C H A P T E R 1. Functions and Models. PDF Created with deskpdf PDF Writer - Trial ::

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1 Licensed to: C H A P T E R A graphical representation of a function here the number of hours of dalight as a function of the time of ear at various latitudes is often the most natural and convenient wa to represent the function. Functions and Models Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Licensed to: jsamuels@bmcc.cun.edu. F o u r W a s t o R e p r e s e n t a F u n c t i o n Year Population (millions) FIGURE Vertical ground acceleration during the Northridge earthquake The fundamental objects that we deal with in calculus are functions. This chapter prepares the wa for calculus b discussing the basic ideas concerning functions, their graphs, and was of transforming and combining them. We stress that a function can be represented in different was: b an equation, in a table, b a graph, or in words. We look at the main tpes 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. Functions arise whenever one quantit 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 b the equation A r. With each positive number r there is associated one value of A, and we sa 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 Pt at time t, for certain ears. For instance, But for each value of the time t there is a corresponding value of P, and we sa that P is a function of t. C. The cost C of mailing a rst-class letter depends on the weight w of the letter. Although there is no simple formula that connects w and C, the post ofce has a rule for determining C when w is known. D. The vertical acceleration a of the ground as measured b a seismograph during an earthquake is a function of the elapsed time t. Figure shows a graph generated b seismic activit during the Northridge earthquake that shook Los Angeles in 994. For a given value of t, the graph provides a corresponding value of a. a {cm/s@} 5 _5 5 P95,56,, 5 5 Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. 3 t (seconds) Calif. Dept. of Mines and Geolog

2 CHAPTER FUNCTIONS AND MODELS SECTION. FOUR WAYS TO REPRESENT A FUNCTION 3 Licensed to: jsamuels@bmcc.cun.edu (input) f ƒ (output) FIGURE Machine diagram for a function a A FIGURE 3 Arrow diagram for f f(a) B ƒ Each of these eamples describes a rule whereb, given a number ( r, t, w, or t), another number ( A, P, C, or a) is assigned. In each case we sa that the second number is a function of the rst number. A function f is a rule that assigns to each element in a set A eactl one element, called f, in a set B. We usuall 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 is the value of f at and is read f of. The range of f is the set of all possible values of f as varies throughout the domain. A smbol that represents an arbitrar number in the domain of a function f is called an independent variable. A smbol that represents a number in the range of f is called a dependent variable. In Eample A, for instance, r is the independent variable and A is the dependent variable. Its helpful to think of a function as a machine (see Figure ). If is in the domain of the function f, then when enters the machine, its accepted as an input and the machine produces an output f 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 eamples of a function as a machine. For eample, the square root ke on our calculator computes such a function. You press the ke labeled s (or s) and enter the input. If, then is not in the domain of this function; that is, is not an acceptable input, and the calculator will indicate an error. If, then an approimation to s will appear in the displa. Thus, the s ke on our calculator is not quite the same as the eact mathematical function f dened b f s. Another wa to picture a function is b an arrow diagram as in Figure 3. Each arrow connects an element of A to an element of B. The arrow indicates that f is associated with, 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 (Notice that these are input-output pairs.) In other words, the graph of f consists of all points, in the coordinate plane such that f and is in the domain of f. The graph of a function f gives us a useful picture of the behavior or life histor of a function. Since the -coordinate of an point, on the graph is f, we can read the value of f from the graph as being the height of the graph above the point (see Figure 4). The graph of f also allows us to picture the domain of f on the -ais and its range on the -ais as in Figure 5. FIGURE 4 f() {, ƒ} f() ƒ, f A range FIGURE 5 ( ) domain Licensed to: jsamuels@bmcc.cun.edu The notation for intervals is given in Appendi A. FIGURE 7 - =- FIGURE 6 FIGURE 8 EXAMPLE The graph of a function f is shown in Figure 6. (a) Find the values of f and f 5. (b) What are the domain and range of f? SOLUTION (a) We see from Figure 6 that the point, 3 lies on the graph of f, so the value of f at is f 3. (In other words, the point on the graph that lies above is 3 units above the -ais.) When 5, the graph lies about.7 unit below the -ais, so we estimate that f 5.7. (b) We see that f is dened when 7, so the domain of f is the closed interval, 7. Notice that f takes on all values from to 4, so the range of f is 4, 4 EXAMPLE Sketch the graph and nd the domain and range of each function. (a) f (b) t SOLUTION (a) The equation of the graph is, and we recognize this as being the equation of a line with slope and -intercept. (Recall the slope-intercept form of the equation of a line: m b. See Appendi B.) This enables us to sketch the graph of f in Figure 7. The epression is dened for all real numbers, so the domain of f is the set of all real numbers, which we denote b. The graph shows that the range is also. (b) Since t 4 and t, we could plot the points, 4 and,, together with a few other points on the graph, and join them to produce the graph (Figure 8). The equation of the graph is, which represents a parabola (see Appendi C). The domain of t is. The range of t consists of all values of t, that is, all numbers of the form. But for all numbers and an positive number is a square. So the range of t is,. This can also be seen from Figure 8. (_,) (,4) = Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

3 verball numericall visuall algebraicall 4 CHAPTER FUNCTIONS AND MODELS Licensed to: jsamuels@bmcc.cun.edu P 6' FIGURE 9 9 R e p r e s e n t a t i o n s o f F u n c t i o n s There are four possible was to represent a function: (b a description in words) (b a table of values) (b a graph) (b an eplicit formula) If a single function can be represented in all four was, it is often useful to go from one representation to another to gain additional insight into the function. (In Eample, for instance, we started with algebraic formulas and then obtained the graphs.) But certain functions are described more naturall b one method than b another. With this in mind, lets reeamine 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 probabl the algebraic formula Ar r, though it is possible to compile a table of values or to sketch a graph (half a parabola). Because a circle has to have a positive radius, the domain is r r,, and the range is also,. B. We are given a description of the function in words: Pt is the human population of the world at time t. The table of values of world population on page 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, its impossible to devise an eplicit formula that gives the eact human population Pt at an time t. But it is possible to nd an e pression for a function that approimates Pt. In fact, using methods eplained in Section.5, we obtain the approimation and Figure shows that it is a reasonabl good t. The function f is called a mathematical model for population growth. In other words, it is a function with an eplicit formula that approimates the behavior of our given function. We will see, however, that the ideas of calculus can be applied to a table of values; an eplicit formula is not necessar. 6' t Pt f t t P FIGURE t Licensed to: jsamuels@bmcc.cun.edu A function defined b a table of values is called a tabular function. w (ounces) w w w 3 3 w 4 4 w 5 a {cm/s@} FIGURE North-south acceleration for the Northridge earthquake T FIGURE 3 Cw (dollars) t SECTION. FOUR WAYS TO REPRESENT A FUNCTION 5 The function P is tpical of the functions that arise whenever we attempt to appl calculus to the real world. We start with a verbal description of a function. Then we ma be able to construct a table of values of the function, perhaps from instrument readings in a scientic e periment. Even though we dont 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: Cw is the cost of mailing a rst-class letter with weight w. The rule that the U.S. Postal Service used as of is as follows: The cost is 37 cents for up to one ounce, plus 3 cents for each successive ounce up to 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 Eample ). D. The graph shown in Figure is the most natural representation of the vertical acceleration function at. Its true that a table of values could be compiled, and it is even possible to devise an approimate formula. But everthing a geologist needs to knowamplitudes and patternscan be seen easil from the graph. (The same is true for the patterns seen in electrocardiograms of heart patients and polgraphs for liedetection.) Figures and show the graphs of the north-south and east-west accelerations for the Northridge earthquake; when used in conjunction with Figure, the provide a great deal of information about the earthquake. 3 t (seconds) Calif. Dept. of Mines and Geolog a {cm/s@} FIGURE East-west acceleration for the Northridge earthquake In the net eample we sketch the graph of a function that is dened v erball t (seconds) Calif. Dept. of Mines and Geolog EXAMPLE 3 When ou 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 quickl. In the net 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 suppl. This enables us to make the rough sketch of T as a function of t in Figure 3. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

4 6 CHAPTER FUNCTIONS AND MODELS SECTION. FOUR WAYS TO REPRESENT A FUNCTION 7 Licensed to: jsamuels@bmcc.cun.edu h FIGURE 6 t Ct w w A more accurate graph of the function in Eample 3 could be obtained b using a thermometer to measure the temperature of the water at -second intervals. In general, scientists collect eperimental data and use them to sketch the graphs of functions, as the net eample illustrates. EXAMPLE 4 The data shown in the margin come from an eperiment on the lactonization of hdrovaleric acid at 5C. The give the concentration Ct of this acid (in moles per liter) after t minutes. Use these data to draw an approimation to the graph of the concentration function. Then use this graph to estimate the concentration after 5 minutes. SOLUTION We plot the v e points corresponding to the data from the table in Figure 4. The curve-tting methods of Section. could be used to choose a model and graph it. But the data points in Figure 4 look quite well behaved, so we simpl draw a smooth curve through them b hand as in Figure 5. C(t) t t FIGURE Then we use the graph to estimate that the concentration after 5 minutes is C5.35 moleliter In the following eample we start with a verbal description of a function in a phsical situation and obtain an eplicit algebraic formula. The abilit to do this is a useful skill in solving calculus problems that ask for the maimum or minimum values of quantities. 3 EXAMPLE 5 A rectangular storage container with an open top has a volume of m. The length of its base is twice its width. Material for the base costs $ per square meter; material for the sides costs $6 per square meter. Epress the cost of materials as a function of the width of the base. SOLUTION We draw a diagram as in Figure 6 and introduce notation b letting w and w be the width and length of the base, respectivel, and h be the height. The area of the base is ww w, so the cost, in dollars, of the material for the base is w. Two of the sides have area wh and the other two have area wh, so the cost of the material for the sides is 6wh wh. The total cost is therefore C w 6wh wh w 36wh To epress C as a function of w alone, we need to eliminate h and we do so b using the 3 fact that the volume is m. Thus C(t) FIGURE 5 wwh which gives h w 5 w Licensed to: jsamuels@bmcc.cun.edu In setting up applied functions as in Eample 5, it ma be useful to review the principles of problem solving as discussed on page 8, particularl Step : Understand the Problem. If a function is given b a formula and the domain is not stated eplicitl, the convention is that the domain is the set of all numbers for which the formula makes sense and defines a real number. FIGURE 7 Substituting this into the epression for C, we have Therefore, the equation epresses C as a function of w. EXAMPLE 6 Find the domain of each function. (b) Since and division b is not allowed, we see that t is not dened when or. Thus, the domain of t is, which could also be written in interval notation as The graph of a function is a curve in the -plane. But the question arises: Which curves in the -plane are graphs of functions? This is answered b the following test. The Vertical Line Test A curve in the -plane is the graph of a function of if and onl 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 7. If each vertical line a intersects a curve onl once, at a, b, then eactl one functional value is dened b f a b. But if a line a intersects the curve twice, at a, b and a, c, then the curve cant represent a function because a function cant assign two different values to a. C w 36w 5 w w 8 w Cw w 8 w (a) f s (b) t SOLUTION (a) Because the square root of a negative number is not dened (as a real number), the domain of f consists of all values of such that. This is equivalent to, so the domain is the interval,. =a (a,b) a t,,, w =a (a,c) (a,b) a Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

5 8 CHAPTER FUNCTIONS AND MODELS SECTION. FOUR WAYS TO REPRESENT A FUNCTION 9 Licensed to: jsamuels@bmcc.cun.edu FIGURE 9 FIGURE 8 For eample, the parabola shown in Figure 8(a) is not the graph of a function of because, as ou can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions of. Notice that the equation implies, so s. Thus, the upper and lower halves of the parabola are the graphs of the functions f s [from Eample 6(a)] and t s. [See Figures 8(b) and (c).] We observe that if we reverse the roles of and, then the equation h does dene as a function of (with as the independent variable and as the dependent variable) and the parabola now appears as the graph of the function h. (_,) (a) = - P i e c e w i s e D e f i n e d F u n c t i o n s The functions in the following four eamples are dened b dif ferent formulas in different parts of their domains. EXAMPLE 7 A function f is dened b _ (b) =œ + f if if Evaluate f, f, and f 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. If it happens that, then the value of f is. On the other hand, if, then the value of f is. Since, we have f. Since, we have f. Since, we have f 4. (c) =_œ + How do we draw the graph of f? We observe that if, then f, so the part of the graph of f that lies to the left of the vertical line must coincide with the line, which has slope and -intercept. If, then f, so the part of the graph of f that lies to the right of the line must coincide with the graph of, which is a parabola. This enables us to sketch the graph in Figure l9. The solid dot indicates that the point, is included on the graph; the open dot indicates that the point, is ecluded from the graph. _ Licensed to: jsamuels@bmcc.cun.edu For a more etensive review of absolute values, see Appendi A. FIGURE FIGURE Point-slope form of the equation of a line: See Appendi B. = m The net eample of a piecewise dened function is the absolute v alue function. Recall that the absolute value of a number a, denoted b a, is the distance from a to on the real number line. Distances are alwas positive or, so we have For eample, 3 3 In general, we have for ever number a (Remember that if a is negative, then a is positive.) EXAMPLE 8 Sketch the graph of the absolute value function. SOLUTION From the preceding discussion we know that Using the same method as in Eample 7, we see that the graph of f coincides with the line to the right of the -ais and coincides with the line to the left of the -ais (see Figure ). EXAMPLE 9 Find a formula for the function f graphed in Figure. SOLUTION The line through, and, has slope m and -intercept b, so its equation is. Thus, for the part of the graph of f that joins, to,, we have The line through, and, has slope m, so its point-slope form is So we have 3 3 a a a a a f f s s if a if a if if if or if f 3 3 Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

6 CHAPTER FUNCTIONS AND MODELS Licensed to: C w FIGURE f(_) _ FIGURE 3 An even function _ FIGURE 4 An odd function ƒ ƒ We also see that the graph of f coincides with the -ais for. Putting this information together, we have the following three-piece formula for f : EXAMPLE In Eample C at the beginning of this section we considered the cost Cw of mailing a rst-class letter with weight w. In effect, this is a piecewise dened function because, from the table of values, we have The graph is shown in Figure. You can see wh functions similar to this one are called step functionsthe jump from one value to the net. Such functions will be studied in Chapter. S m m e t r If a function f satises f f for ever number in its domain, then f is called an even function. For instance, the function f is even because The geometric signicance of an e ven function is that its graph is smmetric with respect to the -ais (see Figure 3). This means that if we have plotted the graph of f for, we obtain the entire graph simpl b reecting about the -ais. If f satises f f for ever number in its domain, then f is called an odd function. For eample, the function f 3 is odd because The graph of an odd function is smmetric about the origin (see Figure 4). If we alread have the graph of f for, we can obtain the entire graph b rotating through 8 about the origin. EXAMPLE Determine whether each of the following functions is even, odd, or neither even nor odd. (a) f 5 (b) t 4 (c) h SOLUTION (a) Therefore, f is an odd function. (b) So t is even. f Cw f f f 3 3 f f f if if if if w if w if w 3 if 3 w 4 t 4 4 t Licensed to: jsamuels@bmcc.cun.edu FIGURE 7 = FIGURE 5 FIGURE 6 (c) SECTION. FOUR WAYS TO REPRESENT A FUNCTION Since h h and h h, we conclude that h is neither even nor odd. The graphs of the functions in Eample are shown in Figure 5. Notice that the graph of h is smmetric neither about the -ais nor about the origin. _ I n c r e a s i n g a n d D e c r e a s i n g F u n c t i o n s The graph shown in Figure 6 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 and are an two numbers between a and b with, then f f. We use this as the dening propert of an increasing function. A _ =ƒ f( ) A function f is called increasing on an interval I if f f It is called decreasing on I if f h f( ) f f (a) (b) (c) B a b c d whenever in I whenever in I In the denition of an increasing function it is important to realize that the inequalit f f must be satised for ever pair of numbers and in I with. You can see from Figure 7 that the function f is decreasing on the interval, and increasing on the interval,. g C D h Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

7 CHAPTER FUNCTIONS AND MODELS SECTION. FOUR WAYS TO REPRESENT A FUNCTION 3 Licensed to: jsamuels@bmcc.cun.edu. Eercises. The graph of a function f is given. (a) State the value of f. (b) Estimate the value of f. (c) For what values of is f? (d) Estimate the values of such that f. (e) State the domain and range of f. (f) On what interval is f increasing?. The graphs of f and t are given. (a) State the values of f 4 and t3. (b) For what values of is f t? (c) Estimate the solution of the equation f. (d) On what interval is f decreasing? (e) State the domain and range of f. (f) State the domain and range of t. f 3. Figures,, and were recorded b an instrument operated b the California Department of Mines and Geolog at the Universit Hospital of the Universit 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 eamples of ordinar, everda functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other eamples of functions from everda life that are described verball. What can ou sa about the domain and range of each of our functions? If possible, sketch a rough graph of each function. g 5 8 Determine whether the curve is the graph of a function of. If it is, state the domain and range of the function The graph shown gives the weight of a certain person as a function of age. Describe in words how this persons weight varies over time. What do ou think happened when this person was 3 ears old? Weight (pounds). The graph shown gives a salesmans distance from his home as a function of time on a certain da. Describe in words what the graph indicates about his travels on this da. Distance from home (miles) Age (ears) 8 A.M. NOON 4 6 P.M. Time (hours). You put some ice cubes in a glass, ll 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. Licensed to: jsamuels@bmcc.cun.edu. Sketch a rough graph of the number of hours of dalight as a function of the time of ear. 3. Sketch a rough graph of the outdoor temperature as a function of time during a tpical spring da. 4. You place a frozen pie in an oven and bake it for an hour. Then ou 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. 5. A homeowner mows the lawn ever Wednesda afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period. 6. An airplane ies from an airport and lands an hour later at another airport, 4 miles awa. If t represents the time in minutes since the plane has left the terminal building, let t be the horizontal distance traveled and t be the altitude of the plane. (a) Sketch a possible graph of t. (b) Sketch a possible graph of t. (c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocit. 7. The number N (in thousands) of cellular phone subscribers in Malasia is shown in the table. (Midear estimates are given.) (a) Use the data to sketch a rough graph of N as a function of t. (b) Use our graph to estimate the number of cell-phone subscribers in Malasia at midear in 994 and Temperature readings T (in F) were recorded ever two hours from midnight to : P.M. in Dallas on June,. The time t was measured in hours from midnight. (a) Use the readings to sketch a rough graph of T as a function of t. (b) Use our graph to estimate the temperature at : A.M. 9. If f 3, nd f, f, f a, f a, f a, f a, f a, f a, [ f a], and f a h.. A spherical balloon with radius r inches has volume Vr 4 3 r 3. Find a function that represents the amount of air required to inate the balloon from a radius of r inches to a radius of r inches. f h f Find f h, f h, and, h where h.. f t N t T f 3 7 Find the domain of the function f 4. f f t st s 3 t 6. tu su s4 u 7. h s Find the domain and range and sketch the graph of the function h s Find the domain and sketch the graph of the function. 9. f 5 3. F 3 3. f t t 6t t s f f G 3 f f Find an epression for the function whose graph is the given curve. 4. The line segment joining the points, and 4, 6 4. The line segment joining the points 3, and 6, The bottom half of the parabola 44. The top half of the circle if if if if if if if if if Ht 4 t t F t Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

8 Licensed to: 47 5 Find a formula for the described function and state its domain. 4 CHAPTER FUNCTIONS AND MODELS 47. A rectangle has perimeter m. Epress the area of the rectangle as a function of the length of one of its sides. 48. A rectangle has area 6 m. Epress the perimeter of the rectangle as a function of the length of one of its sides. 49. Epress the area of an equilateral triangle as a function of the length of a side. 5. Epress the surface area of a cube as a function of its volume An open rectangular bo with volume m has a square base. Epress the surface area of the bo as a function of the length of a side of the base. 5. A Norman window has the shape of a rectangle surmounted b a semicircle. If the perimeter of the window is 3 ft, epress the area A of the window as a function of the width of the window. 53. A bo with an open top is to be constructed from a rectangular piece of cardboard with dimensions in. b in. b cutting out equal squares of side at each corner and then folding up the sides as in the gure. Epress the v olume V of the bo as a function of. 54. A tai compan charges two dollars for the rst mile (or part of a mile) and cents for each succeeding tenth of a mile (or part). Epress the cost C (in dollars) of a ride as a function of the distance traveled (in miles) for, and sketch the graph of this function. Catherine Karnow 55. In a certain countr, income ta is assessed as follows. There is no ta on income up to $,. An income over $, is taed at a rate of %, up to an income of $,. An income over $, is taed at 5%. (a) Sketch the graph of the ta rate R as a function of the income I. (b) How much ta is assessed on an income of $4,? On $6,? (c) Sketch the graph of the total assessed ta T as a function of the income I. 56. The functions in Eample and Eercises 54 and 55(a) are called step functions because their graphs look like stairs. Give two other eamples of step functions that arise in everda life Graphs of f and t are shown. Decide whether each function is even, odd, or neither. Eplain our reasoning g f 59. (a) If the point 5, 3 is on the graph of an even function, what 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? 6. A function f has domain 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. _ Determine whether f is even, odd, or neither. If f is even or odd, use smmetr to sketch its graph. 6. f 6. f f 64. f f f 3 3 f g Licensed to: jsamuels@bmcc.cun.edu. M a t h e m a t i c a l M o d e l s : A C a t a l o g o f E s s e n t i a l F u n c t i o n s FIGURE The modeling process The coordinate geometr of lines is reviewed in Appendi B. A mathematical model is a mathematical description (often b 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 epectanc 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 illustrates the process of mathematical modeling. Given a real-world problem, our rst task is to formulate a mathematical model b identifing and naming the independent and dependent variables and making assumptions that simplif the phenomenon enough to make it mathematicall tractable. We use our knowledge of the phsical situation and our mathematical skills to obtain equations that relate the variables. In situations where there is no phsical law to guide us, we ma need to collect data (either from a librar or the Internet or b conducting our own eperiments) and eamine the data in the form of a table in order to discern patterns. From this numerical representation of a function we ma wish to obtain a graphical representation b plotting the data. The graph might even suggest a suitable algebraic formula in some cases. The second stage is to appl 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 b wa of offering eplanations or making predictions. The nal step is to test our predictions b checking against new real data. If the predictions dont compare well with realit, we need to rene our model or to formulate a ne w model and start the ccle again. A mathematical model is never a completel accurate representation of a phsical situationit is an idealization. A good model simplies realit 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 nal sa. There are man different tpes 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 eamples of situations appropriatel modeled b such functions. L i n e a r M o d e l s Real-world problem Test Real-world predictions SECTION. MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS 5 Formulate Interpret Mathematical model Solve Mathematical conclusions When we sa that is a linear function of, 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 Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

9 6 CHAPTER FUNCTIONS AND MODELS Licensed to: T T=_h+ 3 h FIGURE 3 FIGURE the function as where m is the slope of the line and b is the -intercept. A characteristic feature of linear functions is that the grow at a constant rate. For instance, Figure shows a graph of the linear function f 3 and a table of sample values. Notice that whenever increases b., the value of f increases b.3. So f increases three times as fast as. Thus, the slope of the graph 3, namel 3, can be interpreted as the rate of change of with respect to. _ =3- EXAMPLE (a) As dr air moves upward, it epands and cools. If the ground temperature is C and the temperature at a height of km is C, epress 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.5 km? SOLUTION (a) Because we are assuming that T is a linear function of h, we can write We are given that T when h, so T mh b m b b In other words, the -intercept is b. We are also given that T when h, so m The slope of the line is therefore m and the required linear function is T h (b) The graph is sketched in Figure 3. The slope is m Ckm, and this represents the rate of change of temperature with respect to height. (c) At a height of h.5 km, the temperature is f m b T.5 5C If there is no phsical law or principle to help us formulate a model, we construct an empirical model, which is based entirel on collected data. We seek a curve that ts the data in the sense that it captures the basic trend of the data points. f Licensed to: jsamuels@bmcc.cun.edu TABLE Year CO level (in ppm) FIGURE 4 Scatter plot for the average CO level FIGURE 5 Linear model through rst and last data points SECTION. MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS 7 EXAMPLE Table lists the average carbon dioide level in the atmosphere, measured in parts per million at Mauna Loa Observator from 98 to. Use the data in Table to nd a model for the carbon dioide le vel. SOLUTION We use the data in Table to make the scatter plot in Figure 4, where t represents time (in ears) and C represents the CO level (in parts per million, ppm). Notice that the data points appear to lie close to a straight line, so its natural to choose a linear model in this case. But there are man possible lines that approimate these data points, so which one should we use? From the graph, it appears that one possibilit is the line that passes through the rst and last data points. The slope of this line is and its equation is or C Equation gives one possible linear model for the carbon dioide level; it is graphed in Figure 5. C C t 98 C.535t Although our model ts the data reasonabl well, it gives values higher than most of the actual CO levels. A better linear model is obtained b a procedure from statistics t t Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

10 8 CHAPTER FUNCTIONS AND MODELS Licensed to: A computer or graphing calculator finds the regression line b 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 eplained in Section 4.7. FIGURE 6 The regression line called linear regression. If we use a graphing calculator, we enter the data from Table into the data editor and choose the linear regression command. (With Maple we use the t[leastsquare] command in the stats package; with Mathematica we use the Fit command.) The machine gives the slope and -intercept of the regression line as So our least squares model for the CO level is 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 t than our pre vious linear model EXAMPLE 3 Use the linear model given b Equation to estimate the average CO level for 987 and to predict the level for the ear. According to this model, when will the CO level eceed 4 parts per million? SOLUTION Using Equation with t 987, we estimate that the average CO level in 987 was C This is an eample of interpolation because we have estimated a value between observed values. (In fact, the Mauna Loa Observator reported that the average CO level in 987 was ppm, so our estimate is quite accurate.) With t, we get So we predict that the average CO level in the ear will be ppm. This is an eample of etrapolation because we have predicted a value outside the region of observations. Consequentl, we are far less certain about the accurac of our prediction. Using Equation, we see that the CO level eceeds 4 ppm when Solving this inequalit, we get C 37 m.5388 C.5388t b 77.5 C t t t Licensed to: jsamuels@bmcc.cun.edu FIGURE 7 The graphs of quadratic functions are parabolas. FIGURE 8 SECTION. MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS 9 We therefore predict that the CO level will eceed 4 ppm b the ear. This prediction is somewhat risk because it involves a time quite remote from our observations. P o l n o m i a l s A function P is called a polnomial if where n is a nonnegative integer and the numbers a, a, a,..., an are constants called the coefcients of the polnomial. The domain of an polnomial is,. If the leading coefcient an, then the degree of the polnomial is n. For eample, the function is a polnomial of degree 6. A polnomial of degree is of the form P m b and so it is a linear function. A polnomial of degree is of the form P a b c and is called a quadratic function. Its graph is alwas a parabola obtained b shifting the parabola a, as we will see in the net section. The parabola opens upward if a and downward if a. (See Figure 7.) A polnomial of degree 3 is of the form and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polnomials of degrees 4 and 5 in parts (b) and (c). We will see later wh the graphs have these shapes. (a) = -+ P an n an n a a a (a) = ++ P s P a 3 b c d (b) =$-3 + (b) =_ +3+ (c) =3%-5 +6 Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

11 3 CHAPTER FUNCTIONS AND MODELS Licensed to: Time (seconds) TABLE Height (meters) Polnomials are commonl used to model various quantities that occur in the natural and social sciences. For instance, in Section 3.3 we will eplain wh economists often use a polnomial P to represent the cost of producing units of a commodit. In the following eample we use a quadratic function to model the fall of a ball. EXAMPLE 4 A ball is dropped from the upper observation deck of the CN Tower, 45 m above the ground, and its height h above the ground is recorded at -second intervals in Table. Find a model to t 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 tr a quadratic model instead. Using a graphing calculator or computer algebra sstem (which uses the least squares method), we obtain the following quadratic model: 3 h (meters) FIGURE 9 Scatter plot for a falling ball In Figure we plot the graph of Equation 3 together with the data points and see that the quadratic model gives a ver good t. The ball hits the ground when h, so we solve the quadratic equation The quadratic formula gives The positive root is t 9.67, so we predict that the ball will hit the ground after about 9.7 seconds. P o w e r F u n c t i o n s h t 4.9t t (seconds) 4.9t.96t FIGURE Quadratic model for a falling ball t.96 s A function of the form f a, where a is a constant, is called a power function. We consider several cases. (i) a n, where n is a positive integer The graphs of f n for n,, 3, 4, and 5 are shown in Figure. (These are polnomials with onl one term.) We alread know the shape of the graphs of (a line through the origin with slope ) and [a parabola, see Eample (b) in Section.]. h t Licensed to: jsamuels@bmcc.cun.edu FIGURE = Graphs of ƒ= n for n=,, 3, 4, 5 FIGURE Families of power functions FIGURE 3 Graphs of root functions = SECTION. MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS 3 The general shape of the graph of f n depends on whether n is even or odd. If n is even, then f n is an even function and its graph is similar to the parabola. If n is odd, then f n is an odd function and its graph is similar to that of 3. Notice from Figure, however, that as n increases, the graph of n becomes atter near and steeper when. (If is small, then is smaller, 3 is even smaller, 4 is smaller still, and so on.) (_, ) =^ (ii) a n, where n is a positive integer The function f n s n is a root function. For n it is the square root function f s, whose domain is, and whose graph is the upper half of the parabola. [See Figure 3(a).] For other even values of n, the graph of s n is similar to that of s. For n 3 we have the cube root function f s 3 whose domain is (recall that ever real number has a cube root) and whose graph is shown in Figure 3(b). The graph of s n for n odd n 3 is similar to that of s 3. (,) (a) ƒ=œ =# =$ (, ) = =$ (_, _) (,) (b) ƒ=#œ =# =% (, ) =% Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

12 3 CHAPTER FUNCTIONS AND MODELS Licensed to: = FIGURE 4 The reciprocal function FIGURE 6 FIGURE 5 Volume as a function of pressure at constant temperature ƒ= $ (iii) a The graph of the reciprocal function f is shown in Figure 4. Its graph has the equation, or, and is a hperbola with the coordinate aes as its asmptotes. This function arises in phsics and chemistr in connection with Boles Law, which sas that, when the temperature is constant, the volume V of a gas is inversel proportional to the pressure P: where C is a constant. Thus, the graph of V as a function of P (see Figure 5) has the same general shape as the right half of Figure 4. Another instance in which a power function is used to model a phsical phenomenon is discussed in Eercise. R a t i o n a l F u n c t i o n s A rational function f is a ratio of two polnomials: where P and Q are polnomials. The domain consists of all values of such that Q. A simple eample of a rational function is the function f, whose domain is ; this is the reciprocal function graphed in Figure 4. The function is a rational function with domain. Its graph is shown in Figure 6. A l g e b r a i c F u n c t i o n s 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 polnomials. An rational function is automaticall an algebraic function. Here are two more eamples: f s V V C P f P Q f 4 4 P t 4 6 s s 3 Licensed to: jsamuels@bmcc.cun.edu _ FIGURE 8 FIGURE 7 (a) ƒ=sin 3 SECTION. 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 variet of shapes. Figure 7 illustrates some of the possibilities. An eample of an algebraic function occurs in the theor of relativit. The mass of a particle with velocit v is where m is the rest mass of the particle and c 3. 5 kms is the speed of light in a vacuum. T r i g o n o m e t r i c F u n c t i o n s Trigonometr and the trigonometric functions are reviewed on Reference Page and also in Appendi D. In calculus the convention is that radian measure is alwas used (ecept when otherwise indicated). For eample, when we use the function f sin, it is understood that sin means the sine of the angle whose radian measure is. Thus, the graphs of the sine and cosine functions are as shown in Figure (a) ƒ=œ +3 Notice that for both the sine and cosine functions the domain is, and the range is the closed interval,. Thus, for all values of, we have or, in terms of absolute values, sin sin m f v s v c _ 5 (b) =œ $ -5 m 3 (b) =cos cos cos 5 (c) h()=@?#(-)@ 3 Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

13 34 CHAPTER FUNCTIONS AND MODELS Licensed to: 3 _ FIGURE 9 =tan 3 Also, the zeros of the sine function occur at the integer multiples of ; that is, sin An important propert of the sine and cosine functions is that the are periodic functions and have period. This means that, for all values of, The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves. For instance, in Eample 4 in Section.3 we will see that a reasonable model for the number of hours of dalight in Philadelphia t das after Januar is given b the function Lt.8 sin 365 t 8 The tangent function is related to the sine and cosine functions b the equation and its graph is shown in Figure 9. It is undened whene ver cos, that is, when, 3,.... Its range is,. Notice that the tangent function has period : The remaining three trigonometric functions (cosecant, secant, and cotangent) are the reciprocals of the sine, cosine, and tangent functions. Their graphs are shown in Appendi D. E p o n e n t i a l F u n c t i o n s sin sin The eponential functions are the functions of the form f a, where the base a is a positive constant. The graphs of and.5 are shown in Figure. In both cases the domain is, and the range is,. tan tan FIGURE (a) = (b) =(.5) when tan sin cos n n an integer cos cos for all Eponential functions will be studied in detail in Section.5, and we will see that the are useful for modeling man natural phenomena, such as population growth (if a ) and radioactive deca (if a. Licensed to: jsamuels@bmcc.cun.edu. Eercises FIGURE L o g a r i t h m i c F u n c t i o n s SECTION. MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS 35 The logarithmic functions f loga, where the base a is a positive constant, are the inverse functions of the eponential functions. The will be studied in Section.6. Figure shows the graphs of four logarithmic functions with various bases. In each case the domain is,, the range is,, and the function increases slowl when. T r a n s c e n d e n t a l F u n c t i o n s These are functions that are not algebraic. The set of transcendental functions includes the trigonometric, inverse trigonometric, eponential, and logarithmic functions, but it also includes a vast number of other functions that have never been named. In Chapter we will stud transcendental functions that are dened as sums of innite series. EXAMPLE 5 Classif the following functions as one of the tpes of functions that we have discussed. (a) f 5 (b) t 5 (c) h (d) ut t 5t 4 s SOLUTION (a) f 5 is an eponential function. (The is the eponent.) (b) t 5 is a power function. (The is the base.) We could also consider it to be a polnomial of degree 5. (c) h s Classif each function as a power function, root function, polnomial (state its degree), rational function, algebraic function, trigonometric function, eponential function, or logarithmic function.. (a) f s 5 (b) t s (c) (d) r h is an algebraic function. (d) ut t 5t 4 is a polnomial of degree 4. (e) s tan =log =log =log =log. (a) 6 (b) 6 (c) (e) t 6 t 4 (f) t log s (d) (f) cos sin Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

14 36 CHAPTER FUNCTIONS AND MODELS SECTION. MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS 37 Licensed to: jsamuels@bmcc.cun.edu 3 4 Match each equation with its graph. Eplain our choices. (Dont use a computer or graphing calculator.) 3. (a) (b) 5 (c) 8 4. (a) 3 (b) 3 (c) 3 (d) s 3 5. f f G (a) Find an equation for the famil of linear functions with slope and sketch several members of the famil. (b) Find an equation for the famil of linear functions such that f and sketch several members of the famil. (c) Which function belongs to both families? 6. What do all members of the famil of linear functions f m 3 have in common? Sketch several members of the famil. 7. What do all members of the famil of linear functions f c have in common? Sketch several members of the famil. 8. The manager of a weekend ea mark et knows from past eperience that if he charges dollars for a rental space at the ea market, then the number of spaces he can rent is given b the equation 4. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented cant be negative quantities.) F g g h (b) What do the slope, the -intercept, and the -intercept of the graph represent? 9. The relationship between the Fahrenheit F and Celsius C temperature scales is given b the linear function F 9 5 C 3. (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?. Jason leaves Detroit at : P.M. and drives at a constant speed west along I-96. He passes Ann Arbor, 4 mi from Detroit, at :5 P.M. (a) Epress 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?. Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be ver nearl linear. A cricket produces 3 chirps per minute at 7F and 73 chirps per minute at 8F. (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 5 chirps per minute, estimate the temperature.. The manager of a furniture factor nds that it costs $ to manufacture chairs in one da and $48 to produce 3 chairs in one da. (a) Epress 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 -intercept of the graph and what does it represent? 3. At the surface of the ocean, the water pressure is the same as the air pressure above the water, 5 lbin. Below the surface, the water pressure increases b 4.34 lbin for ever ft of descent. (a) Epress the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure lbin? 4. The monthl cost of driving a car depends on the number of miles driven. Lnn found that in Ma it cost her $38 to drive 48 mi and in June it cost her $46 to drive 8 mi. (a) Epress the monthl 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 5 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the -intercept represent? (e) Wh does a linear function give a suitable model in this situation? Licensed to: jsamuels@bmcc.cun.edu 5 6 For each scatter plot, decide what tpe of function ou might choose as a model for the data. Eplain our choices. 5. (a) (b) 6. (a) (b) ; 7. The table shows (lifetime) peptic ulcer rates (per population) for various famil incomes as reported b the 989 National Health Interview Surve. Income Ulcer rate (per population) $4, 4. $6, 3. $8, 3.4 $,.5 $6,. $,.4 $3,.5 $45, 9.4 $6, 8. (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 rst 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 $5,. (e) According to the model, how likel is someone with an income of $8, to suffer from peptic ulcers? (f) Do ou think it would be reasonable to appl the model to someone with an income of $,? ; 8. 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. Temperature ( F) Chirping rate (chirpsmin) (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 F. ; 9. The table gives the winning heights for the Olmpic pole vault competitions in the th centur. Year Height (ft) Year Height (ft) (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 Olmpics and compare with the winning height of 9.36 feet. (d) Is it reasonable to use the model to predict the winning height at the Olmpics? ;. A stud b the U. S. Ofce of Science and Technolog in 97 estimated the cost (in 97 dollars) to reduce automobile emissions b certain percentages: Reduction in Cost per Reduction in Cost per emissions (%) car (in $) emissions (%) car (in $) Find a model that captures the diminishing returns trend of these data. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

15 38 CHAPTER FUNCTIONS AND MODELS Licensed to: ;. Use the data in the table to model the population of the world in the th centur b a cubic function. Then use our model to estimate the population in the ear 95. Year Population (millions) ;. The table shows the mean (average) distances d of the planets from the Sun (taking the unit of measurement to be the.3 N e w F u n c t i o n s f r o m O l d F u n c t i o n s distance from Earth to the Sun) and their periods T (time of revolution in ears). (a) Fit a power model to the data. (b) Keplers Third Law of Planetar 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 our model corroborate Keplers Third Law? In this section we start with the basic functions we discussed in Section. and obtain new functions b shifting, stretching, and reecting their graphs. We also show how to combine pairs of functions b the standard arithmetic operations and b composition. T r a n s f o r m a t i o n s o f F u n c t i o n s Planet d T Mercur Venus Earth.. Mars Jupiter Saturn Uranus Neptune Pluto B appling certain transformations to the graph of a given function we can obtain the graphs of certain related functions. This will give us the abilit to sketch the graphs of man functions quickl b hand. It will also enable us to write equations for given graphs. Lets rst consider translations. If c is a positive number, then the graph of f c is just the graph of f shifted upward a distance of c units (because each -coordinate is increased b the same number c). Likewise, if t f c, where c, then the value of t at is the same as the value of f at c (c units to the left of ). Therefore, the graph of f c is just the graph of f shifted c units to the right (see Figure ). Vertical and Horizontal Shifts Suppose c. To obtain the graph of f c, shift the graph of f a distance c units upward f c, shift the graph of f a distance c units downward f c, shift the graph of f a distance c units to the right f c, shift the graph of f a distance c units to the left Now lets consider the stretching and reecting transformations. If c, then the graph of cf is the graph of f stretched b a factor of c in the vertical direction (because each -coordinate is multiplied b the same number c). The graph of Licensed to: jsamuels@bmcc.cun.edu =f(+c) c =ƒ FIGURE Translating the graph of In Module.3 ou can see the effect of combining the transformations of this section. c c FIGURE 3 c =ƒ+c =f(-c) =ƒ-c f is the graph of f reected about the -ais because the point, is replaced b the point,. (See Figure and the following chart, where the results of other stretching, compressing, and reecting transformations are also gi ven.) Vertical and Horizontal Stretching and Reflecting Suppose c. To obtain the graph of cf, stretch the graph of f verticall b a factor of c cf, compress the graph of f verticall b a factor of c f c, compress the graph of f horizontall b a factor of c f c, stretch the graph of f horizontall b a factor of c f, reflect the graph of f about the -ais f, reflect the graph of f about the -ais Figure 3 illustrates these stretching transformations when applied to the cosine function with c. For instance, in order to get the graph of cos we multipl the -coordinate of each point on the graph of cos b. This means that the graph of cos gets stretched verticall b a factor of. = Ł =Ł = Ł =f(_) FIGURE Stretching and reecting the graph of SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 39 =Ł =Ł =Ł =cƒ (c>) =ƒ = ƒ c =_ƒ Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

16 4 CHAPTER FUNCTIONS AND MODELS Licensed to: (a) =œ FIGURE 4 FIGURE 6 _ (b) =œ - EXAMPLE Given the graph of s, use transformations to graph s, s, s, s, and s. SOLUTION The graph of the square root function s, obtained from Figure 3 in Section., is shown in Figure 4(a). In the other parts of the gure we sk etch s b shifting units downward, s b shifting units to the right, s b reecting about the -ais, s b stretching verticall b a factor of, and s b reecting about the -ais. EXAMPLE Sketch the graph of the function f () 6. SOLUTION Completing the square, we write the equation of the graph as This means we obtain the desired graph b starting with the parabola and shifting 3 units to the left and then unit upward (see Figure 5). FIGURE 5 (a) = (b) =(+3)@+ EXAMPLE 3 Sketch the graphs of the following functions. (a) sin (b) sin SOLUTION (a) We obtain the graph of sin from that of sin b compressing horizontall b a factor of (see Figures 6 and 7). Thus, whereas the period of sin is, the period of sin is. =sin (c) =œ - (d) =_œ (e) =œ (f) =œ _ 6 3 FIGURE 7 4 (_3,) _3 =sin _ Licensed to: jsamuels@bmcc.cun.edu FIGURE 8 FIGURE 9 Graph of the length of dalight from March through December at various latitudes SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 4 (b) To obtain the graph of sin, we again start with sin. We reect about the -ais to get the graph of sin and then we shift unit upward to get sin. (See Figure 8.) EXAMPLE 4 Figure 9 shows graphs of the number of hours of dalight as functions of the time of the ear at several latitudes. Given that Philadelphia is located at approimatel 4N latitude, nd a function that models the length of dalight at Philadelphia. Hours N 5 N 4 N 3 N N Mar. Apr. Ma June Jul Aug. Sept. Oct. Nov. Dec. Source: Lucia C. Harrison, Dalight, Twilight, Darkness and Time (New York: Silver, Burdett, 935) page 4. SOLUTION Notice that each curve resembles a shifted and stretched sine function. B looking at the blue curve we see that, at the latitude of Philadelphia, dalight lasts about 4.8 hours on June and 9. hours on December, so the amplitude of the curve (the factor b which we have to stretch the sine curve verticall) is B what factor do we need to stretch the sine curve horizontall if we measure the time t in das? Because there are about 365 das in a ear, the period of our model should be 365. But the period of sin t is, so the horizontal stretching factor is c 365. We also notice that the curve begins its ccle on March, the 8th da of the ear, so we have to shift the curve 8 units to the right. In addition, we shift it units upward. Therefore, we model the length of dalight in Philadelphia on the tth da of the ear b the function Lt.8 sin 365 t 8 Another transformation of some interest is taking the absolute value of a function. If f, then according to the denition of absolute v alue, f when f and f when f. This tells us how to get the graph of from the graph =-sin 3 f Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

17 4 CHAPTER FUNCTIONS AND MODELS Licensed to: FIGURE of f : The part of the graph that lies above the -ais remains the same; the part that lies below the -ais is reected about the -ais. EXAMPLE 5 Sketch the graph of the function. SOLUTION We rst graph the parabola in Figure (a) b shifting the parabola downward unit. We see that the graph lies below the -ais when, so we reect that part of the graph about the -ais to obtain the graph of in Figure (b). C o m b i n a t i o n s o f F u n c t i o n s Two functions f and t can be combined to form new functions f t, f t, ft, and ft in a manner similar to the wa we add, subtract, multipl, and divide real numbers. If we dene the sum f t b the equation (a) = - then the right side of Equation makes sense if both f and t are dened, that is, if 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 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 and t. Similarl, we can dene the dif ference f t and the product ft, and their domains are also A B. But in dening the quotient ft we must remember not to divide b. Algebra of Functions Let f and t be functions with domains A and B. Then the functions f t, f t, ft, and ft are dened as follo ws: f t f t f t f t f t f t domain A B domain A B ft f t domain A B t f f t domain A B t (b) = - Licensed to: jsamuels@bmcc.cun.edu Another wa to solve 4 : _ =(f+g)() = =ƒ a f(a) SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 43 EXAMPLE 6 If f s and t s4, nd the functions f t, f t, ft, and ft. SOLUTION The domain of f s is,. The domain of t s4 consists of all numbers such that 4, that is, 4. Taking square roots of both sides, we get, or, so the domain of t is the interval,. The intersection of the domains of f and t is Thus, according to the denitions, we ha ve Notice that the domain of ft is the interval, ; we have to eclude because t. The graph of the function f t is obtained from the graphs of f and t b graphical addition. This means that we add corresponding -coordinates as in Figure. Figure shows the result of using this procedure to graph the function f t from Eample 6. g(a) f(a)+g(a) FIGURE FIGURE f t s s4 f t s s4 ft ss4 s4 3 t f s s4 4 _ C o m p o s i t i o n o f F u n c t i o n s,,, =œ 4- There is another wa of combining two functions to get a new function. For eample, suppose that f u su and u t. Since is a function of u and u is, in turn, a function of, it follows that is ultimatel a function of. We compute this b substitution:.5.5 f u f t f s =(f+g)() ƒ=œ _ Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

18 44 CHAPTER FUNCTIONS AND MODELS Licensed to: FIGURE 3 The f g machine is composed of the g machine (first) and then the f machine. FIGURE 4 Arrow diagram for f g The procedure is called composition because the new function is composed of the two given functions f and t. In general, given an two functions f and t, we start with a number in the domain of t and nd its image t. If this number t is in the domain of f, then we can calculate the value of f t. The result is a new function h f t obtained b substituting t into f. It is called the composition (or composite) of f and t and is denoted b 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 dened b The domain of f t is the set of all in the domain of t such that t is in the domain of f. In other words, f t is dened whene ver both t and f t are dened. The best wa to picture f t is b either a machine diagram (Figure 3) or an arrow diagram (Figure 4). EXAMPLE 7 If f and t 3, nd the composite functions f t and t f. SOLUTION We have NOTE You can see from Eample 7 that, in general, f ttf. Remember, the notation f t means that the function t is applied rst and then f is applied second. In Eample 7, f t is the function that r st subtracts 3 and then squares; t f is the function that r st squares and then subtracts 3. EXAMPLE 8 If f s and t s, nd each function and its domain. (a) f t (b) t f (c) f f (d) t t SOLUTION (a) (input) g g() f g f t f t f g f t f t f 3 3 t f t f t 3 f t f t f (s ) ss s 4 The domain of f t is,. f f{ } (output) f{ } Licensed to: jsamuels@bmcc.cun.edu (b) SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 45 For s to be dened we must ha ve. For s s to be dened we must ha ve If a b, then a b. s, that is, s, or 4. Thus, we have 4, so the domain of t f is the closed interval, 4..3 Eercises (c) The domain of f f is,. (d) This epression is dened when, that is,, and s. This latter inequalit is equivalent to s, or 4, that is,. Thus,, so the domain of t t is the closed interval,. It is possible to take the composition of three or more functions. For instance, the composite function f th is found b rst appling h, then t, and then f as follows: EXAMPLE 9 Find f th if f, t, and h 3. SOLUTION 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 eample. EXAMPLE Given F cos 9, nd functions f, t, and h such that F f th. SOLUTION Since F cos 9, the formula for F sas: First add 9, then take the cosine of the result, and nall square. So we let Then. 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. t f t f t(s) s s f f f f f (s) ss s 4 t t tt t(s ) s s f th f th f t 3 h 9 f th f th f 3 t cos f th f th f t 9 f cos 9 cos 9 F 3 3 f (d) Shift 3 units to the left. (e) Reect about the -ais. (f) Reect about the -ais. (g) Stretch verticall b a factor of 3. (h) Shrink verticall b a factor of 3. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

19 46 CHAPTER FUNCTIONS AND MODELS Licensed to: Eplain how the following graphs are obtained from the graph of f. (a) 5f (b) f 5 (c) f (d) 5f (e) f 5 (f) 5f 3 3. The graph of f is given. Match each equation with its graph and give reasons for our choices. (a) f 4 (b) f 3 (c) 3 f (d) f 4 (e) f 6! _6 _ The graph of f is given. Draw the graphs of the following functions. (a) f 4 (b) f 4 (c) f (d) f 3 5. The graph of f is given. Use it to graph the following functions. (a) f (b) f ( ) (c) f (d) f 6 7 The graph of s3 is given. Use transformations to create a function whose graph is as shown..5 % 3 _3 =œ 3-3 f $ # _4 8. (a) How is the graph of sin related to the graph of sin? Use our answer and Figure 6 to sketch the graph of sin. (b) How is the graph of s related to the graph of s? Use our answer and Figure 4(a) to sketch the graph of s. 9 4 Graph the function, not b plotting points, but b starting with the graph of one of the standard functions given in Section., and then appling the appropriate transformations cos 4. 4 sin 3 5. sin 7. s s tan sin _ The cit of New Orleans is located at latitude 3N. Use Figure 9 to nd a function that models the number of hours of dalight at New Orleans as a function of the time of ear. Use the fact that on March 3 the Sun rises at 5:5 A.M. and sets at 6:8 P.M. in New Orleans to check the accurac of our model. 6. A variable star is one whose brightness alternatel increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maimum brightness is 5.4 das, the average brightness (or magnitude) of the star is 4., and its brightness varies b.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time. SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 47 Licensed to: jsamuels@bmcc.cun.edu 7. (a) How is the graph of f ( ) related to the graph of f? 38. f 3, t 5 3 (b) Sketch the graph of sin. (c) Sketch the graph of s 39. f, t. 8. Use the given graph of f to sketch the graph of f. Which features of f are the most important in sketching f? Eplain how the are used. 4. f s 3, 4 44 Find f th. t 4. f, t, h 9 3 Use graphical addition to sketch the graph of f t f 3 3 Find f t, f t, ft, and ft and state their domains. 3. f 3, t 3 3. f s, t s Use the graphs of f and t and the method of graphical addition to sketch the graph of f t. 33. f, t 34. f 3, t 35 4 Find the functions f t, t f, f f, and t t and their domains. 35. f, t f 3, t 37. f sin, t s g g f 4. f, t, h 43. f s, t, h f, t cos, h s Epress the function in the form f t. 45. F 46. F sin(s) 47. G ut scos t 5 53 Epress the function in the form f th. 5. H 3 5. H s 3 s 53. H sec 4 (s) 54. Use the table to evaluate each epression. (a) f t (b) t f (c) f f (d) tt (e) t f 3 (f) f t6 55. Use the given graphs of f and t to evaluate each epression, or eplain wh it is undened. (a) f t (b) t f (c) f t (d) t f 6 (e) t t (f) f f 4 5. ut tan t tan t g G f t f Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

20 48 CHAPTER FUNCTIONS AND MODELS Licensed to: 56. Use the given graphs of f and t to estimate the value of f t for 5, 4, 3,..., 5. Use these estimates to sketch a rough graph of f t. 57. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 6 cms. (a) Epress 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, nd A r and interpret it. 58. An airplane is ing at a speed of 35 mih at an altitude of one mile and passes directl over a radar station at time t. (a) Epress the horizontal distance d (in miles) that the plane has o wn as a function of t. (b) Epress the distance s between the plane and the radar station as a function of d. (c) Use composition to epress s as a function of t. 59. The Heaviside function H is dened b f Ht if t if t It is used in the stud of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneousl turned on. (a) Sketch the graph of the Heaviside function..4 G r a p h i n g C a l c u l a t o r s a n d C o m p u t e r s g (b) Sketch the graph of the voltage Vt in a circuit if the switch is turned on at time t and volts are applied instantaneousl to the circuit. Write a formula for Vt in terms of Ht. (c) Sketch the graph of the voltage Vt in a circuit if the switch is turned on at time t 5 seconds and 4 volts are applied instantaneousl to the circuit. Write a formula for Vt in terms of Ht. (Note that starting at t 5 corresponds to a translation.) 6. The Heaviside function dened in E ercise 59 can also be used to dene the ramp function ctht, which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function tht. (b) Sketch the graph of the voltage Vt in a circuit if the switch is turned on at time t and the voltage is graduall increased to volts over a 6-second time interval. Write a formula for Vt in terms of Ht for t 6. (c) Sketch the graph of the voltage Vt in a circuit if the switch is turned on at time t 7 seconds and the voltage is graduall increased to volts over a period of 5 seconds. Write a formula for Vt in terms of Ht for t (a) If t and h 4 4 7, nd a function f such that f th. (Think about what operations ou would have to perform on the formula for t to end up with the formula for h.) (b) If f 3 5 and h 3 3, nd a function t such that f th. 6. If f 4 and h 4, nd a function t such that t f h. 63. Suppose t is an even function and let h f t. Is h alwas an even function? 64. Suppose t is an odd function and let h f t. Is h alwas an odd function? What if f is odd? What if f is even? In this section we assume that ou 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 comple 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 ver accurate graphs of functions. But we will see in Chapter 4 that onl 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 displas a rectangular portion of the graph of a function in a displa 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 -values to range from a minimum value of Xmin a to a maimum value of Xma b and the -values to range from Licensed to: jsamuels@bmcc.cun.edu (a, d ) =d (b, d ) =a _ FIGURE Graphs of ƒ= +3 _ (a) _, b _, _4 4 4 _4 (b) _4,4 b _4, 4 =b (a, c) =c (b, c) FIGURE The viewing rectangle a,b b c,d SECTION.4 GRAPHING CALCULATORS AND COMPUTERS 49 a minimum of Ymin c to a maimum of Yma d, then the visible portion of the graph lies in the rectangle a, b c, d, a b, c d shown in Figure. We refer to this rectangle as the a, b b c, d viewing rectangle. The machine draws the graph of a function f much as ou would. It plots points of the form, f for a certain number of equall spaced values of between a and b. If an -value is not in the domain of f, or if f lies outside the viewing rectangle, it moves on to the net -value. The machine connects each point to the preceding plotted point to form a representation of the graph of f. EXAMPLE Draw the graph of the function f 3 in each of the following viewing rectangles. (a), b, (b) 4, 4 b 4, 4 (c), b 5, 3 (d) 5, 5 b, SOLUTION For part (a) we select the range b setting Xmin, Xma, Ymin, and Yma. The resulting graph is shown in Figure (a). The displa window is blank! A moments thought provides the eplanation: Notice that for all, so 3 3 for all. Thus, the range of the function f 3 is 3,. This means that the graph of f lies entirel outside the viewing rectangle, b,. The graphs for the viewing rectangles in parts (b), (c), and (d) are also shown in Figure. Observe that we get a more complete picture in parts (c) and (d), but in part (d) it is not clear that the -intercept is (c) _, b _5, 3 We see from Eample that the choice of a viewing rectangle can make a big difference in the appearance of a graph. Sometimes its necessar to change to a larger viewing rectangle to obtain a more complete picture, a more global view, of the graph. In the net eample 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 Determine an appropriate viewing rectangle for the function f s8 and use it to graph f. SOLUTION The epression for f is dened when _5 5 _ (d) _5,5 b _, 8 &? 8 &? 4 &? &? Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

21 5 CHAPTER FUNCTIONS AND MODELS Licensed to: 4 _3 3 _ FIGURE 3 5 _5 5 _5 FIGURE 4 (a) FIGURE 5 ƒ= -5 Therefore, the domain of f is the interval,. Also, so the range of f is the interval [, s]. We choose the viewing rectangle so that the -interval is somewhat larger than the domain and the -interval is larger than the range. Taking the viewing rectangle to be 3, 3 b, 4, we get the graph shown in Figure 3. EXAMPLE 3 Graph the function 3 5. SOLUTION Here the domain is, the set of all real numbers. That doesnt help us choose a viewing rectangle. Lets eperiment. If we start with the viewing rectangle 5, 5 b 5, 5, we get the graph in Figure 4. It appears blank, but actuall the graph is so nearl vertical that it blends in with the -ais. If we change the viewing rectangle to, b,, we get the picture shown in Figure 5(a). The graph appears to consist of vertical lines, but we know that cant be correct. If we look carefull 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, b 5, 5. The resulting graph is shown in Figure 5(b). It still doesnt quite reveal all the main features of the function, so we tr, b, in Figure 5(c). Now we are more condent that we ha ve 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. 5 5 (b) s8 s8 s.83 EXAMPLE 4 Graph the function f sin 5 in an appropriate viewing rectangle. SOLUTION Figure 6(a) shows the graph of f produced b a graphing calculator using the viewing rectangle, b.5,.5. At rst 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 ver different. Something strange is happening. In order to eplain the big differences in appearance of these graphs and to nd an appropriate viewing rectangle, we need to nd the period of the function sin 5. We know that the function sin has period and the graph of sin 5 is compressed horizontall b a factor of 5, so the period of sin 5 is (c) Licensed to: jsamuels@bmcc.cun.edu The appearance of the graphs in Figure 6 depends on the machine used. The graphs ou get with our own graphing device might not look like these figures, but the will also be quite inaccurate. FIGURE 7 ƒ=sin 5 FIGURE 6 Graphs of ƒ=sin5 in four viewing rectangles.5 _.5.5 _.5 SECTION.4 GRAPHING CALCULATORS AND COMPUTERS 5 This suggests that we should deal onl with small values of in order to show just a few oscillations of the graph. If we choose the viewing rectangle.5,.5 b.5,.5, we get the graph shown in Figure 7. Now we see what went wrong in Figure 6. The oscillations of sin 5 are so rapid that when the calculator plots points and joins them, it misses most of the maimum and minimum points and therefore gives a ver 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 Eamples and 3 we solved the problem b changing to a larger viewing rectangle. In Eample 4 we had to make the viewing rectangle smaller. In the net eample 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 sin cos. SOLUTION Figure 8 shows the graph of f produced b a graphing calculator with viewing rectangle 6.5, 6.5 b.5,.5. It looks much like the graph of sin, but perhaps with some bumps attached. If we zoom in to the viewing rectangle.,. b.,., we can see much more clearl the shape of these bumps in Figure 9. The reason for this behavior is that the second term, cos, is ver small in comparison with the rst term, sin. Thus, we reall need two graphs to see the true nature of this function..5. _ FIGURE 8 _ (a).5 _9 9 _.5 (c) _.. FIGURE 9 _..5.5 (b).5 _6 6 _.5 (d) Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

22 5 CHAPTER FUNCTIONS AND MODELS Licensed to: Another wa to avoid the etraneous line is to change the graphing mode on the calculator so that the dots are not connected. Alternativel, we could zoom in using the Zoom Decimal mode. EXAMPLE 6 Draw the graph of the function. SOLUTION Figure (a) shows the graph produced b a graphing calculator with viewing rectangle 9, 9 b 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 trul part of the graph. Notice that the domain of the function is. We can eliminate the etraneous near-vertical line b eperimenting with a change of scale. When we change to the smaller viewing rectangle 4.7, 4.7 b 4.7, 4.7 on this particular calculator, we obtain the much better graph in Figure (b). FIGURE = _9 _4.7 - (a) (b) EXAMPLE 7 Graph the function s 3. SOLUTION Some graphing devices displa the graph shown in Figure, whereas others produce a graph like that in Figure. We know from Section. (Figure 3) that the graph in Figure is correct, so what happened in Figure? The eplanation is that some machines compute the cube root of using a logarithm, which is not dened if is negative, so onl the right half of the graph is produced. _3 3 FIGURE _9 9 _ 9 You should eperiment with our own machine to see which of these two graphs is produced. If ou get the graph in Figure, ou can obtain the correct picture b graphing the function f _3 3 FIGURE 3 Notice that this function is equal to s 3 (ecept when ). To understand how the epression for a function relates to its graph, its helpful to graph a famil of functions, that is, a collection of functions whose equations are related. In the net eample we graph members of a famil of cubic polnomials. _ 4.7 _ Licensed to: jsamuels@bmcc.cun.edu FIGURE 3 Several members of the famil of functions = +c, all graphed in the viewing rectangle _, b _.5,.5 FIGURE 4 Locating the roots of cos=.5 _5 5 _.5 (a) _5,5 b _.5,.5 -scale= SECTION.4 GRAPHING CALCULATORS AND COMPUTERS 53 EXAMPLE 8 Graph the function 3 c for various values of the number c. How does the graph change when c is changed? SOLUTION Figure 3 shows the graphs of 3 c for c,,,, and. We see that, for positive values of c, the graph increases from left to right with no maimum or minimum points (peaks or valles). When c, the curve is at at the origin. When c is negative, the curve has a maimum point and a minimum point. As c decreases, the maimum point becomes higher and the minimum point lower. (a) = + (b) = + (c) = (d) = - (e) = -.4 ; Eercises EXAMPLE 9 Find the solution of the equation cos correct to two decimal places. SOLUTION The solutions of the equation cos are the -coordinates of the points of intersection of the curves cos and. From Figure 4(a) we see that there is onl one solution and it lies between and. Zooming in to the viewing rectangle, b,, we see from Figure 4(b) that the root lies between.7 and.8. So we zoom in further to the viewing rectangle.7,.8 b.7,.8 in Figure 4(c). B moving the cursor to the intersection point of the two curves, or b inspection and the fact that the -scale is., we see that the root of the equation is about.74. (Man calculators have a built-in intersection feature.) = =Ł. Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f 4. (a), b, (b), 4 b, 4 (c) 4, 4 b 4, 4 (d) 8, 8 b 4, 4 (e) 4, 4 b 8, 8 =Ł = (b), b, -scale=..8.7 = =Ł (c).7,.8 b.7,.8 -scale=.. Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f 7 6. (a) 5, 5 b 5, 5 (b), b, (c) 5, 8 b, (d), 3 b,.8 Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.

23 54 CHAPTER FUNCTIONS AND MODELS Licensed to: 3. Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f 5 3. (a) 4, 4 b 4, 4 (b), b, (c), b, (d), b, 4. Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f s8. (a) 4, 4 b 4, 4 (b) 5, 5 b, (c), b, 4 (d), b, Determine an appropriate viewing rectangle for the given function and use it to draw the graph. 5. f 5 6. f f f f s f s.. f 3. f cos 4. f 3 sin 5. f sin4 6. tan cos 8.. sin 5 9. Graph the ellipse 4 b graphing the functions whose graphs are the upper and lower halves of the ellipse.. Graph the hperbola 9 b graphing the functions whose graphs are the upper and lower branches of the hperbola. 3 Find all solutions of the equation correct to two decimal places sin. f 4. We saw in Eample 9 that the equation cos has eactl one solution. (a) Use a graph to show that the equation cos.3 has three solutions and nd their v alues correct to two decimal places. (b) Find an approimate value of m such that the equation cos m has eactl two solutions. 5. Use graphs to determine which of the functions f and t 3 is eventuall larger (that is, larger when is ver large). 6. Use graphs to determine which of the functions f 4 3 and t 3 is eventuall larger. 7. For what values of is it true that? 8. Graph the polnomials P and Q 3 5 on the same screen, rst using the vie wing rectangle, b [, ] and then changing to, b,,,. What do ou observe from these graphs? 9. In this eercise we consider the famil of root functions f s n, where n is a positive integer. (a) Graph the functions s, s 4, and s 6 on the same screen using the viewing rectangle, 4 b, 3. (b) Graph the functions, s 3, and s 5 on the same screen using the viewing rectangle 3, 3 b,. (See Eample 7.) (c) Graph the functions s, s 3, s 4, and s 5 on the same screen using the viewing rectangle, 3 b,. (d) What conclusions can ou make from these graphs? 3. In this eercise we consider the famil of functions f n, where n is a positive integer. (a) Graph the functions and 3 on the same screen using the viewing rectangle 3, 3 b 3, 3. (b) Graph the functions and 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, 3 b, 3. (d) What conclusions can ou make from these graphs? 3. Graph the function f 4 c for several values of c. How does the graph change when c changes? 3. Graph the function f s c for various values of c. Describe how changing the value of c affects the graph. 33. Graph the function n,, for n,, 3, 4, 5, and 6. How does the graph change as n increases? 34. The curves with equations sc sin. are called bullet-nose curves. Graph some of these curves to see wh. What happens as c increases? 35. What happens to the graph of the equation c 3 as c varies? 36. This eercise eplores the effect of the inner function t on a composite function f t. (a) Graph the function sin(s) using the viewing rectangle, 4 b.5,.5. How does this graph differ from the graph of the sine function? (b) Graph the function sin using the viewing rectangle 5, 5 b.5,.5. How does this graph differ from the graph of the sine function? Licensed to: jsamuels@bmcc.cun.edu 37. The gure sho ws the graphs of sin 96 and sin as displaed b a TI-83 graphing calculator. =sin96 =sin The rst graph is inaccurate. Eplain wh the tw o graphs appear identical. [Hint: The TI-83s graphing window is 95 piels wide. What specic points does the calculator plot?] 38. The rst graph in the gure is that of sin 45 as displaed b a TI-83 graphing calculator. It is inaccurate and so, to help.5 E p o n e n t i a l F u n c t i o n s FIGURE Representation of =, rational SECTION.5 EXPONENTIAL FUNCTIONS 55 eplain its appearance, we replot the curve in dot mode in the second graph. What two sine curves does the calculator appear to be plotting? Show that each point on the graph of sin 45 that the TI-83 chooses to plot is in fact on one of these two curves. (The TI-83s graphing window is 95 piels wide.) The function f is called an eponential function because the variable,, is the eponent. It should not be confused with the power function t, in which the variable is the base. In general, an eponential function is a function of the form where a is a positive constant. Lets recall what this means. If n, a positive integer, then n factors If, then a, and if n, where n is a positive integer, then If is a rational number, pq, where p and q are integers and q, then But what is the meaning of a if is an irrational number? For instance, what is meant b s3 or? To help us answer this question we rst look at the graph of the function, where is rational. A representation of this graph is shown in Figure. We want to enlarge the domain of to include both rational and irrational numbers. There are holes in the graph in Figure corresponding to irrational values of. We want to ll in the holes b dening f, where, so that f is an increasing function. In particular, since the irrational number s3 satises 5 f a a n a a a a n a n a a pq q sa p ( q sa) p.7 s3.8 Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Copright 5 Thomson Learning, Inc. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part.