Preparation for Calculus

Transcription

1 P Preparation for Calculus This chapter reviews several concepts that will help ou prepare for our stud of calculus. These concepts include sketching the graphs of equations and functions, and fitting mathematical models to data. It is important to review these concepts before moving on to calculus. In this chapter, ou should learn the following. How to identif the characteristics of equations and sketch their graphs. (P.) How to find and graph equations of lines, including parallel and perpendicular lines, using the concept of slope. (P.) How to evaluate and graph functions and their transformations. (P.) How to fit mathematical models to real-life data sets. (P.) Jerem Walker/Gett Images In 006, China surpassed the United States as the world s biggest emitter of carbon dioide, the main greenhouse gas. Given the carbon dioide concentrations in the atmosphere for several ears, can older mathematical models still accuratel predict future atmospheric concentrations compared with more recent models? (See Section P., Eample 6.) Mathematical models are commonl used to describe data sets. These models can be represented b man different tpes of functions, such as linear, quadratic, cubic, rational, and trigonometric functions. (See Section P..) Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

2 Chapter P Preparation for Calculus P. 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. Archive Photos RENÉ DESCARTES ( ) 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 (0, 7) (, ) + = 7 (, ) 6 8 (, ) (, 5) 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 was such that within a centur of its introduction, 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 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.. 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. Graphical approach: Figure P. 7 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 = 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 The parabola Figure P. 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. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

3 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.(a). 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.(b). 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. (, ) (0, 0) (, ) Plotting onl a few points can misrepresent a (, ) graph. (a) Figure P. (, ) (b) = (9 0 + ) 0 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 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, or the TI Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

4 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 P.5. No -intercepts One -intercept Figure P.5 Three -intercepts One -intercept One -intercept Two -intercepts No intercepts = (, 0) Intercepts of a graph Figure P.6 (0, 0) (, 0) EXAMPLE Finding - and -intercepts Find the - and -intercepts of the graph of. Solution To find the -intercepts, let be zero and solve for. 0 Let be zero. 0 Factor. 0,, or 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. -intercept (See Figure P.6.) 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. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

5 P. Graphs and Models 5 Smmetr of a Graph (, ) (, ) -ais smmetr (, ) 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 P.7).. 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 (, ) Figure P.7 (, ) (, ) Origin smmetr TESTS FOR SYMMETRY. 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 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 Smmetr Test the graph of for smmetr with respect to the -ais and to the origin. (, ) Origin smmetr Figure P.8 = (, ) Solution -ais Smmetr: Write original equation. Replace b. Simplif. It is not an equivalent equation. Origin Smmetr: Write original equation. Replace b and b. Simplif. Equivalent equation Because replacing both b and b ields an equivalent equation, ou can conclude that the graph of is smmetric with respect to the origin, as shown in Figure P.8. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

6 6 Chapter P Preparation for Calculus EXAMPLE Using Intercepts and Smmetr to Sketch a Graph Sketch the graph of. = (, ) (, 0) -intercept Figure P.9 (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 P.9. 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. Top portion of graph Bottom portion of graph Points of Intersection A point of intersection of the graphs of two equations is a point that satisfies both equations. You can find the point(s) of intersection of two graphs b solving their equations simultaneousl. EXAMPLE 5 Finding Points of Intersection Find all points of intersection of the graphs of and. = (, ) Two points of intersection Figure P.0 (, ) = STUDY TIP You can check the points of intersection in Eample 5 b substituting into both of the original equations or b using the intersect feature of a graphing utilit. Solution Begin b sketching the graphs of both equations on the same rectangular coordinate sstem, as shown in Figure P.0. Having done this, it appears that the graphs have two points of intersection. You can find these two points, as follows. Solve first equation for. Solve second equation for. Equate -values. 0 Write in general form. 0 Factor. or Solve for. The corresponding values of are obtained b substituting and into either of the original equations. Doing this produces two points of intersection:, and,. Points of intersection The icon indicates that ou will find a CAS Investigation on the book s website. The CAS Investigation is a collaborative eploration of this eample using the computer algebra sstems Maple and Mathematica. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

7 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 JG Photograph/Alam The Mauna Loa Observator in Hawaii has been measuring the increasing concentration of carbon dioide in Earth s atmosphere since 958. Carbon dioide is the main greenhouse gas responsible for global climate warming. 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 Quadratic model for data where t 0 represents 960, as shown in Figure P.(a). The data shown in Figure P.(b) represent the ears 980 through 007 and can be modeled b 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 007, does this prediction for the ear 05 seem accurate? CO (in parts per million) t 0.08t 0..6t Year (0 960) t CO (in parts per million) Year (0 960) t (a) Figure P. (b) NOTE The models in Eample 6 were developed using a procedure called least squares regression (see Section.9). The quadratic and linear models have correlations given b r and r 0.99, 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. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

8 8 Chapter P Preparation for Calculus P. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) (b) (d) In Eercises 5, sketch the graph of the equation b point plotting In Eercises 5 and 6, describe the viewing window that ields the figure In Eercises 9 8, find an intercepts In Eercises 9 0, test for smmetr with respect to each ais and to the origin In Eercises 58, sketch the graph of the equation. Identif an intercepts and test for smmetr In Eercises 59 6, 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 (a), (b), (a) 0.5, (b), In Eercises 6 70, find the points of intersection of the graphs of the equations The smbol indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. The solutions of other eercises ma also be facilitated b use of appropriate technolog. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

9 P. Graphs and Models In Eercises 7 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) Year CPI where is the diameter of the wire in mils (0.00 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 79 and 80, write an equation whose graph has the indicated propert. (There ma be more than one correct answer.) 79. The graph has intercepts at,, and The graph has intercepts at,, and (a) 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. (b) 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. (a) 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 5 corresponding to 975. (b) 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 Modeling Data The table shows the numbers of cellular phone subscribers (in millions) in the United States for selected ears. (Source: Cellular Telecommunications and Internet Association) Year Number (a) 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 number of subscribers and t represents the ear, with t 0 corresponding to 990. (b) 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 number of cellular phone subscribers 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 5.5 0,000 Cost equation and the revenue R from selling units is R.9. Revenue equation 78. Copper Wire The resistance in ohms of 000 feet of solid copper wire at 77 F can be approimated b the model 0, , 5 00 CAPSTONE 8. Match the equation or equations with the given characteristic. (i) (ii) (iii) (iv) (v) (vi) (a) Smmetric with respect to the -ais (b) Three -intercepts (c) Smmetric with respect to the -ais (d), is a point on the graph (e) Smmetric with respect to the origin (f) Graph passes through the origin True or False? In Eercises 8 86, 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, 5 is a point on a graph that is smmetric with respect to the -ais, then, 5 is also a point on the graph. 8. If, 5 is a point on a graph that is smmetric with respect to the -ais, then, 5 is also a point on the graph. 85. If b ac > 0 and a 0, then the graph of a b c has two -intercepts. 86. If b ac 0 and a 0, then the graph of a b c has onl one -intercept. In Eercises 87 and 88, find an equation of the graph that consists of all points, having the given distance from the origin. (For a review of the Distance Formula, see Appendi C.) 87. The distance from the origin is twice the distance from 0,. 88. The distance from the origin is K K times the distance from, 0. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

10 0 Chapter P Preparation for Calculus 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. (, ) Δ = (, ) Δ = change in change in Figure P. 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. 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. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

11 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. 0 e. f. g. 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 Δ = Δ = (, ) The line with a slope of passing through the point, Figure P.5 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 Point-slope form Substitute for, for, and for m. Simplif. 5 Solve for. (See Figure P.5.) 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. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

12 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. Population (in millions) Year Population of Colorado Figure P.6 88,000 EXAMPLE Population Growth and Engineering Design a. The population of Colorado was,87,000 in 995 and,665,000 in 005. Over this 0-ear period, the average rate of change of the population was Rate of change If Colorado s population continues to increase at this same rate for the net 0 ears, it will have a 05 population of 5,50,000 (see Figure P.6). (Source: U.S. 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 P.7. 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,665,000,87, ,800 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 P.7 The rate of change found in Eample (a) is an average rate of change. An average rate of change is alwas calculated over an interval. In this case, the interval is 995, 005. In Chapter ou will stud another tpe of rate of change called an instantaneous rate of change. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

13 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. EXAMPLE 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 P.8(a). b. Because b, the -intercept is 0,. Because the slope is m 0, ou know that the line is horizontal, as shown in Figure P.8(b). 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 P.8(c). (0, ) Δ = = + Δ = (0, ) = (0, ) Δ = Δ = = (a) m ; line rises Figure P.8 (b) m 0; line is horizontal (c) m ; line falls Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

14 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. SUMMARY 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 P.9. 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 P.9 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. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

15 P. Linear Models and Rates of Change 5 + = = 5 (, ) = 7 Lines parallel and perpendicular to 5 Figure P.0 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 P.0.) 5, ou can see that the given line has a slope of m. Solution B writing the linear equation 5 in slope-intercept form, a. The line through, that is parallel to the given line also has a slope of. m 7 0 Point-slope form Substitute. Simplif. General form Note the similarit to the original equation. 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 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.(a) and P.(b) both show the lines given b and. Because these lines have slopes that are negative reciprocals, the must be perpendicular. In Figure P.(a), 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.(b), 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 (a) Tick-mark spacing on the -ais is not the same as tick-mark spacing on the -ais. Figure P. (b) Tick-mark spacing on the -ais is the same as tick-mark spacing on the -ais. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

16 6 Chapter P Preparation for Calculus P. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 6, estimate the slope of the line from its graph. To print an enlarged cop of the graph, go to the website 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 Slopes 7., (a) (b) (c) (d) Undefined 8., 5 (a) (b) (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 ,, 5, 0.,,, 7., 6,,..., 5, 5, 5,,, 6 7 8,, 5, Slope Point Slope 5. 6, m 0 6., m is undefined. 7., 7 m 8., m Conveor Design A moving conveor is built to rise meter for each meters of horizontal change. (a) Find the slope of the conveor. (b) 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. (a) m 800 (b) m 50 (c) m 0. Modeling Data The table shows the populations (in millions) of the United States for 000 through 005. The variable t represents the time in ears, with t 0 corresponding to 000. (Source: U.S. Bureau of the Census) t (a) Plot the data b hand and connect adjacent points with a line segment. (b) 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 (a) Plot the data b hand and connect adjacent points with a line segment. (b) Use the slope of each line segment to determine the interval when the vehicle s rate changed most rapidl. How did the rate change? In Eercises 8, find the slope and the -intercept (if possible) of the line In Eercises 9, find an equation of the line that passes through the point and has the indicated slope. Sketch the line. Point Slope Point Slope 9. 0, m 0. 5, m is undefined.. 0, 0 m. 0, m 0., m., m 5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

17 P. Linear Models and Rates of Change 7 In Eercises 5, find an equation of the line that passes through the points, and sketch the line. 5. 0, 0,, , 0,, 5 7.,, 0, 8.,,, 7 9., 8, 5, 0 0., 6,,. 6,, 6, 8.,,,.., 7, 0, 5. Find an equation of the vertical line with -intercept at. 6. Show that the line with intercepts a, 0 and 0, b has the following equation. a 0, b 0 a, b In Eercises 7 50, use the result of Eercise 6 to write an equation of the line in general form. 7. -intercept:, intercept: -intercept: 0, -intercept: 0, 9. Point on line:, 50. Point on line:, -intercept: a, 0 -intercept: a, 0 -intercept: 0, a -intercept: 0, a a 0 a 0 7 8,, 5, In Eercises 5 58, sketch a graph of the equation., Square Setting Use a graphing utilit to graph the lines and in each viewing window. Compare the graphs. Do the lines appear perpendicular? Are the lines perpendicular? Eplain. (a) Xmin = -5 (b) Xmin = -6 Xma = 5 Xma = 6 Xscl = Xscl = Ymin = -5 Ymin = - Yma = 5 Yma = Yscl = Yscl = CAPSTONE 60. A line is represented b the equation a b. (a) When is the line parallel to the -ais? (b) When is the line parallel to the -ais? 5 (c) Give values for a and b such that the line has a slope of 8. (d) Give values for a and b such that the line is perpendicular to 5. (e) Give values for a and b such that the line coincides with the graph of In Eercises 6 66, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point 6. 7, 6., 0 6., 6., , 5, 7 8 Rate of Change In Eercises 67 70, ou are given the dollar value of a product in 008 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.) 008 Value Line Rate 67. $850 $50 increase per ear 68. $56 $.50 increase per ear 69. $7,00 $600 decrease per ear 70. $5,000 $5600 decrease per ear In Eercises 7 and 7, 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 Point Line 7 7 In Eercises 75 77, find the coordinates of the point of intersection of the given segments. Eplain our reasoning. 75. (b, c) ( a, 0) (a, 0) Perpendicular bisectors ( a, 0) (a, 0) Altitudes (b, c) ( a, 0) (a, 0) Medians (b, c) 78. Show that the points of intersection in Eercises 75, 76, and 77 are collinear. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

18 8 Chapter P Preparation for Calculus 79. 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. 80. Reimbursed Epenses A compan reimburses its sales representatives $75 per da for lodging and meals plus 8 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? 8. 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 $0.75 per unit produced. The other pas $.0 per hour plus a unit rate of $.0. (a) Find linear equations for the hourl wages W in terms of, the number of units produced per hour, for each option. (b) 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 (b). How would ou use this information to select the correct option if the goal were to obtain the highest hourl wage? 8. Straight-Line Depreciation A small business purchases a piece of equipment for $875. After 5 ears the equipment will be outdated, having no value. (a) Write a linear equation giving the value of the equipment in terms of the time, 0 5. (b) Find the value of the equipment when. (c) Estimate (to two-decimal-place accurac) the time when the value of the equipment is $ Apartment Rental A real estate office manages an apartment comple with 50 units. When the rent is $780 per month, all 50 units are occupied. However, when the rent is $85, 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.) (a) Write a linear equation giving the demand in terms of the rent p. (b) 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 $855. (c) Linear interpolation Predict the number of units occupied if the rent is lowered to $795. 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. (a) Use the regression capabilities of a graphing utilit to find the least squares regression line for the data. (b) 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 score 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 everone in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line. 85. Tangent Line Find an equation of the line tangent to the circle 69 at the point 5,. 86. Tangent Line Find an equation of the line tangent to the circle 5 at the point,. Distance In Eercises 87 9, find the distance between the point and line, or between the lines, using the formula for the distance between the point, and the line A B C 0. Distance A B C A B 87. Point: 0, Point:, Line: 0 Line: Point:, 90. Point: 6, Line: 0 Line: 9. Line: 9. Line: Line: 5 Line: 0 9. Show that the distance between the point, and the line A B C 0 is Distance A B C A B. 9. Write the distance d between the point, and the line m in terms of m. Use a graphing utilit to graph the equation. When is the distance 0? Eplain the result geometricall. 95. Prove that the diagonals of a rhombus intersect at right angles. (A rhombus is a quadrilateral with sides of equal lengths.) 96. Prove that the figure formed b connecting consecutive midpoints of the sides of an quadrilateral is a parallelogram. 97. Prove that if the points, and, lie on the same line as, and,, then. Assume and. 98. Prove that if the slopes of two nonvertical lines are negative reciprocals of each other, then the lines are perpendicular. True or False? In Eercises 99 and 00, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 99. The lines represented b a b c and b a c are perpendicular. Assume a 0 and b It is possible for two lines with positive slopes to be perpendicular to each other. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

19 P. Functions and Their Graphs 9 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. X Domain f Range = f() A real-valued function f of a real variable Figure P. Y 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? Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

20 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 STUDY TIP In calculus, it is important to specif 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. For the function f defined b f 7, evaluate each epression. a. f a b. f b c. f f, 0 Solution a. f a a 7 9a 7 Substitute a for. Simplif. b. f b b 7 b b 7 b b 8 Substitute b for. Epand binomial. Simplif. c. f f , 0 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. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

21 P. Functions and Their Graphs Range: 0 f() = Domain: (a) The domain of f is, and the range is 0,. Range π π Domain (b) The domain of f is all -values such that n and the range is,. Figure P. Range: 0 f() = tan f() =, <, Domain: all real The domain of f is, and the range is 0,. Figure P. 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 Finding the Domain and Range of a Function a. The domain of the function f 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.(a). b. The domain of the tangent function, as shown in Figure P.(b), f tan is the set of all -values such that n, 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 C. EXAMPLE A Function Defined b More than One Equation Determine the domain and range of the function. f,, 5 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 (a). For <, the values of are positive. So, the range of the function is the interval 0,. (See Figure P..) 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 (a) is one-to-one, whereas the functions given in Eamples (b) and are not one-to-one. A function from X to Y is onto if its range consists of all of Y. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

22 Chapter P Preparation for Calculus = f() (, f()) f() The graph of a function Figure P.5 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 P.5, 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 if and onl if no vertical line intersects the graph at more than one point. For eample, in Figure P.6(a), ou can see that the graph does not define as a function of because a vertical line intersects the graph twice, whereas in Figures P.6(b) and (c), the graphs do define as a function of. (a) Not a function of (b) A function of (c) A function of Figure P.6 Figure P.7 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 C.) 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 P.7 Rational function Sine function Cosine function Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

23 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 P.8. = + 9 = = ( + ) = (a) Vertical shift upward (b) Horizontal shift to the left 9 9 a. 6 6 = = = ( + ) = 5 b. c (c) Reflection Figure P.8 (d) Shift left, reflect, and shift upward Each of the graphs in Figure P.8 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 P.8, the transformations shown can be represented b the following equations. f Vertical shift up units f Horizontal shift to the left units f Reflection about the -ais f Shift left units, reflect about -ais, and shift up unit 6 6 d. BASIC TYPES OF TRANSFORMATIONS c > 0 Original graph: Horizontal shift c units to the right: f f c Horizontal shift c units to the left: f c Vertical shift c units downward: f c Vertical shift c units upward: f c Reflection (about the -ais): f Reflection (about the -ais): f Reflection (about the origin): f Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

24 Chapter P Preparation for Calculus Bettmann/Corbis 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. 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. To view this article, go to the website 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 C. 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 where n is a nonnegative integer. The numbers a i are coefficients, with a n the leading coefficient and a 0 the constant term of the polnomial function. If a n 0, then n is the degree of the polnomial function. The zero polnomial f 0 is not assigned a degree. 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. Note that a 0. Zeroth degree: First degree: Second degree: Third degree: f a n n a n n... a a a 0 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 P.9. 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 P.9 Graphs of polnomial functions of odd degree Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

25 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 g f g g() Domain of f f f(g()) The domain of the composite function f g Figure P.0 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 Sum f g f g Difference fg f g Product f g f Quotient g You can combine two functions in et another wa, called composition. The resulting function is called a composite function. 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 P.0). 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 Definition of g f g Substitute for f. cos Definition of g Note that f g g f. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

26 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. f() = (, 0) (, 0) (0, 0) (a) Odd function π (b) Even function Figure P. g() = + cos π π π EXAMPLE 5 Even and Odd Functions and Zeros of Functions 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 Let f 0. 0 Factor. 0,, Zeros of f See Figure P.(a). b. This function is even because g cos cos g. cos cos The zeros of g are found as shown. cos 0 Let g 0. cos Subtract from each side. n, n is an integer. Zeros of g See Figure P.(b). NOTE Each of the functions in Eample 5 is either even or odd. However, some functions, such as f, are neither even nor odd. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

27 P. Functions and Their Graphs 7 P. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises and, use the graphs of f and g to answer the following. (a) Identif the domains and ranges of f and g. (b) 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... g In Eercises, evaluate (if possible) the function at the given value(s) of the independent variable. Simplif the results.. f 7. f 5 (a) f 0 (a) f (b) f (b) f (c) f b (c) f 8 (d) f (d) f 5. g 5 6. g (a) g 0 (a) g (b) g 5 (b) g (c) g (c) g c (d) g t (d) g t 7. f cos 8. f sin (a) f 0 (a) f (b) f (b) f 5 (c) f (c) f 9. f 0. f f f f f f. f. f f f f f In Eercises 0, find the domain and range of the function.. f. g 5 5. g 6 6. h t 7. f t sec 8. h t cot t f g 9. f 0. In Eercises 6, find the domain of the function.. f. f. g. h cos sin 5. f 6. g In Eercises 7 0, evaluate the function as indicated. Determine its domain and range f,, (a) f (b) f 0 (c) f (d) f t f,, < 0 0 > g (a) f (b) f 0 (c) f (d) f s 9. f, <, (a) f (b) f (c) f (d) f b, 0. f 5 5, > 5 (a) f (b) f 0 (c) f 5 (d) f 0 In Eercises 8, sketch a graph of the function and find its domain and range. Use a graphing utilit to verif our graph.. f. g. h 6. f 5. f 9 6. f 7. g t sin t 8. h 5 cos WRITING ABOUT CONCEPTS 9. 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 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

28 8 Chapter P Preparation for Calculus WRITING ABOUT CONCEPTS (continued) 0. 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, use the Vertical Line Test to determine whether is a function of. To print an enlarged cop of the graph, go to the website , 0.., > 0 In Eercises 5 8, determine whether is a function of In Eercises 9 5, use the graph of f to match the function with its graph. 55. Use the graph of f shown in the figure to sketch the graph of each function. To print an enlarged cop of the graph, go to the website (a) f (b) f (c) f (d) f (e) f (f) f 6 9 f 56. Use the graph of f shown in the figure to sketch the graph of each function. To print an enlarged cop of the graph, go to the website (a) f (b) f (, ) (c) f (d) f (e) f (f) f 6 f 57. Use the graph of f to sketch the graph of each function. In each case, describe the transformation. (a) (b) (c) 58. Specif a sequence of transformations that will ield each graph of h from the graph of the function f sin. (a) h sin (b) h sin 59. Given f and g, evaluate each epression. (a) f g (b) g f (c) g f 0 (d) f g (e) f g (f) g f 60. Given f sin and g, evaluate each epression. (a) f g (b) (c) g f 0 f g (, ) (d) g f (e) f g (f) g f In Eercises 6 6, find the composite functions f g and g f. What is the domain of each composite function? Are the two composite functions equal? 7 5 e d 6 5 f() g 6. f 6. g 6. f 6. g f g cos f g c 5 9. f f 5 5. f 5. f 5. f 6 5. f b a 65. Use the graphs of f and g to evaluate each epression. If the result is undefined, eplain wh. (a) f g (b) g f (c) g f 5 (d) f g (e) g f (f) f g f g Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

29 P. Functions and Their Graphs 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 0.6t, 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. Think About It In Eercises 67 and 68, F f g h. Identif functions for f, g, and h. (There are man correct answers.) 67. F 68. F sin In Eercises 69 7, determine whether the function is even, odd, or neither. Use a graphing utilit to verif our result. 69. f 70. f 7. f cos 7. f sin 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 (a) even and (b) odd , 75. The graphs of f, g, and h are shown in the figure. Decide whether each function is even, odd, or neither. f h g Figure for 75 Figure for The domain of the function f shown in the figure is 6 6. (a) Complete the graph of f given that f is even. (b) Complete the graph of f given that f is odd. Writing Functions In Eercises 77 80, write an equation for a function that has the given graph. 77. Line segment connecting, and 0, Line segment connecting, and 5, The bottom half of the parabola The bottom half of the circle 6 In Eercises 8 8, sketch a possible graph of the situation. 8. The speed of an airplane as a function of time during a 5-hour flight 8. The height of a baseball as a function of horizontal distance during a home run 8. The amount of a certain brand of sneaker sold b a sporting goods store as a function of the price of the sneaker f, The value of a new car as a function of time over a period of 8 ears 85. Find the value of c such that the domain of f c is 5, Find all values of c such that the domain of f is the set of all real numbers. 87. Graphical Reasoning An electronicall controlled thermostat is programmed to lower the temperature during the night automaticall (see figure). The temperature T in degrees Celsius is given in terms of t, the time in hours on a -hour clock. 0 6 T c (a) Approimate T and T 5. (b) 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. CAPSTONE 88. Water runs into a vase of height 0 centimeters at a constant 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). d 0 cm (a) Eplain wh d is a function of t. (b) Determine the domain and range of the function. (c) Sketch a possible graph of the function. (d) Use the graph in part (c) to approimate d. What does this represent? t Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

30 0 Chapter P Preparation for Calculus 89. Modeling Data The table shows the average numbers of acres per farm in the United States for selected ears. (Source: U.S. Department of Agriculture) Year Acreage (a) Plot the data, where A is the acreage and t is the time in ears, with t 5 corresponding to 955. Sketch a freehand curve that approimates the data. (b) Use the curve in part (a) to approimate A 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. (a) Use a graphing utilit to graph H. (b) Rewrite the power function so that represents the speed in kilometers per hour. Find H Think About It Write the function f without using absolute value signs. (For a review of absolute value, see Appendi C.) 9. 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. 9. Prove that the function is odd. f a n n... a a 9. Prove that the function is even f a n n a n n... a a Prove that the product of two even (or two odd) functions is even. 96. Prove that the product of an odd function and an even function is odd. 97. 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). (b) Use a graphing utilit to graph the volume function and approimate the dimensions of the bo that ield a maimum volume. (c) Use the table feature of a graphing utilit to verif our answer in part (b). (The first two rows of the table are shown.) 98. 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. Height, (0, ) (, ) Length and Width (, 0) True or False? In Eercises 99 0, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 99. If f a f b, then a b. 00. A vertical line can intersect the graph of a function at most once. 0. If f f for all in the domain of f, then the graph of f is smmetric with respect to the -ais. 0. If f is a function, then f a af. Volume, V PUTNAM EXAM CHALLENGE 0. Let R be the region consisting of the points, of the Cartesian plane satisfing both and. Sketch the region R and find its area. 0. Consider a polnomial f with real coefficients having the propert f g g f for ever polnomial g with real 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. (a) Write the volume V as a function of, the length of the corner squares. What is the domain of the function? Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

31 P. Fitting Models to Data P. 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. 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 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. 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. 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 0.97, 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. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

32 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 9.8 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 feet. The height of the basketball is recorded times at intervals of about 0.0 second.* The results are shown in the table. 5 Time Height Time Height Time Height Time Height Find a model to fit these data. Then use the model to predict the time when the basketball will hit the ground. Height (in feet) 6 5 s Scatter plot of data Figure P. t 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 5.5t.0t 5.0. 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 t.0t 5.0 Let s 0. t.0 ± Quadratic Formula 5.5 t 0.5 Choose positive solution. The solution is about 0.5 second. In other words, the basketball will continue to fall for about 0. second more before hitting the ground. * Data were collected with a Teas Instruments CBL (Calculator-Based Laborator) Sstem. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

33 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. Dalight (in minutes) d Graph of model Figure P Da (0 December ) NOTE For a review of trigonometric functions, see Appendi C. t 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 The number of hours of dalight on a given da 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 t d 78 7 sin 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, a weather almanac was used 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 Dalight Given b Model 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. * Tet from Dilwn Edwards and Mike Hamson, Guide to Mathematical Modelling (Boca Raton: CRC Press, 990), p.. Used b permission of the authors. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

34 Chapter P Preparation for Calculus P. Eercises See for worked-out solutions to odd-numbered eercises. 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, go to the website F d (a) Use the regression capabilities of a graphing utilit to find a linear model for the data. (b) 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 0 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, 6.7,.6, 0.7,.85, 65.5,.65, 0.7, 7., 8.0, 9.5,. (a) Plot the data. From the graph, do the data appear to be approimatel linear? (b) 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 (a) Plot the data. From the graph, does the relationship between consecutive scores appear to be approimatel linear? (b) 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. (a) Use the regression capabilities of a graphing utilit to find a linear model for the data. (b) 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 energ consumptions (in millions of Btu) and the per capita gross national products (in thousands of U.S. dollars) for several countries in 00. (Source: U.S. Census Bureau) Argentina (7,.5) Chile (75, 0.6) Greece (6,.) Hungar (06, 5.8) Meico (6, 9.6) Portugal (06, 9.) Spain (59,.75) United Kingdom (67,.) Bangladesh (5,.97) Ecuador (9,.77) Hong Kong (59,.56) India (5,.) Poland (95,.7) South Korea (86, 0.5) Turke (5, 7.7) Venezuela (5, 5.8) (a) Use the regression capabilities of a graphing utilit to find a linear model for the data. What is the correlation coefficient? (b) Use a graphing utilit to plot the data and graph the model. (c) Interpret the graph in part (b). Use the graph to identif the four countries that differ most from the linear model. (d) Delete the data for the four countries identified in part (c). Fit a linear model to the remaining data and give the correlation coefficient. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

35 P. Fitting Models to Data 5 0. Brinell Hardness The data in the table show the Brinell hardness H of 0.5 carbon steel when hardened and tempered at temperature t (degrees Fahrenheit). (Source: Standard Handbook for Mechanical Engineers) t H (a) Use the regression capabilities of a graphing utilit to find a linear model for the data. (b) 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 of operating an automobile in the United States for several recent ears. The functions,, and represent the costs in cents per mile for gas, maintenance, and tires, respectivel. (Source: Bureau of Transportation Statistics) Year (a) Use the regression capabilities of a graphing utilit to find cubic models for and and a linear model for. (b) 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 6,50,860 (a) Use the regression capabilities of a graphing utilit to fit a quadratic model to the data. (b) Use a graphing utilit to plot the data and graph the model. (c) Use the model to approimate the breaking strength when.. Car Performance The time t (in seconds) required to attain a speed of s miles per hour from a standing start for a Honda Accord Hbrid is shown in the table. (Source: Car & Driver) (a) Use the regression capabilities of a graphing utilit to find a quadratic model for the data. (b) Use a graphing utilit to plot the data and graph the model. (c) Use the graph in part (b) to state wh the model is not appropriate for determining the times required to attain speeds of less than 0 miles per hour. (d) Because the test began from a standing start, add the point 0, 0 to the data. Fit a quadratic model to the revised data and graph the new model. (e) Does the quadratic model in part (d) more accuratel model the behavior of the car? Eplain. CAPSTONE. 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: HealthLeaders-InterStud) Enrollment (in millions) s t N HMO Enrollment Year (0 990) (a) 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. (b) Use a graphing utilit to graph the data and the linear and cubic models. (c) Use the graphs in part (b) to determine which is the better model. (d) Use a graphing utilit to find and graph a quadratic model for the data. How well does the model fit the data? Eplain our reasoning. (e) Use the linear and cubic models to estimate the number of people receiving care in HMOs in the ear 007. What do ou notice? (f) Use a graphing utilit to find other models for the data. Which models do ou think best represent the data? Eplain. t Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

36 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 (a) Use the regression capabilities of a graphing utilit to find a cubic model for the data. (b) 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 (a) Use the regression capabilities of a graphing utilit to find a cubic model for the data. (b) 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 accurate 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. (a) Is a function of t? Eplain. (b) 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. (0.5,.5) (0.75,.65) t 8. Temperature The table shows the normal dail high temperatures for Miami M and Sracuse S (in degrees Fahrenheit) for month t, with t corresponding to Januar. (Source: NOAA) t 5 6 M S t M S (a) A model for Miami is M t sin 0.9t.95. Find a model for Sracuse. (b) Use a graphing utilit to graph the data and the model for the temperatures in Miami. How well does the model fit? (c) Use a graphing utilit to graph the data and the model for the temperatures in Sracuse. How well does the model fit? (d) Use the models to estimate the average annual temperature in each cit. Which 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 in temperature throughout the ear? Which factor of the models determines this variabilit? Eplain. WRITING ABOUT CONCEPTS In Eercises 9 and 0, describe a possible real-life situation for each data set. Then describe how a model could be used in the real-life setting PUTNAM EXAM CHALLENGE. For i, let T i be a triangle with side lengths a i, b i, c i, and area A i. Suppose that a a, b b, c c, and that T is an acute triangle. Does it follow that A A? This problem was composed b the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

37 Review Eercises 7 P REVIEW EXERCISES See for worked-out solutions to odd-numbered eercises. 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? (a), (b), (c) 0, 0 (d), In Eercises and, plot the points and find the slope of the line passing through the points.,, 5, , 8,, 8 In Eercises and, use the concept of slope to find t such that the three points are collinear.. 8, 5, 0, t,,.,, t,, 8, 6 In Eercises 5 8, find an equation of the line that passes through the point with the indicated slope. Sketch the line. 5., 5, m ,, m is undefined. 7., 0, m 8. 5,, m 0 9. Find equations of the lines passing through, 5 and having the following characteristics. 7 (a) Slope of 6 (b) 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. (a) Slope of (b) Perpendicular to the line 0 (c) Passing through the point 6, (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 $9.5 per hour for fuel and maintenance. The equipment operator is paid $.50 per hour, and customers are charged $0 per hour. (a) Write an equation for the cost C of operating this equipment for t hours. (b) 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 f (a) f 0 (b) 8. Evaluate (if possible) the function at each value of the independent variable. f, < 0, 0 (a) f (b) f 0 (c) f 9. Find the domain and range of each function. (a) (b) (c), < , 0 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

38 8 Chapter P Preparation for Calculus 0. Given f and g, evaluate each epression. (a) f g (b) f g (c) g f. Sketch (on the same set of coordinate aes) graphs of f for c, 0, and. (a) f c (b) f c (c) f c (d) f 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, go to the website (a) 6 (b) (, 5) (, ) g 6 (0, ) g (, ). Conjecture (a) 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: f, g, h 5 Even powers: f, g, h 6 (b) Use the result in part (a) 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 results 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. (a) f 6 (b) g 6 (c) 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. (a) Write the area A of the rectangle as a function of. (b) 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. (a) (b) 00,000 P 00,000 P 7. Think About It What is the minimum degree of the polnomial function whose graph approimates the given graph? What sign must the leading coefficient have? (a) (b) (c) 8. Stress Test A machine part was tested b bending it centimeters 0 times per minute until the time (in hours) of failure. The results are recorded in the table. (a) Use the regression capabilities of a graphing utilit to find a linear model for the data. (b) 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 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 feet and the time t is measured in seconds. (a) Is a function of t? Eplain. (b) 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 (., 0.5) (d) ,000 p 50,000 p (0.5, 0.5).0.0 t Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

39 P.S. Problem Solving 9 P.S. PROBLEM SOLVING. Consider the circle 6 8 0, as shown in the figure. (a) Find the center and radius of the circle. (b) 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? 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 at eactl one point.. The Heaviside function H is widel used in engineering applications., 0 H 0, < 0 Sketch the graph of the Heaviside function and the graphs of the following functions b hand. (a) H (b) H (c) H (d) H (e) H (f) H Institute of Electrical Engineers, London 8 6 OLIVER HEAVISIDE (850 95) 6 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.. 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, go to the website (a) f (b) f (c) f (d) f (e) f (f ) f f (g) f 5. A rancher plans to fence a rectangular pasture adjacent to a river. The rancher has 00 meters of fencing, and no fencing is needed along the river (see figure). (a) 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? (b) 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 fencing to enclose two adjacent pastures. (a) Write the total area A of the two pastures as a function of (see figure). What is the domain of A? (b) Graph the area function and estimate the dimensions that 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. mi mi mi Q Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.