The Graph of an Equation

Transcription

1 CHAPTER P Preparation for Calculus Section P Descartes made man contributions to philosoph, science, and mathematics The idea of representing points in the plane b pairs of real numbers and representing curves in the plane b equations was described b Descartes in his book La Géométrie, published in 67 RENÉ DESCARTES ( ) MathBio Graphs and Models Sketch the graph of an equation Find the intercepts of a graph Test a graph for smmetr with respect to an ais and the origin Find the points of intersection of two graphs Interpret mathematical models for real-life data The Graph of an Equation In 67 the French mathematician René Descartes revolutionized the stud of mathematics b joining its two major fields algebra and geometr With Descartes s coordinate plane, geometric concepts could be formulated analticall and algebraic concepts could be viewed graphicall The power of this approach is such that within a centur, much of calculus had been developed The same approach can be followed in our stud of calculus That is, b viewing calculus from multiple perspectives graphicall, analticall, and numericall ou will increase our understanding of core concepts Consider the equation 7 The point, is a solution point of the equation because the equation is satisfied (is true) when is substituted for and is substituted for This equation has man other solutions, such as, and 0, 7 To find other solutions sstematicall, solve the original equation for 7 Analtic approach Then construct a table of values b substituting several values of (0, 7) (, ) + = 7 (, ) 6 8 (, ) (, 5) Numerical approach From the table, ou can see that 0, 7,,,,,,, and, 5 are solutions of the original equation 7 Like man equations, this equation has an infinite number of solutions The set of all solution points is the graph of the equation, as shown in Figure P Graphical approach: 7 Figure P NOTE Even though we refer to the sketch shown in Figure P as the graph of 7, it reall represents onl a portion of the graph The entire graph would etend beond the page In this course, ou will stud man sketching techniques The simplest is point plotting that is, ou plot points until the basic shape of the graph seems apparent = The parabola Figure P EXAMPLE Sketching a Graph b Point Plotting Sketch the graph of Solution First construct a table of values Then plot the points shown in the table 0 7 Finall, connect the points with a smooth curve, as shown in Figure P This graph is a parabola It is one of the conics ou will stud in Chapter 0 Editable Graph Tr It Eploration A

2 SECTION P Graphs and Models One disadvantage of point plotting is that to get a good idea about the shape of a graph, ou ma need to plot man points With onl a few points, ou could badl misrepresent the graph For instance, suppose that to sketch the graph of ou plotted onl five points:,,,, 0, 0,,, and,, as shown in Figure P From these five points, ou might conclude that the graph is a line This, however, is not correct B plotting several more points, ou can see that the graph is more complicated, as shown in Figure P (, ) Plotting onl a few points can misrepresent a (, ) graph Figure P (0, 0) (, ) (, ) = (9 0 + ) 0 EXPLORATION Comparing Graphical and Analtic Approaches Use a graphing utilit to graph each equation In each case, find a viewing window that shows the important characteristics of the graph a b c d e f A purel graphical approach to this problem would involve a simple guess, check, and revise strateg What tpes of things do ou think an analtic approach might involve? For instance, does the graph have smmetr? Does the graph have turns? If so, where are the? As ou proceed through Chapters,, and of this tet, ou will stud man new analtic tools that will help ou analze graphs of equations such as these TECHNOLOGY Technolog has made sketching of graphs easier Even with technolog, however, it is possible to misrepresent a graph badl For instance, each of the graphing utilit screens in Figure P shows a portion of the graph of 5 From the screen on the left, ou might assume that the graph is a line From the screen on the right, however, ou can see that the graph is not a line So, whether ou are sketching a graph b hand or using a graphing utilit, ou must realize that different viewing windows can produce ver different views of a graph In choosing a viewing window, our goal is to show a view of the graph that fits well in the contet of the problem Graphing utilit screens of Figure P 0 5 NOTE In this tet, the term graphing utilit means either a graphing calculator or computer graphing software such as Maple, Mathematica, Derive, Mathcad, or the TI

3 CHAPTER P Preparation for Calculus Intercepts of a Graph Two tpes of solution points that are especiall useful in graphing an equation are those having zero as their - or -coordinate Such points are called intercepts because the are the points at which the graph intersects the - or -ais The point a, 0 is an -intercept of the graph of an equation if it is a solution point of the equation To find the -intercepts of a graph, let be zero and solve the equation for The point 0, b is a -intercept of the graph of an equation if it is a solution point of the equation To find the -intercepts of a graph, let be zero and solve the equation for NOTE Some tets denote the -intercept as the -coordinate of the point a, 0 rather than the point itself Unless it is necessar to make a distinction, we will use the term intercept to mean either the point or the coordinate It is possible for a graph to have no intercepts, or it might have several For instance, consider the four graphs shown in Figure P5 No -intercepts One -intercept Figure P5 Three -intercepts One -intercept One -intercept Two -intercepts No intercepts EXAMPLE Finding - and -intercepts = (, 0) Intercepts of a graph Figure P6 (0, 0) (, 0) Find the - and -intercepts of the graph of Solution To find the -intercepts, let be zero and solve for 0 Let be zero Factor Solve for Because this equation has three solutions, ou can conclude that the graph has three -intercepts: 0, 0,, 0, and, 0 -intercepts To find the -intercepts, let be zero Doing this produces 0 So, the -intercept is 0, 0 (See Figure P6) 0 0,, or -intercept Editable Graph Tr It Eploration A Video Video TECHNOLOGY Eample uses an analtic approach to finding intercepts When an analtic approach is not possible, ou can use a graphical approach b finding the points at which the graph intersects the aes Use a graphing utilit to approimate the intercepts

4 SECTION P Graphs and Models 5 Smmetr of a Graph (, ) (, ) Knowing the smmetr of a graph before attempting to sketch it is useful because ou need onl half as man points to sketch the graph The following three tpes of smmetr can be used to help sketch the graphs of equations (see Figure P7) -ais smmetr (, ) A graph is smmetric with respect to the -ais if, whenever, is a point on the graph,, is also a point on the graph This means that the portion of the graph to the left of the -ais is a mirror image of the portion to the right of the -ais A graph is smmetric with respect to the -ais if, whenever, is a point on the graph,, is also a point on the graph This means that the portion of the graph above the -ais is a mirror image of the portion below the -ais A graph is smmetric with respect to the origin if, whenever, is a point on the graph,, is also a point on the graph This means that the graph is unchanged b a rotation of 80 about the origin -ais smmetr (, ) Tests for Smmetr (, ) (, ) Origin smmetr The graph of an equation in and is smmetric with respect to the -ais if replacing b ields an equivalent equation The graph of an equation in and is smmetric with respect to the -ais if replacing b ields an equivalent equation The graph of an equation in and is smmetric with respect to the origin if replacing b and b ields an equivalent equation The graph of a polnomial has smmetr with respect to the -ais if each term has an even eponent (or is a constant) For instance, the graph of -ais smmetr Figure P7 has smmetr with respect to the -ais Similarl, the graph of a polnomial has smmetr with respect to the origin if each term has an odd eponent, as illustrated in Eample EXAMPLE Testing for Origin Smmetr = Show that the graph of is smmetric with respect to the origin (, ) Origin smmetr Figure P8 (, ) Solution Write original equation Replace b and b Simplif Equivalent equation Because the replacements ield an equivalent equation, ou can conclude that the graph of is smmetric with respect to the origin, as shown in Figure P8 Editable Graph Tr It Eploration A Video Video

5 6 CHAPTER P Preparation for Calculus EXAMPLE Using Intercepts and Smmetr to Sketch a Graph Sketch the graph of = (, ) (, 0) -intercept Figure P9 (5, ) 5 Solution The graph is smmetric with respect to the -ais because replacing b ields an equivalent equation Write original equation Replace b Equivalent equation This means that the portion of the graph below the -ais is a mirror image of the portion above the -ais To sketch the graph, first plot the -intercept and the points above the -ais Then reflect in the -ais to obtain the entire graph, as shown in Figure P9 Editable Graph Tr It Eploration A Eploration B Open Eploration TECHNOLOGY Graphing utilities are designed so that the most easil graph equations in which is a function of (see Section P for a definition of function) To graph other tpes of equations, ou need to split the graph into two or more parts or ou need to use a different graphing mode For instance, to graph the equation in Eample, ou can split it into two parts Points of Intersection Top portion of graph Bottom portion of graph A point of intersection of the graphs of two equations is a point that satisfies both equations You can find the points of intersection of two graphs b solving their equations simultaneousl = (, ) Two points of intersection Figure P0 Editable Graph (, ) STUDY TIP You can check the points of intersection from Eample 5 b substituting into both of the original equations or b using the intersect feature of a graphing utilit = EXAMPLE 5 Finding Points of Intersection Find all points of intersection of the graphs of and Solution Begin b sketching the graphs of both equations on the same rectangular coordinate sstem, as shown in Figure P0 Having done this, it appears that the graphs have two points of intersection You can find these two points, as follows 0 Solve first equation for Solve second equation for Equate -values Write in general form Factor Solve for The corresponding values of are obtained b substituting and into either of the original equations Doing this produces two points of intersection:, 0 and Tr It or, Eploration A Points of intersection

6 SECTION P Graphs and Models 7 Mathematical Models Real-life applications of mathematics often use equations as mathematical models In developing a mathematical model to represent actual data, ou should strive for two (often conflicting) goals: accurac and simplicit That is, ou want the model to be simple enough to be workable, et accurate enough to produce meaningful results Section P eplores these goals more completel EXAMPLE 6 Comparing Two Mathematical Models The Mauna Loa Observator in Hawaii has been measuring the increasing concentration of carbon dioide in Earth s atmosphere since 958 Video The Mauna Loa Observator in Hawaii records the carbon dioide concentration (in parts per million) in Earth s atmosphere The Januar readings for various ears are shown in Figure P In the Jul 990 issue of Scientific American, these data were used to predict the carbon dioide level in Earth s atmosphere in the ear 05, using the quadratic model 6 070t 008t Quadratic model for data where t 0 represents 960, as shown in Figure P The data shown in Figure P represent the ears 980 through 00 and can be modeled b 06 56t Linear model for data where t 0 represents 960 What was the prediction given in the Scientific American article in 990? Given the new data for 990 through 00, does this prediction for the ear 05 seem accurate? CO (in parts per million) Year (0 960) t CO (in parts per million) Year (0 960) t Figure P NOTE The models in Eample 6 were developed using a procedure called least squares regression (see Section 9) The quadratic and linear models have a correlation given b r 0997 and r 0996, respectivel The closer r is to, the better the model Solution To answer the first question, substitute t 75 (for 05) into the quadratic model Quadratic model So, the prediction in the Scientific American article was that the carbon dioide concentration in Earth s atmosphere would reach about 70 parts per million in the ear 05 Using the linear model for the data, the prediction for the ear 05 is Linear model So, based on the linear model for , it appears that the 990 prediction was too high Tr It Eploration A

7 8 CHAPTER P Preparation for Calculus Eercises for Section P The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises, match the equation with its graph [Graphs are labeled,, (c), and (d)] (c) (d) 9 In Eercises 5, sketch the graph of the equation b point plotting 9 0 In Eercises 5 and 6, describe the viewing window that ields the figure In Eercises 9 6, find an intercepts In Eercises 7 8, test for smmetr with respect to each ais and to the origin In Eercises 9 56, sketch the graph of the equation Identif an intercepts and test for smmetr In Eercises 57 60, use a graphing utilit to graph the equation Identif an intercepts and test for smmetr In Eercises 7 and 8, use a graphing utilit to graph the equation Move the cursor along the curve to approimate the unknown coordinate of each solution point accurate to two decimal places 7 5,, ,, In Eercises 6 68, find the points of intersection of the graphs of the equations

8 SECTION P Graphs and Models In Eercises 69 7, use a graphing utilit to find the points of intersection of the graphs Check our results analticall Modeling Data The table shows the Consumer Price Inde (CPI) for selected ears (Source: Bureau of Labor Statistics) Use the regression capabilities of a graphing utilit to find a mathematical model of the form at bt c for the data In the model, represents the CPI and t represents the ear, with t 0 corresponding to 970 Use a graphing utilit to plot the data and graph the model Compare the data with the model (c) Use the model to predict the CPI for the ear 00 7 Modeling Data The table shows the average numbers of acres per farm in the United States for selected ears (Source: US Department of Agriculture) Use the regression capabilities of a graphing utilit to find a mathematical model of the form at bt c for the data In the model, represents the average acreage and t represents the ear, with t 0 corresponding to 950 Use a graphing utilit to plot the data and graph the model Compare the data with the model (c) Use the model to predict the average number of acres per farm in the United States in the ear Break-Even Point Find the sales necessar to break even R C if the cost C of producing units is C 55 0,000 Cost equation and the revenue R for selling units is R 9 Revenue equation 76 Copper Wire The resistance in ohms of 000 feet of solid copper wire at 77 F can be approimated b the model 0,770 07, 5 00 Year CPI Year Acreage where is the diameter of the wire in mils (000 in) Use a graphing utilit to graph the model If the diameter of the wire is doubled, the resistance is changed b about what factor? Writing About Concepts In Eercises 77 and 78, write an equation whose graph has the indicated propert (There ma be more than one correct answer) 77 The graph has intercepts at,, and 6 78 The graph has intercepts at 5, and, 79 Each table shows solution points for one of the following equations (i) (iii) k (ii) k (iv) k Match each equation with the correct table and find Eplain our reasoning (c) k Prove that if a graph is smmetric with respect to the -ais and to the -ais, then it is smmetric with respect to the origin Give an eample to show that the converse is not true Prove that if a graph is smmetric with respect to one ais and to the origin, then it is smmetric with respect to the other ais True or False? In Eercises 8 8, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 8 If, is a point on a graph that is smmetric with respec to the -ais, then, is also a point on the graph 8 If, is a point on a graph that is smmetric with respec to the -ais, then, is also a point on the graph 8 If b ac > 0 and a 0, then the graph of a b c has two -intercepts 8 If b ac 0 and a 0, then the graph of a b c has onl one -intercept In Eercises 85 and 86, find an equation of the graph tha consists of all points, having the given distance from the origin (For a review of the Distance Formula, see Appendi D 85 The distance from the origin is twice the distance from 0, 86 The distance from the origin is K K times the distance from, 0 (d) k

9 0 CHAPTER P Preparation for Calculus Section P (, ) = (, ) = change in change in Figure P Linear Models and Rates of Change Find the slope of a line passing through two points Write the equation of a line with a given point and slope Interpret slope as a ratio or as a rate in a real-life application Sketch the graph of a linear equation in slope-intercept form Write equations of lines that are parallel or perpendicular to a given line The Slope of a Line The slope of a nonvertical line is a measure of the number of units the line rises (or falls) verticall for each unit of horizontal change from left to right Consider the two points, and, on the line in Figure P As ou move from left to right along this line, a vertical change of Change in units corresponds to a horizontal change of Change in units ( is the Greek uppercase letter delta, and the smbols and are read delta and delta ) Definition of the Slope of a Line The slope m of the nonvertical line passing through, and, is m, Slope is not defined for vertical lines Video Video Video NOTE When using the formula for slope, note that So, it does not matter in which order ou subtract as long as ou are consistent and both subtracted coordinates come from the same point Figure P shows four lines: one has a positive slope, one has a slope of zero, one has a negative slope, and one has an undefined slope In general, the greater the absolute value of the slope of a line, the steeper the line is For instance, in Figure P, the line with a slope of 5 is steeper than the line with a slope of 5 (, 0) m = 5 (, ) (, ) m = 0 (, ) (0, ) m = 5 (, ) (, ) m is undefined (, ) If m is positive, then the line rises from left to right Figure P If m is zero, then the line is horizontal If m is negative, then the line falls from left to right If m is undefined, then the line is vertical

10 SECTION P Linear Models and Rates of Change EXPLORATION Investigating Equations of Lines Use a graphing utilit to graph each of the linear equations Which point is common to all seven lines? Which value in the equation determines the slope of each line? a b c d e f g 0 Use our results to write an equation of a line passing through, with a slope of m Equations of Lines An two points on a nonvertical line can be used to calculate its slope This can be verified from the similar triangles shown in Figure P (Recall that the ratios of corresponding sides of similar triangles are equal) ( *, *) (, ) ( *, *) (, ) m = * * = * * An two points on a nonvertical line can be used to determine its slope Figure P You can write an equation of a nonvertical line if ou know the slope of the line and the coordinates of one point on the line Suppose the slope is m and the point is, If, is an other point on the line, then m This equation, involving the two variables and, can be rewritten in the form m, which is called the point-slope equation of a line = 5 Point-Slope Equation of a Line An equation of the line with slope m passing through the point, is given b m 5 = = (, ) The line with a slope of passing through the point, Figure P5 Editable Graph EXAMPLE Finding an Equation of a Line Find an equation of the line that has a slope of and passes through the point, Solution m (See Figure P5) 5 Point-slope form Substitute for, for, and for m Simplif Solve for Tr It Eploration A Eploration B Eploration C NOTE Remember that onl nonvertical lines have a slope Consequentl, vertical lines cannot be written in point-slope form For instance, the equation of the vertical line passing through the point, is

11 CHAPTER P Preparation for Calculus Ratios and Rates of Change The slope of a line can be interpreted as either a ratio or a rate If the - and -aes have the same unit of measure, the slope has no units and is a ratio If the - and -aes have different units of measure, the slope is a rate or rate of change In our stud of calculus, ou will encounter applications involving both interpretations of slope EXAMPLE Population Growth and Engineering Design Population (in millions) , Year Population of Kentuck in census ears Figure P6 a The population of Kentuck was,687,000 in 990 and,0,000 in 000 Over this 0-ear period, the average rate of change of the population was Rate of change If Kentuck s population continues to increase at this same rate for the net 0 ears, it will have a 00 population of,97,000 (see Figure P6) (Source: US Census Bureau) b In tournament water-ski jumping, the ramp rises to a height of 6 feet on a raft that is feet long, as shown in Figure P7 The slope of the ski ramp is the ratio of its height (the rise) to the length of its base (the run) Slope of ramp rise run change in population change in ears,0,000,687, ,500 people per ear 6 feet feet Rise is vertical change, run is horizontal change 7 In this case, note that the slope is a ratio and has no units 6 ft ft Dimensions of a water-ski ramp Figure P7 Tr It Eploration A Eploration B The rate of change found in Eample is an average rate of change An average rate of change is alwas calculated over an interval In this case, the interval is 990, 000 In Chapter ou will stud another tpe of rate of change called an instantaneous rate of change

12 SECTION P Linear Models and Rates of Change Graphing Linear Models Man problems in analtic geometr can be classified in two basic categories: () Given a graph, what is its equation? and () Given an equation, what is its graph? The point-slope equation of a line can be used to solve problems in the first categor However, this form is not especiall useful for solving problems in the second categor The form that is better suited to sketching the graph of a line is the slopeintercept form of the equation of a line The Slope-Intercept Equation of a Line The graph of the linear equation m b is a line having a slope of m and a -intercept at 0, b Video EXAMPLE Video Sketching Lines in the Plane Sketch the graph of each equation a b c 6 0 Solution a Because b, the -intercept is 0, Because the slope is m, ou know that the line rises two units for each unit it moves to the right, as shown in Figure P8 b Because b, the -intercept is 0, Because the slope is m 0, ou know that the line is horizontal, as shown in Figure P8 c Begin b writing the equation in slope-intercept form Write original equation Isolate -term on the left Slope-intercept form In this form, ou can see that the - intercept is 0, and the slope is m This means that the line falls one unit for ever three units it moves to the right, as shown in Figure P8(c) (0, ) = = + = (0, ) = (0, ) = = + = 5 6 m ; line rises Figure P8 m 0; line is horizontal (c) m ; line falls Editable Graph Editable Graph Editable Graph Tr It Eploration A

13 CHAPTER P Preparation for Calculus Because the slope of a vertical line is not defined, its equation cannot be written in the slope-intercept form However, the equation of an line can be written in the general form A B C 0 General form of the equation of a line where A and B are not both zero For instance, the vertical line given b a can be represented b the general form a 0 Summar of Equations of Lines General form: A B C 0, Vertical line: a Horizontal line: b Point-slope form: m 5 Slope-intercept form: m b A, B 0 Parallel and Perpendicular Lines The slope of a line is a convenient tool for determining whether two lines are parallel or perpendicular, as shown in Figure P9 Specificall, nonvertical lines with the same slope are parallel and nonvertical lines whose slopes are negative reciprocals are perpendicular m = m m m m m m = m Parallel lines Figure P9 Perpendicular lines STUDY TIP In mathematics, the phrase if and onl if is a wa of stating two implications in one statement For instance, the first statement at the right could be rewritten as the following two implications a If two distinct nonvertical lines are parallel, then their slopes are equal b If two distinct nonvertical lines have equal slopes, then the are parallel Parallel and Perpendicular Lines Two distinct nonvertical lines are parallel if and onl if their slopes are equal that is, if and onl if m m Two nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals of each other that is, if and onl if m m

14 SECTION P Linear Models and Rates of Change 5 EXAMPLE Finding Parallel and Perpendicular Lines Find the general forms of the equations of the lines that pass through the point and are, a parallel to the line 5 b perpendicular to the line 5 (See Figure P0) + = = 5 (, ) = 7 Lines parallel and perpendicular to 5 Figure P0 Editable Graph Solution B writing the linear equation 5 in slope-intercept form, 5, ou can see that the given line has a slope of m a The line through, that is parallel to the given line also has a slope of m 7 0 Note the similarit to the original equation Point-slope form Substitute Simplif General form b Using the negative reciprocal of the slope of the given line, ou can determine that the slope of a line perpendicular to the given line is So, the line through the point, that is perpendicular to the given line has the following equation m 0 Point-slope form Substitute Simplif General form Tr It Eploration A Eploration B Eploration C Open Eploration TECHNOLOGY PITFALL The slope of a line will appear distorted if ou use different tick-mark spacing on the - and -aes For instance, the graphing calculator screens in Figures P and P both show the lines given b and Because these lines have slopes that are negative reciprocals, the must be perpendicular In Figure P, however, the lines don t appear to be perpendicular because the tick-mark spacing on the -ais is not the same as that on the -ais In Figure P, the lines appear perpendicular because the tick-mark spacing on the -ais is the same as on the -ais This tpe of viewing window is said to have a square setting Tick-mark spacing on the -ais is not the same as tick-mark spacing on the -ais Figure P 6 Tick-mark spacing on the -ais is the same as tick-mark spacing on the -ais

15 6 CHAPTER P Preparation for Calculus Eercises for Section P The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises 6, estimate the slope of the line from its graph To print an enlarged cop of the graph, select the MathGraph button In Eercises 7 and 8, sketch the lines through the point with the indicated slopes Make the sketches on the same set of coordinate aes Point 7, (c) (d) Undefined 8, (c) (d) 0 In Eercises 9, plot the pair of points and find the slope of the line passing through them In Eercises 5 8, use the point on the line and the slope of the line to find three additional points that the line passes through (There is more than one correct answer) Point Slope Slopes Point 5, m 0 6, m undefined 7, 7 m 8, 9,, 5, 0,,,,,, 5,,,,,, 6 7 8,, 5, Slope m 9 Conveor Design A moving conveor is built to rise meter for each meters of horizontal change Find the slope of the conveor Suppose the conveor runs between two floors in a factor Find the length of the conveor if the vertical distance between floors is 0 feet 0 Rate of Change Each of the following is the slope of a line representing dail revenue in terms of time in das Use the slope to interpret an change in dail revenue for a one-da increase in time m 00 m 00 (c) m 0 Modeling Data The table shows the populations (in millions) of the United States for The variable t represents the time in ears, with t 6 corresponding to 996 (Source: US Bureau of the Census) t Plot the data b hand and connect adjacent points with a line segment Use the slope of each line segment to determine the ear when the population increased least rapidl Modeling Data The table shows the rate r (in miles per hour) that a vehicle is traveling after t seconds t r Plot the data b hand and connect adjacent points with a line segment Use the slope of each line segment to determine the interva when the vehicle s rate changed most rapidl How did the rate change? In Eercises 6, find the slope and the -intercept (if possible) of the line In Eercises 7, find an equation of the line that passes through the point and has the indicated slope Sketch the line Point Slope Point Slope 7 0, m 8, m undefined 9 0, 0 m 0 0, m 0, m, m 5

16 SECTION P Linear Models and Rates of Change 7 In Eercises, find an equation of the line that passes through the points, and sketch the line 0, 0,, 6 0, 0,, 5,, 0, 6,,, 7, 8, 5, 0 8, 6,, 9 5,, 5, 8 0,,,, 7, 0, Find an equation of the vertical line with -intercept at Show that the line with intercepts a, 0 and 0, b has the following equation a, b In Eercises 5 8, use the result of Eercise to write an equation of the line 5 -intercept:, 0 6 -intercept: -intercept: 0, -intercept: a, 0 -intercept: 0, a a 0 a 0, b 0 7 8,, 5, In Eercises 9 56, sketch a graph of the equation , 0 -intercept: 0, 7 Point on line:, 8 Point on line:, -intercept: a, 0 -intercept: 0, a a Square Setting In Eercises 57 and 58, use a graphing utilit to graph both lines in each viewing window Compare the graphs Do the lines appear perpendicular? Are the lines perpendicular? Eplain In Eercises 59 6, write an equation of the line through the point parallel to the given line and perpendicular to the given line Point 59, 60,, , 6, 5 6, 0 Rate of Change In Eercises 65 68, ou are given the dollar value of a product in 00 and the rate at which the value of the product is epected to change during the net 5 ears Write a linear equation that gives the dollar value V of the product in terms of the ear t (Let t 0 represent 000) 00 Value Line Rate Point 65 $50 $5 increase per ear 66 $56 $50 increase per ear 67 $0,00 $000 decrease per ear 68 $5,000 $5600 decrease per ear Line 7 7 In Eercises 69 and 70, use a graphing utilit to graph the parabolas and find their points of intersection Find an equation of the line through the points of intersection and graph the line in the same viewing window In Eercises 7 and 7, determine whether the points are collinear (Three points are collinear if the lie on the same line 7,,, 0,, 7 0,, 7, 6, 5, Writing About Concepts In Eercises 7 75, find the coordinates of the point of intersection of the given segments Eplain our reasoning ,, Xmin = -0 Xma = 0 Xscl = Ymin = -0 Yma = 0 Yscl = Xmin = -5 Xma = 5 Xscl = Ymin = -5 Yma = 5 Yscl = Xmin = -5 Xma = 5 Xscl = Ymin = -0 Yma = 0 Yscl = Xmin = -6 Xma = 6 Xscl = Ymin = - Yma = Yscl = 7 (b, c) 7 75 ( a, 0) (a, 0) Perpendicular bisectors ( a, 0) (a, 0) Altitudes (b, c) ( a, 0) (a, 0) Medians (b, c) 76 Show that the points of intersection in Eercises 7, 7, and 75 are collinear

17 8 CHAPTER P Preparation for Calculus 77 Temperature Conversion Find a linear equation that epresses the relationship between the temperature in degrees Celsius C and degrees Fahrenheit F Use the fact that water freezes at 0 C ( F) and boils at 00 C ( F) Use the equation to convert 7 F to degrees Celsius 78 Reimbursed Epenses A compan reimburses its sales representatives $50 per da for lodging and meals plus per mile driven Write a linear equation giving the dail cost C to the compan in terms of, the number of miles driven How much does it cost the compan if a sales representative drives 7 miles on a given da? 79 Career Choice An emploee has two options for positions in a large corporation One position pas $50 per hour plus an additional unit rate of $075 per unit produced The other pas $90 per hour plus a unit rate of $0 Find linear equations for the hourl wages W in terms of, the number of units produced per hour, for each option Use a graphing utilit to graph the linear equations and find the point of intersection (c) Interpret the meaning of the point of intersection of the graphs in part How would ou use this information to select the correct option if the goal were to obtain the highest hourl wage? 80 Straight-Line Depreciation A small business purchases a piece of equipment for $875 After 5 ears the equipment will be outdated, having no value Write a linear equation giving the value of the equipment in terms of the time, 0 5 Find the value of the equipment when (c) Estimate (to two-decimal-place accurac) the time when the value of the equipment is $00 8 Apartment Rental A real estate office handles an apartment comple with 50 units When the rent is $580 per month, all 50 units are occupied However, when the rent is $65, the average number of occupied units drops to 7 Assume that the relationship between the monthl rent p and the demand is linear (Note: The term demand refers to the number of occupied units) Write a linear equation giving the demand in terms of the rent p Linear etrapolation Use a graphing utilit to graph the demand equation and use the trace feature to predict the number of units occupied if the rent is raised to $655 (c) Linear interpolation Predict the number of units occupied if the rent is lowered to $595 Verif graphicall 8 Modeling Data An instructor gives regular 0-point quizzes and 00-point eams in a mathematics course Average scores for si students, given as ordered pairs, where is the average quiz score and is the average test score, are 8, 87, 0, 55, 9, 96, 6, 79,, 76, and 5, 8 Use the regression capabilities of a graphing utilit to find the least squares regression line for the data Use a graphing utilit to plot the points and graph the regression line in the same viewing window (c) Use the regression line to predict the average eam scor for a student with an average quiz score of 7 (d) Interpret the meaning of the slope of the regression line (e) The instructor adds points to the average test score of ever one in the class Describe the changes in the positions of th plotted points and the change in the equation of the line 8 Tangent Line Find an equation of the line tangent to the circl 69 at the point 5, 8 Tangent Line Find an equation of the line tangent to the circl 5 at the point, Distance In Eercises 85 90, find the distance between th point and line, or between the lines, using the formula for th distance between the point, and the line A B C 0 Distance A B C A B 85 Point: 0, 0 86 Point:, Line: 0 Line: 0 87 Point:, 88 Point: 6, Line: 0 Line: 89 Line: 90 Line: Line: 5 Line: 0 9 Show that the distance between the point, and the lin A B C 0 is Distance A B C A B 9 Write the distance d between the point, and the lin m in terms of m Use a graphing utilit to grap the equation When is the distance 0? Eplain the resul geometricall 9 Prove that the diagonals of a rhombus intersect at right angles (A rhombus is a quadrilateral with sides of equal lengths) 9 Prove that the figure formed b connecting consecutiv midpoints of the sides of an quadrilateral is a parallelogram 95 Prove that if the points, and, lie on the same lin as, and,, then Assume and 96 Prove that if the slopes of two nonvertical lines are negativ reciprocals of each other, then the lines are perpendicular True or False? In Eercises 97 and 98, determine whether th statement is true or false If it is false, eplain wh or give an eample that shows it is false 97 The lines represented b a b c and b a c ar perpendicular Assume a 0 and b 0 98 It is possible for two lines with positive slopes to be perpendi cular to each other

18 SECTION P Functions and Their Graphs 9 Section P X Domain f Range = f() Y A real-valued function f of a real variable Figure P Functions and Their Graphs Use function notation to represent and evaluate a function Find the domain and range of a function Sketch the graph of a function Identif different tpes of transformations of functions Classif functions and recognize combinations of functions Functions and Function Notation A relation between two sets X and Y is a set of ordered pairs, each of the form,, where is a member of X and is a member of Y A function from X to Y is a relation between X and Y that has the propert that an two ordered pairs with the same -value also have the same -value The variable is the independent variable, and the variable is the dependent variable Man real-life situations can be modeled b functions For instance, the area A of a circle is a function of the circle s radius r A r A is a function of r In this case r is the independent variable and A is the dependent variable Definition of a Real-Valued Function of a Real Variable Let X and Y be sets of real numbers A real-valued function f of a real variable from X to Y is a correspondence that assigns to each number in X eactl one number in Y The domain of f is the set X The number is the image of under f and is denoted b f, which is called the value of f at The range of f is a subset of Y and consists of all images of numbers in X (see Figure P) FUNCTION NOTATION The word function was first used b Gottfried Wilhelm Leibniz in 69 as a term to denote an quantit connected with a curve, such as the coordinates of a point on a curve or the slope of a curve Fort ears later, Leonhard Euler used the word function to describe an epression made up of a variable and some constants He introduced the notation f Functions can be specified in a variet of was In this tet, however, we will concentrate primaril on functions that are given b equations involving the dependent and independent variables For instance, the equation Equation in implicit form defines, the dependent variable, as a function of, the independent variable To evaluate this function (that is, to find the -value that corresponds to a given -value), it is convenient to isolate on the left side of the equation Equation in eplicit form Using f as the name of the function, ou can write this equation as f Function notation The original equation,, implicitl defines as a function of When ou solve the equation for, ou are writing the equation in eplicit form Function notation has the advantage of clearl identifing the dependent variable as f while at the same time telling ou that is the independent variable and that the function itself is f The smbol f is read f of Function notation allows ou to be less word Instead of asking What is the value of that corresponds to? ou can ask What is f?

19 0 CHAPTER P Preparation for Calculus In an equation that defines a function, the role of the variable is simpl that of a placeholder For instance, the function given b f can be described b the form f where parentheses are used instead of To evaluate f, simpl place in each set of parentheses f 8 7 Substitute for Simplif Simplif NOTE Although f is often used as a convenient function name and as the independent variable, ou can use other smbols For instance, the following equations all define the same function f 7 f t t t 7 g s s s 7 Function name is f, independent variable is Function name is f, independent variable is t Function name is g, independent variable is s EXAMPLE Evaluating a Function For the function f defined b f 7, evaluate each epression a f a b f b c f f, 0 STUDY TIP In calculus, it is important to communicate clearl the domain of a function or epression For instance, in Eample (c) the two epressions f f 0 and, are equivalent because 0 is ecluded from the domain of each epression Without a stated domain restriction, the two epressions would not be equivalent Solution a f a a 7 Substitute a for Simplif b f b b 7 Substitute b for c 9a 7 b b 8 f f Tr It b b 7, 0 Epand binomial Simplif Eploration A NOTE The epression in Eample (c) is called a difference quotient and has a special significance in calculus You will learn more about this in Chapter

20 SECTION P Functions and Their Graphs Range: 0 f() = The domain of f is, and the range is 0, Figure P Domain: The domain of f is, and the range is 0, Range f() = tan π Domain The domain of f is all -values such that π n and the range is, Figure P Range: 0 Video Editable Graph Editable Graph f() = Editable Graph Domain: all real, <, The Domain and Range of a Function The domain of a function can be described eplicitl, or it ma be described implicitl b an equation used to define the function The implied domain is the set of all real numbers for which the equation is defined, whereas an eplicitl defined domain is one that is given along with the function For eample, the function given b f, has an eplicitl defined domain given b : 5 On the other hand, the function given b g has an implied domain that is the set : ± EXAMPLE a The domain of the function Finding the Domain and Range of a Function is the set of all -values for which 0, which is the interval, To find the range observe that f is never negative So, the range is the interval 0,, as indicated in Figure P b The domain of the tangent function, as shown in Figure P, is the set of all -values such that n is an integer Domain of tangent function The range of this function is the set of all real numbers For a review of the characteristics of this and other trigonometric functions, see Appendi D EXAMPLE f f tan n, Tr It A Function Defined b More than One Equation Determine the domain and range of the function f,, 5 Eploration A if < if Solution Because f is defined for < and, the domain is the entire set of real numbers On the portion of the domain for which, the function behaves as in Eample For <, the values of are positive So, the range of the function is the interval 0, (See Figure P) Tr It Eploration A A function from X to Y is one-to-one if to each -value in the range there corresponds eactl one -value in the domain For instance, the function given in Eample is one-to-one, whereas the functions given in Eamples and are not one-to-one A function from X to Y is onto if its range consists of all of Y

21 CHAPTER P Preparation for Calculus = f() (, f()) f() The graph of a function Figure P5 The Graph of a Function The graph of the function f consists of all points, f, where is in the domain of f In Figure P5, note that the directed distance from the -ais f the directed distance from the -ais A vertical line can intersect the graph of a function of at most once This observation provides a convenient visual test, called the Vertical Line Test, for functions of That is, a graph in the coordinate plane is the graph of a function of f if and onl if no vertical line intersects the graph at more than one point For eample, in Figure P6, ou can see that the graph does not define as a function of because a vertical line intersects the graph twice, whereas in Figures P6 and (c), the graphs do define as a function of Not a function of Figure P6 A function of (c) A function of Figure P7 shows the graphs of eight basic functions You should be able to recognize these graphs (Graphs of the other four basic trigonometric functions are shown in Appendi D) f() = f() = f() = f() = Identit function Squaring function Cubing function Square root function f() = f() = f() = sin f() = cos π π π π π π π Absolute value function The graphs of eight basic functions Figure P7 Rational function Sine function Cosine function

22 SECTION P Functions and Their Graphs EXPLORATION Writing Equations for Functions Each of the graphing utilit screens below shows the graph of one of the eight basic functions shown on page Each screen also shows a transformation of the graph Describe the transformation Then use our description to write an equation for the transformation Transformations of Functions Some families of graphs have the same basic shape For eample, compare the graph of with the graphs of the four other quadratic functions shown in Figure P8 = + 9 = Vertical shift upward = ( + ) Horizontal shift to the left = 9 9 Animation Animation a b = = (c) Reflection Animation Figure P8 = ( + ) = 5 (d) Shift left, reflect, and shift upward Animation 8 0 c Each of the graphs in Figure P8 is a transformation of the graph of The three basic tpes of transformations illustrated b these graphs are vertical shifts, horizontal shifts, and reflections Function notation lends itself well to describing transformations of graphs in the plane For instance, if f is considered to be the original function in Figure P8, the transformations shown can be represented b the following equations f f f f Vertical shift up units Horizontal shift to the left units Reflection about the -ais Shift left units, reflect about -ais, and shift up unit d Video Basic Tpes of Transformations Original graph: Horizontal shift c units to the right: Horizontal shift c units to the left: Vertical shift c units downward: Vertical shift c units upward: Reflection (about the -ais): Reflection (about the -ais): Reflection (about the origin): c > 0 f f c f c f c f c f f f

23 CHAPTER P Preparation for Calculus LEONHARD EULER (707 78) In addition to making major contributions to almost ever branch of mathematics, Euler was one of the first to appl calculus to real-life problems in phsics His etensive published writings include such topics as shipbuilding, acoustics, optics, astronom, mechanics, and magnetism MathBio Classifications and Combinations of Functions The modern notion of a function is derived from the efforts of man seventeenth- and eighteenth-centur mathematicians Of particular note was Leonhard Euler, to whom we are indebted for the function notation f B the end of the eighteenth centur, mathematicians and scientists had concluded that man real-world phenomena could be represented b mathematical models taken from a collection of functions called elementar functions Elementar functions fall into three categories Algebraic functions (polnomial, radical, rational) Trigonometric functions (sine, cosine, tangent, and so on) Eponential and logarithmic functions You can review the trigonometric functions in Appendi D The other nonalgebraic functions, such as the inverse trigonometric functions and the eponential and logarithmic functions, are introduced in Chapter 5 The most common tpe of algebraic function is a polnomial function f a n n a n n a a a 0, a n 0 FOR FURTHER INFORMATION For more on the histor of the concept of a function, see the article Evolution of the Function Concept: A Brief Surve b Israel Kleiner in The College Mathematics Journal MathArticle where the positive integer n is the degree of the polnomial function The constants a i are coefficients, with a n the leading coefficient and a 0 the constant term of the polnomial function It is common practice to use subscript notation for coefficients of general polnomial functions, but for polnomial functions of low degree, the following simpler forms are often used Zeroth degree: First degree: Second degree: Third degree: f a f a b f a b c f a b c d Constant function Linear function Quadratic function Cubic function Although the graph of a nonconstant polnomial function can have several turns, eventuall the graph will rise or fall without bound as moves to the right or left Whether the graph of f a n n a n n a a a 0 eventuall rises or falls can be determined b the function s degree (odd or even) and b the leading coefficient a n, as indicated in Figure P9 Note that the dashed portions of the graphs indicate that the Leading Coefficient Test determines onl the right and left behavior of the graph a n > 0 a n < 0 a n > 0 a n < 0 Up to right Up to left Up to left Up to right Down to left Down to right Down to left Down to right Graphs of polnomial functions of even degree The Leading Coefficient Test for polnomial functions Figure P9 Graphs of polnomial functions of odd degree

24 SECTION P Functions and Their Graphs 5 Just as a rational number can be written as the quotient of two integers, a rational function can be written as the quotient of two polnomials Specificall, a function f is rational if it has the form f p q, q 0 Domain of g f g where p and q are polnomials Polnomial functions and rational functions are eamples of algebraic functions An algebraic function of is one that can be epressed as a finite number of sums, differences, multiples, quotients, and radicals involving n For eample, f is algebraic Functions that are not algebraic are transcendental For instance, the trigonometric functions are transcendental Two functions can be combined in various was to create new functions For eample, given f and g, ou can form the functions shown f g f g f g f g fg f g f g f g Sum Difference Product Quotient You can combine two functions in et another wa, called composition The resulting function is called a composite function g g() Domain of f f f(g()) The domain of the composite function Figure P0 f g Definition of Composite Function Let f and g be functions The function given b f g f g is called the composite of f with g The domain of f g is the set of all in the domain of g such that g is in the domain of f (see Figure P0) The composite of f with g ma not be equal to the composite of g with f EXAMPLE Finding Composite Functions Given f and g cos, find each composite function a f g b g f Solution a f g f g Definition of f g f cos Substitute cos for g cos Definition of f cos Simplif b g f g f g cos Definition of g f Substitute for f Definition of g Note that f g g f Tr It Eploration A Eploration B Eploration C Eploration D Eploration E Open Eploration

25 6 CHAPTER P Preparation for Calculus EXPLORATION Use a graphing utilit to graph each function Determine whether the function is even, odd, or neither f g h 5 j 6 8 k 5 p 9 5 Describe a wa to identif a function as odd or even b inspecting the equation In Section P, an -intercept of a graph was defined to be a point a, 0 at which the graph crosses the -ais If the graph represents a function f, the number a is a zero of f In other words, the zeros of a function f are the solutions of the equation f 0 For eample, the function f has a zero at because f 0 In Section P ou also studied different tpes of smmetr In the terminolog of functions, a function is even if its graph is smmetric with respect to the -ais, and is odd if its graph is smmetric with respect to the origin The smmetr tests in Section P ield the following test for even and odd functions Test for Even and Odd Functions The function f is even if f f The function f is odd if f f NOTE Ecept for the constant function f 0, the graph of a function of cannot have smmetr with respect to the -ais because it then would fail the Vertical Line Test for the graph of the function EXAMPLE 5 Even and Odd Functions and Zeros of Functions (0, 0) Odd function (, 0) Editable Graph π Even function Editable Graph Figure P (, 0) g() = + cos π π π f () = Determine whether each function is even, odd, or neither Then find the zeros of the function a f b g cos Solution a This function is odd because f f The zeros of f are found as shown 0 See Figure P b This function is even because g cos cos g The zeros of g are found as shown cos 0 cos n, n is an integer See Figure P 0 0,, Let f 0 Factor Zeros of f cos cos Let g 0 Subtract from each side Zeros of g Tr It Eploration A Eploration B Eploration C NOTE Each of the functions in Eample 5 is either even or odd However, some functions, such as f, are neither even nor odd

26 SECTION P Functions and Their Graphs 7 Eercises for Section P The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises and, use the graphs of f and g to answer the following Identif the domains and ranges of f and g Identif f and g (c) For what value(s) of is f g? (d) Estimate the solution(s) of f (e) Estimate the solutions of g 0 In Eercises, evaluate (if possible) the function at the given value(s) of the independent variable Simplif the results (c) (d) g 0 g (c) g (d) g t g f 0 f b f 0 f f (c) (d) g (c) g c (d) g t 7 f cos 8 f sin f (c) f (c) f 9 f 0 f f f f f f f f f In Eercises 8, find the domain and range of the function h g 5 t 5 f t sec 6 h t cot t 7 f 8 f f f 5 g 6 g f 6 g f f f 5 f f 5 f f g f g In Eercises 9-, find the domain of the function 9 f 0 f g h cos sin f g In Eercises 5 8, evaluate the function as indicated Determine its domain and range f,, f f 0 (c) f (d) f t f,, f f 0 (c) f (d) f s f, <, f f (c) f (d) f b f, 5, < 0 0 > 5 > 5 f f 0 (c) f 5 (d) f 0 In Eercises 9 6, sketch a graph of the function and find it domain and range Use a graphing utilit to verif our graph 9 f 0 g h f f 9 f 5 g t sin t 6 h 5 cos Writing About Concepts 7 The graph of the distance that a student drives in a 0-minute trip to school is shown in the figure Give a verbal description of characteristics of the student s drive to school Distance (in miles) s (0, 0) (, ) (0, 6) (6, ) Time (in minutes) t

27 8 CHAPTER P Preparation for Calculus Writing About Concepts (continued) 8 A student who commutes 7 miles to attend college remembers, after driving a few minutes, that a term paper that is due has been forgotten Driving faster than usual, the student returns home, picks up the paper, and once again starts toward school Sketch a possible graph of the student s distance from home as a function of time In Eercises 9, use the Vertical Line Test to determine whether is a function of To print an enlarged cop of the graph, select the MathGraph button In Eercises 6, determine whether is a function of In Eercises 7 5, use the graph of function with its graph e d 7 f 5 8 f 5 f to match the 9 f 50 f 5 f 6 5 f c, 0, > b f() g a 5 Use the graph of f shown in the figure to sketch the graph o each function To print an enlarged cop of the graph, select th MathGraph button (c) (e) f (d) (f) 5 Use the graph of f shown in the figure to sketch the graph o each function To print an enlarged cop of the graph, select th MathGraph button (c) (e) f (d) (f) 55 Use the graph of f to sketch the graph of each function In each case, describe the transformation (c) 56 Specif a sequence of transformations that will ield each graph of h from the graph of the function f sin h sin h sin 57 Given f and g, evaluate each epression f g g f (c) g f 0 (d) f g (e) f g (f) g f 58 Given f sin and g, evaluate each epression f g (c) g f 0 (d) g f (e) f g (f) g f In Eercises 59 6, find the composite functions f g and g f What is the domain of each composite function? Are th two composite functions equal? 59 f 60 f g 6 f 6 f g 6 Use the graphs of f and g to evaluate each epression If the result is undefined, eplain wh (c) g f 5 (e) f f f f f g g f f f f f (d) f f f g g f f g (f) f g 6 6 (, ) g cos g f f 7 f (, ) 5 g 9

28 SECTION P Functions and Their Graphs 9 6 Ripples A pebble is dropped into a calm pond, causing ripples in the form of concentric circles The radius (in feet) of the outer ripple is given b r t 06t, where t is the time in seconds after the pebble strikes the water The area of the circle is given b the function A r r Find and interpret A r t Determine the value of the constant c for each function such that the function fits the data shown in the table Think About It In Eercises 65 and 66, F f g h Identif functions for f, g, and h (There are man correct answers) 65 F 66 F sin In Eercises 67 70, determine whether the function is even, odd, or neither Use a graphing utilit to verif our result 67 f 68 f 69 f cos 70 f sin Undef Think About It In Eercises 7 and 7, find the coordinates of a second point on the graph of a function f if the given point is on the graph and the function is even and odd 7 7, 9, 7 The graphs of f, g, and h are shown in the figure Decide whether each function is even, odd, or neither f h Figure for 7 Figure for 7 7 The domain of the function f shown in the figure is 6 6 Complete the graph of f given that f is even Complete the graph of f given that f is odd Writing Functions In Eercises 75 78, write an equation for a function that has the given graph 75 Line segment connecting, and 0, 5 76 Line segment connecting, and 5, 5 77 The bottom half of the parabola 0 78 The bottom half of the circle g f Graphical Reasoning An electronicall controlled thermo stat is programmed to lower the temperature during the nigh automaticall (see figure) The temperature T in degrees Celsius is given in terms of t, the time in hours on a -hou clock Approimate T and T 5 The thermostat is reprogrammed to produce a temperature H t T t How does this change the temperature? Eplain (c) The thermostat is reprogrammed to produce a temperature H t T t How does this change the temperature? Eplain T 0 6 t Water runs into a vase of height 0 centimeters at a constan rate The vase is full after 5 seconds Use this information and the shape of the vase shown to answer the questions if d is the depth of the water in centimeters and t is the time in seconds (see figure) Eplain wh d is a function of t Determine the domain and range of the function (c) Sketch a possible graph of the function Modeling Data In Eercises 79 8, match the data with a function from the following list (i) f c (iii) h c (ii) g c (iv) r c/ d 0 cm

29 0 CHAPTER P Preparation for Calculus 85 Modeling Data The table shows the average numbers of acres per farm in the United States for selected ears (Source: US Department of Agriculture) Year Acreage Plot the data where A is the acreage and t is the time in ears, with t 0 corresponding to 950 Sketch a freehand curve that approimates the data Use the curve in part to approimate A 5 86 Automobile Aerodnamics The horsepower H required to overcome wind drag on a certain automobile is approimated b H , where is the speed of the car in miles per hour Use a graphing utilit to graph H Rewrite the power function so that represents the speed in kilometers per hour Find H 6 87 Think About It Write the function f without using absolute value signs (For a review of absolute value, see Appendi D) 88 Writing Use a graphing utilit to graph the polnomial functions p and p How man zeros does each function have? Is there a cubic polnomial that has no zeros? Eplain 89 Prove that the function is odd f a n n a a 90 Prove that the function is even f a n n a n n a a 0 9 Prove that the product of two even (or two odd) functions is even 9 Prove that the product of an odd function and an even function is odd 9 Volume An open bo of maimum volume is to be made from a square piece of material centimeters on a side b cutting equal squares from the corners and turning up the sides (see figure) 0 00 Use a graphing utilit to graph the volume function and approimate the dimensions of the bo that ield a mai mum volume (c) Use the table feature of a graphing utilit to verif ou answer in part (The first two rows of the table are shown) 9 Length A right triangle is formed in the first quadrant b the - and -aes and a line through the point, (see figure) Write the length L of the hpotenuse as a function of Length Height, and Width Volume, V (0, ) 800 (, ) True or False? In Eercises 95 98, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 95 If f a f b, then a b (, 0) 96 A vertical line can intersect the graph of a function at mos once 97 If f f for all in the domain of f, then the graph o f is smmetric with respect to the -ais 98 If f is a function, then f a af 8 Putnam Eam Challenge 99 Let R be the region consisting of the points of the Cartesian plane satisfing both and Sketch the region R and find its area, 00 Consider a polnomial f with real coefficients having the propert f g g f for ever polnomial g with rea coefficients Determine and prove the nature of f These problems were composed b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserved Write the volume V as a function of, the length of the corner squares What is the domain of the function?

30 SECTION P Fitting Models to Data Section P A computer graphics drawing based on the pen and ink drawing of Leonardo da Vinci s famous stud of human proportions, called Vitruvian Man Fitting Models to Data Fit a linear model to a real-life data set Fit a quadratic model to a real-life data set Fit a trigonometric model to a real-life data set Fitting a Linear Model to Data A basic premise of science is that much of the phsical world can be described mathematicall and that man phsical phenomena are predictable This scientific outlook was part of the scientific revolution that took place in Europe during the late 500s Two earl publications connected with this revolution were On the Revolutions of the Heavenl Spheres b the Polish astronomer Nicolaus Copernicus and On the Structure of the Human Bod b the Belgian anatomist Andreas Vesalius Each of these books was published in 5 and each broke with prior tradition b suggesting the use of a scientific method rather than unquestioned reliance on authorit One basic technique of modern science is gathering data and then describing the data with a mathematical model For instance, the data given in Eample are inspired b Leonardo da Vinci s famous drawing that indicates that a person s height and arm span are equal EXAMPLE Fitting a Linear Model to Data Arm span (in inches) Height (in inches) Linear model and data Figure P Video A class of 8 people collected the following data, which represent their heights and arm spans (rounded to the nearest inch) 60, 6, 65, 65, 68, 67, 7, 7, 6, 6, 6, 6, 70, 7, 75, 7, 7, 7, 6, 60, 65, 65, 66, 68, 6, 6, 7, 7, 70, 70, 69, 68, 69, 70, 60, 6, 6, 6, 6, 6, 7, 7, 68, 67, 69, 70, 70, 7, 65, 65, 6, 6, 7, 70, 67, 67 Find a linear model to represent these data Solution There are different was to model these data with an equation The simplest would be to observe that and are about the same and list the model as simpl A more careful analsis would be to use a procedure from statistics called linear regression (You will stud this procedure in Section 9) The least squares regression line for these data is Least squares regression line The graph of the model and the data are shown in Figure P From this model, ou can see that a person s arm span tends to be about the same as his or her height Tr It Eploration A Open Eploration TECHNOLOGY Man scientific and graphing calculators have built-in least squares regression programs Tpicall, ou enter the data into the calculator and then run the linear regression program The program usuall displas the slope and -intercept of the best-fitting line and the correlation coefficient r The correlation coefficient gives a measure of how well the model fits the data The closer r is to, the better the model fits the data For instance, the correlation coefficient for the model in Eample is r 097, which indicates that the model is a good fit for the data If the r-value is positive, the variables have a positive correlation, as in Eample If the r-value is negative, the variables have a negative correlation

31 CHAPTER P Preparation for Calculus Fitting a Quadratic Model to Data A function that gives the height s of a falling object in terms of the time t is called a position function If air resistance is not considered, the position of a falling object can be modeled b s t gt v 0 t s 0 where g is the acceleration due to gravit, v 0 is the initial velocit, and s 0 is the initial height The value of g depends on where the object is dropped On earth, g is approimatel feet per second per second, or 98 meters per second per second To discover the value of g eperimentall, ou could record the heights of a falling object at several increments, as shown in Eample EXAMPLE Fitting a Quadratic Model to Data A basketball is dropped from a height of about 5 feet The height of the basketball is recorded times at intervals of about 00 second* The results are shown in the table Time Height Time Height Time Height Time Height Height (in feet) s Time (in seconds) Scatter plot of data Figure P t Find a model to fit these data Then use the model to predict the time when the basketball will hit the ground Solution Begin b drawing a scatter plot of the data, as shown in Figure P From the scatter plot, ou can see that the data do not appear to be linear It does appear, however, that the might be quadratic To check this, enter the data into a calculator or computer that has a quadratic regression program You should obtain the model s 55t 0t 5 Least squares regression quadratic Using this model, ou can predict the time when the basketball hits the ground b substituting 0 for s and solving the resulting equation for t 0 55t 0t 5 t 0 ± t 05 Let s 0 Quadratic Formula Choose positive solution The solution is about 05 second In other words, the basketball will continue to fall for about 0 second more before hitting the ground Tr It Eploration A * Data were collected with a Teas Instruments CBL (Calculator-Based Laborator) Sstem

32 SECTION P Fitting Models to Data The plane of Earth s orbit about the sun and its ais of rotation are not perpendicular Instead, Earth s ais is tilted with respect to its orbit The result is that the amount of dalight received b locations on Earth varies with the time of ear That is, it varies with the position of Earth in its orbit Fitting a Trigonometric Model to Data What is mathematical modeling? This is one of the questions that is asked in the book Guide to Mathematical Modelling Here is part of the answer* Mathematical modeling consists of appling our mathematical skills to obtain useful answers to real problems Learning to appl mathematical skills is ver different from learning mathematics itself Models are used in a ver wide range of applications, some of which do not appear initiall to be mathematical in nature Models often allow quick and cheap evaluation of alternatives, leading to optimal solutions that are not otherwise obvious 5 There are no precise rules in mathematical modeling and no correct answers 6 Modeling can be learned onl b doing EXAMPLE Fitting a Trigonometric Model to Data Dalight (in minutes) d Graph of model Figure P 65 Da (0 December ) NOTE For more review of trigonometric functions, see Appendi D t The number of hours of dalight on Earth depends on the latitude and the time of ear Here are the numbers of minutes of dalight at a location of 0 N latitude on the longest and shortest das of the ear: June, 80 minutes; December, 655 minutes Use these data to write a model for the amount of dalight d (in minutes) on each da of the ear at a location of 0 N latitude How could ou check the accurac of our model? Solution Here is one wa to create a model You can hpothesize that the model is a sine function whose period is 65 das Using the given data, ou can conclude that the amplitude of the graph is , or 7 So, one possible model is d 78 7 sin t 65 In this model, t represents the number of each da of the ear, with December represented b t 0 A graph of this model is shown in Figure P To check the accurac of this model, we used a weather almanac to find the numbers of minutes of dalight on different das of the ear at the location of 0 N latitude Date Value of t Actual Dalight Dec min 655 min Jan min 656 min Feb 676 min 67 min Mar min 70 min Apr min 79 min Ma 0 77 min 77 min Jun min 796 min Jun 8 80 min 80 min Jul min 800 min Aug 78 min 785 min Sep 5 75 min 75 min Oct 8 78 min 76 min Nov 685 min 68 min Dec 66 min 660 min You can see that the model is fairl accurate Dalight Given b Model Tr It Eploration A * Tet from Dilwn Edwards and Mike Hamson, Guide to Mathematical Modelling (Boca Raton: CRC Press, 990) Used b permission of the authors

33 CHAPTER P Preparation for Calculus Eercises for Section P The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises, a scatter plot of data is given Determine whether the data can be modeled b a linear function, a quadratic function, or a trigonometric function, or that there appears to be no relationship between and To print an enlarged cop of the graph, select the MathGraph button F d Use the regression capabilities of a graphing utilit to find a linear model for the data Use a graphing utilit to plot the data and graph the model How well does the model fit the data? Eplain our reasoning (c) Use the model to estimate the elongation of the spring when a force of 55 newtons is applied 8 Falling Object In an eperiment, students measured the speed s (in meters per second) of a falling object t seconds after it was released The results are shown in the table t s Carcinogens Each ordered pair gives the eposure inde of a carcinogenic substance and the cancer mortalit per 00,000 people in the population 50, 50, 58,,, 9, 6, 67, 6, 07, 85, 655, 65, 07, 7, 80, 95, Plot the data From the graph, do the data appear to be approimatel linear? Visuall find a linear model for the data Graph the model (c) Use the model to approimate if 6 Quiz Scores The ordered pairs represent the scores on two consecutive 5-point quizzes for a class of 8 students 7,, 9, 7,,, 5, 5, 0, 5, 9, 7,,,, 5, 8, 0, 5, 9, 0,, 9, 0,,, 7,,, 0,,, 0, 5, 9, 6 Plot the data From the graph, does the relationship between consecutive scores appear to be approimatel linear? If the data appear to be approimatel linear, find a linear model for the data If not, give some possible eplanations 7 Hooke s Law Hooke s Law states that the force F required to compress or stretch a spring (within its elastic limits) is proportional to the distance d that the spring is compressed or stretched from its original length That is, F kd, where k is a measure of the stiffness of the spring and is called the spring constant The table shows the elongation d in centimeters of a spring when a force of F newtons is applied Use the regression capabilities of a graphing utilit to find a linear model for the data Use a graphing utilit to plot the data and graph the model How well does the model fit the data? Eplain our reasoning (c) Use the model to estimate the speed of the object after 5 seconds 9 Energ Consumption and Gross National Product The data show the per capita electricit consumptions (in millions of Btu) and the per capita gross national products (in thousands of US dollars) for several countries in 000 (Source: US Census Bureau) Argentina (7, 05) Chile (68, 9) Greece (6, 686) Hungar (05, 99) Meico (6, 879) Portugal (08, 699) Spain (7, 96) United Kingdom (66, 55) Bangladesh (, 59) Egpt (, 67) Hong Kong (8, 559) India (, ) Poland (95, 9) South Korea (67, 7) Turke (7, 70) Venezuela (, 57) Use the regression capabilities of a graphing utilit to find a linear model for the data What is the correlation coefficient? Use a graphing utilit to plot the data and graph the model (c) Interpret the graph in part Use the graph to identif the three countries that differ most from the linear model (d) Delete the data for the three countries identified in par (c) Fit a linear model to the remaining data and give the correlation coefficient

34 SECTION P Fitting Models to Data 5 0 Brinell Hardness The data in the table show the Brinell hardness H of 05 carbon steel when hardened and tempered at temperature t (degrees Fahrenheit) (Source: Standard Handbook for Mechanical Engineers) t H Use the regression capabilities of a graphing utilit to find a linear model for the data Use a graphing utilit to plot the data and graph the model How well does the model fit the data? Eplain our reasoning (c) Use the model to estimate the hardness when t is 500 F Automobile Costs The data in the table show the variable costs for operating an automobile in the United States for several recent ears The functions,, and represent the costs in cents per mile for gas and oil, maintenance, and tires, respectivel (Source: American Automobile Manufacturers Association) Year Use the regression capabilities of a graphing utilit to find a cubic model for and linear models for and Use a graphing utilit to graph,,, and in the same viewing window Use the model to estimate the total variable cost per mile in ear Beam Strength Students in a lab measured the breaking strength S (in pounds) of wood inches thick, inches high, and inches long The results are shown in the table S ,0 6,50, Use the regression capabilities of a graphing utilit to fit a quadratic model to the data Use a graphing utilit to plot the data and graph the model (c) Use the model to approimate the breaking strength when Health Maintenance Organizations The bar graph shows the numbers of people N (in millions) receiving care in HMOs for the ears 990 through 00 (Source: Centers for Disease Control) Enrollment (in millions) N HMO Enrollment Year (0 990) Let t be the time in ears, with t 0 corresponding to 990 Use the regression capabilities of a graphing utilit to find linear and cubic models for the data Use a graphing utilit to graph the data and the linear and cubic models (c) Use the graphs in part to determine which is the better model (d) Use a graphing utilit to find and graph a quadratic mode for the data (e) Use the linear and cubic models to estimate the number of people receiving care in HMOs in the ear 00 (f) Use a graphing utilit to find other models for the data Which models do ou think best represent the data? Eplain Car Performance The time t (in seconds) required to attain a speed of s miles per hour from a standing start for a Dodge Avenger is shown in the table (Source: Road & Track) s t Use the regression capabilities of a graphing utilit to find a quadratic model for the data Use a graphing utilit to plot the data and graph the model (c) Use the graph in part to state wh the model is no appropriate for determining the times required to attain speeds less than 0 miles per hour (d) Because the test began from a standing start, add the poin 0, 0 to the data Fit a quadratic model to the revised data and graph the new model (e) Does the quadratic model more accuratel model the behavior of the car for low speeds? Eplain t

35 6 CHAPTER P Preparation for Calculus 5 Car Performance A V8 car engine is coupled to a dnamometer and the horsepower is measured at different engine speeds (in thousands of revolutions per minute) The results are shown in the table Use the regression capabilities of a graphing utilit to find a cubic model for the data Use a graphing utilit to plot the data and graph the model (c) Use the model to approimate the horsepower when the engine is running at 500 revolutions per minute 6 Boiling Temperature The table shows the temperatures T F at which water boils at selected pressures p (pounds per square inch) (Source: Standard Handbook for Mechanical Engineers) p ( atmosphere) 0 T p T Use the regression capabilities of a graphing utilit to find a cubic model for the data Use a graphing utilit to plot the data and graph the model (c) Use the graph to estimate the pressure required for the boiling point of water to eceed 00 F (d) Eplain wh the model would not be correct for pressures eceeding 00 pounds per square inch 7 Harmonic Motion The motion of an oscillating weight suspended b a spring was measured b a motion detector The data collected and the approimate maimum (positive and negative) displacements from equilibrium are shown in the figure The displacement is measured in centimeters and the time t is measured in seconds Is a function of t? Eplain Approimate the amplitude and period of the oscillations (c) Find a model for the data (d) Use a graphing utilit to graph the model in part (c) Compare the result with the data in the figure 6 50 (05, 5) Temperature The table shows the normal dail high temperatures for Honolulu H and Chicago C (in degrees Fahrenheit) for month t, with t corresponding to Januar (Source: NOAA) A model for Honolulu is H t 80 8 sin t 6 86 Find a model for Chicago Use a graphing utilit to graph the data and the model for the temperatures in Honolulu How well does the model fit? (c) Use a graphing utilit to graph the data and the model for the temperatures in Chicago How well does the model fit? (d) Use the models to estimate the average annual temperature in each cit What term of the model did ou use? Eplain (e) What is the period of each model? Is it what ou epected? Eplain (f) Which cit has a greater variabilit of temperatures throughout the ear? Which factor of the models determines this variabilit? Eplain Writing About Concepts 9 Search for real-life data in a newspaper or magazine Fit the data to a model What does our model impl about the data? 0 Describe a possible real-life situation for each data set Then describe how a model could be used in the real-life setting (c) t 5 6 H C t H C (d) (075, 65) t

36 REVIEW EXERCISES 7 Review Eercises for Chapter P The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises, find the intercepts (if an) In Eercises 5 and 6, check for smmetr with respect to both aes and to the origin In Eercises 7, sketch the graph of the equation In Eercises 5 and 6, describe the viewing window of a graphing utilit that ields the figure In Eercises 7 and 8, use a graphing utilit to find the point(s) of intersection of the graphs of the equations Think About It Write an equation whose graph has intercepts at and and is smmetric with respect to the origin 0 Think About It For what value of k does the graph of k pass through the point?,, (c) 0, 0 (d), In Eercises and, plot the points and find the slope of the line passing through the points,, 5, 5 In Eercises and, use the concept of slope to find t such that the three points are collinear In Eercises 5 8, find an equation of the line that passes through the point with the indicated slope Sketch the line 5 0, 5, m 6, 6, 7 7,, 7,, 5, 0, t,,,, t,, 8, 6 m 0 7, 0, m 8 5,, m is undefined 9 Find equations of the lines passing through, and having the following characteristics Slope of Parallel to the line 5 (c) Passing through the origin (d) Parallel to the -ais 0 Find equations of the lines passing through, and having the following characteristics Slope of Perpendicular to the line 0 (c) Passing through the point, (d) Parallel to the -ais Rate of Change The purchase price of a new machine is $,500, and its value will decrease b $850 per ear Use this information to write a linear equation that gives the value V of the machine t ears after it is purchased Find its value at the end of ears Break-Even Analsis A contractor purchases a piece of equipment for $6,500 that costs an average of $95 per hour for fuel and maintenance The equipment operator is paid $50 per hour, and customers are charged $0 per hour Write an equation for the cost C of operating this equipment for t hours Write an equation for the revenue R derived from t hours of use (c) Find the break-even point for this equipment b finding the time at which R C In Eercises 6, sketch the graph of the equation and use the Vertical Line Test to determine whether the equation epresses as a function of Evaluate (if possible) the function f at the specified values of the independent variable, and simplif the results f Evaluate (if possible) the function at each value of the independent variable f, < 0, 0 f f f f 0 (c) f 9 Find the domain and range of each function (c), < , 0

37 8 CHAPTER P Preparation for Calculus 0 Given f and g, evaluate each epression f g f g (c) g f Sketch (on the same set of coordinate aes) graphs of f for c, 0, and (c) Use a graphing utilit to graph f Use the graph to write a formula for the function g shown in the figure To print an enlarged cop of the graph, select the MathGraph button Conjecture (d) Use a graphing utilit to graph the functions f, g, and h in the same viewing window Write a description of an similarities and differences ou observe among the graphs Odd powers: Even powers: Use the result in part to make a conjecture about the graphs of the functions 7 and 8 Use a graphing utilit to verif our conjecture Think About It Use the result of Eercise to guess the shapes of the graphs of the functions f, g, and h Then use a graphing utilit to graph each function and compare the result with our guess (c) f 6 g 6 h 6 5 Area A wire inches long is to be cut into four pieces to form a rectangle whose shortest side has a length of Write the area A of the rectangle as a function of Determine the domain of the function and use a graphing utilit to graph the function over that domain (c) Use the graph of the function to approimate the maimum area of the rectangle Make a conjecture about the dimensions that ield a maimum area 6 Writing The following graphs give the profits P for two small companies over a period p of ears Create a stor to describe the behavior of each profit function for some hpothetical product the compan produces f c f c g 00,000 00,000 P 6 (0, ) (, 5) p f c f c f, g, h 5 f, g, h 6 00,000 50,000 P (, ) (, ) g 6 p 7 Think About It What is the minimum degree of the polno mial function whose graph approimates the given graph? What sign must the leading coefficient have? (c) 8 Stress Test A machine part was tested b bending it centimeters 0 times per minute until the time (in hours) o failure The results are recorded in the table (d) Use the regression capabilities of a graphing utilit to find a linear model for the data Use a graphing utilit to plot the data and graph the model (c) Use the graph to determine whether there ma have been an error made in conducting one of the tests or in recording the results If so, eliminate the erroneous point and find the model for the remaining data 9 Harmonic Motion The motion of an oscillating weigh suspended b a spring was measured b a motion detector The data collected and the approimate maimum (positive and negative) displacements from equilibrium are shown in the figure The displacement is measured in feet and the time t is measured in seconds Is a function of t? Eplain Approimate the amplitude and period of the oscillations (c) Find a model for the data (d) Use a graphing utilit to graph the model in part (c) Compare the result with the data in the figure (05, 05) (, 05) 0 0 t

38 PS Problem Solving 9 PS Problem Solving The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph Consider the circle 6 8 0, as shown in the figure Find the center and radius of the circle Find an equation of the tangent line to the circle at the point 0, 0 (c) Find an equation of the tangent line to the circle at the point 6, 0 (d) Where do the two tangent lines intersect? 8 6 Figure for Figure for There are two tangent lines from the point 0, to the circle (see figure) Find equations of these two lines b using the fact that each tangent line intersects the circle in eactl one point The Heaviside function H is widel used in engineering applications H, 0, Sketch the graph of the Heaviside function and the graphs of the following functions b hand H H (c) H (d) H (e) H (f) H Consider the graph of the function f shown below Use this graph to sketch the graphs of the following functions To print an enlarged cop of the graph, select the MathGraph button 0 < 0 OLIVER HEAVISIDE (850 95) f 6 (e) f (f) (g) f 8 Heaviside was a British mathematician and phsicist who contributed to the field of applied mathematics, especiall applications of mathematics to electrical engineering The Heaviside function is a classic tpe of on-off function that has applications to electricit and computer science f f f (c) f (d) f 5 A rancher plans to fence a rectangular pasture adjacent to a river The rancher has 00 meters of fence, and no fencing is needed along the river (see figure) Write the area A of the pasture as a function of, the length of the side parallel to the river What is the domain of A? Graph the area function A and estimate the dimensions that ield the maimum amount of area for the pasture (c) Find the dimensions that ield the maimum amount of area for the pasture b completing the square Figure for 5 Figure for 6 6 A rancher has 00 feet of fence to enclose two adjacent pastures Write the total area A of the two pastures as a function of (see figure) What is the domain of A? Graph the area function and estimate the dimensions tha ield the maimum amount of area for the pastures (c) Find the dimensions that ield the maimum amount of area for the pastures b completing the square 7 You are in a boat miles from the nearest point on the coast You are to go to a point Q located miles down the coast and mile inland (see figure) You can row at miles per hour and walk at miles per hour Write the total time T of the trip as a function of 8 You drive to the beach at a rate of 0 kilometers per hour On the return trip, ou drive at a rate of 60 kilometers per hour What is our average speed for the entire trip? Eplain our reasoning 9 One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point To see how this can be done consider the point, on the graph of f (, ) mi 6 mi Find the slope of the line joining, and, 9 Is the slope of the tangent line at, greater than or less than this number? Q mi