Preparation for Calculus

Transcription

1 60_0P00.qd //0 :7 PM Page Horsepower P Two tpes of race cars designed and built b NASCAR teams are short track cars and super-speedwa (long track) cars. The super-speedwa race cars are subjected to etensive testing in wind tunnels like the one shown in the photo. Short track race cars and super-speedwa race cars are designed either to allow for as much downforce as possible or to reduce the amount of drag on the race car. Which design do ou think is used for each tpe of race car? Wh? Horsepower Speed (mph) Preparation for Calculus Horsepower Speed (mph) Speed (mph) 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. In Chapter P, ou will review how to find, graph, and compare mathematical models for different data sets. Carol Anne Petrachenko/Corbis

2 60_0P0.qd //0 :6 PM Page CHAPTER P Preparation for Calculus Archive Photos Section P. 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 67. Graphs and Models Sketch the graph of an equation. Find the intercepts of a graph. Test a graph for smmetr with respect to an ais and the origin. Find the points of intersection of two graphs. Interpret mathematical models for real-life data. The Graph of an Equation In 67 the French mathematician René Descartes revolutionized the stud of mathematics b joining its two major fields algebra and geometr. With Descartes s coordinate plane, geometric concepts could be formulated analticall and algebraic concepts could be viewed graphicall. The power of this approach is such that within a centur, much of calculus had been developed. The same approach can be followed in our stud of calculus. That is, b viewing calculus from multiple perspectives graphicall, analticall, and numericall ou will increase our understanding of core concepts. Consider the equation 7. The point, is a solution point of the equation because the equation is satisfied (is true) when is substituted for and is substituted for. This equation has man other solutions, such as, and 0, 7. To find other solutions sstematicall, solve the original equation for. 7 Analtic approach Then construct a table of values b substituting several values of (0, 7) (, ) + = 7 (, ) 6 8 (, ) (, 5) Numerical approach From the table, ou can see that 0, 7,,,,,,, and, 5 are solutions of the original equation 7. Like man equations, this equation has an infinite number of solutions. The set of all solution points is the graph of the equation, as shown in Figure P.. Graphical approach: 7 Figure P. NOTE Even though we refer to the sketch shown in Figure P. as the graph of 7, it reall represents onl a portion of the graph. The entire graph would etend beond the page. In this course, ou will stud man sketching techniques. The simplest is point plotting that is, ou plot points until the basic shape of the graph seems apparent = 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.

3 60_0P0.qd //0 :6 PM Page SECTION P. Graphs and Models One disadvantage of point plotting is that to get a good idea about the shape of a graph, ou ma need to plot man points. With onl a few points, ou could badl misrepresent the graph. For instance, suppose that to sketch the graph of ou plotted onl five points:,,,, 0, 0,,, and,, as shown in Figure P.(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). (0, 0) (, ) Plotting onl a few points can misrepresent a (, ) graph. (a) Figure P. (, ) (, ) (b) = (9 0 + ) 0 EXPLORATION Comparing Graphical and Analtic Approaches Use a graphing utilit to graph each equation. In each case, find a viewing window that shows the important characteristics of the graph. a. 5 b. 5 c. 0 5 d e. f. 6 A purel graphical approach to this problem would involve a simple guess, check, and revise strateg. What tpes of things do ou think an analtic approach might involve? For instance, does the graph have smmetr? Does the graph have turns? If so, where are the? As ou proceed through Chapters,, and of this tet, ou will stud man new analtic tools that will help ou analze graphs of equations such as these. TECHNOLOGY Technolog has made sketching of graphs easier. Even with technolog, however, it is possible to misrepresent a graph badl. For instance, each of the graphing utilit screens in Figure P. shows a portion of the graph of 5. From the screen on the left, ou might assume that the graph is a line. From the screen on the right, however, ou can see that the graph is not a line. So, whether ou are sketching a graph b hand or using a graphing utilit, ou must realize that different viewing windows can produce ver different views of a graph. In choosing a viewing window, our goal is to show a view of the graph that fits well in the contet of the problem Graphing utilit screens of Figure P. 0 5 NOTE In this tet, the term graphing utilit means either a graphing calculator or computer graphing software such as Maple, Mathematica, Derive, Mathcad, or the TI

4 60_0P0.qd //0 :6 PM Page 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 EXAMPLE Finding - and -intercepts = (, 0) Intercepts of a graph Figure P.6 (0, 0) (, 0) Find the - and -intercepts of the graph of. Solution To find the -intercepts, let be zero and solve for. 0 Let be zero. 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. (See Figure P.6.) -intercept 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.

5 60_0P0.qd //0 :6 PM Page 5 SECTION P. Graphs and Models 5 Smmetr of a Graph (, ) (, ) Knowing the smmetr of a graph before attempting to sketch it is useful because ou need onl half as man points to sketch the graph. The following three tpes of smmetr can be used to help sketch the graphs of equations (see Figure P.7). -ais smmetr (, ). A graph is smmetric with respect to the -ais if, whenever, is a point on the graph,, is also a point on the graph. This means that the portion of the graph to the left of the -ais is a mirror image of the portion to the right of the -ais.. A graph is smmetric with respect to the -ais if, whenever, is a point on the graph,, is also a point on the graph. This means that the portion of the graph above the -ais is a mirror image of the portion below the -ais.. A graph is smmetric with respect to the origin if, whenever, is a point on the graph,, is also a point on the graph. This means that the graph is unchanged b a rotation of 80 about the origin. -ais smmetr (, ) Tests for Smmetr (, ) (, ) Origin smmetr. The graph of an equation in and is smmetric with respect to the -ais if replacing b ields an equivalent equation.. The graph of an equation in and is smmetric with respect to the -ais if replacing b ields an equivalent equation.. The graph of an equation in and is smmetric with respect to the origin if replacing b and b ields an equivalent equation. The graph of a polnomial has smmetr with respect to the -ais if each term has an even eponent (or is a constant). For instance, the graph of -ais smmetr Figure P.7 has smmetr with respect to the -ais. Similarl, the graph of a polnomial has smmetr with respect to the origin if each term has an odd eponent, as illustrated in Eample. EXAMPLE Testing for Origin Smmetr = Show that the graph of is smmetric with respect to the origin. (, ) Origin smmetr Figure P.8 (, ) Solution Write original equation. Replace b and b. Simplif. Equivalent equation Because the replacements ield an equivalent equation, ou can conclude that the graph of is smmetric with respect to the origin, as shown in Figure P.8.

6 60_0P0.qd //0 :6 PM Page 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 points 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 from 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 indicates that in the HM mathspace CD-ROM and the online Eduspace sstem for this tet, ou will find an Open Eploration, which further eplores this eample using the computer algebra sstems Maple, Mathcad, Mathematica, and Derive.

7 60_0P0.qd //0 :6 PM Page 7 SECTION P. Graphs and Models 7 Mathematical Models Real-life applications of mathematics often use equations as mathematical models. In developing a mathematical model to represent actual data, ou should strive for two (often conflicting) goals: accurac and simplicit. That is, ou want the model to be simple enough to be workable, et accurate enough to produce meaningful results. Section P. eplores these goals more completel. EXAMPLE 6 Comparing Two Mathematical Models Gavriel Jecan/Corbis The Mauna Loa Observator in Hawaii has been measuring the increasing concentration of carbon dioide in Earth s atmosphere since 958. 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 t 0.08t 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 00 and can be modeled b t Linear model for data where t 0 represents 960. What was the prediction given in the Scientific American article in 990? Given the new data for 990 through 00, does this prediction for the ear 05 seem accurate? CO (in parts per million) Year (0 960) t CO (in parts per million) Year (0 960) t (a) (b) Figure P. NOTE The models in Eample 6 were developed using a procedure called least squares regression (see Section.9). The quadratic and linear models have a correlation given b r and r 0.996, 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.

8 60_0P0.qd //0 :6 PM Page 8 8 CHAPTER P Preparation for Calculus In Eercises, match the equation with its graph. [Graphs are labeled (a), (b), (c), and (d).] (a) (c) Eercises for Section P. (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 See for worked-out solutions to odd-numbered eercises. In Eercises 9 6, find an intercepts In Eercises 7 8, test for smmetr with respect to each ais and to the origin In Eercises 9 56, sketch the graph of the equation. Identif an intercepts and test for smmetr In Eercises 57 60, use a graphing utilit to graph the equation. Identif an intercepts and test for smmetr In Eercises 7 and 8, use a graphing utilit to graph the equation. Move the cursor along the curve to approimate the unknown coordinate of each solution point accurate to two decimal places (a), (b), (a) 0.5, (b), In Eercises 6 68, 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.

9 60_0P0.qd //0 :6 PM Page 9 SECTION P. Graphs and Models In Eercises 69 7, use a graphing utilit to find the points of intersection of the graphs. Check our results analticall Modeling Data The table shows the Consumer Price Inde (CPI) for selected ears. (Source: Bureau of Labor Statistics) (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 0 corresponding to 970. (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 average numbers of acres per farm in the United States for selected ears. (Source: U.S. Department of Agriculture) (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 average acreage and t represents the ear, with t 0 corresponding to 950. (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 average number of acres per farm in the United States in the ear Break-Even Point Find the sales necessar to break even R C if the cost C of producing units is C 5.5 0,000 Cost equation and the revenue R for selling units is R.9. Revenue equation 76. Copper Wire The resistance in ohms of 000 feet of solid copper wire at 77 F can be approimated b the model 0, , 5 00 Year CPI Year Acreage 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 77 and 78, write an equation whose graph has the indicated propert. (There ma be more than one correct answer.) 77. The graph has intercepts at,, and The graph has intercepts at 5,, and. 79. Each table shows solution points for one of the following equations. (i) (iii) k (ii) k (iv) k Match each equation with the correct table and find Eplain our reasoning. (a) (b) 9 9 (c) k (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. True or False? In Eercises 8 8, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 8. If, is a point on a graph that is smmetric with respect to the -ais, then, is also a point on the graph. 8. If, is a point on a graph that is smmetric with respect to the -ais, then, is also a point on the graph. 8. If b ac > 0 and a 0, then the graph of a b c has two -intercepts. 8. If b ac 0 and a 0, then the graph of a b c has onl one -intercept. In Eercises 85 and 86, find an equation of the graph that consists of all points, having the given distance from the origin. (For a review of the Distance Formula, see Appendi D.) 85. The distance from the origin is twice the distance from 0,. 86. The distance from the origin is K K times the distance from, 0. (d) k.

10 60_0P0.qd //0 : PM Page 0 0 CHAPTER P Preparation for Calculus Section P. (, ) = (, ) = change in change in Figure P. Linear Models and Rates of Change Find the slope of a line passing through two points. Write the equation of a line with a given point and slope. Interpret slope as a ratio or as a rate in a real-life application. Sketch the graph of a linear equation in slope-intercept form. Write equations of lines that are parallel or perpendicular to a given line. The Slope of a Line The slope of a nonvertical line is a measure of the number of units the line rises (or falls) verticall for each unit of horizontal change from left to right. Consider the two points, and, on the line in Figure P.. As ou move from left to right along this line, a vertical change of Change in units corresponds to a horizontal change of Change in units. ( is the Greek uppercase letter delta, and the smbols and are read delta and delta. ) Definition of the Slope of a Line The slope m of the nonvertical line passing through, and, is m,. Slope is not defined for vertical lines. 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.

11 60_0P0.qd //0 : PM Page SECTION P. Linear Models and Rates of Change EXPLORATION Investigating Equations of Lines Use a graphing utilit to graph each of the linear equations. Which point is common to all seven lines? Which value in the equation determines the slope of each line? a. b. c. d. 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 = = (, ) 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 5 (See Figure P.5.) Point-slope form Substitute for, for, and for m. Simplif. Solve for. 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.

12 60_0P0.qd //0 : PM Page CHAPTER P Preparation for Calculus Ratios and Rates of Change The slope of a line can be interpreted as either a ratio or a rate. If the - and -aes have the same unit of measure, the slope has no units and is a ratio. If the - and -aes have different units of measure, the slope is a rate or rate of change. In our stud of calculus, ou will encounter applications involving both interpretations of slope. EXAMPLE Population Growth and Engineering Design Population (in millions) , Year Population of Kentuck in census ears Figure P.6 a. The population of Kentuck was,687,000 in 990 and,0,000 in 000. Over this 0-ear period, the average rate of change of the population was Rate of change If Kentuck s population continues to increase at this same rate for the net 0 ears, it will have a 00 population of,97,000 (see Figure 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,0,000,687, ,500 people per ear. 6 feet feet Rise is vertical change, run is horizontal change. 7 In this case, note that the slope is a ratio and has no units. 6 ft ft Dimensions of a water-ski ramp Figure 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 990, 000. In Chapter ou will stud another tpe of rate of change called an instantaneous rate of change.

13 60_0P0.qd //0 : PM Page SECTION P. Linear Models and Rates of Change Graphing Linear Models Man problems in analtic geometr can be classified in two basic categories: () Given a graph, what is its equation? and () Given an equation, what is its graph? The point-slope equation of a line can be used to solve problems in the first categor. However, this form is not especiall useful for solving problems in the second categor. The form that is better suited to sketching the graph of a line is the slopeintercept form of the equation of a line. The Slope-Intercept Equation of a Line The graph of the linear equation m b is a line having a slope of m and a -intercept at 0, b. 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, ) = = + = (0, ) = = (0, ) = + = 5 6 (a) m ; line rises Figure P.8 (b) m 0; line is horizontal (c) m ; line falls

14 60_0P0.qd //0 : PM Page CHAPTER P Preparation for Calculus Because the slope of a vertical line is not defined, its equation cannot be written in the slope-intercept form. However, the equation of an line can be written in the general form A B C 0 General form of the equation of a line where A and B are not both zero. For instance, the vertical line given b a can be represented b the general form a 0. Summar of Equations of Lines. General form: A B C 0,. Vertical line: a. Horizontal line: b. Point-slope form: m 5. Slope-intercept form: m b A, B 0 Parallel and Perpendicular Lines The slope of a line is a convenient tool for determining whether two lines are parallel or perpendicular, as shown in Figure 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.

15 60_0P0.qd //0 : PM Page 5 SECTION P. Linear Models and Rates of Change 5 EXAMPLE Finding Parallel and Perpendicular Lines + = = 5 (, ) = 7 Lines parallel and perpendicular to 5 Figure P.0 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.) Solution B writing the linear equation 5 in slope-intercept form, 5, ou can see that the given line has a slope of m. a. The line through, that is parallel to the given line also has a slope of. m Point-slope form Substitute. Simplif. 7 0 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 Point-slope form Substitute. Simplif. 0 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. 6 (b) Tick-mark spacing on the -ais is the same as tick-mark spacing on the -ais.

16 60_0P0.qd //0 : PM Page 6 6 CHAPTER P Preparation for Calculus Eercises for Section P. 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 7., (a) (b) (c) (d) Undefined 8., (a) (b) (c) (d) 0 In Eercises 9, plot the pair of points and find the slope of the line passing through them. 9.,, 5, 0.,,,.,,, 5.,,,..,,, 6 In Eercises 5 8, use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point Slope Slopes 7 8,, 5, Point Slope 5., m 0 6., m undefined 7., 7 m 8., m See for worked-out solutions to odd-numbered eercises. 9. 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 00 (b) m 00 (c) m 0. Modeling Data The table shows the populations (in millions) of the United States for The variable t represents the time in ears, with t 6 corresponding to 996. (Source: 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 6, find the slope and the -intercept (if possible) of the line In Eercises 7, find an equation of the line that passes through the point and has the indicated slope. Sketch the line. Point Slope Point Slope 7. 0, m 8., m undefined 9. 0, 0 m 0. 0, m 0., m., m 5

17 60_0P0.qd //0 : PM Page 7 SECTION P. Linear Models and Rates of Change 7 In Eercises, find an equation of the line that passes through the points, and sketch the line.. 0, 0,, 6. 0, 0,, 5.,, 0, 6.,,, 7., 8, 5, 0 8., 6,, 9. 5,, 5, 8 0.,,,.., 7, 0,. Find an equation of the vertical line with -intercept at.. Show that the line with intercepts a, 0 and 0, b has the following equation. a, b In Eercises 5 8, use the result of Eercise to write an equation of the line. 5. -intercept:, intercept: -intercept: 0, -intercept: 0, 7. Point on line:, 8. Point on line:, -intercept: a, 0 -intercept: a, 0 -intercept: 0, a -intercept: 0, a a 0 a 0 a 0, b 0 In Eercises 9 56, sketch a graph of the equation Square Setting In Eercises 57 and 58, use a graphing utilit to graph both lines in each viewing window. Compare the graphs. Do the lines appear perpendicular? Are the lines perpendicular? Eplain , (a) Xmin = -0 Xma = 0 Xscl = Ymin = -0 Yma = 0 Yscl = 58., (a) Xmin = -5 Xma = 5 Xscl = Ymin = -5 Yma = 5 Yscl = (b) (b) 7 8,, 5, Xmin = -5 Xma = 5 Xscl = Ymin = -0 Yma = 0 Yscl = Xmin = -6 Xma = 6 Xscl = Ymin = - Yma = Yscl =, 0 In Eercises 59 6, write an equation of the line through the point (a) parallel to the given line and (b) perpendicular to the given line. Point 59., 60., 6., , 6., 5 6., 0 Rate of Change In Eercises 65 68, ou are given the dollar value of a product in 00 and the rate at which the value of the product is epected to change during the net 5 ears. Write a linear equation that gives the dollar value V of the product in terms of the ear t. (Let t 0 represent 000.) 00 Value Line Rate Point 65. $50 $5 increase per ear 66. $56 $.50 increase per ear 67. $0,00 $000 decrease per ear 68. $5,000 $5600 decrease per ear Line 7 7 In Eercises 69 and 70, use a graphing utilit to graph the parabolas and find their points of intersection. Find an equation of the line through the points of intersection and graph the line in the same viewing window In Eercises 7 and 7, determine whether the points are collinear. (Three points are collinear if the lie on the same line.) 7.,,, 0,, 7. 0,, 7, 6, 5, Writing About Concepts In Eercises 7 75, find the coordinates of the point of intersection of the given segments. Eplain our reasoning. 7. (b, c) ( a, 0) (a, 0) Perpendicular bisectors ( a, 0) (a, 0) Altitudes (b, c) ( a, 0) (a, 0) Medians (b, c) 76. Show that the points of intersection in Eercises 7, 7, and 75 are collinear.

18 60_0P0.qd //0 : PM Page 8 8 CHAPTER P Preparation for Calculus 77. Temperature Conversion Find a linear equation that epresses the relationship between the temperature in degrees Celsius C and degrees Fahrenheit F. Use the fact that water freezes at 0 C ( F) and boils at 00 C ( F). Use the equation to convert 7 F to degrees Celsius. 78. Reimbursed Epenses A compan reimburses its sales representatives $50 per da for lodging and meals plus per mile driven. Write a linear equation giving the dail cost C to the compan in terms of, the number of miles driven. How much does it cost the compan if a sales representative drives 7 miles on a given da? 79. Career Choice An emploee has two options for positions in a large corporation. One position pas $.50 per hour plus an additional unit rate of $0.75 per unit produced. The other pas $9.0 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? 80. Straight-Line Depreciation A small business purchases a piece of equipment for $875. After 5 ears the equipment will be outdated, having no value. (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 handles an apartment comple with 50 units. When the rent is $580 per month, all 50 units are occupied. However, when the rent is $65, the average number of occupied units drops to 7. Assume that the relationship between the monthl rent p and the demand is linear. (Note: The term demand refers to the number of occupied units.) (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 $655. (c) Linear interpolation Predict the number of units occupied if the rent is lowered to $595. Verif graphicall. 8. Modeling Data An instructor gives regular 0-point quizzes and 00-point eams in a mathematics course. Average scores for si students, given as ordered pairs, where is the average quiz score and is the average test score, are 8, 87, 0, 55, 9, 96, 6, 79,, 76, and 5, 8. (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. 8. Tangent Line Find an equation of the line tangent to the circle 69 at the point 5,. 8. Tangent Line Find an equation of the line tangent to the circle 5 at the point,. Distance In Eercises 85 90, 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 85. Point: 0, Point:, Line: 0 Line: Point:, 88. Point: 6, Line: 0 Line: 89. Line: 90. Line: Line: 5 Line: 0 9. Show that the distance between the point, and the 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. 9. Prove that the diagonals of a rhombus intersect at right angles. (A rhombus is a quadrilateral with sides of equal lengths.) 9. Prove that the figure formed b connecting consecutive midpoints of the sides of an quadrilateral is a parallelogram. 95. Prove that if the points, and, lie on the same line as, and,, then. Assume and. 96. 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 97 and 98, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 97. The lines represented b a b c and b a c are perpendicular. Assume a 0 and b It is possible for two lines with positive slopes to be perpendicular to each other.

19 60_0P0.qd //0 :5 PM Page 9 SECTION P. Functions and Their Graphs 9 Section P. X Domain f Range = f() Y A real-valued function f of a real variable Figure P. Functions and Their Graphs Use function notation to represent and evaluate a function. Find the domain and range of a function. Sketch the graph of a function. Identif different tpes of transformations of functions. Classif functions and recognize combinations of functions. Functions and Function Notation A relation between two sets X and Y is a set of ordered pairs, each of the form,, where is a member of X and is a member of Y. A function from X to Y is a relation between X and Y that has the propert that an two ordered pairs with the same -value also have the same -value. The variable is the independent variable, and the variable is the dependent variable. Man real-life situations can be modeled b functions. For instance, the area A of a circle is a function of the circle s radius r. A r A is a function of r. In this case r is the independent variable and A is the dependent variable. Definition of a Real-Valued Function of a Real Variable Let X and Y be sets of real numbers. A real-valued function f of a real variable from X to Y is a correspondence that assigns to each number in X eactl one number in Y. The domain of f is the set X. The number is the image of under f and is denoted b f, which is called the value of f at. The range of f is a subset of Y and consists of all images of numbers in X (see Figure P.). FUNCTION NOTATION The word function was first used b Gottfried Wilhelm Leibniz in 69 as a term to denote an quantit connected with a curve, such as the coordinates of a point on a curve or the slope of a curve. Fort ears later, Leonhard Euler used the word function to describe an epression made up of a variable and some constants. He introduced the notation f. Functions can be specified in a variet of was. In this tet, however, we will concentrate primaril on functions that are given b equations involving the dependent and independent variables. For instance, the equation Equation in implicit form defines, the dependent variable, as a function of, the independent variable. To evaluate this function (that is, to find the -value that corresponds to a given -value), it is convenient to isolate on the left side of the equation. Equation in eplicit form Using f as the name of the function, ou can write this equation as f. Function notation The original equation,, implicitl defines as a function of. When ou solve the equation for, ou are writing the equation in eplicit form. Function notation has the advantage of clearl identifing the dependent variable as f while at the same time telling ou that is the independent variable and that the function itself is f. The smbol f is read f of. Function notation allows ou to be less word. Instead of asking What is the value of that corresponds to? ou can ask What is f?

20 60_0P0.qd //0 :5 PM Page 0 0 CHAPTER P Preparation for Calculus In an equation that defines a function, the role of the variable is simpl that of a placeholder. For instance, the function given b f can be described b the form f where parentheses are used instead of. To evaluate f, simpl place in each set of parentheses. f 8 7 Substitute for. Simplif. Simplif. NOTE Although f is often used as a convenient function name and as the independent variable, ou can use other smbols. For instance, the following equations all define the same function. f 7 f t t t 7 g s s s 7 Function name is f, independent variable is. Function name is f, independent variable is t. Function name is g, independent variable is s. EXAMPLE Evaluating a Function For the function f defined b f 7, evaluate each epression. a. f a b. f b c. f f, 0 STUDY TIP In calculus, it is important to communicate clearl the domain of a function or epression. For instance, in Eample (c) the two epressions f f 0 and, are equivalent because 0 is ecluded from the domain of each epression. Without a stated domain restriction, the two epressions would not be equivalent. Solution a. f a a 7 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.

21 60_0P0.qd //0 :5 PM Page SECTION P. Functions and Their Graphs Range: 0 f() = Domain: (a) The domain of f is, and the range is 0,. Range f() = tan π π Domain (b) The domain of f is all -values such that n and the range is,. 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 a. The domain of the function f Finding the Domain and Range of a Function is the set of all -values for which 0, which is the interval,. To find the range observe that f is never negative. So, the range is the interval 0,, as indicated in Figure P.(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, 5 n is an integer. Domain of tangent function The range of this function is the set of all real numbers. For a review of the characteristics of this and other trigonometric functions, see Appendi D. EXAMPLE A Function Defined b More than One Equation Range: 0 f() = Domain: all real, <, The domain of f is, and the range is 0,. Figure P. Determine the domain and range of the function. f,, 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.

22 60_0P0.qd //0 :5 PM Page 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 f 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 Figure P.6 (b) A function of (c) A function of 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 D.) f() = f() = f() = f() = Identit function Squaring function Cubing function Square root function f() = f() = f() = sin f() = cos π π π π π π π Absolute value function The graphs of eight basic functions Figure P.7 Rational function Sine function Cosine function

23 60_0P0.qd //0 :5 PM Page SECTION P. Functions and Their Graphs EXPLORATION Writing Equations for Functions Each of the graphing utilit screens below shows the graph of one of the eight basic functions shown on page. Each screen also shows a transformation of the graph. Describe the transformation. Then use our description to write an equation for the transformation. Transformations of Functions Some families of graphs have the same basic shape. For eample, compare the graph of with the graphs of the four other quadratic functions shown in Figure 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 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 f f f (d) Shift left, reflect, and shift upward Vertical shift up units Horizontal shift to the left units Reflection about the -ais Shift left units, reflect about -ais, and shift up unit 6 6 d. Basic Tpes of Transformations Original graph: Horizontal shift c units to the right: Horizontal shift c units to the left: Vertical shift c units downward: Vertical shift c units upward: Reflection (about the -ais): Reflection (about the -ais): Reflection (about the origin): c > 0 f f c f c f c f c f f f

24 60_0P0.qd //0 :5 PM Page 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 D. The other nonalgebraic functions, such as the inverse trigonometric functions and the eponential and logarithmic functions, are introduced in Chapter 5. The most common tpe of algebraic function is a polnomial function f a n n a n n... a a a 0, where the positive integer n is the degree of the polnomial function. The constants a i are coefficients, with a n the leading coefficient and a 0 the constant term of the polnomial function. It is common practice to use subscript notation for coefficients of general polnomial functions, but for polnomial functions of low degree, the following simpler forms are often used. Zeroth degree: f a Constant function First degree: f a b Linear function Second degree: f a b c Quadratic function Third degree: f a b c d 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 a n 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