6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 67 CHAPTER Summar 67 69. ƒ() = 3 70. ƒ() = -5 (b) Find the difference quotient. Interpret our result. 7. ƒ() = - 7. ƒ() = 0 73. ƒ() = + 74. ƒ() = -3 + 4 75. ƒ() = 3 + 76. ƒ() = - 77. ƒ() = - + 78. ƒ() = -4 + 79. ƒ() = - + 80. ƒ() = + 3-8. ƒ() = 3 8. ƒ() = - 3 83. Speed of a Car (Refer to Eample 6.) Let the distance in feet that a car travels in t seconds be given b d(t) = 8t for 0 t 6. (a) Find d(t + h). (b) Find the difference quotient for d and simplif. (c) Evaluate the difference quotient when t = 4 h = 0.05. Interpret our result. and 84. Draining a Pool Let the number of gallons G of water in a pool after t hours be given b G(t) = 4000-00t for 0 t 40. (a) Find G(t + h). Writing about Mathematics 85. What does the average rate of change represent for a linear function? What does it represent for a nonlinear function? Eplain our answers. 86. What is the formula for the difference quotient? Given a formula for ƒ(), eplain how to find ƒ( + h). Give an eample. 87. Suppose that a function ƒ has a positive average rate of change from to 4. Is it correct to assume that function ƒ onl increases on the interval [, 4]? Make a sketch to support our answer. 88. If ƒ() = a + b, what does the difference quotient for function ƒ equal? Eplain our reasoning. EXTENDED AND DISCOVERY EXERCISE. Velocit (Refer to Eample 6.) If the distance in feet run b a racehorse in t seconds is given b d(t) = t, then the difference quotient for d is 4t + h. How could ou estimate the velocit of the racehorse at eactl 7 seconds? CHECKING BASIC S FOR SECTION.5. Write each epression in interval notation. (a) 5 (b) 6 6. Determine where ƒ() = - is increasing and where it is decreasing. 3. Find the average rate of change of ƒ() = - 3 from = -3 to = -. 4. Find the difference quotient for ƒ() = 4. Summar SECTION. NUMBERS, DATA, AND PROBLEM SOLVING Sets of Numbers Natural numbers: N = {,, 3, 4, Á} Integers: I = {Á, -3, -, -, 0,,, 3, Á} p Rational numbers: q, where p and q are integers with q Z 0; includes repeating and terminating decimals Irrational numbers: Includes nonrepeating, nonterminating decimals Real numbers: An number that can be epressed in decimal form; includes rational and irrational numbers
6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 68 68 CHAPTER Introduction to Functions and Graphs SECTION. NUMBERS, DATA, AND PROBLEM SOLVING (CONTINUED) Order of Operations Using the following order of operations, first perform all calculations within parentheses, square roots, and absolute value bars and above and below fraction bars. Then use the same order of operations to perform an remaining calculations.. Evaluate all eponents. Then do an negation after evaluating eponents.. Do all multiplication and division from left to right. 3. Do all addition and subtraction from left to right. Eample: 5 + 3 # 3 = 5 + 3 # 8 = 5 + 4 = 9 Scientific Notation A real number r is written as c * 0 n, where ƒ c ƒ 6 0. Eamples: 34 =.34 * 0 3 0.054 = 5.4 * 0 - SECTION. VISUALIZING AND GRAPHING DATA Mean (Average) and Median The mean represents the average of a set of numbers, and the median represents the middle of a sorted list. Eample: 4, 6, 9, 3, 5; Mean = 4 + 6 + 9 + 3 + 5 5 = 9.4; Median = 9 Relation, Domain, and Range A relation S is a set of ordered pairs. The domain D is the set of -values, and the range R is the set of -values. Eample: S = {(-, ), (4, -5), (5, 9)}; D = {-, 4, 5}, R = {-5,, 9} Cartesian (Rectangular) Coordinate Sstem, or -Plane The -plane has four quadrants and is used to graph ordered pairs. Quadrant II Quadrant I (, ) (, ) (, 0) (, ) (, ) Quadrant III Quadrant IV Distance Formula The distance d between the points (, ) and (, ) is d = ( - ) + ( - ). Eample: The distance between (-3, 5) and (, -7) is d = ( - (-3)) + (-7-5) = 5 + (-) = 3.
6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 69 CHAPTER Summar 69 SECTION. VISUALIZING AND GRAPHING DATA (CONTINUED) Midpoint Formula The midpoint M of the line segment with endpoints (, ) and (, ) is M = a + Eample: The midpoint of the line segment connecting (, ) and (-3, 5) is M = a + (-3), + b., + 5 b = a -, 7 b. Standard Equation of a Circle The circle with center (h, k) and radius r has the equation ( - h) + ( - k) = r. Eample: A circle with center (-, 5) and radius 6 has the equation ( + ) + ( - 5) = 36. Scatterplot and Line Graph A scatterplot consists of a set of ordered pairs plotted in the -plane. When consecutive points are connected with line segments, a line graph results. SECTION.3 FUNCTIONS AND THEIR REPRESENTATIONS Function A function computes eactl one output for each valid input. The set of valid inputs is called the domain D, and the set of outputs is called the range R. Eamples: ƒ() = - D = { ƒ }, R = { ƒ Ú 0} g = {(-, 0.5), (0, 4), (, 4), (6, p)} D = {-, 0,, 6}, R = {0.5, p, 4} Function Notation Eamples: ƒ() = - 4; ƒ(3) = 3-4 = 5 ƒ(a + ) = (a + ) - 4 = a + a - 3 Representations of Functions A function can be represented smbolicall (formula), graphicall (graph), numericall (table of values), and verball (words). Other representations are possible. Smbolic Representation ƒ() = - Numerical Representation Graphical Representation - 3-0 0-0 3 3 f () = Verbal Representation ƒ computes the square of the input and then subtracts.
6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 70 70 CHAPTER Introduction to Functions and Graphs SECTION.3 FUNCTIONS AND THEIR REPRESENTATIONS (CONTINUED) Vertical Line Test If an vertical line intersects a graph at most once, then the graph represents a function. SECTION.4 TYPES OF FUNCTIONS Slope The slope m of the line passing through (, ) and (, ) is m = = - -. Eample: The slope of the line passing through (, -) and (-, 3) is m = 3 - (-) - - = -4 3. Constant Function Given b ƒ() = b, where b is a constant; its graph is a horizontal line. Linear Function Given b ƒ() = a + b; its graph is a nonvertical line; the slope of its graph is equal to a, which is also equal to its constant rate of change. Eamples: The graph of ƒ() = -8 + 00 has slope -8. If G(t) = -8t + 00 calculates the number of gallons of water in a tank after t minutes, then water is leaving the tank at 8 gallons per minute. Nonlinear Function Linear and Nonlinear Data The graph of a nonlinear function is not a line and cannot be written as ƒ() = a + b. Eamples: ƒ() = - 4; g() = 3 - ; h(t) = t + If the slopes of the lines passing through consecutive data points are alwas equal (or nearl equal), then the data are linear. Otherwise the data are nonlinear. Eample: For each -unit increase in, the -values increase b 0 units. Consecutive slopes between points are m = 0 = 5 so the data are linear. units units units T T T 4 6 8 0 0 0 30 c c c 0 units 0 units 0 units SECTION.5 FUNCTIONS AND THEIR RATES OF CHANGE Interval Notation A concise wa to epress intervals on the number line Eample: 6 4 is epressed as (- q, 4). -3 6 is epressed as [-3, ). or Ú 5 is epressed as (- q, ] h [5, q).
6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 7 CHAPTER Review Eercises 7 SECTION.5 FUNCTIONS AND THEIR RATES OF CHANGE (CONTINUED) Increasing/Decreasing ƒ increases on interval I if, whenever 6, ƒ( ) 6 ƒ( ). ƒ decreases on interval I if, whenever 6, ƒ( ) 7 ƒ( ). Eample: ƒ() = ƒ ƒ increases on [0, q) and decreases on (- q, 0]. Average Rate of Change If (, ) and (, ) are distinct points on the graph of ƒ, then the average rate of change from to equals the slope of the line passing through these two points and equals - -. Eample: ƒ() = ; because ƒ() = 4 and ƒ(3) = 9, the graph of ƒ passes through the points (, 4) and (3, 9), and the average rate of change from to 3 is 9-4 given b 3 - = 5. Difference Quotient The difference quotient of a function ƒ is an epression of the form ƒ( + h) - ƒ(), h where h Z 0. Eample: Let ƒ() =. The difference quotient of ƒ is ( + h) - h = + h + h - h = + h. Review Eercises Eercises and : Classif each number listed as one or more of the following: natural number, integer, rational number, or real number.. -,, 0,.3, 7, 6. 55,.5, 04 7, 3, 3, -000 Eercises 3 and 4: Write each number in scientific notation. 3.,89,000 4. 0.00000 Eercises 5 and 6: Write each number in standard form. 5..5 * 0 4 6. -7. * 0-3 7. Evaluate each epression with a calculator. Round answers to the nearest hundredth. (a) 3. + p 3 (b) 3. + 5.7 7.9-4.5 (c) 5 +. (d).(6.3) 3. + p - 8. Evaluate each epression. Write our answer in scientific notation and in standard form. (a) (4 * 0 3 )(5 * 0-5 ) (b) 3 * 0-5 6 * 0 -
6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 7 7 CHAPTER Introduction to Functions and Graphs Eercises 9 and 0: Evaluate b hand. 9. 4-3 # 5 0. 3 # 3, 3-5 6 + Eercises and : Sort the list of numbers from smallest to largest and displa the result in a table. (a) Determine the maimum and minimum values. ( b) Calculate the mean and median.. -5, 8, 9, 4, -3 (a) = f() (b) 4 3 4 = f (). Eercises 3 and 4: Complete the following. (a) Epress the data as a relation S. (b) Find the domain and range of S. 3. 4. Eercises 5 and 6: Make a scatterplot of the relation. Determine if the relation is a function. 5. {(0, 3), (-, 40), (-30, -3), (5, -), (0, 0)} 6. 8.9, -., -3.8, 0.8,.7,.7-5 -0 0 5 0-3 - 3 5-0.6-0. 0. 0.5. 0 0 5 30 80 {(.5,.5), (0,.), (-.3, 3.), (0.5, -0.8), (-., 0)} Eercises 7 and 8: Find the distance between the points. 7. (-4, 5), (, -3) 8. (., -4), (0., 6) Eercises 9 and 0: Find the midpoint of the line segment with the given endpoints. 9. (4, -6), (-0, 3) 0. A, 5 4 B, A, -5 B. Determine if the triangle with vertices (, ), (-3, 5), and (0, 9) is isosceles. (Hint: An isosceles triangle has at least two sides with equal measure.). Find the standard equation of a circle with center and radius 9. (-5, 3) 3. A diameter of a circle has endpoints (-, 4) and (6, 6). Find the standard equation of the circle. 4. Use the graph at the top of the net column to determine the domain and range of each function. Evaluate ƒ(-). Eercises 5 3: Graph = ƒ() b first plotting points to determine the shape of the graph. 5. ƒ() = - 6. ƒ() = 3 7. ƒ() = - + 8. ƒ() = - 3 9. ƒ() = 4-30. ƒ() = - 3. ƒ() = ƒ + 3 ƒ 3. ƒ() = 3 - Eercises 33 and 34: Use the verbal representation to epress the function ƒ smbolicall, graphicall, and numericall. Let = ƒ() with 0 00. For the numerical representation, use a table with = 0, 5, 50, 75, 00. 33. To convert pounds to ounces, multipl b 6. 34. To find the area of a square, multipl the length of a side b itself. Eercises 35 4: Complete the following for the function ƒ. (a) Evaluate ƒ() at the indicated values of. ( b) Find the domain of ƒ. 35. ƒ() = 3 for = -8, 36. ƒ() = 3 + for 37. ƒ() = 5 for 38. ƒ() = 4-5 for 39. ƒ() = - 3 for 40. ƒ() = 3-3 for 4. ƒ() = for = -3, a + - 4 4. ƒ() = + 3 for = -, 5 = -3,.5 = -5, 6 = -0, a + = -0, a + =, a - 3 43. Determine if is a function of in = + 5. 44. Write 5 6 0 in interval notation.
6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 73 CHAPTER Eercises 45 and 46: Determine if the graph represents a function. 45. 46. 4 5 4 4 3 3 3 4 3 73 Applications 65. Speed of Light The average distance between the planet Mars and the sun is approimatel 8 million kilometers. Estimate the time required for sunlight, traveling at 300,000 kilometers per second, to reach Mars. (Source: C. Ronan, The Natural Histor of the Universe.) Review Eercises 3 66. Geometr Suppose that 0.5 cubic inch of paint is applied to a circular piece of plastic with a diameter of 0 inches. Estimate the thickness of the paint. 4 Eercises 47 and 48: Determine if S represents a function. 47. S = {(-3, 4), (-, ), (3, -5), (4, )} 48. S = {(-, 3), (0, ), (-, 7), (3, -3)} Eercises 49 and 50: State the slope of the graph of ƒ. 49. ƒ() = 7 50. ƒ() = 3-3 Eercises 5 54: If possible, find the slope of the line passing through each pair of points. 5. (-, 7), (3, 4) 5. (, -4), (, 0) 54. A - 3, 3B, A - 3, - 56B 53. (8, 4), (-, 4) Eercises 55 58: Decide whether the function ƒ is constant, linear, or nonlinear. Support our answer graphicall. 55. ƒ() = 8-3 56. ƒ() = - 3-8 57. ƒ() = + 58. ƒ() = 6 67. Enclosing a Pool A rectangular swimming pool that is 5 feet b 50 feet has a 6-foot-wide sidewalk around it. (a) How much fencing would be needed to enclose the sidewalk? (b) Find the area of the sidewalk. 68. Distance A driver s distance D in miles from a rest stop after t hours is given b D(t) = 80-70t. (a) How far is the driver from the rest stop after hours? (b) Find the slope of the graph of D. Interpret this slope as a rate of change. 69. Survival Rates The survival rates for song sparrows are shown in the table. The values listed are the numbers of song sparrows that attain a given age from 00 eggs. For eample, 6 sparrows reach an age of ears from 00 eggs laid in the wild. (Source: S. Kress, Bird Life.) Age Number 0 3 4 00 0 6 3 59. Sketch a graph for a -hour period showing the distance between two cars meeting on a straight highwa, each traveling 60 miles per hour. Assume that the cars are initiall 0 miles apart. 60. Determine where the graph of ƒ() = - 3 is increasing and where it is decreasing. 6. Determine if the following data are modeled best b a constant, linear, or nonlinear function. - 0 4 50 4 34 6 6. Find the average rate of change of ƒ() = - + from = to = 3. (a) Make a line graph of the data. Interpret the data. Eercises 63 and 64: Find the difference quotient for ƒ(). 63. ƒ() = 5 + 64. ƒ() = 3 - (c) Calculate and interpret the average rate of change for each -ear period. (b) Does this line graph represent a function?
6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 74 74 CHAPTER Introduction to Functions and Graphs 70. Cost of Tuition The graph shows the cost of taking credits at a universit. (a) Wh is it reasonable for the graph to pass through the origin? (b) Find the slope of the graph. (c) Interpret the slope as a rate of change. Tuition (dollars) 900 800 700 600 500 400 300 00 00 0 3 4 5 6 7 8 Credits 7. Average Rate of Change Let ƒ() = 0.5 + 50 represent the outside temperature in degrees Fahrenheit at P.M., where 5. (a) Graph ƒ. Is ƒ linear or nonlinear? (b) Calculate the average rate of change of ƒ from P.M. to 4 P.M. (c) Interpret this average rate of change verball and graphicall. 7. Distance At noon car A is traveling north at 30 miles per hour and is located 0 miles north of car B. Car B is traveling west at 50 miles per hour. Approimate the distance between the cars at :45 P.M. to the nearest mile. EXTENDED AND DISCOVERY EXERCISES Because a parabolic curve becomes sharp graduall, as shown in the first figure, curves designed b engineers for highwas and railroads frequentl have parabolic, rather than circular, shapes. If railroad tracks changed abruptl from straight to circular, the momentum of the locomotive could cause a derailment. The second figure illustrates straight tracks connecting to a circular curve. (Source: F. Mannering and W. Kilareski, Principles of Highwa Engineering and Traffic Analsis.) In order to design a curve and estimate its cost, engineers determine the distance around the curve before it is built. In the third figure the distance along a parabolic curve from A to C is approimated b two line segments AB and BC. The distance formula can be used to calculate the length of each segment. The sum of these two lengths gives a crude estimate of the length of the curve. A Parabolic Curve A B C An Estimate of Curve Length A Circular Curve A better estimate can be made using four line segments, as shown in the fourth figure. As the number of segments increases, so does the accurac of the approimation.. Curve Length Suppose that a curve designed for railroad tracks is represented b the equation = 0., where the units are in kilometers. The points (-3,.8), (-.5, 0.45), (0, 0), (.5, 0.45), and (3,.8) lie on the graph of = 0.. Approimate the length of the curve from = -3 to = 3 b using line segments connecting these points. Eercises 5: Curve Length Use three line segments connecting the four points to estimate the length of the curve on the graph of ƒ from = - to =. Graph ƒ and a line graph of the four points in the indicated viewing rectangle.. ƒ() = ; (-, ), (0, 0), (, ), (, 4); [-4.5, 4.5, ] b [-, 5, ] 3. ƒ() = 3 ; (-, -), (0, 0), (, ), (, 3 ); [-3, 3, ] b [-,, ] 4. ƒ() = 0.5 3 + ; (-,.5), (0, ), (,.5), (, 6); [-4.5, 4.5, ] b [0, 6, ] A B C A Better Estimate D E
6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 75 CHAPTER Review Eercises 75 5. ƒ() = - 0.5 ; (-,.5), (0, ), (,.5), (, 0); [-3, 3, ] b [-, 3, ] 6. The distance along the curve of = from (0, 0) to (3, 9) is about 9.747. Use this fact to estimate the distance along the curve of = 9 - from (0, 9) to (3, 0). 7. Estimate the distance along the curve of = from (, ) to (4, ). (The actual value is approimatel 3.68.) 8. Endangered Species The Florida scrub-ja is an endangered species that prefers to live in open landscape with short vegetation. NASA has attempted to create a habitat for these birds near Kenned Space Center. The following table lists their population for selected ears, where = 0 corresponds to 980, = to 98, = to 98, and so on. (980 4 0) 0 5 9 (population) 3697 5 76 (980 4 0) 5 9 (population) 00 689 7 Source: Mathematics Eplorations II, NASA AMATYC NSF. (a) Make a scatterplot of the data. (b) Find a linear function ƒ that models the data. (c) Graph the data and ƒ in the same viewing rectangle. (d) Estimate the scrub-ja population in 987 and in 003. 9. Global Warming If the global climate were to warm significantl as a result of the greenhouse effect or other climatic change, the Arctic ice cap would start to melt. It is estimated that this ice cap contains the equivalent of 680,000 cubic miles of water. Over 00 million people currentl live on soil that is less than 3 feet above sea level. In the United States, several large cities have low average elevations, such as Boston (4 feet), New Orleans (4 feet), and San Diego (3 feet). (Sources: Department of the Interior, Geological Surve.) (a) Devise a plan to determine how much sea level would rise if the Arctic cap melted. (Hint: The radius of Earth is 3960 miles and 7% of its surface is covered b oceans.) (b) Use our plan to estimate this rise in sea level. (c) Discuss the implications of our calculation. (d) Estimate how much sea level would rise if the 6,300,000 cubic miles of water in the Antarctic ice cap melted. 0. Prove that is irrational b assuming that is rational and arriving at a contradiction.