h) Shrink vertically by a factor of 2. (i) Stretch horizontally by a factor of 2. ( j) Shrink horizontally by a factor of 2. ■ TRUE-FALSE QUIZ ■ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f is a function, then f s 2. If f s 3. If f is a function, then f 3x 4. If x 1 3f x . f x2 . x 2 and f is a decreasing function, then f x 1 t f. t fs f t. 5. A vertical line intersects the graph of a function at most once. 6. If f and t are functions, then f t f t , then s t. 5E-01(pp 56-57) 1/17/06 12:06 PM Page 56 56 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS ■ EXERCISES ■ 1. Let f be the function whose graph is given. 9. Suppose that the graph of f is given. Describe how the graphs (a) (b) (c) (d) (e) (f) Estimate the value of f 2 . Estimate the values of x such that f x 3. State the domain of f. State the range of f. On what interval is f increasing? Is f even, odd, or neither even nor odd? Explain. y of the following functions can be obtained from the graph of f. (a) y f x (b) y f x 8 8 (c) y 1 2 f x (d) y f x 2 2 (e) y (f) y 3 f x fx 10. The graph of f is given. Draw the graphs of the following functions. (a) y f x 8 (c) y 2 f x y (b) y (d) y f 1 1 x 1 2 fx fx 1 1 0 1 x 2. Determine whether each curve is the graph of a function of x. 11–16 |||| Use transformations to sketch the graph of the function. sin 2 x x 1 2 1 2 If it is, state the domain and range of the function. y y (a) (b) 2 0 x 2 0 x 11. y 12. y 13. y 2 2 x3 1 1 14. y 15. f x 16. f x sx 1 x 1 1 ■ 2 x x2 ■ 3. The distance traveled by a car is given by the values in the table. t (seconds) d (feet) 0 0 1 10 2 32 3 70 4 119 5 178 if x if x ■ 0 0 ■ ■ ■ ■ ■ ■ ■ ■ ■ 17. Determine whether f is even, odd, or neither even nor odd. (a) Use the data to sketch the graph of d as a function of t. (b) Use the graph to estimate the distance traveled after 4.5 seconds. 4. Sketch a rough graph of the yield of a crop as a function of the (a) (b) (c) (d) f f f f x x x x 2 x 5 3x 2 x3 x7 cos x 2 1 sin x 2 18. Find an expression for the function whose graph consists of amount of fertilizer used. 5–8 |||| the line segment from the point 2, 2 to the point 1, 0 together with the top half of the circle with center the origin and radius 1. 19. If f x Find the domain and range of the function. s4 1 ■ ■ 5. f x 7. y ■ 3x 2 ■ ■ ■ ■ 6. t x 8. y ■ 1x tan 2 x ■ ■ 1 ■ ■ sin x, find the functions f t, t f , sx and t x f f , t t, and their domains. 1 sx sx as a composition of sin x 20. Express the function F x three functions. 5E-01(pp 56-57) 1/17/06 12:06 PM Page 57 C HAPTER 1 REVIEW ❙❙❙❙ 57 ; 21. Use graphs to discover what members of the family of sin n x have in common, where n is a positive functions f x integer. How do they differ? What happens to the graphs as n becomes large? 22. A small-appliance manufacturer finds that it costs $9000 to Birth year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Life expectancy 48.3 51.1 55.2 57.4 62.5 65.6 66.6 67.1 70.0 71.8 73.0 produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week. (a) Express the cost as a function of the number of toaster ovens produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent? 23. Life expectancy improved dramatically in the 20th century. The table gives the life expectancy at birth (in years) of males born in the United States. Use a scatter plot to choose an appropriate type of model. Use your model to predict the life span of a male born in the year 2010. 5E-01(pp 58-63) 1/17/06 12:00 PM Page 58 PRINCIPLES OF PROBLEM SOLVING There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem-solving process and to give some principles that may be useful in the solution of certain problems. These steps and principles are just common sense made explicit. They have been adapted from George Polya’s book How To Solve It. The first step is to read the problem and make sure that you understand it clearly. Ask yourself the following questions: What is the unknown? What are the given quantities? What are the given conditions? For many problems it is useful to draw a diagram and identify the given and required quantities on the diagram. Usually it is necessary to introduce suitable notation In choosing symbols for the unknown quantities we often use letters such as a, b, c, m, n, x, and y, but in some cases it helps to use initials as suggestive symbols; for instance, V for volume or t for time. 1 UNDERSTAND THE PROBLEM 2 THINK OF A PLAN Find a connection between the given information and the unknown that will enable you to calculate the unknown. It often helps to ask yourself explicitly: “How can I relate the given to the unknown?” If you don’t see a connection immediately, the following ideas may be helpful in devising a plan. Try to Recognize Something Familiar Relate the given situation to previous knowledge. Look at the unknown and try to recall a more familiar problem that has a similar unknown. Try to Recognize Patterns Some problems are solved by recognizing that some kind of pattern is occurring. The pattern could be geometric, or numerical, or algebraic. If you can see regularity or repetition in a problem, you might be able to guess what the continuing pattern is and then prove it. Use Analogy Try to think of an analogous problem, that is, a similar problem, a related problem, but one that is easier than the original problem. If you can solve the similar, simpler problem, then it might give you the clues you need to solve the original, more difficult problem. For instance, if a problem involves very large numbers, you could first try a similar problem with smaller numbers. Or if the problem involves three-dimensional geometry, you could look for a similar problem in two-dimensional geometry. Or if the problem you start with is a general one, you could first try a special case. Introduce Something Extra It may sometimes be necessary to introduce something new, an auxiliary aid, to help make the connection between the given and the unknown. For instance, in a problem where a diagram is useful the auxiliary aid could be a new line drawn in a diagram. In a more algebraic problem it could be a new unknown that is related to the original unknown. 58 5E-01(pp 58-63) 1/17/06 12:00 PM Page 59 Take Cases We may sometimes have to split a problem into several cases and give a different argument for each of the cases. For instance, we often have to use this strategy in dealing with absolute value. Work Backward Sometimes it is useful to imagine that your problem is solved and work backward, step by step, until you arrive at the given data. Then you may be able to reverse your steps and thereby construct a solution to the original problem. This procedure is commonly used in solving equations. For instance, in solving the equation 3x 5 7, we suppose that x is a number that satisfies 3x 5 7 and work backward. We add 5 to each side of the equation and then divide each side by 3 to get x 4. Since each of these steps can be reversed, we have solved the problem. Establish Subgoals In a complex problem it is often useful to set subgoals (in which the desired situation is only partially fulfilled). If we can first reach these subgoals, then we may be able to build on them to reach our final goal. Indirect Reasoning Sometimes it is appropriate to attack a problem indirectly. In using proof by contradiction to prove that P implies Q, we assume that P is true and Q is false and try to see why this can’t happen. Somehow we have to use this information and arrive at a contradiction to what we absolutely know is true. Mathematical Induction In proving statements that involve a positive integer n, it is frequently helpful to use the following principle. Principle of Mathematical Induction Let Sn be a statement about the positive integer n. Suppose that 1. S1 is true. 2. Sk 1 is true whenever Sk is true. Then Sn is true for all positive integers n. This is reasonable because, since S1 is true, it follows from condition 2 (with k 1) that S2 is true. Then, using condition 2 with k 2, we see that S3 is true. Again using condition 2, this time with k 3, we have that S4 is true. This procedure can be followed indefinitely. 3 CARRY OUT THE PLAN 4 LOOK BACK In Step 2 a plan was devised. In carrying out that plan we have to check each stage of the plan and write the details that prove that each stage is correct. Having completed our solution, it is wise to look back over it, partly to see if we have made errors in the solution and partly to see if we can think of an easier way to solve the problem. Another reason for looking back is that it will familiarize us with the method of solution and this may be useful for solving a future problem. Descartes said, “Every problem that I solved became a rule which served afterwards to solve other problems.” These principles of problem solving are illustrated in the following examples. Before you look at the solutions, try to solve these problems yourself, referring to these Principles of Problem Solving if you get stuck. You may find it useful to refer to this section from time to time as you solve the exercises in the remaining chapters of this book. 59 5E-01(pp 58-63) 1/17/06 12:00 PM Page 60 EXAMPLE 1 Express the hypotenuse h of a right triangle with area 25 m2 as a function of its perimeter P. |||| Understand the problem SOLUTION Let’s first sort out the information by identifying the unknown quantity and the data: Unknown: hypotenuse h Given quantities: perimeter P, area 25 m 2 |||| Draw a diagram It helps to draw a diagram and we do so in Figure 1. h b FIGURE 1 a |||| Connect the given with the unknown |||| Introduce something extra In order to connect the given quantities to the unknown, we introduce two extra variables a and b, which are the lengths of the other two sides of the triangle. This enables us to express the given condition, which is that the triangle is right-angled, by the Pythagorean Theorem: h2 a2 b2 The other connections among the variables come by writing expressions for the area and perimeter: 25 1 2 ab P a b h Since P is given, notice that we now have three equations in the three unknowns a, b, and h: 1 2 3 h2 25 P a2 1 2 b2 ab b h a |||| Relate to the familiar Although we have the correct number of equations, they are not easy to solve in a straightforward fashion. But if we use the problem-solving strategy of trying to recognize something familiar, then we can solve these equations by an easier method. Look at the right sides of Equations 1, 2, and 3. Do these expressions remind you of anything familiar? Notice that they contain the ingredients of a familiar formula: a Using this idea, we express a a From Equation 3 we have a Thus b 2 b 2 a2 2ab b2 b 2 in two ways. From Equations 1 and 2 we have 2 b a2 b2 2ab h2 4 25 P 100 2Ph h h 2 P2 2Ph 100 100 2P 2Ph h2 h2 h2 P2 P2 P2 This is the required expression for h as a function of P. 60 5E-01(pp 58-63) 1/17/06 12:00 PM Page 61 As the next example illustrates, it is often necessary to use the problem-solving principle of taking cases when dealing with absolute values. EXAMPLE 2 Solve the inequality x 3 x 2 11. SOLUTION Recall the definition of absolute value: x x x if x x if x 3 if x if x if x if x if x if x if x if x 0 0 3 3 3 3 2 2 2 2 0 0 0 0 It follows that x 3 x x x 3 3 3 2 Similarly x 2 x x x x 2 2 2 |||| Take cases These expressions show that we must consider three cases: x CASE I ■ 2 2 x 3 x 3 If x 2, we have x x 3 3 x x 2 2 2x x 11 11 10 5 CASE II ■ If 2 x 3, the given inequality becomes x 3 x 2 5 11 11 (always true) CASE III ■ If x 3, the inequality becomes x 3 x 2 2x x 11 12 6 5 x 6. Combining cases I, II, and III, we see that the inequality is satisfied when So the solution is the interval 5, 6 . In the following example we first guess the answer by looking at special cases and recognizing a pattern. Then we prove it by mathematical induction. In using the Principle of Mathematical Induction, we follow three steps: STEP 1 Prove that Sn is true when n STEP 2 Assume that Sn is true when n 1. k and deduce that Sn is true when n k 1. S TEP 3 Conclude that Sn is true for all n by the Principle of Mathematical Induction. 61 5E-01(pp 58-63) 1/17/06 12:00 PM Page 62 EXAMPLE 3 If f0 x formula for fn x . |||| Analogy: Try a similar, simpler problem xx 1 and fn 1 f0 fn for n 0, 1, 2, . . . , find a 1, 2, and 3. SOLUTION We start by finding formulas for fn x for the special cases n f1 x f0 f0 x x x x 1 x 1 1 f0 f0 x x x 2x x 1 1 1 f0 x x x 2x 1 1 f2 x f0 f1 x f0 f1 x x 2x 3x 2x 1 1 1 f0 x 2x x 3x 1 1 x 2x 1 x 1 2x 1 |||| Look for a pattern f3 x f0 f2 x 3x x 3x 1 x 1 1 f0 f2 x x 3x 4x 3x 1 1 1 f0 x 3x x 4x 1 1 We notice a pattern: The coefficient of x in the denominator of fn x is n three cases we have computed. So we make the guess that, in general, 4 1 in the fn x n x 1x 1 To prove this, we use the Principle of Mathematical Induction. We have already verified that (4) is true for n 1. Assume that it is true for n k, that is, fk x k f0 k k k x 1x 1 x 1x 1 1 1 Then fk 1 x f0 fk k k x x 1x f0 fk x 1 1 1 x 1x k x 1x 2x 1x k 1 x 2x k 1 This expression shows that (4) is true for n tion, it is true for all positive integers n. 1. Therefore, by mathematical induc- 62 5E-01(pp 58-63) 1/17/06 12:01 PM Page 63 P RO B L E M S 1. One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpen- dicular to the hypotenuse as a function of the length of the hypotenuse. 2. The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse as a function of the perimeter. 3. Solve the equation 2 x 4. Solve the inequality x 1 1 x x 5 3 x x x 4x 2 3. 5. 2 5. Sketch the graph of the function f x 6. Sketch the graph of the function t x 7. Draw the graph of the equation x 8. Draw the graph of the equation x 4 4x 1 x y. 2 2 3. 2 2 4. y xy 4y 2 0. y 1. 9. Sketch the region in the plane consisting of all points x, y such that x 10. Sketch the region in the plane consisting of all points x, y such that x y x y 2 11. A driver sets out on a journey. For the first half of the distance she drives at the leisurely pace of 30 mi h; she drives the second half at 60 mi h. What is her average speed on this trip? 12. Is it true that f t h ft f h? 1 is divisible by 6. n. 0, 1, 2, . . . , find a formula for fn x . 0, 1, 2, . . . , find an expression for fn x and 2 13. Prove that if n is a positive integer, then 7 n 14. Prove that 1 15. If f0 x 16. (a) If f0 x 2 3 5 1 2n x 1 x and fn f0 fn x for n 1 and fn 1 f0 fn for n 2x use mathematical induction to prove it. ; (b) Graph f0 , f1, f2 , f3 on the same screen and describe the effects of repeated composition. 63 ...
View Full Document

Ask a homework question - tutors are online