60_0P0.qd //0 :6 PM Page CHAPTER P Preparation for Calculus Archive Photos Section P. RENÉ DESCARTES (96 60) 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. 8 6 6 (0, 7) (, ) + = 7 (, ) 6 8 (, ) (, ) 0 7 Numerical approach From the table, ou can see that 0, 7,,,,,,, and, 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. 7 6 = 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.
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 0 9 0 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. b. c. 0 d. 0 0 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. 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. 0 0 0 Graphing utilit screens of Figure P. 0 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-89.
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.. No -intercepts One -intercept Figure P. 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.
60_0P0.qd //0 :6 PM Page SECTION P. Graphs and Models 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.
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 (, ) 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 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 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.
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 98. 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 0, using the quadratic model 6. 0.70t 0.08t Quadratic model for 960 990 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 06..6t Linear model for 980 00 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 0 seem accurate? CO (in parts per million) 7 70 6 60 0 0 0 0 0 0 0 0 Year (0 960) t CO (in parts per million) 7 70 6 60 0 0 0 0 0 0 0 0 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 0.997 and r 0.996, respectivel. The closer r is to, the better the model. Solution To answer the first question, substitute t 7 (for 0) into the quadratic model. 6. 0.70 7 0.08 7 69.9 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 0. Using the linear model for the 980 00 data, the prediction for the ear 0 is 06..6 7.. Linear model So, based on the linear model for 980 00, it appears that the 990 prediction was too high.
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).. 9.. In Eercises, sketch the graph of the equation b point plotting.. 6. 6 7. 8. 9. 0..... In Eercises and 6, describe the viewing window that ields the figure.. 6. 0 See www.calcchat.com for worked-out solutions to odd-numbered eercises. In Eercises 9 6, find an intercepts. 9. 0...... 0 6. In Eercises 7 8, test for smmetr with respect to each ais and to the origin. 7. 8. 9. 0... 0.. 0. 6. 7. 8. In Eercises 9 6, sketch the graph of the equation. Identif an intercepts and test for smmetr. 9. 0...... 6. 7. 8. 9. 0. 9..... 6. 6 0 6 In Eercises 7 60, use a graphing utilit to graph the equation. Identif an intercepts and test for smmetr. 7. 9 8. 9. 6 60. 8 In Eercises 7 and 8, use a graphing utilit to graph the equation. Move the cursor along the curve to approimate the unknown coordinate of each solution point accurate to two decimal places. 7. (a), (b), 8. (a) 0., (b), In Eercises 6 68, find the points of intersection of the graphs of the equations. 6. 6. 6. 6 6. 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.
60_0P0.qd //0 :6 PM Page 9 SECTION P. Graphs and Models 9 6. 66. 0 67. 68. In Eercises 69 7, use a graphing utilit to find the points of intersection of the graphs. Check our results analticall. 69. 70. 7. 6 7. 6 6 7. 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 00. 7. 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 90. (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 00. 7. Break-Even Point Find the sales necessar to break even R C if the cost C of producing units is C. 0,000 Cost equation and the revenue R for selling units is R.9. Revenue equation 76. Copper Wire The resistance in ohms of 000 feet of solid copper wire at 77 F can be approimated b the model 0,770 0.7, 00 Year 970 97 980 98 990 99 000 CPI 8.8.8 8. 07.6 0.7. 7. Year 90 960 970 980 990 000 Acreage 97 7 6 60 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 6. 78. The graph has intercepts at,, 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 8 9 6 9 80. (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 8 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.) 8. 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) 7 9 9 6 7 k.